TPTP Problem File: SLH0629^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Clique_and_Monotone_Circuits/0005_Clique_Large_Monotone_Circuits/prob_00835_029481__16250900_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1472 ( 381 unt; 195 typ; 0 def)
% Number of atoms : 4374 ( 990 equ; 0 cnn)
% Maximal formula atoms : 14 ( 3 avg)
% Number of connectives : 13437 ( 238 ~; 25 |; 508 &;10561 @)
% ( 0 <=>;2105 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 15 ( 14 usr)
% Number of type conns : 1488 (1488 >; 0 *; 0 +; 0 <<)
% Number of symbols : 184 ( 181 usr; 21 con; 0-3 aty)
% Number of variables : 4131 ( 354 ^;3476 !; 301 ?;4131 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:49:18.582
%------------------------------------------------------------------------------
% Could-be-implicit typings (14)
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% Explicit typings (181)
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thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_less_set_set_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
ord_le152980574450754630et_nat: set_set_set_nat > set_set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
ord_le8865062304692155706_nat_o: ( ( nat > set_nat ) > $o ) > ( ( nat > set_nat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Nat__Onat_J,type,
ord_less_eq_o_nat: ( $o > nat ) > ( $o > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_It__Nat__Onat_J_J,type,
ord_le7022414076629706543et_nat: ( $o > set_nat ) > ( $o > set_nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
ord_le6539261115178940645et_nat: ( $o > set_set_nat ) > ( $o > set_set_nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
ord_le8326115459943588763et_nat: ( $o > set_set_set_nat ) > ( $o > set_set_set_nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6195038898401538645et_nat: ( nat > set_nat ) > ( nat > set_nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
ord_le3964352015994296041_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
ord_le3616423863276227763_nat_o: ( set_set_nat > $o ) > ( set_set_nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
ord_le1585852046946910987et_nat: set_nat_set_nat > set_nat_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
ord_le9131159989063066194et_nat: set_set_set_nat > set_set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
ord_le572741076514265352et_nat: set_set_set_set_nat > set_set_set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le3495481059733392331at_nat: set_se3873067930692246379at_nat > set_se3873067930692246379at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Sum____Type__Osum_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J_J_J,type,
ord_le4731320016863163777at_nat: set_se8003284279568041249at_nat > set_se8003284279568041249at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Sum____Type__Osum_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_Mt__Nat__Onat_J_J_J,type,
ord_le2853704879392749623at_nat: set_se7521423693449168855at_nat > set_se7521423693449168855at_nat > $o ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J,type,
collect_nat_set_nat: ( ( nat > set_nat ) > $o ) > set_nat_set_nat ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
collect_set_set_nat: ( set_set_nat > $o ) > set_set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
collec7201453139178570183et_nat: ( set_set_set_nat > $o ) > set_set_set_set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J_001t__Nat__Onat,type,
image_970537773860477644at_nat: ( ( nat > set_nat ) > nat ) > set_nat_set_nat > set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_8304670887732450946et_nat: ( ( nat > set_nat ) > set_nat ) > set_nat_set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J,type,
image_4436799006340492620et_nat: ( nat > nat > set_nat ) > set_nat > set_nat_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
image_2803531558198256130et_nat: ( nat > set_nat_set_nat ) > set_nat > set_set_nat_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_2194112158459175443et_nat: ( nat > set_set_nat ) > set_nat > set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
image_5738044413236618185et_nat: ( nat > set_set_set_nat ) > set_nat > set_set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J_J_001_Eo,type,
image_6034715890300748646_nat_o: ( set_nat_set_nat > $o ) > set_set_nat_set_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001_Eo,type,
image_set_nat_o: ( set_nat > $o ) > set_set_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
image_set_nat_nat: ( set_nat > nat ) > set_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_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_6725021117256019401et_nat: ( set_nat > set_set_nat ) > set_set_nat > set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
image_4583741654806091647et_nat: ( set_nat > set_set_set_nat ) > set_set_nat > set_set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001_Eo,type,
image_set_set_nat_o: ( set_set_nat > $o ) > set_set_set_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Nat__Onat,type,
image_1454916318497077779at_nat: ( set_set_nat > nat ) > set_set_set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_5842784325960735177et_nat: ( set_set_nat > set_nat ) > set_set_set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_7884819252390400639et_nat: ( set_set_nat > set_set_nat ) > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
image_8042862740799972405et_nat: ( set_set_nat > set_set_set_nat ) > set_set_set_nat > set_set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_001_Eo,type,
image_3488003393078953823_nat_o: ( set_set_set_nat > $o ) > set_set_set_set_nat > set_o ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
insert_set_nat: set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
insert_set_set_nat: set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J,type,
set_or6252518528881150372et_nat: ( nat > set_nat ) > ( nat > set_nat ) > set_nat_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_It__Nat__Onat_J,type,
set_or3540276404033026485et_nat: set_nat > set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_or5410080298493297259et_nat: set_set_nat > set_set_nat > set_set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_or659464924768625697et_nat: set_set_set_nat > set_set_set_nat > set_set_set_set_nat ).
thf(sy_c_Sunflower_Osunflower_001_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J,type,
sunflo5599553548652064642et_nat: set_set_nat_set_nat > $o ).
thf(sy_c_Sunflower_Osunflower_001t__Nat__Onat,type,
sunflower_nat: set_set_nat > $o ).
thf(sy_c_Sunflower_Osunflower_001t__Set__Oset_It__Nat__Onat_J,type,
sunflower_set_nat: set_set_set_nat > $o ).
thf(sy_c_Sunflower_Osunflower_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
sunflo2680516271513359689et_nat: set_set_set_set_nat > $o ).
thf(sy_c_Sunflower_Osunflower_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
sunflo1841451327523575948at_nat: set_se3873067930692246379at_nat > $o ).
thf(sy_c_Sunflower_Osunflower_001t__Sum____Type__Osum_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J,type,
sunflo6650083805840251970at_nat: set_se8003284279568041249at_nat > $o ).
thf(sy_c_Sunflower_Osunflower_001t__Sum____Type__Osum_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_Mt__Nat__Onat_J,type,
sunflo3853689026006497528at_nat: set_se7521423693449168855at_nat > $o ).
thf(sy_c_fChoice_001_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J,type,
fChoice_nat_set_nat: ( ( nat > set_nat ) > $o ) > nat > set_nat ).
thf(sy_c_fChoice_001t__Nat__Onat,type,
fChoice_nat: ( nat > $o ) > nat ).
thf(sy_c_fChoice_001t__Set__Oset_It__Nat__Onat_J,type,
fChoice_set_nat: ( set_nat > $o ) > set_nat ).
thf(sy_c_fChoice_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
fChoice_set_set_nat: ( set_set_nat > $o ) > set_set_nat ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J,type,
member_nat_set_nat: ( nat > set_nat ) > set_nat_set_nat > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
member6710465769566284994et_nat: set_nat_set_nat > set_set_nat_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
member_set_set_nat: set_set_nat > set_set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
member2946998982187404937et_nat: set_set_set_nat > set_set_set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member1869216328726507724at_nat: set_Sum_sum_nat_nat > set_se3873067930692246379at_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Sum____Type__Osum_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
member5374901640408327554at_nat: set_Su8059080322890262379at_nat > set_se8003284279568041249at_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Sum____Type__Osum_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_Mt__Nat__Onat_J_J,type,
member5638249034155602744at_nat: set_Su1440016900418933025at_nat > set_se7521423693449168855at_nat > $o ).
thf(sy_v_G____,type,
g: nat > set_set_nat ).
thf(sy_v_Gs____,type,
gs: set_set_nat ).
thf(sy_v_S____,type,
s: set_set_nat ).
thf(sy_v_Si____,type,
si: nat > set_nat ).
thf(sy_v_U____,type,
u: set_set_set_nat ).
thf(sy_v_Vs____,type,
vs: set_nat ).
thf(sy_v_W____,type,
w: set_nat ).
thf(sy_v_X,type,
x: set_set_set_nat ).
thf(sy_v_Y,type,
y: set_set_set_nat ).
thf(sy_v_e____,type,
e: nat > set_nat ).
thf(sy_v_fstt____,type,
fstt: set_nat > nat ).
thf(sy_v_l,type,
l: nat ).
thf(sy_v_merge____,type,
merge: ( nat > set_nat ) > ( nat > nat ) > nat > nat ).
thf(sy_v_p,type,
p: nat ).
thf(sy_v_r____,type,
r: nat ).
thf(sy_v_s____,type,
s2: nat ).
thf(sy_v_si____,type,
si2: nat > nat ).
thf(sy_v_sndd____,type,
sndd: set_nat > nat ).
thf(sy_v_thesis____,type,
thesis: $o ).
thf(sy_v_ti____,type,
ti: nat > nat ).
thf(sy_v_v____,type,
v: nat ).
% Relevant facts (1268)
thf(fact_0_p0,axiom,
p != zero_zero_nat ).
% p0
thf(fact_1_True,axiom,
member_nat @ v @ w ).
% True
thf(fact_2_e_I2_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ p )
=> ( member_set_nat @ ( e @ I ) @ ( g @ I ) ) ) ).
% e(2)
thf(fact_3__092_060open_062v_A_092_060in_062_A_I_092_060lambda_062i_O_Afstt_A_Ie_Ai_J_J_A_096_A_1230_O_O_060p_125_092_060close_062,axiom,
( member_nat @ v
@ ( image_nat_nat
@ ^ [I2: nat] : ( fstt @ ( e @ I2 ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ p ) ) ) ).
% \<open>v \<in> (\<lambda>i. fstt (e i)) ` {0..<p}\<close>
thf(fact_4_pl,axiom,
ord_less_nat @ l @ p ).
% pl
thf(fact_5_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_6_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_7_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_8_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_9_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_10_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_11_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_12_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_13_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_14_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_15_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_16_W__def,axiom,
( w
= ( image_nat_nat
@ ^ [I2: nat] : ( fstt @ ( e @ I2 ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ p ) ) ) ).
% W_def
thf(fact_17_e_I1_J,axiom,
member_nat_set_nat @ e @ ( piE_nat_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ p ) @ g ) ).
% e(1)
thf(fact_18_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_19_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_20_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_21_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_22_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_23_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_24_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_25_Lp,axiom,
ord_less_nat @ p @ ( assump1710595444109740301irst_L @ l @ p ) ).
% Lp
thf(fact_26_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_27_i__props_I6_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ p )
=> ( finite1152437895449049373et_nat @ ( g @ I ) ) ) ).
% i_props(6)
thf(fact_28_injG,axiom,
inj_on8105003582846801791et_nat @ g @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ p ) ).
% injG
thf(fact_29_image__ident,axiom,
! [Y2: set_nat] :
( ( image_nat_nat
@ ^ [X2: nat] : X2
@ Y2 )
= Y2 ) ).
% image_ident
thf(fact_30_all__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( P @ M3 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( P @ X2 ) ) ) ) ).
% all_nat_less_eq
thf(fact_31_ex__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M3: nat] :
( ( ord_less_nat @ M3 @ N )
& ( P @ M3 ) ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
& ( P @ X2 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_32_G_I3_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ p )
=> ( member_set_set_nat @ ( g @ I ) @ u ) ) ).
% G(3)
thf(fact_33_image__eqI,axiom,
! [B: nat,F: nat > nat,X: nat,A2: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_34_image__eqI,axiom,
! [B: set_nat,F: nat > set_nat,X: nat,A2: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_set_nat @ B @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_35_image__eqI,axiom,
! [B: nat,F: set_nat > nat,X: set_nat,A2: set_set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_set_nat @ X @ A2 )
=> ( member_nat @ B @ ( image_set_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_36_image__eqI,axiom,
! [B: set_set_nat,F: nat > set_set_nat,X: nat,A2: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_set_set_nat @ B @ ( image_2194112158459175443et_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_37_image__eqI,axiom,
! [B: set_nat,F: set_nat > set_nat,X: set_nat,A2: set_set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_set_nat @ X @ A2 )
=> ( member_set_nat @ B @ ( image_7916887816326733075et_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_38_image__eqI,axiom,
! [B: nat,F: set_set_nat > nat,X: set_set_nat,A2: set_set_set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_set_set_nat @ X @ A2 )
=> ( member_nat @ B @ ( image_1454916318497077779at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_39_image__eqI,axiom,
! [B: nat > set_nat,F: nat > nat > set_nat,X: nat,A2: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat_set_nat @ B @ ( image_4436799006340492620et_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_40_image__eqI,axiom,
! [B: set_set_nat,F: set_nat > set_set_nat,X: set_nat,A2: set_set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_set_nat @ X @ A2 )
=> ( member_set_set_nat @ B @ ( image_6725021117256019401et_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_41_image__eqI,axiom,
! [B: nat,F: ( nat > set_nat ) > nat,X: nat > set_nat,A2: set_nat_set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat_set_nat @ X @ A2 )
=> ( member_nat @ B @ ( image_970537773860477644at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_42_image__eqI,axiom,
! [B: set_nat,F: set_set_nat > set_nat,X: set_set_nat,A2: set_set_set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_set_set_nat @ X @ A2 )
=> ( member_set_nat @ B @ ( image_5842784325960735177et_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_43_G_I1_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ p )
=> ( member_set_set_nat @ ( g @ I ) @ x ) ) ).
% G(1)
thf(fact_44_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_45_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: nat,F: nat > nat] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_46_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: set_nat,F: nat > set_nat] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_set_nat @ B @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_47_rev__image__eqI,axiom,
! [X: set_nat,A2: set_set_nat,B: nat,F: set_nat > nat] :
( ( member_set_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_set_nat_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_48_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: set_set_nat,F: nat > set_set_nat] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_set_set_nat @ B @ ( image_2194112158459175443et_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_49_rev__image__eqI,axiom,
! [X: set_nat,A2: set_set_nat,B: set_nat,F: set_nat > set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_set_nat @ B @ ( image_7916887816326733075et_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_50_rev__image__eqI,axiom,
! [X: set_set_nat,A2: set_set_set_nat,B: nat,F: set_set_nat > nat] :
( ( member_set_set_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_1454916318497077779at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_51_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: nat > set_nat,F: nat > nat > set_nat] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat_set_nat @ B @ ( image_4436799006340492620et_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_52_rev__image__eqI,axiom,
! [X: set_nat,A2: set_set_nat,B: set_set_nat,F: set_nat > set_set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_set_set_nat @ B @ ( image_6725021117256019401et_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_53_rev__image__eqI,axiom,
! [X: nat > set_nat,A2: set_nat_set_nat,B: nat,F: ( nat > set_nat ) > nat] :
( ( member_nat_set_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_970537773860477644at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_54_rev__image__eqI,axiom,
! [X: set_set_nat,A2: set_set_set_nat,B: set_nat,F: set_set_nat > set_nat] :
( ( member_set_set_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_set_nat @ B @ ( image_5842784325960735177et_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_55_mem__Collect__eq,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( member_set_nat @ A @ ( collect_set_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_56_mem__Collect__eq,axiom,
! [A: nat > set_nat,P: ( nat > set_nat ) > $o] :
( ( member_nat_set_nat @ A @ ( collect_nat_set_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_57_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_58_mem__Collect__eq,axiom,
! [A: set_set_nat,P: set_set_nat > $o] :
( ( member_set_set_nat @ A @ ( collect_set_set_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_59_Collect__mem__eq,axiom,
! [A2: set_set_nat] :
( ( collect_set_nat
@ ^ [X2: set_nat] : ( member_set_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_60_Collect__mem__eq,axiom,
! [A2: set_nat_set_nat] :
( ( collect_nat_set_nat
@ ^ [X2: nat > set_nat] : ( member_nat_set_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_61_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_62_Collect__mem__eq,axiom,
! [A2: set_set_set_nat] :
( ( collect_set_set_nat
@ ^ [X2: set_set_nat] : ( member_set_set_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_63_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_64_Collect__cong,axiom,
! [P: set_set_nat > $o,Q: set_set_nat > $o] :
( ! [X3: set_set_nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_set_set_nat @ P )
= ( collect_set_set_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_65_ball__imageD,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( image_nat_nat @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_66_ball__imageD,axiom,
! [F: set_set_nat > set_nat,A2: set_set_set_nat,P: set_nat > $o] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ ( image_5842784325960735177et_nat @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ A2 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_67_ball__imageD,axiom,
! [F: nat > set_set_nat,A2: set_nat,P: set_set_nat > $o] :
( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ ( image_2194112158459175443et_nat @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_68_image__cong,axiom,
! [M4: set_nat,N3: set_nat,F: nat > nat,G: nat > nat] :
( ( M4 = N3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N3 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_nat_nat @ F @ M4 )
= ( image_nat_nat @ G @ N3 ) ) ) ) ).
% image_cong
thf(fact_69_image__cong,axiom,
! [M4: set_nat,N3: set_nat,F: nat > set_set_nat,G: nat > set_set_nat] :
( ( M4 = N3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N3 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_2194112158459175443et_nat @ F @ M4 )
= ( image_2194112158459175443et_nat @ G @ N3 ) ) ) ) ).
% image_cong
thf(fact_70_image__cong,axiom,
! [M4: set_set_set_nat,N3: set_set_set_nat,F: set_set_nat > set_nat,G: set_set_nat > set_nat] :
( ( M4 = N3 )
=> ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ N3 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_5842784325960735177et_nat @ F @ M4 )
= ( image_5842784325960735177et_nat @ G @ N3 ) ) ) ) ).
% image_cong
thf(fact_71_bex__imageD,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ X4 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_72_bex__imageD,axiom,
! [F: set_set_nat > set_nat,A2: set_set_set_nat,P: set_nat > $o] :
( ? [X4: set_nat] :
( ( member_set_nat @ X4 @ ( image_5842784325960735177et_nat @ F @ A2 ) )
& ( P @ X4 ) )
=> ? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_73_bex__imageD,axiom,
! [F: nat > set_set_nat,A2: set_nat,P: set_set_nat > $o] :
( ? [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ ( image_2194112158459175443et_nat @ F @ A2 ) )
& ( P @ X4 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_74_image__iff,axiom,
! [Z: nat,F: nat > nat,A2: set_nat] :
( ( member_nat @ Z @ ( image_nat_nat @ F @ A2 ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_75_image__iff,axiom,
! [Z: set_nat,F: set_set_nat > set_nat,A2: set_set_set_nat] :
( ( member_set_nat @ Z @ ( image_5842784325960735177et_nat @ F @ A2 ) )
= ( ? [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A2 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_76_image__iff,axiom,
! [Z: set_set_nat,F: nat > set_set_nat,A2: set_nat] :
( ( member_set_set_nat @ Z @ ( image_2194112158459175443et_nat @ F @ A2 ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_77_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_78_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > set_nat] :
( ( member_nat @ X @ A2 )
=> ( member_set_nat @ ( F @ X ) @ ( image_nat_set_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_79_imageI,axiom,
! [X: set_nat,A2: set_set_nat,F: set_nat > nat] :
( ( member_set_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( image_set_nat_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_80_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > set_set_nat] :
( ( member_nat @ X @ A2 )
=> ( member_set_set_nat @ ( F @ X ) @ ( image_2194112158459175443et_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_81_imageI,axiom,
! [X: set_nat,A2: set_set_nat,F: set_nat > set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( member_set_nat @ ( F @ X ) @ ( image_7916887816326733075et_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_82_imageI,axiom,
! [X: set_set_nat,A2: set_set_set_nat,F: set_set_nat > nat] :
( ( member_set_set_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( image_1454916318497077779at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_83_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > nat > set_nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat_set_nat @ ( F @ X ) @ ( image_4436799006340492620et_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_84_imageI,axiom,
! [X: set_nat,A2: set_set_nat,F: set_nat > set_set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( member_set_set_nat @ ( F @ X ) @ ( image_6725021117256019401et_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_85_imageI,axiom,
! [X: nat > set_nat,A2: set_nat_set_nat,F: ( nat > set_nat ) > nat] :
( ( member_nat_set_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( image_970537773860477644at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_86_imageI,axiom,
! [X: set_set_nat,A2: set_set_set_nat,F: set_set_nat > set_nat] :
( ( member_set_set_nat @ X @ A2 )
=> ( member_set_nat @ ( F @ X ) @ ( image_5842784325960735177et_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_87_Compr__image__eq,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_88_Compr__image__eq,axiom,
! [F: nat > set_nat,A2: set_nat,P: set_nat > $o] :
( ( collect_set_nat
@ ^ [X2: set_nat] :
( ( member_set_nat @ X2 @ ( image_nat_set_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_nat_set_nat @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_89_Compr__image__eq,axiom,
! [F: set_nat > nat,A2: set_set_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_set_nat_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_set_nat_nat @ F
@ ( collect_set_nat
@ ^ [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_90_Compr__image__eq,axiom,
! [F: set_nat > set_nat,A2: set_set_nat,P: set_nat > $o] :
( ( collect_set_nat
@ ^ [X2: set_nat] :
( ( member_set_nat @ X2 @ ( image_7916887816326733075et_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_7916887816326733075et_nat @ F
@ ( collect_set_nat
@ ^ [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_91_Compr__image__eq,axiom,
! [F: set_set_nat > nat,A2: set_set_set_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_1454916318497077779at_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_1454916318497077779at_nat @ F
@ ( collect_set_set_nat
@ ^ [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_92_Compr__image__eq,axiom,
! [F: nat > set_set_nat,A2: set_nat,P: set_set_nat > $o] :
( ( collect_set_set_nat
@ ^ [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ ( image_2194112158459175443et_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_2194112158459175443et_nat @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_93_Compr__image__eq,axiom,
! [F: nat > nat > set_nat,A2: set_nat,P: ( nat > set_nat ) > $o] :
( ( collect_nat_set_nat
@ ^ [X2: nat > set_nat] :
( ( member_nat_set_nat @ X2 @ ( image_4436799006340492620et_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_4436799006340492620et_nat @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_94_Compr__image__eq,axiom,
! [F: set_set_nat > set_nat,A2: set_set_set_nat,P: set_nat > $o] :
( ( collect_set_nat
@ ^ [X2: set_nat] :
( ( member_set_nat @ X2 @ ( image_5842784325960735177et_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_5842784325960735177et_nat @ F
@ ( collect_set_set_nat
@ ^ [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_95_Compr__image__eq,axiom,
! [F: ( nat > set_nat ) > nat,A2: set_nat_set_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_970537773860477644at_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_970537773860477644at_nat @ F
@ ( collect_nat_set_nat
@ ^ [X2: nat > set_nat] :
( ( member_nat_set_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_96_Compr__image__eq,axiom,
! [F: set_nat > set_set_nat,A2: set_set_nat,P: set_set_nat > $o] :
( ( collect_set_set_nat
@ ^ [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ ( image_6725021117256019401et_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_6725021117256019401et_nat @ F
@ ( collect_set_nat
@ ^ [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_97_image__image,axiom,
! [F: set_nat > set_nat,G: set_set_nat > set_nat,A2: set_set_set_nat] :
( ( image_7916887816326733075et_nat @ F @ ( image_5842784325960735177et_nat @ G @ A2 ) )
= ( image_5842784325960735177et_nat
@ ^ [X2: set_set_nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_98_image__image,axiom,
! [F: set_set_nat > nat,G: nat > set_set_nat,A2: set_nat] :
( ( image_1454916318497077779at_nat @ F @ ( image_2194112158459175443et_nat @ G @ A2 ) )
= ( image_nat_nat
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_99_image__image,axiom,
! [F: set_set_nat > set_set_nat,G: nat > set_set_nat,A2: set_nat] :
( ( image_7884819252390400639et_nat @ F @ ( image_2194112158459175443et_nat @ G @ A2 ) )
= ( image_2194112158459175443et_nat
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_100_image__image,axiom,
! [F: nat > nat,G: nat > nat,A2: set_nat] :
( ( image_nat_nat @ F @ ( image_nat_nat @ G @ A2 ) )
= ( image_nat_nat
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_101_image__image,axiom,
! [F: set_set_nat > set_nat,G: set_set_nat > set_set_nat,A2: set_set_set_nat] :
( ( image_5842784325960735177et_nat @ F @ ( image_7884819252390400639et_nat @ G @ A2 ) )
= ( image_5842784325960735177et_nat
@ ^ [X2: set_set_nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_102_image__image,axiom,
! [F: set_set_nat > set_nat,G: nat > set_set_nat,A2: set_nat] :
( ( image_5842784325960735177et_nat @ F @ ( image_2194112158459175443et_nat @ G @ A2 ) )
= ( image_nat_set_nat
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_103_image__image,axiom,
! [F: nat > set_set_nat,G: nat > nat,A2: set_nat] :
( ( image_2194112158459175443et_nat @ F @ ( image_nat_nat @ G @ A2 ) )
= ( image_2194112158459175443et_nat
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_104_imageE,axiom,
! [B: nat,F: nat > nat,A2: set_nat] :
( ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [X3: nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_105_imageE,axiom,
! [B: nat,F: set_nat > nat,A2: set_set_nat] :
( ( member_nat @ B @ ( image_set_nat_nat @ F @ A2 ) )
=> ~ ! [X3: set_nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_set_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_106_imageE,axiom,
! [B: set_nat,F: nat > set_nat,A2: set_nat] :
( ( member_set_nat @ B @ ( image_nat_set_nat @ F @ A2 ) )
=> ~ ! [X3: nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_107_imageE,axiom,
! [B: nat,F: set_set_nat > nat,A2: set_set_set_nat] :
( ( member_nat @ B @ ( image_1454916318497077779at_nat @ F @ A2 ) )
=> ~ ! [X3: set_set_nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_set_set_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_108_imageE,axiom,
! [B: set_nat,F: set_nat > set_nat,A2: set_set_nat] :
( ( member_set_nat @ B @ ( image_7916887816326733075et_nat @ F @ A2 ) )
=> ~ ! [X3: set_nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_set_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_109_imageE,axiom,
! [B: set_set_nat,F: nat > set_set_nat,A2: set_nat] :
( ( member_set_set_nat @ B @ ( image_2194112158459175443et_nat @ F @ A2 ) )
=> ~ ! [X3: nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_110_imageE,axiom,
! [B: nat,F: ( nat > set_nat ) > nat,A2: set_nat_set_nat] :
( ( member_nat @ B @ ( image_970537773860477644at_nat @ F @ A2 ) )
=> ~ ! [X3: nat > set_nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat_set_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_111_imageE,axiom,
! [B: set_nat,F: set_set_nat > set_nat,A2: set_set_set_nat] :
( ( member_set_nat @ B @ ( image_5842784325960735177et_nat @ F @ A2 ) )
=> ~ ! [X3: set_set_nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_set_set_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_112_imageE,axiom,
! [B: nat > set_nat,F: nat > nat > set_nat,A2: set_nat] :
( ( member_nat_set_nat @ B @ ( image_4436799006340492620et_nat @ F @ A2 ) )
=> ~ ! [X3: nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_113_imageE,axiom,
! [B: set_set_nat,F: set_nat > set_set_nat,A2: set_set_nat] :
( ( member_set_set_nat @ B @ ( image_6725021117256019401et_nat @ F @ A2 ) )
=> ~ ! [X3: set_nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_set_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_114_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_115_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_116_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_117_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_118_assms_I3_J,axiom,
( y
= ( clique4095374090462327202g_step @ p @ x ) ) ).
% assms(3)
thf(fact_119_finite__imageI,axiom,
! [F2: set_set_nat,H: set_nat > set_nat] :
( ( finite1152437895449049373et_nat @ F2 )
=> ( finite1152437895449049373et_nat @ ( image_7916887816326733075et_nat @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_120_finite__imageI,axiom,
! [F2: set_set_nat,H: set_nat > set_set_nat] :
( ( finite1152437895449049373et_nat @ F2 )
=> ( finite6739761609112101331et_nat @ ( image_6725021117256019401et_nat @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_121_finite__imageI,axiom,
! [F2: set_set_nat,H: set_nat > nat] :
( ( finite1152437895449049373et_nat @ F2 )
=> ( finite_finite_nat @ ( image_set_nat_nat @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_122_finite__imageI,axiom,
! [F2: set_set_set_nat,H: set_set_nat > set_nat] :
( ( finite6739761609112101331et_nat @ F2 )
=> ( finite1152437895449049373et_nat @ ( image_5842784325960735177et_nat @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_123_finite__imageI,axiom,
! [F2: set_set_set_nat,H: set_set_nat > set_set_nat] :
( ( finite6739761609112101331et_nat @ F2 )
=> ( finite6739761609112101331et_nat @ ( image_7884819252390400639et_nat @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_124_finite__imageI,axiom,
! [F2: set_set_set_nat,H: set_set_nat > nat] :
( ( finite6739761609112101331et_nat @ F2 )
=> ( finite_finite_nat @ ( image_1454916318497077779at_nat @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_125_finite__imageI,axiom,
! [F2: set_nat,H: nat > set_nat] :
( ( finite_finite_nat @ F2 )
=> ( finite1152437895449049373et_nat @ ( image_nat_set_nat @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_126_finite__imageI,axiom,
! [F2: set_nat,H: nat > set_set_nat] :
( ( finite_finite_nat @ F2 )
=> ( finite6739761609112101331et_nat @ ( image_2194112158459175443et_nat @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_127_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_128_finite__imp__nat__seg__image__inj__on,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ? [N2: nat,F3: nat > set_nat] :
( ( A2
= ( image_nat_set_nat @ F3
@ ( collect_nat
@ ^ [I2: nat] : ( ord_less_nat @ I2 @ N2 ) ) ) )
& ( inj_on_nat_set_nat @ F3
@ ( collect_nat
@ ^ [I2: nat] : ( ord_less_nat @ I2 @ N2 ) ) ) ) ) ).
% finite_imp_nat_seg_image_inj_on
thf(fact_129_finite__imp__nat__seg__image__inj__on,axiom,
! [A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ? [N2: nat,F3: nat > set_set_nat] :
( ( A2
= ( image_2194112158459175443et_nat @ F3
@ ( collect_nat
@ ^ [I2: nat] : ( ord_less_nat @ I2 @ N2 ) ) ) )
& ( inj_on8105003582846801791et_nat @ F3
@ ( collect_nat
@ ^ [I2: nat] : ( ord_less_nat @ I2 @ N2 ) ) ) ) ) ).
% finite_imp_nat_seg_image_inj_on
thf(fact_130_finite__imp__nat__seg__image__inj__on,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ? [N2: nat,F3: nat > nat] :
( ( A2
= ( image_nat_nat @ F3
@ ( collect_nat
@ ^ [I2: nat] : ( ord_less_nat @ I2 @ N2 ) ) ) )
& ( inj_on_nat_nat @ F3
@ ( collect_nat
@ ^ [I2: nat] : ( ord_less_nat @ I2 @ N2 ) ) ) ) ) ).
% finite_imp_nat_seg_image_inj_on
thf(fact_131_finite__Collect__conjI,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
| ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X2: set_nat] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_132_finite__Collect__conjI,axiom,
! [P: set_set_nat > $o,Q: set_set_nat > $o] :
( ( ( finite6739761609112101331et_nat @ ( collect_set_set_nat @ P ) )
| ( finite6739761609112101331et_nat @ ( collect_set_set_nat @ Q ) ) )
=> ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [X2: set_set_nat] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_133_finite__Collect__conjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P ) )
| ( finite_finite_nat @ ( collect_nat @ Q ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_134_finite__Collect__disjI,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X2: set_nat] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
& ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_135_finite__Collect__disjI,axiom,
! [P: set_set_nat > $o,Q: set_set_nat > $o] :
( ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [X2: set_set_nat] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite6739761609112101331et_nat @ ( collect_set_set_nat @ P ) )
& ( finite6739761609112101331et_nat @ ( collect_set_set_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_136_finite__Collect__disjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_137_finite__conv__nat__seg__image,axiom,
( finite1152437895449049373et_nat
= ( ^ [A3: set_set_nat] :
? [N4: nat,F4: nat > set_nat] :
( A3
= ( image_nat_set_nat @ F4
@ ( collect_nat
@ ^ [I2: nat] : ( ord_less_nat @ I2 @ N4 ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_138_finite__conv__nat__seg__image,axiom,
( finite6739761609112101331et_nat
= ( ^ [A3: set_set_set_nat] :
? [N4: nat,F4: nat > set_set_nat] :
( A3
= ( image_2194112158459175443et_nat @ F4
@ ( collect_nat
@ ^ [I2: nat] : ( ord_less_nat @ I2 @ N4 ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_139_finite__conv__nat__seg__image,axiom,
( finite_finite_nat
= ( ^ [A3: set_nat] :
? [N4: nat,F4: nat > nat] :
( A3
= ( image_nat_nat @ F4
@ ( collect_nat
@ ^ [I2: nat] : ( ord_less_nat @ I2 @ N4 ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_140_nat__seg__image__imp__finite,axiom,
! [A2: set_set_nat,F: nat > set_nat,N: nat] :
( ( A2
= ( image_nat_set_nat @ F
@ ( collect_nat
@ ^ [I2: nat] : ( ord_less_nat @ I2 @ N ) ) ) )
=> ( finite1152437895449049373et_nat @ A2 ) ) ).
% nat_seg_image_imp_finite
thf(fact_141_nat__seg__image__imp__finite,axiom,
! [A2: set_set_set_nat,F: nat > set_set_nat,N: nat] :
( ( A2
= ( image_2194112158459175443et_nat @ F
@ ( collect_nat
@ ^ [I2: nat] : ( ord_less_nat @ I2 @ N ) ) ) )
=> ( finite6739761609112101331et_nat @ A2 ) ) ).
% nat_seg_image_imp_finite
thf(fact_142_nat__seg__image__imp__finite,axiom,
! [A2: set_nat,F: nat > nat,N: nat] :
( ( A2
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [I2: nat] : ( ord_less_nat @ I2 @ N ) ) ) )
=> ( finite_finite_nat @ A2 ) ) ).
% nat_seg_image_imp_finite
thf(fact_143_filter__preserves__multiset,axiom,
! [M4: set_nat > nat,P: set_nat > $o] :
( ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X2: set_nat] : ( ord_less_nat @ zero_zero_nat @ ( M4 @ X2 ) ) ) )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X2: set_nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X2 ) @ ( M4 @ X2 ) @ zero_zero_nat ) ) ) ) ) ).
% filter_preserves_multiset
thf(fact_144_filter__preserves__multiset,axiom,
! [M4: set_set_nat > nat,P: set_set_nat > $o] :
( ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [X2: set_set_nat] : ( ord_less_nat @ zero_zero_nat @ ( M4 @ X2 ) ) ) )
=> ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [X2: set_set_nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X2 ) @ ( M4 @ X2 ) @ zero_zero_nat ) ) ) ) ) ).
% filter_preserves_multiset
thf(fact_145_filter__preserves__multiset,axiom,
! [M4: nat > nat,P: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] : ( ord_less_nat @ zero_zero_nat @ ( M4 @ X2 ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X2 ) @ ( M4 @ X2 ) @ zero_zero_nat ) ) ) ) ) ).
% filter_preserves_multiset
thf(fact_146_UX,axiom,
ord_le9131159989063066194et_nat @ u @ x ).
% UX
thf(fact_147_finX,axiom,
finite6739761609112101331et_nat @ x ).
% finX
thf(fact_148_finU,axiom,
finite6739761609112101331et_nat @ u ).
% finU
thf(fact_149__092_060open_062_092_060And_062A_O_AA_A_092_060subseteq_062_AX_A_092_060Longrightarrow_062_Afinite_AA_092_060close_062,axiom,
! [A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ x )
=> ( finite6739761609112101331et_nat @ A2 ) ) ).
% \<open>\<And>A. A \<subseteq> X \<Longrightarrow> finite A\<close>
thf(fact_150_subset__antisym,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_151_subset__antisym,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_152_subset__antisym,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_153_psubsetI,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_le152980574450754630et_nat @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_154_psubsetI,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_set_nat @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_155_psubsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_nat @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_156_subsetI,axiom,
! [A2: set_nat_set_nat,B2: set_nat_set_nat] :
( ! [X3: nat > set_nat] :
( ( member_nat_set_nat @ X3 @ A2 )
=> ( member_nat_set_nat @ X3 @ B2 ) )
=> ( ord_le1585852046946910987et_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_157_subsetI,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A2 )
=> ( member_set_set_nat @ X3 @ B2 ) )
=> ( ord_le9131159989063066194et_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_158_subsetI,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( member_set_nat @ X3 @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_159_subsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ X3 @ B2 ) )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_160_finite__atLeastLessThan,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or4665077453230672383an_nat @ L @ U ) ) ).
% finite_atLeastLessThan
thf(fact_161_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_nat @ N4 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_162_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_163_atLeastLessThan__iff,axiom,
! [I: nat > set_nat,L: nat > set_nat,U: nat > set_nat] :
( ( member_nat_set_nat @ I @ ( set_or6252518528881150372et_nat @ L @ U ) )
= ( ( ord_le6195038898401538645et_nat @ L @ I )
& ( ord_less_nat_set_nat @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_164_atLeastLessThan__iff,axiom,
! [I: set_set_set_nat,L: set_set_set_nat,U: set_set_set_nat] :
( ( member2946998982187404937et_nat @ I @ ( set_or659464924768625697et_nat @ L @ U ) )
= ( ( ord_le9131159989063066194et_nat @ L @ I )
& ( ord_le152980574450754630et_nat @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_165_atLeastLessThan__iff,axiom,
! [I: set_set_nat,L: set_set_nat,U: set_set_nat] :
( ( member_set_set_nat @ I @ ( set_or5410080298493297259et_nat @ L @ U ) )
= ( ( ord_le6893508408891458716et_nat @ L @ I )
& ( ord_less_set_set_nat @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_166_atLeastLessThan__iff,axiom,
! [I: set_nat,L: set_nat,U: set_nat] :
( ( member_set_nat @ I @ ( set_or3540276404033026485et_nat @ L @ U ) )
= ( ( ord_less_eq_set_nat @ L @ I )
& ( ord_less_set_nat @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_167_atLeastLessThan__iff,axiom,
! [I: nat,L: nat,U: nat] :
( ( member_nat @ I @ ( set_or4665077453230672383an_nat @ L @ U ) )
= ( ( ord_less_eq_nat @ L @ I )
& ( ord_less_nat @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_168_ivl__subset,axiom,
! [I: nat,J: nat,M: nat,N: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ J @ I )
| ( ( ord_less_eq_nat @ M @ I )
& ( ord_less_eq_nat @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_169_finite__Collect__subsets,axiom,
! [A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( finite5926941155766903689et_nat
@ ( collec7201453139178570183et_nat
@ ^ [B3: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ B3 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_170_finite__Collect__subsets,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [B3: set_set_nat] : ( ord_le6893508408891458716et_nat @ B3 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_171_finite__Collect__subsets,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B3: set_nat] : ( ord_less_eq_set_nat @ B3 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_172_subset__iff__psubset__eq,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_173_subset__iff__psubset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ( ord_less_set_set_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_174_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_175_subset__psubset__trans,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( ord_le152980574450754630et_nat @ B2 @ C )
=> ( ord_le152980574450754630et_nat @ A2 @ C ) ) ) ).
% subset_psubset_trans
thf(fact_176_subset__psubset__trans,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ord_less_set_set_nat @ B2 @ C )
=> ( ord_less_set_set_nat @ A2 @ C ) ) ) ).
% subset_psubset_trans
thf(fact_177_subset__psubset__trans,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_set_nat @ B2 @ C )
=> ( ord_less_set_nat @ A2 @ C ) ) ) ).
% subset_psubset_trans
thf(fact_178_subset__not__subset__eq,axiom,
( ord_le152980574450754630et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A3 @ B3 )
& ~ ( ord_le9131159989063066194et_nat @ B3 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_179_subset__not__subset__eq,axiom,
( ord_less_set_set_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A3 @ B3 )
& ~ ( ord_le6893508408891458716et_nat @ B3 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_180_subset__not__subset__eq,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ~ ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_181_psubset__subset__trans,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A2 @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ C )
=> ( ord_le152980574450754630et_nat @ A2 @ C ) ) ) ).
% psubset_subset_trans
thf(fact_182_psubset__subset__trans,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_set_nat] :
( ( ord_less_set_set_nat @ A2 @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ C )
=> ( ord_less_set_set_nat @ A2 @ C ) ) ) ).
% psubset_subset_trans
thf(fact_183_psubset__subset__trans,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ord_less_set_nat @ A2 @ C ) ) ) ).
% psubset_subset_trans
thf(fact_184_psubset__imp__subset,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A2 @ B2 )
=> ( ord_le9131159989063066194et_nat @ A2 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_185_psubset__imp__subset,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_less_set_set_nat @ A2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_186_psubset__imp__subset,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_187_Collect__mono__iff,axiom,
! [P: set_set_nat > $o,Q: set_set_nat > $o] :
( ( ord_le9131159989063066194et_nat @ ( collect_set_set_nat @ P ) @ ( collect_set_set_nat @ Q ) )
= ( ! [X2: set_set_nat] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_188_Collect__mono__iff,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) )
= ( ! [X2: set_nat] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% 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 ) )
= ( ! [X2: nat] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_190_set__eq__subset,axiom,
( ( ^ [Y3: set_set_set_nat,Z2: set_set_set_nat] : ( Y3 = Z2 ) )
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A3 @ B3 )
& ( ord_le9131159989063066194et_nat @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_191_set__eq__subset,axiom,
( ( ^ [Y3: set_set_nat,Z2: set_set_nat] : ( Y3 = Z2 ) )
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A3 @ B3 )
& ( ord_le6893508408891458716et_nat @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_192_set__eq__subset,axiom,
( ( ^ [Y3: set_nat,Z2: set_nat] : ( Y3 = Z2 ) )
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_193_subset__trans,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ C )
=> ( ord_le9131159989063066194et_nat @ A2 @ C ) ) ) ).
% subset_trans
thf(fact_194_subset__trans,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ C )
=> ( ord_le6893508408891458716et_nat @ A2 @ C ) ) ) ).
% subset_trans
thf(fact_195_subset__trans,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% subset_trans
thf(fact_196_Collect__mono,axiom,
! [P: set_set_nat > $o,Q: set_set_nat > $o] :
( ! [X3: set_set_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le9131159989063066194et_nat @ ( collect_set_set_nat @ P ) @ ( collect_set_set_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_197_Collect__mono,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ! [X3: set_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_198_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_199_subset__refl,axiom,
! [A2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_200_subset__refl,axiom,
! [A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_201_subset__refl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_202_subset__iff,axiom,
( ord_le1585852046946910987et_nat
= ( ^ [A3: set_nat_set_nat,B3: set_nat_set_nat] :
! [T2: nat > set_nat] :
( ( member_nat_set_nat @ T2 @ A3 )
=> ( member_nat_set_nat @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_203_subset__iff,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
! [T2: set_set_nat] :
( ( member_set_set_nat @ T2 @ A3 )
=> ( member_set_set_nat @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_204_subset__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
! [T2: set_nat] :
( ( member_set_nat @ T2 @ A3 )
=> ( member_set_nat @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_205_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A3 )
=> ( member_nat @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_206_psubset__eq,axiom,
( ord_le152980574450754630et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_207_psubset__eq,axiom,
( ord_less_set_set_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_208_psubset__eq,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_209_equalityD2,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( A2 = B2 )
=> ( ord_le9131159989063066194et_nat @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_210_equalityD2,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( A2 = B2 )
=> ( ord_le6893508408891458716et_nat @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_211_equalityD2,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 = B2 )
=> ( ord_less_eq_set_nat @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_212_equalityD1,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( A2 = B2 )
=> ( ord_le9131159989063066194et_nat @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_213_equalityD1,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( A2 = B2 )
=> ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_214_equalityD1,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 = B2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_215_subset__eq,axiom,
( ord_le1585852046946910987et_nat
= ( ^ [A3: set_nat_set_nat,B3: set_nat_set_nat] :
! [X2: nat > set_nat] :
( ( member_nat_set_nat @ X2 @ A3 )
=> ( member_nat_set_nat @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_216_subset__eq,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A3 )
=> ( member_set_set_nat @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_217_subset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
! [X2: set_nat] :
( ( member_set_nat @ X2 @ A3 )
=> ( member_set_nat @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_218_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A3 )
=> ( member_nat @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_219_equalityE,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( A2 = B2 )
=> ~ ( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ~ ( ord_le9131159989063066194et_nat @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_220_equalityE,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( A2 = B2 )
=> ~ ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ~ ( ord_le6893508408891458716et_nat @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_221_equalityE,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ~ ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_222_psubsetE,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A2 @ B2 )
=> ~ ( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ord_le9131159989063066194et_nat @ B2 @ A2 ) ) ) ).
% psubsetE
thf(fact_223_psubsetE,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_less_set_set_nat @ A2 @ B2 )
=> ~ ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ B2 @ A2 ) ) ) ).
% psubsetE
thf(fact_224_psubsetE,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ).
% psubsetE
thf(fact_225_subsetD,axiom,
! [A2: set_nat_set_nat,B2: set_nat_set_nat,C2: nat > set_nat] :
( ( ord_le1585852046946910987et_nat @ A2 @ B2 )
=> ( ( member_nat_set_nat @ C2 @ A2 )
=> ( member_nat_set_nat @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_226_subsetD,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C2: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( member_set_set_nat @ C2 @ A2 )
=> ( member_set_set_nat @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_227_subsetD,axiom,
! [A2: set_set_nat,B2: set_set_nat,C2: set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( member_set_nat @ C2 @ A2 )
=> ( member_set_nat @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_228_subsetD,axiom,
! [A2: set_nat,B2: set_nat,C2: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C2 @ A2 )
=> ( member_nat @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_229_in__mono,axiom,
! [A2: set_nat_set_nat,B2: set_nat_set_nat,X: nat > set_nat] :
( ( ord_le1585852046946910987et_nat @ A2 @ B2 )
=> ( ( member_nat_set_nat @ X @ A2 )
=> ( member_nat_set_nat @ X @ B2 ) ) ) ).
% in_mono
thf(fact_230_in__mono,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,X: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( member_set_set_nat @ X @ A2 )
=> ( member_set_set_nat @ X @ B2 ) ) ) ).
% in_mono
thf(fact_231_in__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat,X: set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( member_set_nat @ X @ A2 )
=> ( member_set_nat @ X @ B2 ) ) ) ).
% in_mono
thf(fact_232_in__mono,axiom,
! [A2: set_nat,B2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B2 ) ) ) ).
% in_mono
thf(fact_233_atLeastLessThan__subset__iff,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ A @ B ) @ ( set_or4665077453230672383an_nat @ C2 @ D ) )
=> ( ( ord_less_eq_nat @ B @ A )
| ( ( ord_less_eq_nat @ C2 @ A )
& ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_234_all__subset__image,axiom,
! [F: set_set_nat > set_set_nat,A2: set_set_set_nat,P: set_set_set_nat > $o] :
( ( ! [B3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B3 @ ( image_7884819252390400639et_nat @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B3 @ A2 )
=> ( P @ ( image_7884819252390400639et_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_235_all__subset__image,axiom,
! [F: set_nat > set_set_nat,A2: set_set_nat,P: set_set_set_nat > $o] :
( ( ! [B3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B3 @ ( image_6725021117256019401et_nat @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B3 @ A2 )
=> ( P @ ( image_6725021117256019401et_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_236_all__subset__image,axiom,
! [F: nat > set_set_nat,A2: set_nat,P: set_set_set_nat > $o] :
( ( ! [B3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B3 @ ( image_2194112158459175443et_nat @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( P @ ( image_2194112158459175443et_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_237_all__subset__image,axiom,
! [F: set_set_nat > set_nat,A2: set_set_set_nat,P: set_set_nat > $o] :
( ( ! [B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B3 @ ( image_5842784325960735177et_nat @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B3 @ A2 )
=> ( P @ ( image_5842784325960735177et_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_238_all__subset__image,axiom,
! [F: set_nat > set_nat,A2: set_set_nat,P: set_set_nat > $o] :
( ( ! [B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B3 @ ( image_7916887816326733075et_nat @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B3 @ A2 )
=> ( P @ ( image_7916887816326733075et_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_239_all__subset__image,axiom,
! [F: nat > set_nat,A2: set_nat,P: set_set_nat > $o] :
( ( ! [B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B3 @ ( image_nat_set_nat @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( P @ ( image_nat_set_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_240_all__subset__image,axiom,
! [F: set_set_nat > nat,A2: set_set_set_nat,P: set_nat > $o] :
( ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ ( image_1454916318497077779at_nat @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B3 @ A2 )
=> ( P @ ( image_1454916318497077779at_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_241_all__subset__image,axiom,
! [F: set_nat > nat,A2: set_set_nat,P: set_nat > $o] :
( ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ ( image_set_nat_nat @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B3 @ A2 )
=> ( P @ ( image_set_nat_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_242_all__subset__image,axiom,
! [F: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( P @ ( image_nat_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_243_finite__has__maximal2,axiom,
! [A2: set_nat_set_nat,A: nat > set_nat] :
( ( finite722436868047473932et_nat @ A2 )
=> ( ( member_nat_set_nat @ A @ A2 )
=> ? [X3: nat > set_nat] :
( ( member_nat_set_nat @ X3 @ A2 )
& ( ord_le6195038898401538645et_nat @ A @ X3 )
& ! [Xa: nat > set_nat] :
( ( member_nat_set_nat @ Xa @ A2 )
=> ( ( ord_le6195038898401538645et_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_244_finite__has__maximal2,axiom,
! [A2: set_set_set_set_nat,A: set_set_set_nat] :
( ( finite5926941155766903689et_nat @ A2 )
=> ( ( member2946998982187404937et_nat @ A @ A2 )
=> ? [X3: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X3 @ A2 )
& ( ord_le9131159989063066194et_nat @ A @ X3 )
& ! [Xa: set_set_set_nat] :
( ( member2946998982187404937et_nat @ Xa @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_245_finite__has__maximal2,axiom,
! [A2: set_set_set_nat,A: set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( member_set_set_nat @ A @ A2 )
=> ? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A2 )
& ( ord_le6893508408891458716et_nat @ A @ X3 )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_246_finite__has__maximal2,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ A @ A2 )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
& ( ord_less_eq_set_nat @ A @ X3 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_247_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ A @ X3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_248_finite__has__minimal2,axiom,
! [A2: set_nat_set_nat,A: nat > set_nat] :
( ( finite722436868047473932et_nat @ A2 )
=> ( ( member_nat_set_nat @ A @ A2 )
=> ? [X3: nat > set_nat] :
( ( member_nat_set_nat @ X3 @ A2 )
& ( ord_le6195038898401538645et_nat @ X3 @ A )
& ! [Xa: nat > set_nat] :
( ( member_nat_set_nat @ Xa @ A2 )
=> ( ( ord_le6195038898401538645et_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_249_finite__has__minimal2,axiom,
! [A2: set_set_set_set_nat,A: set_set_set_nat] :
( ( finite5926941155766903689et_nat @ A2 )
=> ( ( member2946998982187404937et_nat @ A @ A2 )
=> ? [X3: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X3 @ A2 )
& ( ord_le9131159989063066194et_nat @ X3 @ A )
& ! [Xa: set_set_set_nat] :
( ( member2946998982187404937et_nat @ Xa @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_250_finite__has__minimal2,axiom,
! [A2: set_set_set_nat,A: set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( member_set_set_nat @ A @ A2 )
=> ? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A2 )
& ( ord_le6893508408891458716et_nat @ X3 @ A )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_251_finite__has__minimal2,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ A @ A2 )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
& ( ord_less_eq_set_nat @ X3 @ A )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_252_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ X3 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_253_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M3: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N5 )
=> ( ord_less_nat @ X2 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_254_bounded__nat__set__is__finite,axiom,
! [N3: set_nat,N: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ N3 )
=> ( ord_less_nat @ X3 @ N ) )
=> ( finite_finite_nat @ N3 ) ) ).
% bounded_nat_set_is_finite
thf(fact_255_finite__subset,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( finite6739761609112101331et_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_256_finite__subset,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( finite1152437895449049373et_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_257_finite__subset,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_258_infinite__super,axiom,
! [S2: set_set_set_nat,T3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ S2 @ T3 )
=> ( ~ ( finite6739761609112101331et_nat @ S2 )
=> ~ ( finite6739761609112101331et_nat @ T3 ) ) ) ).
% infinite_super
thf(fact_259_infinite__super,axiom,
! [S2: set_set_nat,T3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ S2 @ T3 )
=> ( ~ ( finite1152437895449049373et_nat @ S2 )
=> ~ ( finite1152437895449049373et_nat @ T3 ) ) ) ).
% infinite_super
thf(fact_260_infinite__super,axiom,
! [S2: set_nat,T3: set_nat] :
( ( ord_less_eq_set_nat @ S2 @ T3 )
=> ( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ T3 ) ) ) ).
% infinite_super
thf(fact_261_rev__finite__subset,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( finite6739761609112101331et_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_262_rev__finite__subset,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( finite1152437895449049373et_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_263_rev__finite__subset,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_264_Collect__subset,axiom,
! [A2: set_nat_set_nat,P: ( nat > set_nat ) > $o] :
( ord_le1585852046946910987et_nat
@ ( collect_nat_set_nat
@ ^ [X2: nat > set_nat] :
( ( member_nat_set_nat @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_265_Collect__subset,axiom,
! [A2: set_set_set_nat,P: set_set_nat > $o] :
( ord_le9131159989063066194et_nat
@ ( collect_set_set_nat
@ ^ [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_266_Collect__subset,axiom,
! [A2: set_set_nat,P: set_nat > $o] :
( ord_le6893508408891458716et_nat
@ ( collect_set_nat
@ ^ [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_267_Collect__subset,axiom,
! [A2: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_268_first__assumptions_Oplucking__step_Ocong,axiom,
clique4095374090462327202g_step = clique4095374090462327202g_step ).
% first_assumptions.plucking_step.cong
thf(fact_269_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K2: nat] :
( ( P @ K2 )
& ( ord_less_nat @ K2 @ I ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_270_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_271_image__mono,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,F: set_set_nat > set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ord_le9131159989063066194et_nat @ ( image_7884819252390400639et_nat @ F @ A2 ) @ ( image_7884819252390400639et_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_272_image__mono,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,F: set_set_nat > set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( image_5842784325960735177et_nat @ F @ A2 ) @ ( image_5842784325960735177et_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_273_image__mono,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,F: set_set_nat > nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ ( image_1454916318497077779at_nat @ F @ A2 ) @ ( image_1454916318497077779at_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_274_image__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat,F: set_nat > set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ord_le9131159989063066194et_nat @ ( image_6725021117256019401et_nat @ F @ A2 ) @ ( image_6725021117256019401et_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_275_image__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat,F: set_nat > set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) @ ( image_7916887816326733075et_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_276_image__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat,F: set_nat > nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ ( image_set_nat_nat @ F @ A2 ) @ ( image_set_nat_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_277_image__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > set_set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_le9131159989063066194et_nat @ ( image_2194112158459175443et_nat @ F @ A2 ) @ ( image_2194112158459175443et_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_278_image__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A2 ) @ ( image_nat_set_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_279_image__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_280_image__subsetI,axiom,
! [A2: set_nat,F: nat > nat,B2: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_281_image__subsetI,axiom,
! [A2: set_nat,F: nat > set_nat,B2: set_set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_set_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_282_image__subsetI,axiom,
! [A2: set_set_nat,F: set_nat > nat,B2: set_nat] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_set_nat_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_283_image__subsetI,axiom,
! [A2: set_nat,F: nat > set_set_nat,B2: set_set_set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_set_set_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_le9131159989063066194et_nat @ ( image_2194112158459175443et_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_284_image__subsetI,axiom,
! [A2: set_set_nat,F: set_nat > set_nat,B2: set_set_nat] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( member_set_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_285_image__subsetI,axiom,
! [A2: set_set_set_nat,F: set_set_nat > nat,B2: set_nat] :
( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_1454916318497077779at_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_286_image__subsetI,axiom,
! [A2: set_nat,F: nat > nat > set_nat,B2: set_nat_set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat_set_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_le1585852046946910987et_nat @ ( image_4436799006340492620et_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_287_image__subsetI,axiom,
! [A2: set_set_nat,F: set_nat > set_set_nat,B2: set_set_set_nat] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( member_set_set_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_le9131159989063066194et_nat @ ( image_6725021117256019401et_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_288_image__subsetI,axiom,
! [A2: set_set_set_nat,F: set_set_nat > set_nat,B2: set_set_nat] :
( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A2 )
=> ( member_set_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ ( image_5842784325960735177et_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_289_image__subsetI,axiom,
! [A2: set_nat_set_nat,F: ( nat > set_nat ) > nat,B2: set_nat] :
( ! [X3: nat > set_nat] :
( ( member_nat_set_nat @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_970537773860477644at_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_290_subset__imageE,axiom,
! [B2: set_set_set_nat,F: set_set_nat > set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ ( image_7884819252390400639et_nat @ F @ A2 ) )
=> ~ ! [C3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C3 @ A2 )
=> ( B2
!= ( image_7884819252390400639et_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_291_subset__imageE,axiom,
! [B2: set_set_set_nat,F: set_nat > set_set_nat,A2: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ ( image_6725021117256019401et_nat @ F @ A2 ) )
=> ~ ! [C3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C3 @ A2 )
=> ( B2
!= ( image_6725021117256019401et_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_292_subset__imageE,axiom,
! [B2: set_set_set_nat,F: nat > set_set_nat,A2: set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ ( image_2194112158459175443et_nat @ F @ A2 ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A2 )
=> ( B2
!= ( image_2194112158459175443et_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_293_subset__imageE,axiom,
! [B2: set_set_nat,F: set_set_nat > set_nat,A2: set_set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_5842784325960735177et_nat @ F @ A2 ) )
=> ~ ! [C3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C3 @ A2 )
=> ( B2
!= ( image_5842784325960735177et_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_294_subset__imageE,axiom,
! [B2: set_set_nat,F: set_nat > set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_7916887816326733075et_nat @ F @ A2 ) )
=> ~ ! [C3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C3 @ A2 )
=> ( B2
!= ( image_7916887816326733075et_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_295_subset__imageE,axiom,
! [B2: set_set_nat,F: nat > set_nat,A2: set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_nat_set_nat @ F @ A2 ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A2 )
=> ( B2
!= ( image_nat_set_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_296_subset__imageE,axiom,
! [B2: set_nat,F: set_set_nat > nat,A2: set_set_set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_1454916318497077779at_nat @ F @ A2 ) )
=> ~ ! [C3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C3 @ A2 )
=> ( B2
!= ( image_1454916318497077779at_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_297_subset__imageE,axiom,
! [B2: set_nat,F: set_nat > nat,A2: set_set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_set_nat_nat @ F @ A2 ) )
=> ~ ! [C3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C3 @ A2 )
=> ( B2
!= ( image_set_nat_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_298_subset__imageE,axiom,
! [B2: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A2 )
=> ( B2
!= ( image_nat_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_299_image__subset__iff,axiom,
! [F: nat > set_set_nat,A2: set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( image_2194112158459175443et_nat @ F @ A2 ) @ B2 )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_set_set_nat @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_300_image__subset__iff,axiom,
! [F: set_set_nat > set_nat,A2: set_set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_5842784325960735177et_nat @ F @ A2 ) @ B2 )
= ( ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A2 )
=> ( member_set_nat @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_301_image__subset__iff,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B2 )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_302_subset__image__iff,axiom,
! [B2: set_set_set_nat,F: set_set_nat > set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ ( image_7884819252390400639et_nat @ F @ A2 ) )
= ( ? [AA: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ AA @ A2 )
& ( B2
= ( image_7884819252390400639et_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_303_subset__image__iff,axiom,
! [B2: set_set_set_nat,F: set_nat > set_set_nat,A2: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ ( image_6725021117256019401et_nat @ F @ A2 ) )
= ( ? [AA: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ AA @ A2 )
& ( B2
= ( image_6725021117256019401et_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_304_subset__image__iff,axiom,
! [B2: set_set_set_nat,F: nat > set_set_nat,A2: set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ ( image_2194112158459175443et_nat @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B2
= ( image_2194112158459175443et_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_305_subset__image__iff,axiom,
! [B2: set_set_nat,F: set_set_nat > set_nat,A2: set_set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_5842784325960735177et_nat @ F @ A2 ) )
= ( ? [AA: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ AA @ A2 )
& ( B2
= ( image_5842784325960735177et_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_306_subset__image__iff,axiom,
! [B2: set_set_nat,F: set_nat > set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_7916887816326733075et_nat @ F @ A2 ) )
= ( ? [AA: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ AA @ A2 )
& ( B2
= ( image_7916887816326733075et_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_307_subset__image__iff,axiom,
! [B2: set_set_nat,F: nat > set_nat,A2: set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_nat_set_nat @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B2
= ( image_nat_set_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_308_subset__image__iff,axiom,
! [B2: set_nat,F: set_set_nat > nat,A2: set_set_set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_1454916318497077779at_nat @ F @ A2 ) )
= ( ? [AA: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ AA @ A2 )
& ( B2
= ( image_1454916318497077779at_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_309_subset__image__iff,axiom,
! [B2: set_nat,F: set_nat > nat,A2: set_set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_set_nat_nat @ F @ A2 ) )
= ( ? [AA: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ AA @ A2 )
& ( B2
= ( image_set_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_310_subset__image__iff,axiom,
! [B2: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B2
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_311_all__finite__subset__image,axiom,
! [F: set_set_nat > set_set_nat,A2: set_set_set_nat,P: set_set_set_nat > $o] :
( ( ! [B3: set_set_set_nat] :
( ( ( finite6739761609112101331et_nat @ B3 )
& ( ord_le9131159989063066194et_nat @ B3 @ ( image_7884819252390400639et_nat @ F @ A2 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_set_set_nat] :
( ( ( finite6739761609112101331et_nat @ B3 )
& ( ord_le9131159989063066194et_nat @ B3 @ A2 ) )
=> ( P @ ( image_7884819252390400639et_nat @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_312_all__finite__subset__image,axiom,
! [F: set_nat > set_set_nat,A2: set_set_nat,P: set_set_set_nat > $o] :
( ( ! [B3: set_set_set_nat] :
( ( ( finite6739761609112101331et_nat @ B3 )
& ( ord_le9131159989063066194et_nat @ B3 @ ( image_6725021117256019401et_nat @ F @ A2 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_set_nat] :
( ( ( finite1152437895449049373et_nat @ B3 )
& ( ord_le6893508408891458716et_nat @ B3 @ A2 ) )
=> ( P @ ( image_6725021117256019401et_nat @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_313_all__finite__subset__image,axiom,
! [F: nat > set_set_nat,A2: set_nat,P: set_set_set_nat > $o] :
( ( ! [B3: set_set_set_nat] :
( ( ( finite6739761609112101331et_nat @ B3 )
& ( ord_le9131159989063066194et_nat @ B3 @ ( image_2194112158459175443et_nat @ F @ A2 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A2 ) )
=> ( P @ ( image_2194112158459175443et_nat @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_314_all__finite__subset__image,axiom,
! [F: set_set_nat > set_nat,A2: set_set_set_nat,P: set_set_nat > $o] :
( ( ! [B3: set_set_nat] :
( ( ( finite1152437895449049373et_nat @ B3 )
& ( ord_le6893508408891458716et_nat @ B3 @ ( image_5842784325960735177et_nat @ F @ A2 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_set_set_nat] :
( ( ( finite6739761609112101331et_nat @ B3 )
& ( ord_le9131159989063066194et_nat @ B3 @ A2 ) )
=> ( P @ ( image_5842784325960735177et_nat @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_315_all__finite__subset__image,axiom,
! [F: set_nat > set_nat,A2: set_set_nat,P: set_set_nat > $o] :
( ( ! [B3: set_set_nat] :
( ( ( finite1152437895449049373et_nat @ B3 )
& ( ord_le6893508408891458716et_nat @ B3 @ ( image_7916887816326733075et_nat @ F @ A2 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_set_nat] :
( ( ( finite1152437895449049373et_nat @ B3 )
& ( ord_le6893508408891458716et_nat @ B3 @ A2 ) )
=> ( P @ ( image_7916887816326733075et_nat @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_316_all__finite__subset__image,axiom,
! [F: nat > set_nat,A2: set_nat,P: set_set_nat > $o] :
( ( ! [B3: set_set_nat] :
( ( ( finite1152437895449049373et_nat @ B3 )
& ( ord_le6893508408891458716et_nat @ B3 @ ( image_nat_set_nat @ F @ A2 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A2 ) )
=> ( P @ ( image_nat_set_nat @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_317_all__finite__subset__image,axiom,
! [F: set_set_nat > nat,A2: set_set_set_nat,P: set_nat > $o] :
( ( ! [B3: set_nat] :
( ( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ ( image_1454916318497077779at_nat @ F @ A2 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_set_set_nat] :
( ( ( finite6739761609112101331et_nat @ B3 )
& ( ord_le9131159989063066194et_nat @ B3 @ A2 ) )
=> ( P @ ( image_1454916318497077779at_nat @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_318_all__finite__subset__image,axiom,
! [F: set_nat > nat,A2: set_set_nat,P: set_nat > $o] :
( ( ! [B3: set_nat] :
( ( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ ( image_set_nat_nat @ F @ A2 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_set_nat] :
( ( ( finite1152437895449049373et_nat @ B3 )
& ( ord_le6893508408891458716et_nat @ B3 @ A2 ) )
=> ( P @ ( image_set_nat_nat @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_319_all__finite__subset__image,axiom,
! [F: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ! [B3: set_nat] :
( ( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A2 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A2 ) )
=> ( P @ ( image_nat_nat @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_320_ex__finite__subset__image,axiom,
! [F: set_set_nat > set_set_nat,A2: set_set_set_nat,P: set_set_set_nat > $o] :
( ( ? [B3: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B3 )
& ( ord_le9131159989063066194et_nat @ B3 @ ( image_7884819252390400639et_nat @ F @ A2 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B3 )
& ( ord_le9131159989063066194et_nat @ B3 @ A2 )
& ( P @ ( image_7884819252390400639et_nat @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_321_ex__finite__subset__image,axiom,
! [F: set_nat > set_set_nat,A2: set_set_nat,P: set_set_set_nat > $o] :
( ( ? [B3: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B3 )
& ( ord_le9131159989063066194et_nat @ B3 @ ( image_6725021117256019401et_nat @ F @ A2 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_set_nat] :
( ( finite1152437895449049373et_nat @ B3 )
& ( ord_le6893508408891458716et_nat @ B3 @ A2 )
& ( P @ ( image_6725021117256019401et_nat @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_322_ex__finite__subset__image,axiom,
! [F: nat > set_set_nat,A2: set_nat,P: set_set_set_nat > $o] :
( ( ? [B3: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B3 )
& ( ord_le9131159989063066194et_nat @ B3 @ ( image_2194112158459175443et_nat @ F @ A2 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A2 )
& ( P @ ( image_2194112158459175443et_nat @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_323_ex__finite__subset__image,axiom,
! [F: set_set_nat > set_nat,A2: set_set_set_nat,P: set_set_nat > $o] :
( ( ? [B3: set_set_nat] :
( ( finite1152437895449049373et_nat @ B3 )
& ( ord_le6893508408891458716et_nat @ B3 @ ( image_5842784325960735177et_nat @ F @ A2 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B3 )
& ( ord_le9131159989063066194et_nat @ B3 @ A2 )
& ( P @ ( image_5842784325960735177et_nat @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_324_ex__finite__subset__image,axiom,
! [F: set_nat > set_nat,A2: set_set_nat,P: set_set_nat > $o] :
( ( ? [B3: set_set_nat] :
( ( finite1152437895449049373et_nat @ B3 )
& ( ord_le6893508408891458716et_nat @ B3 @ ( image_7916887816326733075et_nat @ F @ A2 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_set_nat] :
( ( finite1152437895449049373et_nat @ B3 )
& ( ord_le6893508408891458716et_nat @ B3 @ A2 )
& ( P @ ( image_7916887816326733075et_nat @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_325_ex__finite__subset__image,axiom,
! [F: nat > set_nat,A2: set_nat,P: set_set_nat > $o] :
( ( ? [B3: set_set_nat] :
( ( finite1152437895449049373et_nat @ B3 )
& ( ord_le6893508408891458716et_nat @ B3 @ ( image_nat_set_nat @ F @ A2 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A2 )
& ( P @ ( image_nat_set_nat @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_326_ex__finite__subset__image,axiom,
! [F: set_set_nat > nat,A2: set_set_set_nat,P: set_nat > $o] :
( ( ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ ( image_1454916318497077779at_nat @ F @ A2 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B3 )
& ( ord_le9131159989063066194et_nat @ B3 @ A2 )
& ( P @ ( image_1454916318497077779at_nat @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_327_ex__finite__subset__image,axiom,
! [F: set_nat > nat,A2: set_set_nat,P: set_nat > $o] :
( ( ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ ( image_set_nat_nat @ F @ A2 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_set_nat] :
( ( finite1152437895449049373et_nat @ B3 )
& ( ord_le6893508408891458716et_nat @ B3 @ A2 )
& ( P @ ( image_set_nat_nat @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_328_ex__finite__subset__image,axiom,
! [F: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A2 )
& ( P @ ( image_nat_nat @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_329_finite__subset__image,axiom,
! [B2: set_set_set_nat,F: set_set_nat > set_set_nat,A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ ( image_7884819252390400639et_nat @ F @ A2 ) )
=> ? [C3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C3 @ A2 )
& ( finite6739761609112101331et_nat @ C3 )
& ( B2
= ( image_7884819252390400639et_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_330_finite__subset__image,axiom,
! [B2: set_set_set_nat,F: set_nat > set_set_nat,A2: set_set_nat] :
( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ ( image_6725021117256019401et_nat @ F @ A2 ) )
=> ? [C3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C3 @ A2 )
& ( finite1152437895449049373et_nat @ C3 )
& ( B2
= ( image_6725021117256019401et_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_331_finite__subset__image,axiom,
! [B2: set_set_set_nat,F: nat > set_set_nat,A2: set_nat] :
( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ ( image_2194112158459175443et_nat @ F @ A2 ) )
=> ? [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A2 )
& ( finite_finite_nat @ C3 )
& ( B2
= ( image_2194112158459175443et_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_332_finite__subset__image,axiom,
! [B2: set_set_nat,F: set_set_nat > set_nat,A2: set_set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ ( image_5842784325960735177et_nat @ F @ A2 ) )
=> ? [C3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C3 @ A2 )
& ( finite6739761609112101331et_nat @ C3 )
& ( B2
= ( image_5842784325960735177et_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_333_finite__subset__image,axiom,
! [B2: set_set_nat,F: set_nat > set_nat,A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ ( image_7916887816326733075et_nat @ F @ A2 ) )
=> ? [C3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C3 @ A2 )
& ( finite1152437895449049373et_nat @ C3 )
& ( B2
= ( image_7916887816326733075et_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_334_finite__subset__image,axiom,
! [B2: set_set_nat,F: nat > set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ ( image_nat_set_nat @ F @ A2 ) )
=> ? [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A2 )
& ( finite_finite_nat @ C3 )
& ( B2
= ( image_nat_set_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_335_finite__subset__image,axiom,
! [B2: set_nat,F: set_set_nat > nat,A2: set_set_set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_1454916318497077779at_nat @ F @ A2 ) )
=> ? [C3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C3 @ A2 )
& ( finite6739761609112101331et_nat @ C3 )
& ( B2
= ( image_1454916318497077779at_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_336_finite__subset__image,axiom,
! [B2: set_nat,F: set_nat > nat,A2: set_set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_set_nat_nat @ F @ A2 ) )
=> ? [C3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C3 @ A2 )
& ( finite1152437895449049373et_nat @ C3 )
& ( B2
= ( image_set_nat_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_337_finite__subset__image,axiom,
! [B2: set_nat,F: nat > nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
=> ? [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A2 )
& ( finite_finite_nat @ C3 )
& ( B2
= ( image_nat_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_338_finite__surj,axiom,
! [A2: set_set_nat,B2: set_set_set_nat,F: set_nat > set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ ( image_6725021117256019401et_nat @ F @ A2 ) )
=> ( finite6739761609112101331et_nat @ B2 ) ) ) ).
% finite_surj
thf(fact_339_finite__surj,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,F: set_set_nat > set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ ( image_7884819252390400639et_nat @ F @ A2 ) )
=> ( finite6739761609112101331et_nat @ B2 ) ) ) ).
% finite_surj
thf(fact_340_finite__surj,axiom,
! [A2: set_nat,B2: set_set_set_nat,F: nat > set_set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ ( image_2194112158459175443et_nat @ F @ A2 ) )
=> ( finite6739761609112101331et_nat @ B2 ) ) ) ).
% finite_surj
thf(fact_341_finite__surj,axiom,
! [A2: set_set_nat,B2: set_set_nat,F: set_nat > set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ ( image_7916887816326733075et_nat @ F @ A2 ) )
=> ( finite1152437895449049373et_nat @ B2 ) ) ) ).
% finite_surj
thf(fact_342_finite__surj,axiom,
! [A2: set_set_set_nat,B2: set_set_nat,F: set_set_nat > set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ ( image_5842784325960735177et_nat @ F @ A2 ) )
=> ( finite1152437895449049373et_nat @ B2 ) ) ) ).
% finite_surj
thf(fact_343_finite__surj,axiom,
! [A2: set_nat,B2: set_set_nat,F: nat > set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ ( image_nat_set_nat @ F @ A2 ) )
=> ( finite1152437895449049373et_nat @ B2 ) ) ) ).
% finite_surj
thf(fact_344_finite__surj,axiom,
! [A2: set_set_nat,B2: set_nat,F: set_nat > nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_set_nat_nat @ F @ A2 ) )
=> ( finite_finite_nat @ B2 ) ) ) ).
% finite_surj
thf(fact_345_finite__surj,axiom,
! [A2: set_set_set_nat,B2: set_nat,F: set_set_nat > nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_1454916318497077779at_nat @ F @ A2 ) )
=> ( finite_finite_nat @ B2 ) ) ) ).
% finite_surj
thf(fact_346_finite__surj,axiom,
! [A2: set_nat,B2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
=> ( finite_finite_nat @ B2 ) ) ) ).
% finite_surj
thf(fact_347_finite__surj__inj,axiom,
! [A2: set_set_set_nat,F: set_set_nat > set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ A2 @ ( image_7884819252390400639et_nat @ F @ A2 ) )
=> ( inj_on2040386338155636715et_nat @ F @ A2 ) ) ) ).
% finite_surj_inj
thf(fact_348_finite__surj__inj,axiom,
! [A2: set_set_nat,F: set_nat > set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ ( image_7916887816326733075et_nat @ F @ A2 ) )
=> ( inj_on4604407203859583615et_nat @ F @ A2 ) ) ) ).
% finite_surj_inj
thf(fact_349_finite__surj__inj,axiom,
! [A2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ ( image_nat_nat @ F @ A2 ) )
=> ( inj_on_nat_nat @ F @ A2 ) ) ) ).
% finite_surj_inj
thf(fact_350_inj__on__finite,axiom,
! [F: set_nat > set_set_nat,A2: set_set_nat,B2: set_set_set_nat] :
( ( inj_on2776966659131765557et_nat @ F @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ ( image_6725021117256019401et_nat @ F @ A2 ) @ B2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( finite1152437895449049373et_nat @ A2 ) ) ) ) ).
% inj_on_finite
thf(fact_351_inj__on__finite,axiom,
! [F: set_set_nat > set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( inj_on2040386338155636715et_nat @ F @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ ( image_7884819252390400639et_nat @ F @ A2 ) @ B2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( finite6739761609112101331et_nat @ A2 ) ) ) ) ).
% inj_on_finite
thf(fact_352_inj__on__finite,axiom,
! [F: nat > set_set_nat,A2: set_nat,B2: set_set_set_nat] :
( ( inj_on8105003582846801791et_nat @ F @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ ( image_2194112158459175443et_nat @ F @ A2 ) @ B2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ) ).
% inj_on_finite
thf(fact_353_inj__on__finite,axiom,
! [F: set_nat > set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( inj_on4604407203859583615et_nat @ F @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) @ B2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( finite1152437895449049373et_nat @ A2 ) ) ) ) ).
% inj_on_finite
thf(fact_354_inj__on__finite,axiom,
! [F: set_set_nat > set_nat,A2: set_set_set_nat,B2: set_set_nat] :
( ( inj_on1894729867836481333et_nat @ F @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ ( image_5842784325960735177et_nat @ F @ A2 ) @ B2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( finite6739761609112101331et_nat @ A2 ) ) ) ) ).
% inj_on_finite
thf(fact_355_inj__on__finite,axiom,
! [F: nat > set_nat,A2: set_nat,B2: set_set_nat] :
( ( inj_on_nat_set_nat @ F @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A2 ) @ B2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ) ).
% inj_on_finite
thf(fact_356_inj__on__finite,axiom,
! [F: set_nat > nat,A2: set_set_nat,B2: set_nat] :
( ( inj_on_set_nat_nat @ F @ A2 )
=> ( ( ord_less_eq_set_nat @ ( image_set_nat_nat @ F @ A2 ) @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( finite1152437895449049373et_nat @ A2 ) ) ) ) ).
% inj_on_finite
thf(fact_357_inj__on__finite,axiom,
! [F: set_set_nat > nat,A2: set_set_set_nat,B2: set_nat] :
( ( inj_on7365807742884704127at_nat @ F @ A2 )
=> ( ( ord_less_eq_set_nat @ ( image_1454916318497077779at_nat @ F @ A2 ) @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( finite6739761609112101331et_nat @ A2 ) ) ) ) ).
% inj_on_finite
thf(fact_358_inj__on__finite,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( inj_on_nat_nat @ F @ A2 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ) ).
% inj_on_finite
thf(fact_359_endo__inj__surj,axiom,
! [A2: set_set_set_nat,F: set_set_nat > set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ ( image_7884819252390400639et_nat @ F @ A2 ) @ A2 )
=> ( ( inj_on2040386338155636715et_nat @ F @ A2 )
=> ( ( image_7884819252390400639et_nat @ F @ A2 )
= A2 ) ) ) ) ).
% endo_inj_surj
thf(fact_360_endo__inj__surj,axiom,
! [A2: set_set_nat,F: set_nat > set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) @ A2 )
=> ( ( inj_on4604407203859583615et_nat @ F @ A2 )
=> ( ( image_7916887816326733075et_nat @ F @ A2 )
= A2 ) ) ) ) ).
% endo_inj_surj
thf(fact_361_endo__inj__surj,axiom,
! [A2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ A2 )
=> ( ( inj_on_nat_nat @ F @ A2 )
=> ( ( image_nat_nat @ F @ A2 )
= A2 ) ) ) ) ).
% endo_inj_surj
thf(fact_362_finite__psubset__induct,axiom,
! [A2: set_set_nat,P: set_set_nat > $o] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ! [A4: set_set_nat] :
( ( finite1152437895449049373et_nat @ A4 )
=> ( ! [B4: set_set_nat] :
( ( ord_less_set_set_nat @ B4 @ A4 )
=> ( P @ B4 ) )
=> ( P @ A4 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_363_finite__psubset__induct,axiom,
! [A2: set_set_set_nat,P: set_set_set_nat > $o] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ! [A4: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A4 )
=> ( ! [B4: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ B4 @ A4 )
=> ( P @ B4 ) )
=> ( P @ A4 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_364_finite__psubset__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ! [B4: set_nat] :
( ( ord_less_set_nat @ B4 @ A4 )
=> ( P @ B4 ) )
=> ( P @ A4 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_365_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B2: set_nat,R: nat > nat > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A5: nat] :
( ( member_nat @ A5 @ A2 )
& ( R @ A5 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_366_pigeonhole__infinite__rel,axiom,
! [A2: set_set_nat,B2: set_nat,R: set_nat > nat > $o] :
( ~ ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B2 )
& ~ ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [A5: set_nat] :
( ( member_set_nat @ A5 @ A2 )
& ( R @ A5 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_367_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B2: set_set_nat,R: nat > set_nat > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ? [Xa: set_nat] :
( ( member_set_nat @ Xa @ B2 )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ B2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A5: nat] :
( ( member_nat @ A5 @ A2 )
& ( R @ A5 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_368_pigeonhole__infinite__rel,axiom,
! [A2: set_set_nat,B2: set_set_nat,R: set_nat > set_nat > $o] :
( ~ ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ? [Xa: set_nat] :
( ( member_set_nat @ Xa @ B2 )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ B2 )
& ~ ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [A5: set_nat] :
( ( member_set_nat @ A5 @ A2 )
& ( R @ A5 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_369_pigeonhole__infinite__rel,axiom,
! [A2: set_set_set_nat,B2: set_nat,R: set_set_nat > nat > $o] :
( ~ ( finite6739761609112101331et_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B2 )
& ~ ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [A5: set_set_nat] :
( ( member_set_set_nat @ A5 @ A2 )
& ( R @ A5 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_370_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B2: set_set_set_nat,R: nat > set_set_nat > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ? [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ B2 )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ B2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A5: nat] :
( ( member_nat @ A5 @ A2 )
& ( R @ A5 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_371_pigeonhole__infinite__rel,axiom,
! [A2: set_nat_set_nat,B2: set_nat,R: ( nat > set_nat ) > nat > $o] :
( ~ ( finite722436868047473932et_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X3: nat > set_nat] :
( ( member_nat_set_nat @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B2 )
& ~ ( finite722436868047473932et_nat
@ ( collect_nat_set_nat
@ ^ [A5: nat > set_nat] :
( ( member_nat_set_nat @ A5 @ A2 )
& ( R @ A5 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_372_pigeonhole__infinite__rel,axiom,
! [A2: set_set_nat,B2: set_set_set_nat,R: set_nat > set_set_nat > $o] :
( ~ ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ? [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ B2 )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ B2 )
& ~ ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [A5: set_nat] :
( ( member_set_nat @ A5 @ A2 )
& ( R @ A5 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_373_pigeonhole__infinite__rel,axiom,
! [A2: set_set_set_nat,B2: set_set_nat,R: set_set_nat > set_nat > $o] :
( ~ ( finite6739761609112101331et_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A2 )
=> ? [Xa: set_nat] :
( ( member_set_nat @ Xa @ B2 )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ B2 )
& ~ ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [A5: set_set_nat] :
( ( member_set_set_nat @ A5 @ A2 )
& ( R @ A5 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_374_pigeonhole__infinite__rel,axiom,
! [A2: set_nat_set_nat,B2: set_set_nat,R: ( nat > set_nat ) > set_nat > $o] :
( ~ ( finite722436868047473932et_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ! [X3: nat > set_nat] :
( ( member_nat_set_nat @ X3 @ A2 )
=> ? [Xa: set_nat] :
( ( member_set_nat @ Xa @ B2 )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ B2 )
& ~ ( finite722436868047473932et_nat
@ ( collect_nat_set_nat
@ ^ [A5: nat > set_nat] :
( ( member_nat_set_nat @ A5 @ A2 )
& ( R @ A5 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_375_not__finite__existsD,axiom,
! [P: set_nat > $o] :
( ~ ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
=> ? [X_1: set_nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_376_not__finite__existsD,axiom,
! [P: set_set_nat > $o] :
( ~ ( finite6739761609112101331et_nat @ ( collect_set_set_nat @ P ) )
=> ? [X_1: set_set_nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_377_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_1: nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_378_atLeastLessThan__eq__iff,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C2 @ D ) )
= ( ( A = C2 )
& ( B = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_379_Ico__eq__Ico,axiom,
! [L: nat,H: nat,L2: nat,H2: nat] :
( ( ( set_or4665077453230672383an_nat @ L @ H )
= ( set_or4665077453230672383an_nat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_nat @ L @ H )
& ~ ( ord_less_nat @ L2 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_380_atLeastLessThan__inj_I1_J,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C2 @ D ) )
=> ( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D )
=> ( A = C2 ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_381_atLeastLessThan__inj_I2_J,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C2 @ D ) )
=> ( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D )
=> ( B = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_382_pigeonhole__infinite,axiom,
! [A2: set_nat,F: nat > nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ ( image_nat_nat @ F @ A2 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A5: nat] :
( ( member_nat @ A5 @ A2 )
& ( ( F @ A5 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_383_pigeonhole__infinite,axiom,
! [A2: set_set_nat,F: set_nat > nat] :
( ~ ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite_finite_nat @ ( image_set_nat_nat @ F @ A2 ) )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
& ~ ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [A5: set_nat] :
( ( member_set_nat @ A5 @ A2 )
& ( ( F @ A5 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_384_pigeonhole__infinite,axiom,
! [A2: set_nat,F: nat > set_nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ ( image_nat_set_nat @ F @ A2 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A5: nat] :
( ( member_nat @ A5 @ A2 )
& ( ( F @ A5 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_385_pigeonhole__infinite,axiom,
! [A2: set_set_nat,F: set_nat > set_nat] :
( ~ ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
& ~ ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [A5: set_nat] :
( ( member_set_nat @ A5 @ A2 )
& ( ( F @ A5 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_386_pigeonhole__infinite,axiom,
! [A2: set_set_set_nat,F: set_set_nat > nat] :
( ~ ( finite6739761609112101331et_nat @ A2 )
=> ( ( finite_finite_nat @ ( image_1454916318497077779at_nat @ F @ A2 ) )
=> ? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A2 )
& ~ ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [A5: set_set_nat] :
( ( member_set_set_nat @ A5 @ A2 )
& ( ( F @ A5 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_387_pigeonhole__infinite,axiom,
! [A2: set_nat,F: nat > set_set_nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite6739761609112101331et_nat @ ( image_2194112158459175443et_nat @ F @ A2 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A5: nat] :
( ( member_nat @ A5 @ A2 )
& ( ( F @ A5 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_388_pigeonhole__infinite,axiom,
! [A2: set_nat_set_nat,F: ( nat > set_nat ) > nat] :
( ~ ( finite722436868047473932et_nat @ A2 )
=> ( ( finite_finite_nat @ ( image_970537773860477644at_nat @ F @ A2 ) )
=> ? [X3: nat > set_nat] :
( ( member_nat_set_nat @ X3 @ A2 )
& ~ ( finite722436868047473932et_nat
@ ( collect_nat_set_nat
@ ^ [A5: nat > set_nat] :
( ( member_nat_set_nat @ A5 @ A2 )
& ( ( F @ A5 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_389_pigeonhole__infinite,axiom,
! [A2: set_set_nat,F: set_nat > set_set_nat] :
( ~ ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite6739761609112101331et_nat @ ( image_6725021117256019401et_nat @ F @ A2 ) )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
& ~ ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [A5: set_nat] :
( ( member_set_nat @ A5 @ A2 )
& ( ( F @ A5 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_390_pigeonhole__infinite,axiom,
! [A2: set_set_set_nat,F: set_set_nat > set_nat] :
( ~ ( finite6739761609112101331et_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ ( image_5842784325960735177et_nat @ F @ A2 ) )
=> ? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A2 )
& ~ ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [A5: set_set_nat] :
( ( member_set_set_nat @ A5 @ A2 )
& ( ( F @ A5 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_391_pigeonhole__infinite,axiom,
! [A2: set_nat_set_nat,F: ( nat > set_nat ) > set_nat] :
( ~ ( finite722436868047473932et_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ ( image_8304670887732450946et_nat @ F @ A2 ) )
=> ? [X3: nat > set_nat] :
( ( member_nat_set_nat @ X3 @ A2 )
& ~ ( finite722436868047473932et_nat
@ ( collect_nat_set_nat
@ ^ [A5: nat > set_nat] :
( ( member_nat_set_nat @ A5 @ A2 )
& ( ( F @ A5 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_392_finite__inverse__image__gen,axiom,
! [A2: set_nat,F: nat > nat,D2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( inj_on_nat_nat @ F @ D2 )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [J2: nat] :
( ( member_nat @ J2 @ D2 )
& ( member_nat @ ( F @ J2 ) @ A2 ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_393_finite__inverse__image__gen,axiom,
! [A2: set_set_nat,F: nat > set_nat,D2: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( inj_on_nat_set_nat @ F @ D2 )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [J2: nat] :
( ( member_nat @ J2 @ D2 )
& ( member_set_nat @ ( F @ J2 ) @ A2 ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_394_finite__inverse__image__gen,axiom,
! [A2: set_nat,F: set_nat > nat,D2: set_set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( inj_on_set_nat_nat @ F @ D2 )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [J2: set_nat] :
( ( member_set_nat @ J2 @ D2 )
& ( member_nat @ ( F @ J2 ) @ A2 ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_395_finite__inverse__image__gen,axiom,
! [A2: set_set_nat,F: set_nat > set_nat,D2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( inj_on4604407203859583615et_nat @ F @ D2 )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [J2: set_nat] :
( ( member_set_nat @ J2 @ D2 )
& ( member_set_nat @ ( F @ J2 ) @ A2 ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_396_finite__inverse__image__gen,axiom,
! [A2: set_set_set_nat,F: nat > set_set_nat,D2: set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( inj_on8105003582846801791et_nat @ F @ D2 )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [J2: nat] :
( ( member_nat @ J2 @ D2 )
& ( member_set_set_nat @ ( F @ J2 ) @ A2 ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_397_finite__inverse__image__gen,axiom,
! [A2: set_nat,F: set_set_nat > nat,D2: set_set_set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( inj_on7365807742884704127at_nat @ F @ D2 )
=> ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [J2: set_set_nat] :
( ( member_set_set_nat @ J2 @ D2 )
& ( member_nat @ ( F @ J2 ) @ A2 ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_398_finite__inverse__image__gen,axiom,
! [A2: set_nat_set_nat,F: nat > nat > set_nat,D2: set_nat] :
( ( finite722436868047473932et_nat @ A2 )
=> ( ( inj_on236506435458908344et_nat @ F @ D2 )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [J2: nat] :
( ( member_nat @ J2 @ D2 )
& ( member_nat_set_nat @ ( F @ J2 ) @ A2 ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_399_finite__inverse__image__gen,axiom,
! [A2: set_set_nat,F: set_set_nat > set_nat,D2: set_set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( inj_on1894729867836481333et_nat @ F @ D2 )
=> ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [J2: set_set_nat] :
( ( member_set_set_nat @ J2 @ D2 )
& ( member_set_nat @ ( F @ J2 ) @ A2 ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_400_finite__inverse__image__gen,axiom,
! [A2: set_set_set_nat,F: set_nat > set_set_nat,D2: set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( inj_on2776966659131765557et_nat @ F @ D2 )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [J2: set_nat] :
( ( member_set_nat @ J2 @ D2 )
& ( member_set_set_nat @ ( F @ J2 ) @ A2 ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_401_finite__inverse__image__gen,axiom,
! [A2: set_nat,F: ( nat > set_nat ) > nat,D2: set_nat_set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( inj_on5993617239833669176at_nat @ F @ D2 )
=> ( finite722436868047473932et_nat
@ ( collect_nat_set_nat
@ ^ [J2: nat > set_nat] :
( ( member_nat_set_nat @ J2 @ D2 )
& ( member_nat @ ( F @ J2 ) @ A2 ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_402_finite__image__iff,axiom,
! [F: set_nat > set_nat,A2: set_set_nat] :
( ( inj_on4604407203859583615et_nat @ F @ A2 )
=> ( ( finite1152437895449049373et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) )
= ( finite1152437895449049373et_nat @ A2 ) ) ) ).
% finite_image_iff
thf(fact_403_finite__image__iff,axiom,
! [F: set_set_nat > set_nat,A2: set_set_set_nat] :
( ( inj_on1894729867836481333et_nat @ F @ A2 )
=> ( ( finite1152437895449049373et_nat @ ( image_5842784325960735177et_nat @ F @ A2 ) )
= ( finite6739761609112101331et_nat @ A2 ) ) ) ).
% finite_image_iff
thf(fact_404_finite__image__iff,axiom,
! [F: nat > set_nat,A2: set_nat] :
( ( inj_on_nat_set_nat @ F @ A2 )
=> ( ( finite1152437895449049373et_nat @ ( image_nat_set_nat @ F @ A2 ) )
= ( finite_finite_nat @ A2 ) ) ) ).
% finite_image_iff
thf(fact_405_finite__image__iff,axiom,
! [F: set_nat > set_set_nat,A2: set_set_nat] :
( ( inj_on2776966659131765557et_nat @ F @ A2 )
=> ( ( finite6739761609112101331et_nat @ ( image_6725021117256019401et_nat @ F @ A2 ) )
= ( finite1152437895449049373et_nat @ A2 ) ) ) ).
% finite_image_iff
thf(fact_406_finite__image__iff,axiom,
! [F: set_set_nat > set_set_nat,A2: set_set_set_nat] :
( ( inj_on2040386338155636715et_nat @ F @ A2 )
=> ( ( finite6739761609112101331et_nat @ ( image_7884819252390400639et_nat @ F @ A2 ) )
= ( finite6739761609112101331et_nat @ A2 ) ) ) ).
% finite_image_iff
thf(fact_407_finite__image__iff,axiom,
! [F: nat > set_set_nat,A2: set_nat] :
( ( inj_on8105003582846801791et_nat @ F @ A2 )
=> ( ( finite6739761609112101331et_nat @ ( image_2194112158459175443et_nat @ F @ A2 ) )
= ( finite_finite_nat @ A2 ) ) ) ).
% finite_image_iff
thf(fact_408_finite__image__iff,axiom,
! [F: set_nat > nat,A2: set_set_nat] :
( ( inj_on_set_nat_nat @ F @ A2 )
=> ( ( finite_finite_nat @ ( image_set_nat_nat @ F @ A2 ) )
= ( finite1152437895449049373et_nat @ A2 ) ) ) ).
% finite_image_iff
thf(fact_409_finite__image__iff,axiom,
! [F: set_set_nat > nat,A2: set_set_set_nat] :
( ( inj_on7365807742884704127at_nat @ F @ A2 )
=> ( ( finite_finite_nat @ ( image_1454916318497077779at_nat @ F @ A2 ) )
= ( finite6739761609112101331et_nat @ A2 ) ) ) ).
% finite_image_iff
thf(fact_410_finite__image__iff,axiom,
! [F: nat > nat,A2: set_nat] :
( ( inj_on_nat_nat @ F @ A2 )
=> ( ( finite_finite_nat @ ( image_nat_nat @ F @ A2 ) )
= ( finite_finite_nat @ A2 ) ) ) ).
% finite_image_iff
thf(fact_411_finite__imageD,axiom,
! [F: set_nat > set_nat,A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) )
=> ( ( inj_on4604407203859583615et_nat @ F @ A2 )
=> ( finite1152437895449049373et_nat @ A2 ) ) ) ).
% finite_imageD
thf(fact_412_finite__imageD,axiom,
! [F: set_set_nat > set_nat,A2: set_set_set_nat] :
( ( finite1152437895449049373et_nat @ ( image_5842784325960735177et_nat @ F @ A2 ) )
=> ( ( inj_on1894729867836481333et_nat @ F @ A2 )
=> ( finite6739761609112101331et_nat @ A2 ) ) ) ).
% finite_imageD
thf(fact_413_finite__imageD,axiom,
! [F: nat > set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ ( image_nat_set_nat @ F @ A2 ) )
=> ( ( inj_on_nat_set_nat @ F @ A2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_imageD
thf(fact_414_finite__imageD,axiom,
! [F: set_nat > set_set_nat,A2: set_set_nat] :
( ( finite6739761609112101331et_nat @ ( image_6725021117256019401et_nat @ F @ A2 ) )
=> ( ( inj_on2776966659131765557et_nat @ F @ A2 )
=> ( finite1152437895449049373et_nat @ A2 ) ) ) ).
% finite_imageD
thf(fact_415_finite__imageD,axiom,
! [F: set_set_nat > set_set_nat,A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ ( image_7884819252390400639et_nat @ F @ A2 ) )
=> ( ( inj_on2040386338155636715et_nat @ F @ A2 )
=> ( finite6739761609112101331et_nat @ A2 ) ) ) ).
% finite_imageD
thf(fact_416_finite__imageD,axiom,
! [F: nat > set_set_nat,A2: set_nat] :
( ( finite6739761609112101331et_nat @ ( image_2194112158459175443et_nat @ F @ A2 ) )
=> ( ( inj_on8105003582846801791et_nat @ F @ A2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_imageD
thf(fact_417_finite__imageD,axiom,
! [F: set_nat > nat,A2: set_set_nat] :
( ( finite_finite_nat @ ( image_set_nat_nat @ F @ A2 ) )
=> ( ( inj_on_set_nat_nat @ F @ A2 )
=> ( finite1152437895449049373et_nat @ A2 ) ) ) ).
% finite_imageD
thf(fact_418_finite__imageD,axiom,
! [F: set_set_nat > nat,A2: set_set_set_nat] :
( ( finite_finite_nat @ ( image_1454916318497077779at_nat @ F @ A2 ) )
=> ( ( inj_on7365807742884704127at_nat @ F @ A2 )
=> ( finite6739761609112101331et_nat @ A2 ) ) ) ).
% finite_imageD
thf(fact_419_finite__imageD,axiom,
! [F: nat > nat,A2: set_nat] :
( ( finite_finite_nat @ ( image_nat_nat @ F @ A2 ) )
=> ( ( inj_on_nat_nat @ F @ A2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_imageD
thf(fact_420_i__props_I2_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ p )
=> ( finite_finite_nat @ ( clique5033774636164728513irst_v @ ( g @ I ) ) ) ) ).
% i_props(2)
thf(fact_421_inj__on__image__subset__iff,axiom,
! [F: set_set_nat > set_set_nat,C: set_set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( inj_on2040386338155636715et_nat @ F @ C )
=> ( ( ord_le9131159989063066194et_nat @ A2 @ C )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ C )
=> ( ( ord_le9131159989063066194et_nat @ ( image_7884819252390400639et_nat @ F @ A2 ) @ ( image_7884819252390400639et_nat @ F @ B2 ) )
= ( ord_le9131159989063066194et_nat @ A2 @ B2 ) ) ) ) ) ).
% inj_on_image_subset_iff
thf(fact_422_inj__on__image__subset__iff,axiom,
! [F: set_set_nat > set_nat,C: set_set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( inj_on1894729867836481333et_nat @ F @ C )
=> ( ( ord_le9131159989063066194et_nat @ A2 @ C )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ C )
=> ( ( ord_le6893508408891458716et_nat @ ( image_5842784325960735177et_nat @ F @ A2 ) @ ( image_5842784325960735177et_nat @ F @ B2 ) )
= ( ord_le9131159989063066194et_nat @ A2 @ B2 ) ) ) ) ) ).
% inj_on_image_subset_iff
thf(fact_423_inj__on__image__subset__iff,axiom,
! [F: set_set_nat > nat,C: set_set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( inj_on7365807742884704127at_nat @ F @ C )
=> ( ( ord_le9131159989063066194et_nat @ A2 @ C )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ C )
=> ( ( ord_less_eq_set_nat @ ( image_1454916318497077779at_nat @ F @ A2 ) @ ( image_1454916318497077779at_nat @ F @ B2 ) )
= ( ord_le9131159989063066194et_nat @ A2 @ B2 ) ) ) ) ) ).
% inj_on_image_subset_iff
thf(fact_424_inj__on__image__subset__iff,axiom,
! [F: set_nat > set_set_nat,C: set_set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( inj_on2776966659131765557et_nat @ F @ C )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ C )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ C )
=> ( ( ord_le9131159989063066194et_nat @ ( image_6725021117256019401et_nat @ F @ A2 ) @ ( image_6725021117256019401et_nat @ F @ B2 ) )
= ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ) ) ).
% inj_on_image_subset_iff
thf(fact_425_inj__on__image__subset__iff,axiom,
! [F: set_nat > set_nat,C: set_set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( inj_on4604407203859583615et_nat @ F @ C )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ C )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ C )
=> ( ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) @ ( image_7916887816326733075et_nat @ F @ B2 ) )
= ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ) ) ).
% inj_on_image_subset_iff
thf(fact_426_inj__on__image__subset__iff,axiom,
! [F: set_nat > nat,C: set_set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( inj_on_set_nat_nat @ F @ C )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ C )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ C )
=> ( ( ord_less_eq_set_nat @ ( image_set_nat_nat @ F @ A2 ) @ ( image_set_nat_nat @ F @ B2 ) )
= ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ) ) ).
% inj_on_image_subset_iff
thf(fact_427_inj__on__image__subset__iff,axiom,
! [F: nat > set_set_nat,C: set_nat,A2: set_nat,B2: set_nat] :
( ( inj_on8105003582846801791et_nat @ F @ C )
=> ( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ( ord_le9131159989063066194et_nat @ ( image_2194112158459175443et_nat @ F @ A2 ) @ ( image_2194112158459175443et_nat @ F @ B2 ) )
= ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ) ).
% inj_on_image_subset_iff
thf(fact_428_inj__on__image__subset__iff,axiom,
! [F: nat > set_nat,C: set_nat,A2: set_nat,B2: set_nat] :
( ( inj_on_nat_set_nat @ F @ C )
=> ( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A2 ) @ ( image_nat_set_nat @ F @ B2 ) )
= ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ) ).
% inj_on_image_subset_iff
thf(fact_429_inj__on__image__subset__iff,axiom,
! [F: nat > nat,C: set_nat,A2: set_nat,B2: set_nat] :
( ( inj_on_nat_nat @ F @ C )
=> ( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B2 ) )
= ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ) ).
% inj_on_image_subset_iff
thf(fact_430_all__subset__image__inj,axiom,
! [F: set_set_nat > set_set_nat,S2: set_set_set_nat,P: set_set_set_nat > $o] :
( ( ! [T4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ T4 @ ( image_7884819252390400639et_nat @ F @ S2 ) )
=> ( P @ T4 ) ) )
= ( ! [T4: set_set_set_nat] :
( ( ( ord_le9131159989063066194et_nat @ T4 @ S2 )
& ( inj_on2040386338155636715et_nat @ F @ T4 ) )
=> ( P @ ( image_7884819252390400639et_nat @ F @ T4 ) ) ) ) ) ).
% all_subset_image_inj
thf(fact_431_all__subset__image__inj,axiom,
! [F: set_nat > set_set_nat,S2: set_set_nat,P: set_set_set_nat > $o] :
( ( ! [T4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ T4 @ ( image_6725021117256019401et_nat @ F @ S2 ) )
=> ( P @ T4 ) ) )
= ( ! [T4: set_set_nat] :
( ( ( ord_le6893508408891458716et_nat @ T4 @ S2 )
& ( inj_on2776966659131765557et_nat @ F @ T4 ) )
=> ( P @ ( image_6725021117256019401et_nat @ F @ T4 ) ) ) ) ) ).
% all_subset_image_inj
thf(fact_432_all__subset__image__inj,axiom,
! [F: nat > set_set_nat,S2: set_nat,P: set_set_set_nat > $o] :
( ( ! [T4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ T4 @ ( image_2194112158459175443et_nat @ F @ S2 ) )
=> ( P @ T4 ) ) )
= ( ! [T4: set_nat] :
( ( ( ord_less_eq_set_nat @ T4 @ S2 )
& ( inj_on8105003582846801791et_nat @ F @ T4 ) )
=> ( P @ ( image_2194112158459175443et_nat @ F @ T4 ) ) ) ) ) ).
% all_subset_image_inj
thf(fact_433_all__subset__image__inj,axiom,
! [F: set_set_nat > set_nat,S2: set_set_set_nat,P: set_set_nat > $o] :
( ( ! [T4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ T4 @ ( image_5842784325960735177et_nat @ F @ S2 ) )
=> ( P @ T4 ) ) )
= ( ! [T4: set_set_set_nat] :
( ( ( ord_le9131159989063066194et_nat @ T4 @ S2 )
& ( inj_on1894729867836481333et_nat @ F @ T4 ) )
=> ( P @ ( image_5842784325960735177et_nat @ F @ T4 ) ) ) ) ) ).
% all_subset_image_inj
thf(fact_434_all__subset__image__inj,axiom,
! [F: set_nat > set_nat,S2: set_set_nat,P: set_set_nat > $o] :
( ( ! [T4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ T4 @ ( image_7916887816326733075et_nat @ F @ S2 ) )
=> ( P @ T4 ) ) )
= ( ! [T4: set_set_nat] :
( ( ( ord_le6893508408891458716et_nat @ T4 @ S2 )
& ( inj_on4604407203859583615et_nat @ F @ T4 ) )
=> ( P @ ( image_7916887816326733075et_nat @ F @ T4 ) ) ) ) ) ).
% all_subset_image_inj
thf(fact_435_all__subset__image__inj,axiom,
! [F: nat > set_nat,S2: set_nat,P: set_set_nat > $o] :
( ( ! [T4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ T4 @ ( image_nat_set_nat @ F @ S2 ) )
=> ( P @ T4 ) ) )
= ( ! [T4: set_nat] :
( ( ( ord_less_eq_set_nat @ T4 @ S2 )
& ( inj_on_nat_set_nat @ F @ T4 ) )
=> ( P @ ( image_nat_set_nat @ F @ T4 ) ) ) ) ) ).
% all_subset_image_inj
thf(fact_436_all__subset__image__inj,axiom,
! [F: set_set_nat > nat,S2: set_set_set_nat,P: set_nat > $o] :
( ( ! [T4: set_nat] :
( ( ord_less_eq_set_nat @ T4 @ ( image_1454916318497077779at_nat @ F @ S2 ) )
=> ( P @ T4 ) ) )
= ( ! [T4: set_set_set_nat] :
( ( ( ord_le9131159989063066194et_nat @ T4 @ S2 )
& ( inj_on7365807742884704127at_nat @ F @ T4 ) )
=> ( P @ ( image_1454916318497077779at_nat @ F @ T4 ) ) ) ) ) ).
% all_subset_image_inj
thf(fact_437_all__subset__image__inj,axiom,
! [F: set_nat > nat,S2: set_set_nat,P: set_nat > $o] :
( ( ! [T4: set_nat] :
( ( ord_less_eq_set_nat @ T4 @ ( image_set_nat_nat @ F @ S2 ) )
=> ( P @ T4 ) ) )
= ( ! [T4: set_set_nat] :
( ( ( ord_le6893508408891458716et_nat @ T4 @ S2 )
& ( inj_on_set_nat_nat @ F @ T4 ) )
=> ( P @ ( image_set_nat_nat @ F @ T4 ) ) ) ) ) ).
% all_subset_image_inj
thf(fact_438_all__subset__image__inj,axiom,
! [F: nat > nat,S2: set_nat,P: set_nat > $o] :
( ( ! [T4: set_nat] :
( ( ord_less_eq_set_nat @ T4 @ ( image_nat_nat @ F @ S2 ) )
=> ( P @ T4 ) ) )
= ( ! [T4: set_nat] :
( ( ( ord_less_eq_set_nat @ T4 @ S2 )
& ( inj_on_nat_nat @ F @ T4 ) )
=> ( P @ ( image_nat_nat @ F @ T4 ) ) ) ) ) ).
% all_subset_image_inj
thf(fact_439_ex__subset__image__inj,axiom,
! [F: set_set_nat > set_set_nat,S2: set_set_set_nat,P: set_set_set_nat > $o] :
( ( ? [T4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ T4 @ ( image_7884819252390400639et_nat @ F @ S2 ) )
& ( P @ T4 ) ) )
= ( ? [T4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ T4 @ S2 )
& ( inj_on2040386338155636715et_nat @ F @ T4 )
& ( P @ ( image_7884819252390400639et_nat @ F @ T4 ) ) ) ) ) ).
% ex_subset_image_inj
thf(fact_440_ex__subset__image__inj,axiom,
! [F: set_nat > set_set_nat,S2: set_set_nat,P: set_set_set_nat > $o] :
( ( ? [T4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ T4 @ ( image_6725021117256019401et_nat @ F @ S2 ) )
& ( P @ T4 ) ) )
= ( ? [T4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ T4 @ S2 )
& ( inj_on2776966659131765557et_nat @ F @ T4 )
& ( P @ ( image_6725021117256019401et_nat @ F @ T4 ) ) ) ) ) ).
% ex_subset_image_inj
thf(fact_441_ex__subset__image__inj,axiom,
! [F: nat > set_set_nat,S2: set_nat,P: set_set_set_nat > $o] :
( ( ? [T4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ T4 @ ( image_2194112158459175443et_nat @ F @ S2 ) )
& ( P @ T4 ) ) )
= ( ? [T4: set_nat] :
( ( ord_less_eq_set_nat @ T4 @ S2 )
& ( inj_on8105003582846801791et_nat @ F @ T4 )
& ( P @ ( image_2194112158459175443et_nat @ F @ T4 ) ) ) ) ) ).
% ex_subset_image_inj
thf(fact_442_ex__subset__image__inj,axiom,
! [F: set_set_nat > set_nat,S2: set_set_set_nat,P: set_set_nat > $o] :
( ( ? [T4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ T4 @ ( image_5842784325960735177et_nat @ F @ S2 ) )
& ( P @ T4 ) ) )
= ( ? [T4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ T4 @ S2 )
& ( inj_on1894729867836481333et_nat @ F @ T4 )
& ( P @ ( image_5842784325960735177et_nat @ F @ T4 ) ) ) ) ) ).
% ex_subset_image_inj
thf(fact_443_ex__subset__image__inj,axiom,
! [F: set_nat > set_nat,S2: set_set_nat,P: set_set_nat > $o] :
( ( ? [T4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ T4 @ ( image_7916887816326733075et_nat @ F @ S2 ) )
& ( P @ T4 ) ) )
= ( ? [T4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ T4 @ S2 )
& ( inj_on4604407203859583615et_nat @ F @ T4 )
& ( P @ ( image_7916887816326733075et_nat @ F @ T4 ) ) ) ) ) ).
% ex_subset_image_inj
thf(fact_444_ex__subset__image__inj,axiom,
! [F: nat > set_nat,S2: set_nat,P: set_set_nat > $o] :
( ( ? [T4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ T4 @ ( image_nat_set_nat @ F @ S2 ) )
& ( P @ T4 ) ) )
= ( ? [T4: set_nat] :
( ( ord_less_eq_set_nat @ T4 @ S2 )
& ( inj_on_nat_set_nat @ F @ T4 )
& ( P @ ( image_nat_set_nat @ F @ T4 ) ) ) ) ) ).
% ex_subset_image_inj
thf(fact_445_ex__subset__image__inj,axiom,
! [F: set_set_nat > nat,S2: set_set_set_nat,P: set_nat > $o] :
( ( ? [T4: set_nat] :
( ( ord_less_eq_set_nat @ T4 @ ( image_1454916318497077779at_nat @ F @ S2 ) )
& ( P @ T4 ) ) )
= ( ? [T4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ T4 @ S2 )
& ( inj_on7365807742884704127at_nat @ F @ T4 )
& ( P @ ( image_1454916318497077779at_nat @ F @ T4 ) ) ) ) ) ).
% ex_subset_image_inj
thf(fact_446_ex__subset__image__inj,axiom,
! [F: set_nat > nat,S2: set_set_nat,P: set_nat > $o] :
( ( ? [T4: set_nat] :
( ( ord_less_eq_set_nat @ T4 @ ( image_set_nat_nat @ F @ S2 ) )
& ( P @ T4 ) ) )
= ( ? [T4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ T4 @ S2 )
& ( inj_on_set_nat_nat @ F @ T4 )
& ( P @ ( image_set_nat_nat @ F @ T4 ) ) ) ) ) ).
% ex_subset_image_inj
thf(fact_447_ex__subset__image__inj,axiom,
! [F: nat > nat,S2: set_nat,P: set_nat > $o] :
( ( ? [T4: set_nat] :
( ( ord_less_eq_set_nat @ T4 @ ( image_nat_nat @ F @ S2 ) )
& ( P @ T4 ) ) )
= ( ? [T4: set_nat] :
( ( ord_less_eq_set_nat @ T4 @ S2 )
& ( inj_on_nat_nat @ F @ T4 )
& ( P @ ( image_nat_nat @ F @ T4 ) ) ) ) ) ).
% ex_subset_image_inj
thf(fact_448_subset__image__inj,axiom,
! [S2: set_set_set_nat,F: set_set_nat > set_set_nat,T3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ S2 @ ( image_7884819252390400639et_nat @ F @ T3 ) )
= ( ? [U2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ U2 @ T3 )
& ( inj_on2040386338155636715et_nat @ F @ U2 )
& ( S2
= ( image_7884819252390400639et_nat @ F @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_449_subset__image__inj,axiom,
! [S2: set_set_set_nat,F: set_nat > set_set_nat,T3: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ S2 @ ( image_6725021117256019401et_nat @ F @ T3 ) )
= ( ? [U2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ U2 @ T3 )
& ( inj_on2776966659131765557et_nat @ F @ U2 )
& ( S2
= ( image_6725021117256019401et_nat @ F @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_450_subset__image__inj,axiom,
! [S2: set_set_set_nat,F: nat > set_set_nat,T3: set_nat] :
( ( ord_le9131159989063066194et_nat @ S2 @ ( image_2194112158459175443et_nat @ F @ T3 ) )
= ( ? [U2: set_nat] :
( ( ord_less_eq_set_nat @ U2 @ T3 )
& ( inj_on8105003582846801791et_nat @ F @ U2 )
& ( S2
= ( image_2194112158459175443et_nat @ F @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_451_subset__image__inj,axiom,
! [S2: set_set_nat,F: set_set_nat > set_nat,T3: set_set_set_nat] :
( ( ord_le6893508408891458716et_nat @ S2 @ ( image_5842784325960735177et_nat @ F @ T3 ) )
= ( ? [U2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ U2 @ T3 )
& ( inj_on1894729867836481333et_nat @ F @ U2 )
& ( S2
= ( image_5842784325960735177et_nat @ F @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_452_subset__image__inj,axiom,
! [S2: set_set_nat,F: set_nat > set_nat,T3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ S2 @ ( image_7916887816326733075et_nat @ F @ T3 ) )
= ( ? [U2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ U2 @ T3 )
& ( inj_on4604407203859583615et_nat @ F @ U2 )
& ( S2
= ( image_7916887816326733075et_nat @ F @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_453_subset__image__inj,axiom,
! [S2: set_set_nat,F: nat > set_nat,T3: set_nat] :
( ( ord_le6893508408891458716et_nat @ S2 @ ( image_nat_set_nat @ F @ T3 ) )
= ( ? [U2: set_nat] :
( ( ord_less_eq_set_nat @ U2 @ T3 )
& ( inj_on_nat_set_nat @ F @ U2 )
& ( S2
= ( image_nat_set_nat @ F @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_454_subset__image__inj,axiom,
! [S2: set_nat,F: set_set_nat > nat,T3: set_set_set_nat] :
( ( ord_less_eq_set_nat @ S2 @ ( image_1454916318497077779at_nat @ F @ T3 ) )
= ( ? [U2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ U2 @ T3 )
& ( inj_on7365807742884704127at_nat @ F @ U2 )
& ( S2
= ( image_1454916318497077779at_nat @ F @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_455_subset__image__inj,axiom,
! [S2: set_nat,F: set_nat > nat,T3: set_set_nat] :
( ( ord_less_eq_set_nat @ S2 @ ( image_set_nat_nat @ F @ T3 ) )
= ( ? [U2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ U2 @ T3 )
& ( inj_on_set_nat_nat @ F @ U2 )
& ( S2
= ( image_set_nat_nat @ F @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_456_subset__image__inj,axiom,
! [S2: set_nat,F: nat > nat,T3: set_nat] :
( ( ord_less_eq_set_nat @ S2 @ ( image_nat_nat @ F @ T3 ) )
= ( ? [U2: set_nat] :
( ( ord_less_eq_set_nat @ U2 @ T3 )
& ( inj_on_nat_nat @ F @ U2 )
& ( S2
= ( image_nat_nat @ F @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_457_inj__on__image__mem__iff,axiom,
! [F: nat > nat,B2: set_nat,A: nat,A2: set_nat] :
( ( inj_on_nat_nat @ F @ B2 )
=> ( ( member_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ ( F @ A ) @ ( image_nat_nat @ F @ A2 ) )
= ( member_nat @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_458_inj__on__image__mem__iff,axiom,
! [F: set_nat > nat,B2: set_set_nat,A: set_nat,A2: set_set_nat] :
( ( inj_on_set_nat_nat @ F @ B2 )
=> ( ( member_set_nat @ A @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( member_nat @ ( F @ A ) @ ( image_set_nat_nat @ F @ A2 ) )
= ( member_set_nat @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_459_inj__on__image__mem__iff,axiom,
! [F: nat > set_nat,B2: set_nat,A: nat,A2: set_nat] :
( ( inj_on_nat_set_nat @ F @ B2 )
=> ( ( member_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_set_nat @ ( F @ A ) @ ( image_nat_set_nat @ F @ A2 ) )
= ( member_nat @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_460_inj__on__image__mem__iff,axiom,
! [F: set_set_nat > nat,B2: set_set_set_nat,A: set_set_nat,A2: set_set_set_nat] :
( ( inj_on7365807742884704127at_nat @ F @ B2 )
=> ( ( member_set_set_nat @ A @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( member_nat @ ( F @ A ) @ ( image_1454916318497077779at_nat @ F @ A2 ) )
= ( member_set_set_nat @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_461_inj__on__image__mem__iff,axiom,
! [F: set_nat > set_nat,B2: set_set_nat,A: set_nat,A2: set_set_nat] :
( ( inj_on4604407203859583615et_nat @ F @ B2 )
=> ( ( member_set_nat @ A @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( member_set_nat @ ( F @ A ) @ ( image_7916887816326733075et_nat @ F @ A2 ) )
= ( member_set_nat @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_462_inj__on__image__mem__iff,axiom,
! [F: nat > set_set_nat,B2: set_nat,A: nat,A2: set_nat] :
( ( inj_on8105003582846801791et_nat @ F @ B2 )
=> ( ( member_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_set_set_nat @ ( F @ A ) @ ( image_2194112158459175443et_nat @ F @ A2 ) )
= ( member_nat @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_463_inj__on__image__mem__iff,axiom,
! [F: ( nat > set_nat ) > nat,B2: set_nat_set_nat,A: nat > set_nat,A2: set_nat_set_nat] :
( ( inj_on5993617239833669176at_nat @ F @ B2 )
=> ( ( member_nat_set_nat @ A @ B2 )
=> ( ( ord_le1585852046946910987et_nat @ A2 @ B2 )
=> ( ( member_nat @ ( F @ A ) @ ( image_970537773860477644at_nat @ F @ A2 ) )
= ( member_nat_set_nat @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_464_inj__on__image__mem__iff,axiom,
! [F: set_set_nat > set_nat,B2: set_set_set_nat,A: set_set_nat,A2: set_set_set_nat] :
( ( inj_on1894729867836481333et_nat @ F @ B2 )
=> ( ( member_set_set_nat @ A @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( member_set_nat @ ( F @ A ) @ ( image_5842784325960735177et_nat @ F @ A2 ) )
= ( member_set_set_nat @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_465_inj__on__image__mem__iff,axiom,
! [F: set_nat > set_set_nat,B2: set_set_nat,A: set_nat,A2: set_set_nat] :
( ( inj_on2776966659131765557et_nat @ F @ B2 )
=> ( ( member_set_nat @ A @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( member_set_set_nat @ ( F @ A ) @ ( image_6725021117256019401et_nat @ F @ A2 ) )
= ( member_set_nat @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_466_inj__on__image__mem__iff,axiom,
! [F: nat > nat > set_nat,B2: set_nat,A: nat,A2: set_nat] :
( ( inj_on236506435458908344et_nat @ F @ B2 )
=> ( ( member_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat_set_nat @ ( F @ A ) @ ( image_4436799006340492620et_nat @ F @ A2 ) )
= ( member_nat @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_467_inj__on__image__eq__iff,axiom,
! [F: set_set_nat > set_nat,C: set_set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( inj_on1894729867836481333et_nat @ F @ C )
=> ( ( ord_le9131159989063066194et_nat @ A2 @ C )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ C )
=> ( ( ( image_5842784325960735177et_nat @ F @ A2 )
= ( image_5842784325960735177et_nat @ F @ B2 ) )
= ( A2 = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_468_inj__on__image__eq__iff,axiom,
! [F: nat > set_set_nat,C: set_nat,A2: set_nat,B2: set_nat] :
( ( inj_on8105003582846801791et_nat @ F @ C )
=> ( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ( ( image_2194112158459175443et_nat @ F @ A2 )
= ( image_2194112158459175443et_nat @ F @ B2 ) )
= ( A2 = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_469_inj__on__image__eq__iff,axiom,
! [F: nat > nat,C: set_nat,A2: set_nat,B2: set_nat] :
( ( inj_on_nat_nat @ F @ C )
=> ( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ( ( image_nat_nat @ F @ A2 )
= ( image_nat_nat @ F @ B2 ) )
= ( A2 = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_470_linorder__inj__onI,axiom,
! [A2: set_nat,F: nat > set_set_nat] :
( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ( member_nat @ X3 @ A2 )
=> ( ( member_nat @ Y4 @ A2 )
=> ( ( F @ X3 )
!= ( F @ Y4 ) ) ) ) )
=> ( ! [X3: nat,Y4: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_nat @ Y4 @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Y4 )
| ( ord_less_eq_nat @ Y4 @ X3 ) ) ) )
=> ( inj_on8105003582846801791et_nat @ F @ A2 ) ) ) ).
% linorder_inj_onI
thf(fact_471_linorder__inj__onI,axiom,
! [A2: set_nat,F: nat > nat] :
( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ( member_nat @ X3 @ A2 )
=> ( ( member_nat @ Y4 @ A2 )
=> ( ( F @ X3 )
!= ( F @ Y4 ) ) ) ) )
=> ( ! [X3: nat,Y4: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_nat @ Y4 @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Y4 )
| ( ord_less_eq_nat @ Y4 @ X3 ) ) ) )
=> ( inj_on_nat_nat @ F @ A2 ) ) ) ).
% linorder_inj_onI
thf(fact_472_L,axiom,
ord_less_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ x ) ) ).
% L
thf(fact_473_fin1,axiom,
finite1152437895449049373et_nat @ ( clique8462013130872731469t_v_gs @ x ) ).
% fin1
thf(fact_474_v__mono,axiom,
! [G2: set_set_nat,H3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ G2 @ H3 )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ G2 ) @ ( clique5033774636164728513irst_v @ H3 ) ) ) ).
% v_mono
thf(fact_475_v__gs__def,axiom,
( clique8462013130872731469t_v_gs
= ( image_5842784325960735177et_nat @ clique5033774636164728513irst_v ) ) ).
% v_gs_def
thf(fact_476_v__gs__mono,axiom,
! [X5: set_set_set_nat,Y2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X5 @ Y2 )
=> ( ord_le6893508408891458716et_nat @ ( clique8462013130872731469t_v_gs @ X5 ) @ ( clique8462013130872731469t_v_gs @ Y2 ) ) ) ).
% v_gs_mono
thf(fact_477_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_478_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_479_rq,axiom,
ord_less_eq_nat @ p @ r ).
% rq
thf(fact_480_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_eq_nat @ N4 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_481_card_Oinfinite,axiom,
! [A2: set_set_nat] :
( ~ ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite_card_set_nat @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_482_card_Oinfinite,axiom,
! [A2: set_set_set_nat] :
( ~ ( finite6739761609112101331et_nat @ A2 )
=> ( ( finite1149291290879098388et_nat @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_483_card_Oinfinite,axiom,
! [A2: set_nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_card_nat @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_484_G_I2_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ p )
=> ( ( clique5033774636164728513irst_v @ ( g @ I ) )
= ( si @ I ) ) ) ).
% G(2)
thf(fact_485_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M3: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N5 )
=> ( ord_less_eq_nat @ X2 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_486_psubsetD,axiom,
! [A2: set_nat,B2: set_nat,C2: nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C2 @ A2 )
=> ( member_nat @ C2 @ B2 ) ) ) ).
% psubsetD
thf(fact_487_psubsetD,axiom,
! [A2: set_set_nat,B2: set_set_nat,C2: set_nat] :
( ( ord_less_set_set_nat @ A2 @ B2 )
=> ( ( member_set_nat @ C2 @ A2 )
=> ( member_set_nat @ C2 @ B2 ) ) ) ).
% psubsetD
thf(fact_488_psubsetD,axiom,
! [A2: set_nat_set_nat,B2: set_nat_set_nat,C2: nat > set_nat] :
( ( ord_le7745323766158300927et_nat @ A2 @ B2 )
=> ( ( member_nat_set_nat @ C2 @ A2 )
=> ( member_nat_set_nat @ C2 @ B2 ) ) ) ).
% psubsetD
thf(fact_489_psubsetD,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C2: set_set_nat] :
( ( ord_le152980574450754630et_nat @ A2 @ B2 )
=> ( ( member_set_set_nat @ C2 @ A2 )
=> ( member_set_set_nat @ C2 @ B2 ) ) ) ).
% psubsetD
thf(fact_490_less__set__def,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ord_less_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A3 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B3 ) ) ) ) ).
% less_set_def
thf(fact_491_less__set__def,axiom,
( ord_less_set_set_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ord_less_set_nat_o
@ ^ [X2: set_nat] : ( member_set_nat @ X2 @ A3 )
@ ^ [X2: set_nat] : ( member_set_nat @ X2 @ B3 ) ) ) ) ).
% less_set_def
thf(fact_492_less__set__def,axiom,
( ord_le7745323766158300927et_nat
= ( ^ [A3: set_nat_set_nat,B3: set_nat_set_nat] :
( ord_le2363938258001488454_nat_o
@ ^ [X2: nat > set_nat] : ( member_nat_set_nat @ X2 @ A3 )
@ ^ [X2: nat > set_nat] : ( member_nat_set_nat @ X2 @ B3 ) ) ) ) ).
% less_set_def
thf(fact_493_less__set__def,axiom,
( ord_le152980574450754630et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
( ord_le466346588697744319_nat_o
@ ^ [X2: set_set_nat] : ( member_set_set_nat @ X2 @ A3 )
@ ^ [X2: set_set_nat] : ( member_set_set_nat @ X2 @ B3 ) ) ) ) ).
% less_set_def
thf(fact_494_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A2: set_nat,R2: nat > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ A2 )
=> ? [B5: nat] :
( ( member_nat @ B5 @ B2 )
& ( R2 @ A6 @ B5 ) ) )
=> ( ! [A1: nat,A22: nat,B6: nat] :
( ( member_nat @ A1 @ A2 )
=> ( ( member_nat @ A22 @ A2 )
=> ( ( member_nat @ B6 @ B2 )
=> ( ( R2 @ A1 @ B6 )
=> ( ( R2 @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_495_card__le__if__inj__on__rel,axiom,
! [B2: set_set_nat,A2: set_nat,R2: nat > set_nat > $o] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ A2 )
=> ? [B5: set_nat] :
( ( member_set_nat @ B5 @ B2 )
& ( R2 @ A6 @ B5 ) ) )
=> ( ! [A1: nat,A22: nat,B6: set_nat] :
( ( member_nat @ A1 @ A2 )
=> ( ( member_nat @ A22 @ A2 )
=> ( ( member_set_nat @ B6 @ B2 )
=> ( ( R2 @ A1 @ B6 )
=> ( ( R2 @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_496_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A2: set_set_nat,R2: set_nat > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A6: set_nat] :
( ( member_set_nat @ A6 @ A2 )
=> ? [B5: nat] :
( ( member_nat @ B5 @ B2 )
& ( R2 @ A6 @ B5 ) ) )
=> ( ! [A1: set_nat,A22: set_nat,B6: nat] :
( ( member_set_nat @ A1 @ A2 )
=> ( ( member_set_nat @ A22 @ A2 )
=> ( ( member_nat @ B6 @ B2 )
=> ( ( R2 @ A1 @ B6 )
=> ( ( R2 @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_497_card__le__if__inj__on__rel,axiom,
! [B2: set_set_nat,A2: set_set_nat,R2: set_nat > set_nat > $o] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ! [A6: set_nat] :
( ( member_set_nat @ A6 @ A2 )
=> ? [B5: set_nat] :
( ( member_set_nat @ B5 @ B2 )
& ( R2 @ A6 @ B5 ) ) )
=> ( ! [A1: set_nat,A22: set_nat,B6: set_nat] :
( ( member_set_nat @ A1 @ A2 )
=> ( ( member_set_nat @ A22 @ A2 )
=> ( ( member_set_nat @ B6 @ B2 )
=> ( ( R2 @ A1 @ B6 )
=> ( ( R2 @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_498_card__le__if__inj__on__rel,axiom,
! [B2: set_set_set_nat,A2: set_nat,R2: nat > set_set_nat > $o] :
( ( finite6739761609112101331et_nat @ B2 )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ A2 )
=> ? [B5: set_set_nat] :
( ( member_set_set_nat @ B5 @ B2 )
& ( R2 @ A6 @ B5 ) ) )
=> ( ! [A1: nat,A22: nat,B6: set_set_nat] :
( ( member_nat @ A1 @ A2 )
=> ( ( member_nat @ A22 @ A2 )
=> ( ( member_set_set_nat @ B6 @ B2 )
=> ( ( R2 @ A1 @ B6 )
=> ( ( R2 @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite1149291290879098388et_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_499_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A2: set_set_set_nat,R2: set_set_nat > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A6: set_set_nat] :
( ( member_set_set_nat @ A6 @ A2 )
=> ? [B5: nat] :
( ( member_nat @ B5 @ B2 )
& ( R2 @ A6 @ B5 ) ) )
=> ( ! [A1: set_set_nat,A22: set_set_nat,B6: nat] :
( ( member_set_set_nat @ A1 @ A2 )
=> ( ( member_set_set_nat @ A22 @ A2 )
=> ( ( member_nat @ B6 @ B2 )
=> ( ( R2 @ A1 @ B6 )
=> ( ( R2 @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_500_card__le__if__inj__on__rel,axiom,
! [B2: set_nat_set_nat,A2: set_nat,R2: nat > ( nat > set_nat ) > $o] :
( ( finite722436868047473932et_nat @ B2 )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ A2 )
=> ? [B5: nat > set_nat] :
( ( member_nat_set_nat @ B5 @ B2 )
& ( R2 @ A6 @ B5 ) ) )
=> ( ! [A1: nat,A22: nat,B6: nat > set_nat] :
( ( member_nat @ A1 @ A2 )
=> ( ( member_nat @ A22 @ A2 )
=> ( ( member_nat_set_nat @ B6 @ B2 )
=> ( ( R2 @ A1 @ B6 )
=> ( ( R2 @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite3028741397543221197et_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_501_card__le__if__inj__on__rel,axiom,
! [B2: set_set_nat,A2: set_set_set_nat,R2: set_set_nat > set_nat > $o] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ! [A6: set_set_nat] :
( ( member_set_set_nat @ A6 @ A2 )
=> ? [B5: set_nat] :
( ( member_set_nat @ B5 @ B2 )
& ( R2 @ A6 @ B5 ) ) )
=> ( ! [A1: set_set_nat,A22: set_set_nat,B6: set_nat] :
( ( member_set_set_nat @ A1 @ A2 )
=> ( ( member_set_set_nat @ A22 @ A2 )
=> ( ( member_set_nat @ B6 @ B2 )
=> ( ( R2 @ A1 @ B6 )
=> ( ( R2 @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_502_card__le__if__inj__on__rel,axiom,
! [B2: set_set_set_nat,A2: set_set_nat,R2: set_nat > set_set_nat > $o] :
( ( finite6739761609112101331et_nat @ B2 )
=> ( ! [A6: set_nat] :
( ( member_set_nat @ A6 @ A2 )
=> ? [B5: set_set_nat] :
( ( member_set_set_nat @ B5 @ B2 )
& ( R2 @ A6 @ B5 ) ) )
=> ( ! [A1: set_nat,A22: set_nat,B6: set_set_nat] :
( ( member_set_nat @ A1 @ A2 )
=> ( ( member_set_nat @ A22 @ A2 )
=> ( ( member_set_set_nat @ B6 @ B2 )
=> ( ( R2 @ A1 @ B6 )
=> ( ( R2 @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ ( finite1149291290879098388et_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_503_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A2: set_nat_set_nat,R2: ( nat > set_nat ) > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A6: nat > set_nat] :
( ( member_nat_set_nat @ A6 @ A2 )
=> ? [B5: nat] :
( ( member_nat @ B5 @ B2 )
& ( R2 @ A6 @ B5 ) ) )
=> ( ! [A1: nat > set_nat,A22: nat > set_nat,B6: nat] :
( ( member_nat_set_nat @ A1 @ A2 )
=> ( ( member_nat_set_nat @ A22 @ A2 )
=> ( ( member_nat @ B6 @ B2 )
=> ( ( R2 @ A1 @ B6 )
=> ( ( R2 @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite3028741397543221197et_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_504_finite__less__ub,axiom,
! [F: nat > nat,U: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( F @ N2 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_505_card__image__le,axiom,
! [A2: set_set_nat,F: set_nat > set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) ) @ ( finite_card_set_nat @ A2 ) ) ) ).
% card_image_le
thf(fact_506_card__image__le,axiom,
! [A2: set_set_nat,F: set_nat > nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_set_nat_nat @ F @ A2 ) ) @ ( finite_card_set_nat @ A2 ) ) ) ).
% card_image_le
thf(fact_507_card__image__le,axiom,
! [A2: set_set_nat,F: set_nat > set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ ( image_6725021117256019401et_nat @ F @ A2 ) ) @ ( finite_card_set_nat @ A2 ) ) ) ).
% card_image_le
thf(fact_508_card__image__le,axiom,
! [A2: set_set_set_nat,F: set_set_nat > set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ ( image_5842784325960735177et_nat @ F @ A2 ) ) @ ( finite1149291290879098388et_nat @ A2 ) ) ) ).
% card_image_le
thf(fact_509_card__image__le,axiom,
! [A2: set_set_set_nat,F: set_set_nat > nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_1454916318497077779at_nat @ F @ A2 ) ) @ ( finite1149291290879098388et_nat @ A2 ) ) ) ).
% card_image_le
thf(fact_510_card__image__le,axiom,
! [A2: set_set_set_nat,F: set_set_nat > set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ ( image_7884819252390400639et_nat @ F @ A2 ) ) @ ( finite1149291290879098388et_nat @ A2 ) ) ) ).
% card_image_le
thf(fact_511_card__image__le,axiom,
! [A2: set_nat,F: nat > set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ ( finite_card_nat @ A2 ) ) ) ).
% card_image_le
thf(fact_512_card__image__le,axiom,
! [A2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_nat_nat @ F @ A2 ) ) @ ( finite_card_nat @ A2 ) ) ) ).
% card_image_le
thf(fact_513_card__image__le,axiom,
! [A2: set_nat,F: nat > set_set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ ( image_2194112158459175443et_nat @ F @ A2 ) ) @ ( finite_card_nat @ A2 ) ) ) ).
% card_image_le
thf(fact_514_card__mono,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite1149291290879098388et_nat @ B2 ) ) ) ) ).
% card_mono
thf(fact_515_card__mono,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) ) ) ) ).
% card_mono
thf(fact_516_card__mono,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ).
% card_mono
thf(fact_517_card__seteq,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ B2 ) @ ( finite1149291290879098388et_nat @ A2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_seteq
thf(fact_518_card__seteq,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_set_nat @ B2 ) @ ( finite_card_set_nat @ A2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_seteq
thf(fact_519_card__seteq,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B2 ) @ ( finite_card_nat @ A2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_seteq
thf(fact_520_exists__subset__between,axiom,
! [A2: set_set_set_nat,N: nat,C: set_set_set_nat] :
( ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite1149291290879098388et_nat @ C ) )
=> ( ( ord_le9131159989063066194et_nat @ A2 @ C )
=> ( ( finite6739761609112101331et_nat @ C )
=> ? [B7: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B7 )
& ( ord_le9131159989063066194et_nat @ B7 @ C )
& ( ( finite1149291290879098388et_nat @ B7 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_521_exists__subset__between,axiom,
! [A2: set_set_nat,N: nat,C: set_set_nat] :
( ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_set_nat @ C ) )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ C )
=> ( ( finite1152437895449049373et_nat @ C )
=> ? [B7: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B7 )
& ( ord_le6893508408891458716et_nat @ B7 @ C )
& ( ( finite_card_set_nat @ B7 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_522_exists__subset__between,axiom,
! [A2: set_nat,N: nat,C: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ C ) )
=> ( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( finite_finite_nat @ C )
=> ? [B7: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B7 )
& ( ord_less_eq_set_nat @ B7 @ C )
& ( ( finite_card_nat @ B7 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_523_obtain__subset__with__card__n,axiom,
! [N: nat,S2: set_set_set_nat] :
( ( ord_less_eq_nat @ N @ ( finite1149291290879098388et_nat @ S2 ) )
=> ~ ! [T5: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ T5 @ S2 )
=> ( ( ( finite1149291290879098388et_nat @ T5 )
= N )
=> ~ ( finite6739761609112101331et_nat @ T5 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_524_obtain__subset__with__card__n,axiom,
! [N: nat,S2: set_set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_set_nat @ S2 ) )
=> ~ ! [T5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ T5 @ S2 )
=> ( ( ( finite_card_set_nat @ T5 )
= N )
=> ~ ( finite1152437895449049373et_nat @ T5 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_525_obtain__subset__with__card__n,axiom,
! [N: nat,S2: set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S2 ) )
=> ~ ! [T5: set_nat] :
( ( ord_less_eq_set_nat @ T5 @ S2 )
=> ( ( ( finite_card_nat @ T5 )
= N )
=> ~ ( finite_finite_nat @ T5 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_526_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_set_set_nat,C: nat] :
( ! [G3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ G3 @ F2 )
=> ( ( finite6739761609112101331et_nat @ G3 )
=> ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ G3 ) @ C ) ) )
=> ( ( finite6739761609112101331et_nat @ F2 )
& ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ F2 ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_527_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_set_nat,C: nat] :
( ! [G3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ G3 @ F2 )
=> ( ( finite1152437895449049373et_nat @ G3 )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ G3 ) @ C ) ) )
=> ( ( finite1152437895449049373et_nat @ F2 )
& ( ord_less_eq_nat @ ( finite_card_set_nat @ F2 ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_528_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_nat,C: nat] :
( ! [G3: set_nat] :
( ( ord_less_eq_set_nat @ G3 @ F2 )
=> ( ( finite_finite_nat @ G3 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G3 ) @ C ) ) )
=> ( ( finite_finite_nat @ F2 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F2 ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_529_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_530_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_531_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_532_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_533_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N4: nat] :
( ( ord_less_eq_nat @ M3 @ N4 )
& ( M3 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_534_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_535_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N4: nat] :
( ( ord_less_nat @ M3 @ N4 )
| ( M3 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_536_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_537_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_538_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_539_surj__card__le,axiom,
! [A2: set_set_nat,B2: set_set_set_nat,F: set_nat > set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ ( image_6725021117256019401et_nat @ F @ A2 ) )
=> ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ B2 ) @ ( finite_card_set_nat @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_540_surj__card__le,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,F: set_set_nat > set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ ( image_7884819252390400639et_nat @ F @ A2 ) )
=> ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ B2 ) @ ( finite1149291290879098388et_nat @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_541_surj__card__le,axiom,
! [A2: set_nat,B2: set_set_set_nat,F: nat > set_set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ ( image_2194112158459175443et_nat @ F @ A2 ) )
=> ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ B2 ) @ ( finite_card_nat @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_542_surj__card__le,axiom,
! [A2: set_set_nat,B2: set_set_nat,F: set_nat > set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ ( image_7916887816326733075et_nat @ F @ A2 ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ B2 ) @ ( finite_card_set_nat @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_543_surj__card__le,axiom,
! [A2: set_set_set_nat,B2: set_set_nat,F: set_set_nat > set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ ( image_5842784325960735177et_nat @ F @ A2 ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ B2 ) @ ( finite1149291290879098388et_nat @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_544_surj__card__le,axiom,
! [A2: set_nat,B2: set_set_nat,F: nat > set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ ( image_nat_set_nat @ F @ A2 ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ B2 ) @ ( finite_card_nat @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_545_surj__card__le,axiom,
! [A2: set_set_nat,B2: set_nat,F: set_nat > nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_set_nat_nat @ F @ A2 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B2 ) @ ( finite_card_set_nat @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_546_surj__card__le,axiom,
! [A2: set_set_set_nat,B2: set_nat,F: set_set_nat > nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_1454916318497077779at_nat @ F @ A2 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B2 ) @ ( finite1149291290879098388et_nat @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_547_surj__card__le,axiom,
! [A2: set_nat,B2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B2 ) @ ( finite_card_nat @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_548_less__eq__set__def,axiom,
( ord_le1585852046946910987et_nat
= ( ^ [A3: set_nat_set_nat,B3: set_nat_set_nat] :
( ord_le8865062304692155706_nat_o
@ ^ [X2: nat > set_nat] : ( member_nat_set_nat @ X2 @ A3 )
@ ^ [X2: nat > set_nat] : ( member_nat_set_nat @ X2 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_549_less__eq__set__def,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
( ord_le3616423863276227763_nat_o
@ ^ [X2: set_set_nat] : ( member_set_set_nat @ X2 @ A3 )
@ ^ [X2: set_set_nat] : ( member_set_set_nat @ X2 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_550_less__eq__set__def,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ord_le3964352015994296041_nat_o
@ ^ [X2: set_nat] : ( member_set_nat @ X2 @ A3 )
@ ^ [X2: set_nat] : ( member_set_nat @ X2 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_551_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ord_less_eq_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A3 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_552_card__le__inj,axiom,
! [A2: set_set_nat,B2: set_set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ ( finite1149291290879098388et_nat @ B2 ) )
=> ? [F3: set_nat > set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( image_6725021117256019401et_nat @ F3 @ A2 ) @ B2 )
& ( inj_on2776966659131765557et_nat @ F3 @ A2 ) ) ) ) ) ).
% card_le_inj
thf(fact_553_card__le__inj,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite1149291290879098388et_nat @ B2 ) )
=> ? [F3: set_set_nat > set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( image_7884819252390400639et_nat @ F3 @ A2 ) @ B2 )
& ( inj_on2040386338155636715et_nat @ F3 @ A2 ) ) ) ) ) ).
% card_le_inj
thf(fact_554_card__le__inj,axiom,
! [A2: set_nat,B2: set_set_set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite1149291290879098388et_nat @ B2 ) )
=> ? [F3: nat > set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( image_2194112158459175443et_nat @ F3 @ A2 ) @ B2 )
& ( inj_on8105003582846801791et_nat @ F3 @ A2 ) ) ) ) ) ).
% card_le_inj
thf(fact_555_card__le__inj,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) )
=> ? [F3: set_nat > set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ F3 @ A2 ) @ B2 )
& ( inj_on4604407203859583615et_nat @ F3 @ A2 ) ) ) ) ) ).
% card_le_inj
thf(fact_556_card__le__inj,axiom,
! [A2: set_set_set_nat,B2: set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) )
=> ? [F3: set_set_nat > set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_5842784325960735177et_nat @ F3 @ A2 ) @ B2 )
& ( inj_on1894729867836481333et_nat @ F3 @ A2 ) ) ) ) ) ).
% card_le_inj
thf(fact_557_card__le__inj,axiom,
! [A2: set_nat,B2: set_set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) )
=> ? [F3: nat > set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F3 @ A2 ) @ B2 )
& ( inj_on_nat_set_nat @ F3 @ A2 ) ) ) ) ) ).
% card_le_inj
thf(fact_558_card__le__inj,axiom,
! [A2: set_set_nat,B2: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
=> ? [F3: set_nat > nat] :
( ( ord_less_eq_set_nat @ ( image_set_nat_nat @ F3 @ A2 ) @ B2 )
& ( inj_on_set_nat_nat @ F3 @ A2 ) ) ) ) ) ).
% card_le_inj
thf(fact_559_card__le__inj,axiom,
! [A2: set_set_set_nat,B2: set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
=> ? [F3: set_set_nat > nat] :
( ( ord_less_eq_set_nat @ ( image_1454916318497077779at_nat @ F3 @ A2 ) @ B2 )
& ( inj_on7365807742884704127at_nat @ F3 @ A2 ) ) ) ) ) ).
% card_le_inj
thf(fact_560_card__le__inj,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
=> ? [F3: nat > nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ A2 ) @ B2 )
& ( inj_on_nat_nat @ F3 @ A2 ) ) ) ) ) ).
% card_le_inj
thf(fact_561_card__inj__on__le,axiom,
! [F: set_nat > set_set_nat,A2: set_set_nat,B2: set_set_set_nat] :
( ( inj_on2776966659131765557et_nat @ F @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ ( image_6725021117256019401et_nat @ F @ A2 ) @ B2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ ( finite1149291290879098388et_nat @ B2 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_562_card__inj__on__le,axiom,
! [F: nat > set_set_nat,A2: set_nat,B2: set_set_set_nat] :
( ( inj_on8105003582846801791et_nat @ F @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ ( image_2194112158459175443et_nat @ F @ A2 ) @ B2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite1149291290879098388et_nat @ B2 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_563_card__inj__on__le,axiom,
! [F: set_set_nat > set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( inj_on2040386338155636715et_nat @ F @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ ( image_7884819252390400639et_nat @ F @ A2 ) @ B2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite1149291290879098388et_nat @ B2 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_564_card__inj__on__le,axiom,
! [F: set_nat > set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( inj_on4604407203859583615et_nat @ F @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) @ B2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_565_card__inj__on__le,axiom,
! [F: nat > set_nat,A2: set_nat,B2: set_set_nat] :
( ( inj_on_nat_set_nat @ F @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A2 ) @ B2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_566_card__inj__on__le,axiom,
! [F: set_set_nat > set_nat,A2: set_set_set_nat,B2: set_set_nat] :
( ( inj_on1894729867836481333et_nat @ F @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ ( image_5842784325960735177et_nat @ F @ A2 ) @ B2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_567_card__inj__on__le,axiom,
! [F: set_nat > nat,A2: set_set_nat,B2: set_nat] :
( ( inj_on_set_nat_nat @ F @ A2 )
=> ( ( ord_less_eq_set_nat @ ( image_set_nat_nat @ F @ A2 ) @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_568_card__inj__on__le,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( inj_on_nat_nat @ F @ A2 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_569_card__inj__on__le,axiom,
! [F: set_set_nat > nat,A2: set_set_set_nat,B2: set_nat] :
( ( inj_on7365807742884704127at_nat @ F @ A2 )
=> ( ( ord_less_eq_set_nat @ ( image_1454916318497077779at_nat @ F @ A2 ) @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_570_inj__on__iff__card__le,axiom,
! [A2: set_set_nat,B2: set_set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ? [F4: set_nat > set_set_nat] :
( ( inj_on2776966659131765557et_nat @ F4 @ A2 )
& ( ord_le9131159989063066194et_nat @ ( image_6725021117256019401et_nat @ F4 @ A2 ) @ B2 ) ) )
= ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ ( finite1149291290879098388et_nat @ B2 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_571_inj__on__iff__card__le,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ? [F4: set_set_nat > set_set_nat] :
( ( inj_on2040386338155636715et_nat @ F4 @ A2 )
& ( ord_le9131159989063066194et_nat @ ( image_7884819252390400639et_nat @ F4 @ A2 ) @ B2 ) ) )
= ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite1149291290879098388et_nat @ B2 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_572_inj__on__iff__card__le,axiom,
! [A2: set_nat,B2: set_set_set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ? [F4: nat > set_set_nat] :
( ( inj_on8105003582846801791et_nat @ F4 @ A2 )
& ( ord_le9131159989063066194et_nat @ ( image_2194112158459175443et_nat @ F4 @ A2 ) @ B2 ) ) )
= ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite1149291290879098388et_nat @ B2 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_573_inj__on__iff__card__le,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ? [F4: set_nat > set_nat] :
( ( inj_on4604407203859583615et_nat @ F4 @ A2 )
& ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ F4 @ A2 ) @ B2 ) ) )
= ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_574_inj__on__iff__card__le,axiom,
! [A2: set_set_set_nat,B2: set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ? [F4: set_set_nat > set_nat] :
( ( inj_on1894729867836481333et_nat @ F4 @ A2 )
& ( ord_le6893508408891458716et_nat @ ( image_5842784325960735177et_nat @ F4 @ A2 ) @ B2 ) ) )
= ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_575_inj__on__iff__card__le,axiom,
! [A2: set_nat,B2: set_set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ? [F4: nat > set_nat] :
( ( inj_on_nat_set_nat @ F4 @ A2 )
& ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F4 @ A2 ) @ B2 ) ) )
= ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_576_inj__on__iff__card__le,axiom,
! [A2: set_set_nat,B2: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ? [F4: set_nat > nat] :
( ( inj_on_set_nat_nat @ F4 @ A2 )
& ( ord_less_eq_set_nat @ ( image_set_nat_nat @ F4 @ A2 ) @ B2 ) ) )
= ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_577_inj__on__iff__card__le,axiom,
! [A2: set_set_set_nat,B2: set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ? [F4: set_set_nat > nat] :
( ( inj_on7365807742884704127at_nat @ F4 @ A2 )
& ( ord_less_eq_set_nat @ ( image_1454916318497077779at_nat @ F4 @ A2 ) @ B2 ) ) )
= ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_578_inj__on__iff__card__le,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ? [F4: nat > nat] :
( ( inj_on_nat_nat @ F4 @ A2 )
& ( ord_less_eq_set_nat @ ( image_nat_nat @ F4 @ A2 ) @ B2 ) ) )
= ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_579_card__subset__eq,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( ( finite1149291290879098388et_nat @ A2 )
= ( finite1149291290879098388et_nat @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_580_card__subset__eq,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ( finite_card_set_nat @ A2 )
= ( finite_card_set_nat @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_581_card__subset__eq,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_582_infinite__arbitrarily__large,axiom,
! [A2: set_set_set_nat,N: nat] :
( ~ ( finite6739761609112101331et_nat @ A2 )
=> ? [B7: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B7 )
& ( ( finite1149291290879098388et_nat @ B7 )
= N )
& ( ord_le9131159989063066194et_nat @ B7 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_583_infinite__arbitrarily__large,axiom,
! [A2: set_set_nat,N: nat] :
( ~ ( finite1152437895449049373et_nat @ A2 )
=> ? [B7: set_set_nat] :
( ( finite1152437895449049373et_nat @ B7 )
& ( ( finite_card_set_nat @ B7 )
= N )
& ( ord_le6893508408891458716et_nat @ B7 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_584_infinite__arbitrarily__large,axiom,
! [A2: set_nat,N: nat] :
( ~ ( finite_finite_nat @ A2 )
=> ? [B7: set_nat] :
( ( finite_finite_nat @ B7 )
& ( ( finite_card_nat @ B7 )
= N )
& ( ord_less_eq_set_nat @ B7 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_585_psubset__card__mono,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_less_set_set_nat @ A2 @ B2 )
=> ( ord_less_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_586_psubset__card__mono,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ord_le152980574450754630et_nat @ A2 @ B2 )
=> ( ord_less_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite1149291290879098388et_nat @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_587_psubset__card__mono,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_set_nat @ A2 @ B2 )
=> ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_588_card__image,axiom,
! [F: set_nat > set_nat,A2: set_set_nat] :
( ( inj_on4604407203859583615et_nat @ F @ A2 )
=> ( ( finite_card_set_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) )
= ( finite_card_set_nat @ A2 ) ) ) ).
% card_image
thf(fact_589_card__image,axiom,
! [F: nat > set_nat,A2: set_nat] :
( ( inj_on_nat_set_nat @ F @ A2 )
=> ( ( finite_card_set_nat @ ( image_nat_set_nat @ F @ A2 ) )
= ( finite_card_nat @ A2 ) ) ) ).
% card_image
thf(fact_590_card__image,axiom,
! [F: set_set_nat > set_nat,A2: set_set_set_nat] :
( ( inj_on1894729867836481333et_nat @ F @ A2 )
=> ( ( finite_card_set_nat @ ( image_5842784325960735177et_nat @ F @ A2 ) )
= ( finite1149291290879098388et_nat @ A2 ) ) ) ).
% card_image
thf(fact_591_card__image,axiom,
! [F: set_nat > nat,A2: set_set_nat] :
( ( inj_on_set_nat_nat @ F @ A2 )
=> ( ( finite_card_nat @ ( image_set_nat_nat @ F @ A2 ) )
= ( finite_card_set_nat @ A2 ) ) ) ).
% card_image
thf(fact_592_card__image,axiom,
! [F: nat > nat,A2: set_nat] :
( ( inj_on_nat_nat @ F @ A2 )
=> ( ( finite_card_nat @ ( image_nat_nat @ F @ A2 ) )
= ( finite_card_nat @ A2 ) ) ) ).
% card_image
thf(fact_593_card__image,axiom,
! [F: set_set_nat > nat,A2: set_set_set_nat] :
( ( inj_on7365807742884704127at_nat @ F @ A2 )
=> ( ( finite_card_nat @ ( image_1454916318497077779at_nat @ F @ A2 ) )
= ( finite1149291290879098388et_nat @ A2 ) ) ) ).
% card_image
thf(fact_594_card__image,axiom,
! [F: set_nat > set_set_nat,A2: set_set_nat] :
( ( inj_on2776966659131765557et_nat @ F @ A2 )
=> ( ( finite1149291290879098388et_nat @ ( image_6725021117256019401et_nat @ F @ A2 ) )
= ( finite_card_set_nat @ A2 ) ) ) ).
% card_image
thf(fact_595_card__image,axiom,
! [F: nat > set_set_nat,A2: set_nat] :
( ( inj_on8105003582846801791et_nat @ F @ A2 )
=> ( ( finite1149291290879098388et_nat @ ( image_2194112158459175443et_nat @ F @ A2 ) )
= ( finite_card_nat @ A2 ) ) ) ).
% card_image
thf(fact_596_card__image,axiom,
! [F: set_set_nat > set_set_nat,A2: set_set_set_nat] :
( ( inj_on2040386338155636715et_nat @ F @ A2 )
=> ( ( finite1149291290879098388et_nat @ ( image_7884819252390400639et_nat @ F @ A2 ) )
= ( finite1149291290879098388et_nat @ A2 ) ) ) ).
% card_image
thf(fact_597_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K3 )
=> ~ ( P @ I4 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_598_subset__eq__atLeast0__lessThan__finite,axiom,
! [N3: set_nat,N: nat] :
( ( ord_less_eq_set_nat @ N3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( finite_finite_nat @ N3 ) ) ).
% subset_eq_atLeast0_lessThan_finite
thf(fact_599_card__psubset,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite1149291290879098388et_nat @ B2 ) )
=> ( ord_le152980574450754630et_nat @ A2 @ B2 ) ) ) ) ).
% card_psubset
thf(fact_600_card__psubset,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) )
=> ( ord_less_set_set_nat @ A2 @ B2 ) ) ) ) ).
% card_psubset
thf(fact_601_card__psubset,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
=> ( ord_less_set_nat @ A2 @ B2 ) ) ) ) ).
% card_psubset
thf(fact_602_card__ge__0__finite,axiom,
! [A2: set_set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_set_nat @ A2 ) )
=> ( finite1152437895449049373et_nat @ A2 ) ) ).
% card_ge_0_finite
thf(fact_603_card__ge__0__finite,axiom,
! [A2: set_set_set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite1149291290879098388et_nat @ A2 ) )
=> ( finite6739761609112101331et_nat @ A2 ) ) ).
% card_ge_0_finite
thf(fact_604_card__ge__0__finite,axiom,
! [A2: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A2 ) )
=> ( finite_finite_nat @ A2 ) ) ).
% card_ge_0_finite
thf(fact_605_eq__card__imp__inj__on,axiom,
! [A2: set_set_nat,F: set_nat > set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( ( finite_card_set_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) )
= ( finite_card_set_nat @ A2 ) )
=> ( inj_on4604407203859583615et_nat @ F @ A2 ) ) ) ).
% eq_card_imp_inj_on
thf(fact_606_eq__card__imp__inj__on,axiom,
! [A2: set_set_nat,F: set_nat > nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( ( finite_card_nat @ ( image_set_nat_nat @ F @ A2 ) )
= ( finite_card_set_nat @ A2 ) )
=> ( inj_on_set_nat_nat @ F @ A2 ) ) ) ).
% eq_card_imp_inj_on
thf(fact_607_eq__card__imp__inj__on,axiom,
! [A2: set_set_nat,F: set_nat > set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( ( finite1149291290879098388et_nat @ ( image_6725021117256019401et_nat @ F @ A2 ) )
= ( finite_card_set_nat @ A2 ) )
=> ( inj_on2776966659131765557et_nat @ F @ A2 ) ) ) ).
% eq_card_imp_inj_on
thf(fact_608_eq__card__imp__inj__on,axiom,
! [A2: set_set_set_nat,F: set_set_nat > set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( ( finite_card_set_nat @ ( image_5842784325960735177et_nat @ F @ A2 ) )
= ( finite1149291290879098388et_nat @ A2 ) )
=> ( inj_on1894729867836481333et_nat @ F @ A2 ) ) ) ).
% eq_card_imp_inj_on
thf(fact_609_eq__card__imp__inj__on,axiom,
! [A2: set_set_set_nat,F: set_set_nat > nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( ( finite_card_nat @ ( image_1454916318497077779at_nat @ F @ A2 ) )
= ( finite1149291290879098388et_nat @ A2 ) )
=> ( inj_on7365807742884704127at_nat @ F @ A2 ) ) ) ).
% eq_card_imp_inj_on
thf(fact_610_eq__card__imp__inj__on,axiom,
! [A2: set_set_set_nat,F: set_set_nat > set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( ( finite1149291290879098388et_nat @ ( image_7884819252390400639et_nat @ F @ A2 ) )
= ( finite1149291290879098388et_nat @ A2 ) )
=> ( inj_on2040386338155636715et_nat @ F @ A2 ) ) ) ).
% eq_card_imp_inj_on
thf(fact_611_eq__card__imp__inj__on,axiom,
! [A2: set_nat,F: nat > set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( finite_card_set_nat @ ( image_nat_set_nat @ F @ A2 ) )
= ( finite_card_nat @ A2 ) )
=> ( inj_on_nat_set_nat @ F @ A2 ) ) ) ).
% eq_card_imp_inj_on
thf(fact_612_eq__card__imp__inj__on,axiom,
! [A2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( finite_card_nat @ ( image_nat_nat @ F @ A2 ) )
= ( finite_card_nat @ A2 ) )
=> ( inj_on_nat_nat @ F @ A2 ) ) ) ).
% eq_card_imp_inj_on
thf(fact_613_eq__card__imp__inj__on,axiom,
! [A2: set_nat,F: nat > set_set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( finite1149291290879098388et_nat @ ( image_2194112158459175443et_nat @ F @ A2 ) )
= ( finite_card_nat @ A2 ) )
=> ( inj_on8105003582846801791et_nat @ F @ A2 ) ) ) ).
% eq_card_imp_inj_on
thf(fact_614_inj__on__iff__eq__card,axiom,
! [A2: set_set_nat,F: set_nat > set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( inj_on4604407203859583615et_nat @ F @ A2 )
= ( ( finite_card_set_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) )
= ( finite_card_set_nat @ A2 ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_615_inj__on__iff__eq__card,axiom,
! [A2: set_set_nat,F: set_nat > nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( inj_on_set_nat_nat @ F @ A2 )
= ( ( finite_card_nat @ ( image_set_nat_nat @ F @ A2 ) )
= ( finite_card_set_nat @ A2 ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_616_inj__on__iff__eq__card,axiom,
! [A2: set_set_nat,F: set_nat > set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( inj_on2776966659131765557et_nat @ F @ A2 )
= ( ( finite1149291290879098388et_nat @ ( image_6725021117256019401et_nat @ F @ A2 ) )
= ( finite_card_set_nat @ A2 ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_617_inj__on__iff__eq__card,axiom,
! [A2: set_set_set_nat,F: set_set_nat > set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( inj_on1894729867836481333et_nat @ F @ A2 )
= ( ( finite_card_set_nat @ ( image_5842784325960735177et_nat @ F @ A2 ) )
= ( finite1149291290879098388et_nat @ A2 ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_618_inj__on__iff__eq__card,axiom,
! [A2: set_set_set_nat,F: set_set_nat > nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( inj_on7365807742884704127at_nat @ F @ A2 )
= ( ( finite_card_nat @ ( image_1454916318497077779at_nat @ F @ A2 ) )
= ( finite1149291290879098388et_nat @ A2 ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_619_inj__on__iff__eq__card,axiom,
! [A2: set_set_set_nat,F: set_set_nat > set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( inj_on2040386338155636715et_nat @ F @ A2 )
= ( ( finite1149291290879098388et_nat @ ( image_7884819252390400639et_nat @ F @ A2 ) )
= ( finite1149291290879098388et_nat @ A2 ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_620_inj__on__iff__eq__card,axiom,
! [A2: set_nat,F: nat > set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( inj_on_nat_set_nat @ F @ A2 )
= ( ( finite_card_set_nat @ ( image_nat_set_nat @ F @ A2 ) )
= ( finite_card_nat @ A2 ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_621_inj__on__iff__eq__card,axiom,
! [A2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( inj_on_nat_nat @ F @ A2 )
= ( ( finite_card_nat @ ( image_nat_nat @ F @ A2 ) )
= ( finite_card_nat @ A2 ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_622_inj__on__iff__eq__card,axiom,
! [A2: set_nat,F: nat > set_set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( inj_on8105003582846801791et_nat @ F @ A2 )
= ( ( finite1149291290879098388et_nat @ ( image_2194112158459175443et_nat @ F @ A2 ) )
= ( finite_card_nat @ A2 ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_623_pigeonhole,axiom,
! [F: set_nat > set_nat,A2: set_set_nat] :
( ( ord_less_nat @ ( finite_card_set_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) ) @ ( finite_card_set_nat @ A2 ) )
=> ~ ( inj_on4604407203859583615et_nat @ F @ A2 ) ) ).
% pigeonhole
thf(fact_624_pigeonhole,axiom,
! [F: nat > set_nat,A2: set_nat] :
( ( ord_less_nat @ ( finite_card_set_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ ( finite_card_nat @ A2 ) )
=> ~ ( inj_on_nat_set_nat @ F @ A2 ) ) ).
% pigeonhole
thf(fact_625_pigeonhole,axiom,
! [F: set_set_nat > set_nat,A2: set_set_set_nat] :
( ( ord_less_nat @ ( finite_card_set_nat @ ( image_5842784325960735177et_nat @ F @ A2 ) ) @ ( finite1149291290879098388et_nat @ A2 ) )
=> ~ ( inj_on1894729867836481333et_nat @ F @ A2 ) ) ).
% pigeonhole
thf(fact_626_pigeonhole,axiom,
! [F: set_nat > nat,A2: set_set_nat] :
( ( ord_less_nat @ ( finite_card_nat @ ( image_set_nat_nat @ F @ A2 ) ) @ ( finite_card_set_nat @ A2 ) )
=> ~ ( inj_on_set_nat_nat @ F @ A2 ) ) ).
% pigeonhole
thf(fact_627_pigeonhole,axiom,
! [F: nat > nat,A2: set_nat] :
( ( ord_less_nat @ ( finite_card_nat @ ( image_nat_nat @ F @ A2 ) ) @ ( finite_card_nat @ A2 ) )
=> ~ ( inj_on_nat_nat @ F @ A2 ) ) ).
% pigeonhole
thf(fact_628_pigeonhole,axiom,
! [F: set_set_nat > nat,A2: set_set_set_nat] :
( ( ord_less_nat @ ( finite_card_nat @ ( image_1454916318497077779at_nat @ F @ A2 ) ) @ ( finite1149291290879098388et_nat @ A2 ) )
=> ~ ( inj_on7365807742884704127at_nat @ F @ A2 ) ) ).
% pigeonhole
thf(fact_629_pigeonhole,axiom,
! [F: set_nat > set_set_nat,A2: set_set_nat] :
( ( ord_less_nat @ ( finite1149291290879098388et_nat @ ( image_6725021117256019401et_nat @ F @ A2 ) ) @ ( finite_card_set_nat @ A2 ) )
=> ~ ( inj_on2776966659131765557et_nat @ F @ A2 ) ) ).
% pigeonhole
thf(fact_630_pigeonhole,axiom,
! [F: nat > set_set_nat,A2: set_nat] :
( ( ord_less_nat @ ( finite1149291290879098388et_nat @ ( image_2194112158459175443et_nat @ F @ A2 ) ) @ ( finite_card_nat @ A2 ) )
=> ~ ( inj_on8105003582846801791et_nat @ F @ A2 ) ) ).
% pigeonhole
thf(fact_631_pigeonhole,axiom,
! [F: set_set_nat > set_set_nat,A2: set_set_set_nat] :
( ( ord_less_nat @ ( finite1149291290879098388et_nat @ ( image_7884819252390400639et_nat @ F @ A2 ) ) @ ( finite1149291290879098388et_nat @ A2 ) )
=> ~ ( inj_on2040386338155636715et_nat @ F @ A2 ) ) ).
% pigeonhole
thf(fact_632_finite__imp__inj__to__nat__seg,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ? [F3: set_nat > nat,N2: nat] :
( ( ( image_set_nat_nat @ F3 @ A2 )
= ( collect_nat
@ ^ [I2: nat] : ( ord_less_nat @ I2 @ N2 ) ) )
& ( inj_on_set_nat_nat @ F3 @ A2 ) ) ) ).
% finite_imp_inj_to_nat_seg
thf(fact_633_finite__imp__inj__to__nat__seg,axiom,
! [A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ? [F3: set_set_nat > nat,N2: nat] :
( ( ( image_1454916318497077779at_nat @ F3 @ A2 )
= ( collect_nat
@ ^ [I2: nat] : ( ord_less_nat @ I2 @ N2 ) ) )
& ( inj_on7365807742884704127at_nat @ F3 @ A2 ) ) ) ).
% finite_imp_inj_to_nat_seg
thf(fact_634_finite__imp__inj__to__nat__seg,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ? [F3: nat > nat,N2: nat] :
( ( ( image_nat_nat @ F3 @ A2 )
= ( collect_nat
@ ^ [I2: nat] : ( ord_less_nat @ I2 @ N2 ) ) )
& ( inj_on_nat_nat @ F3 @ A2 ) ) ) ).
% finite_imp_inj_to_nat_seg
thf(fact_635_inj__on__inverseI,axiom,
! [A2: set_nat,G: set_set_nat > nat,F: nat > set_set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( G @ ( F @ X3 ) )
= X3 ) )
=> ( inj_on8105003582846801791et_nat @ F @ A2 ) ) ).
% inj_on_inverseI
thf(fact_636_inj__on__inverseI,axiom,
! [A2: set_nat,G: nat > nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( G @ ( F @ X3 ) )
= X3 ) )
=> ( inj_on_nat_nat @ F @ A2 ) ) ).
% inj_on_inverseI
thf(fact_637_inj__on__contraD,axiom,
! [F: nat > set_set_nat,A2: set_nat,X: nat,Y: nat] :
( ( inj_on8105003582846801791et_nat @ F @ A2 )
=> ( ( X != Y )
=> ( ( member_nat @ X @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( ( F @ X )
!= ( F @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_638_inj__on__contraD,axiom,
! [F: nat > nat,A2: set_nat,X: nat,Y: nat] :
( ( inj_on_nat_nat @ F @ A2 )
=> ( ( X != Y )
=> ( ( member_nat @ X @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( ( F @ X )
!= ( F @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_639_inj__on__eq__iff,axiom,
! [F: nat > set_set_nat,A2: set_nat,X: nat,Y: nat] :
( ( inj_on8105003582846801791et_nat @ F @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_640_inj__on__eq__iff,axiom,
! [F: nat > nat,A2: set_nat,X: nat,Y: nat] :
( ( inj_on_nat_nat @ F @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_641_inj__on__cong,axiom,
! [A2: set_nat,F: nat > set_set_nat,G: nat > set_set_nat] :
( ! [A6: nat] :
( ( member_nat @ A6 @ A2 )
=> ( ( F @ A6 )
= ( G @ A6 ) ) )
=> ( ( inj_on8105003582846801791et_nat @ F @ A2 )
= ( inj_on8105003582846801791et_nat @ G @ A2 ) ) ) ).
% inj_on_cong
thf(fact_642_inj__on__cong,axiom,
! [A2: set_nat,F: nat > nat,G: nat > nat] :
( ! [A6: nat] :
( ( member_nat @ A6 @ A2 )
=> ( ( F @ A6 )
= ( G @ A6 ) ) )
=> ( ( inj_on_nat_nat @ F @ A2 )
= ( inj_on_nat_nat @ G @ A2 ) ) ) ).
% inj_on_cong
thf(fact_643_inj__on__def,axiom,
( inj_on8105003582846801791et_nat
= ( ^ [F4: nat > set_set_nat,A3: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A3 )
=> ! [Y5: nat] :
( ( member_nat @ Y5 @ A3 )
=> ( ( ( F4 @ X2 )
= ( F4 @ Y5 ) )
=> ( X2 = Y5 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_644_inj__on__def,axiom,
( inj_on_nat_nat
= ( ^ [F4: nat > nat,A3: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A3 )
=> ! [Y5: nat] :
( ( member_nat @ Y5 @ A3 )
=> ( ( ( F4 @ X2 )
= ( F4 @ Y5 ) )
=> ( X2 = Y5 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_645_inj__onI,axiom,
! [A2: set_nat,F: nat > set_set_nat] :
( ! [X3: nat,Y4: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_nat @ Y4 @ A2 )
=> ( ( ( F @ X3 )
= ( F @ Y4 ) )
=> ( X3 = Y4 ) ) ) )
=> ( inj_on8105003582846801791et_nat @ F @ A2 ) ) ).
% inj_onI
thf(fact_646_inj__onI,axiom,
! [A2: set_nat,F: nat > nat] :
( ! [X3: nat,Y4: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_nat @ Y4 @ A2 )
=> ( ( ( F @ X3 )
= ( F @ Y4 ) )
=> ( X3 = Y4 ) ) ) )
=> ( inj_on_nat_nat @ F @ A2 ) ) ).
% inj_onI
thf(fact_647_inj__onD,axiom,
! [F: nat > set_set_nat,A2: set_nat,X: nat,Y: nat] :
( ( inj_on8105003582846801791et_nat @ F @ A2 )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_nat @ X @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( X = Y ) ) ) ) ) ).
% inj_onD
thf(fact_648_inj__onD,axiom,
! [F: nat > nat,A2: set_nat,X: nat,Y: nat] :
( ( inj_on_nat_nat @ F @ A2 )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_nat @ X @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( X = Y ) ) ) ) ) ).
% inj_onD
thf(fact_649_card__bij__eq,axiom,
! [F: set_set_nat > set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat,G: set_set_nat > set_set_nat] :
( ( inj_on2040386338155636715et_nat @ F @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ ( image_7884819252390400639et_nat @ F @ A2 ) @ B2 )
=> ( ( inj_on2040386338155636715et_nat @ G @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ ( image_7884819252390400639et_nat @ G @ B2 ) @ A2 )
=> ( ( finite6739761609112101331et_nat @ A2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( ( finite1149291290879098388et_nat @ A2 )
= ( finite1149291290879098388et_nat @ B2 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_650_card__bij__eq,axiom,
! [F: set_nat > set_set_nat,A2: set_set_nat,B2: set_set_set_nat,G: set_set_nat > set_nat] :
( ( inj_on2776966659131765557et_nat @ F @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ ( image_6725021117256019401et_nat @ F @ A2 ) @ B2 )
=> ( ( inj_on1894729867836481333et_nat @ G @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ ( image_5842784325960735177et_nat @ G @ B2 ) @ A2 )
=> ( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( ( finite_card_set_nat @ A2 )
= ( finite1149291290879098388et_nat @ B2 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_651_card__bij__eq,axiom,
! [F: nat > set_set_nat,A2: set_nat,B2: set_set_set_nat,G: set_set_nat > nat] :
( ( inj_on8105003582846801791et_nat @ F @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ ( image_2194112158459175443et_nat @ F @ A2 ) @ B2 )
=> ( ( inj_on7365807742884704127at_nat @ G @ B2 )
=> ( ( ord_less_eq_set_nat @ ( image_1454916318497077779at_nat @ G @ B2 ) @ A2 )
=> ( ( finite_finite_nat @ A2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( ( finite_card_nat @ A2 )
= ( finite1149291290879098388et_nat @ B2 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_652_card__bij__eq,axiom,
! [F: set_set_nat > set_nat,A2: set_set_set_nat,B2: set_set_nat,G: set_nat > set_set_nat] :
( ( inj_on1894729867836481333et_nat @ F @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ ( image_5842784325960735177et_nat @ F @ A2 ) @ B2 )
=> ( ( inj_on2776966659131765557et_nat @ G @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ ( image_6725021117256019401et_nat @ G @ B2 ) @ A2 )
=> ( ( finite6739761609112101331et_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ( finite1149291290879098388et_nat @ A2 )
= ( finite_card_set_nat @ B2 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_653_card__bij__eq,axiom,
! [F: set_nat > set_nat,A2: set_set_nat,B2: set_set_nat,G: set_nat > set_nat] :
( ( inj_on4604407203859583615et_nat @ F @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) @ B2 )
=> ( ( inj_on4604407203859583615et_nat @ G @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ G @ B2 ) @ A2 )
=> ( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ( finite_card_set_nat @ A2 )
= ( finite_card_set_nat @ B2 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_654_card__bij__eq,axiom,
! [F: nat > set_nat,A2: set_nat,B2: set_set_nat,G: set_nat > nat] :
( ( inj_on_nat_set_nat @ F @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A2 ) @ B2 )
=> ( ( inj_on_set_nat_nat @ G @ B2 )
=> ( ( ord_less_eq_set_nat @ ( image_set_nat_nat @ G @ B2 ) @ A2 )
=> ( ( finite_finite_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ( finite_card_nat @ A2 )
= ( finite_card_set_nat @ B2 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_655_card__bij__eq,axiom,
! [F: set_set_nat > nat,A2: set_set_set_nat,B2: set_nat,G: nat > set_set_nat] :
( ( inj_on7365807742884704127at_nat @ F @ A2 )
=> ( ( ord_less_eq_set_nat @ ( image_1454916318497077779at_nat @ F @ A2 ) @ B2 )
=> ( ( inj_on8105003582846801791et_nat @ G @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ ( image_2194112158459175443et_nat @ G @ B2 ) @ A2 )
=> ( ( finite6739761609112101331et_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( finite1149291290879098388et_nat @ A2 )
= ( finite_card_nat @ B2 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_656_card__bij__eq,axiom,
! [F: set_nat > nat,A2: set_set_nat,B2: set_nat,G: nat > set_nat] :
( ( inj_on_set_nat_nat @ F @ A2 )
=> ( ( ord_less_eq_set_nat @ ( image_set_nat_nat @ F @ A2 ) @ B2 )
=> ( ( inj_on_nat_set_nat @ G @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ G @ B2 ) @ A2 )
=> ( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( finite_card_set_nat @ A2 )
= ( finite_card_nat @ B2 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_657_card__bij__eq,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat,G: nat > nat] :
( ( inj_on_nat_nat @ F @ A2 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B2 )
=> ( ( inj_on_nat_nat @ G @ B2 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ G @ B2 ) @ A2 )
=> ( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B2 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_658_surjective__iff__injective__gen,axiom,
! [S2: set_set_nat,T3: set_set_set_nat,F: set_nat > set_set_nat] :
( ( finite1152437895449049373et_nat @ S2 )
=> ( ( finite6739761609112101331et_nat @ T3 )
=> ( ( ( finite_card_set_nat @ S2 )
= ( finite1149291290879098388et_nat @ T3 ) )
=> ( ( ord_le9131159989063066194et_nat @ ( image_6725021117256019401et_nat @ F @ S2 ) @ T3 )
=> ( ( ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ T3 )
=> ? [Y5: set_nat] :
( ( member_set_nat @ Y5 @ S2 )
& ( ( F @ Y5 )
= X2 ) ) ) )
= ( inj_on2776966659131765557et_nat @ F @ S2 ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_659_surjective__iff__injective__gen,axiom,
! [S2: set_set_set_nat,T3: set_set_set_nat,F: set_set_nat > set_set_nat] :
( ( finite6739761609112101331et_nat @ S2 )
=> ( ( finite6739761609112101331et_nat @ T3 )
=> ( ( ( finite1149291290879098388et_nat @ S2 )
= ( finite1149291290879098388et_nat @ T3 ) )
=> ( ( ord_le9131159989063066194et_nat @ ( image_7884819252390400639et_nat @ F @ S2 ) @ T3 )
=> ( ( ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ T3 )
=> ? [Y5: set_set_nat] :
( ( member_set_set_nat @ Y5 @ S2 )
& ( ( F @ Y5 )
= X2 ) ) ) )
= ( inj_on2040386338155636715et_nat @ F @ S2 ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_660_surjective__iff__injective__gen,axiom,
! [S2: set_nat,T3: set_set_set_nat,F: nat > set_set_nat] :
( ( finite_finite_nat @ S2 )
=> ( ( finite6739761609112101331et_nat @ T3 )
=> ( ( ( finite_card_nat @ S2 )
= ( finite1149291290879098388et_nat @ T3 ) )
=> ( ( ord_le9131159989063066194et_nat @ ( image_2194112158459175443et_nat @ F @ S2 ) @ T3 )
=> ( ( ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ T3 )
=> ? [Y5: nat] :
( ( member_nat @ Y5 @ S2 )
& ( ( F @ Y5 )
= X2 ) ) ) )
= ( inj_on8105003582846801791et_nat @ F @ S2 ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_661_surjective__iff__injective__gen,axiom,
! [S2: set_set_nat,T3: set_set_nat,F: set_nat > set_nat] :
( ( finite1152437895449049373et_nat @ S2 )
=> ( ( finite1152437895449049373et_nat @ T3 )
=> ( ( ( finite_card_set_nat @ S2 )
= ( finite_card_set_nat @ T3 ) )
=> ( ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ F @ S2 ) @ T3 )
=> ( ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ T3 )
=> ? [Y5: set_nat] :
( ( member_set_nat @ Y5 @ S2 )
& ( ( F @ Y5 )
= X2 ) ) ) )
= ( inj_on4604407203859583615et_nat @ F @ S2 ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_662_surjective__iff__injective__gen,axiom,
! [S2: set_set_set_nat,T3: set_set_nat,F: set_set_nat > set_nat] :
( ( finite6739761609112101331et_nat @ S2 )
=> ( ( finite1152437895449049373et_nat @ T3 )
=> ( ( ( finite1149291290879098388et_nat @ S2 )
= ( finite_card_set_nat @ T3 ) )
=> ( ( ord_le6893508408891458716et_nat @ ( image_5842784325960735177et_nat @ F @ S2 ) @ T3 )
=> ( ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ T3 )
=> ? [Y5: set_set_nat] :
( ( member_set_set_nat @ Y5 @ S2 )
& ( ( F @ Y5 )
= X2 ) ) ) )
= ( inj_on1894729867836481333et_nat @ F @ S2 ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_663_surjective__iff__injective__gen,axiom,
! [S2: set_nat,T3: set_set_nat,F: nat > set_nat] :
( ( finite_finite_nat @ S2 )
=> ( ( finite1152437895449049373et_nat @ T3 )
=> ( ( ( finite_card_nat @ S2 )
= ( finite_card_set_nat @ T3 ) )
=> ( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ S2 ) @ T3 )
=> ( ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ T3 )
=> ? [Y5: nat] :
( ( member_nat @ Y5 @ S2 )
& ( ( F @ Y5 )
= X2 ) ) ) )
= ( inj_on_nat_set_nat @ F @ S2 ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_664_surjective__iff__injective__gen,axiom,
! [S2: set_set_nat,T3: set_nat,F: set_nat > nat] :
( ( finite1152437895449049373et_nat @ S2 )
=> ( ( finite_finite_nat @ T3 )
=> ( ( ( finite_card_set_nat @ S2 )
= ( finite_card_nat @ T3 ) )
=> ( ( ord_less_eq_set_nat @ ( image_set_nat_nat @ F @ S2 ) @ T3 )
=> ( ( ! [X2: nat] :
( ( member_nat @ X2 @ T3 )
=> ? [Y5: set_nat] :
( ( member_set_nat @ Y5 @ S2 )
& ( ( F @ Y5 )
= X2 ) ) ) )
= ( inj_on_set_nat_nat @ F @ S2 ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_665_surjective__iff__injective__gen,axiom,
! [S2: set_set_set_nat,T3: set_nat,F: set_set_nat > nat] :
( ( finite6739761609112101331et_nat @ S2 )
=> ( ( finite_finite_nat @ T3 )
=> ( ( ( finite1149291290879098388et_nat @ S2 )
= ( finite_card_nat @ T3 ) )
=> ( ( ord_less_eq_set_nat @ ( image_1454916318497077779at_nat @ F @ S2 ) @ T3 )
=> ( ( ! [X2: nat] :
( ( member_nat @ X2 @ T3 )
=> ? [Y5: set_set_nat] :
( ( member_set_set_nat @ Y5 @ S2 )
& ( ( F @ Y5 )
= X2 ) ) ) )
= ( inj_on7365807742884704127at_nat @ F @ S2 ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_666_surjective__iff__injective__gen,axiom,
! [S2: set_nat,T3: set_nat,F: nat > nat] :
( ( finite_finite_nat @ S2 )
=> ( ( finite_finite_nat @ T3 )
=> ( ( ( finite_card_nat @ S2 )
= ( finite_card_nat @ T3 ) )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ S2 ) @ T3 )
=> ( ( ! [X2: nat] :
( ( member_nat @ X2 @ T3 )
=> ? [Y5: nat] :
( ( member_nat @ Y5 @ S2 )
& ( ( F @ Y5 )
= X2 ) ) ) )
= ( inj_on_nat_nat @ F @ S2 ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_667_inj__on__id2,axiom,
! [A2: set_nat] :
( inj_on_nat_nat
@ ^ [X2: nat] : X2
@ A2 ) ).
% inj_on_id2
thf(fact_668_linorder__inj__onI_H,axiom,
! [A2: set_nat,F: nat > set_set_nat] :
( ! [I3: nat,J3: nat] :
( ( member_nat @ I3 @ A2 )
=> ( ( member_nat @ J3 @ A2 )
=> ( ( ord_less_nat @ I3 @ J3 )
=> ( ( F @ I3 )
!= ( F @ J3 ) ) ) ) )
=> ( inj_on8105003582846801791et_nat @ F @ A2 ) ) ).
% linorder_inj_onI'
thf(fact_669_linorder__inj__onI_H,axiom,
! [A2: set_nat,F: nat > nat] :
( ! [I3: nat,J3: nat] :
( ( member_nat @ I3 @ A2 )
=> ( ( member_nat @ J3 @ A2 )
=> ( ( ord_less_nat @ I3 @ J3 )
=> ( ( F @ I3 )
!= ( F @ J3 ) ) ) ) )
=> ( inj_on_nat_nat @ F @ A2 ) ) ).
% linorder_inj_onI'
thf(fact_670_image__strict__mono,axiom,
! [F: set_set_nat > set_nat,B2: set_set_set_nat,A2: set_set_set_nat] :
( ( inj_on1894729867836481333et_nat @ F @ B2 )
=> ( ( ord_le152980574450754630et_nat @ A2 @ B2 )
=> ( ord_less_set_set_nat @ ( image_5842784325960735177et_nat @ F @ A2 ) @ ( image_5842784325960735177et_nat @ F @ B2 ) ) ) ) ).
% image_strict_mono
thf(fact_671_image__strict__mono,axiom,
! [F: nat > set_set_nat,B2: set_nat,A2: set_nat] :
( ( inj_on8105003582846801791et_nat @ F @ B2 )
=> ( ( ord_less_set_nat @ A2 @ B2 )
=> ( ord_le152980574450754630et_nat @ ( image_2194112158459175443et_nat @ F @ A2 ) @ ( image_2194112158459175443et_nat @ F @ B2 ) ) ) ) ).
% image_strict_mono
thf(fact_672_image__strict__mono,axiom,
! [F: nat > nat,B2: set_nat,A2: set_nat] :
( ( inj_on_nat_nat @ F @ B2 )
=> ( ( ord_less_set_nat @ A2 @ B2 )
=> ( ord_less_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B2 ) ) ) ) ).
% image_strict_mono
thf(fact_673_inj__on__image__iff,axiom,
! [A2: set_nat,G: nat > set_set_nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( ( G @ ( F @ X3 ) )
= ( G @ ( F @ Xa2 ) ) )
= ( ( G @ X3 )
= ( G @ Xa2 ) ) ) ) )
=> ( ( inj_on_nat_nat @ F @ A2 )
=> ( ( inj_on8105003582846801791et_nat @ G @ ( image_nat_nat @ F @ A2 ) )
= ( inj_on8105003582846801791et_nat @ G @ A2 ) ) ) ) ).
% inj_on_image_iff
thf(fact_674_inj__on__image__iff,axiom,
! [A2: set_nat,G: nat > nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( ( G @ ( F @ X3 ) )
= ( G @ ( F @ Xa2 ) ) )
= ( ( G @ X3 )
= ( G @ Xa2 ) ) ) ) )
=> ( ( inj_on_nat_nat @ F @ A2 )
=> ( ( inj_on_nat_nat @ G @ ( image_nat_nat @ F @ A2 ) )
= ( inj_on_nat_nat @ G @ A2 ) ) ) ) ).
% inj_on_image_iff
thf(fact_675_subset__inj__on,axiom,
! [F: nat > set_set_nat,B2: set_nat,A2: set_nat] :
( ( inj_on8105003582846801791et_nat @ F @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( inj_on8105003582846801791et_nat @ F @ A2 ) ) ) ).
% subset_inj_on
thf(fact_676_subset__inj__on,axiom,
! [F: nat > nat,B2: set_nat,A2: set_nat] :
( ( inj_on_nat_nat @ F @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( inj_on_nat_nat @ F @ A2 ) ) ) ).
% subset_inj_on
thf(fact_677_inj__on__subset,axiom,
! [F: nat > set_set_nat,A2: set_nat,B2: set_nat] :
( ( inj_on8105003582846801791et_nat @ F @ A2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( inj_on8105003582846801791et_nat @ F @ B2 ) ) ) ).
% inj_on_subset
thf(fact_678_inj__on__subset,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( inj_on_nat_nat @ F @ A2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( inj_on_nat_nat @ F @ B2 ) ) ) ).
% inj_on_subset
thf(fact_679_i__props_I5_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ p )
=> ( ord_less_eq_nat @ ( si2 @ I ) @ l ) ) ).
% i_props(5)
thf(fact_680_accepts__def,axiom,
( clique3686358387679108662ccepts
= ( ^ [X6: set_set_set_nat,G4: set_set_nat] :
? [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ X6 )
& ( ord_le6893508408891458716et_nat @ X2 @ G4 ) ) ) ) ).
% accepts_def
thf(fact_681_sunflower,axiom,
? [S3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ S3 @ ( clique8462013130872731469t_v_gs @ x ) )
& ( sunflower_nat @ S3 )
& ( ( finite_card_set_nat @ S3 )
= p ) ) ).
% sunflower
thf(fact_682_SvG,axiom,
( s
= ( image_5842784325960735177et_nat @ clique5033774636164728513irst_v @ ( image_2194112158459175443et_nat @ g @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ p ) ) ) ) ).
% SvG
thf(fact_683_sf__precond,axiom,
! [X4: set_nat] :
( ( member_set_nat @ X4 @ ( clique8462013130872731469t_v_gs @ x ) )
=> ( ( finite_finite_nat @ X4 )
& ( ord_less_eq_nat @ ( finite_card_nat @ X4 ) @ l ) ) ) ).
% sf_precond
thf(fact_684_i__props_I1_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ p )
=> ( ord_less_eq_set_nat @ vs @ ( clique5033774636164728513irst_v @ ( g @ I ) ) ) ) ).
% i_props(1)
thf(fact_685_G_I4_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ p )
=> ( member_set_nat @ ( clique5033774636164728513irst_v @ ( g @ I ) ) @ s ) ) ).
% G(4)
thf(fact_686_U__def,axiom,
( u
= ( collect_set_set_nat
@ ^ [E: set_set_nat] :
( ( member_set_set_nat @ E @ x )
& ( member_set_nat @ ( clique5033774636164728513irst_v @ E ) @ s ) ) ) ) ).
% U_def
thf(fact_687__092_060open_062card_A_Iv__gs_AU_J_A_061_Acard_AS_092_060close_062,axiom,
( ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ u ) )
= ( finite_card_set_nat @ s ) ) ).
% \<open>card (v_gs U) = card S\<close>
thf(fact_688_fin__Vs,axiom,
finite_finite_nat @ vs ).
% fin_Vs
thf(fact_689__092_060open_062si_A_092_060equiv_062_A_092_060lambda_062i_O_Acard_A_Iv_A_IG_Ai_J_J_092_060close_062,axiom,
( si2
= ( ^ [I2: nat] : ( finite_card_nat @ ( clique5033774636164728513irst_v @ ( g @ I2 ) ) ) ) ) ).
% \<open>si \<equiv> \<lambda>i. card (v (G i))\<close>
thf(fact_690_finS,axiom,
finite1152437895449049373et_nat @ s ).
% finS
thf(fact_691_S_I2_J,axiom,
sunflower_nat @ s ).
% S(2)
thf(fact_692_S_I3_J,axiom,
( ( finite_card_set_nat @ s )
= p ) ).
% S(3)
thf(fact_693_vplus__dsU,axiom,
( ( clique8462013130872731469t_v_gs @ u )
= s ) ).
% vplus_dsU
thf(fact_694_r__def,axiom,
( r
= ( finite1149291290879098388et_nat @ u ) ) ).
% r_def
thf(fact_695_card__Vs,axiom,
ord_less_eq_nat @ ( finite_card_nat @ vs ) @ l ).
% card_Vs
thf(fact_696_S_I1_J,axiom,
ord_le6893508408891458716et_nat @ s @ ( clique8462013130872731469t_v_gs @ x ) ).
% S(1)
thf(fact_697_si__def,axiom,
! [I: nat] :
( ( si2 @ I )
= ( finite_card_nat @ ( clique5033774636164728513irst_v @ ( g @ I ) ) ) ) ).
% si_def
thf(fact_698_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I2: nat] : ( ord_less_nat @ I2 @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_699__092_060open_062S_A_092_060subseteq_062_Av__gs_AX_A_092_060and_062_Asunflower_AS_A_092_060and_062_Acard_AS_A_061_Ap_092_060close_062,axiom,
( ( ord_le6893508408891458716et_nat @ s @ ( clique8462013130872731469t_v_gs @ x ) )
& ( sunflower_nat @ s )
& ( ( finite_card_set_nat @ s )
= p ) ) ).
% \<open>S \<subseteq> v_gs X \<and> sunflower S \<and> card S = p\<close>
thf(fact_700_acceptsI,axiom,
! [D2: set_set_nat,G2: set_set_nat,X5: set_set_set_nat] :
( ( ord_le6893508408891458716et_nat @ D2 @ G2 )
=> ( ( member_set_set_nat @ D2 @ X5 )
=> ( clique3686358387679108662ccepts @ X5 @ G2 ) ) ) ).
% acceptsI
thf(fact_701_s__def,axiom,
( s2
= ( finite_card_nat @ vs ) ) ).
% s_def
thf(fact_702__092_060open_062fstt_A_092_060equiv_062_A_092_060lambda_062e_O_ASOME_Ax_O_Ax_A_092_060in_062_Ae_A_092_060and_062_Ax_A_092_060notin_062_AVs_092_060close_062,axiom,
( fstt
= ( ^ [E2: set_nat] :
( fChoice_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ E2 )
& ~ ( member_nat @ X2 @ vs ) ) ) ) ) ).
% \<open>fstt \<equiv> \<lambda>e. SOME x. x \<in> e \<and> x \<notin> Vs\<close>
thf(fact_703_i__props_I4_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ p )
=> ( ord_less_eq_nat @ s2 @ ( si2 @ I ) ) ) ).
% i_props(4)
thf(fact_704_Si,axiom,
bij_betw_nat_set_nat @ si @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ p ) @ s ).
% Si
thf(fact_705_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M4: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M4 ) )
=> ~ ! [M5: nat] :
( ( P @ M5 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M5 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_706_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_707_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_708_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_709_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_710_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_711_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_712_subset__eq__atLeast0__lessThan__card,axiom,
! [N3: set_nat,N: nat] :
( ( ord_less_eq_set_nat @ N3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ N3 ) @ N ) ) ).
% subset_eq_atLeast0_lessThan_card
thf(fact_713_sunflower__card__subset__lift,axiom,
! [K: nat,C2: nat,R2: nat,F2: set_set_set_set_nat] :
( ! [G3: set_se7521423693449168855at_nat] :
( ! [X4: set_Su1440016900418933025at_nat] :
( ( member5638249034155602744at_nat @ X4 @ G3 )
=> ( ( finite8770298478261192322at_nat @ X4 )
& ( ( finite8251389301641259331at_nat @ X4 )
= K ) ) )
=> ( ( ord_less_nat @ C2 @ ( finite7696428214769936121at_nat @ G3 ) )
=> ? [S4: set_se7521423693449168855at_nat] :
( ( ord_le2853704879392749623at_nat @ S4 @ G3 )
& ( sunflo3853689026006497528at_nat @ S4 )
& ( ( finite7696428214769936121at_nat @ S4 )
= R2 ) ) ) )
=> ( ! [X3: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X3 @ F2 )
=> ( ( finite6739761609112101331et_nat @ X3 )
& ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ X3 ) @ K ) ) )
=> ( ( ord_less_nat @ C2 @ ( finite8805468973633305546et_nat @ F2 ) )
=> ? [S3: set_set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ S3 @ F2 )
& ( sunflo2680516271513359689et_nat @ S3 )
& ( ( finite8805468973633305546et_nat @ S3 )
= R2 ) ) ) ) ) ).
% sunflower_card_subset_lift
thf(fact_714_sunflower__card__subset__lift,axiom,
! [K: nat,C2: nat,R2: nat,F2: set_set_set_nat] :
( ! [G3: set_se8003284279568041249at_nat] :
( ! [X4: set_Su8059080322890262379at_nat] :
( ( member5374901640408327554at_nat @ X4 @ G3 )
=> ( ( finite2491568536608231884at_nat @ X4 )
& ( ( finite8413070326521870477at_nat @ X4 )
= K ) ) )
=> ( ( ord_less_nat @ C2 @ ( finite7758422657562484035at_nat @ G3 ) )
=> ? [S4: set_se8003284279568041249at_nat] :
( ( ord_le4731320016863163777at_nat @ S4 @ G3 )
& ( sunflo6650083805840251970at_nat @ S4 )
& ( ( finite7758422657562484035at_nat @ S4 )
= R2 ) ) ) )
=> ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ F2 )
=> ( ( finite1152437895449049373et_nat @ X3 )
& ( ord_less_eq_nat @ ( finite_card_set_nat @ X3 ) @ K ) ) )
=> ( ( ord_less_nat @ C2 @ ( finite1149291290879098388et_nat @ F2 ) )
=> ? [S3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ S3 @ F2 )
& ( sunflower_set_nat @ S3 )
& ( ( finite1149291290879098388et_nat @ S3 )
= R2 ) ) ) ) ) ).
% sunflower_card_subset_lift
thf(fact_715_sunflower__card__subset__lift,axiom,
! [K: nat,C2: nat,R2: nat,F2: set_set_nat] :
( ! [G3: set_se3873067930692246379at_nat] :
( ! [X4: set_Sum_sum_nat_nat] :
( ( member1869216328726507724at_nat @ X4 @ G3 )
=> ( ( finite6187706683773761046at_nat @ X4 )
& ( ( finite8494011213269508311at_nat @ X4 )
= K ) ) )
=> ( ( ord_less_nat @ C2 @ ( finite2024029949821234317at_nat @ G3 ) )
=> ? [S4: set_se3873067930692246379at_nat] :
( ( ord_le3495481059733392331at_nat @ S4 @ G3 )
& ( sunflo1841451327523575948at_nat @ S4 )
& ( ( finite2024029949821234317at_nat @ S4 )
= R2 ) ) ) )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ F2 )
=> ( ( finite_finite_nat @ X3 )
& ( ord_less_eq_nat @ ( finite_card_nat @ X3 ) @ K ) ) )
=> ( ( ord_less_nat @ C2 @ ( finite_card_set_nat @ F2 ) )
=> ? [S3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ S3 @ F2 )
& ( sunflower_nat @ S3 )
& ( ( finite_card_set_nat @ S3 )
= R2 ) ) ) ) ) ).
% sunflower_card_subset_lift
thf(fact_716_S__def,axiom,
( s
= ( fChoice_set_set_nat
@ ^ [S5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ S5 @ ( clique8462013130872731469t_v_gs @ x ) )
& ( sunflower_nat @ S5 )
& ( ( finite_card_set_nat @ S5 )
= p ) ) ) ) ).
% S_def
thf(fact_717__092_060open_062ti_A_092_060equiv_062_A_092_060lambda_062i_O_Acard_A_Iv_A_IG_Ai_J_A_N_AVs_J_092_060close_062,axiom,
( ti
= ( ^ [I2: nat] : ( finite_card_nat @ ( minus_minus_set_nat @ ( clique5033774636164728513irst_v @ ( g @ I2 ) ) @ vs ) ) ) ) ).
% \<open>ti \<equiv> \<lambda>i. card (v (G i) - Vs)\<close>
thf(fact_718_G__def,axiom,
( g
= ( ^ [I2: nat] :
( fChoice_set_set_nat
@ ^ [Gb: set_set_nat] :
( ( member_set_set_nat @ Gb @ x )
& ( ( clique5033774636164728513irst_v @ Gb )
= ( si @ I2 ) ) ) ) ) ) ).
% G_def
thf(fact_719_vGs,axiom,
ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ gs ) @ vs ).
% vGs
thf(fact_720__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062Si_O_Abij__betw_ASi_A_1230_O_O_060p_125_AS_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Si: nat > set_nat] :
~ ( bij_betw_nat_set_nat @ Si @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ p ) @ s ) ).
% \<open>\<And>thesis. (\<And>Si. bij_betw Si {0..<p} S \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_721__092_060open_062_092_060exists_062h_O_Abij__betw_Ah_A_1230_O_O_060p_125_AS_092_060close_062,axiom,
? [H4: nat > set_nat] : ( bij_betw_nat_set_nat @ H4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ p ) @ s ) ).
% \<open>\<exists>h. bij_betw h {0..<p} S\<close>
thf(fact_722_DiffI,axiom,
! [C2: nat > set_nat,A2: set_nat_set_nat,B2: set_nat_set_nat] :
( ( member_nat_set_nat @ C2 @ A2 )
=> ( ~ ( member_nat_set_nat @ C2 @ B2 )
=> ( member_nat_set_nat @ C2 @ ( minus_8060664002660188164et_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_723_DiffI,axiom,
! [C2: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ A2 )
=> ( ~ ( member_nat @ C2 @ B2 )
=> ( member_nat @ C2 @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_724_DiffI,axiom,
! [C2: set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( member_set_set_nat @ C2 @ A2 )
=> ( ~ ( member_set_set_nat @ C2 @ B2 )
=> ( member_set_set_nat @ C2 @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_725_DiffI,axiom,
! [C2: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C2 @ A2 )
=> ( ~ ( member_set_nat @ C2 @ B2 )
=> ( member_set_nat @ C2 @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_726_Diff__iff,axiom,
! [C2: nat > set_nat,A2: set_nat_set_nat,B2: set_nat_set_nat] :
( ( member_nat_set_nat @ C2 @ ( minus_8060664002660188164et_nat @ A2 @ B2 ) )
= ( ( member_nat_set_nat @ C2 @ A2 )
& ~ ( member_nat_set_nat @ C2 @ B2 ) ) ) ).
% Diff_iff
thf(fact_727_Diff__iff,axiom,
! [C2: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( ( member_nat @ C2 @ A2 )
& ~ ( member_nat @ C2 @ B2 ) ) ) ).
% Diff_iff
thf(fact_728_Diff__iff,axiom,
! [C2: set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( member_set_set_nat @ C2 @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) )
= ( ( member_set_set_nat @ C2 @ A2 )
& ~ ( member_set_set_nat @ C2 @ B2 ) ) ) ).
% Diff_iff
thf(fact_729_Diff__iff,axiom,
! [C2: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C2 @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
= ( ( member_set_nat @ C2 @ A2 )
& ~ ( member_set_nat @ C2 @ B2 ) ) ) ).
% Diff_iff
thf(fact_730_Diff__idemp,axiom,
! [A2: set_nat,B2: set_nat] :
( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ B2 )
= ( minus_minus_set_nat @ A2 @ B2 ) ) ).
% Diff_idemp
thf(fact_731_Diff__idemp,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( minus_2447799839930672331et_nat @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) @ B2 )
= ( minus_2447799839930672331et_nat @ A2 @ B2 ) ) ).
% Diff_idemp
thf(fact_732_Diff__idemp,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( minus_2163939370556025621et_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) @ B2 )
= ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ).
% Diff_idemp
thf(fact_733_fstt__def,axiom,
! [E3: set_nat] :
( ( fstt @ E3 )
= ( fChoice_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ E3 )
& ~ ( member_nat @ X2 @ vs ) ) ) ) ).
% fstt_def
thf(fact_734_some__sym__eq__trivial,axiom,
! [X: nat] :
( ( fChoice_nat
@ ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 )
@ X ) )
= X ) ).
% some_sym_eq_trivial
thf(fact_735_some__sym__eq__trivial,axiom,
! [X: set_set_nat] :
( ( fChoice_set_set_nat
@ ( ^ [Y3: set_set_nat,Z2: set_set_nat] : ( Y3 = Z2 )
@ X ) )
= X ) ).
% some_sym_eq_trivial
thf(fact_736_some__eq__trivial,axiom,
! [X: nat] :
( ( fChoice_nat
@ ^ [Y5: nat] : ( Y5 = X ) )
= X ) ).
% some_eq_trivial
thf(fact_737_some__eq__trivial,axiom,
! [X: set_set_nat] :
( ( fChoice_set_set_nat
@ ^ [Y5: set_set_nat] : ( Y5 = X ) )
= X ) ).
% some_eq_trivial
thf(fact_738_some__equality,axiom,
! [P: nat > $o,A: nat] :
( ( P @ A )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( X3 = A ) )
=> ( ( fChoice_nat @ P )
= A ) ) ) ).
% some_equality
thf(fact_739_some__equality,axiom,
! [P: set_set_nat > $o,A: set_set_nat] :
( ( P @ A )
=> ( ! [X3: set_set_nat] :
( ( P @ X3 )
=> ( X3 = A ) )
=> ( ( fChoice_set_set_nat @ P )
= A ) ) ) ).
% some_equality
thf(fact_740_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_741_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_742_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_743_finite__Diff2,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( finite_finite_nat @ A2 ) ) ) ).
% finite_Diff2
thf(fact_744_finite__Diff2,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B2 )
=> ( ( finite6739761609112101331et_nat @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) )
= ( finite6739761609112101331et_nat @ A2 ) ) ) ).
% finite_Diff2
thf(fact_745_finite__Diff2,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( finite1152437895449049373et_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
= ( finite1152437895449049373et_nat @ A2 ) ) ) ).
% finite_Diff2
thf(fact_746_finite__Diff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_747_finite__Diff,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( finite6739761609112101331et_nat @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_748_finite__Diff,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( finite1152437895449049373et_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_749_ti__def,axiom,
! [I: nat] :
( ( ti @ I )
= ( finite_card_nat @ ( minus_minus_set_nat @ ( clique5033774636164728513irst_v @ ( g @ I ) ) @ vs ) ) ) ).
% ti_def
thf(fact_750__092_060open_062sndd_A_092_060equiv_062_A_092_060lambda_062e_O_ASOME_Ax_O_Ax_A_092_060in_062_Ae_A_092_060and_062_Ax_A_092_060noteq_062_Afstt_Ae_092_060close_062,axiom,
( sndd
= ( ^ [E2: set_nat] :
( fChoice_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ E2 )
& ( X2
!= ( fstt @ E2 ) ) ) ) ) ) ).
% \<open>sndd \<equiv> \<lambda>e. SOME x. x \<in> e \<and> x \<noteq> fstt e\<close>
thf(fact_751_ivl__diff,axiom,
! [I: nat,N: nat,M: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ I @ M ) @ ( set_or4665077453230672383an_nat @ I @ N ) )
= ( set_or4665077453230672383an_nat @ N @ M ) ) ) ).
% ivl_diff
thf(fact_752_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C2: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C2 ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C2 ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_753_DiffE,axiom,
! [C2: nat > set_nat,A2: set_nat_set_nat,B2: set_nat_set_nat] :
( ( member_nat_set_nat @ C2 @ ( minus_8060664002660188164et_nat @ A2 @ B2 ) )
=> ~ ( ( member_nat_set_nat @ C2 @ A2 )
=> ( member_nat_set_nat @ C2 @ B2 ) ) ) ).
% DiffE
thf(fact_754_DiffE,axiom,
! [C2: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ~ ( ( member_nat @ C2 @ A2 )
=> ( member_nat @ C2 @ B2 ) ) ) ).
% DiffE
thf(fact_755_DiffE,axiom,
! [C2: set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( member_set_set_nat @ C2 @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) )
=> ~ ( ( member_set_set_nat @ C2 @ A2 )
=> ( member_set_set_nat @ C2 @ B2 ) ) ) ).
% DiffE
thf(fact_756_DiffE,axiom,
! [C2: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C2 @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
=> ~ ( ( member_set_nat @ C2 @ A2 )
=> ( member_set_nat @ C2 @ B2 ) ) ) ).
% DiffE
thf(fact_757_DiffD1,axiom,
! [C2: nat > set_nat,A2: set_nat_set_nat,B2: set_nat_set_nat] :
( ( member_nat_set_nat @ C2 @ ( minus_8060664002660188164et_nat @ A2 @ B2 ) )
=> ( member_nat_set_nat @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_758_DiffD1,axiom,
! [C2: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ( member_nat @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_759_DiffD1,axiom,
! [C2: set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( member_set_set_nat @ C2 @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) )
=> ( member_set_set_nat @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_760_DiffD1,axiom,
! [C2: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C2 @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
=> ( member_set_nat @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_761_DiffD2,axiom,
! [C2: nat > set_nat,A2: set_nat_set_nat,B2: set_nat_set_nat] :
( ( member_nat_set_nat @ C2 @ ( minus_8060664002660188164et_nat @ A2 @ B2 ) )
=> ~ ( member_nat_set_nat @ C2 @ B2 ) ) ).
% DiffD2
thf(fact_762_DiffD2,axiom,
! [C2: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ~ ( member_nat @ C2 @ B2 ) ) ).
% DiffD2
thf(fact_763_DiffD2,axiom,
! [C2: set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( member_set_set_nat @ C2 @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) )
=> ~ ( member_set_set_nat @ C2 @ B2 ) ) ).
% DiffD2
thf(fact_764_DiffD2,axiom,
! [C2: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C2 @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
=> ~ ( member_set_nat @ C2 @ B2 ) ) ).
% DiffD2
thf(fact_765_set__diff__eq,axiom,
( minus_8060664002660188164et_nat
= ( ^ [A3: set_nat_set_nat,B3: set_nat_set_nat] :
( collect_nat_set_nat
@ ^ [X2: nat > set_nat] :
( ( member_nat_set_nat @ X2 @ A3 )
& ~ ( member_nat_set_nat @ X2 @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_766_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ~ ( member_nat @ X2 @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_767_set__diff__eq,axiom,
( minus_2447799839930672331et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
( collect_set_set_nat
@ ^ [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A3 )
& ~ ( member_set_set_nat @ X2 @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_768_set__diff__eq,axiom,
( minus_2163939370556025621et_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( collect_set_nat
@ ^ [X2: set_nat] :
( ( member_set_nat @ X2 @ A3 )
& ~ ( member_set_nat @ X2 @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_769_bij__betwE,axiom,
! [F: nat > set_nat,A2: set_nat,B2: set_set_nat] :
( ( bij_betw_nat_set_nat @ F @ A2 @ B2 )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( member_set_nat @ ( F @ X4 ) @ B2 ) ) ) ).
% bij_betwE
thf(fact_770_bij__betw__inv,axiom,
! [F: set_nat > nat,A2: set_set_nat,B2: set_nat] :
( ( bij_betw_set_nat_nat @ F @ A2 @ B2 )
=> ? [G5: nat > set_nat] : ( bij_betw_nat_set_nat @ G5 @ B2 @ A2 ) ) ).
% bij_betw_inv
thf(fact_771_bij__betw__inv,axiom,
! [F: nat > set_nat,A2: set_nat,B2: set_set_nat] :
( ( bij_betw_nat_set_nat @ F @ A2 @ B2 )
=> ? [G5: set_nat > nat] : ( bij_betw_set_nat_nat @ G5 @ B2 @ A2 ) ) ).
% bij_betw_inv
thf(fact_772_bij__betw__ball,axiom,
! [F: nat > set_nat,A2: set_nat,B2: set_set_nat,Phi: set_nat > $o] :
( ( bij_betw_nat_set_nat @ F @ A2 @ B2 )
=> ( ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ B2 )
=> ( Phi @ X2 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( Phi @ ( F @ X2 ) ) ) ) ) ) ).
% bij_betw_ball
thf(fact_773_bij__betw__cong,axiom,
! [A2: set_nat,F: nat > set_nat,G: nat > set_nat,A7: set_set_nat] :
( ! [A6: nat] :
( ( member_nat @ A6 @ A2 )
=> ( ( F @ A6 )
= ( G @ A6 ) ) )
=> ( ( bij_betw_nat_set_nat @ F @ A2 @ A7 )
= ( bij_betw_nat_set_nat @ G @ A2 @ A7 ) ) ) ).
% bij_betw_cong
thf(fact_774_bij__betw__apply,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat,A: nat] :
( ( bij_betw_nat_nat @ F @ A2 @ B2 )
=> ( ( member_nat @ A @ A2 )
=> ( member_nat @ ( F @ A ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_775_bij__betw__apply,axiom,
! [F: set_nat > nat,A2: set_set_nat,B2: set_nat,A: set_nat] :
( ( bij_betw_set_nat_nat @ F @ A2 @ B2 )
=> ( ( member_set_nat @ A @ A2 )
=> ( member_nat @ ( F @ A ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_776_bij__betw__apply,axiom,
! [F: nat > set_nat,A2: set_nat,B2: set_set_nat,A: nat] :
( ( bij_betw_nat_set_nat @ F @ A2 @ B2 )
=> ( ( member_nat @ A @ A2 )
=> ( member_set_nat @ ( F @ A ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_777_bij__betw__apply,axiom,
! [F: nat > set_set_nat,A2: set_nat,B2: set_set_set_nat,A: nat] :
( ( bij_be6938610931847138308et_nat @ F @ A2 @ B2 )
=> ( ( member_nat @ A @ A2 )
=> ( member_set_set_nat @ ( F @ A ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_778_bij__betw__apply,axiom,
! [F: set_nat > set_nat,A2: set_set_nat,B2: set_set_nat,A: set_nat] :
( ( bij_be3438014552859920132et_nat @ F @ A2 @ B2 )
=> ( ( member_set_nat @ A @ A2 )
=> ( member_set_nat @ ( F @ A ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_779_bij__betw__apply,axiom,
! [F: set_set_nat > nat,A2: set_set_set_nat,B2: set_nat,A: set_set_nat] :
( ( bij_be6199415091885040644at_nat @ F @ A2 @ B2 )
=> ( ( member_set_set_nat @ A @ A2 )
=> ( member_nat @ ( F @ A ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_780_bij__betw__apply,axiom,
! [F: nat > nat > set_nat,A2: set_nat,B2: set_nat_set_nat,A: nat] :
( ( bij_be8549092308015455677et_nat @ F @ A2 @ B2 )
=> ( ( member_nat @ A @ A2 )
=> ( member_nat_set_nat @ ( F @ A ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_781_bij__betw__apply,axiom,
! [F: set_nat > set_set_nat,A2: set_set_nat,B2: set_set_set_nat,A: set_nat] :
( ( bij_be5767359585022399418et_nat @ F @ A2 @ B2 )
=> ( ( member_set_nat @ A @ A2 )
=> ( member_set_set_nat @ ( F @ A ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_782_bij__betw__apply,axiom,
! [F: ( nat > set_nat ) > nat,A2: set_nat_set_nat,B2: set_nat,A: nat > set_nat] :
( ( bij_be5082831075535440701at_nat @ F @ A2 @ B2 )
=> ( ( member_nat_set_nat @ A @ A2 )
=> ( member_nat @ ( F @ A ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_783_bij__betw__apply,axiom,
! [F: set_set_nat > set_nat,A2: set_set_set_nat,B2: set_set_nat,A: set_set_nat] :
( ( bij_be4885122793727115194et_nat @ F @ A2 @ B2 )
=> ( ( member_set_set_nat @ A @ A2 )
=> ( member_set_nat @ ( F @ A ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_784_someI,axiom,
! [P: nat > $o,X: nat] :
( ( P @ X )
=> ( P @ ( fChoice_nat @ P ) ) ) ).
% someI
thf(fact_785_someI,axiom,
! [P: set_set_nat > $o,X: set_set_nat] :
( ( P @ X )
=> ( P @ ( fChoice_set_set_nat @ P ) ) ) ).
% someI
thf(fact_786_Eps__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( fChoice_nat @ P )
= ( fChoice_nat @ Q ) ) ) ).
% Eps_cong
thf(fact_787_Eps__cong,axiom,
! [P: set_set_nat > $o,Q: set_set_nat > $o] :
( ! [X3: set_set_nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( fChoice_set_set_nat @ P )
= ( fChoice_set_set_nat @ Q ) ) ) ).
% Eps_cong
thf(fact_788_tfl__some,axiom,
! [P2: nat > $o,X4: nat] :
( ( P2 @ X4 )
=> ( P2 @ ( fChoice_nat @ P2 ) ) ) ).
% tfl_some
thf(fact_789_tfl__some,axiom,
! [P2: set_set_nat > $o,X4: set_set_nat] :
( ( P2 @ X4 )
=> ( P2 @ ( fChoice_set_set_nat @ P2 ) ) ) ).
% tfl_some
thf(fact_790_some__eq__imp,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( ( fChoice_nat @ P )
= A )
=> ( ( P @ B )
=> ( P @ A ) ) ) ).
% some_eq_imp
thf(fact_791_some__eq__imp,axiom,
! [P: set_set_nat > $o,A: set_set_nat,B: set_set_nat] :
( ( ( fChoice_set_set_nat @ P )
= A )
=> ( ( P @ B )
=> ( P @ A ) ) ) ).
% some_eq_imp
thf(fact_792_bij__betw__iff__bijections,axiom,
( bij_betw_nat_nat
= ( ^ [F4: nat > nat,A3: set_nat,B3: set_nat] :
? [G6: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A3 )
=> ( ( member_nat @ ( F4 @ X2 ) @ B3 )
& ( ( G6 @ ( F4 @ X2 ) )
= X2 ) ) )
& ! [X2: nat] :
( ( member_nat @ X2 @ B3 )
=> ( ( member_nat @ ( G6 @ X2 ) @ A3 )
& ( ( F4 @ ( G6 @ X2 ) )
= X2 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_793_bij__betw__iff__bijections,axiom,
( bij_betw_set_nat_nat
= ( ^ [F4: set_nat > nat,A3: set_set_nat,B3: set_nat] :
? [G6: nat > set_nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A3 )
=> ( ( member_nat @ ( F4 @ X2 ) @ B3 )
& ( ( G6 @ ( F4 @ X2 ) )
= X2 ) ) )
& ! [X2: nat] :
( ( member_nat @ X2 @ B3 )
=> ( ( member_set_nat @ ( G6 @ X2 ) @ A3 )
& ( ( F4 @ ( G6 @ X2 ) )
= X2 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_794_bij__betw__iff__bijections,axiom,
( bij_betw_nat_set_nat
= ( ^ [F4: nat > set_nat,A3: set_nat,B3: set_set_nat] :
? [G6: set_nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A3 )
=> ( ( member_set_nat @ ( F4 @ X2 ) @ B3 )
& ( ( G6 @ ( F4 @ X2 ) )
= X2 ) ) )
& ! [X2: set_nat] :
( ( member_set_nat @ X2 @ B3 )
=> ( ( member_nat @ ( G6 @ X2 ) @ A3 )
& ( ( F4 @ ( G6 @ X2 ) )
= X2 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_795_bij__betw__iff__bijections,axiom,
( bij_be6199415091885040644at_nat
= ( ^ [F4: set_set_nat > nat,A3: set_set_set_nat,B3: set_nat] :
? [G6: nat > set_set_nat] :
( ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A3 )
=> ( ( member_nat @ ( F4 @ X2 ) @ B3 )
& ( ( G6 @ ( F4 @ X2 ) )
= X2 ) ) )
& ! [X2: nat] :
( ( member_nat @ X2 @ B3 )
=> ( ( member_set_set_nat @ ( G6 @ X2 ) @ A3 )
& ( ( F4 @ ( G6 @ X2 ) )
= X2 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_796_bij__betw__iff__bijections,axiom,
( bij_be3438014552859920132et_nat
= ( ^ [F4: set_nat > set_nat,A3: set_set_nat,B3: set_set_nat] :
? [G6: set_nat > set_nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A3 )
=> ( ( member_set_nat @ ( F4 @ X2 ) @ B3 )
& ( ( G6 @ ( F4 @ X2 ) )
= X2 ) ) )
& ! [X2: set_nat] :
( ( member_set_nat @ X2 @ B3 )
=> ( ( member_set_nat @ ( G6 @ X2 ) @ A3 )
& ( ( F4 @ ( G6 @ X2 ) )
= X2 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_797_bij__betw__iff__bijections,axiom,
( bij_be6938610931847138308et_nat
= ( ^ [F4: nat > set_set_nat,A3: set_nat,B3: set_set_set_nat] :
? [G6: set_set_nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A3 )
=> ( ( member_set_set_nat @ ( F4 @ X2 ) @ B3 )
& ( ( G6 @ ( F4 @ X2 ) )
= X2 ) ) )
& ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ B3 )
=> ( ( member_nat @ ( G6 @ X2 ) @ A3 )
& ( ( F4 @ ( G6 @ X2 ) )
= X2 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_798_bij__betw__iff__bijections,axiom,
( bij_be5082831075535440701at_nat
= ( ^ [F4: ( nat > set_nat ) > nat,A3: set_nat_set_nat,B3: set_nat] :
? [G6: nat > nat > set_nat] :
( ! [X2: nat > set_nat] :
( ( member_nat_set_nat @ X2 @ A3 )
=> ( ( member_nat @ ( F4 @ X2 ) @ B3 )
& ( ( G6 @ ( F4 @ X2 ) )
= X2 ) ) )
& ! [X2: nat] :
( ( member_nat @ X2 @ B3 )
=> ( ( member_nat_set_nat @ ( G6 @ X2 ) @ A3 )
& ( ( F4 @ ( G6 @ X2 ) )
= X2 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_799_bij__betw__iff__bijections,axiom,
( bij_be4885122793727115194et_nat
= ( ^ [F4: set_set_nat > set_nat,A3: set_set_set_nat,B3: set_set_nat] :
? [G6: set_nat > set_set_nat] :
( ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A3 )
=> ( ( member_set_nat @ ( F4 @ X2 ) @ B3 )
& ( ( G6 @ ( F4 @ X2 ) )
= X2 ) ) )
& ! [X2: set_nat] :
( ( member_set_nat @ X2 @ B3 )
=> ( ( member_set_set_nat @ ( G6 @ X2 ) @ A3 )
& ( ( F4 @ ( G6 @ X2 ) )
= X2 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_800_bij__betw__iff__bijections,axiom,
( bij_be8549092308015455677et_nat
= ( ^ [F4: nat > nat > set_nat,A3: set_nat,B3: set_nat_set_nat] :
? [G6: ( nat > set_nat ) > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A3 )
=> ( ( member_nat_set_nat @ ( F4 @ X2 ) @ B3 )
& ( ( G6 @ ( F4 @ X2 ) )
= X2 ) ) )
& ! [X2: nat > set_nat] :
( ( member_nat_set_nat @ X2 @ B3 )
=> ( ( member_nat @ ( G6 @ X2 ) @ A3 )
& ( ( F4 @ ( G6 @ X2 ) )
= X2 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_801_bij__betw__iff__bijections,axiom,
( bij_be5767359585022399418et_nat
= ( ^ [F4: set_nat > set_set_nat,A3: set_set_nat,B3: set_set_set_nat] :
? [G6: set_set_nat > set_nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A3 )
=> ( ( member_set_set_nat @ ( F4 @ X2 ) @ B3 )
& ( ( G6 @ ( F4 @ X2 ) )
= X2 ) ) )
& ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ B3 )
=> ( ( member_set_nat @ ( G6 @ X2 ) @ A3 )
& ( ( F4 @ ( G6 @ X2 ) )
= X2 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_802_some1__equality,axiom,
! [P: nat > $o,A: nat] :
( ? [X4: nat] :
( ( P @ X4 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( Y4 = X4 ) ) )
=> ( ( P @ A )
=> ( ( fChoice_nat @ P )
= A ) ) ) ).
% some1_equality
thf(fact_803_some1__equality,axiom,
! [P: set_set_nat > $o,A: set_set_nat] :
( ? [X4: set_set_nat] :
( ( P @ X4 )
& ! [Y4: set_set_nat] :
( ( P @ Y4 )
=> ( Y4 = X4 ) ) )
=> ( ( P @ A )
=> ( ( fChoice_set_set_nat @ P )
= A ) ) ) ).
% some1_equality
thf(fact_804_some__eq__ex,axiom,
! [P: nat > $o] :
( ( P @ ( fChoice_nat @ P ) )
= ( ? [X6: nat] : ( P @ X6 ) ) ) ).
% some_eq_ex
thf(fact_805_some__eq__ex,axiom,
! [P: set_set_nat > $o] :
( ( P @ ( fChoice_set_set_nat @ P ) )
= ( ? [X6: set_set_nat] : ( P @ X6 ) ) ) ).
% some_eq_ex
thf(fact_806_someI2__bex,axiom,
! [A2: set_set_nat,P: set_nat > $o,Q: set_nat > $o] :
( ? [X4: set_nat] :
( ( member_set_nat @ X4 @ A2 )
& ( P @ X4 ) )
=> ( ! [X3: set_nat] :
( ( ( member_set_nat @ X3 @ A2 )
& ( P @ X3 ) )
=> ( Q @ X3 ) )
=> ( Q
@ ( fChoice_set_nat
@ ^ [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
& ( P @ X2 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_807_someI2__bex,axiom,
! [A2: set_nat_set_nat,P: ( nat > set_nat ) > $o,Q: ( nat > set_nat ) > $o] :
( ? [X4: nat > set_nat] :
( ( member_nat_set_nat @ X4 @ A2 )
& ( P @ X4 ) )
=> ( ! [X3: nat > set_nat] :
( ( ( member_nat_set_nat @ X3 @ A2 )
& ( P @ X3 ) )
=> ( Q @ X3 ) )
=> ( Q
@ ( fChoice_nat_set_nat
@ ^ [X2: nat > set_nat] :
( ( member_nat_set_nat @ X2 @ A2 )
& ( P @ X2 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_808_someI2__bex,axiom,
! [A2: set_nat,P: nat > $o,Q: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( P @ X4 ) )
=> ( ! [X3: nat] :
( ( ( member_nat @ X3 @ A2 )
& ( P @ X3 ) )
=> ( Q @ X3 ) )
=> ( Q
@ ( fChoice_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ X2 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_809_someI2__bex,axiom,
! [A2: set_set_set_nat,P: set_set_nat > $o,Q: set_set_nat > $o] :
( ? [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ A2 )
& ( P @ X4 ) )
=> ( ! [X3: set_set_nat] :
( ( ( member_set_set_nat @ X3 @ A2 )
& ( P @ X3 ) )
=> ( Q @ X3 ) )
=> ( Q
@ ( fChoice_set_set_nat
@ ^ [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A2 )
& ( P @ X2 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_810_someI2__ex,axiom,
! [P: nat > $o,Q: nat > $o] :
( ? [X_12: nat] : ( P @ X_12 )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( fChoice_nat @ P ) ) ) ) ).
% someI2_ex
thf(fact_811_someI2__ex,axiom,
! [P: set_set_nat > $o,Q: set_set_nat > $o] :
( ? [X_12: set_set_nat] : ( P @ X_12 )
=> ( ! [X3: set_set_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( fChoice_set_set_nat @ P ) ) ) ) ).
% someI2_ex
thf(fact_812_someI__ex,axiom,
! [P: nat > $o] :
( ? [X_12: nat] : ( P @ X_12 )
=> ( P @ ( fChoice_nat @ P ) ) ) ).
% someI_ex
thf(fact_813_someI__ex,axiom,
! [P: set_set_nat > $o] :
( ? [X_12: set_set_nat] : ( P @ X_12 )
=> ( P @ ( fChoice_set_set_nat @ P ) ) ) ).
% someI_ex
thf(fact_814_someI2,axiom,
! [P: nat > $o,A: nat,Q: nat > $o] :
( ( P @ A )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( fChoice_nat @ P ) ) ) ) ).
% someI2
thf(fact_815_someI2,axiom,
! [P: set_set_nat > $o,A: set_set_nat,Q: set_set_nat > $o] :
( ( P @ A )
=> ( ! [X3: set_set_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( fChoice_set_set_nat @ P ) ) ) ) ).
% someI2
thf(fact_816_bij__betw__imp__surj__on,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( bij_betw_nat_nat @ F @ A2 @ B2 )
=> ( ( image_nat_nat @ F @ A2 )
= B2 ) ) ).
% bij_betw_imp_surj_on
thf(fact_817_bij__betw__imp__surj__on,axiom,
! [F: set_set_nat > set_nat,A2: set_set_set_nat,B2: set_set_nat] :
( ( bij_be4885122793727115194et_nat @ F @ A2 @ B2 )
=> ( ( image_5842784325960735177et_nat @ F @ A2 )
= B2 ) ) ).
% bij_betw_imp_surj_on
thf(fact_818_bij__betw__imp__surj__on,axiom,
! [F: nat > set_set_nat,A2: set_nat,B2: set_set_set_nat] :
( ( bij_be6938610931847138308et_nat @ F @ A2 @ B2 )
=> ( ( image_2194112158459175443et_nat @ F @ A2 )
= B2 ) ) ).
% bij_betw_imp_surj_on
thf(fact_819_bij__betw__imp__surj__on,axiom,
! [F: nat > set_nat,A2: set_nat,B2: set_set_nat] :
( ( bij_betw_nat_set_nat @ F @ A2 @ B2 )
=> ( ( image_nat_set_nat @ F @ A2 )
= B2 ) ) ).
% bij_betw_imp_surj_on
thf(fact_820_bij__betw__finite,axiom,
! [F: set_nat > set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( bij_be3438014552859920132et_nat @ F @ A2 @ B2 )
=> ( ( finite1152437895449049373et_nat @ A2 )
= ( finite1152437895449049373et_nat @ B2 ) ) ) ).
% bij_betw_finite
thf(fact_821_bij__betw__finite,axiom,
! [F: set_nat > set_set_nat,A2: set_set_nat,B2: set_set_set_nat] :
( ( bij_be5767359585022399418et_nat @ F @ A2 @ B2 )
=> ( ( finite1152437895449049373et_nat @ A2 )
= ( finite6739761609112101331et_nat @ B2 ) ) ) ).
% bij_betw_finite
thf(fact_822_bij__betw__finite,axiom,
! [F: set_nat > nat,A2: set_set_nat,B2: set_nat] :
( ( bij_betw_set_nat_nat @ F @ A2 @ B2 )
=> ( ( finite1152437895449049373et_nat @ A2 )
= ( finite_finite_nat @ B2 ) ) ) ).
% bij_betw_finite
thf(fact_823_bij__betw__finite,axiom,
! [F: set_set_nat > set_nat,A2: set_set_set_nat,B2: set_set_nat] :
( ( bij_be4885122793727115194et_nat @ F @ A2 @ B2 )
=> ( ( finite6739761609112101331et_nat @ A2 )
= ( finite1152437895449049373et_nat @ B2 ) ) ) ).
% bij_betw_finite
thf(fact_824_bij__betw__finite,axiom,
! [F: set_set_nat > set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( bij_be1917187662166652016et_nat @ F @ A2 @ B2 )
=> ( ( finite6739761609112101331et_nat @ A2 )
= ( finite6739761609112101331et_nat @ B2 ) ) ) ).
% bij_betw_finite
thf(fact_825_bij__betw__finite,axiom,
! [F: set_set_nat > nat,A2: set_set_set_nat,B2: set_nat] :
( ( bij_be6199415091885040644at_nat @ F @ A2 @ B2 )
=> ( ( finite6739761609112101331et_nat @ A2 )
= ( finite_finite_nat @ B2 ) ) ) ).
% bij_betw_finite
thf(fact_826_bij__betw__finite,axiom,
! [F: nat > set_nat,A2: set_nat,B2: set_set_nat] :
( ( bij_betw_nat_set_nat @ F @ A2 @ B2 )
=> ( ( finite_finite_nat @ A2 )
= ( finite1152437895449049373et_nat @ B2 ) ) ) ).
% bij_betw_finite
thf(fact_827_bij__betw__finite,axiom,
! [F: nat > set_set_nat,A2: set_nat,B2: set_set_set_nat] :
( ( bij_be6938610931847138308et_nat @ F @ A2 @ B2 )
=> ( ( finite_finite_nat @ A2 )
= ( finite6739761609112101331et_nat @ B2 ) ) ) ).
% bij_betw_finite
thf(fact_828_bij__betw__finite,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( bij_betw_nat_nat @ F @ A2 @ B2 )
=> ( ( finite_finite_nat @ A2 )
= ( finite_finite_nat @ B2 ) ) ) ).
% bij_betw_finite
thf(fact_829_bij__betw__same__card,axiom,
! [F: set_nat > set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( bij_be3438014552859920132et_nat @ F @ A2 @ B2 )
=> ( ( finite_card_set_nat @ A2 )
= ( finite_card_set_nat @ B2 ) ) ) ).
% bij_betw_same_card
thf(fact_830_bij__betw__same__card,axiom,
! [F: set_nat > nat,A2: set_set_nat,B2: set_nat] :
( ( bij_betw_set_nat_nat @ F @ A2 @ B2 )
=> ( ( finite_card_set_nat @ A2 )
= ( finite_card_nat @ B2 ) ) ) ).
% bij_betw_same_card
thf(fact_831_bij__betw__same__card,axiom,
! [F: set_nat > set_set_nat,A2: set_set_nat,B2: set_set_set_nat] :
( ( bij_be5767359585022399418et_nat @ F @ A2 @ B2 )
=> ( ( finite_card_set_nat @ A2 )
= ( finite1149291290879098388et_nat @ B2 ) ) ) ).
% bij_betw_same_card
thf(fact_832_bij__betw__same__card,axiom,
! [F: nat > set_nat,A2: set_nat,B2: set_set_nat] :
( ( bij_betw_nat_set_nat @ F @ A2 @ B2 )
=> ( ( finite_card_nat @ A2 )
= ( finite_card_set_nat @ B2 ) ) ) ).
% bij_betw_same_card
thf(fact_833_bij__betw__same__card,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( bij_betw_nat_nat @ F @ A2 @ B2 )
=> ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B2 ) ) ) ).
% bij_betw_same_card
thf(fact_834_bij__betw__same__card,axiom,
! [F: nat > set_set_nat,A2: set_nat,B2: set_set_set_nat] :
( ( bij_be6938610931847138308et_nat @ F @ A2 @ B2 )
=> ( ( finite_card_nat @ A2 )
= ( finite1149291290879098388et_nat @ B2 ) ) ) ).
% bij_betw_same_card
thf(fact_835_bij__betw__same__card,axiom,
! [F: set_set_nat > set_nat,A2: set_set_set_nat,B2: set_set_nat] :
( ( bij_be4885122793727115194et_nat @ F @ A2 @ B2 )
=> ( ( finite1149291290879098388et_nat @ A2 )
= ( finite_card_set_nat @ B2 ) ) ) ).
% bij_betw_same_card
thf(fact_836_bij__betw__same__card,axiom,
! [F: set_set_nat > nat,A2: set_set_set_nat,B2: set_nat] :
( ( bij_be6199415091885040644at_nat @ F @ A2 @ B2 )
=> ( ( finite1149291290879098388et_nat @ A2 )
= ( finite_card_nat @ B2 ) ) ) ).
% bij_betw_same_card
thf(fact_837_bij__betw__same__card,axiom,
! [F: set_set_nat > set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( bij_be1917187662166652016et_nat @ F @ A2 @ B2 )
=> ( ( finite1149291290879098388et_nat @ A2 )
= ( finite1149291290879098388et_nat @ B2 ) ) ) ).
% bij_betw_same_card
thf(fact_838_bij__betw__imp__inj__on,axiom,
! [F: nat > set_set_nat,A2: set_nat,B2: set_set_set_nat] :
( ( bij_be6938610931847138308et_nat @ F @ A2 @ B2 )
=> ( inj_on8105003582846801791et_nat @ F @ A2 ) ) ).
% bij_betw_imp_inj_on
thf(fact_839_bij__betw__imp__inj__on,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( bij_betw_nat_nat @ F @ A2 @ B2 )
=> ( inj_on_nat_nat @ F @ A2 ) ) ).
% bij_betw_imp_inj_on
thf(fact_840_bij__betw__imp__inj__on,axiom,
! [F: nat > set_nat,A2: set_nat,B2: set_set_nat] :
( ( bij_betw_nat_set_nat @ F @ A2 @ B2 )
=> ( inj_on_nat_set_nat @ F @ A2 ) ) ).
% bij_betw_imp_inj_on
thf(fact_841_Diff__mono,axiom,
! [A2: set_set_set_nat,C: set_set_set_nat,D2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ C )
=> ( ( ord_le9131159989063066194et_nat @ D2 @ B2 )
=> ( ord_le9131159989063066194et_nat @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) @ ( minus_2447799839930672331et_nat @ C @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_842_Diff__mono,axiom,
! [A2: set_set_nat,C: set_set_nat,D2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ C )
=> ( ( ord_le6893508408891458716et_nat @ D2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) @ ( minus_2163939370556025621et_nat @ C @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_843_Diff__mono,axiom,
! [A2: set_nat,C: set_nat,D2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( ord_less_eq_set_nat @ D2 @ B2 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( minus_minus_set_nat @ C @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_844_Diff__subset,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_845_Diff__subset,axiom,
! [A2: set_set_nat,B2: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_846_Diff__subset,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_847_double__diff,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ C )
=> ( ( minus_2447799839930672331et_nat @ B2 @ ( minus_2447799839930672331et_nat @ C @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_848_double__diff,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ C )
=> ( ( minus_2163939370556025621et_nat @ B2 @ ( minus_2163939370556025621et_nat @ C @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_849_double__diff,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ( minus_minus_set_nat @ B2 @ ( minus_minus_set_nat @ C @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_850_Diff__infinite__finite,axiom,
! [T3: set_nat,S2: set_nat] :
( ( finite_finite_nat @ T3 )
=> ( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S2 @ T3 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_851_Diff__infinite__finite,axiom,
! [T3: set_set_set_nat,S2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ T3 )
=> ( ~ ( finite6739761609112101331et_nat @ S2 )
=> ~ ( finite6739761609112101331et_nat @ ( minus_2447799839930672331et_nat @ S2 @ T3 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_852_Diff__infinite__finite,axiom,
! [T3: set_set_nat,S2: set_set_nat] :
( ( finite1152437895449049373et_nat @ T3 )
=> ( ~ ( finite1152437895449049373et_nat @ S2 )
=> ~ ( finite1152437895449049373et_nat @ ( minus_2163939370556025621et_nat @ S2 @ T3 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_853_inj__on__diff,axiom,
! [F: nat > set_set_nat,A2: set_nat,B2: set_nat] :
( ( inj_on8105003582846801791et_nat @ F @ A2 )
=> ( inj_on8105003582846801791et_nat @ F @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% inj_on_diff
thf(fact_854_inj__on__diff,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( inj_on_nat_nat @ F @ A2 )
=> ( inj_on_nat_nat @ F @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% inj_on_diff
thf(fact_855_psubset__imp__ex__mem,axiom,
! [A2: set_nat_set_nat,B2: set_nat_set_nat] :
( ( ord_le7745323766158300927et_nat @ A2 @ B2 )
=> ? [B6: nat > set_nat] : ( member_nat_set_nat @ B6 @ ( minus_8060664002660188164et_nat @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_856_psubset__imp__ex__mem,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ? [B6: nat] : ( member_nat @ B6 @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_857_psubset__imp__ex__mem,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A2 @ B2 )
=> ? [B6: set_set_nat] : ( member_set_set_nat @ B6 @ ( minus_2447799839930672331et_nat @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_858_psubset__imp__ex__mem,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_less_set_set_nat @ A2 @ B2 )
=> ? [B6: set_nat] : ( member_set_nat @ B6 @ ( minus_2163939370556025621et_nat @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_859_bij__betw__byWitness,axiom,
! [A2: set_set_set_nat,F5: set_set_nat > set_set_nat,F: set_set_nat > set_set_nat,A7: set_set_set_nat] :
( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A2 )
=> ( ( F5 @ ( F @ X3 ) )
= X3 ) )
=> ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A7 )
=> ( ( F @ ( F5 @ X3 ) )
= X3 ) )
=> ( ( ord_le9131159989063066194et_nat @ ( image_7884819252390400639et_nat @ F @ A2 ) @ A7 )
=> ( ( ord_le9131159989063066194et_nat @ ( image_7884819252390400639et_nat @ F5 @ A7 ) @ A2 )
=> ( bij_be1917187662166652016et_nat @ F @ A2 @ A7 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_860_bij__betw__byWitness,axiom,
! [A2: set_set_nat,F5: set_set_nat > set_nat,F: set_nat > set_set_nat,A7: set_set_set_nat] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( ( F5 @ ( F @ X3 ) )
= X3 ) )
=> ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A7 )
=> ( ( F @ ( F5 @ X3 ) )
= X3 ) )
=> ( ( ord_le9131159989063066194et_nat @ ( image_6725021117256019401et_nat @ F @ A2 ) @ A7 )
=> ( ( ord_le6893508408891458716et_nat @ ( image_5842784325960735177et_nat @ F5 @ A7 ) @ A2 )
=> ( bij_be5767359585022399418et_nat @ F @ A2 @ A7 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_861_bij__betw__byWitness,axiom,
! [A2: set_nat,F5: set_set_nat > nat,F: nat > set_set_nat,A7: set_set_set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( F5 @ ( F @ X3 ) )
= X3 ) )
=> ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A7 )
=> ( ( F @ ( F5 @ X3 ) )
= X3 ) )
=> ( ( ord_le9131159989063066194et_nat @ ( image_2194112158459175443et_nat @ F @ A2 ) @ A7 )
=> ( ( ord_less_eq_set_nat @ ( image_1454916318497077779at_nat @ F5 @ A7 ) @ A2 )
=> ( bij_be6938610931847138308et_nat @ F @ A2 @ A7 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_862_bij__betw__byWitness,axiom,
! [A2: set_set_set_nat,F5: set_nat > set_set_nat,F: set_set_nat > set_nat,A7: set_set_nat] :
( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A2 )
=> ( ( F5 @ ( F @ X3 ) )
= X3 ) )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A7 )
=> ( ( F @ ( F5 @ X3 ) )
= X3 ) )
=> ( ( ord_le6893508408891458716et_nat @ ( image_5842784325960735177et_nat @ F @ A2 ) @ A7 )
=> ( ( ord_le9131159989063066194et_nat @ ( image_6725021117256019401et_nat @ F5 @ A7 ) @ A2 )
=> ( bij_be4885122793727115194et_nat @ F @ A2 @ A7 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_863_bij__betw__byWitness,axiom,
! [A2: set_set_nat,F5: set_nat > set_nat,F: set_nat > set_nat,A7: set_set_nat] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( ( F5 @ ( F @ X3 ) )
= X3 ) )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A7 )
=> ( ( F @ ( F5 @ X3 ) )
= X3 ) )
=> ( ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) @ A7 )
=> ( ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ F5 @ A7 ) @ A2 )
=> ( bij_be3438014552859920132et_nat @ F @ A2 @ A7 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_864_bij__betw__byWitness,axiom,
! [A2: set_nat,F5: set_nat > nat,F: nat > set_nat,A7: set_set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( F5 @ ( F @ X3 ) )
= X3 ) )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A7 )
=> ( ( F @ ( F5 @ X3 ) )
= X3 ) )
=> ( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A2 ) @ A7 )
=> ( ( ord_less_eq_set_nat @ ( image_set_nat_nat @ F5 @ A7 ) @ A2 )
=> ( bij_betw_nat_set_nat @ F @ A2 @ A7 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_865_bij__betw__byWitness,axiom,
! [A2: set_set_set_nat,F5: nat > set_set_nat,F: set_set_nat > nat,A7: set_nat] :
( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A2 )
=> ( ( F5 @ ( F @ X3 ) )
= X3 ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A7 )
=> ( ( F @ ( F5 @ X3 ) )
= X3 ) )
=> ( ( ord_less_eq_set_nat @ ( image_1454916318497077779at_nat @ F @ A2 ) @ A7 )
=> ( ( ord_le9131159989063066194et_nat @ ( image_2194112158459175443et_nat @ F5 @ A7 ) @ A2 )
=> ( bij_be6199415091885040644at_nat @ F @ A2 @ A7 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_866_bij__betw__byWitness,axiom,
! [A2: set_set_nat,F5: nat > set_nat,F: set_nat > nat,A7: set_nat] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( ( F5 @ ( F @ X3 ) )
= X3 ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A7 )
=> ( ( F @ ( F5 @ X3 ) )
= X3 ) )
=> ( ( ord_less_eq_set_nat @ ( image_set_nat_nat @ F @ A2 ) @ A7 )
=> ( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F5 @ A7 ) @ A2 )
=> ( bij_betw_set_nat_nat @ F @ A2 @ A7 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_867_bij__betw__byWitness,axiom,
! [A2: set_nat,F5: nat > nat,F: nat > nat,A7: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( F5 @ ( F @ X3 ) )
= X3 ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A7 )
=> ( ( F @ ( F5 @ X3 ) )
= X3 ) )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ A7 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F5 @ A7 ) @ A2 )
=> ( bij_betw_nat_nat @ F @ A2 @ A7 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_868_bij__betw__subset,axiom,
! [F: set_set_nat > set_nat,A2: set_set_set_nat,A7: set_set_nat,B2: set_set_set_nat,B8: set_set_nat] :
( ( bij_be4885122793727115194et_nat @ F @ A2 @ A7 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
=> ( ( ( image_5842784325960735177et_nat @ F @ B2 )
= B8 )
=> ( bij_be4885122793727115194et_nat @ F @ B2 @ B8 ) ) ) ) ).
% bij_betw_subset
thf(fact_869_bij__betw__subset,axiom,
! [F: nat > nat,A2: set_nat,A7: set_nat,B2: set_nat,B8: set_nat] :
( ( bij_betw_nat_nat @ F @ A2 @ A7 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ( image_nat_nat @ F @ B2 )
= B8 )
=> ( bij_betw_nat_nat @ F @ B2 @ B8 ) ) ) ) ).
% bij_betw_subset
thf(fact_870_bij__betw__subset,axiom,
! [F: nat > set_set_nat,A2: set_nat,A7: set_set_set_nat,B2: set_nat,B8: set_set_set_nat] :
( ( bij_be6938610931847138308et_nat @ F @ A2 @ A7 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ( image_2194112158459175443et_nat @ F @ B2 )
= B8 )
=> ( bij_be6938610931847138308et_nat @ F @ B2 @ B8 ) ) ) ) ).
% bij_betw_subset
thf(fact_871_bij__betw__subset,axiom,
! [F: nat > set_nat,A2: set_nat,A7: set_set_nat,B2: set_nat,B8: set_set_nat] :
( ( bij_betw_nat_set_nat @ F @ A2 @ A7 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ( image_nat_set_nat @ F @ B2 )
= B8 )
=> ( bij_betw_nat_set_nat @ F @ B2 @ B8 ) ) ) ) ).
% bij_betw_subset
thf(fact_872_finite__same__card__bij,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ( finite_card_set_nat @ A2 )
= ( finite_card_set_nat @ B2 ) )
=> ? [H4: set_nat > set_nat] : ( bij_be3438014552859920132et_nat @ H4 @ A2 @ B2 ) ) ) ) ).
% finite_same_card_bij
thf(fact_873_finite__same__card__bij,axiom,
! [A2: set_set_nat,B2: set_set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ( finite_card_set_nat @ A2 )
= ( finite1149291290879098388et_nat @ B2 ) )
=> ? [H4: set_nat > set_set_nat] : ( bij_be5767359585022399418et_nat @ H4 @ A2 @ B2 ) ) ) ) ).
% finite_same_card_bij
thf(fact_874_finite__same__card__bij,axiom,
! [A2: set_set_nat,B2: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ( finite_card_set_nat @ A2 )
= ( finite_card_nat @ B2 ) )
=> ? [H4: set_nat > nat] : ( bij_betw_set_nat_nat @ H4 @ A2 @ B2 ) ) ) ) ).
% finite_same_card_bij
thf(fact_875_finite__same__card__bij,axiom,
! [A2: set_set_set_nat,B2: set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ( finite1149291290879098388et_nat @ A2 )
= ( finite_card_set_nat @ B2 ) )
=> ? [H4: set_set_nat > set_nat] : ( bij_be4885122793727115194et_nat @ H4 @ A2 @ B2 ) ) ) ) ).
% finite_same_card_bij
thf(fact_876_finite__same__card__bij,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ( finite1149291290879098388et_nat @ A2 )
= ( finite1149291290879098388et_nat @ B2 ) )
=> ? [H4: set_set_nat > set_set_nat] : ( bij_be1917187662166652016et_nat @ H4 @ A2 @ B2 ) ) ) ) ).
% finite_same_card_bij
thf(fact_877_finite__same__card__bij,axiom,
! [A2: set_set_set_nat,B2: set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ( finite1149291290879098388et_nat @ A2 )
= ( finite_card_nat @ B2 ) )
=> ? [H4: set_set_nat > nat] : ( bij_be6199415091885040644at_nat @ H4 @ A2 @ B2 ) ) ) ) ).
% finite_same_card_bij
thf(fact_878_finite__same__card__bij,axiom,
! [A2: set_nat,B2: set_set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ( finite_card_nat @ A2 )
= ( finite_card_set_nat @ B2 ) )
=> ? [H4: nat > set_nat] : ( bij_betw_nat_set_nat @ H4 @ A2 @ B2 ) ) ) ) ).
% finite_same_card_bij
thf(fact_879_finite__same__card__bij,axiom,
! [A2: set_nat,B2: set_set_set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ( finite_card_nat @ A2 )
= ( finite1149291290879098388et_nat @ B2 ) )
=> ? [H4: nat > set_set_nat] : ( bij_be6938610931847138308et_nat @ H4 @ A2 @ B2 ) ) ) ) ).
% finite_same_card_bij
thf(fact_880_finite__same__card__bij,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B2 ) )
=> ? [H4: nat > nat] : ( bij_betw_nat_nat @ H4 @ A2 @ B2 ) ) ) ) ).
% finite_same_card_bij
thf(fact_881_bij__betw__iff__card,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ? [F4: set_nat > set_nat] : ( bij_be3438014552859920132et_nat @ F4 @ A2 @ B2 ) )
= ( ( finite_card_set_nat @ A2 )
= ( finite_card_set_nat @ B2 ) ) ) ) ) ).
% bij_betw_iff_card
thf(fact_882_bij__betw__iff__card,axiom,
! [A2: set_set_nat,B2: set_set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ? [F4: set_nat > set_set_nat] : ( bij_be5767359585022399418et_nat @ F4 @ A2 @ B2 ) )
= ( ( finite_card_set_nat @ A2 )
= ( finite1149291290879098388et_nat @ B2 ) ) ) ) ) ).
% bij_betw_iff_card
thf(fact_883_bij__betw__iff__card,axiom,
! [A2: set_set_nat,B2: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ? [F4: set_nat > nat] : ( bij_betw_set_nat_nat @ F4 @ A2 @ B2 ) )
= ( ( finite_card_set_nat @ A2 )
= ( finite_card_nat @ B2 ) ) ) ) ) ).
% bij_betw_iff_card
thf(fact_884_bij__betw__iff__card,axiom,
! [A2: set_set_set_nat,B2: set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ? [F4: set_set_nat > set_nat] : ( bij_be4885122793727115194et_nat @ F4 @ A2 @ B2 ) )
= ( ( finite1149291290879098388et_nat @ A2 )
= ( finite_card_set_nat @ B2 ) ) ) ) ) ).
% bij_betw_iff_card
thf(fact_885_bij__betw__iff__card,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ? [F4: set_set_nat > set_set_nat] : ( bij_be1917187662166652016et_nat @ F4 @ A2 @ B2 ) )
= ( ( finite1149291290879098388et_nat @ A2 )
= ( finite1149291290879098388et_nat @ B2 ) ) ) ) ) ).
% bij_betw_iff_card
thf(fact_886_bij__betw__iff__card,axiom,
! [A2: set_set_set_nat,B2: set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ? [F4: set_set_nat > nat] : ( bij_be6199415091885040644at_nat @ F4 @ A2 @ B2 ) )
= ( ( finite1149291290879098388et_nat @ A2 )
= ( finite_card_nat @ B2 ) ) ) ) ) ).
% bij_betw_iff_card
thf(fact_887_bij__betw__iff__card,axiom,
! [A2: set_nat,B2: set_set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ? [F4: nat > set_nat] : ( bij_betw_nat_set_nat @ F4 @ A2 @ B2 ) )
= ( ( finite_card_nat @ A2 )
= ( finite_card_set_nat @ B2 ) ) ) ) ) ).
% bij_betw_iff_card
thf(fact_888_bij__betw__iff__card,axiom,
! [A2: set_nat,B2: set_set_set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ? [F4: nat > set_set_nat] : ( bij_be6938610931847138308et_nat @ F4 @ A2 @ B2 ) )
= ( ( finite_card_nat @ A2 )
= ( finite1149291290879098388et_nat @ B2 ) ) ) ) ) ).
% bij_betw_iff_card
thf(fact_889_bij__betw__iff__card,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ? [F4: nat > nat] : ( bij_betw_nat_nat @ F4 @ A2 @ B2 ) )
= ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B2 ) ) ) ) ) ).
% bij_betw_iff_card
thf(fact_890_bij__betw__def,axiom,
( bij_be4885122793727115194et_nat
= ( ^ [F4: set_set_nat > set_nat,A3: set_set_set_nat,B3: set_set_nat] :
( ( inj_on1894729867836481333et_nat @ F4 @ A3 )
& ( ( image_5842784325960735177et_nat @ F4 @ A3 )
= B3 ) ) ) ) ).
% bij_betw_def
thf(fact_891_bij__betw__def,axiom,
( bij_be6938610931847138308et_nat
= ( ^ [F4: nat > set_set_nat,A3: set_nat,B3: set_set_set_nat] :
( ( inj_on8105003582846801791et_nat @ F4 @ A3 )
& ( ( image_2194112158459175443et_nat @ F4 @ A3 )
= B3 ) ) ) ) ).
% bij_betw_def
thf(fact_892_bij__betw__def,axiom,
( bij_betw_nat_nat
= ( ^ [F4: nat > nat,A3: set_nat,B3: set_nat] :
( ( inj_on_nat_nat @ F4 @ A3 )
& ( ( image_nat_nat @ F4 @ A3 )
= B3 ) ) ) ) ).
% bij_betw_def
thf(fact_893_bij__betw__def,axiom,
( bij_betw_nat_set_nat
= ( ^ [F4: nat > set_nat,A3: set_nat,B3: set_set_nat] :
( ( inj_on_nat_set_nat @ F4 @ A3 )
& ( ( image_nat_set_nat @ F4 @ A3 )
= B3 ) ) ) ) ).
% bij_betw_def
thf(fact_894_bij__betw__imageI,axiom,
! [F: set_set_nat > set_nat,A2: set_set_set_nat,B2: set_set_nat] :
( ( inj_on1894729867836481333et_nat @ F @ A2 )
=> ( ( ( image_5842784325960735177et_nat @ F @ A2 )
= B2 )
=> ( bij_be4885122793727115194et_nat @ F @ A2 @ B2 ) ) ) ).
% bij_betw_imageI
thf(fact_895_bij__betw__imageI,axiom,
! [F: nat > set_set_nat,A2: set_nat,B2: set_set_set_nat] :
( ( inj_on8105003582846801791et_nat @ F @ A2 )
=> ( ( ( image_2194112158459175443et_nat @ F @ A2 )
= B2 )
=> ( bij_be6938610931847138308et_nat @ F @ A2 @ B2 ) ) ) ).
% bij_betw_imageI
thf(fact_896_bij__betw__imageI,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( inj_on_nat_nat @ F @ A2 )
=> ( ( ( image_nat_nat @ F @ A2 )
= B2 )
=> ( bij_betw_nat_nat @ F @ A2 @ B2 ) ) ) ).
% bij_betw_imageI
thf(fact_897_bij__betw__imageI,axiom,
! [F: nat > set_nat,A2: set_nat,B2: set_set_nat] :
( ( inj_on_nat_set_nat @ F @ A2 )
=> ( ( ( image_nat_set_nat @ F @ A2 )
= B2 )
=> ( bij_betw_nat_set_nat @ F @ A2 @ B2 ) ) ) ).
% bij_betw_imageI
thf(fact_898_inj__on__imp__bij__betw,axiom,
! [F: set_set_nat > set_nat,A2: set_set_set_nat] :
( ( inj_on1894729867836481333et_nat @ F @ A2 )
=> ( bij_be4885122793727115194et_nat @ F @ A2 @ ( image_5842784325960735177et_nat @ F @ A2 ) ) ) ).
% inj_on_imp_bij_betw
thf(fact_899_inj__on__imp__bij__betw,axiom,
! [F: nat > set_set_nat,A2: set_nat] :
( ( inj_on8105003582846801791et_nat @ F @ A2 )
=> ( bij_be6938610931847138308et_nat @ F @ A2 @ ( image_2194112158459175443et_nat @ F @ A2 ) ) ) ).
% inj_on_imp_bij_betw
thf(fact_900_inj__on__imp__bij__betw,axiom,
! [F: nat > nat,A2: set_nat] :
( ( inj_on_nat_nat @ F @ A2 )
=> ( bij_betw_nat_nat @ F @ A2 @ ( image_nat_nat @ F @ A2 ) ) ) ).
% inj_on_imp_bij_betw
thf(fact_901_inj__on__imp__bij__betw,axiom,
! [F: nat > set_nat,A2: set_nat] :
( ( inj_on_nat_set_nat @ F @ A2 )
=> ( bij_betw_nat_set_nat @ F @ A2 @ ( image_nat_set_nat @ F @ A2 ) ) ) ).
% inj_on_imp_bij_betw
thf(fact_902_image__diff__subset,axiom,
! [F: nat > set_set_nat,A2: set_nat,B2: set_nat] : ( ord_le9131159989063066194et_nat @ ( minus_2447799839930672331et_nat @ ( image_2194112158459175443et_nat @ F @ A2 ) @ ( image_2194112158459175443et_nat @ F @ B2 ) ) @ ( image_2194112158459175443et_nat @ F @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_903_image__diff__subset,axiom,
! [F: set_set_nat > set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( minus_2447799839930672331et_nat @ ( image_7884819252390400639et_nat @ F @ A2 ) @ ( image_7884819252390400639et_nat @ F @ B2 ) ) @ ( image_7884819252390400639et_nat @ F @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_904_image__diff__subset,axiom,
! [F: set_nat > set_set_nat,A2: set_set_nat,B2: set_set_nat] : ( ord_le9131159989063066194et_nat @ ( minus_2447799839930672331et_nat @ ( image_6725021117256019401et_nat @ F @ A2 ) @ ( image_6725021117256019401et_nat @ F @ B2 ) ) @ ( image_6725021117256019401et_nat @ F @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_905_image__diff__subset,axiom,
! [F: nat > set_nat,A2: set_nat,B2: set_nat] : ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ ( image_nat_set_nat @ F @ A2 ) @ ( image_nat_set_nat @ F @ B2 ) ) @ ( image_nat_set_nat @ F @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_906_image__diff__subset,axiom,
! [F: set_set_nat > set_nat,A2: set_set_set_nat,B2: set_set_set_nat] : ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ ( image_5842784325960735177et_nat @ F @ A2 ) @ ( image_5842784325960735177et_nat @ F @ B2 ) ) @ ( image_5842784325960735177et_nat @ F @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_907_image__diff__subset,axiom,
! [F: set_nat > set_nat,A2: set_set_nat,B2: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) @ ( image_7916887816326733075et_nat @ F @ B2 ) ) @ ( image_7916887816326733075et_nat @ F @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_908_image__diff__subset,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B2 ) ) @ ( image_nat_nat @ F @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_909_image__diff__subset,axiom,
! [F: set_set_nat > nat,A2: set_set_set_nat,B2: set_set_set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_1454916318497077779at_nat @ F @ A2 ) @ ( image_1454916318497077779at_nat @ F @ B2 ) ) @ ( image_1454916318497077779at_nat @ F @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_910_image__diff__subset,axiom,
! [F: set_nat > nat,A2: set_set_nat,B2: set_set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_set_nat_nat @ F @ A2 ) @ ( image_set_nat_nat @ F @ B2 ) ) @ ( image_set_nat_nat @ F @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_911_inj__on__image__set__diff,axiom,
! [F: set_set_nat > nat,C: set_set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( inj_on7365807742884704127at_nat @ F @ C )
=> ( ( ord_le9131159989063066194et_nat @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) @ C )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ C )
=> ( ( image_1454916318497077779at_nat @ F @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) )
= ( minus_minus_set_nat @ ( image_1454916318497077779at_nat @ F @ A2 ) @ ( image_1454916318497077779at_nat @ F @ B2 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_912_inj__on__image__set__diff,axiom,
! [F: set_set_nat > set_set_nat,C: set_set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( inj_on2040386338155636715et_nat @ F @ C )
=> ( ( ord_le9131159989063066194et_nat @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) @ C )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ C )
=> ( ( image_7884819252390400639et_nat @ F @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) )
= ( minus_2447799839930672331et_nat @ ( image_7884819252390400639et_nat @ F @ A2 ) @ ( image_7884819252390400639et_nat @ F @ B2 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_913_inj__on__image__set__diff,axiom,
! [F: set_set_nat > set_nat,C: set_set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( inj_on1894729867836481333et_nat @ F @ C )
=> ( ( ord_le9131159989063066194et_nat @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) @ C )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ C )
=> ( ( image_5842784325960735177et_nat @ F @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) )
= ( minus_2163939370556025621et_nat @ ( image_5842784325960735177et_nat @ F @ A2 ) @ ( image_5842784325960735177et_nat @ F @ B2 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_914_inj__on__image__set__diff,axiom,
! [F: set_nat > nat,C: set_set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( inj_on_set_nat_nat @ F @ C )
=> ( ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) @ C )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ C )
=> ( ( image_set_nat_nat @ F @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
= ( minus_minus_set_nat @ ( image_set_nat_nat @ F @ A2 ) @ ( image_set_nat_nat @ F @ B2 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_915_inj__on__image__set__diff,axiom,
! [F: set_nat > set_set_nat,C: set_set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( inj_on2776966659131765557et_nat @ F @ C )
=> ( ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) @ C )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ C )
=> ( ( image_6725021117256019401et_nat @ F @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
= ( minus_2447799839930672331et_nat @ ( image_6725021117256019401et_nat @ F @ A2 ) @ ( image_6725021117256019401et_nat @ F @ B2 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_916_inj__on__image__set__diff,axiom,
! [F: set_nat > set_nat,C: set_set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( inj_on4604407203859583615et_nat @ F @ C )
=> ( ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) @ C )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ C )
=> ( ( image_7916887816326733075et_nat @ F @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
= ( minus_2163939370556025621et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) @ ( image_7916887816326733075et_nat @ F @ B2 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_917_inj__on__image__set__diff,axiom,
! [F: nat > nat,C: set_nat,A2: set_nat,B2: set_nat] :
( ( inj_on_nat_nat @ F @ C )
=> ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ C )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ( image_nat_nat @ F @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( minus_minus_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B2 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_918_inj__on__image__set__diff,axiom,
! [F: nat > set_set_nat,C: set_nat,A2: set_nat,B2: set_nat] :
( ( inj_on8105003582846801791et_nat @ F @ C )
=> ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ C )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ( image_2194112158459175443et_nat @ F @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( minus_2447799839930672331et_nat @ ( image_2194112158459175443et_nat @ F @ A2 ) @ ( image_2194112158459175443et_nat @ F @ B2 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_919_inj__on__image__set__diff,axiom,
! [F: nat > set_nat,C: set_nat,A2: set_nat,B2: set_nat] :
( ( inj_on_nat_set_nat @ F @ C )
=> ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ C )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ( image_nat_set_nat @ F @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( minus_2163939370556025621et_nat @ ( image_nat_set_nat @ F @ A2 ) @ ( image_nat_set_nat @ F @ B2 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_920_card__le__sym__Diff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_921_card__le__sym__Diff,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite1149291290879098388et_nat @ B2 ) )
=> ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) ) @ ( finite1149291290879098388et_nat @ ( minus_2447799839930672331et_nat @ B2 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_922_card__le__sym__Diff,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ B2 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_923_card__less__sym__Diff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_924_card__less__sym__Diff,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ord_less_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite1149291290879098388et_nat @ B2 ) )
=> ( ord_less_nat @ ( finite1149291290879098388et_nat @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) ) @ ( finite1149291290879098388et_nat @ ( minus_2447799839930672331et_nat @ B2 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_925_card__less__sym__Diff,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_less_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) )
=> ( ord_less_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ B2 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_926_ex__bij__betw__nat__finite,axiom,
! [M4: set_set_nat] :
( ( finite1152437895449049373et_nat @ M4 )
=> ? [H4: nat > set_nat] : ( bij_betw_nat_set_nat @ H4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_set_nat @ M4 ) ) @ M4 ) ) ).
% ex_bij_betw_nat_finite
thf(fact_927_ex__bij__betw__nat__finite,axiom,
! [M4: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ M4 )
=> ? [H4: nat > set_set_nat] : ( bij_be6938610931847138308et_nat @ H4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite1149291290879098388et_nat @ M4 ) ) @ M4 ) ) ).
% ex_bij_betw_nat_finite
thf(fact_928_ex__bij__betw__nat__finite,axiom,
! [M4: set_nat] :
( ( finite_finite_nat @ M4 )
=> ? [H4: nat > nat] : ( bij_betw_nat_nat @ H4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ M4 ) ) @ M4 ) ) ).
% ex_bij_betw_nat_finite
thf(fact_929_sunflower__def,axiom,
( sunflo5599553548652064642et_nat
= ( ^ [S5: set_set_nat_set_nat] :
! [X2: nat > set_nat] :
( ? [A3: set_nat_set_nat,B3: set_nat_set_nat] :
( ( member6710465769566284994et_nat @ A3 @ S5 )
& ( member6710465769566284994et_nat @ B3 @ S5 )
& ( A3 != B3 )
& ( member_nat_set_nat @ X2 @ A3 )
& ( member_nat_set_nat @ X2 @ B3 ) )
=> ! [A3: set_nat_set_nat] :
( ( member6710465769566284994et_nat @ A3 @ S5 )
=> ( member_nat_set_nat @ X2 @ A3 ) ) ) ) ) ).
% sunflower_def
thf(fact_930_sunflower__def,axiom,
( sunflo2680516271513359689et_nat
= ( ^ [S5: set_set_set_set_nat] :
! [X2: set_set_nat] :
( ? [A3: set_set_set_nat,B3: set_set_set_nat] :
( ( member2946998982187404937et_nat @ A3 @ S5 )
& ( member2946998982187404937et_nat @ B3 @ S5 )
& ( A3 != B3 )
& ( member_set_set_nat @ X2 @ A3 )
& ( member_set_set_nat @ X2 @ B3 ) )
=> ! [A3: set_set_set_nat] :
( ( member2946998982187404937et_nat @ A3 @ S5 )
=> ( member_set_set_nat @ X2 @ A3 ) ) ) ) ) ).
% sunflower_def
thf(fact_931_sunflower__def,axiom,
( sunflower_set_nat
= ( ^ [S5: set_set_set_nat] :
! [X2: set_nat] :
( ? [A3: set_set_nat,B3: set_set_nat] :
( ( member_set_set_nat @ A3 @ S5 )
& ( member_set_set_nat @ B3 @ S5 )
& ( A3 != B3 )
& ( member_set_nat @ X2 @ A3 )
& ( member_set_nat @ X2 @ B3 ) )
=> ! [A3: set_set_nat] :
( ( member_set_set_nat @ A3 @ S5 )
=> ( member_set_nat @ X2 @ A3 ) ) ) ) ) ).
% sunflower_def
thf(fact_932_sunflower__def,axiom,
( sunflower_nat
= ( ^ [S5: set_set_nat] :
! [X2: nat] :
( ? [A3: set_nat,B3: set_nat] :
( ( member_set_nat @ A3 @ S5 )
& ( member_set_nat @ B3 @ S5 )
& ( A3 != B3 )
& ( member_nat @ X2 @ A3 )
& ( member_nat @ X2 @ B3 ) )
=> ! [A3: set_nat] :
( ( member_set_nat @ A3 @ S5 )
=> ( member_nat @ X2 @ A3 ) ) ) ) ) ).
% sunflower_def
thf(fact_933_sunflower__subset,axiom,
! [F2: set_set_set_nat,G2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ F2 @ G2 )
=> ( ( sunflower_set_nat @ G2 )
=> ( sunflower_set_nat @ F2 ) ) ) ).
% sunflower_subset
thf(fact_934_sunflower__subset,axiom,
! [F2: set_set_nat,G2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ F2 @ G2 )
=> ( ( sunflower_nat @ G2 )
=> ( sunflower_nat @ F2 ) ) ) ).
% sunflower_subset
thf(fact_935_i__props_I3_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ p )
=> ( ( ti @ I )
= ( minus_minus_nat @ ( si2 @ I ) @ s2 ) ) ) ).
% i_props(3)
thf(fact_936_Gs__def,axiom,
( gs
= ( clique6722202388162463298od_nat @ vs @ vs ) ) ).
% Gs_def
thf(fact_937_local_Omerge__def,axiom,
( merge
= ( ^ [E2: nat > set_nat,G6: nat > nat,V: nat] :
( if_nat
@ ( member_nat @ V
@ ( image_nat_nat
@ ^ [I2: nat] : ( fstt @ ( E2 @ I2 ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ p ) ) )
@ ( G6
@ ( sndd
@ ( E2
@ ( fChoice_nat
@ ^ [I2: nat] :
( ( ord_less_nat @ I2 @ p )
& ( V
= ( fstt @ ( E2 @ I2 ) ) ) ) ) ) ) )
@ ( G6 @ V ) ) ) ) ).
% local.merge_def
thf(fact_938_Schroeder__Bernstein,axiom,
! [F: set_set_nat > set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat,G: set_set_nat > set_set_nat] :
( ( inj_on2040386338155636715et_nat @ F @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ ( image_7884819252390400639et_nat @ F @ A2 ) @ B2 )
=> ( ( inj_on2040386338155636715et_nat @ G @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ ( image_7884819252390400639et_nat @ G @ B2 ) @ A2 )
=> ? [H4: set_set_nat > set_set_nat] : ( bij_be1917187662166652016et_nat @ H4 @ A2 @ B2 ) ) ) ) ) ).
% Schroeder_Bernstein
thf(fact_939_Schroeder__Bernstein,axiom,
! [F: set_nat > set_set_nat,A2: set_set_nat,B2: set_set_set_nat,G: set_set_nat > set_nat] :
( ( inj_on2776966659131765557et_nat @ F @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ ( image_6725021117256019401et_nat @ F @ A2 ) @ B2 )
=> ( ( inj_on1894729867836481333et_nat @ G @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ ( image_5842784325960735177et_nat @ G @ B2 ) @ A2 )
=> ? [H4: set_nat > set_set_nat] : ( bij_be5767359585022399418et_nat @ H4 @ A2 @ B2 ) ) ) ) ) ).
% Schroeder_Bernstein
thf(fact_940_Schroeder__Bernstein,axiom,
! [F: nat > set_set_nat,A2: set_nat,B2: set_set_set_nat,G: set_set_nat > nat] :
( ( inj_on8105003582846801791et_nat @ F @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ ( image_2194112158459175443et_nat @ F @ A2 ) @ B2 )
=> ( ( inj_on7365807742884704127at_nat @ G @ B2 )
=> ( ( ord_less_eq_set_nat @ ( image_1454916318497077779at_nat @ G @ B2 ) @ A2 )
=> ? [H4: nat > set_set_nat] : ( bij_be6938610931847138308et_nat @ H4 @ A2 @ B2 ) ) ) ) ) ).
% Schroeder_Bernstein
thf(fact_941_Schroeder__Bernstein,axiom,
! [F: set_set_nat > set_nat,A2: set_set_set_nat,B2: set_set_nat,G: set_nat > set_set_nat] :
( ( inj_on1894729867836481333et_nat @ F @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ ( image_5842784325960735177et_nat @ F @ A2 ) @ B2 )
=> ( ( inj_on2776966659131765557et_nat @ G @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ ( image_6725021117256019401et_nat @ G @ B2 ) @ A2 )
=> ? [H4: set_set_nat > set_nat] : ( bij_be4885122793727115194et_nat @ H4 @ A2 @ B2 ) ) ) ) ) ).
% Schroeder_Bernstein
thf(fact_942_Schroeder__Bernstein,axiom,
! [F: set_nat > set_nat,A2: set_set_nat,B2: set_set_nat,G: set_nat > set_nat] :
( ( inj_on4604407203859583615et_nat @ F @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) @ B2 )
=> ( ( inj_on4604407203859583615et_nat @ G @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ G @ B2 ) @ A2 )
=> ? [H4: set_nat > set_nat] : ( bij_be3438014552859920132et_nat @ H4 @ A2 @ B2 ) ) ) ) ) ).
% Schroeder_Bernstein
thf(fact_943_Schroeder__Bernstein,axiom,
! [F: nat > set_nat,A2: set_nat,B2: set_set_nat,G: set_nat > nat] :
( ( inj_on_nat_set_nat @ F @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A2 ) @ B2 )
=> ( ( inj_on_set_nat_nat @ G @ B2 )
=> ( ( ord_less_eq_set_nat @ ( image_set_nat_nat @ G @ B2 ) @ A2 )
=> ? [H4: nat > set_nat] : ( bij_betw_nat_set_nat @ H4 @ A2 @ B2 ) ) ) ) ) ).
% Schroeder_Bernstein
thf(fact_944_Schroeder__Bernstein,axiom,
! [F: set_set_nat > nat,A2: set_set_set_nat,B2: set_nat,G: nat > set_set_nat] :
( ( inj_on7365807742884704127at_nat @ F @ A2 )
=> ( ( ord_less_eq_set_nat @ ( image_1454916318497077779at_nat @ F @ A2 ) @ B2 )
=> ( ( inj_on8105003582846801791et_nat @ G @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ ( image_2194112158459175443et_nat @ G @ B2 ) @ A2 )
=> ? [H4: set_set_nat > nat] : ( bij_be6199415091885040644at_nat @ H4 @ A2 @ B2 ) ) ) ) ) ).
% Schroeder_Bernstein
thf(fact_945_Schroeder__Bernstein,axiom,
! [F: set_nat > nat,A2: set_set_nat,B2: set_nat,G: nat > set_nat] :
( ( inj_on_set_nat_nat @ F @ A2 )
=> ( ( ord_less_eq_set_nat @ ( image_set_nat_nat @ F @ A2 ) @ B2 )
=> ( ( inj_on_nat_set_nat @ G @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ G @ B2 ) @ A2 )
=> ? [H4: set_nat > nat] : ( bij_betw_set_nat_nat @ H4 @ A2 @ B2 ) ) ) ) ) ).
% Schroeder_Bernstein
thf(fact_946_Schroeder__Bernstein,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat,G: nat > nat] :
( ( inj_on_nat_nat @ F @ A2 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B2 )
=> ( ( inj_on_nat_nat @ G @ B2 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ G @ B2 ) @ A2 )
=> ? [H4: nat > nat] : ( bij_betw_nat_nat @ H4 @ A2 @ B2 ) ) ) ) ) ).
% Schroeder_Bernstein
thf(fact_947_sndd__def,axiom,
! [E3: set_nat] :
( ( sndd @ E3 )
= ( fChoice_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ E3 )
& ( X2
!= ( fstt @ E3 ) ) ) ) ) ).
% sndd_def
thf(fact_948_v__sameprod__subset,axiom,
! [Vs: set_nat] : ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ ( clique6722202388162463298od_nat @ Vs @ Vs ) ) @ Vs ) ).
% v_sameprod_subset
thf(fact_949_Vs__def,axiom,
( vs
= ( comple7806235888213564991et_nat @ s ) ) ).
% Vs_def
thf(fact_950_vplus__dsXU,axiom,
( ( clique8462013130872731469t_v_gs @ ( minus_2447799839930672331et_nat @ x @ u ) )
= ( minus_2163939370556025621et_nat @ ( clique8462013130872731469t_v_gs @ x ) @ ( clique8462013130872731469t_v_gs @ u ) ) ) ).
% vplus_dsXU
thf(fact_951__092_060open_062card_A_Iv__gs_AX_A_N_Av__gs_AU_J_A_061_Acard_A_Iv__gs_AX_J_A_N_Acard_A_Iv__gs_AU_J_092_060close_062,axiom,
( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ ( clique8462013130872731469t_v_gs @ x ) @ ( clique8462013130872731469t_v_gs @ u ) ) )
= ( minus_minus_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ x ) ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ u ) ) ) ) ).
% \<open>card (v_gs X - v_gs U) = card (v_gs X) - card (v_gs U)\<close>
thf(fact_952_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_953_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_954_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_955_finite__Inter,axiom,
! [M4: set_set_set_nat] :
( ? [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ M4 )
& ( finite1152437895449049373et_nat @ X4 ) )
=> ( finite1152437895449049373et_nat @ ( comple1065008630642458357et_nat @ M4 ) ) ) ).
% finite_Inter
thf(fact_956_finite__Inter,axiom,
! [M4: set_set_set_set_nat] :
( ? [X4: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X4 @ M4 )
& ( finite6739761609112101331et_nat @ X4 ) )
=> ( finite6739761609112101331et_nat @ ( comple8067742441731897515et_nat @ M4 ) ) ) ).
% finite_Inter
thf(fact_957_finite__Inter,axiom,
! [M4: set_set_nat] :
( ? [X4: set_nat] :
( ( member_set_nat @ X4 @ M4 )
& ( finite_finite_nat @ X4 ) )
=> ( finite_finite_nat @ ( comple7806235888213564991et_nat @ M4 ) ) ) ).
% finite_Inter
thf(fact_958_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_959_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_960_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_961_Inf__atLeastLessThan,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( ( comple7806235888213564991et_nat @ ( set_or3540276404033026485et_nat @ X @ Y ) )
= X ) ) ).
% Inf_atLeastLessThan
thf(fact_962_card__atLeastLessThan,axiom,
! [L: nat,U: nat] :
( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ L @ U ) )
= ( minus_minus_nat @ U @ L ) ) ).
% card_atLeastLessThan
thf(fact_963_finite__INT,axiom,
! [I5: set_nat,A2: nat > set_set_nat] :
( ? [X4: nat] :
( ( member_nat @ X4 @ I5 )
& ( finite1152437895449049373et_nat @ ( A2 @ X4 ) ) )
=> ( finite1152437895449049373et_nat @ ( comple1065008630642458357et_nat @ ( image_2194112158459175443et_nat @ A2 @ I5 ) ) ) ) ).
% finite_INT
thf(fact_964_finite__INT,axiom,
! [I5: set_set_set_nat,A2: set_set_nat > set_nat] :
( ? [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ I5 )
& ( finite_finite_nat @ ( A2 @ X4 ) ) )
=> ( finite_finite_nat @ ( comple7806235888213564991et_nat @ ( image_5842784325960735177et_nat @ A2 @ I5 ) ) ) ) ).
% finite_INT
thf(fact_965_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_966_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_967_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_968_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_969_le__diff__iff_H,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A ) @ ( minus_minus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_970_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_971_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_972_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_973_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_974_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_975_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_976_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_977_minus__set__def,axiom,
( minus_8060664002660188164et_nat
= ( ^ [A3: set_nat_set_nat,B3: set_nat_set_nat] :
( collect_nat_set_nat
@ ( minus_3205255705857990017_nat_o
@ ^ [X2: nat > set_nat] : ( member_nat_set_nat @ X2 @ A3 )
@ ^ [X2: nat > set_nat] : ( member_nat_set_nat @ X2 @ B3 ) ) ) ) ) ).
% minus_set_def
thf(fact_978_minus__set__def,axiom,
( minus_minus_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( collect_nat
@ ( minus_minus_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A3 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B3 ) ) ) ) ) ).
% minus_set_def
thf(fact_979_minus__set__def,axiom,
( minus_2447799839930672331et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
( collect_set_set_nat
@ ( minus_463385787819020154_nat_o
@ ^ [X2: set_set_nat] : ( member_set_set_nat @ X2 @ A3 )
@ ^ [X2: set_set_nat] : ( member_set_set_nat @ X2 @ B3 ) ) ) ) ) ).
% minus_set_def
thf(fact_980_minus__set__def,axiom,
( minus_2163939370556025621et_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( collect_set_nat
@ ( minus_6910147592129066416_nat_o
@ ^ [X2: set_nat] : ( member_set_nat @ X2 @ A3 )
@ ^ [X2: set_nat] : ( member_set_nat @ X2 @ B3 ) ) ) ) ) ).
% minus_set_def
thf(fact_981_inj__on__diff__nat,axiom,
! [N3: set_nat,K: nat] :
( ! [N2: nat] :
( ( member_nat @ N2 @ N3 )
=> ( ord_less_eq_nat @ K @ N2 ) )
=> ( inj_on_nat_nat
@ ^ [N4: nat] : ( minus_minus_nat @ N4 @ K )
@ N3 ) ) ).
% inj_on_diff_nat
thf(fact_982_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_983_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_984_diff__less__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C2 ) @ ( minus_minus_nat @ B @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_985_le__rel__bool__arg__iff,axiom,
( ord_le8326115459943588763et_nat
= ( ^ [X6: $o > set_set_set_nat,Y7: $o > set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( X6 @ $false ) @ ( Y7 @ $false ) )
& ( ord_le9131159989063066194et_nat @ ( X6 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_986_le__rel__bool__arg__iff,axiom,
( ord_le6539261115178940645et_nat
= ( ^ [X6: $o > set_set_nat,Y7: $o > set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( X6 @ $false ) @ ( Y7 @ $false ) )
& ( ord_le6893508408891458716et_nat @ ( X6 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_987_le__rel__bool__arg__iff,axiom,
( ord_le7022414076629706543et_nat
= ( ^ [X6: $o > set_nat,Y7: $o > set_nat] :
( ( ord_less_eq_set_nat @ ( X6 @ $false ) @ ( Y7 @ $false ) )
& ( ord_less_eq_set_nat @ ( X6 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_988_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_nat
= ( ^ [X6: $o > nat,Y7: $o > nat] :
( ( ord_less_eq_nat @ ( X6 @ $false ) @ ( Y7 @ $false ) )
& ( ord_less_eq_nat @ ( X6 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_989_diff__preserves__multiset,axiom,
! [M4: set_nat > nat,N3: set_nat > nat] :
( ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X2: set_nat] : ( ord_less_nat @ zero_zero_nat @ ( M4 @ X2 ) ) ) )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X2: set_nat] : ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( M4 @ X2 ) @ ( N3 @ X2 ) ) ) ) ) ) ).
% diff_preserves_multiset
thf(fact_990_diff__preserves__multiset,axiom,
! [M4: set_set_nat > nat,N3: set_set_nat > nat] :
( ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [X2: set_set_nat] : ( ord_less_nat @ zero_zero_nat @ ( M4 @ X2 ) ) ) )
=> ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [X2: set_set_nat] : ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( M4 @ X2 ) @ ( N3 @ X2 ) ) ) ) ) ) ).
% diff_preserves_multiset
thf(fact_991_diff__preserves__multiset,axiom,
! [M4: nat > nat,N3: nat > nat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] : ( ord_less_nat @ zero_zero_nat @ ( M4 @ X2 ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] : ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( M4 @ X2 ) @ ( N3 @ X2 ) ) ) ) ) ) ).
% diff_preserves_multiset
thf(fact_992_ex__bij__betw__finite__nat,axiom,
! [M4: set_set_nat] :
( ( finite1152437895449049373et_nat @ M4 )
=> ? [H4: set_nat > nat] : ( bij_betw_set_nat_nat @ H4 @ M4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_set_nat @ M4 ) ) ) ) ).
% ex_bij_betw_finite_nat
thf(fact_993_ex__bij__betw__finite__nat,axiom,
! [M4: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ M4 )
=> ? [H4: set_set_nat > nat] : ( bij_be6199415091885040644at_nat @ H4 @ M4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite1149291290879098388et_nat @ M4 ) ) ) ) ).
% ex_bij_betw_finite_nat
thf(fact_994_ex__bij__betw__finite__nat,axiom,
! [M4: set_nat] :
( ( finite_finite_nat @ M4 )
=> ? [H4: nat > nat] : ( bij_betw_nat_nat @ H4 @ M4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ M4 ) ) ) ) ).
% ex_bij_betw_finite_nat
thf(fact_995_card__Diff__subset,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
=> ( ( finite1149291290879098388et_nat @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) )
= ( minus_minus_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite1149291290879098388et_nat @ B2 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_996_card__Diff__subset,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
= ( minus_minus_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_997_card__Diff__subset,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_998_diff__card__le__card__Diff,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_999_diff__card__le__card__Diff,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite1149291290879098388et_nat @ B2 ) ) @ ( finite1149291290879098388et_nat @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_1000_diff__card__le__card__Diff,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) ) @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_1001_sameprod__mono,axiom,
! [X5: set_set_set_nat,Y2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X5 @ Y2 )
=> ( ord_le572741076514265352et_nat @ ( clique1181040904276305582et_nat @ X5 @ X5 ) @ ( clique1181040904276305582et_nat @ Y2 @ Y2 ) ) ) ).
% sameprod_mono
thf(fact_1002_sameprod__mono,axiom,
! [X5: set_set_nat,Y2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X5 @ Y2 )
=> ( ord_le9131159989063066194et_nat @ ( clique8906516429304539640et_nat @ X5 @ X5 ) @ ( clique8906516429304539640et_nat @ Y2 @ Y2 ) ) ) ).
% sameprod_mono
thf(fact_1003_sameprod__mono,axiom,
! [X5: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X5 @ Y2 )
=> ( ord_le6893508408891458716et_nat @ ( clique6722202388162463298od_nat @ X5 @ X5 ) @ ( clique6722202388162463298od_nat @ Y2 @ Y2 ) ) ) ).
% sameprod_mono
thf(fact_1004_sameprod__finite,axiom,
! [X5: set_set_nat] :
( ( finite1152437895449049373et_nat @ X5 )
=> ( finite6739761609112101331et_nat @ ( clique8906516429304539640et_nat @ X5 @ X5 ) ) ) ).
% sameprod_finite
thf(fact_1005_sameprod__finite,axiom,
! [X5: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ X5 )
=> ( finite5926941155766903689et_nat @ ( clique1181040904276305582et_nat @ X5 @ X5 ) ) ) ).
% sameprod_finite
thf(fact_1006_sameprod__finite,axiom,
! [X5: set_nat] :
( ( finite_finite_nat @ X5 )
=> ( finite1152437895449049373et_nat @ ( clique6722202388162463298od_nat @ X5 @ X5 ) ) ) ).
% sameprod_finite
thf(fact_1007_cInf__atLeastLessThan,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_set_nat @ Y @ X )
=> ( ( comple7806235888213564991et_nat @ ( set_or3540276404033026485et_nat @ Y @ X ) )
= Y ) ) ).
% cInf_atLeastLessThan
thf(fact_1008_cInf__atLeastLessThan,axiom,
! [Y: nat,X: nat] :
( ( ord_less_nat @ Y @ X )
=> ( ( complete_Inf_Inf_nat @ ( set_or4665077453230672383an_nat @ Y @ X ) )
= Y ) ) ).
% cInf_atLeastLessThan
thf(fact_1009_INT__iff,axiom,
! [B: set_nat,B2: nat > set_set_nat,A2: set_nat] :
( ( member_set_nat @ B @ ( comple1065008630642458357et_nat @ ( image_2194112158459175443et_nat @ B2 @ A2 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_set_nat @ B @ ( B2 @ X2 ) ) ) ) ) ).
% INT_iff
thf(fact_1010_INT__iff,axiom,
! [B: nat,B2: set_set_nat > set_nat,A2: set_set_set_nat] :
( ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_5842784325960735177et_nat @ B2 @ A2 ) ) )
= ( ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A2 )
=> ( member_nat @ B @ ( B2 @ X2 ) ) ) ) ) ).
% INT_iff
thf(fact_1011_INT__I,axiom,
! [A2: set_nat,B: nat,B2: nat > set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ B @ ( B2 @ X3 ) ) )
=> ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) ) ) ).
% INT_I
thf(fact_1012_INT__I,axiom,
! [A2: set_nat,B: set_nat,B2: nat > set_set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_set_nat @ B @ ( B2 @ X3 ) ) )
=> ( member_set_nat @ B @ ( comple1065008630642458357et_nat @ ( image_2194112158459175443et_nat @ B2 @ A2 ) ) ) ) ).
% INT_I
thf(fact_1013_INT__I,axiom,
! [A2: set_set_nat,B: nat,B2: set_nat > set_nat] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( member_nat @ B @ ( B2 @ X3 ) ) )
=> ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_7916887816326733075et_nat @ B2 @ A2 ) ) ) ) ).
% INT_I
thf(fact_1014_INT__I,axiom,
! [A2: set_nat,B: set_set_nat,B2: nat > set_set_set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_set_set_nat @ B @ ( B2 @ X3 ) ) )
=> ( member_set_set_nat @ B @ ( comple8067742441731897515et_nat @ ( image_5738044413236618185et_nat @ B2 @ A2 ) ) ) ) ).
% INT_I
thf(fact_1015_INT__I,axiom,
! [A2: set_set_nat,B: set_nat,B2: set_nat > set_set_nat] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( member_set_nat @ B @ ( B2 @ X3 ) ) )
=> ( member_set_nat @ B @ ( comple1065008630642458357et_nat @ ( image_6725021117256019401et_nat @ B2 @ A2 ) ) ) ) ).
% INT_I
thf(fact_1016_INT__I,axiom,
! [A2: set_set_set_nat,B: nat,B2: set_set_nat > set_nat] :
( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A2 )
=> ( member_nat @ B @ ( B2 @ X3 ) ) )
=> ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_5842784325960735177et_nat @ B2 @ A2 ) ) ) ) ).
% INT_I
thf(fact_1017_INT__I,axiom,
! [A2: set_nat,B: nat > set_nat,B2: nat > set_nat_set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat_set_nat @ B @ ( B2 @ X3 ) ) )
=> ( member_nat_set_nat @ B @ ( comple5153742063261271012et_nat @ ( image_2803531558198256130et_nat @ B2 @ A2 ) ) ) ) ).
% INT_I
thf(fact_1018_INT__I,axiom,
! [A2: set_set_nat,B: set_set_nat,B2: set_nat > set_set_set_nat] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( member_set_set_nat @ B @ ( B2 @ X3 ) ) )
=> ( member_set_set_nat @ B @ ( comple8067742441731897515et_nat @ ( image_4583741654806091647et_nat @ B2 @ A2 ) ) ) ) ).
% INT_I
thf(fact_1019_INT__I,axiom,
! [A2: set_set_set_nat,B: set_nat,B2: set_set_nat > set_set_nat] :
( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A2 )
=> ( member_set_nat @ B @ ( B2 @ X3 ) ) )
=> ( member_set_nat @ B @ ( comple1065008630642458357et_nat @ ( image_7884819252390400639et_nat @ B2 @ A2 ) ) ) ) ).
% INT_I
thf(fact_1020_INT__I,axiom,
! [A2: set_nat_set_nat,B: nat,B2: ( nat > set_nat ) > set_nat] :
( ! [X3: nat > set_nat] :
( ( member_nat_set_nat @ X3 @ A2 )
=> ( member_nat @ B @ ( B2 @ X3 ) ) )
=> ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_8304670887732450946et_nat @ B2 @ A2 ) ) ) ) ).
% INT_I
thf(fact_1021_INF__identity__eq,axiom,
! [A2: set_nat] :
( ( complete_Inf_Inf_nat
@ ( image_nat_nat
@ ^ [X2: nat] : X2
@ A2 ) )
= ( complete_Inf_Inf_nat @ A2 ) ) ).
% INF_identity_eq
thf(fact_1022_INF__identity__eq,axiom,
! [A2: set_set_nat] :
( ( comple7806235888213564991et_nat
@ ( image_7916887816326733075et_nat
@ ^ [X2: set_nat] : X2
@ A2 ) )
= ( comple7806235888213564991et_nat @ A2 ) ) ).
% INF_identity_eq
thf(fact_1023_image__INT,axiom,
! [F: nat > nat,C: set_nat,A2: set_nat,B2: nat > set_nat,J: nat] :
( ( inj_on_nat_nat @ F @ C )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ X3 ) @ C ) )
=> ( ( member_nat @ J @ A2 )
=> ( ( image_nat_nat @ F @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
= ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( image_nat_nat @ F @ ( B2 @ X2 ) )
@ A2 ) ) ) ) ) ) ).
% image_INT
thf(fact_1024_image__INT,axiom,
! [F: set_nat > nat,C: set_set_nat,A2: set_nat,B2: nat > set_set_nat,J: nat] :
( ( inj_on_set_nat_nat @ F @ C )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( B2 @ X3 ) @ C ) )
=> ( ( member_nat @ J @ A2 )
=> ( ( image_set_nat_nat @ F @ ( comple1065008630642458357et_nat @ ( image_2194112158459175443et_nat @ B2 @ A2 ) ) )
= ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( image_set_nat_nat @ F @ ( B2 @ X2 ) )
@ A2 ) ) ) ) ) ) ).
% image_INT
thf(fact_1025_image__INT,axiom,
! [F: nat > set_nat,C: set_nat,A2: set_nat,B2: nat > set_nat,J: nat] :
( ( inj_on_nat_set_nat @ F @ C )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ X3 ) @ C ) )
=> ( ( member_nat @ J @ A2 )
=> ( ( image_nat_set_nat @ F @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
= ( comple1065008630642458357et_nat
@ ( image_2194112158459175443et_nat
@ ^ [X2: nat] : ( image_nat_set_nat @ F @ ( B2 @ X2 ) )
@ A2 ) ) ) ) ) ) ).
% image_INT
thf(fact_1026_image__INT,axiom,
! [F: nat > nat,C: set_nat,A2: set_set_nat,B2: set_nat > set_nat,J: set_nat] :
( ( inj_on_nat_nat @ F @ C )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ X3 ) @ C ) )
=> ( ( member_set_nat @ J @ A2 )
=> ( ( image_nat_nat @ F @ ( comple7806235888213564991et_nat @ ( image_7916887816326733075et_nat @ B2 @ A2 ) ) )
= ( comple7806235888213564991et_nat
@ ( image_7916887816326733075et_nat
@ ^ [X2: set_nat] : ( image_nat_nat @ F @ ( B2 @ X2 ) )
@ A2 ) ) ) ) ) ) ).
% image_INT
thf(fact_1027_image__INT,axiom,
! [F: set_nat > set_nat,C: set_set_nat,A2: set_nat,B2: nat > set_set_nat,J: nat] :
( ( inj_on4604407203859583615et_nat @ F @ C )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( B2 @ X3 ) @ C ) )
=> ( ( member_nat @ J @ A2 )
=> ( ( image_7916887816326733075et_nat @ F @ ( comple1065008630642458357et_nat @ ( image_2194112158459175443et_nat @ B2 @ A2 ) ) )
= ( comple1065008630642458357et_nat
@ ( image_2194112158459175443et_nat
@ ^ [X2: nat] : ( image_7916887816326733075et_nat @ F @ ( B2 @ X2 ) )
@ A2 ) ) ) ) ) ) ).
% image_INT
thf(fact_1028_image__INT,axiom,
! [F: set_set_nat > nat,C: set_set_set_nat,A2: set_nat,B2: nat > set_set_set_nat,J: nat] :
( ( inj_on7365807742884704127at_nat @ F @ C )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_le9131159989063066194et_nat @ ( B2 @ X3 ) @ C ) )
=> ( ( member_nat @ J @ A2 )
=> ( ( image_1454916318497077779at_nat @ F @ ( comple8067742441731897515et_nat @ ( image_5738044413236618185et_nat @ B2 @ A2 ) ) )
= ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( image_1454916318497077779at_nat @ F @ ( B2 @ X2 ) )
@ A2 ) ) ) ) ) ) ).
% image_INT
thf(fact_1029_image__INT,axiom,
! [F: set_nat > nat,C: set_set_nat,A2: set_set_nat,B2: set_nat > set_set_nat,J: set_nat] :
( ( inj_on_set_nat_nat @ F @ C )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( B2 @ X3 ) @ C ) )
=> ( ( member_set_nat @ J @ A2 )
=> ( ( image_set_nat_nat @ F @ ( comple1065008630642458357et_nat @ ( image_6725021117256019401et_nat @ B2 @ A2 ) ) )
= ( comple7806235888213564991et_nat
@ ( image_7916887816326733075et_nat
@ ^ [X2: set_nat] : ( image_set_nat_nat @ F @ ( B2 @ X2 ) )
@ A2 ) ) ) ) ) ) ).
% image_INT
thf(fact_1030_image__INT,axiom,
! [F: nat > set_set_nat,C: set_nat,A2: set_nat,B2: nat > set_nat,J: nat] :
( ( inj_on8105003582846801791et_nat @ F @ C )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ X3 ) @ C ) )
=> ( ( member_nat @ J @ A2 )
=> ( ( image_2194112158459175443et_nat @ F @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
= ( comple8067742441731897515et_nat
@ ( image_5738044413236618185et_nat
@ ^ [X2: nat] : ( image_2194112158459175443et_nat @ F @ ( B2 @ X2 ) )
@ A2 ) ) ) ) ) ) ).
% image_INT
thf(fact_1031_image__INT,axiom,
! [F: nat > nat,C: set_nat,A2: set_set_set_nat,B2: set_set_nat > set_nat,J: set_set_nat] :
( ( inj_on_nat_nat @ F @ C )
=> ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ X3 ) @ C ) )
=> ( ( member_set_set_nat @ J @ A2 )
=> ( ( image_nat_nat @ F @ ( comple7806235888213564991et_nat @ ( image_5842784325960735177et_nat @ B2 @ A2 ) ) )
= ( comple7806235888213564991et_nat
@ ( image_5842784325960735177et_nat
@ ^ [X2: set_set_nat] : ( image_nat_nat @ F @ ( B2 @ X2 ) )
@ A2 ) ) ) ) ) ) ).
% image_INT
thf(fact_1032_image__INT,axiom,
! [F: set_set_nat > set_nat,C: set_set_set_nat,A2: set_nat,B2: nat > set_set_set_nat,J: nat] :
( ( inj_on1894729867836481333et_nat @ F @ C )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_le9131159989063066194et_nat @ ( B2 @ X3 ) @ C ) )
=> ( ( member_nat @ J @ A2 )
=> ( ( image_5842784325960735177et_nat @ F @ ( comple8067742441731897515et_nat @ ( image_5738044413236618185et_nat @ B2 @ A2 ) ) )
= ( comple1065008630642458357et_nat
@ ( image_2194112158459175443et_nat
@ ^ [X2: nat] : ( image_5842784325960735177et_nat @ F @ ( B2 @ X2 ) )
@ A2 ) ) ) ) ) ) ).
% image_INT
thf(fact_1033_Inf__set__def,axiom,
( comple1065008630642458357et_nat
= ( ^ [A3: set_set_set_nat] :
( collect_set_nat
@ ^ [X2: set_nat] : ( complete_Inf_Inf_o @ ( image_set_set_nat_o @ ( member_set_nat @ X2 ) @ A3 ) ) ) ) ) ).
% Inf_set_def
thf(fact_1034_Inf__set__def,axiom,
( comple5153742063261271012et_nat
= ( ^ [A3: set_set_nat_set_nat] :
( collect_nat_set_nat
@ ^ [X2: nat > set_nat] : ( complete_Inf_Inf_o @ ( image_6034715890300748646_nat_o @ ( member_nat_set_nat @ X2 ) @ A3 ) ) ) ) ) ).
% Inf_set_def
thf(fact_1035_Inf__set__def,axiom,
( comple8067742441731897515et_nat
= ( ^ [A3: set_set_set_set_nat] :
( collect_set_set_nat
@ ^ [X2: set_set_nat] : ( complete_Inf_Inf_o @ ( image_3488003393078953823_nat_o @ ( member_set_set_nat @ X2 ) @ A3 ) ) ) ) ) ).
% Inf_set_def
thf(fact_1036_Inf__set__def,axiom,
( comple7806235888213564991et_nat
= ( ^ [A3: set_set_nat] :
( collect_nat
@ ^ [X2: nat] : ( complete_Inf_Inf_o @ ( image_set_nat_o @ ( member_nat @ X2 ) @ A3 ) ) ) ) ) ).
% Inf_set_def
thf(fact_1037_Sup_OSUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C: nat > nat,D2: nat > nat,Sup: set_nat > nat] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( Sup @ ( image_nat_nat @ C @ A2 ) )
= ( Sup @ ( image_nat_nat @ D2 @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_1038_Sup_OSUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C: nat > set_set_nat,D2: nat > set_set_nat,Sup: set_set_set_nat > set_set_nat] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( Sup @ ( image_2194112158459175443et_nat @ C @ A2 ) )
= ( Sup @ ( image_2194112158459175443et_nat @ D2 @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_1039_Sup_OSUP__cong,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C: set_set_nat > set_nat,D2: set_set_nat > set_nat,Sup: set_set_nat > set_nat] :
( ( A2 = B2 )
=> ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( Sup @ ( image_5842784325960735177et_nat @ C @ A2 ) )
= ( Sup @ ( image_5842784325960735177et_nat @ D2 @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_1040_Inf_OINF__cong,axiom,
! [A2: set_nat,B2: set_nat,C: nat > nat,D2: nat > nat,Inf: set_nat > nat] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( Inf @ ( image_nat_nat @ C @ A2 ) )
= ( Inf @ ( image_nat_nat @ D2 @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_1041_Inf_OINF__cong,axiom,
! [A2: set_nat,B2: set_nat,C: nat > set_set_nat,D2: nat > set_set_nat,Inf: set_set_set_nat > set_set_nat] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( Inf @ ( image_2194112158459175443et_nat @ C @ A2 ) )
= ( Inf @ ( image_2194112158459175443et_nat @ D2 @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_1042_Inf_OINF__cong,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C: set_set_nat > set_nat,D2: set_set_nat > set_nat,Inf: set_set_nat > set_nat] :
( ( A2 = B2 )
=> ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( Inf @ ( image_5842784325960735177et_nat @ C @ A2 ) )
= ( Inf @ ( image_5842784325960735177et_nat @ D2 @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_1043_Sup_OSUP__identity__eq,axiom,
! [Sup: set_nat > nat,A2: set_nat] :
( ( Sup
@ ( image_nat_nat
@ ^ [X2: nat] : X2
@ A2 ) )
= ( Sup @ A2 ) ) ).
% Sup.SUP_identity_eq
thf(fact_1044_Inf_OINF__identity__eq,axiom,
! [Inf: set_nat > nat,A2: set_nat] :
( ( Inf
@ ( image_nat_nat
@ ^ [X2: nat] : X2
@ A2 ) )
= ( Inf @ A2 ) ) ).
% Inf.INF_identity_eq
thf(fact_1045_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C4: nat] :
( ( ord_less_eq_nat @ A @ C4 )
& ( ord_less_eq_nat @ C4 @ B )
& ! [X4: nat] :
( ( ( ord_less_eq_nat @ A @ X4 )
& ( ord_less_nat @ X4 @ C4 ) )
=> ( P @ X4 ) )
& ! [D3: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A @ X3 )
& ( ord_less_nat @ X3 @ D3 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_nat @ D3 @ C4 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1046_Inf__eqI,axiom,
! [A2: set_nat_set_nat,X: nat > set_nat] :
( ! [I3: nat > set_nat] :
( ( member_nat_set_nat @ I3 @ A2 )
=> ( ord_le6195038898401538645et_nat @ X @ I3 ) )
=> ( ! [Y4: nat > set_nat] :
( ! [I4: nat > set_nat] :
( ( member_nat_set_nat @ I4 @ A2 )
=> ( ord_le6195038898401538645et_nat @ Y4 @ I4 ) )
=> ( ord_le6195038898401538645et_nat @ Y4 @ X ) )
=> ( ( comple6797894177231197998et_nat @ A2 )
= X ) ) ) ).
% Inf_eqI
thf(fact_1047_Inf__eqI,axiom,
! [A2: set_set_set_set_nat,X: set_set_set_nat] :
( ! [I3: set_set_set_nat] :
( ( member2946998982187404937et_nat @ I3 @ A2 )
=> ( ord_le9131159989063066194et_nat @ X @ I3 ) )
=> ( ! [Y4: set_set_set_nat] :
( ! [I4: set_set_set_nat] :
( ( member2946998982187404937et_nat @ I4 @ A2 )
=> ( ord_le9131159989063066194et_nat @ Y4 @ I4 ) )
=> ( ord_le9131159989063066194et_nat @ Y4 @ X ) )
=> ( ( comple8067742441731897515et_nat @ A2 )
= X ) ) ) ).
% Inf_eqI
thf(fact_1048_Inf__eqI,axiom,
! [A2: set_set_set_nat,X: set_set_nat] :
( ! [I3: set_set_nat] :
( ( member_set_set_nat @ I3 @ A2 )
=> ( ord_le6893508408891458716et_nat @ X @ I3 ) )
=> ( ! [Y4: set_set_nat] :
( ! [I4: set_set_nat] :
( ( member_set_set_nat @ I4 @ A2 )
=> ( ord_le6893508408891458716et_nat @ Y4 @ I4 ) )
=> ( ord_le6893508408891458716et_nat @ Y4 @ X ) )
=> ( ( comple1065008630642458357et_nat @ A2 )
= X ) ) ) ).
% Inf_eqI
thf(fact_1049_Inf__eqI,axiom,
! [A2: set_set_nat,X: set_nat] :
( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ A2 )
=> ( ord_less_eq_set_nat @ X @ I3 ) )
=> ( ! [Y4: set_nat] :
( ! [I4: set_nat] :
( ( member_set_nat @ I4 @ A2 )
=> ( ord_less_eq_set_nat @ Y4 @ I4 ) )
=> ( ord_less_eq_set_nat @ Y4 @ X ) )
=> ( ( comple7806235888213564991et_nat @ A2 )
= X ) ) ) ).
% Inf_eqI
thf(fact_1050_Inf__mono,axiom,
! [B2: set_nat_set_nat,A2: set_nat_set_nat] :
( ! [B6: nat > set_nat] :
( ( member_nat_set_nat @ B6 @ B2 )
=> ? [X4: nat > set_nat] :
( ( member_nat_set_nat @ X4 @ A2 )
& ( ord_le6195038898401538645et_nat @ X4 @ B6 ) ) )
=> ( ord_le6195038898401538645et_nat @ ( comple6797894177231197998et_nat @ A2 ) @ ( comple6797894177231197998et_nat @ B2 ) ) ) ).
% Inf_mono
thf(fact_1051_Inf__mono,axiom,
! [B2: set_set_set_set_nat,A2: set_set_set_set_nat] :
( ! [B6: set_set_set_nat] :
( ( member2946998982187404937et_nat @ B6 @ B2 )
=> ? [X4: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X4 @ A2 )
& ( ord_le9131159989063066194et_nat @ X4 @ B6 ) ) )
=> ( ord_le9131159989063066194et_nat @ ( comple8067742441731897515et_nat @ A2 ) @ ( comple8067742441731897515et_nat @ B2 ) ) ) ).
% Inf_mono
thf(fact_1052_Inf__mono,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ! [B6: set_set_nat] :
( ( member_set_set_nat @ B6 @ B2 )
=> ? [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ A2 )
& ( ord_le6893508408891458716et_nat @ X4 @ B6 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( comple1065008630642458357et_nat @ A2 ) @ ( comple1065008630642458357et_nat @ B2 ) ) ) ).
% Inf_mono
thf(fact_1053_Inf__mono,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ! [B6: set_nat] :
( ( member_set_nat @ B6 @ B2 )
=> ? [X4: set_nat] :
( ( member_set_nat @ X4 @ A2 )
& ( ord_less_eq_set_nat @ X4 @ B6 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A2 ) @ ( comple7806235888213564991et_nat @ B2 ) ) ) ).
% Inf_mono
thf(fact_1054_Inf__lower,axiom,
! [X: nat > set_nat,A2: set_nat_set_nat] :
( ( member_nat_set_nat @ X @ A2 )
=> ( ord_le6195038898401538645et_nat @ ( comple6797894177231197998et_nat @ A2 ) @ X ) ) ).
% Inf_lower
thf(fact_1055_Inf__lower,axiom,
! [X: set_set_set_nat,A2: set_set_set_set_nat] :
( ( member2946998982187404937et_nat @ X @ A2 )
=> ( ord_le9131159989063066194et_nat @ ( comple8067742441731897515et_nat @ A2 ) @ X ) ) ).
% Inf_lower
thf(fact_1056_Inf__lower,axiom,
! [X: set_set_nat,A2: set_set_set_nat] :
( ( member_set_set_nat @ X @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( comple1065008630642458357et_nat @ A2 ) @ X ) ) ).
% Inf_lower
thf(fact_1057_Inf__lower,axiom,
! [X: set_nat,A2: set_set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A2 ) @ X ) ) ).
% Inf_lower
thf(fact_1058_Inf__lower2,axiom,
! [U: nat > set_nat,A2: set_nat_set_nat,V2: nat > set_nat] :
( ( member_nat_set_nat @ U @ A2 )
=> ( ( ord_le6195038898401538645et_nat @ U @ V2 )
=> ( ord_le6195038898401538645et_nat @ ( comple6797894177231197998et_nat @ A2 ) @ V2 ) ) ) ).
% Inf_lower2
thf(fact_1059_Inf__lower2,axiom,
! [U: set_set_set_nat,A2: set_set_set_set_nat,V2: set_set_set_nat] :
( ( member2946998982187404937et_nat @ U @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ U @ V2 )
=> ( ord_le9131159989063066194et_nat @ ( comple8067742441731897515et_nat @ A2 ) @ V2 ) ) ) ).
% Inf_lower2
thf(fact_1060_Inf__lower2,axiom,
! [U: set_set_nat,A2: set_set_set_nat,V2: set_set_nat] :
( ( member_set_set_nat @ U @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ U @ V2 )
=> ( ord_le6893508408891458716et_nat @ ( comple1065008630642458357et_nat @ A2 ) @ V2 ) ) ) ).
% Inf_lower2
thf(fact_1061_Inf__lower2,axiom,
! [U: set_nat,A2: set_set_nat,V2: set_nat] :
( ( member_set_nat @ U @ A2 )
=> ( ( ord_less_eq_set_nat @ U @ V2 )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A2 ) @ V2 ) ) ) ).
% Inf_lower2
thf(fact_1062_le__Inf__iff,axiom,
! [B: set_set_set_nat,A2: set_set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B @ ( comple8067742441731897515et_nat @ A2 ) )
= ( ! [X2: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X2 @ A2 )
=> ( ord_le9131159989063066194et_nat @ B @ X2 ) ) ) ) ).
% le_Inf_iff
thf(fact_1063_le__Inf__iff,axiom,
! [B: set_set_nat,A2: set_set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ ( comple1065008630642458357et_nat @ A2 ) )
= ( ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A2 )
=> ( ord_le6893508408891458716et_nat @ B @ X2 ) ) ) ) ).
% le_Inf_iff
thf(fact_1064_le__Inf__iff,axiom,
! [B: set_nat,A2: set_set_nat] :
( ( ord_less_eq_set_nat @ B @ ( comple7806235888213564991et_nat @ A2 ) )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ B @ X2 ) ) ) ) ).
% le_Inf_iff
thf(fact_1065_Inf__greatest,axiom,
! [A2: set_nat_set_nat,Z: nat > set_nat] :
( ! [X3: nat > set_nat] :
( ( member_nat_set_nat @ X3 @ A2 )
=> ( ord_le6195038898401538645et_nat @ Z @ X3 ) )
=> ( ord_le6195038898401538645et_nat @ Z @ ( comple6797894177231197998et_nat @ A2 ) ) ) ).
% Inf_greatest
thf(fact_1066_Inf__greatest,axiom,
! [A2: set_set_set_set_nat,Z: set_set_set_nat] :
( ! [X3: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X3 @ A2 )
=> ( ord_le9131159989063066194et_nat @ Z @ X3 ) )
=> ( ord_le9131159989063066194et_nat @ Z @ ( comple8067742441731897515et_nat @ A2 ) ) ) ).
% Inf_greatest
thf(fact_1067_Inf__greatest,axiom,
! [A2: set_set_set_nat,Z: set_set_nat] :
( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A2 )
=> ( ord_le6893508408891458716et_nat @ Z @ X3 ) )
=> ( ord_le6893508408891458716et_nat @ Z @ ( comple1065008630642458357et_nat @ A2 ) ) ) ).
% Inf_greatest
thf(fact_1068_Inf__greatest,axiom,
! [A2: set_set_nat,Z: set_nat] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ Z @ X3 ) )
=> ( ord_less_eq_set_nat @ Z @ ( comple7806235888213564991et_nat @ A2 ) ) ) ).
% Inf_greatest
thf(fact_1069_cInf__eq__minimum,axiom,
! [Z: nat > set_nat,X5: set_nat_set_nat] :
( ( member_nat_set_nat @ Z @ X5 )
=> ( ! [X3: nat > set_nat] :
( ( member_nat_set_nat @ X3 @ X5 )
=> ( ord_le6195038898401538645et_nat @ Z @ X3 ) )
=> ( ( comple6797894177231197998et_nat @ X5 )
= Z ) ) ) ).
% cInf_eq_minimum
thf(fact_1070_cInf__eq__minimum,axiom,
! [Z: set_set_set_nat,X5: set_set_set_set_nat] :
( ( member2946998982187404937et_nat @ Z @ X5 )
=> ( ! [X3: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X3 @ X5 )
=> ( ord_le9131159989063066194et_nat @ Z @ X3 ) )
=> ( ( comple8067742441731897515et_nat @ X5 )
= Z ) ) ) ).
% cInf_eq_minimum
thf(fact_1071_cInf__eq__minimum,axiom,
! [Z: set_set_nat,X5: set_set_set_nat] :
( ( member_set_set_nat @ Z @ X5 )
=> ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ X5 )
=> ( ord_le6893508408891458716et_nat @ Z @ X3 ) )
=> ( ( comple1065008630642458357et_nat @ X5 )
= Z ) ) ) ).
% cInf_eq_minimum
thf(fact_1072_cInf__eq__minimum,axiom,
! [Z: nat,X5: set_nat] :
( ( member_nat @ Z @ X5 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X5 )
=> ( ord_less_eq_nat @ Z @ X3 ) )
=> ( ( complete_Inf_Inf_nat @ X5 )
= Z ) ) ) ).
% cInf_eq_minimum
thf(fact_1073_cInf__eq__minimum,axiom,
! [Z: set_nat,X5: set_set_nat] :
( ( member_set_nat @ Z @ X5 )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ X5 )
=> ( ord_less_eq_set_nat @ Z @ X3 ) )
=> ( ( comple7806235888213564991et_nat @ X5 )
= Z ) ) ) ).
% cInf_eq_minimum
thf(fact_1074_cInf__eq,axiom,
! [X5: set_nat,A: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ X5 )
=> ( ord_less_eq_nat @ A @ X3 ) )
=> ( ! [Y4: nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ X5 )
=> ( ord_less_eq_nat @ Y4 @ X4 ) )
=> ( ord_less_eq_nat @ Y4 @ A ) )
=> ( ( complete_Inf_Inf_nat @ X5 )
= A ) ) ) ).
% cInf_eq
thf(fact_1075_wellorder__Inf__le1,axiom,
! [K: nat,A2: set_nat] :
( ( member_nat @ K @ A2 )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ A2 ) @ K ) ) ).
% wellorder_Inf_le1
thf(fact_1076_INF__cong,axiom,
! [A2: set_nat,B2: set_nat,C: nat > nat,D2: nat > nat] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( complete_Inf_Inf_nat @ ( image_nat_nat @ C @ A2 ) )
= ( complete_Inf_Inf_nat @ ( image_nat_nat @ D2 @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_1077_INF__cong,axiom,
! [A2: set_nat,B2: set_nat,C: nat > set_set_nat,D2: nat > set_set_nat] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( comple1065008630642458357et_nat @ ( image_2194112158459175443et_nat @ C @ A2 ) )
= ( comple1065008630642458357et_nat @ ( image_2194112158459175443et_nat @ D2 @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_1078_INF__cong,axiom,
! [A2: set_nat,B2: set_nat,C: nat > set_nat,D2: nat > set_nat] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ C @ A2 ) )
= ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ D2 @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_1079_INF__cong,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_nat > set_nat,D2: set_nat > set_nat] :
( ( A2 = B2 )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( comple7806235888213564991et_nat @ ( image_7916887816326733075et_nat @ C @ A2 ) )
= ( comple7806235888213564991et_nat @ ( image_7916887816326733075et_nat @ D2 @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_1080_INF__cong,axiom,
! [A2: set_nat_set_nat,B2: set_nat_set_nat,C: ( nat > set_nat ) > set_nat,D2: ( nat > set_nat ) > set_nat] :
( ( A2 = B2 )
=> ( ! [X3: nat > set_nat] :
( ( member_nat_set_nat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( comple7806235888213564991et_nat @ ( image_8304670887732450946et_nat @ C @ A2 ) )
= ( comple7806235888213564991et_nat @ ( image_8304670887732450946et_nat @ D2 @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_1081_INF__cong,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C: set_set_nat > set_nat,D2: set_set_nat > set_nat] :
( ( A2 = B2 )
=> ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( comple7806235888213564991et_nat @ ( image_5842784325960735177et_nat @ C @ A2 ) )
= ( comple7806235888213564991et_nat @ ( image_5842784325960735177et_nat @ D2 @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_1082_Inter__lower,axiom,
! [B2: set_set_set_nat,A2: set_set_set_set_nat] :
( ( member2946998982187404937et_nat @ B2 @ A2 )
=> ( ord_le9131159989063066194et_nat @ ( comple8067742441731897515et_nat @ A2 ) @ B2 ) ) ).
% Inter_lower
thf(fact_1083_Inter__lower,axiom,
! [B2: set_set_nat,A2: set_set_set_nat] :
( ( member_set_set_nat @ B2 @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( comple1065008630642458357et_nat @ A2 ) @ B2 ) ) ).
% Inter_lower
thf(fact_1084_Inter__lower,axiom,
! [B2: set_nat,A2: set_set_nat] :
( ( member_set_nat @ B2 @ A2 )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A2 ) @ B2 ) ) ).
% Inter_lower
thf(fact_1085_Inter__greatest,axiom,
! [A2: set_set_set_set_nat,C: set_set_set_nat] :
( ! [X7: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X7 @ A2 )
=> ( ord_le9131159989063066194et_nat @ C @ X7 ) )
=> ( ord_le9131159989063066194et_nat @ C @ ( comple8067742441731897515et_nat @ A2 ) ) ) ).
% Inter_greatest
thf(fact_1086_Inter__greatest,axiom,
! [A2: set_set_set_nat,C: set_set_nat] :
( ! [X7: set_set_nat] :
( ( member_set_set_nat @ X7 @ A2 )
=> ( ord_le6893508408891458716et_nat @ C @ X7 ) )
=> ( ord_le6893508408891458716et_nat @ C @ ( comple1065008630642458357et_nat @ A2 ) ) ) ).
% Inter_greatest
thf(fact_1087_Inter__greatest,axiom,
! [A2: set_set_nat,C: set_nat] :
( ! [X7: set_nat] :
( ( member_set_nat @ X7 @ A2 )
=> ( ord_less_eq_set_nat @ C @ X7 ) )
=> ( ord_less_eq_set_nat @ C @ ( comple7806235888213564991et_nat @ A2 ) ) ) ).
% Inter_greatest
thf(fact_1088_INF__commute,axiom,
! [F: nat > nat > set_set_nat,B2: set_nat,A2: set_nat] :
( ( comple1065008630642458357et_nat
@ ( image_2194112158459175443et_nat
@ ^ [I2: nat] : ( comple1065008630642458357et_nat @ ( image_2194112158459175443et_nat @ ( F @ I2 ) @ B2 ) )
@ A2 ) )
= ( comple1065008630642458357et_nat
@ ( image_2194112158459175443et_nat
@ ^ [J2: nat] :
( comple1065008630642458357et_nat
@ ( image_2194112158459175443et_nat
@ ^ [I2: nat] : ( F @ I2 @ J2 )
@ A2 ) )
@ B2 ) ) ) ).
% INF_commute
thf(fact_1089_INF__commute,axiom,
! [F: set_set_nat > set_set_nat > set_nat,B2: set_set_set_nat,A2: set_set_set_nat] :
( ( comple7806235888213564991et_nat
@ ( image_5842784325960735177et_nat
@ ^ [I2: set_set_nat] : ( comple7806235888213564991et_nat @ ( image_5842784325960735177et_nat @ ( F @ I2 ) @ B2 ) )
@ A2 ) )
= ( comple7806235888213564991et_nat
@ ( image_5842784325960735177et_nat
@ ^ [J2: set_set_nat] :
( comple7806235888213564991et_nat
@ ( image_5842784325960735177et_nat
@ ^ [I2: set_set_nat] : ( F @ I2 @ J2 )
@ A2 ) )
@ B2 ) ) ) ).
% INF_commute
thf(fact_1090_INT__D,axiom,
! [B: nat,B2: nat > set_nat,A2: set_nat,A: nat] :
( ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
=> ( ( member_nat @ A @ A2 )
=> ( member_nat @ B @ ( B2 @ A ) ) ) ) ).
% INT_D
thf(fact_1091_INT__D,axiom,
! [B: set_nat,B2: nat > set_set_nat,A2: set_nat,A: nat] :
( ( member_set_nat @ B @ ( comple1065008630642458357et_nat @ ( image_2194112158459175443et_nat @ B2 @ A2 ) ) )
=> ( ( member_nat @ A @ A2 )
=> ( member_set_nat @ B @ ( B2 @ A ) ) ) ) ).
% INT_D
thf(fact_1092_INT__D,axiom,
! [B: nat,B2: set_nat > set_nat,A2: set_set_nat,A: set_nat] :
( ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_7916887816326733075et_nat @ B2 @ A2 ) ) )
=> ( ( member_set_nat @ A @ A2 )
=> ( member_nat @ B @ ( B2 @ A ) ) ) ) ).
% INT_D
thf(fact_1093_INT__D,axiom,
! [B: set_nat,B2: set_nat > set_set_nat,A2: set_set_nat,A: set_nat] :
( ( member_set_nat @ B @ ( comple1065008630642458357et_nat @ ( image_6725021117256019401et_nat @ B2 @ A2 ) ) )
=> ( ( member_set_nat @ A @ A2 )
=> ( member_set_nat @ B @ ( B2 @ A ) ) ) ) ).
% INT_D
thf(fact_1094_INT__D,axiom,
! [B: set_set_nat,B2: nat > set_set_set_nat,A2: set_nat,A: nat] :
( ( member_set_set_nat @ B @ ( comple8067742441731897515et_nat @ ( image_5738044413236618185et_nat @ B2 @ A2 ) ) )
=> ( ( member_nat @ A @ A2 )
=> ( member_set_set_nat @ B @ ( B2 @ A ) ) ) ) ).
% INT_D
thf(fact_1095_INT__D,axiom,
! [B: nat,B2: set_set_nat > set_nat,A2: set_set_set_nat,A: set_set_nat] :
( ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_5842784325960735177et_nat @ B2 @ A2 ) ) )
=> ( ( member_set_set_nat @ A @ A2 )
=> ( member_nat @ B @ ( B2 @ A ) ) ) ) ).
% INT_D
thf(fact_1096_INT__D,axiom,
! [B: set_nat,B2: set_set_nat > set_set_nat,A2: set_set_set_nat,A: set_set_nat] :
( ( member_set_nat @ B @ ( comple1065008630642458357et_nat @ ( image_7884819252390400639et_nat @ B2 @ A2 ) ) )
=> ( ( member_set_set_nat @ A @ A2 )
=> ( member_set_nat @ B @ ( B2 @ A ) ) ) ) ).
% INT_D
thf(fact_1097_INT__D,axiom,
! [B: nat > set_nat,B2: nat > set_nat_set_nat,A2: set_nat,A: nat] :
( ( member_nat_set_nat @ B @ ( comple5153742063261271012et_nat @ ( image_2803531558198256130et_nat @ B2 @ A2 ) ) )
=> ( ( member_nat @ A @ A2 )
=> ( member_nat_set_nat @ B @ ( B2 @ A ) ) ) ) ).
% INT_D
thf(fact_1098_INT__D,axiom,
! [B: set_set_nat,B2: set_nat > set_set_set_nat,A2: set_set_nat,A: set_nat] :
( ( member_set_set_nat @ B @ ( comple8067742441731897515et_nat @ ( image_4583741654806091647et_nat @ B2 @ A2 ) ) )
=> ( ( member_set_nat @ A @ A2 )
=> ( member_set_set_nat @ B @ ( B2 @ A ) ) ) ) ).
% INT_D
thf(fact_1099_INT__D,axiom,
! [B: nat,B2: ( nat > set_nat ) > set_nat,A2: set_nat_set_nat,A: nat > set_nat] :
( ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_8304670887732450946et_nat @ B2 @ A2 ) ) )
=> ( ( member_nat_set_nat @ A @ A2 )
=> ( member_nat @ B @ ( B2 @ A ) ) ) ) ).
% INT_D
thf(fact_1100_INT__E,axiom,
! [B: nat,B2: nat > set_nat,A2: set_nat,A: nat] :
( ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
=> ( ~ ( member_nat @ B @ ( B2 @ A ) )
=> ~ ( member_nat @ A @ A2 ) ) ) ).
% INT_E
thf(fact_1101_INT__E,axiom,
! [B: set_nat,B2: nat > set_set_nat,A2: set_nat,A: nat] :
( ( member_set_nat @ B @ ( comple1065008630642458357et_nat @ ( image_2194112158459175443et_nat @ B2 @ A2 ) ) )
=> ( ~ ( member_set_nat @ B @ ( B2 @ A ) )
=> ~ ( member_nat @ A @ A2 ) ) ) ).
% INT_E
thf(fact_1102_INT__E,axiom,
! [B: nat,B2: set_nat > set_nat,A2: set_set_nat,A: set_nat] :
( ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_7916887816326733075et_nat @ B2 @ A2 ) ) )
=> ( ~ ( member_nat @ B @ ( B2 @ A ) )
=> ~ ( member_set_nat @ A @ A2 ) ) ) ).
% INT_E
thf(fact_1103_INT__E,axiom,
! [B: set_nat,B2: set_nat > set_set_nat,A2: set_set_nat,A: set_nat] :
( ( member_set_nat @ B @ ( comple1065008630642458357et_nat @ ( image_6725021117256019401et_nat @ B2 @ A2 ) ) )
=> ( ~ ( member_set_nat @ B @ ( B2 @ A ) )
=> ~ ( member_set_nat @ A @ A2 ) ) ) ).
% INT_E
thf(fact_1104_INT__E,axiom,
! [B: set_set_nat,B2: nat > set_set_set_nat,A2: set_nat,A: nat] :
( ( member_set_set_nat @ B @ ( comple8067742441731897515et_nat @ ( image_5738044413236618185et_nat @ B2 @ A2 ) ) )
=> ( ~ ( member_set_set_nat @ B @ ( B2 @ A ) )
=> ~ ( member_nat @ A @ A2 ) ) ) ).
% INT_E
thf(fact_1105_INT__E,axiom,
! [B: nat,B2: set_set_nat > set_nat,A2: set_set_set_nat,A: set_set_nat] :
( ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_5842784325960735177et_nat @ B2 @ A2 ) ) )
=> ( ~ ( member_nat @ B @ ( B2 @ A ) )
=> ~ ( member_set_set_nat @ A @ A2 ) ) ) ).
% INT_E
thf(fact_1106_INT__E,axiom,
! [B: set_nat,B2: set_set_nat > set_set_nat,A2: set_set_set_nat,A: set_set_nat] :
( ( member_set_nat @ B @ ( comple1065008630642458357et_nat @ ( image_7884819252390400639et_nat @ B2 @ A2 ) ) )
=> ( ~ ( member_set_nat @ B @ ( B2 @ A ) )
=> ~ ( member_set_set_nat @ A @ A2 ) ) ) ).
% INT_E
thf(fact_1107_INT__E,axiom,
! [B: nat > set_nat,B2: nat > set_nat_set_nat,A2: set_nat,A: nat] :
( ( member_nat_set_nat @ B @ ( comple5153742063261271012et_nat @ ( image_2803531558198256130et_nat @ B2 @ A2 ) ) )
=> ( ~ ( member_nat_set_nat @ B @ ( B2 @ A ) )
=> ~ ( member_nat @ A @ A2 ) ) ) ).
% INT_E
thf(fact_1108_INT__E,axiom,
! [B: set_set_nat,B2: set_nat > set_set_set_nat,A2: set_set_nat,A: set_nat] :
( ( member_set_set_nat @ B @ ( comple8067742441731897515et_nat @ ( image_4583741654806091647et_nat @ B2 @ A2 ) ) )
=> ( ~ ( member_set_set_nat @ B @ ( B2 @ A ) )
=> ~ ( member_set_nat @ A @ A2 ) ) ) ).
% INT_E
thf(fact_1109_INT__E,axiom,
! [B: nat,B2: ( nat > set_nat ) > set_nat,A2: set_nat_set_nat,A: nat > set_nat] :
( ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_8304670887732450946et_nat @ B2 @ A2 ) ) )
=> ( ~ ( member_nat @ B @ ( B2 @ A ) )
=> ~ ( member_nat_set_nat @ A @ A2 ) ) ) ).
% INT_E
thf(fact_1110_INF__eq,axiom,
! [A2: set_nat,B2: set_nat,G: nat > set_nat,F: nat > set_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B2 )
& ( ord_less_eq_set_nat @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J3: nat] :
( ( member_nat @ J3 @ B2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( G @ J3 ) ) ) )
=> ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A2 ) )
= ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_1111_INF__eq,axiom,
! [A2: set_nat,B2: set_nat,G: nat > set_set_nat,F: nat > set_set_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B2 )
& ( ord_le6893508408891458716et_nat @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J3: nat] :
( ( member_nat @ J3 @ B2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_le6893508408891458716et_nat @ ( F @ X4 ) @ ( G @ J3 ) ) ) )
=> ( ( comple1065008630642458357et_nat @ ( image_2194112158459175443et_nat @ F @ A2 ) )
= ( comple1065008630642458357et_nat @ ( image_2194112158459175443et_nat @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_1112_INF__eq,axiom,
! [A2: set_nat,B2: set_set_nat,G: set_nat > set_nat,F: nat > set_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ? [X4: set_nat] :
( ( member_set_nat @ X4 @ B2 )
& ( ord_less_eq_set_nat @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J3: set_nat] :
( ( member_set_nat @ J3 @ B2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( G @ J3 ) ) ) )
=> ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A2 ) )
= ( comple7806235888213564991et_nat @ ( image_7916887816326733075et_nat @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_1113_INF__eq,axiom,
! [A2: set_set_nat,B2: set_nat,G: nat > set_nat,F: set_nat > set_nat] :
( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B2 )
& ( ord_less_eq_set_nat @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J3: nat] :
( ( member_nat @ J3 @ B2 )
=> ? [X4: set_nat] :
( ( member_set_nat @ X4 @ A2 )
& ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( G @ J3 ) ) ) )
=> ( ( comple7806235888213564991et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) )
= ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_1114_INF__eq,axiom,
! [A2: set_nat,B2: set_nat,G: nat > set_set_set_nat,F: nat > set_set_set_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B2 )
& ( ord_le9131159989063066194et_nat @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J3: nat] :
( ( member_nat @ J3 @ B2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_le9131159989063066194et_nat @ ( F @ X4 ) @ ( G @ J3 ) ) ) )
=> ( ( comple8067742441731897515et_nat @ ( image_5738044413236618185et_nat @ F @ A2 ) )
= ( comple8067742441731897515et_nat @ ( image_5738044413236618185et_nat @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_1115_INF__eq,axiom,
! [A2: set_nat,B2: set_set_nat,G: set_nat > set_set_nat,F: nat > set_set_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ? [X4: set_nat] :
( ( member_set_nat @ X4 @ B2 )
& ( ord_le6893508408891458716et_nat @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J3: set_nat] :
( ( member_set_nat @ J3 @ B2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_le6893508408891458716et_nat @ ( F @ X4 ) @ ( G @ J3 ) ) ) )
=> ( ( comple1065008630642458357et_nat @ ( image_2194112158459175443et_nat @ F @ A2 ) )
= ( comple1065008630642458357et_nat @ ( image_6725021117256019401et_nat @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_1116_INF__eq,axiom,
! [A2: set_set_nat,B2: set_nat,G: nat > set_set_nat,F: set_nat > set_set_nat] :
( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B2 )
& ( ord_le6893508408891458716et_nat @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J3: nat] :
( ( member_nat @ J3 @ B2 )
=> ? [X4: set_nat] :
( ( member_set_nat @ X4 @ A2 )
& ( ord_le6893508408891458716et_nat @ ( F @ X4 ) @ ( G @ J3 ) ) ) )
=> ( ( comple1065008630642458357et_nat @ ( image_6725021117256019401et_nat @ F @ A2 ) )
= ( comple1065008630642458357et_nat @ ( image_2194112158459175443et_nat @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_1117_INF__eq,axiom,
! [A2: set_nat,B2: set_set_set_nat,G: set_set_nat > set_nat,F: nat > set_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ? [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ B2 )
& ( ord_less_eq_set_nat @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J3: set_set_nat] :
( ( member_set_set_nat @ J3 @ B2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( G @ J3 ) ) ) )
=> ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A2 ) )
= ( comple7806235888213564991et_nat @ ( image_5842784325960735177et_nat @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_1118_INF__eq,axiom,
! [A2: set_set_nat,B2: set_set_nat,G: set_nat > set_nat,F: set_nat > set_nat] :
( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ A2 )
=> ? [X4: set_nat] :
( ( member_set_nat @ X4 @ B2 )
& ( ord_less_eq_set_nat @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J3: set_nat] :
( ( member_set_nat @ J3 @ B2 )
=> ? [X4: set_nat] :
( ( member_set_nat @ X4 @ A2 )
& ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( G @ J3 ) ) ) )
=> ( ( comple7806235888213564991et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) )
= ( comple7806235888213564991et_nat @ ( image_7916887816326733075et_nat @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_1119_INF__eq,axiom,
! [A2: set_set_set_nat,B2: set_nat,G: nat > set_nat,F: set_set_nat > set_nat] :
( ! [I3: set_set_nat] :
( ( member_set_set_nat @ I3 @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B2 )
& ( ord_less_eq_set_nat @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J3: nat] :
( ( member_nat @ J3 @ B2 )
=> ? [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ A2 )
& ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( G @ J3 ) ) ) )
=> ( ( comple7806235888213564991et_nat @ ( image_5842784325960735177et_nat @ F @ A2 ) )
= ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_1120_Inf__superset__mono,axiom,
! [B2: set_set_set_set_nat,A2: set_set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ B2 @ A2 )
=> ( ord_le9131159989063066194et_nat @ ( comple8067742441731897515et_nat @ A2 ) @ ( comple8067742441731897515et_nat @ B2 ) ) ) ).
% Inf_superset_mono
thf(fact_1121_Inf__superset__mono,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( comple1065008630642458357et_nat @ A2 ) @ ( comple1065008630642458357et_nat @ B2 ) ) ) ).
% Inf_superset_mono
thf(fact_1122_Inf__superset__mono,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A2 ) @ ( comple7806235888213564991et_nat @ B2 ) ) ) ).
% Inf_superset_mono
thf(fact_1123_cInf__le__finite,axiom,
! [X5: set_nat_set_nat,X: nat > set_nat] :
( ( finite722436868047473932et_nat @ X5 )
=> ( ( member_nat_set_nat @ X @ X5 )
=> ( ord_le6195038898401538645et_nat @ ( comple6797894177231197998et_nat @ X5 ) @ X ) ) ) ).
% cInf_le_finite
thf(fact_1124_cInf__le__finite,axiom,
! [X5: set_set_set_set_nat,X: set_set_set_nat] :
( ( finite5926941155766903689et_nat @ X5 )
=> ( ( member2946998982187404937et_nat @ X @ X5 )
=> ( ord_le9131159989063066194et_nat @ ( comple8067742441731897515et_nat @ X5 ) @ X ) ) ) ).
% cInf_le_finite
thf(fact_1125_cInf__le__finite,axiom,
! [X5: set_set_set_nat,X: set_set_nat] :
( ( finite6739761609112101331et_nat @ X5 )
=> ( ( member_set_set_nat @ X @ X5 )
=> ( ord_le6893508408891458716et_nat @ ( comple1065008630642458357et_nat @ X5 ) @ X ) ) ) ).
% cInf_le_finite
thf(fact_1126_cInf__le__finite,axiom,
! [X5: set_nat,X: nat] :
( ( finite_finite_nat @ X5 )
=> ( ( member_nat @ X @ X5 )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ X5 ) @ X ) ) ) ).
% cInf_le_finite
thf(fact_1127_cInf__le__finite,axiom,
! [X5: set_set_nat,X: set_nat] :
( ( finite1152437895449049373et_nat @ X5 )
=> ( ( member_set_nat @ X @ X5 )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ X5 ) @ X ) ) ) ).
% cInf_le_finite
thf(fact_1128_finite__imp__less__Inf,axiom,
! [X5: set_nat,X: nat,A: nat] :
( ( finite_finite_nat @ X5 )
=> ( ( member_nat @ X @ X5 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X5 )
=> ( ord_less_nat @ A @ X3 ) )
=> ( ord_less_nat @ A @ ( complete_Inf_Inf_nat @ X5 ) ) ) ) ) ).
% finite_imp_less_Inf
thf(fact_1129_Inter__anti__mono,axiom,
! [B2: set_set_set_set_nat,A2: set_set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ B2 @ A2 )
=> ( ord_le9131159989063066194et_nat @ ( comple8067742441731897515et_nat @ A2 ) @ ( comple8067742441731897515et_nat @ B2 ) ) ) ).
% Inter_anti_mono
thf(fact_1130_Inter__anti__mono,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( comple1065008630642458357et_nat @ A2 ) @ ( comple1065008630642458357et_nat @ B2 ) ) ) ).
% Inter_anti_mono
thf(fact_1131_Inter__anti__mono,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A2 ) @ ( comple7806235888213564991et_nat @ B2 ) ) ) ).
% Inter_anti_mono
thf(fact_1132_INF__greatest,axiom,
! [A2: set_nat,U: set_nat,F: nat > set_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ( ord_less_eq_set_nat @ U @ ( F @ I3 ) ) )
=> ( ord_less_eq_set_nat @ U @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ).
% INF_greatest
thf(fact_1133_INF__greatest,axiom,
! [A2: set_nat,U: set_set_nat,F: nat > set_set_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ( ord_le6893508408891458716et_nat @ U @ ( F @ I3 ) ) )
=> ( ord_le6893508408891458716et_nat @ U @ ( comple1065008630642458357et_nat @ ( image_2194112158459175443et_nat @ F @ A2 ) ) ) ) ).
% INF_greatest
thf(fact_1134_INF__greatest,axiom,
! [A2: set_set_nat,U: set_nat,F: set_nat > set_nat] :
( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ A2 )
=> ( ord_less_eq_set_nat @ U @ ( F @ I3 ) ) )
=> ( ord_less_eq_set_nat @ U @ ( comple7806235888213564991et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) ) ) ) ).
% INF_greatest
thf(fact_1135_INF__greatest,axiom,
! [A2: set_nat,U: set_set_set_nat,F: nat > set_set_set_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ( ord_le9131159989063066194et_nat @ U @ ( F @ I3 ) ) )
=> ( ord_le9131159989063066194et_nat @ U @ ( comple8067742441731897515et_nat @ ( image_5738044413236618185et_nat @ F @ A2 ) ) ) ) ).
% INF_greatest
thf(fact_1136_INF__greatest,axiom,
! [A2: set_set_nat,U: set_set_nat,F: set_nat > set_set_nat] :
( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ A2 )
=> ( ord_le6893508408891458716et_nat @ U @ ( F @ I3 ) ) )
=> ( ord_le6893508408891458716et_nat @ U @ ( comple1065008630642458357et_nat @ ( image_6725021117256019401et_nat @ F @ A2 ) ) ) ) ).
% INF_greatest
thf(fact_1137_INF__greatest,axiom,
! [A2: set_set_set_nat,U: set_nat,F: set_set_nat > set_nat] :
( ! [I3: set_set_nat] :
( ( member_set_set_nat @ I3 @ A2 )
=> ( ord_less_eq_set_nat @ U @ ( F @ I3 ) ) )
=> ( ord_less_eq_set_nat @ U @ ( comple7806235888213564991et_nat @ ( image_5842784325960735177et_nat @ F @ A2 ) ) ) ) ).
% INF_greatest
thf(fact_1138_INF__greatest,axiom,
! [A2: set_set_nat,U: set_set_set_nat,F: set_nat > set_set_set_nat] :
( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ A2 )
=> ( ord_le9131159989063066194et_nat @ U @ ( F @ I3 ) ) )
=> ( ord_le9131159989063066194et_nat @ U @ ( comple8067742441731897515et_nat @ ( image_4583741654806091647et_nat @ F @ A2 ) ) ) ) ).
% INF_greatest
thf(fact_1139_INF__greatest,axiom,
! [A2: set_set_set_nat,U: set_set_nat,F: set_set_nat > set_set_nat] :
( ! [I3: set_set_nat] :
( ( member_set_set_nat @ I3 @ A2 )
=> ( ord_le6893508408891458716et_nat @ U @ ( F @ I3 ) ) )
=> ( ord_le6893508408891458716et_nat @ U @ ( comple1065008630642458357et_nat @ ( image_7884819252390400639et_nat @ F @ A2 ) ) ) ) ).
% INF_greatest
thf(fact_1140_INF__greatest,axiom,
! [A2: set_nat_set_nat,U: set_nat,F: ( nat > set_nat ) > set_nat] :
( ! [I3: nat > set_nat] :
( ( member_nat_set_nat @ I3 @ A2 )
=> ( ord_less_eq_set_nat @ U @ ( F @ I3 ) ) )
=> ( ord_less_eq_set_nat @ U @ ( comple7806235888213564991et_nat @ ( image_8304670887732450946et_nat @ F @ A2 ) ) ) ) ).
% INF_greatest
thf(fact_1141_INF__greatest,axiom,
! [A2: set_set_set_nat,U: set_set_set_nat,F: set_set_nat > set_set_set_nat] :
( ! [I3: set_set_nat] :
( ( member_set_set_nat @ I3 @ A2 )
=> ( ord_le9131159989063066194et_nat @ U @ ( F @ I3 ) ) )
=> ( ord_le9131159989063066194et_nat @ U @ ( comple8067742441731897515et_nat @ ( image_8042862740799972405et_nat @ F @ A2 ) ) ) ) ).
% INF_greatest
thf(fact_1142_le__INF__iff,axiom,
! [U: set_set_nat,F: nat > set_set_nat,A2: set_nat] :
( ( ord_le6893508408891458716et_nat @ U @ ( comple1065008630642458357et_nat @ ( image_2194112158459175443et_nat @ F @ A2 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_le6893508408891458716et_nat @ U @ ( F @ X2 ) ) ) ) ) ).
% le_INF_iff
thf(fact_1143_le__INF__iff,axiom,
! [U: set_nat,F: set_set_nat > set_nat,A2: set_set_set_nat] :
( ( ord_less_eq_set_nat @ U @ ( comple7806235888213564991et_nat @ ( image_5842784325960735177et_nat @ F @ A2 ) ) )
= ( ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ U @ ( F @ X2 ) ) ) ) ) ).
% le_INF_iff
thf(fact_1144_INF__lower2,axiom,
! [I: nat,A2: set_nat,F: nat > set_nat,U: set_nat] :
( ( member_nat @ I @ A2 )
=> ( ( ord_less_eq_set_nat @ ( F @ I ) @ U )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ U ) ) ) ).
% INF_lower2
thf(fact_1145_INF__lower2,axiom,
! [I: nat,A2: set_nat,F: nat > set_set_nat,U: set_set_nat] :
( ( member_nat @ I @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ ( F @ I ) @ U )
=> ( ord_le6893508408891458716et_nat @ ( comple1065008630642458357et_nat @ ( image_2194112158459175443et_nat @ F @ A2 ) ) @ U ) ) ) ).
% INF_lower2
thf(fact_1146_INF__lower2,axiom,
! [I: set_nat,A2: set_set_nat,F: set_nat > set_nat,U: set_nat] :
( ( member_set_nat @ I @ A2 )
=> ( ( ord_less_eq_set_nat @ ( F @ I ) @ U )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) ) @ U ) ) ) ).
% INF_lower2
thf(fact_1147_INF__lower2,axiom,
! [I: nat,A2: set_nat,F: nat > set_set_set_nat,U: set_set_set_nat] :
( ( member_nat @ I @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ ( F @ I ) @ U )
=> ( ord_le9131159989063066194et_nat @ ( comple8067742441731897515et_nat @ ( image_5738044413236618185et_nat @ F @ A2 ) ) @ U ) ) ) ).
% INF_lower2
thf(fact_1148_INF__lower2,axiom,
! [I: set_nat,A2: set_set_nat,F: set_nat > set_set_nat,U: set_set_nat] :
( ( member_set_nat @ I @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ ( F @ I ) @ U )
=> ( ord_le6893508408891458716et_nat @ ( comple1065008630642458357et_nat @ ( image_6725021117256019401et_nat @ F @ A2 ) ) @ U ) ) ) ).
% INF_lower2
thf(fact_1149_INF__lower2,axiom,
! [I: set_set_nat,A2: set_set_set_nat,F: set_set_nat > set_nat,U: set_nat] :
( ( member_set_set_nat @ I @ A2 )
=> ( ( ord_less_eq_set_nat @ ( F @ I ) @ U )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_5842784325960735177et_nat @ F @ A2 ) ) @ U ) ) ) ).
% INF_lower2
thf(fact_1150_INF__lower2,axiom,
! [I: set_nat,A2: set_set_nat,F: set_nat > set_set_set_nat,U: set_set_set_nat] :
( ( member_set_nat @ I @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ ( F @ I ) @ U )
=> ( ord_le9131159989063066194et_nat @ ( comple8067742441731897515et_nat @ ( image_4583741654806091647et_nat @ F @ A2 ) ) @ U ) ) ) ).
% INF_lower2
thf(fact_1151_INF__lower2,axiom,
! [I: set_set_nat,A2: set_set_set_nat,F: set_set_nat > set_set_nat,U: set_set_nat] :
( ( member_set_set_nat @ I @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ ( F @ I ) @ U )
=> ( ord_le6893508408891458716et_nat @ ( comple1065008630642458357et_nat @ ( image_7884819252390400639et_nat @ F @ A2 ) ) @ U ) ) ) ).
% INF_lower2
thf(fact_1152_INF__lower2,axiom,
! [I: nat > set_nat,A2: set_nat_set_nat,F: ( nat > set_nat ) > set_nat,U: set_nat] :
( ( member_nat_set_nat @ I @ A2 )
=> ( ( ord_less_eq_set_nat @ ( F @ I ) @ U )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_8304670887732450946et_nat @ F @ A2 ) ) @ U ) ) ) ).
% INF_lower2
thf(fact_1153_INF__lower2,axiom,
! [I: set_set_nat,A2: set_set_set_nat,F: set_set_nat > set_set_set_nat,U: set_set_set_nat] :
( ( member_set_set_nat @ I @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ ( F @ I ) @ U )
=> ( ord_le9131159989063066194et_nat @ ( comple8067742441731897515et_nat @ ( image_8042862740799972405et_nat @ F @ A2 ) ) @ U ) ) ) ).
% INF_lower2
thf(fact_1154_INF__mono_H,axiom,
! [F: nat > set_set_nat,G: nat > set_set_nat,A2: set_nat] :
( ! [X3: nat] : ( ord_le6893508408891458716et_nat @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_le6893508408891458716et_nat @ ( comple1065008630642458357et_nat @ ( image_2194112158459175443et_nat @ F @ A2 ) ) @ ( comple1065008630642458357et_nat @ ( image_2194112158459175443et_nat @ G @ A2 ) ) ) ) ).
% INF_mono'
thf(fact_1155_INF__mono_H,axiom,
! [F: set_set_nat > set_nat,G: set_set_nat > set_nat,A2: set_set_set_nat] :
( ! [X3: set_set_nat] : ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_5842784325960735177et_nat @ F @ A2 ) ) @ ( comple7806235888213564991et_nat @ ( image_5842784325960735177et_nat @ G @ A2 ) ) ) ) ).
% INF_mono'
thf(fact_1156_INF__lower,axiom,
! [I: nat > set_nat,A2: set_nat_set_nat,F: ( nat > set_nat ) > set_nat] :
( ( member_nat_set_nat @ I @ A2 )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_8304670887732450946et_nat @ F @ A2 ) ) @ ( F @ I ) ) ) ).
% INF_lower
thf(fact_1157_INF__lower,axiom,
! [I: set_set_nat,A2: set_set_set_nat,F: set_set_nat > set_nat] :
( ( member_set_set_nat @ I @ A2 )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_5842784325960735177et_nat @ F @ A2 ) ) @ ( F @ I ) ) ) ).
% INF_lower
thf(fact_1158__092_060open_062Y_A_092_060noteq_062_A_123_125_092_060close_062,axiom,
y != bot_bo7198184520161983622et_nat ).
% \<open>Y \<noteq> {}\<close>
thf(fact_1159_Unempty,axiom,
u != bot_bo7198184520161983622et_nat ).
% Unempty
thf(fact_1160_plucking__step__def,axiom,
! [X5: set_set_set_nat] :
( ( clique4095374090462327202g_step @ p @ X5 )
= ( sup_su4213647025997063966et_nat
@ ( minus_2447799839930672331et_nat @ X5
@ ( collect_set_set_nat
@ ^ [E: set_set_nat] :
( ( member_set_set_nat @ E @ X5 )
& ( member_set_nat @ ( clique5033774636164728513irst_v @ E )
@ ( fChoice_set_set_nat
@ ^ [S5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ S5 @ ( clique8462013130872731469t_v_gs @ X5 ) )
& ( sunflower_nat @ S5 )
& ( ( finite_card_set_nat @ S5 )
= p ) ) ) ) ) ) )
@ ( insert_set_set_nat
@ ( clique6722202388162463298od_nat
@ ( comple7806235888213564991et_nat
@ ( fChoice_set_set_nat
@ ^ [S5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ S5 @ ( clique8462013130872731469t_v_gs @ X5 ) )
& ( sunflower_nat @ S5 )
& ( ( finite_card_set_nat @ S5 )
= p ) ) ) )
@ ( comple7806235888213564991et_nat
@ ( fChoice_set_set_nat
@ ^ [S5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ S5 @ ( clique8462013130872731469t_v_gs @ X5 ) )
& ( sunflower_nat @ S5 )
& ( ( finite_card_set_nat @ S5 )
= p ) ) ) ) )
@ bot_bo7198184520161983622et_nat ) ) ) ).
% plucking_step_def
thf(fact_1161_Snempty,axiom,
s != bot_bot_set_set_nat ).
% Snempty
thf(fact_1162_v__empty,axiom,
( ( clique5033774636164728513irst_v @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% v_empty
thf(fact_1163_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1164_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1165_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1166_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1167_v__gs__empty,axiom,
( ( clique8462013130872731469t_v_gs @ bot_bo7198184520161983622et_nat )
= bot_bot_set_set_nat ) ).
% v_gs_empty
thf(fact_1168_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1169_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1170_Y,axiom,
( y
= ( sup_su4213647025997063966et_nat @ ( minus_2447799839930672331et_nat @ x @ u ) @ ( insert_set_set_nat @ gs @ bot_bo7198184520161983622et_nat ) ) ) ).
% Y
thf(fact_1171_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1172__092_060open_062card_A_Iv__gs_AY_J_A_061_Acard_A_Iv__gs_A_IX_A_N_AU_A_092_060union_062_A_123Gs_125_J_J_092_060close_062,axiom,
( ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ y ) )
= ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ ( sup_su4213647025997063966et_nat @ ( minus_2447799839930672331et_nat @ x @ u ) @ ( insert_set_set_nat @ gs @ bot_bo7198184520161983622et_nat ) ) ) ) ) ).
% \<open>card (v_gs Y) = card (v_gs (X - U \<union> {Gs}))\<close>
thf(fact_1173_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1174_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_1175_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1176_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1177_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_1178_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1179_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_1180_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_1181_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_1182_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1183_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1184_atLeastLessThan0,axiom,
! [M: nat] :
( ( set_or4665077453230672383an_nat @ M @ zero_zero_nat )
= bot_bot_set_nat ) ).
% atLeastLessThan0
thf(fact_1185_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1186__092_060open_062v__gs_A_IX_A_N_AU_A_092_060union_062_A_123Gs_125_J_A_061_Av__gs_A_IX_A_N_AU_J_A_092_060union_062_Av__gs_A_123Gs_125_092_060close_062,axiom,
( ( clique8462013130872731469t_v_gs @ ( sup_su4213647025997063966et_nat @ ( minus_2447799839930672331et_nat @ x @ u ) @ ( insert_set_set_nat @ gs @ bot_bo7198184520161983622et_nat ) ) )
= ( sup_sup_set_set_nat @ ( clique8462013130872731469t_v_gs @ ( minus_2447799839930672331et_nat @ x @ u ) ) @ ( clique8462013130872731469t_v_gs @ ( insert_set_set_nat @ gs @ bot_bo7198184520161983622et_nat ) ) ) ) ).
% \<open>v_gs (X - U \<union> {Gs}) = v_gs (X - U) \<union> v_gs {Gs}\<close>
thf(fact_1187_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1188_v__union,axiom,
! [G2: set_set_nat,H3: set_set_nat] :
( ( clique5033774636164728513irst_v @ ( sup_sup_set_set_nat @ G2 @ H3 ) )
= ( sup_sup_set_nat @ ( clique5033774636164728513irst_v @ G2 ) @ ( clique5033774636164728513irst_v @ H3 ) ) ) ).
% v_union
thf(fact_1189_v__gs__union,axiom,
! [X5: set_set_set_nat,Y2: set_set_set_nat] :
( ( clique8462013130872731469t_v_gs @ ( sup_su4213647025997063966et_nat @ X5 @ Y2 ) )
= ( sup_sup_set_set_nat @ ( clique8462013130872731469t_v_gs @ X5 ) @ ( clique8462013130872731469t_v_gs @ Y2 ) ) ) ).
% v_gs_union
thf(fact_1190__092_060open_062v__gs_A_123Gs_125_A_061_A_123v_AGs_125_092_060close_062,axiom,
( ( clique8462013130872731469t_v_gs @ ( insert_set_set_nat @ gs @ bot_bo7198184520161983622et_nat ) )
= ( insert_set_nat @ ( clique5033774636164728513irst_v @ gs ) @ bot_bot_set_set_nat ) ) ).
% \<open>v_gs {Gs} = {v Gs}\<close>
thf(fact_1191_image__minus__const__atLeastLessThan__nat,axiom,
! [C2: nat,Y: nat,X: nat] :
( ( ( ord_less_nat @ C2 @ Y )
=> ( ( image_nat_nat
@ ^ [I2: nat] : ( minus_minus_nat @ I2 @ C2 )
@ ( set_or4665077453230672383an_nat @ X @ Y ) )
= ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X @ C2 ) @ ( minus_minus_nat @ Y @ C2 ) ) ) )
& ( ~ ( ord_less_nat @ C2 @ Y )
=> ( ( ( ord_less_nat @ X @ Y )
=> ( ( image_nat_nat
@ ^ [I2: nat] : ( minus_minus_nat @ I2 @ C2 )
@ ( set_or4665077453230672383an_nat @ X @ Y ) )
= ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
& ( ~ ( ord_less_nat @ X @ Y )
=> ( ( image_nat_nat
@ ^ [I2: nat] : ( minus_minus_nat @ I2 @ C2 )
@ ( set_or4665077453230672383an_nat @ X @ Y ) )
= bot_bot_set_nat ) ) ) ) ) ).
% image_minus_const_atLeastLessThan_nat
thf(fact_1192_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1193_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1194_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1195_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1196__092_060open_062card_A_Iv__gs_A_IX_A_N_AU_J_A_092_060union_062_A_123v_AGs_125_J_A_092_060le_062_Acard_A_Iv__gs_A_IX_A_N_AU_J_J_A_L_Acard_A_123v_AGs_125_092_060close_062,axiom,
ord_less_eq_nat @ ( finite_card_set_nat @ ( sup_sup_set_set_nat @ ( clique8462013130872731469t_v_gs @ ( minus_2447799839930672331et_nat @ x @ u ) ) @ ( insert_set_nat @ ( clique5033774636164728513irst_v @ gs ) @ bot_bot_set_set_nat ) ) ) @ ( plus_plus_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ ( minus_2447799839930672331et_nat @ x @ u ) ) ) @ ( finite_card_set_nat @ ( insert_set_nat @ ( clique5033774636164728513irst_v @ gs ) @ bot_bot_set_set_nat ) ) ) ).
% \<open>card (v_gs (X - U) \<union> {v Gs}) \<le> card (v_gs (X - U)) + card {v Gs}\<close>
thf(fact_1197_first__assumptions_Oplucking__step__def,axiom,
! [L: nat,P3: nat,K: nat,X5: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( clique4095374090462327202g_step @ P3 @ X5 )
= ( sup_su4213647025997063966et_nat
@ ( minus_2447799839930672331et_nat @ X5
@ ( collect_set_set_nat
@ ^ [E: set_set_nat] :
( ( member_set_set_nat @ E @ X5 )
& ( member_set_nat @ ( clique5033774636164728513irst_v @ E )
@ ( fChoice_set_set_nat
@ ^ [S5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ S5 @ ( clique8462013130872731469t_v_gs @ X5 ) )
& ( sunflower_nat @ S5 )
& ( ( finite_card_set_nat @ S5 )
= P3 ) ) ) ) ) ) )
@ ( insert_set_set_nat
@ ( clique6722202388162463298od_nat
@ ( comple7806235888213564991et_nat
@ ( fChoice_set_set_nat
@ ^ [S5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ S5 @ ( clique8462013130872731469t_v_gs @ X5 ) )
& ( sunflower_nat @ S5 )
& ( ( finite_card_set_nat @ S5 )
= P3 ) ) ) )
@ ( comple7806235888213564991et_nat
@ ( fChoice_set_set_nat
@ ^ [S5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ S5 @ ( clique8462013130872731469t_v_gs @ X5 ) )
& ( sunflower_nat @ S5 )
& ( ( finite_card_set_nat @ S5 )
= P3 ) ) ) ) )
@ bot_bo7198184520161983622et_nat ) ) ) ) ).
% first_assumptions.plucking_step_def
thf(fact_1198_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1199_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1200_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1201_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1202_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1203_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1204_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1205_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1206_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1207_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_1208_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_1209_atLeastLessThan__add__Un,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( set_or4665077453230672383an_nat @ I @ ( plus_plus_nat @ J @ K ) )
= ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% atLeastLessThan_add_Un
thf(fact_1210_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1211_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1212_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1213_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M5: nat,N2: nat] :
( ( ord_less_nat @ M5 @ N2 )
=> ( ord_less_nat @ ( F @ M5 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1214_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1215_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1216_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1217_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1218_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1219_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1220_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1221_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_1222_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1223_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1224_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_1225_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_1226_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_1227_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1228_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1229_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1230_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1231_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_1232_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_1233_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_1234_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N4: nat] :
? [K2: nat] :
( N4
= ( plus_plus_nat @ M3 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1235_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_1236_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_1237_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1238_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1239_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_1240_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1241_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_1242_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1243_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1244_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1245_first__assumptions_Ov__union,axiom,
! [L: nat,P3: nat,K: nat,G2: set_set_nat,H3: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( clique5033774636164728513irst_v @ ( sup_sup_set_set_nat @ G2 @ H3 ) )
= ( sup_sup_set_nat @ ( clique5033774636164728513irst_v @ G2 ) @ ( clique5033774636164728513irst_v @ H3 ) ) ) ) ).
% first_assumptions.v_union
thf(fact_1246_first__assumptions_OacceptsI,axiom,
! [L: nat,P3: nat,K: nat,D2: set_set_nat,G2: set_set_nat,X5: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( ord_le6893508408891458716et_nat @ D2 @ G2 )
=> ( ( member_set_set_nat @ D2 @ X5 )
=> ( clique3686358387679108662ccepts @ X5 @ G2 ) ) ) ) ).
% first_assumptions.acceptsI
thf(fact_1247_first__assumptions_Oaccepts__def,axiom,
! [L: nat,P3: nat,K: nat,X5: set_set_set_nat,G2: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( clique3686358387679108662ccepts @ X5 @ G2 )
= ( ? [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ X5 )
& ( ord_le6893508408891458716et_nat @ X2 @ G2 ) ) ) ) ) ).
% first_assumptions.accepts_def
thf(fact_1248_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D4: nat] :
( ( A
= ( plus_plus_nat @ B @ D4 ) )
=> ( P @ D4 ) ) ) ) ).
% nat_diff_split
thf(fact_1249_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D4: nat] :
( ( A
= ( plus_plus_nat @ B @ D4 ) )
& ~ ( P @ D4 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1250_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1251_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1252_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_1253_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1254_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_1255_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1256_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_1257_first__assumptions_Ov__gs__mono,axiom,
! [L: nat,P3: nat,K: nat,X5: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( ord_le9131159989063066194et_nat @ X5 @ Y2 )
=> ( ord_le6893508408891458716et_nat @ ( clique8462013130872731469t_v_gs @ X5 ) @ ( clique8462013130872731469t_v_gs @ Y2 ) ) ) ) ).
% first_assumptions.v_gs_mono
thf(fact_1258_first__assumptions_Ov__mono,axiom,
! [L: nat,P3: nat,K: nat,G2: set_set_nat,H3: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( ord_le6893508408891458716et_nat @ G2 @ H3 )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ G2 ) @ ( clique5033774636164728513irst_v @ H3 ) ) ) ) ).
% first_assumptions.v_mono
thf(fact_1259_first__assumptions_Ov__empty,axiom,
! [L: nat,P3: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( clique5033774636164728513irst_v @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ) ).
% first_assumptions.v_empty
thf(fact_1260_first__assumptions_Ov__gs__empty,axiom,
! [L: nat,P3: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( clique8462013130872731469t_v_gs @ bot_bo7198184520161983622et_nat )
= bot_bot_set_set_nat ) ) ).
% first_assumptions.v_gs_empty
thf(fact_1261_subset__card__intvl__is__intvl,axiom,
! [A2: set_nat,K: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A2 ) ) ) )
=> ( A2
= ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A2 ) ) ) ) ) ).
% subset_card_intvl_is_intvl
thf(fact_1262_first__assumptions_Ov__gs__union,axiom,
! [L: nat,P3: nat,K: nat,X5: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( clique8462013130872731469t_v_gs @ ( sup_su4213647025997063966et_nat @ X5 @ Y2 ) )
= ( sup_sup_set_set_nat @ ( clique8462013130872731469t_v_gs @ X5 ) @ ( clique8462013130872731469t_v_gs @ Y2 ) ) ) ) ).
% first_assumptions.v_gs_union
thf(fact_1263_first__assumptions_Ov__gs__def,axiom,
! [L: nat,P3: nat,K: nat,X5: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( clique8462013130872731469t_v_gs @ X5 )
= ( image_5842784325960735177et_nat @ clique5033774636164728513irst_v @ X5 ) ) ) ).
% first_assumptions.v_gs_def
thf(fact_1264_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_1265_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1266__092_060open_062card_A_Iv__gs_A_IX_A_N_AU_J_J_A_L_Acard_A_123v_AGs_125_A_092_060le_062_Acard_A_Iv__gs_A_IX_A_N_AU_J_J_A_L_A1_092_060close_062,axiom,
ord_less_eq_nat @ ( plus_plus_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ ( minus_2447799839930672331et_nat @ x @ u ) ) ) @ ( finite_card_set_nat @ ( insert_set_nat @ ( clique5033774636164728513irst_v @ gs ) @ bot_bot_set_set_nat ) ) ) @ ( plus_plus_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ ( minus_2447799839930672331et_nat @ x @ u ) ) ) @ one_one_nat ) ).
% \<open>card (v_gs (X - U)) + card {v Gs} \<le> card (v_gs (X - U)) + 1\<close>
thf(fact_1267__092_060open_062card_A_Iv__gs_AY_J_A_092_060le_062_Acard_A_Iv__gs_AX_J_A_N_Ap_A_L_A1_092_060close_062,axiom,
ord_less_eq_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ y ) ) @ ( plus_plus_nat @ ( minus_minus_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ x ) ) @ p ) @ one_one_nat ) ).
% \<open>card (v_gs Y) \<le> card (v_gs X) - p + 1\<close>
% Helper facts (7)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_fChoice_1_1_fChoice_001t__Nat__Onat_T,axiom,
! [P: nat > $o] :
( ( P @ ( fChoice_nat @ P ) )
= ( ? [X6: nat] : ( P @ X6 ) ) ) ).
thf(help_fChoice_1_1_fChoice_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [P: set_nat > $o] :
( ( P @ ( fChoice_set_nat @ P ) )
= ( ? [X6: set_nat] : ( P @ X6 ) ) ) ).
thf(help_fChoice_1_1_fChoice_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_T,axiom,
! [P: set_set_nat > $o] :
( ( P @ ( fChoice_set_set_nat @ P ) )
= ( ? [X6: set_set_nat] : ( P @ X6 ) ) ) ).
thf(help_fChoice_1_1_fChoice_001_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J_T,axiom,
! [P: ( nat > set_nat ) > $o] :
( ( P @ ( fChoice_nat_set_nat @ P ) )
= ( ? [X6: nat > set_nat] : ( P @ X6 ) ) ) ).
% Conjectures (2)
thf(conj_0,hypothesis,
! [I4: nat] :
( ( ord_less_nat @ I4 @ p )
=> ( ( v
= ( fstt @ ( e @ I4 ) ) )
=> thesis ) ) ).
thf(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------