TPTP Problem File: SLH0357^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Safe_Range_RC/0021_Relational_Calculus/prob_01729_065852__17542468_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1594 ( 430 unt; 323 typ; 0 def)
% Number of atoms : 3856 ( 972 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 10628 ( 263 ~; 42 |; 209 &;8274 @)
% ( 0 <=>;1840 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 56 ( 55 usr)
% Number of type conns : 1118 (1118 >; 0 *; 0 +; 0 <<)
% Number of symbols : 271 ( 268 usr; 52 con; 0-4 aty)
% Number of variables : 3105 ( 113 ^;2863 !; 129 ?;3105 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:26:23.695
%------------------------------------------------------------------------------
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top_to9085566620183473993m0_nat: set_Pr4875302081019572377m0_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Numeral____Type__Onum0_Mt__Numeral____Type__Onum0_J_J,type,
top_to2553066108271082967l_num0: set_Pr5812241219738026119l_num0 ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Numeral____Type__Onum0_Mt__Numeral____Type__Onum1_J_J,type,
top_to2624100148317428952l_num1: set_Pr5883275259784372104l_num1 ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Numeral____Type__Onum0_Mt__Product____Type__Ounit_J_J,type,
top_to886564821216933632t_unit: set_Pr4145739932683876784t_unit ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Numeral____Type__Onum0_Mt__String__Oliteral_J_J,type,
top_to7655489177542131015iteral: set_Pr7950633868761106423iteral ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Ounit_J,type,
top_to1996260823553986621t_unit: set_Product_unit ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
top_top_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Numeral____Type__Onum0_J_J,type,
top_to3433798952193014772l_num0: set_set_Numeral_num0 ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Numeral____Type__Onum1_J_J,type,
top_to3504832992239360757l_num1: set_set_Numeral_num1 ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Product____Type__Ounit_J_J,type,
top_to1767297665138865437t_unit: set_set_Product_unit ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__String__Oliteral_J_J,type,
top_to5694933271948605156iteral: set_set_literal ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__Oliteral_J,type,
top_top_set_literal: set_literal ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
top_to6661820994512907621at_nat: set_Sum_sum_nat_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Numeral____Type__Onum0_J_J,type,
top_to7131751369954024123l_num0: set_Su9206079544978634091l_num0 ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Numeral____Type__Onum1_J_J,type,
top_to7202785410000370108l_num1: set_Su53741548170204268l_num1 ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Product____Type__Ounit_J_J,type,
top_to5465250082899874788t_unit: set_Su7539578257924484756t_unit ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__String__Oliteral_J_J,type,
top_to148093990134820907iteral: set_Su4835947104832017115iteral ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Numeral____Type__Onum0_Mt__Nat__Onat_J_J,type,
top_to6006073747853177493m0_nat: set_Su8080401922877787461m0_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Numeral____Type__Onum0_Mt__Numeral____Type__Onum0_J_J,type,
top_to3489177893006913931l_num0: set_Su4827871808400282331l_num0 ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Numeral____Type__Onum0_Mt__Numeral____Type__Onum1_J_J,type,
top_to3560211933053259916l_num1: set_Su4898905848446628316l_num1 ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Numeral____Type__Onum0_Mt__Product____Type__Ounit_J_J,type,
top_to1822676605952764596t_unit: set_Su3161370521346132996t_unit ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Numeral____Type__Onum0_Mt__String__Oliteral_J_J,type,
top_to5151745856415547131iteral: set_Su6928962721346476107iteral ).
thf(sy_c_Relational__Calculus_Oadom_001tf__b_001tf__a,type,
relational_adom_b_a: ( product_prod_b_nat > set_list_a ) > set_a ).
thf(sy_c_Relational__Calculus_Oap_001tf__a_001tf__b,type,
relational_ap_a_b: relational_fmla_a_b > $o ).
thf(sy_c_Relational__Calculus_Ocsts_001tf__a_001tf__b,type,
relational_csts_a_b: relational_fmla_a_b > set_a ).
thf(sy_c_Relational__Calculus_Oequiv_001tf__a_001tf__b,type,
relational_equiv_a_b: relational_fmla_a_b > relational_fmla_a_b > $o ).
thf(sy_c_Relational__Calculus_Oeval_001tf__a_001tf__b,type,
relational_eval_a_b: relational_fmla_a_b > ( product_prod_b_nat > set_list_a ) > set_list_a ).
thf(sy_c_Relational__Calculus_Oeval__on_001tf__a_001tf__b,type,
relati8814510239606734169on_a_b: set_nat > relational_fmla_a_b > ( product_prod_b_nat > set_list_a ) > set_list_a ).
thf(sy_c_Relational__Calculus_Ofmla_ODisj_001tf__a_001tf__b,type,
relational_Disj_a_b: relational_fmla_a_b > relational_fmla_a_b > relational_fmla_a_b ).
thf(sy_c_Relational__Calculus_Ofv_001tf__a_001tf__b,type,
relational_fv_a_b: relational_fmla_a_b > set_nat ).
thf(sy_c_Relational__Calculus_Oqp_001tf__a_001tf__b,type,
relational_qp_a_b: relational_fmla_a_b > $o ).
thf(sy_c_Relational__Calculus_Osat_001tf__a_001tf__b,type,
relational_sat_a_b: relational_fmla_a_b > ( product_prod_b_nat > set_list_a ) > ( nat > a ) > $o ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Numeral____Type__Onum0,type,
collect_Numeral_num0: ( numeral_num0 > $o ) > set_Numeral_num0 ).
thf(sy_c_Set_OCollect_001t__Numeral____Type__Onum1,type,
collect_Numeral_num1: ( numeral_num1 > $o ) > set_Numeral_num1 ).
thf(sy_c_Set_OCollect_001t__Product____Type__Ounit,type,
collect_Product_unit: ( product_unit > $o ) > set_Product_unit ).
thf(sy_c_Set_OCollect_001t__String__Oliteral,type,
collect_literal: ( literal > $o ) > set_literal ).
thf(sy_c_Set_OPow_001t__Nat__Onat,type,
pow_nat: set_nat > set_set_nat ).
thf(sy_c_Set_OPow_001t__Numeral____Type__Onum0,type,
pow_Numeral_num0: set_Numeral_num0 > set_set_Numeral_num0 ).
thf(sy_c_Set_OPow_001t__Numeral____Type__Onum1,type,
pow_Numeral_num1: set_Numeral_num1 > set_set_Numeral_num1 ).
thf(sy_c_Set_OPow_001t__Product____Type__Ounit,type,
pow_Product_unit: set_Product_unit > set_set_Product_unit ).
thf(sy_c_Set_OPow_001t__String__Oliteral,type,
pow_literal: set_literal > set_set_literal ).
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__Numeral____Type__Onum0,type,
image_5550796612950789325l_num0: ( nat > numeral_num0 ) > set_nat > set_Numeral_num0 ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Numeral____Type__Onum1,type,
image_5550796612950789326l_num1: ( nat > numeral_num1 ) > set_nat > set_Numeral_num1 ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Product____Type__Ounit,type,
image_8730104196221521654t_unit: ( nat > product_unit ) > set_nat > set_Product_unit ).
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__String__Oliteral,type,
image_nat_literal: ( nat > literal ) > set_nat > set_literal ).
thf(sy_c_Set_Oimage_001t__Numeral____Type__Onum0_001t__Nat__Onat,type,
image_8797574156932312687m0_nat: ( numeral_num0 > nat ) > set_Numeral_num0 > set_nat ).
thf(sy_c_Set_Oimage_001t__Numeral____Type__Onum0_001t__Numeral____Type__Onum0,type,
image_2832974300507296261l_num0: ( numeral_num0 > numeral_num0 ) > set_Numeral_num0 > set_Numeral_num0 ).
thf(sy_c_Set_Oimage_001t__Numeral____Type__Onum0_001t__Numeral____Type__Onum1,type,
image_2832974300507296262l_num1: ( numeral_num0 > numeral_num1 ) > set_Numeral_num0 > set_Numeral_num1 ).
thf(sy_c_Set_Oimage_001t__Numeral____Type__Onum0_001t__Product____Type__Ounit,type,
image_6012281883778028590t_unit: ( numeral_num0 > product_unit ) > set_Numeral_num0 > set_Product_unit ).
thf(sy_c_Set_Oimage_001t__Numeral____Type__Onum0_001t__String__Oliteral,type,
image_8737817577461598069iteral: ( numeral_num0 > literal ) > set_Numeral_num0 > set_literal ).
thf(sy_c_Set_Oimage_001t__Numeral____Type__Onum1_001t__Nat__Onat,type,
image_809646449033931376m1_nat: ( numeral_num1 > nat ) > set_Numeral_num1 > set_nat ).
thf(sy_c_Set_Oimage_001t__Numeral____Type__Onum1_001t__Numeral____Type__Onum0,type,
image_6783865936125884740l_num0: ( numeral_num1 > numeral_num0 ) > set_Numeral_num1 > set_Numeral_num0 ).
thf(sy_c_Set_Oimage_001t__Numeral____Type__Onum1_001t__Numeral____Type__Onum1,type,
image_6783865936125884741l_num1: ( numeral_num1 > numeral_num1 ) > set_Numeral_num1 > set_Numeral_num1 ).
thf(sy_c_Set_Oimage_001t__Numeral____Type__Onum1_001t__String__Oliteral,type,
image_5852747068178070836iteral: ( numeral_num1 > literal ) > set_Numeral_num1 > set_literal ).
thf(sy_c_Set_Oimage_001t__Product____Type__Ounit_001t__Nat__Onat,type,
image_875570014554754200it_nat: ( product_unit > nat ) > set_Product_unit > set_nat ).
thf(sy_c_Set_Oimage_001t__Product____Type__Ounit_001t__Numeral____Type__Onum0,type,
image_6449127158079674652l_num0: ( product_unit > numeral_num0 ) > set_Product_unit > set_Numeral_num0 ).
thf(sy_c_Set_Oimage_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
image_405062704495631173t_unit: ( product_unit > product_unit ) > set_Product_unit > set_Product_unit ).
thf(sy_c_Set_Oimage_001t__Product____Type__Ounit_001t__String__Oliteral,type,
image_5876984745897992460iteral: ( product_unit > literal ) > set_Product_unit > set_literal ).
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__String__Oliteral_001t__Nat__Onat,type,
image_literal_nat: ( literal > nat ) > set_literal > set_nat ).
thf(sy_c_Set_Oimage_001t__String__Oliteral_001t__Numeral____Type__Onum0,type,
image_3608546570274595605l_num0: ( literal > numeral_num0 ) > set_literal > set_Numeral_num0 ).
thf(sy_c_Set_Oimage_001t__String__Oliteral_001t__Numeral____Type__Onum1,type,
image_3608546570274595606l_num1: ( literal > numeral_num1 ) > set_literal > set_Numeral_num1 ).
thf(sy_c_Set_Oimage_001t__String__Oliteral_001t__Product____Type__Ounit,type,
image_6787854153545327934t_unit: ( literal > product_unit ) > set_literal > set_Product_unit ).
thf(sy_c_Set_Oimage_001t__String__Oliteral_001t__String__Oliteral,type,
image_8195128725298311301iteral: ( literal > literal ) > set_literal > set_literal ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Nat__Onat,type,
vimage_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Numeral____Type__Onum0,type,
vimage1705183035017847703l_num0: ( nat > numeral_num0 ) > set_Numeral_num0 > set_nat ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Numeral____Type__Onum1,type,
vimage1705183035017847704l_num1: ( nat > numeral_num1 ) > set_Numeral_num1 > set_nat ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Product____Type__Ounit,type,
vimage4884490618288580032t_unit: ( nat > product_unit ) > set_Product_unit > set_nat ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__String__Oliteral,type,
vimage_nat_literal: ( nat > literal ) > set_literal > set_nat ).
thf(sy_c_Set_Ovimage_001t__Numeral____Type__Onum0_001t__Nat__Onat,type,
vimage4951960578999371065m0_nat: ( numeral_num0 > nat ) > set_nat > set_Numeral_num0 ).
thf(sy_c_Set_Ovimage_001t__Numeral____Type__Onum0_001t__Numeral____Type__Onum0,type,
vimage1199591675108543419l_num0: ( numeral_num0 > numeral_num0 ) > set_Numeral_num0 > set_Numeral_num0 ).
thf(sy_c_Set_Ovimage_001t__Numeral____Type__Onum0_001t__Numeral____Type__Onum1,type,
vimage1199591675108543420l_num1: ( numeral_num0 > numeral_num1 ) > set_Numeral_num1 > set_Numeral_num0 ).
thf(sy_c_Set_Ovimage_001t__Numeral____Type__Onum0_001t__Product____Type__Ounit,type,
vimage4378899258379275748t_unit: ( numeral_num0 > product_unit ) > set_Product_unit > set_Numeral_num0 ).
thf(sy_c_Set_Ovimage_001t__Numeral____Type__Onum0_001t__String__Oliteral,type,
vimage7997836657330795051iteral: ( numeral_num0 > literal ) > set_literal > set_Numeral_num0 ).
thf(sy_c_Set_Ovimage_001t__Numeral____Type__Onum1_001t__Nat__Onat,type,
vimage6187404907955765562m1_nat: ( numeral_num1 > nat ) > set_nat > set_Numeral_num1 ).
thf(sy_c_Set_Ovimage_001t__Numeral____Type__Onum1_001t__Numeral____Type__Onum0,type,
vimage5150483310727131898l_num0: ( numeral_num1 > numeral_num0 ) > set_Numeral_num0 > set_Numeral_num1 ).
thf(sy_c_Set_Ovimage_001t__Numeral____Type__Onum1_001t__String__Oliteral,type,
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thf(sy_c_Set_Ovimage_001t__Product____Type__Ounit_001t__Nat__Onat,type,
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thf(sy_c_Set_Ovimage_001t__Product____Type__Ounit_001t__Numeral____Type__Onum0,type,
vimage4815744532680921810l_num0: ( product_unit > numeral_num0 ) > set_Numeral_num0 > set_Product_unit ).
thf(sy_c_Set_Ovimage_001t__String__Oliteral_001t__Nat__Onat,type,
vimage_literal_nat: ( literal > nat ) > set_nat > set_literal ).
thf(sy_c_Set_Ovimage_001t__String__Oliteral_001t__Numeral____Type__Onum0,type,
vimage2868565650143792587l_num0: ( literal > numeral_num0 ) > set_Numeral_num0 > set_literal ).
thf(sy_c_Set_Ovimage_001t__String__Oliteral_001t__Numeral____Type__Onum1,type,
vimage2868565650143792588l_num1: ( literal > numeral_num1 ) > set_Numeral_num1 > set_literal ).
thf(sy_c_Set_Ovimage_001t__String__Oliteral_001t__Product____Type__Ounit,type,
vimage6047873233414524916t_unit: ( literal > product_unit ) > set_Product_unit > set_literal ).
thf(sy_c_Set_Ovimage_001t__String__Oliteral_001t__String__Oliteral,type,
vimage8238609917233974331iteral: ( literal > literal ) > set_literal > set_literal ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
set_or1210151606488870762an_nat: nat > set_nat ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Numeral____Type__Onum0,type,
member_Numeral_num0: numeral_num0 > set_Numeral_num0 > $o ).
thf(sy_c_member_001t__Numeral____Type__Onum1,type,
member_Numeral_num1: numeral_num1 > set_Numeral_num1 > $o ).
thf(sy_c_member_001t__Product____Type__Ounit,type,
member_Product_unit: product_unit > set_Product_unit > $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__String__Oliteral,type,
member_literal: literal > set_literal > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_I,type,
i: product_prod_b_nat > set_list_a ).
thf(sy_v_Q,type,
q: relational_fmla_a_b ).
thf(sy_v_X,type,
x: set_nat ).
thf(sy_v_thesis____,type,
thesis: $o ).
% Relevant facts (1266)
thf(fact_0_assms_I1_J,axiom,
finite_finite_nat @ x ).
% assms(1)
thf(fact_1_assms_I3_J,axiom,
? [X_1: nat > a] : ( relational_sat_a_b @ q @ i @ X_1 ) ).
% assms(3)
thf(fact_2_assms_I2_J,axiom,
ord_less_set_nat @ ( relational_fv_a_b @ q ) @ x ).
% assms(2)
thf(fact_3_sat__fv__cong,axiom,
! [Phi: relational_fmla_a_b,Sigma: nat > a,Sigma2: nat > a,I: product_prod_b_nat > set_list_a] :
( ! [N: nat] :
( ( member_nat @ N @ ( relational_fv_a_b @ Phi ) )
=> ( ( Sigma @ N )
= ( Sigma2 @ N ) ) )
=> ( ( relational_sat_a_b @ Phi @ I @ Sigma )
= ( relational_sat_a_b @ Phi @ I @ Sigma2 ) ) ) ).
% sat_fv_cong
thf(fact_4_DiffI,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ A )
=> ( ~ ( member_nat @ C @ B )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) ) ) ) ).
% DiffI
thf(fact_5_Diff__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
= ( ( member_nat @ C @ A )
& ~ ( member_nat @ C @ B ) ) ) ).
% Diff_iff
thf(fact_6_Diff__idemp,axiom,
! [A: set_nat,B: set_nat] :
( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ B ) @ B )
= ( minus_minus_set_nat @ A @ B ) ) ).
% Diff_idemp
thf(fact_7_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_nat @ X @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_8_subset__antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_9_order__refl,axiom,
! [X2: set_nat] : ( ord_less_eq_set_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_10_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_11_dual__order_Orefl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_12_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_13_Diff__mono,axiom,
! [A: set_nat,C2: set_nat,D: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ D @ B )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B ) @ ( minus_minus_set_nat @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_14_Diff__subset,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_15_double__diff,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ( minus_minus_set_nat @ B @ ( minus_minus_set_nat @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_16_psubsetI,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_nat @ A @ B ) ) ) ).
% psubsetI
thf(fact_17_gt__ex,axiom,
! [X2: nat] :
? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% gt_ex
thf(fact_18_less__imp__neq,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_19_less__imp__neq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_20_order_Oasym,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ~ ( ord_less_set_nat @ B2 @ A2 ) ) ).
% order.asym
thf(fact_21_order_Oasym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% order.asym
thf(fact_22_ord__eq__less__trans,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( A2 = B2 )
=> ( ( ord_less_set_nat @ B2 @ C )
=> ( ord_less_set_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_23_ord__eq__less__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 = B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_24_ord__less__eq__trans,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_set_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_25_ord__less__eq__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_26_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X: nat] :
( ! [Y2: nat] :
( ( ord_less_nat @ Y2 @ X )
=> ( P @ Y2 ) )
=> ( P @ X ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_27_antisym__conv3,axiom,
! [Y: nat,X2: nat] :
( ~ ( ord_less_nat @ Y @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_28_linorder__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_29_dual__order_Oasym,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ B2 @ A2 )
=> ~ ( ord_less_set_nat @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_30_dual__order_Oasym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ~ ( ord_less_nat @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_31_dual__order_Oirrefl,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_32_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_33_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X3: nat] : ( P2 @ X3 ) )
= ( ^ [P3: nat > $o] :
? [N2: nat] :
( ( P3 @ N2 )
& ! [M: nat] :
( ( ord_less_nat @ M @ N2 )
=> ~ ( P3 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_34_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat] : ( P @ A3 @ A3 )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_35_order_Ostrict__trans,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ( ord_less_set_nat @ B2 @ C )
=> ( ord_less_set_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_36_order_Ostrict__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_37_not__less__iff__gr__or__eq,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( ( ord_less_nat @ Y @ X2 )
| ( X2 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_38_dual__order_Ostrict__trans,axiom,
! [B2: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_set_nat @ B2 @ A2 )
=> ( ( ord_less_set_nat @ C @ B2 )
=> ( ord_less_set_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_39_dual__order_Ostrict__trans,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_40_order_Ostrict__implies__not__eq,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_41_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_42_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_43_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_44_psubsetD,axiom,
! [A: set_nat,B: set_nat,C: nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% psubsetD
thf(fact_45_psubset__trans,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ B @ C2 )
=> ( ord_less_set_nat @ A @ C2 ) ) ) ).
% psubset_trans
thf(fact_46_linorder__neqE,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neqE
thf(fact_47_order__less__asym,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X2 @ Y )
=> ~ ( ord_less_set_nat @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_48_order__less__asym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_49_linorder__neq__iff,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
= ( ( ord_less_nat @ X2 @ Y )
| ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_50_order__less__asym_H,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ~ ( ord_less_set_nat @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_51_order__less__asym_H,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_52_order__less__trans,axiom,
! [X2: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_set_nat @ X2 @ Y )
=> ( ( ord_less_set_nat @ Y @ Z )
=> ( ord_less_set_nat @ X2 @ Z ) ) ) ).
% order_less_trans
thf(fact_53_order__less__trans,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X2 @ Z ) ) ) ).
% order_less_trans
thf(fact_54_ord__eq__less__subst,axiom,
! [A2: set_nat,F: set_nat > set_nat,B2: set_nat,C: set_nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_set_nat @ B2 @ C )
=> ( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X @ Y3 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_55_ord__eq__less__subst,axiom,
! [A2: nat,F: set_nat > nat,B2: set_nat,C: set_nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_set_nat @ B2 @ C )
=> ( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_56_ord__eq__less__subst,axiom,
! [A2: set_nat,F: nat > set_nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_57_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_58_ord__less__eq__subst,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X @ Y3 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_59_ord__less__eq__subst,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_60_ord__less__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_61_ord__less__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_62_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_63_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X4: nat] : ( member_nat @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_64_order__less__irrefl,axiom,
! [X2: set_nat] :
~ ( ord_less_set_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_65_order__less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_66_order__less__subst1,axiom,
! [A2: set_nat,F: set_nat > set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_nat @ B2 @ C )
=> ( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X @ Y3 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_67_order__less__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B2: nat,C: nat] :
( ( ord_less_set_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_68_order__less__subst1,axiom,
! [A2: nat,F: set_nat > nat,B2: set_nat,C: set_nat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_nat @ B2 @ C )
=> ( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_69_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_70_order__less__subst2,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ( ord_less_set_nat @ ( F @ B2 ) @ C )
=> ( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X @ Y3 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_71_order__less__subst2,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_72_order__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_set_nat @ ( F @ B2 ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_73_order__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_74_order__less__not__sym,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X2 @ Y )
=> ~ ( ord_less_set_nat @ Y @ X2 ) ) ).
% order_less_not_sym
thf(fact_75_order__less__not__sym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_not_sym
thf(fact_76_order__less__imp__triv,axiom,
! [X2: set_nat,Y: set_nat,P: $o] :
( ( ord_less_set_nat @ X2 @ Y )
=> ( ( ord_less_set_nat @ Y @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_77_order__less__imp__triv,axiom,
! [X2: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_78_linorder__less__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
| ( X2 = Y )
| ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_less_linear
thf(fact_79_order__less__imp__not__eq,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_80_order__less__imp__not__eq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_81_order__less__imp__not__eq2,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_82_order__less__imp__not__eq2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_83_order__less__imp__not__less,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X2 @ Y )
=> ~ ( ord_less_set_nat @ Y @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_84_order__less__imp__not__less,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_85_order__le__imp__less__or__eq,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ( ord_less_set_nat @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_86_order__le__imp__less__or__eq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_nat @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_87_linorder__le__less__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_88_order__less__le__subst2,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ ( F @ B2 ) @ C )
=> ( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X @ Y3 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_89_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ ( F @ B2 ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_90_order__less__le__subst2,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_91_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_92_order__less__le__subst1,axiom,
! [A2: set_nat,F: set_nat > set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_93_order__less__le__subst1,axiom,
! [A2: nat,F: set_nat > nat,B2: set_nat,C: set_nat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_94_order__less__le__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B2: nat,C: nat] :
( ( ord_less_set_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_95_order__less__le__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_96_order__le__less__subst2,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_set_nat @ ( F @ B2 ) @ C )
=> ( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_97_order__le__less__subst2,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_98_order__le__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_set_nat @ ( F @ B2 ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_99_order__le__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_100_order__le__less__subst1,axiom,
! [A2: set_nat,F: set_nat > set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_nat @ B2 @ C )
=> ( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X @ Y3 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_101_order__le__less__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B2: nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_102_order__le__less__subst1,axiom,
! [A2: nat,F: set_nat > nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_nat @ B2 @ C )
=> ( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_103_order__le__less__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_104_order__less__le__trans,axiom,
! [X2: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_set_nat @ X2 @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z )
=> ( ord_less_set_nat @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_105_order__less__le__trans,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_nat @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_106_order__le__less__trans,axiom,
! [X2: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ( ord_less_set_nat @ Y @ Z )
=> ( ord_less_set_nat @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_107_order__le__less__trans,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_108_order__neq__le__trans,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 != B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_set_nat @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_109_order__neq__le__trans,axiom,
! [A2: nat,B2: nat] :
( ( A2 != B2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_110_order__le__neq__trans,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_nat @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_111_order__le__neq__trans,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_112_order__less__imp__le,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X2 @ Y )
=> ( ord_less_eq_set_nat @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_113_order__less__imp__le,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_114_linorder__not__less,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_not_less
thf(fact_115_linorder__not__le,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y ) )
= ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_not_le
thf(fact_116_order__less__le,axiom,
( ord_less_set_nat
= ( ^ [X4: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_117_order__less__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_118_order__le__less,axiom,
( ord_less_eq_set_nat
= ( ^ [X4: set_nat,Y4: set_nat] :
( ( ord_less_set_nat @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_119_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_120_dual__order_Ostrict__implies__order,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ B2 @ A2 )
=> ( ord_less_eq_set_nat @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_121_dual__order_Ostrict__implies__order,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_122_order_Ostrict__implies__order,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_123_order_Ostrict__implies__order,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_124_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [B4: set_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A4 )
& ~ ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_125_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_126_dual__order_Ostrict__trans2,axiom,
! [B2: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_set_nat @ B2 @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ B2 )
=> ( ord_less_set_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_127_dual__order_Ostrict__trans2,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_128_dual__order_Ostrict__trans1,axiom,
! [B2: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ord_less_set_nat @ C @ B2 )
=> ( ord_less_set_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_129_dual__order_Ostrict__trans1,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_130_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [B4: set_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_131_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_132_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [B4: set_nat,A4: set_nat] :
( ( ord_less_set_nat @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_133_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_nat @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_134_order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
& ~ ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_135_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ~ ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_136_order_Ostrict__trans2,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 ) ) ) ).
% order.strict_trans2
thf(fact_137_order_Ostrict__trans2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_138_order_Ostrict__trans1,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 ) ) ) ).
% order.strict_trans1
thf(fact_139_order_Ostrict__trans1,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_140_order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_141_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_142_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_set_nat @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_143_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_144_not__le__imp__less,axiom,
! [Y: nat,X2: nat] :
( ~ ( ord_less_eq_nat @ Y @ X2 )
=> ( ord_less_nat @ X2 @ Y ) ) ).
% not_le_imp_less
thf(fact_145_less__le__not__le,axiom,
( ord_less_set_nat
= ( ^ [X4: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y4 )
& ~ ( ord_less_eq_set_nat @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_146_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_147_antisym__conv2,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ( ~ ( ord_less_set_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_148_antisym__conv2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_149_antisym__conv1,axiom,
! [X2: set_nat,Y: set_nat] :
( ~ ( ord_less_set_nat @ X2 @ Y )
=> ( ( ord_less_eq_set_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_150_antisym__conv1,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_151_nless__le,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ~ ( ord_less_set_nat @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_set_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_152_nless__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_nat @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_153_leI,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% leI
thf(fact_154_leD,axiom,
! [Y: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X2 )
=> ~ ( ord_less_set_nat @ X2 @ Y ) ) ).
% leD
thf(fact_155_leD,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_nat @ X2 @ Y ) ) ).
% leD
thf(fact_156_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_set_nat @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_157_subset__psubset__trans,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ B @ C2 )
=> ( ord_less_set_nat @ A @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_158_subset__not__subset__eq,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
& ~ ( ord_less_eq_set_nat @ B5 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_159_psubset__subset__trans,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_set_nat @ A @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_160_psubset__imp__subset,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_161_psubset__eq,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% psubset_eq
thf(fact_162_psubsetE,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ~ ( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ B @ A ) ) ) ).
% psubsetE
thf(fact_163_psubset__imp__ex__mem,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ? [B3: nat] : ( member_nat @ B3 @ ( minus_minus_set_nat @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_164_order__antisym__conv,axiom,
! [Y: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_165_order__antisym__conv,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_166_linorder__le__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_167_ord__le__eq__subst,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_168_ord__le__eq__subst,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_169_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_170_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_171_ord__eq__le__subst,axiom,
! [A2: set_nat,F: set_nat > set_nat,B2: set_nat,C: set_nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_172_ord__eq__le__subst,axiom,
! [A2: nat,F: set_nat > nat,B2: set_nat,C: set_nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_173_ord__eq__le__subst,axiom,
! [A2: set_nat,F: nat > set_nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_174_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_175_linorder__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_176_order__eq__refl,axiom,
! [X2: set_nat,Y: set_nat] :
( ( X2 = Y )
=> ( ord_less_eq_set_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_177_order__eq__refl,axiom,
! [X2: nat,Y: nat] :
( ( X2 = Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_178_order__subst2,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ ( F @ B2 ) @ C )
=> ( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_179_order__subst2,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_180_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ ( F @ B2 ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_181_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_182_order__subst1,axiom,
! [A2: set_nat,F: set_nat > set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_183_order__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B2: nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_184_order__subst1,axiom,
! [A2: nat,F: set_nat > nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_185_order__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_186_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_nat,Z2: set_nat] : ( Y5 = Z2 ) )
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_187_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_188_antisym,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_189_antisym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_190_dual__order_Otrans,axiom,
! [B2: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ B2 )
=> ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_191_dual__order_Otrans,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_192_dual__order_Oantisym,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_193_dual__order_Oantisym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_194_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_nat,Z2: set_nat] : ( Y5 = Z2 ) )
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A4 )
& ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_195_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_196_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_197_order__trans,axiom,
! [X2: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z )
=> ( ord_less_eq_set_nat @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_198_order__trans,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_199_order_Otrans,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_200_order_Otrans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_201_order__antisym,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_202_order__antisym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_203_ord__le__eq__trans,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_204_ord__le__eq__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_205_ord__eq__le__trans,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_206_ord__eq__le__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_207_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_nat,Z2: set_nat] : ( Y5 = Z2 ) )
= ( ^ [X4: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y4 )
& ( ord_less_eq_set_nat @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_208_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_209_le__cases3,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_210_nle__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_211_diff__right__commute,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ C ) @ B2 )
= ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B2 ) @ C ) ) ).
% diff_right_commute
thf(fact_212_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X4: nat] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_213_set__eq__subset,axiom,
( ( ^ [Y5: set_nat,Z2: set_nat] : ( Y5 = Z2 ) )
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_214_subset__trans,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_215_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X: nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_216_subset__refl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% subset_refl
thf(fact_217_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A5 )
=> ( member_nat @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_218_Set_OequalityD2,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% Set.equalityD2
thf(fact_219_equalityD1,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% equalityD1
thf(fact_220_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [X4: nat] :
( ( member_nat @ X4 @ A5 )
=> ( member_nat @ X4 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_221_equalityE,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_nat @ A @ B )
=> ~ ( ord_less_eq_set_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_222_subsetD,axiom,
! [A: set_nat,B: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_223_in__mono,axiom,
! [A: set_nat,B: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X2 @ A )
=> ( member_nat @ X2 @ B ) ) ) ).
% in_mono
thf(fact_224_DiffD2,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ~ ( member_nat @ C @ B ) ) ).
% DiffD2
thf(fact_225_DiffD1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ( member_nat @ C @ A ) ) ).
% DiffD1
thf(fact_226_DiffE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ~ ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% DiffE
thf(fact_227_finite__fv,axiom,
! [Phi: relational_fmla_a_b] : ( finite_finite_nat @ ( relational_fv_a_b @ Phi ) ) ).
% finite_fv
thf(fact_228_finite__Diff2,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) )
= ( finite_finite_nat @ A ) ) ) ).
% finite_Diff2
thf(fact_229_finite__Diff,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% finite_Diff
thf(fact_230_finite__psubset__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ! [B6: set_nat] :
( ( ord_less_set_nat @ B6 @ A6 )
=> ( P @ B6 ) )
=> ( P @ A6 ) ) )
=> ( P @ A ) ) ) ).
% finite_psubset_induct
thf(fact_231_Diff__infinite__finite,axiom,
! [T2: set_nat,S: set_nat] :
( ( finite_finite_nat @ T2 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_232_rev__finite__subset,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_233_infinite__super,axiom,
! [S: set_nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S @ T2 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_234_finite__subset,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_subset
thf(fact_235_finite__has__minimal2,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_set_nat @ A2 @ A )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A )
& ( ord_less_eq_set_nat @ X @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_236_finite__has__minimal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ( ord_less_eq_nat @ X @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_237_finite__has__maximal2,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_set_nat @ A2 @ A )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A )
& ( ord_less_eq_set_nat @ A2 @ X )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_238_finite__has__maximal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ( ord_less_eq_nat @ A2 @ X )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_239_complete__interval,axiom,
! [A2: nat,B2: nat,P: nat > $o] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B2 )
=> ? [C3: nat] :
( ( ord_less_eq_nat @ A2 @ C3 )
& ( ord_less_eq_nat @ C3 @ B2 )
& ! [X5: nat] :
( ( ( ord_less_eq_nat @ A2 @ X5 )
& ( ord_less_nat @ X5 @ C3 ) )
=> ( P @ X5 ) )
& ! [D2: nat] :
( ! [X: nat] :
( ( ( ord_less_eq_nat @ A2 @ X )
& ( ord_less_nat @ X @ D2 ) )
=> ( P @ X ) )
=> ( ord_less_eq_nat @ D2 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_240_verit__comp__simplify1_I3_J,axiom,
! [B7: nat,A7: nat] :
( ( ~ ( ord_less_eq_nat @ B7 @ A7 ) )
= ( ord_less_nat @ A7 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_241_verit__comp__simplify1_I2_J,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_242_verit__comp__simplify1_I2_J,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_243_verit__la__disequality,axiom,
! [A2: nat,B2: nat] :
( ( A2 = B2 )
| ~ ( ord_less_eq_nat @ A2 @ B2 )
| ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% verit_la_disequality
thf(fact_244_verit__comp__simplify1_I1_J,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_245_verit__comp__simplify1_I1_J,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_246_minf_I8_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ~ ( ord_less_eq_nat @ T3 @ X5 ) ) ).
% minf(8)
thf(fact_247_minf_I6_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ord_less_eq_nat @ X5 @ T3 ) ) ).
% minf(6)
thf(fact_248_pinf_I8_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ord_less_eq_nat @ T3 @ X5 ) ) ).
% pinf(8)
thf(fact_249_pinf_I6_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ~ ( ord_less_eq_nat @ X5 @ T3 ) ) ).
% pinf(6)
thf(fact_250_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N3: set_nat] :
? [M: nat] :
! [X4: nat] :
( ( member_nat @ X4 @ N3 )
=> ( ord_less_nat @ X4 @ M ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_251_bounded__nat__set__is__finite,axiom,
! [N4: set_nat,N5: nat] :
( ! [X: nat] :
( ( member_nat @ X @ N4 )
=> ( ord_less_nat @ X @ N5 ) )
=> ( finite_finite_nat @ N4 ) ) ).
% bounded_nat_set_is_finite
thf(fact_252_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N3: set_nat] :
? [M: nat] :
! [X4: nat] :
( ( member_nat @ X4 @ N3 )
=> ( ord_less_eq_nat @ X4 @ M ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_253_bounded__Max__nat,axiom,
! [P: nat > $o,X2: nat,M2: nat] :
( ( P @ X2 )
=> ( ! [X: nat] :
( ( P @ X )
=> ( ord_less_eq_nat @ X @ M2 ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_254_sat_Osimps_I6_J,axiom,
! [Phi: relational_fmla_a_b,Psi: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
( ( relational_sat_a_b @ ( relational_Disj_a_b @ Phi @ Psi ) @ I @ Sigma )
= ( ( relational_sat_a_b @ Phi @ I @ Sigma )
| ( relational_sat_a_b @ Psi @ I @ Sigma ) ) ) ).
% sat.simps(6)
thf(fact_255_eval__on__cong,axiom,
! [Q: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Q2: relational_fmla_a_b,X6: set_nat] :
( ! [Sigma3: nat > a] :
( ( relational_sat_a_b @ Q @ I @ Sigma3 )
= ( relational_sat_a_b @ Q2 @ I @ Sigma3 ) )
=> ( ( relati8814510239606734169on_a_b @ X6 @ Q @ I )
= ( relati8814510239606734169on_a_b @ X6 @ Q2 @ I ) ) ) ).
% eval_on_cong
thf(fact_256_minf_I7_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ~ ( ord_less_nat @ T3 @ X5 ) ) ).
% minf(7)
thf(fact_257_minf_I5_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ord_less_nat @ X5 @ T3 ) ) ).
% minf(5)
thf(fact_258_minf_I4_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( X5 != T3 ) ) ).
% minf(4)
thf(fact_259_minf_I3_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( X5 != T3 ) ) ).
% minf(3)
thf(fact_260_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z4 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z4 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_261_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z4 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z4 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_262_pinf_I7_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ord_less_nat @ T3 @ X5 ) ) ).
% pinf(7)
thf(fact_263_pinf_I5_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ~ ( ord_less_nat @ X5 @ T3 ) ) ).
% pinf(5)
thf(fact_264_pinf_I4_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( X5 != T3 ) ) ).
% pinf(4)
thf(fact_265_pinf_I3_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( X5 != T3 ) ) ).
% pinf(3)
thf(fact_266_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ Z4 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ Z4 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_267_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ Z4 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ Z4 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_268_diff__diff__cancel,axiom,
! [I2: nat,N5: nat] :
( ( ord_less_eq_nat @ I2 @ N5 )
=> ( ( minus_minus_nat @ N5 @ ( minus_minus_nat @ N5 @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_269_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
& ( M != N2 ) ) ) ) ).
% nat_less_le
thf(fact_270_less__diff__iff,axiom,
! [K: nat,M4: nat,N5: nat] :
( ( ord_less_eq_nat @ K @ M4 )
=> ( ( ord_less_eq_nat @ K @ N5 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M4 @ K ) @ ( minus_minus_nat @ N5 @ K ) )
= ( ord_less_nat @ M4 @ N5 ) ) ) ) ).
% less_diff_iff
thf(fact_271_diff__less__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_nat @ ( minus_minus_nat @ A2 @ C ) @ ( minus_minus_nat @ B2 @ C ) ) ) ) ).
% diff_less_mono
thf(fact_272_less__imp__le__nat,axiom,
! [M4: nat,N5: nat] :
( ( ord_less_nat @ M4 @ N5 )
=> ( ord_less_eq_nat @ M4 @ N5 ) ) ).
% less_imp_le_nat
thf(fact_273_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_274_less__or__eq__imp__le,axiom,
! [M4: nat,N5: nat] :
( ( ( ord_less_nat @ M4 @ N5 )
| ( M4 = N5 ) )
=> ( ord_less_eq_nat @ M4 @ N5 ) ) ).
% less_or_eq_imp_le
thf(fact_275_diff__commute,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J ) ) ).
% diff_commute
thf(fact_276_eval__cong,axiom,
! [Q: relational_fmla_a_b,Q2: relational_fmla_a_b,I: product_prod_b_nat > set_list_a] :
( ( ( relational_fv_a_b @ Q )
= ( relational_fv_a_b @ Q2 ) )
=> ( ! [Sigma3: nat > a] :
( ( relational_sat_a_b @ Q @ I @ Sigma3 )
= ( relational_sat_a_b @ Q2 @ I @ Sigma3 ) )
=> ( ( relational_eval_a_b @ Q @ I )
= ( relational_eval_a_b @ Q2 @ I ) ) ) ) ).
% eval_cong
thf(fact_277_less__imp__diff__less,axiom,
! [J: nat,K: nat,N5: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N5 ) @ K ) ) ).
% less_imp_diff_less
thf(fact_278_linorder__neqE__nat,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_279_infinite__descent,axiom,
! [P: nat > $o,N5: nat] :
( ! [N: nat] :
( ~ ( P @ N )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N )
& ~ ( P @ M5 ) ) )
=> ( P @ N5 ) ) ).
% infinite_descent
thf(fact_280_nat__less__induct,axiom,
! [P: nat > $o,N5: nat] :
( ! [N: nat] :
( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N )
=> ( P @ M5 ) )
=> ( P @ N ) )
=> ( P @ N5 ) ) ).
% nat_less_induct
thf(fact_281_less__irrefl__nat,axiom,
! [N5: nat] :
~ ( ord_less_nat @ N5 @ N5 ) ).
% less_irrefl_nat
thf(fact_282_diff__less__mono2,axiom,
! [M4: nat,N5: nat,L: nat] :
( ( ord_less_nat @ M4 @ N5 )
=> ( ( ord_less_nat @ M4 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N5 ) @ ( minus_minus_nat @ L @ M4 ) ) ) ) ).
% diff_less_mono2
thf(fact_283_less__not__refl3,axiom,
! [S2: nat,T3: nat] :
( ( ord_less_nat @ S2 @ T3 )
=> ( S2 != T3 ) ) ).
% less_not_refl3
thf(fact_284_less__not__refl2,axiom,
! [N5: nat,M4: nat] :
( ( ord_less_nat @ N5 @ M4 )
=> ( M4 != N5 ) ) ).
% less_not_refl2
thf(fact_285_less__not__refl,axiom,
! [N5: nat] :
~ ( ord_less_nat @ N5 @ N5 ) ).
% less_not_refl
thf(fact_286_nat__neq__iff,axiom,
! [M4: nat,N5: nat] :
( ( M4 != N5 )
= ( ( ord_less_nat @ M4 @ N5 )
| ( ord_less_nat @ N5 @ M4 ) ) ) ).
% nat_neq_iff
thf(fact_287_Relational__Calculus_Oeval__def,axiom,
( relational_eval_a_b
= ( ^ [Q3: relational_fmla_a_b] : ( relati8814510239606734169on_a_b @ ( relational_fv_a_b @ Q3 ) @ Q3 ) ) ) ).
% Relational_Calculus.eval_def
thf(fact_288_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B2 ) )
=> ? [X: nat] :
( ( P @ X )
& ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_289_nat__le__linear,axiom,
! [M4: nat,N5: nat] :
( ( ord_less_eq_nat @ M4 @ N5 )
| ( ord_less_eq_nat @ N5 @ M4 ) ) ).
% nat_le_linear
thf(fact_290_diff__le__mono2,axiom,
! [M4: nat,N5: nat,L: nat] :
( ( ord_less_eq_nat @ M4 @ N5 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N5 ) @ ( minus_minus_nat @ L @ M4 ) ) ) ).
% diff_le_mono2
thf(fact_291_le__diff__iff_H,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_292_diff__le__self,axiom,
! [M4: nat,N5: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M4 @ N5 ) @ M4 ) ).
% diff_le_self
thf(fact_293_diff__le__mono,axiom,
! [M4: nat,N5: nat,L: nat] :
( ( ord_less_eq_nat @ M4 @ N5 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M4 @ L ) @ ( minus_minus_nat @ N5 @ L ) ) ) ).
% diff_le_mono
thf(fact_294_Nat_Odiff__diff__eq,axiom,
! [K: nat,M4: nat,N5: nat] :
( ( ord_less_eq_nat @ K @ M4 )
=> ( ( ord_less_eq_nat @ K @ N5 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M4 @ K ) @ ( minus_minus_nat @ N5 @ K ) )
= ( minus_minus_nat @ M4 @ N5 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_295_le__diff__iff,axiom,
! [K: nat,M4: nat,N5: nat] :
( ( ord_less_eq_nat @ K @ M4 )
=> ( ( ord_less_eq_nat @ K @ N5 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M4 @ K ) @ ( minus_minus_nat @ N5 @ K ) )
= ( ord_less_eq_nat @ M4 @ N5 ) ) ) ) ).
% le_diff_iff
thf(fact_296_eq__diff__iff,axiom,
! [K: nat,M4: nat,N5: nat] :
( ( ord_less_eq_nat @ K @ M4 )
=> ( ( ord_less_eq_nat @ K @ N5 )
=> ( ( ( minus_minus_nat @ M4 @ K )
= ( minus_minus_nat @ N5 @ K ) )
= ( M4 = N5 ) ) ) ) ).
% eq_diff_iff
thf(fact_297_le__antisym,axiom,
! [M4: nat,N5: nat] :
( ( ord_less_eq_nat @ M4 @ N5 )
=> ( ( ord_less_eq_nat @ N5 @ M4 )
=> ( M4 = N5 ) ) ) ).
% le_antisym
thf(fact_298_eq__imp__le,axiom,
! [M4: nat,N5: nat] :
( ( M4 = N5 )
=> ( ord_less_eq_nat @ M4 @ N5 ) ) ).
% eq_imp_le
thf(fact_299_le__trans,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% le_trans
thf(fact_300_le__refl,axiom,
! [N5: nat] : ( ord_less_eq_nat @ N5 @ N5 ) ).
% le_refl
thf(fact_301_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I2: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_302_le__neq__implies__less,axiom,
! [M4: nat,N5: nat] :
( ( ord_less_eq_nat @ M4 @ N5 )
=> ( ( M4 != N5 )
=> ( ord_less_nat @ M4 @ N5 ) ) ) ).
% le_neq_implies_less
thf(fact_303_diff__diff__less,axiom,
! [I2: nat,M4: nat,N5: nat] :
( ( ord_less_nat @ I2 @ ( minus_minus_nat @ M4 @ ( minus_minus_nat @ M4 @ N5 ) ) )
= ( ( ord_less_nat @ I2 @ M4 )
& ( ord_less_nat @ I2 @ N5 ) ) ) ).
% diff_diff_less
thf(fact_304_nat__descend__induct,axiom,
! [N5: nat,P: nat > $o,M4: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N5 @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N5 )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K2 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K2 ) ) )
=> ( P @ M4 ) ) ) ).
% nat_descend_induct
thf(fact_305_card__psubset,axiom,
! [B: set_Numeral_num0,A: set_Numeral_num0] :
( ( finite1111429032697314573l_num0 @ B )
=> ( ( ord_le5200684351691877604l_num0 @ A @ B )
=> ( ( ord_less_nat @ ( finite6454714172617411596l_num0 @ A ) @ ( finite6454714172617411596l_num0 @ B ) )
=> ( ord_le526730871819019248l_num0 @ A @ B ) ) ) ) ).
% card_psubset
thf(fact_306_card__psubset,axiom,
! [B: set_literal,A: set_literal] :
( ( finite5847741373460823677iteral @ B )
=> ( ( ord_le7307670543136651348iteral @ A @ B )
=> ( ( ord_less_nat @ ( finite_card_literal @ A ) @ ( finite_card_literal @ B ) )
=> ( ord_less_set_literal @ A @ B ) ) ) ) ).
% card_psubset
thf(fact_307_card__psubset,axiom,
! [B: set_Numeral_num1,A: set_Numeral_num1] :
( ( finite1111429032697314574l_num1 @ B )
=> ( ( ord_le5200684355995106405l_num1 @ A @ B )
=> ( ( ord_less_nat @ ( finite6454714172617411597l_num1 @ A ) @ ( finite6454714172617411597l_num1 @ B ) )
=> ( ord_le526730876122248049l_num1 @ A @ B ) ) ) ) ).
% card_psubset
thf(fact_308_card__psubset,axiom,
! [B: set_Product_unit,A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ B )
=> ( ( ord_le3507040750410214029t_unit @ A @ B )
=> ( ( ord_less_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite410649719033368117t_unit @ B ) )
=> ( ord_le8056459307392131481t_unit @ A @ B ) ) ) ) ).
% card_psubset
thf(fact_309_card__psubset,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) )
=> ( ord_less_set_nat @ A @ B ) ) ) ) ).
% card_psubset
thf(fact_310_diff__card__le__card__Diff,axiom,
! [B: set_Numeral_num0,A: set_Numeral_num0] :
( ( finite1111429032697314573l_num0 @ B )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite6454714172617411596l_num0 @ A ) @ ( finite6454714172617411596l_num0 @ B ) ) @ ( finite6454714172617411596l_num0 @ ( minus_8146479927826647979l_num0 @ A @ B ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_311_diff__card__le__card__Diff,axiom,
! [B: set_literal,A: set_literal] :
( ( finite5847741373460823677iteral @ B )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_literal @ A ) @ ( finite_card_literal @ B ) ) @ ( finite_card_literal @ ( minus_7832829386415567259iteral @ A @ B ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_312_diff__card__le__card__Diff,axiom,
! [B: set_Numeral_num1,A: set_Numeral_num1] :
( ( finite1111429032697314574l_num1 @ B )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite6454714172617411597l_num1 @ A ) @ ( finite6454714172617411597l_num1 @ B ) ) @ ( finite6454714172617411597l_num1 @ ( minus_8146479932129876780l_num1 @ A @ B ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_313_diff__card__le__card__Diff,axiom,
! [B: set_Product_unit,A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ B )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite410649719033368117t_unit @ B ) ) @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ B ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_314_diff__card__le__card__Diff,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ B ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_315_card__Diff__subset,axiom,
! [B: set_Numeral_num0,A: set_Numeral_num0] :
( ( finite1111429032697314573l_num0 @ B )
=> ( ( ord_le5200684351691877604l_num0 @ B @ A )
=> ( ( finite6454714172617411596l_num0 @ ( minus_8146479927826647979l_num0 @ A @ B ) )
= ( minus_minus_nat @ ( finite6454714172617411596l_num0 @ A ) @ ( finite6454714172617411596l_num0 @ B ) ) ) ) ) ).
% card_Diff_subset
thf(fact_316_card__Diff__subset,axiom,
! [B: set_literal,A: set_literal] :
( ( finite5847741373460823677iteral @ B )
=> ( ( ord_le7307670543136651348iteral @ B @ A )
=> ( ( finite_card_literal @ ( minus_7832829386415567259iteral @ A @ B ) )
= ( minus_minus_nat @ ( finite_card_literal @ A ) @ ( finite_card_literal @ B ) ) ) ) ) ).
% card_Diff_subset
thf(fact_317_card__Diff__subset,axiom,
! [B: set_Numeral_num1,A: set_Numeral_num1] :
( ( finite1111429032697314574l_num1 @ B )
=> ( ( ord_le5200684355995106405l_num1 @ B @ A )
=> ( ( finite6454714172617411597l_num1 @ ( minus_8146479932129876780l_num1 @ A @ B ) )
= ( minus_minus_nat @ ( finite6454714172617411597l_num1 @ A ) @ ( finite6454714172617411597l_num1 @ B ) ) ) ) ) ).
% card_Diff_subset
thf(fact_318_card__Diff__subset,axiom,
! [B: set_Product_unit,A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ B )
=> ( ( ord_le3507040750410214029t_unit @ B @ A )
=> ( ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ B ) )
= ( minus_minus_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite410649719033368117t_unit @ B ) ) ) ) ) ).
% card_Diff_subset
thf(fact_319_card__Diff__subset,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ B ) )
= ( minus_minus_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_Diff_subset
thf(fact_320_GreatestI2__order,axiom,
! [P: set_nat > $o,X2: set_nat,Q: set_nat > $o] :
( ( P @ X2 )
=> ( ! [Y3: set_nat] :
( ( P @ Y3 )
=> ( ord_less_eq_set_nat @ Y3 @ X2 ) )
=> ( ! [X: set_nat] :
( ( P @ X )
=> ( ! [Y2: set_nat] :
( ( P @ Y2 )
=> ( ord_less_eq_set_nat @ Y2 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_5724808138429204845et_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_321_GreatestI2__order,axiom,
! [P: nat > $o,X2: nat,Q: nat > $o] :
( ( P @ X2 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X2 ) )
=> ( ! [X: nat] :
( ( P @ X )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_Greatest_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_322_Greatest__equality,axiom,
! [P: set_nat > $o,X2: set_nat] :
( ( P @ X2 )
=> ( ! [Y3: set_nat] :
( ( P @ Y3 )
=> ( ord_less_eq_set_nat @ Y3 @ X2 ) )
=> ( ( order_5724808138429204845et_nat @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_323_Greatest__equality,axiom,
! [P: nat > $o,X2: nat] :
( ( P @ X2 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X2 ) )
=> ( ( order_Greatest_nat @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_324_infinite__arbitrarily__large,axiom,
! [A: set_Numeral_num0,N5: nat] :
( ~ ( finite1111429032697314573l_num0 @ A )
=> ? [B8: set_Numeral_num0] :
( ( finite1111429032697314573l_num0 @ B8 )
& ( ( finite6454714172617411596l_num0 @ B8 )
= N5 )
& ( ord_le5200684351691877604l_num0 @ B8 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_325_infinite__arbitrarily__large,axiom,
! [A: set_literal,N5: nat] :
( ~ ( finite5847741373460823677iteral @ A )
=> ? [B8: set_literal] :
( ( finite5847741373460823677iteral @ B8 )
& ( ( finite_card_literal @ B8 )
= N5 )
& ( ord_le7307670543136651348iteral @ B8 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_326_infinite__arbitrarily__large,axiom,
! [A: set_Numeral_num1,N5: nat] :
( ~ ( finite1111429032697314574l_num1 @ A )
=> ? [B8: set_Numeral_num1] :
( ( finite1111429032697314574l_num1 @ B8 )
& ( ( finite6454714172617411597l_num1 @ B8 )
= N5 )
& ( ord_le5200684355995106405l_num1 @ B8 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_327_infinite__arbitrarily__large,axiom,
! [A: set_Product_unit,N5: nat] :
( ~ ( finite4290736615968046902t_unit @ A )
=> ? [B8: set_Product_unit] :
( ( finite4290736615968046902t_unit @ B8 )
& ( ( finite410649719033368117t_unit @ B8 )
= N5 )
& ( ord_le3507040750410214029t_unit @ B8 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_328_infinite__arbitrarily__large,axiom,
! [A: set_nat,N5: nat] :
( ~ ( finite_finite_nat @ A )
=> ? [B8: set_nat] :
( ( finite_finite_nat @ B8 )
& ( ( finite_card_nat @ B8 )
= N5 )
& ( ord_less_eq_set_nat @ B8 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_329_card__subset__eq,axiom,
! [B: set_Numeral_num0,A: set_Numeral_num0] :
( ( finite1111429032697314573l_num0 @ B )
=> ( ( ord_le5200684351691877604l_num0 @ A @ B )
=> ( ( ( finite6454714172617411596l_num0 @ A )
= ( finite6454714172617411596l_num0 @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_330_card__subset__eq,axiom,
! [B: set_literal,A: set_literal] :
( ( finite5847741373460823677iteral @ B )
=> ( ( ord_le7307670543136651348iteral @ A @ B )
=> ( ( ( finite_card_literal @ A )
= ( finite_card_literal @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_331_card__subset__eq,axiom,
! [B: set_Numeral_num1,A: set_Numeral_num1] :
( ( finite1111429032697314574l_num1 @ B )
=> ( ( ord_le5200684355995106405l_num1 @ A @ B )
=> ( ( ( finite6454714172617411597l_num1 @ A )
= ( finite6454714172617411597l_num1 @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_332_card__subset__eq,axiom,
! [B: set_Product_unit,A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ B )
=> ( ( ord_le3507040750410214029t_unit @ A @ B )
=> ( ( ( finite410649719033368117t_unit @ A )
= ( finite410649719033368117t_unit @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_333_card__subset__eq,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( finite_card_nat @ A )
= ( finite_card_nat @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_334_card__le__if__inj__on__rel,axiom,
! [B: set_Numeral_num0,A: set_nat,R: nat > numeral_num0 > $o] :
( ( finite1111429032697314573l_num0 @ B )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ? [B9: numeral_num0] :
( ( member_Numeral_num0 @ B9 @ B )
& ( R @ A3 @ B9 ) ) )
=> ( ! [A1: nat,A22: nat,B3: numeral_num0] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_Numeral_num0 @ B3 @ B )
=> ( ( R @ A1 @ B3 )
=> ( ( R @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite6454714172617411596l_num0 @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_335_card__le__if__inj__on__rel,axiom,
! [B: set_literal,A: set_nat,R: nat > literal > $o] :
( ( finite5847741373460823677iteral @ B )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ? [B9: literal] :
( ( member_literal @ B9 @ B )
& ( R @ A3 @ B9 ) ) )
=> ( ! [A1: nat,A22: nat,B3: literal] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_literal @ B3 @ B )
=> ( ( R @ A1 @ B3 )
=> ( ( R @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_literal @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_336_card__le__if__inj__on__rel,axiom,
! [B: set_Numeral_num1,A: set_nat,R: nat > numeral_num1 > $o] :
( ( finite1111429032697314574l_num1 @ B )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ? [B9: numeral_num1] :
( ( member_Numeral_num1 @ B9 @ B )
& ( R @ A3 @ B9 ) ) )
=> ( ! [A1: nat,A22: nat,B3: numeral_num1] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_Numeral_num1 @ B3 @ B )
=> ( ( R @ A1 @ B3 )
=> ( ( R @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite6454714172617411597l_num1 @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_337_card__le__if__inj__on__rel,axiom,
! [B: set_Product_unit,A: set_nat,R: nat > product_unit > $o] :
( ( finite4290736615968046902t_unit @ B )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ? [B9: product_unit] :
( ( member_Product_unit @ B9 @ B )
& ( R @ A3 @ B9 ) ) )
=> ( ! [A1: nat,A22: nat,B3: product_unit] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_Product_unit @ B3 @ B )
=> ( ( R @ A1 @ B3 )
=> ( ( R @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite410649719033368117t_unit @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_338_card__le__if__inj__on__rel,axiom,
! [B: set_Numeral_num0,A: set_Numeral_num0,R: numeral_num0 > numeral_num0 > $o] :
( ( finite1111429032697314573l_num0 @ B )
=> ( ! [A3: numeral_num0] :
( ( member_Numeral_num0 @ A3 @ A )
=> ? [B9: numeral_num0] :
( ( member_Numeral_num0 @ B9 @ B )
& ( R @ A3 @ B9 ) ) )
=> ( ! [A1: numeral_num0,A22: numeral_num0,B3: numeral_num0] :
( ( member_Numeral_num0 @ A1 @ A )
=> ( ( member_Numeral_num0 @ A22 @ A )
=> ( ( member_Numeral_num0 @ B3 @ B )
=> ( ( R @ A1 @ B3 )
=> ( ( R @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite6454714172617411596l_num0 @ A ) @ ( finite6454714172617411596l_num0 @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_339_card__le__if__inj__on__rel,axiom,
! [B: set_literal,A: set_Numeral_num0,R: numeral_num0 > literal > $o] :
( ( finite5847741373460823677iteral @ B )
=> ( ! [A3: numeral_num0] :
( ( member_Numeral_num0 @ A3 @ A )
=> ? [B9: literal] :
( ( member_literal @ B9 @ B )
& ( R @ A3 @ B9 ) ) )
=> ( ! [A1: numeral_num0,A22: numeral_num0,B3: literal] :
( ( member_Numeral_num0 @ A1 @ A )
=> ( ( member_Numeral_num0 @ A22 @ A )
=> ( ( member_literal @ B3 @ B )
=> ( ( R @ A1 @ B3 )
=> ( ( R @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite6454714172617411596l_num0 @ A ) @ ( finite_card_literal @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_340_card__le__if__inj__on__rel,axiom,
! [B: set_Numeral_num1,A: set_Numeral_num0,R: numeral_num0 > numeral_num1 > $o] :
( ( finite1111429032697314574l_num1 @ B )
=> ( ! [A3: numeral_num0] :
( ( member_Numeral_num0 @ A3 @ A )
=> ? [B9: numeral_num1] :
( ( member_Numeral_num1 @ B9 @ B )
& ( R @ A3 @ B9 ) ) )
=> ( ! [A1: numeral_num0,A22: numeral_num0,B3: numeral_num1] :
( ( member_Numeral_num0 @ A1 @ A )
=> ( ( member_Numeral_num0 @ A22 @ A )
=> ( ( member_Numeral_num1 @ B3 @ B )
=> ( ( R @ A1 @ B3 )
=> ( ( R @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite6454714172617411596l_num0 @ A ) @ ( finite6454714172617411597l_num1 @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_341_card__le__if__inj__on__rel,axiom,
! [B: set_Product_unit,A: set_Numeral_num0,R: numeral_num0 > product_unit > $o] :
( ( finite4290736615968046902t_unit @ B )
=> ( ! [A3: numeral_num0] :
( ( member_Numeral_num0 @ A3 @ A )
=> ? [B9: product_unit] :
( ( member_Product_unit @ B9 @ B )
& ( R @ A3 @ B9 ) ) )
=> ( ! [A1: numeral_num0,A22: numeral_num0,B3: product_unit] :
( ( member_Numeral_num0 @ A1 @ A )
=> ( ( member_Numeral_num0 @ A22 @ A )
=> ( ( member_Product_unit @ B3 @ B )
=> ( ( R @ A1 @ B3 )
=> ( ( R @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite6454714172617411596l_num0 @ A ) @ ( finite410649719033368117t_unit @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_342_card__le__if__inj__on__rel,axiom,
! [B: set_Numeral_num0,A: set_literal,R: literal > numeral_num0 > $o] :
( ( finite1111429032697314573l_num0 @ B )
=> ( ! [A3: literal] :
( ( member_literal @ A3 @ A )
=> ? [B9: numeral_num0] :
( ( member_Numeral_num0 @ B9 @ B )
& ( R @ A3 @ B9 ) ) )
=> ( ! [A1: literal,A22: literal,B3: numeral_num0] :
( ( member_literal @ A1 @ A )
=> ( ( member_literal @ A22 @ A )
=> ( ( member_Numeral_num0 @ B3 @ B )
=> ( ( R @ A1 @ B3 )
=> ( ( R @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_literal @ A ) @ ( finite6454714172617411596l_num0 @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_343_card__le__if__inj__on__rel,axiom,
! [B: set_literal,A: set_literal,R: literal > literal > $o] :
( ( finite5847741373460823677iteral @ B )
=> ( ! [A3: literal] :
( ( member_literal @ A3 @ A )
=> ? [B9: literal] :
( ( member_literal @ B9 @ B )
& ( R @ A3 @ B9 ) ) )
=> ( ! [A1: literal,A22: literal,B3: literal] :
( ( member_literal @ A1 @ A )
=> ( ( member_literal @ A22 @ A )
=> ( ( member_literal @ B3 @ B )
=> ( ( R @ A1 @ B3 )
=> ( ( R @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_literal @ A ) @ ( finite_card_literal @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_344_GreatestI__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B2 ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_nat
thf(fact_345_Greatest__le__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B2 ) )
=> ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).
% Greatest_le_nat
thf(fact_346_GreatestI__ex__nat,axiom,
! [P: nat > $o,B2: nat] :
( ? [X_12: nat] : ( P @ X_12 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B2 ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_347_card__mono,axiom,
! [B: set_Numeral_num0,A: set_Numeral_num0] :
( ( finite1111429032697314573l_num0 @ B )
=> ( ( ord_le5200684351691877604l_num0 @ A @ B )
=> ( ord_less_eq_nat @ ( finite6454714172617411596l_num0 @ A ) @ ( finite6454714172617411596l_num0 @ B ) ) ) ) ).
% card_mono
thf(fact_348_card__mono,axiom,
! [B: set_literal,A: set_literal] :
( ( finite5847741373460823677iteral @ B )
=> ( ( ord_le7307670543136651348iteral @ A @ B )
=> ( ord_less_eq_nat @ ( finite_card_literal @ A ) @ ( finite_card_literal @ B ) ) ) ) ).
% card_mono
thf(fact_349_card__mono,axiom,
! [B: set_Numeral_num1,A: set_Numeral_num1] :
( ( finite1111429032697314574l_num1 @ B )
=> ( ( ord_le5200684355995106405l_num1 @ A @ B )
=> ( ord_less_eq_nat @ ( finite6454714172617411597l_num1 @ A ) @ ( finite6454714172617411597l_num1 @ B ) ) ) ) ).
% card_mono
thf(fact_350_card__mono,axiom,
! [B: set_Product_unit,A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ B )
=> ( ( ord_le3507040750410214029t_unit @ A @ B )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite410649719033368117t_unit @ B ) ) ) ) ).
% card_mono
thf(fact_351_card__mono,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ).
% card_mono
thf(fact_352_card__seteq,axiom,
! [B: set_Numeral_num0,A: set_Numeral_num0] :
( ( finite1111429032697314573l_num0 @ B )
=> ( ( ord_le5200684351691877604l_num0 @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite6454714172617411596l_num0 @ B ) @ ( finite6454714172617411596l_num0 @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_353_card__seteq,axiom,
! [B: set_literal,A: set_literal] :
( ( finite5847741373460823677iteral @ B )
=> ( ( ord_le7307670543136651348iteral @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_literal @ B ) @ ( finite_card_literal @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_354_card__seteq,axiom,
! [B: set_Numeral_num1,A: set_Numeral_num1] :
( ( finite1111429032697314574l_num1 @ B )
=> ( ( ord_le5200684355995106405l_num1 @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite6454714172617411597l_num1 @ B ) @ ( finite6454714172617411597l_num1 @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_355_card__seteq,axiom,
! [B: set_Product_unit,A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ B )
=> ( ( ord_le3507040750410214029t_unit @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ B ) @ ( finite410649719033368117t_unit @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_356_card__seteq,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_nat @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_357_exists__subset__between,axiom,
! [A: set_Numeral_num0,N5: nat,C2: set_Numeral_num0] :
( ( ord_less_eq_nat @ ( finite6454714172617411596l_num0 @ A ) @ N5 )
=> ( ( ord_less_eq_nat @ N5 @ ( finite6454714172617411596l_num0 @ C2 ) )
=> ( ( ord_le5200684351691877604l_num0 @ A @ C2 )
=> ( ( finite1111429032697314573l_num0 @ C2 )
=> ? [B8: set_Numeral_num0] :
( ( ord_le5200684351691877604l_num0 @ A @ B8 )
& ( ord_le5200684351691877604l_num0 @ B8 @ C2 )
& ( ( finite6454714172617411596l_num0 @ B8 )
= N5 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_358_exists__subset__between,axiom,
! [A: set_literal,N5: nat,C2: set_literal] :
( ( ord_less_eq_nat @ ( finite_card_literal @ A ) @ N5 )
=> ( ( ord_less_eq_nat @ N5 @ ( finite_card_literal @ C2 ) )
=> ( ( ord_le7307670543136651348iteral @ A @ C2 )
=> ( ( finite5847741373460823677iteral @ C2 )
=> ? [B8: set_literal] :
( ( ord_le7307670543136651348iteral @ A @ B8 )
& ( ord_le7307670543136651348iteral @ B8 @ C2 )
& ( ( finite_card_literal @ B8 )
= N5 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_359_exists__subset__between,axiom,
! [A: set_Numeral_num1,N5: nat,C2: set_Numeral_num1] :
( ( ord_less_eq_nat @ ( finite6454714172617411597l_num1 @ A ) @ N5 )
=> ( ( ord_less_eq_nat @ N5 @ ( finite6454714172617411597l_num1 @ C2 ) )
=> ( ( ord_le5200684355995106405l_num1 @ A @ C2 )
=> ( ( finite1111429032697314574l_num1 @ C2 )
=> ? [B8: set_Numeral_num1] :
( ( ord_le5200684355995106405l_num1 @ A @ B8 )
& ( ord_le5200684355995106405l_num1 @ B8 @ C2 )
& ( ( finite6454714172617411597l_num1 @ B8 )
= N5 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_360_exists__subset__between,axiom,
! [A: set_Product_unit,N5: nat,C2: set_Product_unit] :
( ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ N5 )
=> ( ( ord_less_eq_nat @ N5 @ ( finite410649719033368117t_unit @ C2 ) )
=> ( ( ord_le3507040750410214029t_unit @ A @ C2 )
=> ( ( finite4290736615968046902t_unit @ C2 )
=> ? [B8: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A @ B8 )
& ( ord_le3507040750410214029t_unit @ B8 @ C2 )
& ( ( finite410649719033368117t_unit @ B8 )
= N5 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_361_exists__subset__between,axiom,
! [A: set_nat,N5: nat,C2: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ N5 )
=> ( ( ord_less_eq_nat @ N5 @ ( finite_card_nat @ C2 ) )
=> ( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( finite_finite_nat @ C2 )
=> ? [B8: set_nat] :
( ( ord_less_eq_set_nat @ A @ B8 )
& ( ord_less_eq_set_nat @ B8 @ C2 )
& ( ( finite_card_nat @ B8 )
= N5 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_362_obtain__subset__with__card__n,axiom,
! [N5: nat,S: set_Numeral_num0] :
( ( ord_less_eq_nat @ N5 @ ( finite6454714172617411596l_num0 @ S ) )
=> ~ ! [T4: set_Numeral_num0] :
( ( ord_le5200684351691877604l_num0 @ T4 @ S )
=> ( ( ( finite6454714172617411596l_num0 @ T4 )
= N5 )
=> ~ ( finite1111429032697314573l_num0 @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_363_obtain__subset__with__card__n,axiom,
! [N5: nat,S: set_literal] :
( ( ord_less_eq_nat @ N5 @ ( finite_card_literal @ S ) )
=> ~ ! [T4: set_literal] :
( ( ord_le7307670543136651348iteral @ T4 @ S )
=> ( ( ( finite_card_literal @ T4 )
= N5 )
=> ~ ( finite5847741373460823677iteral @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_364_obtain__subset__with__card__n,axiom,
! [N5: nat,S: set_Numeral_num1] :
( ( ord_less_eq_nat @ N5 @ ( finite6454714172617411597l_num1 @ S ) )
=> ~ ! [T4: set_Numeral_num1] :
( ( ord_le5200684355995106405l_num1 @ T4 @ S )
=> ( ( ( finite6454714172617411597l_num1 @ T4 )
= N5 )
=> ~ ( finite1111429032697314574l_num1 @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_365_obtain__subset__with__card__n,axiom,
! [N5: nat,S: set_Product_unit] :
( ( ord_less_eq_nat @ N5 @ ( finite410649719033368117t_unit @ S ) )
=> ~ ! [T4: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ T4 @ S )
=> ( ( ( finite410649719033368117t_unit @ T4 )
= N5 )
=> ~ ( finite4290736615968046902t_unit @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_366_obtain__subset__with__card__n,axiom,
! [N5: nat,S: set_nat] :
( ( ord_less_eq_nat @ N5 @ ( finite_card_nat @ S ) )
=> ~ ! [T4: set_nat] :
( ( ord_less_eq_set_nat @ T4 @ S )
=> ( ( ( finite_card_nat @ T4 )
= N5 )
=> ~ ( finite_finite_nat @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_367_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_Numeral_num0,C2: nat] :
( ! [G: set_Numeral_num0] :
( ( ord_le5200684351691877604l_num0 @ G @ F2 )
=> ( ( finite1111429032697314573l_num0 @ G )
=> ( ord_less_eq_nat @ ( finite6454714172617411596l_num0 @ G ) @ C2 ) ) )
=> ( ( finite1111429032697314573l_num0 @ F2 )
& ( ord_less_eq_nat @ ( finite6454714172617411596l_num0 @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_368_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_literal,C2: nat] :
( ! [G: set_literal] :
( ( ord_le7307670543136651348iteral @ G @ F2 )
=> ( ( finite5847741373460823677iteral @ G )
=> ( ord_less_eq_nat @ ( finite_card_literal @ G ) @ C2 ) ) )
=> ( ( finite5847741373460823677iteral @ F2 )
& ( ord_less_eq_nat @ ( finite_card_literal @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_369_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_Numeral_num1,C2: nat] :
( ! [G: set_Numeral_num1] :
( ( ord_le5200684355995106405l_num1 @ G @ F2 )
=> ( ( finite1111429032697314574l_num1 @ G )
=> ( ord_less_eq_nat @ ( finite6454714172617411597l_num1 @ G ) @ C2 ) ) )
=> ( ( finite1111429032697314574l_num1 @ F2 )
& ( ord_less_eq_nat @ ( finite6454714172617411597l_num1 @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_370_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_Product_unit,C2: nat] :
( ! [G: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ G @ F2 )
=> ( ( finite4290736615968046902t_unit @ G )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ G ) @ C2 ) ) )
=> ( ( finite4290736615968046902t_unit @ F2 )
& ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_371_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_nat,C2: nat] :
( ! [G: set_nat] :
( ( ord_less_eq_set_nat @ G @ F2 )
=> ( ( finite_finite_nat @ G )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G ) @ C2 ) ) )
=> ( ( finite_finite_nat @ F2 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_372_card__less__sym__Diff,axiom,
! [A: set_Numeral_num0,B: set_Numeral_num0] :
( ( finite1111429032697314573l_num0 @ A )
=> ( ( finite1111429032697314573l_num0 @ B )
=> ( ( ord_less_nat @ ( finite6454714172617411596l_num0 @ A ) @ ( finite6454714172617411596l_num0 @ B ) )
=> ( ord_less_nat @ ( finite6454714172617411596l_num0 @ ( minus_8146479927826647979l_num0 @ A @ B ) ) @ ( finite6454714172617411596l_num0 @ ( minus_8146479927826647979l_num0 @ B @ A ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_373_card__less__sym__Diff,axiom,
! [A: set_literal,B: set_literal] :
( ( finite5847741373460823677iteral @ A )
=> ( ( finite5847741373460823677iteral @ B )
=> ( ( ord_less_nat @ ( finite_card_literal @ A ) @ ( finite_card_literal @ B ) )
=> ( ord_less_nat @ ( finite_card_literal @ ( minus_7832829386415567259iteral @ A @ B ) ) @ ( finite_card_literal @ ( minus_7832829386415567259iteral @ B @ A ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_374_card__less__sym__Diff,axiom,
! [A: set_Numeral_num1,B: set_Numeral_num1] :
( ( finite1111429032697314574l_num1 @ A )
=> ( ( finite1111429032697314574l_num1 @ B )
=> ( ( ord_less_nat @ ( finite6454714172617411597l_num1 @ A ) @ ( finite6454714172617411597l_num1 @ B ) )
=> ( ord_less_nat @ ( finite6454714172617411597l_num1 @ ( minus_8146479932129876780l_num1 @ A @ B ) ) @ ( finite6454714172617411597l_num1 @ ( minus_8146479932129876780l_num1 @ B @ A ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_375_card__less__sym__Diff,axiom,
! [A: set_Product_unit,B: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( finite4290736615968046902t_unit @ B )
=> ( ( ord_less_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite410649719033368117t_unit @ B ) )
=> ( ord_less_nat @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ B ) ) @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ B @ A ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_376_card__less__sym__Diff,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( ord_less_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ B ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B @ A ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_377_card__le__sym__Diff,axiom,
! [A: set_Numeral_num0,B: set_Numeral_num0] :
( ( finite1111429032697314573l_num0 @ A )
=> ( ( finite1111429032697314573l_num0 @ B )
=> ( ( ord_less_eq_nat @ ( finite6454714172617411596l_num0 @ A ) @ ( finite6454714172617411596l_num0 @ B ) )
=> ( ord_less_eq_nat @ ( finite6454714172617411596l_num0 @ ( minus_8146479927826647979l_num0 @ A @ B ) ) @ ( finite6454714172617411596l_num0 @ ( minus_8146479927826647979l_num0 @ B @ A ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_378_card__le__sym__Diff,axiom,
! [A: set_literal,B: set_literal] :
( ( finite5847741373460823677iteral @ A )
=> ( ( finite5847741373460823677iteral @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_literal @ A ) @ ( finite_card_literal @ B ) )
=> ( ord_less_eq_nat @ ( finite_card_literal @ ( minus_7832829386415567259iteral @ A @ B ) ) @ ( finite_card_literal @ ( minus_7832829386415567259iteral @ B @ A ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_379_card__le__sym__Diff,axiom,
! [A: set_Numeral_num1,B: set_Numeral_num1] :
( ( finite1111429032697314574l_num1 @ A )
=> ( ( finite1111429032697314574l_num1 @ B )
=> ( ( ord_less_eq_nat @ ( finite6454714172617411597l_num1 @ A ) @ ( finite6454714172617411597l_num1 @ B ) )
=> ( ord_less_eq_nat @ ( finite6454714172617411597l_num1 @ ( minus_8146479932129876780l_num1 @ A @ B ) ) @ ( finite6454714172617411597l_num1 @ ( minus_8146479932129876780l_num1 @ B @ A ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_380_card__le__sym__Diff,axiom,
! [A: set_Product_unit,B: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( finite4290736615968046902t_unit @ B )
=> ( ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite410649719033368117t_unit @ B ) )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ B ) ) @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ B @ A ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_381_card__le__sym__Diff,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ B ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B @ A ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_382_psubset__card__mono,axiom,
! [B: set_Numeral_num0,A: set_Numeral_num0] :
( ( finite1111429032697314573l_num0 @ B )
=> ( ( ord_le526730871819019248l_num0 @ A @ B )
=> ( ord_less_nat @ ( finite6454714172617411596l_num0 @ A ) @ ( finite6454714172617411596l_num0 @ B ) ) ) ) ).
% psubset_card_mono
thf(fact_383_psubset__card__mono,axiom,
! [B: set_literal,A: set_literal] :
( ( finite5847741373460823677iteral @ B )
=> ( ( ord_less_set_literal @ A @ B )
=> ( ord_less_nat @ ( finite_card_literal @ A ) @ ( finite_card_literal @ B ) ) ) ) ).
% psubset_card_mono
thf(fact_384_psubset__card__mono,axiom,
! [B: set_Numeral_num1,A: set_Numeral_num1] :
( ( finite1111429032697314574l_num1 @ B )
=> ( ( ord_le526730876122248049l_num1 @ A @ B )
=> ( ord_less_nat @ ( finite6454714172617411597l_num1 @ A ) @ ( finite6454714172617411597l_num1 @ B ) ) ) ) ).
% psubset_card_mono
thf(fact_385_psubset__card__mono,axiom,
! [B: set_Product_unit,A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ B )
=> ( ( ord_le8056459307392131481t_unit @ A @ B )
=> ( ord_less_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite410649719033368117t_unit @ B ) ) ) ) ).
% psubset_card_mono
thf(fact_386_psubset__card__mono,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_set_nat @ A @ B )
=> ( ord_less_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ).
% psubset_card_mono
thf(fact_387_card__Diff__subset__Int,axiom,
! [A: set_Numeral_num0,B: set_Numeral_num0] :
( ( finite1111429032697314573l_num0 @ ( inf_in6354261966906920242l_num0 @ A @ B ) )
=> ( ( finite6454714172617411596l_num0 @ ( minus_8146479927826647979l_num0 @ A @ B ) )
= ( minus_minus_nat @ ( finite6454714172617411596l_num0 @ A ) @ ( finite6454714172617411596l_num0 @ ( inf_in6354261966906920242l_num0 @ A @ B ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_388_card__Diff__subset__Int,axiom,
! [A: set_literal,B: set_literal] :
( ( finite5847741373460823677iteral @ ( inf_inf_set_literal @ A @ B ) )
=> ( ( finite_card_literal @ ( minus_7832829386415567259iteral @ A @ B ) )
= ( minus_minus_nat @ ( finite_card_literal @ A ) @ ( finite_card_literal @ ( inf_inf_set_literal @ A @ B ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_389_card__Diff__subset__Int,axiom,
! [A: set_Numeral_num1,B: set_Numeral_num1] :
( ( finite1111429032697314574l_num1 @ ( inf_in6354261971210149043l_num1 @ A @ B ) )
=> ( ( finite6454714172617411597l_num1 @ ( minus_8146479932129876780l_num1 @ A @ B ) )
= ( minus_minus_nat @ ( finite6454714172617411597l_num1 @ A ) @ ( finite6454714172617411597l_num1 @ ( inf_in6354261971210149043l_num1 @ A @ B ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_390_card__Diff__subset__Int,axiom,
! [A: set_Product_unit,B: set_Product_unit] :
( ( finite4290736615968046902t_unit @ ( inf_in4660618365625256667t_unit @ A @ B ) )
=> ( ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ B ) )
= ( minus_minus_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite410649719033368117t_unit @ ( inf_in4660618365625256667t_unit @ A @ B ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_391_card__Diff__subset__Int,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ ( inf_inf_set_nat @ A @ B ) )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ B ) )
= ( minus_minus_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ ( inf_inf_set_nat @ A @ B ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_392_surj__card__le,axiom,
! [A: set_Numeral_num0,B: set_Numeral_num0,F: numeral_num0 > numeral_num0] :
( ( finite1111429032697314573l_num0 @ A )
=> ( ( ord_le5200684351691877604l_num0 @ B @ ( image_2832974300507296261l_num0 @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite6454714172617411596l_num0 @ B ) @ ( finite6454714172617411596l_num0 @ A ) ) ) ) ).
% surj_card_le
thf(fact_393_surj__card__le,axiom,
! [A: set_literal,B: set_Numeral_num0,F: literal > numeral_num0] :
( ( finite5847741373460823677iteral @ A )
=> ( ( ord_le5200684351691877604l_num0 @ B @ ( image_3608546570274595605l_num0 @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite6454714172617411596l_num0 @ B ) @ ( finite_card_literal @ A ) ) ) ) ).
% surj_card_le
thf(fact_394_surj__card__le,axiom,
! [A: set_Numeral_num1,B: set_Numeral_num0,F: numeral_num1 > numeral_num0] :
( ( finite1111429032697314574l_num1 @ A )
=> ( ( ord_le5200684351691877604l_num0 @ B @ ( image_6783865936125884740l_num0 @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite6454714172617411596l_num0 @ B ) @ ( finite6454714172617411597l_num1 @ A ) ) ) ) ).
% surj_card_le
thf(fact_395_surj__card__le,axiom,
! [A: set_Product_unit,B: set_Numeral_num0,F: product_unit > numeral_num0] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( ord_le5200684351691877604l_num0 @ B @ ( image_6449127158079674652l_num0 @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite6454714172617411596l_num0 @ B ) @ ( finite410649719033368117t_unit @ A ) ) ) ) ).
% surj_card_le
thf(fact_396_surj__card__le,axiom,
! [A: set_Numeral_num0,B: set_literal,F: numeral_num0 > literal] :
( ( finite1111429032697314573l_num0 @ A )
=> ( ( ord_le7307670543136651348iteral @ B @ ( image_8737817577461598069iteral @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_literal @ B ) @ ( finite6454714172617411596l_num0 @ A ) ) ) ) ).
% surj_card_le
thf(fact_397_surj__card__le,axiom,
! [A: set_literal,B: set_literal,F: literal > literal] :
( ( finite5847741373460823677iteral @ A )
=> ( ( ord_le7307670543136651348iteral @ B @ ( image_8195128725298311301iteral @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_literal @ B ) @ ( finite_card_literal @ A ) ) ) ) ).
% surj_card_le
thf(fact_398_surj__card__le,axiom,
! [A: set_Numeral_num1,B: set_literal,F: numeral_num1 > literal] :
( ( finite1111429032697314574l_num1 @ A )
=> ( ( ord_le7307670543136651348iteral @ B @ ( image_5852747068178070836iteral @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_literal @ B ) @ ( finite6454714172617411597l_num1 @ A ) ) ) ) ).
% surj_card_le
thf(fact_399_surj__card__le,axiom,
! [A: set_Product_unit,B: set_literal,F: product_unit > literal] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( ord_le7307670543136651348iteral @ B @ ( image_5876984745897992460iteral @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_literal @ B ) @ ( finite410649719033368117t_unit @ A ) ) ) ) ).
% surj_card_le
thf(fact_400_surj__card__le,axiom,
! [A: set_Numeral_num0,B: set_Numeral_num1,F: numeral_num0 > numeral_num1] :
( ( finite1111429032697314573l_num0 @ A )
=> ( ( ord_le5200684355995106405l_num1 @ B @ ( image_2832974300507296262l_num1 @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite6454714172617411597l_num1 @ B ) @ ( finite6454714172617411596l_num0 @ A ) ) ) ) ).
% surj_card_le
thf(fact_401_surj__card__le,axiom,
! [A: set_literal,B: set_Numeral_num1,F: literal > numeral_num1] :
( ( finite5847741373460823677iteral @ A )
=> ( ( ord_le5200684355995106405l_num1 @ B @ ( image_3608546570274595606l_num1 @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite6454714172617411597l_num1 @ B ) @ ( finite_card_literal @ A ) ) ) ) ).
% surj_card_le
thf(fact_402_equiv__eval__on__eval__eqI,axiom,
! [I: product_prod_b_nat > set_list_a,Q: relational_fmla_a_b,Q2: relational_fmla_a_b] :
( ( finite_finite_a @ ( relational_adom_b_a @ I ) )
=> ( ( ord_less_eq_set_nat @ ( relational_fv_a_b @ Q ) @ ( relational_fv_a_b @ Q2 ) )
=> ( ( relational_equiv_a_b @ Q @ Q2 )
=> ( ( relati8814510239606734169on_a_b @ ( relational_fv_a_b @ Q2 ) @ Q @ I )
= ( relational_eval_a_b @ Q2 @ I ) ) ) ) ) ).
% equiv_eval_on_eval_eqI
thf(fact_403_image__eqI,axiom,
! [B2: nat,F: nat > nat,X2: nat,A: set_nat] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_404_IntI,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ A )
=> ( ( member_nat @ C @ B )
=> ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_405_Int__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
= ( ( member_nat @ C @ A )
& ( member_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_406_le__inf__iff,axiom,
! [X2: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ ( inf_inf_set_nat @ Y @ Z ) )
= ( ( ord_less_eq_set_nat @ X2 @ Y )
& ( ord_less_eq_set_nat @ X2 @ Z ) ) ) ).
% le_inf_iff
thf(fact_407_le__inf__iff,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ Y @ Z ) )
= ( ( ord_less_eq_nat @ X2 @ Y )
& ( ord_less_eq_nat @ X2 @ Z ) ) ) ).
% le_inf_iff
thf(fact_408_inf_Obounded__iff,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C ) )
= ( ( ord_less_eq_set_nat @ A2 @ B2 )
& ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_409_inf_Obounded__iff,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C ) )
= ( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_410_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_411_finite__Int,axiom,
! [F2: set_nat,G2: set_nat] :
( ( ( finite_finite_nat @ F2 )
| ( finite_finite_nat @ G2 ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F2 @ G2 ) ) ) ).
% finite_Int
thf(fact_412_Int__subset__iff,axiom,
! [C2: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) )
= ( ( ord_less_eq_set_nat @ C2 @ A )
& ( ord_less_eq_set_nat @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_413_image__Int__subset,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( inf_inf_set_nat @ A @ B ) ) @ ( inf_inf_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_414_IntE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ~ ( ( member_nat @ C @ A )
=> ~ ( member_nat @ C @ B ) ) ) ).
% IntE
thf(fact_415_IntD1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ( member_nat @ C @ A ) ) ).
% IntD1
thf(fact_416_IntD2,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ( member_nat @ C @ B ) ) ).
% IntD2
thf(fact_417_imageI,axiom,
! [X2: nat,A: set_nat,F: nat > nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ ( image_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_418_image__iff,axiom,
! [Z: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ Z @ ( image_nat_nat @ F @ A ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( Z
= ( F @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_419_bex__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( image_nat_nat @ F @ A ) )
& ( P @ X5 ) )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_420_image__cong,axiom,
! [M2: set_nat,N4: set_nat,F: nat > nat,G3: nat > nat] :
( ( M2 = N4 )
=> ( ! [X: nat] :
( ( member_nat @ X @ N4 )
=> ( ( F @ X )
= ( G3 @ X ) ) )
=> ( ( image_nat_nat @ F @ M2 )
= ( image_nat_nat @ G3 @ N4 ) ) ) ) ).
% image_cong
thf(fact_421_ball__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ! [X: nat] :
( ( member_nat @ X @ ( image_nat_nat @ F @ A ) )
=> ( P @ X ) )
=> ! [X5: nat] :
( ( member_nat @ X5 @ A )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_422_rev__image__eqI,axiom,
! [X2: nat,A: set_nat,B2: nat,F: nat > nat] :
( ( member_nat @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_423_inf__sup__ord_I2_J,axiom,
! [X2: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_424_inf__sup__ord_I2_J,axiom,
! [X2: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_425_inf__sup__ord_I1_J,axiom,
! [X2: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_426_inf__sup__ord_I1_J,axiom,
! [X2: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_427_inf__le1,axiom,
! [X2: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y ) @ X2 ) ).
% inf_le1
thf(fact_428_inf__le1,axiom,
! [X2: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y ) @ X2 ) ).
% inf_le1
thf(fact_429_inf__le2,axiom,
! [X2: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y ) @ Y ) ).
% inf_le2
thf(fact_430_inf__le2,axiom,
! [X2: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y ) @ Y ) ).
% inf_le2
thf(fact_431_le__infE,axiom,
! [X2: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_set_nat @ X2 @ A2 )
=> ~ ( ord_less_eq_set_nat @ X2 @ B2 ) ) ) ).
% le_infE
thf(fact_432_le__infE,axiom,
! [X2: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_nat @ X2 @ A2 )
=> ~ ( ord_less_eq_nat @ X2 @ B2 ) ) ) ).
% le_infE
thf(fact_433_le__infI,axiom,
! [X2: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ A2 )
=> ( ( ord_less_eq_set_nat @ X2 @ B2 )
=> ( ord_less_eq_set_nat @ X2 @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_434_le__infI,axiom,
! [X2: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X2 @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ B2 )
=> ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_435_inf__mono,axiom,
! [A2: set_nat,C: set_nat,B2: set_nat,D3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( ord_less_eq_set_nat @ B2 @ D3 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ ( inf_inf_set_nat @ C @ D3 ) ) ) ) ).
% inf_mono
thf(fact_436_inf__mono,axiom,
! [A2: nat,C: nat,B2: nat,D3: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ D3 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ ( inf_inf_nat @ C @ D3 ) ) ) ) ).
% inf_mono
thf(fact_437_le__infI1,axiom,
! [A2: set_nat,X2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ X2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ X2 ) ) ).
% le_infI1
thf(fact_438_le__infI1,axiom,
! [A2: nat,X2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ X2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X2 ) ) ).
% le_infI1
thf(fact_439_le__infI2,axiom,
! [B2: set_nat,X2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ X2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ X2 ) ) ).
% le_infI2
thf(fact_440_le__infI2,axiom,
! [B2: nat,X2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ X2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X2 ) ) ).
% le_infI2
thf(fact_441_inf_OorderE,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( A2
= ( inf_inf_set_nat @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_442_inf_OorderE,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2
= ( inf_inf_nat @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_443_inf_OorderI,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2
= ( inf_inf_set_nat @ A2 @ B2 ) )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_444_inf_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( inf_inf_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_445_inf__unique,axiom,
! [F: set_nat > set_nat > set_nat,X2: set_nat,Y: set_nat] :
( ! [X: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( F @ X @ Y3 ) @ X )
=> ( ! [X: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( F @ X @ Y3 ) @ Y3 )
=> ( ! [X: set_nat,Y3: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y3 )
=> ( ( ord_less_eq_set_nat @ X @ Z3 )
=> ( ord_less_eq_set_nat @ X @ ( F @ Y3 @ Z3 ) ) ) )
=> ( ( inf_inf_set_nat @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_446_inf__unique,axiom,
! [F: nat > nat > nat,X2: nat,Y: nat] :
( ! [X: nat,Y3: nat] : ( ord_less_eq_nat @ ( F @ X @ Y3 ) @ X )
=> ( ! [X: nat,Y3: nat] : ( ord_less_eq_nat @ ( F @ X @ Y3 ) @ Y3 )
=> ( ! [X: nat,Y3: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ( ord_less_eq_nat @ X @ Z3 )
=> ( ord_less_eq_nat @ X @ ( F @ Y3 @ Z3 ) ) ) )
=> ( ( inf_inf_nat @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_447_le__iff__inf,axiom,
( ord_less_eq_set_nat
= ( ^ [X4: set_nat,Y4: set_nat] :
( ( inf_inf_set_nat @ X4 @ Y4 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_448_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y4: nat] :
( ( inf_inf_nat @ X4 @ Y4 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_449_inf_Oabsorb1,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( inf_inf_set_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_450_inf_Oabsorb1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_451_inf_Oabsorb2,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_452_inf_Oabsorb2,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_453_inf__absorb1,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ( inf_inf_set_nat @ X2 @ Y )
= X2 ) ) ).
% inf_absorb1
thf(fact_454_inf__absorb1,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( inf_inf_nat @ X2 @ Y )
= X2 ) ) ).
% inf_absorb1
thf(fact_455_inf__absorb2,axiom,
! [Y: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X2 )
=> ( ( inf_inf_set_nat @ X2 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_456_inf__absorb2,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( inf_inf_nat @ X2 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_457_inf_OboundedE,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C ) )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ~ ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_458_inf_OboundedE,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C ) )
=> ~ ( ( ord_less_eq_nat @ A2 @ B2 )
=> ~ ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_459_inf_OboundedI,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_460_inf_OboundedI,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ A2 @ C )
=> ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_461_inf__greatest,axiom,
! [X2: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ( ord_less_eq_set_nat @ X2 @ Z )
=> ( ord_less_eq_set_nat @ X2 @ ( inf_inf_set_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_462_inf__greatest,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ X2 @ Z )
=> ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_463_inf_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( A4
= ( inf_inf_set_nat @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_464_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( A4
= ( inf_inf_nat @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_465_inf_Ocobounded1,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_466_inf_Ocobounded1,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_467_inf_Ocobounded2,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_468_inf_Ocobounded2,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_469_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( inf_inf_set_nat @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_470_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( inf_inf_nat @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_471_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [B4: set_nat,A4: set_nat] :
( ( inf_inf_set_nat @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_472_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( inf_inf_nat @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_473_inf_OcoboundedI1,axiom,
! [A2: set_nat,C: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_474_inf_OcoboundedI1,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_475_inf_OcoboundedI2,axiom,
! [B2: set_nat,C: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_476_inf_OcoboundedI2,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_477_inf_Ostrict__coboundedI2,axiom,
! [B2: set_nat,C: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ B2 @ C )
=> ( ord_less_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_478_inf_Ostrict__coboundedI2,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_479_inf_Ostrict__coboundedI1,axiom,
! [A2: set_nat,C: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ C )
=> ( ord_less_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_480_inf_Ostrict__coboundedI1,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_nat @ A2 @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_481_inf_Ostrict__order__iff,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( A4
= ( inf_inf_set_nat @ A4 @ B4 ) )
& ( A4 != B4 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_482_inf_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] :
( ( A4
= ( inf_inf_nat @ A4 @ B4 ) )
& ( A4 != B4 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_483_inf_Ostrict__boundedE,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C ) )
=> ~ ( ( ord_less_set_nat @ A2 @ B2 )
=> ~ ( ord_less_set_nat @ A2 @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_484_inf_Ostrict__boundedE,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( inf_inf_nat @ B2 @ C ) )
=> ~ ( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ A2 @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_485_inf_Oabsorb4,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ B2 @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb4
thf(fact_486_inf_Oabsorb4,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb4
thf(fact_487_inf_Oabsorb3,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ( inf_inf_set_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb3
thf(fact_488_inf_Oabsorb3,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb3
thf(fact_489_less__infI2,axiom,
! [B2: set_nat,X2: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ B2 @ X2 )
=> ( ord_less_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ X2 ) ) ).
% less_infI2
thf(fact_490_less__infI2,axiom,
! [B2: nat,X2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ X2 )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X2 ) ) ).
% less_infI2
thf(fact_491_less__infI1,axiom,
! [A2: set_nat,X2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ X2 )
=> ( ord_less_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ X2 ) ) ).
% less_infI1
thf(fact_492_less__infI1,axiom,
! [A2: nat,X2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ X2 )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X2 ) ) ).
% less_infI1
thf(fact_493_all__subset__image,axiom,
! [F: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ! [B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A )
=> ( P @ ( image_nat_nat @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_494_image__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_495_image__subsetI,axiom,
! [A: set_nat,F: nat > nat,B: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_nat @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_496_subset__imageE,axiom,
! [B: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ~ ! [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A )
=> ( B
!= ( image_nat_nat @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_497_image__subset__iff,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_nat @ ( F @ X4 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_498_subset__image__iff,axiom,
! [B: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_499_Int__mono,axiom,
! [A: set_nat,C2: set_nat,B: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ D )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( inf_inf_set_nat @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_500_Int__lower1,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_501_Int__lower2,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_502_Int__absorb1,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( inf_inf_set_nat @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_503_Int__absorb2,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( inf_inf_set_nat @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_504_Int__greatest,axiom,
! [C2: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A )
=> ( ( ord_less_eq_set_nat @ C2 @ B )
=> ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_505_Int__Collect__mono,axiom,
! [A: set_nat,B: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_506_Int__Diff,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C2 )
= ( inf_inf_set_nat @ A @ ( minus_minus_set_nat @ B @ C2 ) ) ) ).
% Int_Diff
thf(fact_507_Diff__Int2,axiom,
! [A: set_nat,C2: set_nat,B: set_nat] :
( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A @ C2 ) @ ( inf_inf_set_nat @ B @ C2 ) )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ A @ C2 ) @ B ) ) ).
% Diff_Int2
thf(fact_508_Diff__Diff__Int,axiom,
! [A: set_nat,B: set_nat] :
( ( minus_minus_set_nat @ A @ ( minus_minus_set_nat @ A @ B ) )
= ( inf_inf_set_nat @ A @ B ) ) ).
% Diff_Diff_Int
thf(fact_509_Diff__Int__distrib,axiom,
! [C2: set_nat,A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ C2 @ ( minus_minus_set_nat @ A @ B ) )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ C2 @ A ) @ ( inf_inf_set_nat @ C2 @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_510_Diff__Int__distrib2,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ ( minus_minus_set_nat @ A @ B ) @ C2 )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ A @ C2 ) @ ( inf_inf_set_nat @ B @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_511_Relational__Calculus_Oequiv__def,axiom,
( relational_equiv_a_b
= ( ^ [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
! [I5: product_prod_b_nat > set_list_a,Sigma4: nat > a] :
( ( finite_finite_a @ ( relational_adom_b_a @ I5 ) )
=> ( ( relational_sat_a_b @ Q1 @ I5 @ Sigma4 )
= ( relational_sat_a_b @ Q22 @ I5 @ Sigma4 ) ) ) ) ) ).
% Relational_Calculus.equiv_def
thf(fact_512_equiv__eval__eqI,axiom,
! [I: product_prod_b_nat > set_list_a,Q: relational_fmla_a_b,Q2: relational_fmla_a_b] :
( ( finite_finite_a @ ( relational_adom_b_a @ I ) )
=> ( ( ( relational_fv_a_b @ Q )
= ( relational_fv_a_b @ Q2 ) )
=> ( ( relational_equiv_a_b @ Q @ Q2 )
=> ( ( relational_eval_a_b @ Q @ I )
= ( relational_eval_a_b @ Q2 @ I ) ) ) ) ) ).
% equiv_eval_eqI
thf(fact_513_finite__surj,axiom,
! [A: set_nat,B: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ( finite_finite_nat @ B ) ) ) ).
% finite_surj
thf(fact_514_finite__subset__image,axiom,
! [B: set_nat,F: nat > nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ? [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A )
& ( finite_finite_nat @ C4 )
& ( B
= ( image_nat_nat @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_515_ex__finite__subset__image,axiom,
! [F: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ? [B5: set_nat] :
( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A ) )
& ( P @ B5 ) ) )
= ( ? [B5: set_nat] :
( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A )
& ( P @ ( image_nat_nat @ F @ B5 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_516_all__finite__subset__image,axiom,
! [F: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ! [B5: set_nat] :
( ( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A ) ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_nat] :
( ( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A ) )
=> ( P @ ( image_nat_nat @ F @ B5 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_517_image__diff__subset,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) @ ( image_nat_nat @ F @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% image_diff_subset
thf(fact_518_card__image__le,axiom,
! [A: set_Numeral_num0,F: numeral_num0 > nat] :
( ( finite1111429032697314573l_num0 @ A )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_8797574156932312687m0_nat @ F @ A ) ) @ ( finite6454714172617411596l_num0 @ A ) ) ) ).
% card_image_le
thf(fact_519_card__image__le,axiom,
! [A: set_literal,F: literal > nat] :
( ( finite5847741373460823677iteral @ A )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_literal_nat @ F @ A ) ) @ ( finite_card_literal @ A ) ) ) ).
% card_image_le
thf(fact_520_card__image__le,axiom,
! [A: set_Numeral_num1,F: numeral_num1 > nat] :
( ( finite1111429032697314574l_num1 @ A )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_809646449033931376m1_nat @ F @ A ) ) @ ( finite6454714172617411597l_num1 @ A ) ) ) ).
% card_image_le
thf(fact_521_card__image__le,axiom,
! [A: set_Product_unit,F: product_unit > nat] :
( ( finite4290736615968046902t_unit @ A )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_875570014554754200it_nat @ F @ A ) ) @ ( finite410649719033368117t_unit @ A ) ) ) ).
% card_image_le
thf(fact_522_card__image__le,axiom,
! [A: set_Numeral_num0,F: numeral_num0 > numeral_num0] :
( ( finite1111429032697314573l_num0 @ A )
=> ( ord_less_eq_nat @ ( finite6454714172617411596l_num0 @ ( image_2832974300507296261l_num0 @ F @ A ) ) @ ( finite6454714172617411596l_num0 @ A ) ) ) ).
% card_image_le
thf(fact_523_card__image__le,axiom,
! [A: set_literal,F: literal > numeral_num0] :
( ( finite5847741373460823677iteral @ A )
=> ( ord_less_eq_nat @ ( finite6454714172617411596l_num0 @ ( image_3608546570274595605l_num0 @ F @ A ) ) @ ( finite_card_literal @ A ) ) ) ).
% card_image_le
thf(fact_524_card__image__le,axiom,
! [A: set_Numeral_num1,F: numeral_num1 > numeral_num0] :
( ( finite1111429032697314574l_num1 @ A )
=> ( ord_less_eq_nat @ ( finite6454714172617411596l_num0 @ ( image_6783865936125884740l_num0 @ F @ A ) ) @ ( finite6454714172617411597l_num1 @ A ) ) ) ).
% card_image_le
thf(fact_525_card__image__le,axiom,
! [A: set_Product_unit,F: product_unit > numeral_num0] :
( ( finite4290736615968046902t_unit @ A )
=> ( ord_less_eq_nat @ ( finite6454714172617411596l_num0 @ ( image_6449127158079674652l_num0 @ F @ A ) ) @ ( finite410649719033368117t_unit @ A ) ) ) ).
% card_image_le
thf(fact_526_card__image__le,axiom,
! [A: set_Numeral_num0,F: numeral_num0 > literal] :
( ( finite1111429032697314573l_num0 @ A )
=> ( ord_less_eq_nat @ ( finite_card_literal @ ( image_8737817577461598069iteral @ F @ A ) ) @ ( finite6454714172617411596l_num0 @ A ) ) ) ).
% card_image_le
thf(fact_527_card__image__le,axiom,
! [A: set_literal,F: literal > literal] :
( ( finite5847741373460823677iteral @ A )
=> ( ord_less_eq_nat @ ( finite_card_literal @ ( image_8195128725298311301iteral @ F @ A ) ) @ ( finite_card_literal @ A ) ) ) ).
% card_image_le
thf(fact_528_infinite__surj,axiom,
! [A: set_nat,F: nat > nat,B: set_nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( image_nat_nat @ F @ B ) )
=> ~ ( finite_finite_nat @ B ) ) ) ).
% infinite_surj
thf(fact_529_Inf__fin_Osemilattice__order__set__axioms,axiom,
lattic3109210760196336428et_nat @ inf_inf_set_nat @ ord_less_eq_set_nat @ ord_less_set_nat ).
% Inf_fin.semilattice_order_set_axioms
thf(fact_530_Inf__fin_Osemilattice__order__set__axioms,axiom,
lattic6009151579333465974et_nat @ inf_inf_nat @ ord_less_eq_nat @ ord_less_nat ).
% Inf_fin.semilattice_order_set_axioms
thf(fact_531_inj__on__iff__card__le,axiom,
! [A: set_Numeral_num0,B: set_Numeral_num0] :
( ( finite1111429032697314573l_num0 @ A )
=> ( ( finite1111429032697314573l_num0 @ B )
=> ( ( ? [F3: numeral_num0 > numeral_num0] :
( ( inj_on1355912882866956657l_num0 @ F3 @ A )
& ( ord_le5200684351691877604l_num0 @ ( image_2832974300507296261l_num0 @ F3 @ A ) @ B ) ) )
= ( ord_less_eq_nat @ ( finite6454714172617411596l_num0 @ A ) @ ( finite6454714172617411596l_num0 @ B ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_532_inj__on__iff__card__le,axiom,
! [A: set_Numeral_num0,B: set_literal] :
( ( finite1111429032697314573l_num0 @ A )
=> ( ( finite5847741373460823677iteral @ B )
=> ( ( ? [F3: numeral_num0 > literal] :
( ( inj_on6217666562437480673iteral @ F3 @ A )
& ( ord_le7307670543136651348iteral @ ( image_8737817577461598069iteral @ F3 @ A ) @ B ) ) )
= ( ord_less_eq_nat @ ( finite6454714172617411596l_num0 @ A ) @ ( finite_card_literal @ B ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_533_inj__on__iff__card__le,axiom,
! [A: set_Numeral_num0,B: set_Numeral_num1] :
( ( finite1111429032697314573l_num0 @ A )
=> ( ( finite1111429032697314574l_num1 @ B )
=> ( ( ? [F3: numeral_num0 > numeral_num1] :
( ( inj_on1355912882866956658l_num1 @ F3 @ A )
& ( ord_le5200684355995106405l_num1 @ ( image_2832974300507296262l_num1 @ F3 @ A ) @ B ) ) )
= ( ord_less_eq_nat @ ( finite6454714172617411596l_num0 @ A ) @ ( finite6454714172617411597l_num1 @ B ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_534_inj__on__iff__card__le,axiom,
! [A: set_Numeral_num0,B: set_Product_unit] :
( ( finite1111429032697314573l_num0 @ A )
=> ( ( finite4290736615968046902t_unit @ B )
=> ( ( ? [F3: numeral_num0 > product_unit] :
( ( inj_on4535220466137688986t_unit @ F3 @ A )
& ( ord_le3507040750410214029t_unit @ ( image_6012281883778028590t_unit @ F3 @ A ) @ B ) ) )
= ( ord_less_eq_nat @ ( finite6454714172617411596l_num0 @ A ) @ ( finite410649719033368117t_unit @ B ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_535_inj__on__iff__card__le,axiom,
! [A: set_literal,B: set_Numeral_num0] :
( ( finite5847741373460823677iteral @ A )
=> ( ( finite1111429032697314573l_num0 @ B )
=> ( ( ? [F3: literal > numeral_num0] :
( ( inj_on1088395555250478209l_num0 @ F3 @ A )
& ( ord_le5200684351691877604l_num0 @ ( image_3608546570274595605l_num0 @ F3 @ A ) @ B ) ) )
= ( ord_less_eq_nat @ ( finite_card_literal @ A ) @ ( finite6454714172617411596l_num0 @ B ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_536_inj__on__iff__card__le,axiom,
! [A: set_literal,B: set_literal] :
( ( finite5847741373460823677iteral @ A )
=> ( ( finite5847741373460823677iteral @ B )
=> ( ( ? [F3: literal > literal] :
( ( inj_on602069361295035377iteral @ F3 @ A )
& ( ord_le7307670543136651348iteral @ ( image_8195128725298311301iteral @ F3 @ A ) @ B ) ) )
= ( ord_less_eq_nat @ ( finite_card_literal @ A ) @ ( finite_card_literal @ B ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_537_inj__on__iff__card__le,axiom,
! [A: set_literal,B: set_Numeral_num1] :
( ( finite5847741373460823677iteral @ A )
=> ( ( finite1111429032697314574l_num1 @ B )
=> ( ( ? [F3: literal > numeral_num1] :
( ( inj_on1088395555250478210l_num1 @ F3 @ A )
& ( ord_le5200684355995106405l_num1 @ ( image_3608546570274595606l_num1 @ F3 @ A ) @ B ) ) )
= ( ord_less_eq_nat @ ( finite_card_literal @ A ) @ ( finite6454714172617411597l_num1 @ B ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_538_inj__on__iff__card__le,axiom,
! [A: set_literal,B: set_Product_unit] :
( ( finite5847741373460823677iteral @ A )
=> ( ( finite4290736615968046902t_unit @ B )
=> ( ( ? [F3: literal > product_unit] :
( ( inj_on4267703138521210538t_unit @ F3 @ A )
& ( ord_le3507040750410214029t_unit @ ( image_6787854153545327934t_unit @ F3 @ A ) @ B ) ) )
= ( ord_less_eq_nat @ ( finite_card_literal @ A ) @ ( finite410649719033368117t_unit @ B ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_539_inj__on__iff__card__le,axiom,
! [A: set_Numeral_num1,B: set_Numeral_num0] :
( ( finite1111429032697314574l_num1 @ A )
=> ( ( finite1111429032697314573l_num0 @ B )
=> ( ( ? [F3: numeral_num1 > numeral_num0] :
( ( inj_on5306804518485545136l_num0 @ F3 @ A )
& ( ord_le5200684351691877604l_num0 @ ( image_6783865936125884740l_num0 @ F3 @ A ) @ B ) ) )
= ( ord_less_eq_nat @ ( finite6454714172617411597l_num1 @ A ) @ ( finite6454714172617411596l_num0 @ B ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_540_inj__on__iff__card__le,axiom,
! [A: set_Numeral_num1,B: set_literal] :
( ( finite1111429032697314574l_num1 @ A )
=> ( ( finite5847741373460823677iteral @ B )
=> ( ( ? [F3: numeral_num1 > literal] :
( ( inj_on3332596053153953440iteral @ F3 @ A )
& ( ord_le7307670543136651348iteral @ ( image_5852747068178070836iteral @ F3 @ A ) @ B ) ) )
= ( ord_less_eq_nat @ ( finite6454714172617411597l_num1 @ A ) @ ( finite_card_literal @ B ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_541_card__inj__on__le,axiom,
! [F: nat > numeral_num0,A: set_nat,B: set_Numeral_num0] :
( ( inj_on3882293653322094177l_num0 @ F @ A )
=> ( ( ord_le5200684351691877604l_num0 @ ( image_5550796612950789325l_num0 @ F @ A ) @ B )
=> ( ( finite1111429032697314573l_num0 @ B )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite6454714172617411596l_num0 @ B ) ) ) ) ) ).
% card_inj_on_le
thf(fact_542_card__inj__on__le,axiom,
! [F: nat > literal,A: set_nat,B: set_literal] :
( ( inj_on_nat_literal @ F @ A )
=> ( ( ord_le7307670543136651348iteral @ ( image_nat_literal @ F @ A ) @ B )
=> ( ( finite5847741373460823677iteral @ B )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_literal @ B ) ) ) ) ) ).
% card_inj_on_le
thf(fact_543_card__inj__on__le,axiom,
! [F: nat > numeral_num1,A: set_nat,B: set_Numeral_num1] :
( ( inj_on3882293653322094178l_num1 @ F @ A )
=> ( ( ord_le5200684355995106405l_num1 @ ( image_5550796612950789326l_num1 @ F @ A ) @ B )
=> ( ( finite1111429032697314574l_num1 @ B )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite6454714172617411597l_num1 @ B ) ) ) ) ) ).
% card_inj_on_le
thf(fact_544_card__inj__on__le,axiom,
! [F: nat > product_unit,A: set_nat,B: set_Product_unit] :
( ( inj_on7061601236592826506t_unit @ F @ A )
=> ( ( ord_le3507040750410214029t_unit @ ( image_8730104196221521654t_unit @ F @ A ) @ B )
=> ( ( finite4290736615968046902t_unit @ B )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite410649719033368117t_unit @ B ) ) ) ) ) ).
% card_inj_on_le
thf(fact_545_card__inj__on__le,axiom,
! [F: numeral_num0 > numeral_num0,A: set_Numeral_num0,B: set_Numeral_num0] :
( ( inj_on1355912882866956657l_num0 @ F @ A )
=> ( ( ord_le5200684351691877604l_num0 @ ( image_2832974300507296261l_num0 @ F @ A ) @ B )
=> ( ( finite1111429032697314573l_num0 @ B )
=> ( ord_less_eq_nat @ ( finite6454714172617411596l_num0 @ A ) @ ( finite6454714172617411596l_num0 @ B ) ) ) ) ) ).
% card_inj_on_le
thf(fact_546_card__inj__on__le,axiom,
! [F: numeral_num0 > literal,A: set_Numeral_num0,B: set_literal] :
( ( inj_on6217666562437480673iteral @ F @ A )
=> ( ( ord_le7307670543136651348iteral @ ( image_8737817577461598069iteral @ F @ A ) @ B )
=> ( ( finite5847741373460823677iteral @ B )
=> ( ord_less_eq_nat @ ( finite6454714172617411596l_num0 @ A ) @ ( finite_card_literal @ B ) ) ) ) ) ).
% card_inj_on_le
thf(fact_547_card__inj__on__le,axiom,
! [F: numeral_num0 > numeral_num1,A: set_Numeral_num0,B: set_Numeral_num1] :
( ( inj_on1355912882866956658l_num1 @ F @ A )
=> ( ( ord_le5200684355995106405l_num1 @ ( image_2832974300507296262l_num1 @ F @ A ) @ B )
=> ( ( finite1111429032697314574l_num1 @ B )
=> ( ord_less_eq_nat @ ( finite6454714172617411596l_num0 @ A ) @ ( finite6454714172617411597l_num1 @ B ) ) ) ) ) ).
% card_inj_on_le
thf(fact_548_card__inj__on__le,axiom,
! [F: numeral_num0 > product_unit,A: set_Numeral_num0,B: set_Product_unit] :
( ( inj_on4535220466137688986t_unit @ F @ A )
=> ( ( ord_le3507040750410214029t_unit @ ( image_6012281883778028590t_unit @ F @ A ) @ B )
=> ( ( finite4290736615968046902t_unit @ B )
=> ( ord_less_eq_nat @ ( finite6454714172617411596l_num0 @ A ) @ ( finite410649719033368117t_unit @ B ) ) ) ) ) ).
% card_inj_on_le
thf(fact_549_card__inj__on__le,axiom,
! [F: literal > numeral_num0,A: set_literal,B: set_Numeral_num0] :
( ( inj_on1088395555250478209l_num0 @ F @ A )
=> ( ( ord_le5200684351691877604l_num0 @ ( image_3608546570274595605l_num0 @ F @ A ) @ B )
=> ( ( finite1111429032697314573l_num0 @ B )
=> ( ord_less_eq_nat @ ( finite_card_literal @ A ) @ ( finite6454714172617411596l_num0 @ B ) ) ) ) ) ).
% card_inj_on_le
thf(fact_550_card__inj__on__le,axiom,
! [F: literal > literal,A: set_literal,B: set_literal] :
( ( inj_on602069361295035377iteral @ F @ A )
=> ( ( ord_le7307670543136651348iteral @ ( image_8195128725298311301iteral @ F @ A ) @ B )
=> ( ( finite5847741373460823677iteral @ B )
=> ( ord_less_eq_nat @ ( finite_card_literal @ A ) @ ( finite_card_literal @ B ) ) ) ) ) ).
% card_inj_on_le
thf(fact_551_card__le__inj,axiom,
! [A: set_Numeral_num0,B: set_Numeral_num0] :
( ( finite1111429032697314573l_num0 @ A )
=> ( ( finite1111429032697314573l_num0 @ B )
=> ( ( ord_less_eq_nat @ ( finite6454714172617411596l_num0 @ A ) @ ( finite6454714172617411596l_num0 @ B ) )
=> ? [F4: numeral_num0 > numeral_num0] :
( ( ord_le5200684351691877604l_num0 @ ( image_2832974300507296261l_num0 @ F4 @ A ) @ B )
& ( inj_on1355912882866956657l_num0 @ F4 @ A ) ) ) ) ) ).
% card_le_inj
thf(fact_552_card__le__inj,axiom,
! [A: set_Numeral_num0,B: set_literal] :
( ( finite1111429032697314573l_num0 @ A )
=> ( ( finite5847741373460823677iteral @ B )
=> ( ( ord_less_eq_nat @ ( finite6454714172617411596l_num0 @ A ) @ ( finite_card_literal @ B ) )
=> ? [F4: numeral_num0 > literal] :
( ( ord_le7307670543136651348iteral @ ( image_8737817577461598069iteral @ F4 @ A ) @ B )
& ( inj_on6217666562437480673iteral @ F4 @ A ) ) ) ) ) ).
% card_le_inj
thf(fact_553_card__le__inj,axiom,
! [A: set_Numeral_num0,B: set_Numeral_num1] :
( ( finite1111429032697314573l_num0 @ A )
=> ( ( finite1111429032697314574l_num1 @ B )
=> ( ( ord_less_eq_nat @ ( finite6454714172617411596l_num0 @ A ) @ ( finite6454714172617411597l_num1 @ B ) )
=> ? [F4: numeral_num0 > numeral_num1] :
( ( ord_le5200684355995106405l_num1 @ ( image_2832974300507296262l_num1 @ F4 @ A ) @ B )
& ( inj_on1355912882866956658l_num1 @ F4 @ A ) ) ) ) ) ).
% card_le_inj
thf(fact_554_card__le__inj,axiom,
! [A: set_Numeral_num0,B: set_Product_unit] :
( ( finite1111429032697314573l_num0 @ A )
=> ( ( finite4290736615968046902t_unit @ B )
=> ( ( ord_less_eq_nat @ ( finite6454714172617411596l_num0 @ A ) @ ( finite410649719033368117t_unit @ B ) )
=> ? [F4: numeral_num0 > product_unit] :
( ( ord_le3507040750410214029t_unit @ ( image_6012281883778028590t_unit @ F4 @ A ) @ B )
& ( inj_on4535220466137688986t_unit @ F4 @ A ) ) ) ) ) ).
% card_le_inj
thf(fact_555_card__le__inj,axiom,
! [A: set_literal,B: set_Numeral_num0] :
( ( finite5847741373460823677iteral @ A )
=> ( ( finite1111429032697314573l_num0 @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_literal @ A ) @ ( finite6454714172617411596l_num0 @ B ) )
=> ? [F4: literal > numeral_num0] :
( ( ord_le5200684351691877604l_num0 @ ( image_3608546570274595605l_num0 @ F4 @ A ) @ B )
& ( inj_on1088395555250478209l_num0 @ F4 @ A ) ) ) ) ) ).
% card_le_inj
thf(fact_556_card__le__inj,axiom,
! [A: set_literal,B: set_literal] :
( ( finite5847741373460823677iteral @ A )
=> ( ( finite5847741373460823677iteral @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_literal @ A ) @ ( finite_card_literal @ B ) )
=> ? [F4: literal > literal] :
( ( ord_le7307670543136651348iteral @ ( image_8195128725298311301iteral @ F4 @ A ) @ B )
& ( inj_on602069361295035377iteral @ F4 @ A ) ) ) ) ) ).
% card_le_inj
thf(fact_557_card__le__inj,axiom,
! [A: set_literal,B: set_Numeral_num1] :
( ( finite5847741373460823677iteral @ A )
=> ( ( finite1111429032697314574l_num1 @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_literal @ A ) @ ( finite6454714172617411597l_num1 @ B ) )
=> ? [F4: literal > numeral_num1] :
( ( ord_le5200684355995106405l_num1 @ ( image_3608546570274595606l_num1 @ F4 @ A ) @ B )
& ( inj_on1088395555250478210l_num1 @ F4 @ A ) ) ) ) ) ).
% card_le_inj
thf(fact_558_card__le__inj,axiom,
! [A: set_literal,B: set_Product_unit] :
( ( finite5847741373460823677iteral @ A )
=> ( ( finite4290736615968046902t_unit @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_literal @ A ) @ ( finite410649719033368117t_unit @ B ) )
=> ? [F4: literal > product_unit] :
( ( ord_le3507040750410214029t_unit @ ( image_6787854153545327934t_unit @ F4 @ A ) @ B )
& ( inj_on4267703138521210538t_unit @ F4 @ A ) ) ) ) ) ).
% card_le_inj
thf(fact_559_card__le__inj,axiom,
! [A: set_Numeral_num1,B: set_Numeral_num0] :
( ( finite1111429032697314574l_num1 @ A )
=> ( ( finite1111429032697314573l_num0 @ B )
=> ( ( ord_less_eq_nat @ ( finite6454714172617411597l_num1 @ A ) @ ( finite6454714172617411596l_num0 @ B ) )
=> ? [F4: numeral_num1 > numeral_num0] :
( ( ord_le5200684351691877604l_num0 @ ( image_6783865936125884740l_num0 @ F4 @ A ) @ B )
& ( inj_on5306804518485545136l_num0 @ F4 @ A ) ) ) ) ) ).
% card_le_inj
thf(fact_560_card__le__inj,axiom,
! [A: set_Numeral_num1,B: set_literal] :
( ( finite1111429032697314574l_num1 @ A )
=> ( ( finite5847741373460823677iteral @ B )
=> ( ( ord_less_eq_nat @ ( finite6454714172617411597l_num1 @ A ) @ ( finite_card_literal @ B ) )
=> ? [F4: numeral_num1 > literal] :
( ( ord_le7307670543136651348iteral @ ( image_5852747068178070836iteral @ F4 @ A ) @ B )
& ( inj_on3332596053153953440iteral @ F4 @ A ) ) ) ) ) ).
% card_le_inj
thf(fact_561_finite__image__iff,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( finite_finite_nat @ ( image_nat_nat @ F @ A ) )
= ( finite_finite_nat @ A ) ) ) ).
% finite_image_iff
thf(fact_562_finite__imageD,axiom,
! [F: nat > nat,A: set_nat] :
( ( finite_finite_nat @ ( image_nat_nat @ F @ A ) )
=> ( ( inj_on_nat_nat @ F @ A )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_imageD
thf(fact_563_card__image,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( finite_card_nat @ ( image_nat_nat @ F @ A ) )
= ( finite_card_nat @ A ) ) ) ).
% card_image
thf(fact_564_card__image,axiom,
! [F: numeral_num0 > nat,A: set_Numeral_num0] :
( ( inj_on7129071197303617539m0_nat @ F @ A )
=> ( ( finite_card_nat @ ( image_8797574156932312687m0_nat @ F @ A ) )
= ( finite6454714172617411596l_num0 @ A ) ) ) ).
% card_image
thf(fact_565_card__image,axiom,
! [F: literal > nat,A: set_literal] :
( ( inj_on_literal_nat @ F @ A )
=> ( ( finite_card_nat @ ( image_literal_nat @ F @ A ) )
= ( finite_card_literal @ A ) ) ) ).
% card_image
thf(fact_566_card__image,axiom,
! [F: numeral_num1 > nat,A: set_Numeral_num1] :
( ( inj_on8364515526260012036m1_nat @ F @ A )
=> ( ( finite_card_nat @ ( image_809646449033931376m1_nat @ F @ A ) )
= ( finite6454714172617411597l_num1 @ A ) ) ) ).
% card_image
thf(fact_567_card__image,axiom,
! [F: product_unit > nat,A: set_Product_unit] :
( ( inj_on8430439091780834860it_nat @ F @ A )
=> ( ( finite_card_nat @ ( image_875570014554754200it_nat @ F @ A ) )
= ( finite410649719033368117t_unit @ A ) ) ) ).
% card_image
thf(fact_568_card__image,axiom,
! [F: nat > numeral_num0,A: set_nat] :
( ( inj_on3882293653322094177l_num0 @ F @ A )
=> ( ( finite6454714172617411596l_num0 @ ( image_5550796612950789325l_num0 @ F @ A ) )
= ( finite_card_nat @ A ) ) ) ).
% card_image
thf(fact_569_card__image,axiom,
! [F: numeral_num0 > numeral_num0,A: set_Numeral_num0] :
( ( inj_on1355912882866956657l_num0 @ F @ A )
=> ( ( finite6454714172617411596l_num0 @ ( image_2832974300507296261l_num0 @ F @ A ) )
= ( finite6454714172617411596l_num0 @ A ) ) ) ).
% card_image
thf(fact_570_card__image,axiom,
! [F: literal > numeral_num0,A: set_literal] :
( ( inj_on1088395555250478209l_num0 @ F @ A )
=> ( ( finite6454714172617411596l_num0 @ ( image_3608546570274595605l_num0 @ F @ A ) )
= ( finite_card_literal @ A ) ) ) ).
% card_image
thf(fact_571_card__image,axiom,
! [F: numeral_num1 > numeral_num0,A: set_Numeral_num1] :
( ( inj_on5306804518485545136l_num0 @ F @ A )
=> ( ( finite6454714172617411596l_num0 @ ( image_6783865936125884740l_num0 @ F @ A ) )
= ( finite6454714172617411597l_num1 @ A ) ) ) ).
% card_image
thf(fact_572_card__image,axiom,
! [F: product_unit > numeral_num0,A: set_Product_unit] :
( ( inj_on4972065740439335048l_num0 @ F @ A )
=> ( ( finite6454714172617411596l_num0 @ ( image_6449127158079674652l_num0 @ F @ A ) )
= ( finite410649719033368117t_unit @ A ) ) ) ).
% card_image
thf(fact_573_endo__inj__surj,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ A )
=> ( ( inj_on_nat_nat @ F @ A )
=> ( ( image_nat_nat @ F @ A )
= A ) ) ) ) ).
% endo_inj_surj
thf(fact_574_inj__on__finite,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A ) ) ) ) ).
% inj_on_finite
thf(fact_575_finite__surj__inj,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( image_nat_nat @ F @ A ) )
=> ( inj_on_nat_nat @ F @ A ) ) ) ).
% finite_surj_inj
thf(fact_576_inj__on__iff__eq__card,axiom,
! [A: set_Numeral_num0,F: numeral_num0 > nat] :
( ( finite1111429032697314573l_num0 @ A )
=> ( ( inj_on7129071197303617539m0_nat @ F @ A )
= ( ( finite_card_nat @ ( image_8797574156932312687m0_nat @ F @ A ) )
= ( finite6454714172617411596l_num0 @ A ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_577_inj__on__iff__eq__card,axiom,
! [A: set_literal,F: literal > nat] :
( ( finite5847741373460823677iteral @ A )
=> ( ( inj_on_literal_nat @ F @ A )
= ( ( finite_card_nat @ ( image_literal_nat @ F @ A ) )
= ( finite_card_literal @ A ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_578_inj__on__iff__eq__card,axiom,
! [A: set_Numeral_num1,F: numeral_num1 > nat] :
( ( finite1111429032697314574l_num1 @ A )
=> ( ( inj_on8364515526260012036m1_nat @ F @ A )
= ( ( finite_card_nat @ ( image_809646449033931376m1_nat @ F @ A ) )
= ( finite6454714172617411597l_num1 @ A ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_579_inj__on__iff__eq__card,axiom,
! [A: set_Product_unit,F: product_unit > nat] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( inj_on8430439091780834860it_nat @ F @ A )
= ( ( finite_card_nat @ ( image_875570014554754200it_nat @ F @ A ) )
= ( finite410649719033368117t_unit @ A ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_580_inj__on__iff__eq__card,axiom,
! [A: set_Numeral_num0,F: numeral_num0 > numeral_num0] :
( ( finite1111429032697314573l_num0 @ A )
=> ( ( inj_on1355912882866956657l_num0 @ F @ A )
= ( ( finite6454714172617411596l_num0 @ ( image_2832974300507296261l_num0 @ F @ A ) )
= ( finite6454714172617411596l_num0 @ A ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_581_inj__on__iff__eq__card,axiom,
! [A: set_literal,F: literal > numeral_num0] :
( ( finite5847741373460823677iteral @ A )
=> ( ( inj_on1088395555250478209l_num0 @ F @ A )
= ( ( finite6454714172617411596l_num0 @ ( image_3608546570274595605l_num0 @ F @ A ) )
= ( finite_card_literal @ A ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_582_inj__on__iff__eq__card,axiom,
! [A: set_Numeral_num1,F: numeral_num1 > numeral_num0] :
( ( finite1111429032697314574l_num1 @ A )
=> ( ( inj_on5306804518485545136l_num0 @ F @ A )
= ( ( finite6454714172617411596l_num0 @ ( image_6783865936125884740l_num0 @ F @ A ) )
= ( finite6454714172617411597l_num1 @ A ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_583_inj__on__iff__eq__card,axiom,
! [A: set_Product_unit,F: product_unit > numeral_num0] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( inj_on4972065740439335048l_num0 @ F @ A )
= ( ( finite6454714172617411596l_num0 @ ( image_6449127158079674652l_num0 @ F @ A ) )
= ( finite410649719033368117t_unit @ A ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_584_inj__on__iff__eq__card,axiom,
! [A: set_Numeral_num0,F: numeral_num0 > literal] :
( ( finite1111429032697314573l_num0 @ A )
=> ( ( inj_on6217666562437480673iteral @ F @ A )
= ( ( finite_card_literal @ ( image_8737817577461598069iteral @ F @ A ) )
= ( finite6454714172617411596l_num0 @ A ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_585_inj__on__iff__eq__card,axiom,
! [A: set_literal,F: literal > literal] :
( ( finite5847741373460823677iteral @ A )
=> ( ( inj_on602069361295035377iteral @ F @ A )
= ( ( finite_card_literal @ ( image_8195128725298311301iteral @ F @ A ) )
= ( finite_card_literal @ A ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_586_eq__card__imp__inj__on,axiom,
! [A: set_Numeral_num0,F: numeral_num0 > nat] :
( ( finite1111429032697314573l_num0 @ A )
=> ( ( ( finite_card_nat @ ( image_8797574156932312687m0_nat @ F @ A ) )
= ( finite6454714172617411596l_num0 @ A ) )
=> ( inj_on7129071197303617539m0_nat @ F @ A ) ) ) ).
% eq_card_imp_inj_on
thf(fact_587_eq__card__imp__inj__on,axiom,
! [A: set_literal,F: literal > nat] :
( ( finite5847741373460823677iteral @ A )
=> ( ( ( finite_card_nat @ ( image_literal_nat @ F @ A ) )
= ( finite_card_literal @ A ) )
=> ( inj_on_literal_nat @ F @ A ) ) ) ).
% eq_card_imp_inj_on
thf(fact_588_eq__card__imp__inj__on,axiom,
! [A: set_Numeral_num1,F: numeral_num1 > nat] :
( ( finite1111429032697314574l_num1 @ A )
=> ( ( ( finite_card_nat @ ( image_809646449033931376m1_nat @ F @ A ) )
= ( finite6454714172617411597l_num1 @ A ) )
=> ( inj_on8364515526260012036m1_nat @ F @ A ) ) ) ).
% eq_card_imp_inj_on
thf(fact_589_eq__card__imp__inj__on,axiom,
! [A: set_Product_unit,F: product_unit > nat] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( ( finite_card_nat @ ( image_875570014554754200it_nat @ F @ A ) )
= ( finite410649719033368117t_unit @ A ) )
=> ( inj_on8430439091780834860it_nat @ F @ A ) ) ) ).
% eq_card_imp_inj_on
thf(fact_590_eq__card__imp__inj__on,axiom,
! [A: set_Numeral_num0,F: numeral_num0 > numeral_num0] :
( ( finite1111429032697314573l_num0 @ A )
=> ( ( ( finite6454714172617411596l_num0 @ ( image_2832974300507296261l_num0 @ F @ A ) )
= ( finite6454714172617411596l_num0 @ A ) )
=> ( inj_on1355912882866956657l_num0 @ F @ A ) ) ) ).
% eq_card_imp_inj_on
thf(fact_591_eq__card__imp__inj__on,axiom,
! [A: set_literal,F: literal > numeral_num0] :
( ( finite5847741373460823677iteral @ A )
=> ( ( ( finite6454714172617411596l_num0 @ ( image_3608546570274595605l_num0 @ F @ A ) )
= ( finite_card_literal @ A ) )
=> ( inj_on1088395555250478209l_num0 @ F @ A ) ) ) ).
% eq_card_imp_inj_on
thf(fact_592_eq__card__imp__inj__on,axiom,
! [A: set_Numeral_num1,F: numeral_num1 > numeral_num0] :
( ( finite1111429032697314574l_num1 @ A )
=> ( ( ( finite6454714172617411596l_num0 @ ( image_6783865936125884740l_num0 @ F @ A ) )
= ( finite6454714172617411597l_num1 @ A ) )
=> ( inj_on5306804518485545136l_num0 @ F @ A ) ) ) ).
% eq_card_imp_inj_on
thf(fact_593_eq__card__imp__inj__on,axiom,
! [A: set_Product_unit,F: product_unit > numeral_num0] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( ( finite6454714172617411596l_num0 @ ( image_6449127158079674652l_num0 @ F @ A ) )
= ( finite410649719033368117t_unit @ A ) )
=> ( inj_on4972065740439335048l_num0 @ F @ A ) ) ) ).
% eq_card_imp_inj_on
thf(fact_594_eq__card__imp__inj__on,axiom,
! [A: set_Numeral_num0,F: numeral_num0 > literal] :
( ( finite1111429032697314573l_num0 @ A )
=> ( ( ( finite_card_literal @ ( image_8737817577461598069iteral @ F @ A ) )
= ( finite6454714172617411596l_num0 @ A ) )
=> ( inj_on6217666562437480673iteral @ F @ A ) ) ) ).
% eq_card_imp_inj_on
thf(fact_595_eq__card__imp__inj__on,axiom,
! [A: set_literal,F: literal > literal] :
( ( finite5847741373460823677iteral @ A )
=> ( ( ( finite_card_literal @ ( image_8195128725298311301iteral @ F @ A ) )
= ( finite_card_literal @ A ) )
=> ( inj_on602069361295035377iteral @ F @ A ) ) ) ).
% eq_card_imp_inj_on
thf(fact_596_pigeonhole,axiom,
! [F: nat > nat,A: set_nat] :
( ( ord_less_nat @ ( finite_card_nat @ ( image_nat_nat @ F @ A ) ) @ ( finite_card_nat @ A ) )
=> ~ ( inj_on_nat_nat @ F @ A ) ) ).
% pigeonhole
thf(fact_597_pigeonhole,axiom,
! [F: numeral_num0 > nat,A: set_Numeral_num0] :
( ( ord_less_nat @ ( finite_card_nat @ ( image_8797574156932312687m0_nat @ F @ A ) ) @ ( finite6454714172617411596l_num0 @ A ) )
=> ~ ( inj_on7129071197303617539m0_nat @ F @ A ) ) ).
% pigeonhole
thf(fact_598_pigeonhole,axiom,
! [F: literal > nat,A: set_literal] :
( ( ord_less_nat @ ( finite_card_nat @ ( image_literal_nat @ F @ A ) ) @ ( finite_card_literal @ A ) )
=> ~ ( inj_on_literal_nat @ F @ A ) ) ).
% pigeonhole
thf(fact_599_pigeonhole,axiom,
! [F: numeral_num1 > nat,A: set_Numeral_num1] :
( ( ord_less_nat @ ( finite_card_nat @ ( image_809646449033931376m1_nat @ F @ A ) ) @ ( finite6454714172617411597l_num1 @ A ) )
=> ~ ( inj_on8364515526260012036m1_nat @ F @ A ) ) ).
% pigeonhole
thf(fact_600_pigeonhole,axiom,
! [F: product_unit > nat,A: set_Product_unit] :
( ( ord_less_nat @ ( finite_card_nat @ ( image_875570014554754200it_nat @ F @ A ) ) @ ( finite410649719033368117t_unit @ A ) )
=> ~ ( inj_on8430439091780834860it_nat @ F @ A ) ) ).
% pigeonhole
thf(fact_601_pigeonhole,axiom,
! [F: nat > numeral_num0,A: set_nat] :
( ( ord_less_nat @ ( finite6454714172617411596l_num0 @ ( image_5550796612950789325l_num0 @ F @ A ) ) @ ( finite_card_nat @ A ) )
=> ~ ( inj_on3882293653322094177l_num0 @ F @ A ) ) ).
% pigeonhole
thf(fact_602_pigeonhole,axiom,
! [F: numeral_num0 > numeral_num0,A: set_Numeral_num0] :
( ( ord_less_nat @ ( finite6454714172617411596l_num0 @ ( image_2832974300507296261l_num0 @ F @ A ) ) @ ( finite6454714172617411596l_num0 @ A ) )
=> ~ ( inj_on1355912882866956657l_num0 @ F @ A ) ) ).
% pigeonhole
thf(fact_603_pigeonhole,axiom,
! [F: literal > numeral_num0,A: set_literal] :
( ( ord_less_nat @ ( finite6454714172617411596l_num0 @ ( image_3608546570274595605l_num0 @ F @ A ) ) @ ( finite_card_literal @ A ) )
=> ~ ( inj_on1088395555250478209l_num0 @ F @ A ) ) ).
% pigeonhole
thf(fact_604_pigeonhole,axiom,
! [F: numeral_num1 > numeral_num0,A: set_Numeral_num1] :
( ( ord_less_nat @ ( finite6454714172617411596l_num0 @ ( image_6783865936125884740l_num0 @ F @ A ) ) @ ( finite6454714172617411597l_num1 @ A ) )
=> ~ ( inj_on5306804518485545136l_num0 @ F @ A ) ) ).
% pigeonhole
thf(fact_605_pigeonhole,axiom,
! [F: product_unit > numeral_num0,A: set_Product_unit] :
( ( ord_less_nat @ ( finite6454714172617411596l_num0 @ ( image_6449127158079674652l_num0 @ F @ A ) ) @ ( finite410649719033368117t_unit @ A ) )
=> ~ ( inj_on4972065740439335048l_num0 @ F @ A ) ) ).
% pigeonhole
thf(fact_606_surjective__iff__injective__gen,axiom,
! [S: set_Numeral_num0,T2: set_Numeral_num0,F: numeral_num0 > numeral_num0] :
( ( finite1111429032697314573l_num0 @ S )
=> ( ( finite1111429032697314573l_num0 @ T2 )
=> ( ( ( finite6454714172617411596l_num0 @ S )
= ( finite6454714172617411596l_num0 @ T2 ) )
=> ( ( ord_le5200684351691877604l_num0 @ ( image_2832974300507296261l_num0 @ F @ S ) @ T2 )
=> ( ( ! [X4: numeral_num0] :
( ( member_Numeral_num0 @ X4 @ T2 )
=> ? [Y4: numeral_num0] :
( ( member_Numeral_num0 @ Y4 @ S )
& ( ( F @ Y4 )
= X4 ) ) ) )
= ( inj_on1355912882866956657l_num0 @ F @ S ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_607_surjective__iff__injective__gen,axiom,
! [S: set_Numeral_num0,T2: set_literal,F: numeral_num0 > literal] :
( ( finite1111429032697314573l_num0 @ S )
=> ( ( finite5847741373460823677iteral @ T2 )
=> ( ( ( finite6454714172617411596l_num0 @ S )
= ( finite_card_literal @ T2 ) )
=> ( ( ord_le7307670543136651348iteral @ ( image_8737817577461598069iteral @ F @ S ) @ T2 )
=> ( ( ! [X4: literal] :
( ( member_literal @ X4 @ T2 )
=> ? [Y4: numeral_num0] :
( ( member_Numeral_num0 @ Y4 @ S )
& ( ( F @ Y4 )
= X4 ) ) ) )
= ( inj_on6217666562437480673iteral @ F @ S ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_608_surjective__iff__injective__gen,axiom,
! [S: set_Numeral_num0,T2: set_Numeral_num1,F: numeral_num0 > numeral_num1] :
( ( finite1111429032697314573l_num0 @ S )
=> ( ( finite1111429032697314574l_num1 @ T2 )
=> ( ( ( finite6454714172617411596l_num0 @ S )
= ( finite6454714172617411597l_num1 @ T2 ) )
=> ( ( ord_le5200684355995106405l_num1 @ ( image_2832974300507296262l_num1 @ F @ S ) @ T2 )
=> ( ( ! [X4: numeral_num1] :
( ( member_Numeral_num1 @ X4 @ T2 )
=> ? [Y4: numeral_num0] :
( ( member_Numeral_num0 @ Y4 @ S )
& ( ( F @ Y4 )
= X4 ) ) ) )
= ( inj_on1355912882866956658l_num1 @ F @ S ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_609_surjective__iff__injective__gen,axiom,
! [S: set_Numeral_num0,T2: set_Product_unit,F: numeral_num0 > product_unit] :
( ( finite1111429032697314573l_num0 @ S )
=> ( ( finite4290736615968046902t_unit @ T2 )
=> ( ( ( finite6454714172617411596l_num0 @ S )
= ( finite410649719033368117t_unit @ T2 ) )
=> ( ( ord_le3507040750410214029t_unit @ ( image_6012281883778028590t_unit @ F @ S ) @ T2 )
=> ( ( ! [X4: product_unit] :
( ( member_Product_unit @ X4 @ T2 )
=> ? [Y4: numeral_num0] :
( ( member_Numeral_num0 @ Y4 @ S )
& ( ( F @ Y4 )
= X4 ) ) ) )
= ( inj_on4535220466137688986t_unit @ F @ S ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_610_surjective__iff__injective__gen,axiom,
! [S: set_literal,T2: set_Numeral_num0,F: literal > numeral_num0] :
( ( finite5847741373460823677iteral @ S )
=> ( ( finite1111429032697314573l_num0 @ T2 )
=> ( ( ( finite_card_literal @ S )
= ( finite6454714172617411596l_num0 @ T2 ) )
=> ( ( ord_le5200684351691877604l_num0 @ ( image_3608546570274595605l_num0 @ F @ S ) @ T2 )
=> ( ( ! [X4: numeral_num0] :
( ( member_Numeral_num0 @ X4 @ T2 )
=> ? [Y4: literal] :
( ( member_literal @ Y4 @ S )
& ( ( F @ Y4 )
= X4 ) ) ) )
= ( inj_on1088395555250478209l_num0 @ F @ S ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_611_surjective__iff__injective__gen,axiom,
! [S: set_literal,T2: set_literal,F: literal > literal] :
( ( finite5847741373460823677iteral @ S )
=> ( ( finite5847741373460823677iteral @ T2 )
=> ( ( ( finite_card_literal @ S )
= ( finite_card_literal @ T2 ) )
=> ( ( ord_le7307670543136651348iteral @ ( image_8195128725298311301iteral @ F @ S ) @ T2 )
=> ( ( ! [X4: literal] :
( ( member_literal @ X4 @ T2 )
=> ? [Y4: literal] :
( ( member_literal @ Y4 @ S )
& ( ( F @ Y4 )
= X4 ) ) ) )
= ( inj_on602069361295035377iteral @ F @ S ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_612_surjective__iff__injective__gen,axiom,
! [S: set_literal,T2: set_Numeral_num1,F: literal > numeral_num1] :
( ( finite5847741373460823677iteral @ S )
=> ( ( finite1111429032697314574l_num1 @ T2 )
=> ( ( ( finite_card_literal @ S )
= ( finite6454714172617411597l_num1 @ T2 ) )
=> ( ( ord_le5200684355995106405l_num1 @ ( image_3608546570274595606l_num1 @ F @ S ) @ T2 )
=> ( ( ! [X4: numeral_num1] :
( ( member_Numeral_num1 @ X4 @ T2 )
=> ? [Y4: literal] :
( ( member_literal @ Y4 @ S )
& ( ( F @ Y4 )
= X4 ) ) ) )
= ( inj_on1088395555250478210l_num1 @ F @ S ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_613_surjective__iff__injective__gen,axiom,
! [S: set_literal,T2: set_Product_unit,F: literal > product_unit] :
( ( finite5847741373460823677iteral @ S )
=> ( ( finite4290736615968046902t_unit @ T2 )
=> ( ( ( finite_card_literal @ S )
= ( finite410649719033368117t_unit @ T2 ) )
=> ( ( ord_le3507040750410214029t_unit @ ( image_6787854153545327934t_unit @ F @ S ) @ T2 )
=> ( ( ! [X4: product_unit] :
( ( member_Product_unit @ X4 @ T2 )
=> ? [Y4: literal] :
( ( member_literal @ Y4 @ S )
& ( ( F @ Y4 )
= X4 ) ) ) )
= ( inj_on4267703138521210538t_unit @ F @ S ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_614_surjective__iff__injective__gen,axiom,
! [S: set_Numeral_num1,T2: set_Numeral_num0,F: numeral_num1 > numeral_num0] :
( ( finite1111429032697314574l_num1 @ S )
=> ( ( finite1111429032697314573l_num0 @ T2 )
=> ( ( ( finite6454714172617411597l_num1 @ S )
= ( finite6454714172617411596l_num0 @ T2 ) )
=> ( ( ord_le5200684351691877604l_num0 @ ( image_6783865936125884740l_num0 @ F @ S ) @ T2 )
=> ( ( ! [X4: numeral_num0] :
( ( member_Numeral_num0 @ X4 @ T2 )
=> ? [Y4: numeral_num1] :
( ( member_Numeral_num1 @ Y4 @ S )
& ( ( F @ Y4 )
= X4 ) ) ) )
= ( inj_on5306804518485545136l_num0 @ F @ S ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_615_surjective__iff__injective__gen,axiom,
! [S: set_Numeral_num1,T2: set_literal,F: numeral_num1 > literal] :
( ( finite1111429032697314574l_num1 @ S )
=> ( ( finite5847741373460823677iteral @ T2 )
=> ( ( ( finite6454714172617411597l_num1 @ S )
= ( finite_card_literal @ T2 ) )
=> ( ( ord_le7307670543136651348iteral @ ( image_5852747068178070836iteral @ F @ S ) @ T2 )
=> ( ( ! [X4: literal] :
( ( member_literal @ X4 @ T2 )
=> ? [Y4: numeral_num1] :
( ( member_Numeral_num1 @ Y4 @ S )
& ( ( F @ Y4 )
= X4 ) ) ) )
= ( inj_on3332596053153953440iteral @ F @ S ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_616_card__bij__eq,axiom,
! [F: numeral_num0 > numeral_num0,A: set_Numeral_num0,B: set_Numeral_num0,G3: numeral_num0 > numeral_num0] :
( ( inj_on1355912882866956657l_num0 @ F @ A )
=> ( ( ord_le5200684351691877604l_num0 @ ( image_2832974300507296261l_num0 @ F @ A ) @ B )
=> ( ( inj_on1355912882866956657l_num0 @ G3 @ B )
=> ( ( ord_le5200684351691877604l_num0 @ ( image_2832974300507296261l_num0 @ G3 @ B ) @ A )
=> ( ( finite1111429032697314573l_num0 @ A )
=> ( ( finite1111429032697314573l_num0 @ B )
=> ( ( finite6454714172617411596l_num0 @ A )
= ( finite6454714172617411596l_num0 @ B ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_617_card__bij__eq,axiom,
! [F: numeral_num0 > literal,A: set_Numeral_num0,B: set_literal,G3: literal > numeral_num0] :
( ( inj_on6217666562437480673iteral @ F @ A )
=> ( ( ord_le7307670543136651348iteral @ ( image_8737817577461598069iteral @ F @ A ) @ B )
=> ( ( inj_on1088395555250478209l_num0 @ G3 @ B )
=> ( ( ord_le5200684351691877604l_num0 @ ( image_3608546570274595605l_num0 @ G3 @ B ) @ A )
=> ( ( finite1111429032697314573l_num0 @ A )
=> ( ( finite5847741373460823677iteral @ B )
=> ( ( finite6454714172617411596l_num0 @ A )
= ( finite_card_literal @ B ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_618_card__bij__eq,axiom,
! [F: numeral_num0 > numeral_num1,A: set_Numeral_num0,B: set_Numeral_num1,G3: numeral_num1 > numeral_num0] :
( ( inj_on1355912882866956658l_num1 @ F @ A )
=> ( ( ord_le5200684355995106405l_num1 @ ( image_2832974300507296262l_num1 @ F @ A ) @ B )
=> ( ( inj_on5306804518485545136l_num0 @ G3 @ B )
=> ( ( ord_le5200684351691877604l_num0 @ ( image_6783865936125884740l_num0 @ G3 @ B ) @ A )
=> ( ( finite1111429032697314573l_num0 @ A )
=> ( ( finite1111429032697314574l_num1 @ B )
=> ( ( finite6454714172617411596l_num0 @ A )
= ( finite6454714172617411597l_num1 @ B ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_619_card__bij__eq,axiom,
! [F: numeral_num0 > product_unit,A: set_Numeral_num0,B: set_Product_unit,G3: product_unit > numeral_num0] :
( ( inj_on4535220466137688986t_unit @ F @ A )
=> ( ( ord_le3507040750410214029t_unit @ ( image_6012281883778028590t_unit @ F @ A ) @ B )
=> ( ( inj_on4972065740439335048l_num0 @ G3 @ B )
=> ( ( ord_le5200684351691877604l_num0 @ ( image_6449127158079674652l_num0 @ G3 @ B ) @ A )
=> ( ( finite1111429032697314573l_num0 @ A )
=> ( ( finite4290736615968046902t_unit @ B )
=> ( ( finite6454714172617411596l_num0 @ A )
= ( finite410649719033368117t_unit @ B ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_620_card__bij__eq,axiom,
! [F: literal > numeral_num0,A: set_literal,B: set_Numeral_num0,G3: numeral_num0 > literal] :
( ( inj_on1088395555250478209l_num0 @ F @ A )
=> ( ( ord_le5200684351691877604l_num0 @ ( image_3608546570274595605l_num0 @ F @ A ) @ B )
=> ( ( inj_on6217666562437480673iteral @ G3 @ B )
=> ( ( ord_le7307670543136651348iteral @ ( image_8737817577461598069iteral @ G3 @ B ) @ A )
=> ( ( finite5847741373460823677iteral @ A )
=> ( ( finite1111429032697314573l_num0 @ B )
=> ( ( finite_card_literal @ A )
= ( finite6454714172617411596l_num0 @ B ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_621_card__bij__eq,axiom,
! [F: literal > literal,A: set_literal,B: set_literal,G3: literal > literal] :
( ( inj_on602069361295035377iteral @ F @ A )
=> ( ( ord_le7307670543136651348iteral @ ( image_8195128725298311301iteral @ F @ A ) @ B )
=> ( ( inj_on602069361295035377iteral @ G3 @ B )
=> ( ( ord_le7307670543136651348iteral @ ( image_8195128725298311301iteral @ G3 @ B ) @ A )
=> ( ( finite5847741373460823677iteral @ A )
=> ( ( finite5847741373460823677iteral @ B )
=> ( ( finite_card_literal @ A )
= ( finite_card_literal @ B ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_622_card__bij__eq,axiom,
! [F: literal > numeral_num1,A: set_literal,B: set_Numeral_num1,G3: numeral_num1 > literal] :
( ( inj_on1088395555250478210l_num1 @ F @ A )
=> ( ( ord_le5200684355995106405l_num1 @ ( image_3608546570274595606l_num1 @ F @ A ) @ B )
=> ( ( inj_on3332596053153953440iteral @ G3 @ B )
=> ( ( ord_le7307670543136651348iteral @ ( image_5852747068178070836iteral @ G3 @ B ) @ A )
=> ( ( finite5847741373460823677iteral @ A )
=> ( ( finite1111429032697314574l_num1 @ B )
=> ( ( finite_card_literal @ A )
= ( finite6454714172617411597l_num1 @ B ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_623_card__bij__eq,axiom,
! [F: literal > product_unit,A: set_literal,B: set_Product_unit,G3: product_unit > literal] :
( ( inj_on4267703138521210538t_unit @ F @ A )
=> ( ( ord_le3507040750410214029t_unit @ ( image_6787854153545327934t_unit @ F @ A ) @ B )
=> ( ( inj_on3356833730873875064iteral @ G3 @ B )
=> ( ( ord_le7307670543136651348iteral @ ( image_5876984745897992460iteral @ G3 @ B ) @ A )
=> ( ( finite5847741373460823677iteral @ A )
=> ( ( finite4290736615968046902t_unit @ B )
=> ( ( finite_card_literal @ A )
= ( finite410649719033368117t_unit @ B ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_624_card__bij__eq,axiom,
! [F: numeral_num1 > numeral_num0,A: set_Numeral_num1,B: set_Numeral_num0,G3: numeral_num0 > numeral_num1] :
( ( inj_on5306804518485545136l_num0 @ F @ A )
=> ( ( ord_le5200684351691877604l_num0 @ ( image_6783865936125884740l_num0 @ F @ A ) @ B )
=> ( ( inj_on1355912882866956658l_num1 @ G3 @ B )
=> ( ( ord_le5200684355995106405l_num1 @ ( image_2832974300507296262l_num1 @ G3 @ B ) @ A )
=> ( ( finite1111429032697314574l_num1 @ A )
=> ( ( finite1111429032697314573l_num0 @ B )
=> ( ( finite6454714172617411597l_num1 @ A )
= ( finite6454714172617411596l_num0 @ B ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_625_card__bij__eq,axiom,
! [F: numeral_num1 > literal,A: set_Numeral_num1,B: set_literal,G3: literal > numeral_num1] :
( ( inj_on3332596053153953440iteral @ F @ A )
=> ( ( ord_le7307670543136651348iteral @ ( image_5852747068178070836iteral @ F @ A ) @ B )
=> ( ( inj_on1088395555250478210l_num1 @ G3 @ B )
=> ( ( ord_le5200684355995106405l_num1 @ ( image_3608546570274595606l_num1 @ G3 @ B ) @ A )
=> ( ( finite1111429032697314574l_num1 @ A )
=> ( ( finite5847741373460823677iteral @ B )
=> ( ( finite6454714172617411597l_num1 @ A )
= ( finite_card_literal @ B ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_626_inj__on__image__set__diff,axiom,
! [F: nat > nat,C2: set_nat,A: set_nat,B: set_nat] :
( ( inj_on_nat_nat @ F @ C2 )
=> ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B ) @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ( image_nat_nat @ F @ ( minus_minus_set_nat @ A @ B ) )
= ( minus_minus_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_627_inj__on__image__Int,axiom,
! [F: nat > nat,C2: set_nat,A: set_nat,B: set_nat] :
( ( inj_on_nat_nat @ F @ C2 )
=> ( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ( image_nat_nat @ F @ ( inf_inf_set_nat @ A @ B ) )
= ( inf_inf_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_628_image__strict__mono,axiom,
! [F: nat > nat,B: set_nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ B )
=> ( ( ord_less_set_nat @ A @ B )
=> ( ord_less_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ) ).
% image_strict_mono
thf(fact_629_subset__image__inj,axiom,
! [S: set_nat,F: nat > nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S @ ( image_nat_nat @ F @ T2 ) )
= ( ? [U: set_nat] :
( ( ord_less_eq_set_nat @ U @ T2 )
& ( inj_on_nat_nat @ F @ U )
& ( S
= ( image_nat_nat @ F @ U ) ) ) ) ) ).
% subset_image_inj
thf(fact_630_inj__on__image__mem__iff,axiom,
! [F: nat > nat,B: set_nat,A2: nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ B )
=> ( ( member_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ ( F @ A2 ) @ ( image_nat_nat @ F @ A ) )
= ( member_nat @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_631_linorder__inj__onI_H,axiom,
! [A: set_nat,F: nat > nat] :
( ! [I3: nat,J2: nat] :
( ( member_nat @ I3 @ A )
=> ( ( member_nat @ J2 @ A )
=> ( ( ord_less_nat @ I3 @ J2 )
=> ( ( F @ I3 )
!= ( F @ J2 ) ) ) ) )
=> ( inj_on_nat_nat @ F @ A ) ) ).
% linorder_inj_onI'
thf(fact_632_inj__on__subset,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( inj_on_nat_nat @ F @ B ) ) ) ).
% inj_on_subset
thf(fact_633_subset__inj__on,axiom,
! [F: nat > nat,B: set_nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( inj_on_nat_nat @ F @ A ) ) ) ).
% subset_inj_on
thf(fact_634_inj__on__diff,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( inj_on_nat_nat @ F @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% inj_on_diff
thf(fact_635_linorder__inj__onI,axiom,
! [A: set_nat,F: nat > nat] :
( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y3 @ A )
=> ( ( F @ X )
!= ( F @ Y3 ) ) ) ) )
=> ( ! [X: nat,Y3: nat] :
( ( member_nat @ X @ A )
=> ( ( member_nat @ Y3 @ A )
=> ( ( ord_less_eq_nat @ X @ Y3 )
| ( ord_less_eq_nat @ Y3 @ X ) ) ) )
=> ( inj_on_nat_nat @ F @ A ) ) ) ).
% linorder_inj_onI
thf(fact_636_inj__on__image__eq__iff,axiom,
! [F: nat > nat,C2: set_nat,A: set_nat,B: set_nat] :
( ( inj_on_nat_nat @ F @ C2 )
=> ( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ( ( image_nat_nat @ F @ A )
= ( image_nat_nat @ F @ B ) )
= ( A = B ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_637_card__vimage__inj__on__le,axiom,
! [F: nat > numeral_num0,D: set_nat,A: set_Numeral_num0] :
( ( inj_on3882293653322094177l_num0 @ F @ D )
=> ( ( finite1111429032697314573l_num0 @ A )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( inf_inf_set_nat @ ( vimage1705183035017847703l_num0 @ F @ A ) @ D ) ) @ ( finite6454714172617411596l_num0 @ A ) ) ) ) ).
% card_vimage_inj_on_le
thf(fact_638_card__vimage__inj__on__le,axiom,
! [F: nat > literal,D: set_nat,A: set_literal] :
( ( inj_on_nat_literal @ F @ D )
=> ( ( finite5847741373460823677iteral @ A )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( inf_inf_set_nat @ ( vimage_nat_literal @ F @ A ) @ D ) ) @ ( finite_card_literal @ A ) ) ) ) ).
% card_vimage_inj_on_le
thf(fact_639_card__vimage__inj__on__le,axiom,
! [F: nat > numeral_num1,D: set_nat,A: set_Numeral_num1] :
( ( inj_on3882293653322094178l_num1 @ F @ D )
=> ( ( finite1111429032697314574l_num1 @ A )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( inf_inf_set_nat @ ( vimage1705183035017847704l_num1 @ F @ A ) @ D ) ) @ ( finite6454714172617411597l_num1 @ A ) ) ) ) ).
% card_vimage_inj_on_le
thf(fact_640_card__vimage__inj__on__le,axiom,
! [F: nat > product_unit,D: set_nat,A: set_Product_unit] :
( ( inj_on7061601236592826506t_unit @ F @ D )
=> ( ( finite4290736615968046902t_unit @ A )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( inf_inf_set_nat @ ( vimage4884490618288580032t_unit @ F @ A ) @ D ) ) @ ( finite410649719033368117t_unit @ A ) ) ) ) ).
% card_vimage_inj_on_le
thf(fact_641_card__vimage__inj__on__le,axiom,
! [F: numeral_num0 > numeral_num0,D: set_Numeral_num0,A: set_Numeral_num0] :
( ( inj_on1355912882866956657l_num0 @ F @ D )
=> ( ( finite1111429032697314573l_num0 @ A )
=> ( ord_less_eq_nat @ ( finite6454714172617411596l_num0 @ ( inf_in6354261966906920242l_num0 @ ( vimage1199591675108543419l_num0 @ F @ A ) @ D ) ) @ ( finite6454714172617411596l_num0 @ A ) ) ) ) ).
% card_vimage_inj_on_le
thf(fact_642_card__vimage__inj__on__le,axiom,
! [F: numeral_num0 > literal,D: set_Numeral_num0,A: set_literal] :
( ( inj_on6217666562437480673iteral @ F @ D )
=> ( ( finite5847741373460823677iteral @ A )
=> ( ord_less_eq_nat @ ( finite6454714172617411596l_num0 @ ( inf_in6354261966906920242l_num0 @ ( vimage7997836657330795051iteral @ F @ A ) @ D ) ) @ ( finite_card_literal @ A ) ) ) ) ).
% card_vimage_inj_on_le
thf(fact_643_card__vimage__inj__on__le,axiom,
! [F: numeral_num0 > numeral_num1,D: set_Numeral_num0,A: set_Numeral_num1] :
( ( inj_on1355912882866956658l_num1 @ F @ D )
=> ( ( finite1111429032697314574l_num1 @ A )
=> ( ord_less_eq_nat @ ( finite6454714172617411596l_num0 @ ( inf_in6354261966906920242l_num0 @ ( vimage1199591675108543420l_num1 @ F @ A ) @ D ) ) @ ( finite6454714172617411597l_num1 @ A ) ) ) ) ).
% card_vimage_inj_on_le
thf(fact_644_card__vimage__inj__on__le,axiom,
! [F: numeral_num0 > product_unit,D: set_Numeral_num0,A: set_Product_unit] :
( ( inj_on4535220466137688986t_unit @ F @ D )
=> ( ( finite4290736615968046902t_unit @ A )
=> ( ord_less_eq_nat @ ( finite6454714172617411596l_num0 @ ( inf_in6354261966906920242l_num0 @ ( vimage4378899258379275748t_unit @ F @ A ) @ D ) ) @ ( finite410649719033368117t_unit @ A ) ) ) ) ).
% card_vimage_inj_on_le
thf(fact_645_card__vimage__inj__on__le,axiom,
! [F: literal > numeral_num0,D: set_literal,A: set_Numeral_num0] :
( ( inj_on1088395555250478209l_num0 @ F @ D )
=> ( ( finite1111429032697314573l_num0 @ A )
=> ( ord_less_eq_nat @ ( finite_card_literal @ ( inf_inf_set_literal @ ( vimage2868565650143792587l_num0 @ F @ A ) @ D ) ) @ ( finite6454714172617411596l_num0 @ A ) ) ) ) ).
% card_vimage_inj_on_le
thf(fact_646_card__vimage__inj__on__le,axiom,
! [F: literal > literal,D: set_literal,A: set_literal] :
( ( inj_on602069361295035377iteral @ F @ D )
=> ( ( finite5847741373460823677iteral @ A )
=> ( ord_less_eq_nat @ ( finite_card_literal @ ( inf_inf_set_literal @ ( vimage8238609917233974331iteral @ F @ A ) @ D ) ) @ ( finite_card_literal @ A ) ) ) ) ).
% card_vimage_inj_on_le
thf(fact_647_the__inv__into__into,axiom,
! [F: nat > nat,A: set_nat,X2: nat,B: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( member_nat @ X2 @ ( image_nat_nat @ F @ A ) )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( member_nat @ ( the_inv_into_nat_nat @ A @ F @ X2 ) @ B ) ) ) ) ).
% the_inv_into_into
thf(fact_648_inj__on__image__Fpow,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( inj_on4604407203859583615et_nat @ ( image_nat_nat @ F ) @ ( finite_Fpow_nat @ A ) ) ) ).
% inj_on_image_Fpow
thf(fact_649_image__Fpow__mono,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ ( finite_Fpow_nat @ A ) ) @ ( finite_Fpow_nat @ B ) ) ) ).
% image_Fpow_mono
thf(fact_650_vimage__eq,axiom,
! [A2: nat,F: nat > nat,B: set_nat] :
( ( member_nat @ A2 @ ( vimage_nat_nat @ F @ B ) )
= ( member_nat @ ( F @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_651_vimageI,axiom,
! [F: nat > nat,A2: nat,B2: nat,B: set_nat] :
( ( ( F @ A2 )
= B2 )
=> ( ( member_nat @ B2 @ B )
=> ( member_nat @ A2 @ ( vimage_nat_nat @ F @ B ) ) ) ) ).
% vimageI
thf(fact_652_vimage__Int,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( vimage_nat_nat @ F @ ( inf_inf_set_nat @ A @ B ) )
= ( inf_inf_set_nat @ ( vimage_nat_nat @ F @ A ) @ ( vimage_nat_nat @ F @ B ) ) ) ).
% vimage_Int
thf(fact_653_vimage__Collect,axiom,
! [P: nat > $o,F: nat > nat,Q: nat > $o] :
( ! [X: nat] :
( ( P @ ( F @ X ) )
= ( Q @ X ) )
=> ( ( vimage_nat_nat @ F @ ( collect_nat @ P ) )
= ( collect_nat @ Q ) ) ) ).
% vimage_Collect
thf(fact_654_vimageI2,axiom,
! [F: nat > nat,A2: nat,A: set_nat] :
( ( member_nat @ ( F @ A2 ) @ A )
=> ( member_nat @ A2 @ ( vimage_nat_nat @ F @ A ) ) ) ).
% vimageI2
thf(fact_655_vimageE,axiom,
! [A2: nat,F: nat > nat,B: set_nat] :
( ( member_nat @ A2 @ ( vimage_nat_nat @ F @ B ) )
=> ( member_nat @ ( F @ A2 ) @ B ) ) ).
% vimageE
thf(fact_656_vimageD,axiom,
! [A2: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ A2 @ ( vimage_nat_nat @ F @ A ) )
=> ( member_nat @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_657_vimage__Diff,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( vimage_nat_nat @ F @ ( minus_minus_set_nat @ A @ B ) )
= ( minus_minus_set_nat @ ( vimage_nat_nat @ F @ A ) @ ( vimage_nat_nat @ F @ B ) ) ) ).
% vimage_Diff
thf(fact_658_vimage__inter__cong,axiom,
! [S: set_nat,F: nat > nat,G3: nat > nat,Y: set_nat] :
( ! [W: nat] :
( ( member_nat @ W @ S )
=> ( ( F @ W )
= ( G3 @ W ) ) )
=> ( ( inf_inf_set_nat @ ( vimage_nat_nat @ F @ Y ) @ S )
= ( inf_inf_set_nat @ ( vimage_nat_nat @ G3 @ Y ) @ S ) ) ) ).
% vimage_inter_cong
thf(fact_659_vimage__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( vimage_nat_nat @ F @ A ) @ ( vimage_nat_nat @ F @ B ) ) ) ).
% vimage_mono
thf(fact_660_subset__vimage__iff,axiom,
! [A: set_nat,F: nat > nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( vimage_nat_nat @ F @ B ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_nat @ ( F @ X4 ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_661_image__vimage__subset,axiom,
! [F: nat > nat,A: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( vimage_nat_nat @ F @ A ) ) @ A ) ).
% image_vimage_subset
thf(fact_662_image__subset__iff__subset__vimage,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B )
= ( ord_less_eq_set_nat @ A @ ( vimage_nat_nat @ F @ B ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_663_Fpow__mono,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_le6893508408891458716et_nat @ ( finite_Fpow_nat @ A ) @ ( finite_Fpow_nat @ B ) ) ) ).
% Fpow_mono
thf(fact_664_finite__vimage__IntI,axiom,
! [F2: set_nat,H: nat > nat,A: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ( inj_on_nat_nat @ H @ A )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_nat @ H @ F2 ) @ A ) ) ) ) ).
% finite_vimage_IntI
thf(fact_665_card__vimage__inj,axiom,
! [F: nat > numeral_num0,A: set_Numeral_num0] :
( ( inj_on3882293653322094177l_num0 @ F @ top_top_set_nat )
=> ( ( ord_le5200684351691877604l_num0 @ A @ ( image_5550796612950789325l_num0 @ F @ top_top_set_nat ) )
=> ( ( finite_card_nat @ ( vimage1705183035017847703l_num0 @ F @ A ) )
= ( finite6454714172617411596l_num0 @ A ) ) ) ) ).
% card_vimage_inj
thf(fact_666_card__vimage__inj,axiom,
! [F: nat > literal,A: set_literal] :
( ( inj_on_nat_literal @ F @ top_top_set_nat )
=> ( ( ord_le7307670543136651348iteral @ A @ ( image_nat_literal @ F @ top_top_set_nat ) )
=> ( ( finite_card_nat @ ( vimage_nat_literal @ F @ A ) )
= ( finite_card_literal @ A ) ) ) ) ).
% card_vimage_inj
thf(fact_667_card__vimage__inj,axiom,
! [F: nat > numeral_num1,A: set_Numeral_num1] :
( ( inj_on3882293653322094178l_num1 @ F @ top_top_set_nat )
=> ( ( ord_le5200684355995106405l_num1 @ A @ ( image_5550796612950789326l_num1 @ F @ top_top_set_nat ) )
=> ( ( finite_card_nat @ ( vimage1705183035017847704l_num1 @ F @ A ) )
= ( finite6454714172617411597l_num1 @ A ) ) ) ) ).
% card_vimage_inj
thf(fact_668_card__vimage__inj,axiom,
! [F: nat > product_unit,A: set_Product_unit] :
( ( inj_on7061601236592826506t_unit @ F @ top_top_set_nat )
=> ( ( ord_le3507040750410214029t_unit @ A @ ( image_8730104196221521654t_unit @ F @ top_top_set_nat ) )
=> ( ( finite_card_nat @ ( vimage4884490618288580032t_unit @ F @ A ) )
= ( finite410649719033368117t_unit @ A ) ) ) ) ).
% card_vimage_inj
thf(fact_669_card__vimage__inj,axiom,
! [F: numeral_num0 > numeral_num0,A: set_Numeral_num0] :
( ( inj_on1355912882866956657l_num0 @ F @ top_to3689904424835650196l_num0 )
=> ( ( ord_le5200684351691877604l_num0 @ A @ ( image_2832974300507296261l_num0 @ F @ top_to3689904424835650196l_num0 ) )
=> ( ( finite6454714172617411596l_num0 @ ( vimage1199591675108543419l_num0 @ F @ A ) )
= ( finite6454714172617411596l_num0 @ A ) ) ) ) ).
% card_vimage_inj
thf(fact_670_card__vimage__inj,axiom,
! [F: numeral_num0 > literal,A: set_literal] :
( ( inj_on6217666562437480673iteral @ F @ top_to3689904424835650196l_num0 )
=> ( ( ord_le7307670543136651348iteral @ A @ ( image_8737817577461598069iteral @ F @ top_to3689904424835650196l_num0 ) )
=> ( ( finite6454714172617411596l_num0 @ ( vimage7997836657330795051iteral @ F @ A ) )
= ( finite_card_literal @ A ) ) ) ) ).
% card_vimage_inj
thf(fact_671_card__vimage__inj,axiom,
! [F: numeral_num0 > numeral_num1,A: set_Numeral_num1] :
( ( inj_on1355912882866956658l_num1 @ F @ top_to3689904424835650196l_num0 )
=> ( ( ord_le5200684355995106405l_num1 @ A @ ( image_2832974300507296262l_num1 @ F @ top_to3689904424835650196l_num0 ) )
=> ( ( finite6454714172617411596l_num0 @ ( vimage1199591675108543420l_num1 @ F @ A ) )
= ( finite6454714172617411597l_num1 @ A ) ) ) ) ).
% card_vimage_inj
thf(fact_672_card__vimage__inj,axiom,
! [F: numeral_num0 > product_unit,A: set_Product_unit] :
( ( inj_on4535220466137688986t_unit @ F @ top_to3689904424835650196l_num0 )
=> ( ( ord_le3507040750410214029t_unit @ A @ ( image_6012281883778028590t_unit @ F @ top_to3689904424835650196l_num0 ) )
=> ( ( finite6454714172617411596l_num0 @ ( vimage4378899258379275748t_unit @ F @ A ) )
= ( finite410649719033368117t_unit @ A ) ) ) ) ).
% card_vimage_inj
thf(fact_673_card__vimage__inj,axiom,
! [F: literal > numeral_num0,A: set_Numeral_num0] :
( ( inj_on1088395555250478209l_num0 @ F @ top_top_set_literal )
=> ( ( ord_le5200684351691877604l_num0 @ A @ ( image_3608546570274595605l_num0 @ F @ top_top_set_literal ) )
=> ( ( finite_card_literal @ ( vimage2868565650143792587l_num0 @ F @ A ) )
= ( finite6454714172617411596l_num0 @ A ) ) ) ) ).
% card_vimage_inj
thf(fact_674_card__vimage__inj,axiom,
! [F: literal > literal,A: set_literal] :
( ( inj_on602069361295035377iteral @ F @ top_top_set_literal )
=> ( ( ord_le7307670543136651348iteral @ A @ ( image_8195128725298311301iteral @ F @ top_top_set_literal ) )
=> ( ( finite_card_literal @ ( vimage8238609917233974331iteral @ F @ A ) )
= ( finite_card_literal @ A ) ) ) ) ).
% card_vimage_inj
thf(fact_675_ap__fresh__val,axiom,
! [Q: relational_fmla_a_b,Sigma: nat > a,X2: nat,I: product_prod_b_nat > set_list_a] :
( ( relational_ap_a_b @ Q )
=> ( ~ ( member_a @ ( Sigma @ X2 ) @ ( relational_adom_b_a @ I ) )
=> ( ~ ( member_a @ ( Sigma @ X2 ) @ ( relational_csts_a_b @ Q ) )
=> ( ( relational_sat_a_b @ Q @ I @ Sigma )
=> ~ ( member_nat @ X2 @ ( relational_fv_a_b @ Q ) ) ) ) ) ) ).
% ap_fresh_val
thf(fact_676_qp__fresh__val,axiom,
! [Q: relational_fmla_a_b,Sigma: nat > a,X2: nat,I: product_prod_b_nat > set_list_a] :
( ( relational_qp_a_b @ Q )
=> ( ~ ( member_a @ ( Sigma @ X2 ) @ ( relational_adom_b_a @ I ) )
=> ( ~ ( member_a @ ( Sigma @ X2 ) @ ( relational_csts_a_b @ Q ) )
=> ( ( relational_sat_a_b @ Q @ I @ Sigma )
=> ~ ( member_nat @ X2 @ ( relational_fv_a_b @ Q ) ) ) ) ) ) ).
% qp_fresh_val
thf(fact_677_image__Pow__mono,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ ( pow_nat @ A ) ) @ ( pow_nat @ B ) ) ) ).
% image_Pow_mono
thf(fact_678_vimage__subsetI,axiom,
! [F: numeral_num0 > nat,B: set_nat,A: set_Numeral_num0] :
( ( inj_on7129071197303617539m0_nat @ F @ top_to3689904424835650196l_num0 )
=> ( ( ord_less_eq_set_nat @ B @ ( image_8797574156932312687m0_nat @ F @ A ) )
=> ( ord_le5200684351691877604l_num0 @ ( vimage4951960578999371065m0_nat @ F @ B ) @ A ) ) ) ).
% vimage_subsetI
thf(fact_679_vimage__subsetI,axiom,
! [F: literal > nat,B: set_nat,A: set_literal] :
( ( inj_on_literal_nat @ F @ top_top_set_literal )
=> ( ( ord_less_eq_set_nat @ B @ ( image_literal_nat @ F @ A ) )
=> ( ord_le7307670543136651348iteral @ ( vimage_literal_nat @ F @ B ) @ A ) ) ) ).
% vimage_subsetI
thf(fact_680_vimage__subsetI,axiom,
! [F: numeral_num1 > nat,B: set_nat,A: set_Numeral_num1] :
( ( inj_on8364515526260012036m1_nat @ F @ top_to3689904429138878997l_num1 )
=> ( ( ord_less_eq_set_nat @ B @ ( image_809646449033931376m1_nat @ F @ A ) )
=> ( ord_le5200684355995106405l_num1 @ ( vimage6187404907955765562m1_nat @ F @ B ) @ A ) ) ) ).
% vimage_subsetI
thf(fact_681_vimage__subsetI,axiom,
! [F: product_unit > nat,B: set_nat,A: set_Product_unit] :
( ( inj_on8430439091780834860it_nat @ F @ top_to1996260823553986621t_unit )
=> ( ( ord_less_eq_set_nat @ B @ ( image_875570014554754200it_nat @ F @ A ) )
=> ( ord_le3507040750410214029t_unit @ ( vimage6253328473476588386it_nat @ F @ B ) @ A ) ) ) ).
% vimage_subsetI
thf(fact_682_vimage__subsetI,axiom,
! [F: nat > nat,B: set_nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ( ord_less_eq_set_nat @ ( vimage_nat_nat @ F @ B ) @ A ) ) ) ).
% vimage_subsetI
thf(fact_683_UNIV__I,axiom,
! [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).
% UNIV_I
thf(fact_684_UNIV__I,axiom,
! [X2: numeral_num0] : ( member_Numeral_num0 @ X2 @ top_to3689904424835650196l_num0 ) ).
% UNIV_I
thf(fact_685_UNIV__I,axiom,
! [X2: literal] : ( member_literal @ X2 @ top_top_set_literal ) ).
% UNIV_I
thf(fact_686_UNIV__I,axiom,
! [X2: numeral_num1] : ( member_Numeral_num1 @ X2 @ top_to3689904429138878997l_num1 ) ).
% UNIV_I
thf(fact_687_UNIV__I,axiom,
! [X2: product_unit] : ( member_Product_unit @ X2 @ top_to1996260823553986621t_unit ) ).
% UNIV_I
thf(fact_688_finite__Plus__UNIV__iff,axiom,
( ( finite6187706683773761046at_nat @ top_to6661820994512907621at_nat )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_689_finite__Plus__UNIV__iff,axiom,
( ( finite6021156207414448820l_num0 @ top_to7131751369954024123l_num0 )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite1111429032697314573l_num0 @ top_to3689904424835650196l_num0 ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_690_finite__Plus__UNIV__iff,axiom,
( ( finite7336130560110450212iteral @ top_to148093990134820907iteral )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite5847741373460823677iteral @ top_top_set_literal ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_691_finite__Plus__UNIV__iff,axiom,
( ( finite6021156211717677621l_num1 @ top_to7202785410000370108l_num1 )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_692_finite__Plus__UNIV__iff,axiom,
( ( finite4327512606132785245t_unit @ top_to5465250082899874788t_unit )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_693_finite__Plus__UNIV__iff,axiom,
( ( finite3100585017152042894m0_nat @ top_to6006073747853177493m0_nat )
= ( ( finite1111429032697314573l_num0 @ top_to3689904424835650196l_num0 )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_694_finite__Plus__UNIV__iff,axiom,
( ( finite1224042245303146300l_num0 @ top_to3489177893006913931l_num0 )
= ( ( finite1111429032697314573l_num0 @ top_to3689904424835650196l_num0 )
& ( finite1111429032697314573l_num0 @ top_to3689904424835650196l_num0 ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_695_finite__Plus__UNIV__iff,axiom,
( ( finite5188522788368459436iteral @ top_to5151745856415547131iteral )
= ( ( finite1111429032697314573l_num0 @ top_to3689904424835650196l_num0 )
& ( finite5847741373460823677iteral @ top_top_set_literal ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_696_finite__Plus__UNIV__iff,axiom,
( ( finite1224042249606375101l_num1 @ top_to3560211933053259916l_num1 )
= ( ( finite1111429032697314573l_num0 @ top_to3689904424835650196l_num0 )
& ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_697_finite__Plus__UNIV__iff,axiom,
( ( finite8753770680876258533t_unit @ top_to1822676605952764596t_unit )
= ( ( finite1111429032697314573l_num0 @ top_to3689904424835650196l_num0 )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_698_inf__top__left,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ X2 )
= X2 ) ).
% inf_top_left
thf(fact_699_inf__top__left,axiom,
! [X2: set_Numeral_num0] :
( ( inf_in6354261966906920242l_num0 @ top_to3689904424835650196l_num0 @ X2 )
= X2 ) ).
% inf_top_left
thf(fact_700_inf__top__left,axiom,
! [X2: set_literal] :
( ( inf_inf_set_literal @ top_top_set_literal @ X2 )
= X2 ) ).
% inf_top_left
thf(fact_701_inf__top__left,axiom,
! [X2: set_Numeral_num1] :
( ( inf_in6354261971210149043l_num1 @ top_to3689904429138878997l_num1 @ X2 )
= X2 ) ).
% inf_top_left
thf(fact_702_inf__top__left,axiom,
! [X2: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ top_to1996260823553986621t_unit @ X2 )
= X2 ) ).
% inf_top_left
thf(fact_703_inf__top__right,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ X2 @ top_top_set_nat )
= X2 ) ).
% inf_top_right
thf(fact_704_inf__top__right,axiom,
! [X2: set_Numeral_num0] :
( ( inf_in6354261966906920242l_num0 @ X2 @ top_to3689904424835650196l_num0 )
= X2 ) ).
% inf_top_right
thf(fact_705_inf__top__right,axiom,
! [X2: set_literal] :
( ( inf_inf_set_literal @ X2 @ top_top_set_literal )
= X2 ) ).
% inf_top_right
thf(fact_706_inf__top__right,axiom,
! [X2: set_Numeral_num1] :
( ( inf_in6354261971210149043l_num1 @ X2 @ top_to3689904429138878997l_num1 )
= X2 ) ).
% inf_top_right
thf(fact_707_inf__top__right,axiom,
! [X2: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ X2 @ top_to1996260823553986621t_unit )
= X2 ) ).
% inf_top_right
thf(fact_708_inf__eq__top__iff,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ( inf_inf_set_nat @ X2 @ Y )
= top_top_set_nat )
= ( ( X2 = top_top_set_nat )
& ( Y = top_top_set_nat ) ) ) ).
% inf_eq_top_iff
thf(fact_709_inf__eq__top__iff,axiom,
! [X2: set_Numeral_num0,Y: set_Numeral_num0] :
( ( ( inf_in6354261966906920242l_num0 @ X2 @ Y )
= top_to3689904424835650196l_num0 )
= ( ( X2 = top_to3689904424835650196l_num0 )
& ( Y = top_to3689904424835650196l_num0 ) ) ) ).
% inf_eq_top_iff
thf(fact_710_inf__eq__top__iff,axiom,
! [X2: set_literal,Y: set_literal] :
( ( ( inf_inf_set_literal @ X2 @ Y )
= top_top_set_literal )
= ( ( X2 = top_top_set_literal )
& ( Y = top_top_set_literal ) ) ) ).
% inf_eq_top_iff
thf(fact_711_inf__eq__top__iff,axiom,
! [X2: set_Numeral_num1,Y: set_Numeral_num1] :
( ( ( inf_in6354261971210149043l_num1 @ X2 @ Y )
= top_to3689904429138878997l_num1 )
= ( ( X2 = top_to3689904429138878997l_num1 )
& ( Y = top_to3689904429138878997l_num1 ) ) ) ).
% inf_eq_top_iff
thf(fact_712_inf__eq__top__iff,axiom,
! [X2: set_Product_unit,Y: set_Product_unit] :
( ( ( inf_in4660618365625256667t_unit @ X2 @ Y )
= top_to1996260823553986621t_unit )
= ( ( X2 = top_to1996260823553986621t_unit )
& ( Y = top_to1996260823553986621t_unit ) ) ) ).
% inf_eq_top_iff
thf(fact_713_top__eq__inf__iff,axiom,
! [X2: set_nat,Y: set_nat] :
( ( top_top_set_nat
= ( inf_inf_set_nat @ X2 @ Y ) )
= ( ( X2 = top_top_set_nat )
& ( Y = top_top_set_nat ) ) ) ).
% top_eq_inf_iff
thf(fact_714_top__eq__inf__iff,axiom,
! [X2: set_Numeral_num0,Y: set_Numeral_num0] :
( ( top_to3689904424835650196l_num0
= ( inf_in6354261966906920242l_num0 @ X2 @ Y ) )
= ( ( X2 = top_to3689904424835650196l_num0 )
& ( Y = top_to3689904424835650196l_num0 ) ) ) ).
% top_eq_inf_iff
thf(fact_715_top__eq__inf__iff,axiom,
! [X2: set_literal,Y: set_literal] :
( ( top_top_set_literal
= ( inf_inf_set_literal @ X2 @ Y ) )
= ( ( X2 = top_top_set_literal )
& ( Y = top_top_set_literal ) ) ) ).
% top_eq_inf_iff
thf(fact_716_top__eq__inf__iff,axiom,
! [X2: set_Numeral_num1,Y: set_Numeral_num1] :
( ( top_to3689904429138878997l_num1
= ( inf_in6354261971210149043l_num1 @ X2 @ Y ) )
= ( ( X2 = top_to3689904429138878997l_num1 )
& ( Y = top_to3689904429138878997l_num1 ) ) ) ).
% top_eq_inf_iff
thf(fact_717_top__eq__inf__iff,axiom,
! [X2: set_Product_unit,Y: set_Product_unit] :
( ( top_to1996260823553986621t_unit
= ( inf_in4660618365625256667t_unit @ X2 @ Y ) )
= ( ( X2 = top_to1996260823553986621t_unit )
& ( Y = top_to1996260823553986621t_unit ) ) ) ).
% top_eq_inf_iff
thf(fact_718_inf__top_Oeq__neutr__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B2 )
= top_top_set_nat )
= ( ( A2 = top_top_set_nat )
& ( B2 = top_top_set_nat ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_719_inf__top_Oeq__neutr__iff,axiom,
! [A2: set_Numeral_num0,B2: set_Numeral_num0] :
( ( ( inf_in6354261966906920242l_num0 @ A2 @ B2 )
= top_to3689904424835650196l_num0 )
= ( ( A2 = top_to3689904424835650196l_num0 )
& ( B2 = top_to3689904424835650196l_num0 ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_720_inf__top_Oeq__neutr__iff,axiom,
! [A2: set_literal,B2: set_literal] :
( ( ( inf_inf_set_literal @ A2 @ B2 )
= top_top_set_literal )
= ( ( A2 = top_top_set_literal )
& ( B2 = top_top_set_literal ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_721_inf__top_Oeq__neutr__iff,axiom,
! [A2: set_Numeral_num1,B2: set_Numeral_num1] :
( ( ( inf_in6354261971210149043l_num1 @ A2 @ B2 )
= top_to3689904429138878997l_num1 )
= ( ( A2 = top_to3689904429138878997l_num1 )
& ( B2 = top_to3689904429138878997l_num1 ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_722_inf__top_Oeq__neutr__iff,axiom,
! [A2: set_Product_unit,B2: set_Product_unit] :
( ( ( inf_in4660618365625256667t_unit @ A2 @ B2 )
= top_to1996260823553986621t_unit )
= ( ( A2 = top_to1996260823553986621t_unit )
& ( B2 = top_to1996260823553986621t_unit ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_723_inf__top_Oleft__neutral,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ A2 )
= A2 ) ).
% inf_top.left_neutral
thf(fact_724_inf__top_Oleft__neutral,axiom,
! [A2: set_Numeral_num0] :
( ( inf_in6354261966906920242l_num0 @ top_to3689904424835650196l_num0 @ A2 )
= A2 ) ).
% inf_top.left_neutral
thf(fact_725_inf__top_Oleft__neutral,axiom,
! [A2: set_literal] :
( ( inf_inf_set_literal @ top_top_set_literal @ A2 )
= A2 ) ).
% inf_top.left_neutral
thf(fact_726_inf__top_Oleft__neutral,axiom,
! [A2: set_Numeral_num1] :
( ( inf_in6354261971210149043l_num1 @ top_to3689904429138878997l_num1 @ A2 )
= A2 ) ).
% inf_top.left_neutral
thf(fact_727_inf__top_Oleft__neutral,axiom,
! [A2: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ top_to1996260823553986621t_unit @ A2 )
= A2 ) ).
% inf_top.left_neutral
thf(fact_728_inf__top_Oneutr__eq__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( top_top_set_nat
= ( inf_inf_set_nat @ A2 @ B2 ) )
= ( ( A2 = top_top_set_nat )
& ( B2 = top_top_set_nat ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_729_inf__top_Oneutr__eq__iff,axiom,
! [A2: set_Numeral_num0,B2: set_Numeral_num0] :
( ( top_to3689904424835650196l_num0
= ( inf_in6354261966906920242l_num0 @ A2 @ B2 ) )
= ( ( A2 = top_to3689904424835650196l_num0 )
& ( B2 = top_to3689904424835650196l_num0 ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_730_inf__top_Oneutr__eq__iff,axiom,
! [A2: set_literal,B2: set_literal] :
( ( top_top_set_literal
= ( inf_inf_set_literal @ A2 @ B2 ) )
= ( ( A2 = top_top_set_literal )
& ( B2 = top_top_set_literal ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_731_inf__top_Oneutr__eq__iff,axiom,
! [A2: set_Numeral_num1,B2: set_Numeral_num1] :
( ( top_to3689904429138878997l_num1
= ( inf_in6354261971210149043l_num1 @ A2 @ B2 ) )
= ( ( A2 = top_to3689904429138878997l_num1 )
& ( B2 = top_to3689904429138878997l_num1 ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_732_inf__top_Oneutr__eq__iff,axiom,
! [A2: set_Product_unit,B2: set_Product_unit] :
( ( top_to1996260823553986621t_unit
= ( inf_in4660618365625256667t_unit @ A2 @ B2 ) )
= ( ( A2 = top_to1996260823553986621t_unit )
& ( B2 = top_to1996260823553986621t_unit ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_733_inf__top_Oright__neutral,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ A2 @ top_top_set_nat )
= A2 ) ).
% inf_top.right_neutral
thf(fact_734_inf__top_Oright__neutral,axiom,
! [A2: set_Numeral_num0] :
( ( inf_in6354261966906920242l_num0 @ A2 @ top_to3689904424835650196l_num0 )
= A2 ) ).
% inf_top.right_neutral
thf(fact_735_inf__top_Oright__neutral,axiom,
! [A2: set_literal] :
( ( inf_inf_set_literal @ A2 @ top_top_set_literal )
= A2 ) ).
% inf_top.right_neutral
thf(fact_736_inf__top_Oright__neutral,axiom,
! [A2: set_Numeral_num1] :
( ( inf_in6354261971210149043l_num1 @ A2 @ top_to3689904429138878997l_num1 )
= A2 ) ).
% inf_top.right_neutral
thf(fact_737_inf__top_Oright__neutral,axiom,
! [A2: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ A2 @ top_to1996260823553986621t_unit )
= A2 ) ).
% inf_top.right_neutral
thf(fact_738_Int__UNIV,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= top_top_set_nat )
= ( ( A = top_top_set_nat )
& ( B = top_top_set_nat ) ) ) ).
% Int_UNIV
thf(fact_739_Int__UNIV,axiom,
! [A: set_Numeral_num0,B: set_Numeral_num0] :
( ( ( inf_in6354261966906920242l_num0 @ A @ B )
= top_to3689904424835650196l_num0 )
= ( ( A = top_to3689904424835650196l_num0 )
& ( B = top_to3689904424835650196l_num0 ) ) ) ).
% Int_UNIV
thf(fact_740_Int__UNIV,axiom,
! [A: set_literal,B: set_literal] :
( ( ( inf_inf_set_literal @ A @ B )
= top_top_set_literal )
= ( ( A = top_top_set_literal )
& ( B = top_top_set_literal ) ) ) ).
% Int_UNIV
thf(fact_741_Int__UNIV,axiom,
! [A: set_Numeral_num1,B: set_Numeral_num1] :
( ( ( inf_in6354261971210149043l_num1 @ A @ B )
= top_to3689904429138878997l_num1 )
= ( ( A = top_to3689904429138878997l_num1 )
& ( B = top_to3689904429138878997l_num1 ) ) ) ).
% Int_UNIV
thf(fact_742_Int__UNIV,axiom,
! [A: set_Product_unit,B: set_Product_unit] :
( ( ( inf_in4660618365625256667t_unit @ A @ B )
= top_to1996260823553986621t_unit )
= ( ( A = top_to1996260823553986621t_unit )
& ( B = top_to1996260823553986621t_unit ) ) ) ).
% Int_UNIV
thf(fact_743_vimage__UNIV,axiom,
! [F: nat > nat] :
( ( vimage_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat ) ).
% vimage_UNIV
thf(fact_744_vimage__UNIV,axiom,
! [F: numeral_num0 > nat] :
( ( vimage4951960578999371065m0_nat @ F @ top_top_set_nat )
= top_to3689904424835650196l_num0 ) ).
% vimage_UNIV
thf(fact_745_vimage__UNIV,axiom,
! [F: literal > nat] :
( ( vimage_literal_nat @ F @ top_top_set_nat )
= top_top_set_literal ) ).
% vimage_UNIV
thf(fact_746_vimage__UNIV,axiom,
! [F: numeral_num1 > nat] :
( ( vimage6187404907955765562m1_nat @ F @ top_top_set_nat )
= top_to3689904429138878997l_num1 ) ).
% vimage_UNIV
thf(fact_747_vimage__UNIV,axiom,
! [F: product_unit > nat] :
( ( vimage6253328473476588386it_nat @ F @ top_top_set_nat )
= top_to1996260823553986621t_unit ) ).
% vimage_UNIV
thf(fact_748_vimage__UNIV,axiom,
! [F: nat > numeral_num0] :
( ( vimage1705183035017847703l_num0 @ F @ top_to3689904424835650196l_num0 )
= top_top_set_nat ) ).
% vimage_UNIV
thf(fact_749_vimage__UNIV,axiom,
! [F: numeral_num0 > numeral_num0] :
( ( vimage1199591675108543419l_num0 @ F @ top_to3689904424835650196l_num0 )
= top_to3689904424835650196l_num0 ) ).
% vimage_UNIV
thf(fact_750_vimage__UNIV,axiom,
! [F: literal > numeral_num0] :
( ( vimage2868565650143792587l_num0 @ F @ top_to3689904424835650196l_num0 )
= top_top_set_literal ) ).
% vimage_UNIV
thf(fact_751_vimage__UNIV,axiom,
! [F: numeral_num1 > numeral_num0] :
( ( vimage5150483310727131898l_num0 @ F @ top_to3689904424835650196l_num0 )
= top_to3689904429138878997l_num1 ) ).
% vimage_UNIV
thf(fact_752_vimage__UNIV,axiom,
! [F: product_unit > numeral_num0] :
( ( vimage4815744532680921810l_num0 @ F @ top_to3689904424835650196l_num0 )
= top_to1996260823553986621t_unit ) ).
% vimage_UNIV
thf(fact_753_Pow__UNIV,axiom,
( ( pow_nat @ top_top_set_nat )
= top_top_set_set_nat ) ).
% Pow_UNIV
thf(fact_754_Pow__UNIV,axiom,
( ( pow_Numeral_num0 @ top_to3689904424835650196l_num0 )
= top_to3433798952193014772l_num0 ) ).
% Pow_UNIV
thf(fact_755_Pow__UNIV,axiom,
( ( pow_literal @ top_top_set_literal )
= top_to5694933271948605156iteral ) ).
% Pow_UNIV
thf(fact_756_Pow__UNIV,axiom,
( ( pow_Numeral_num1 @ top_to3689904429138878997l_num1 )
= top_to3504832992239360757l_num1 ) ).
% Pow_UNIV
thf(fact_757_Pow__UNIV,axiom,
( ( pow_Product_unit @ top_to1996260823553986621t_unit )
= top_to1767297665138865437t_unit ) ).
% Pow_UNIV
thf(fact_758_finite__Pow__iff,axiom,
! [A: set_nat] :
( ( finite1152437895449049373et_nat @ ( pow_nat @ A ) )
= ( finite_finite_nat @ A ) ) ).
% finite_Pow_iff
thf(fact_759_PowI,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( member_set_nat @ A @ ( pow_nat @ B ) ) ) ).
% PowI
thf(fact_760_Pow__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( member_set_nat @ A @ ( pow_nat @ B ) )
= ( ord_less_eq_set_nat @ A @ B ) ) ).
% Pow_iff
thf(fact_761_surj__diff,axiom,
! [A2: numeral_num1] :
( ( image_6783865936125884741l_num1 @ ( minus_344844880397253196l_num1 @ A2 ) @ top_to3689904429138878997l_num1 )
= top_to3689904429138878997l_num1 ) ).
% surj_diff
thf(fact_762_image__vimage__eq,axiom,
! [F: nat > nat,A: set_nat] :
( ( image_nat_nat @ F @ ( vimage_nat_nat @ F @ A ) )
= ( inf_inf_set_nat @ A @ ( image_nat_nat @ F @ top_top_set_nat ) ) ) ).
% image_vimage_eq
thf(fact_763_top__greatest,axiom,
! [A2: set_Numeral_num0] : ( ord_le5200684351691877604l_num0 @ A2 @ top_to3689904424835650196l_num0 ) ).
% top_greatest
thf(fact_764_top__greatest,axiom,
! [A2: set_literal] : ( ord_le7307670543136651348iteral @ A2 @ top_top_set_literal ) ).
% top_greatest
thf(fact_765_top__greatest,axiom,
! [A2: set_Numeral_num1] : ( ord_le5200684355995106405l_num1 @ A2 @ top_to3689904429138878997l_num1 ) ).
% top_greatest
thf(fact_766_top__greatest,axiom,
! [A2: set_Product_unit] : ( ord_le3507040750410214029t_unit @ A2 @ top_to1996260823553986621t_unit ) ).
% top_greatest
thf(fact_767_top__greatest,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ top_top_set_nat ) ).
% top_greatest
thf(fact_768_top_Oextremum__unique,axiom,
! [A2: set_Numeral_num0] :
( ( ord_le5200684351691877604l_num0 @ top_to3689904424835650196l_num0 @ A2 )
= ( A2 = top_to3689904424835650196l_num0 ) ) ).
% top.extremum_unique
thf(fact_769_top_Oextremum__unique,axiom,
! [A2: set_literal] :
( ( ord_le7307670543136651348iteral @ top_top_set_literal @ A2 )
= ( A2 = top_top_set_literal ) ) ).
% top.extremum_unique
thf(fact_770_top_Oextremum__unique,axiom,
! [A2: set_Numeral_num1] :
( ( ord_le5200684355995106405l_num1 @ top_to3689904429138878997l_num1 @ A2 )
= ( A2 = top_to3689904429138878997l_num1 ) ) ).
% top.extremum_unique
thf(fact_771_top_Oextremum__unique,axiom,
! [A2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ top_to1996260823553986621t_unit @ A2 )
= ( A2 = top_to1996260823553986621t_unit ) ) ).
% top.extremum_unique
thf(fact_772_top_Oextremum__unique,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
= ( A2 = top_top_set_nat ) ) ).
% top.extremum_unique
thf(fact_773_top_Oextremum__uniqueI,axiom,
! [A2: set_Numeral_num0] :
( ( ord_le5200684351691877604l_num0 @ top_to3689904424835650196l_num0 @ A2 )
=> ( A2 = top_to3689904424835650196l_num0 ) ) ).
% top.extremum_uniqueI
thf(fact_774_top_Oextremum__uniqueI,axiom,
! [A2: set_literal] :
( ( ord_le7307670543136651348iteral @ top_top_set_literal @ A2 )
=> ( A2 = top_top_set_literal ) ) ).
% top.extremum_uniqueI
thf(fact_775_top_Oextremum__uniqueI,axiom,
! [A2: set_Numeral_num1] :
( ( ord_le5200684355995106405l_num1 @ top_to3689904429138878997l_num1 @ A2 )
=> ( A2 = top_to3689904429138878997l_num1 ) ) ).
% top.extremum_uniqueI
thf(fact_776_top_Oextremum__uniqueI,axiom,
! [A2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ top_to1996260823553986621t_unit @ A2 )
=> ( A2 = top_to1996260823553986621t_unit ) ) ).
% top.extremum_uniqueI
thf(fact_777_top_Oextremum__uniqueI,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
=> ( A2 = top_top_set_nat ) ) ).
% top.extremum_uniqueI
thf(fact_778_top_Oextremum__strict,axiom,
! [A2: set_Numeral_num0] :
~ ( ord_le526730871819019248l_num0 @ top_to3689904424835650196l_num0 @ A2 ) ).
% top.extremum_strict
thf(fact_779_top_Oextremum__strict,axiom,
! [A2: set_literal] :
~ ( ord_less_set_literal @ top_top_set_literal @ A2 ) ).
% top.extremum_strict
thf(fact_780_top_Oextremum__strict,axiom,
! [A2: set_Numeral_num1] :
~ ( ord_le526730876122248049l_num1 @ top_to3689904429138878997l_num1 @ A2 ) ).
% top.extremum_strict
thf(fact_781_top_Oextremum__strict,axiom,
! [A2: set_Product_unit] :
~ ( ord_le8056459307392131481t_unit @ top_to1996260823553986621t_unit @ A2 ) ).
% top.extremum_strict
thf(fact_782_top_Oextremum__strict,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ top_top_set_nat @ A2 ) ).
% top.extremum_strict
thf(fact_783_top_Onot__eq__extremum,axiom,
! [A2: set_Numeral_num0] :
( ( A2 != top_to3689904424835650196l_num0 )
= ( ord_le526730871819019248l_num0 @ A2 @ top_to3689904424835650196l_num0 ) ) ).
% top.not_eq_extremum
thf(fact_784_top_Onot__eq__extremum,axiom,
! [A2: set_literal] :
( ( A2 != top_top_set_literal )
= ( ord_less_set_literal @ A2 @ top_top_set_literal ) ) ).
% top.not_eq_extremum
thf(fact_785_top_Onot__eq__extremum,axiom,
! [A2: set_Numeral_num1] :
( ( A2 != top_to3689904429138878997l_num1 )
= ( ord_le526730876122248049l_num1 @ A2 @ top_to3689904429138878997l_num1 ) ) ).
% top.not_eq_extremum
thf(fact_786_top_Onot__eq__extremum,axiom,
! [A2: set_Product_unit] :
( ( A2 != top_to1996260823553986621t_unit )
= ( ord_le8056459307392131481t_unit @ A2 @ top_to1996260823553986621t_unit ) ) ).
% top.not_eq_extremum
thf(fact_787_top_Onot__eq__extremum,axiom,
! [A2: set_nat] :
( ( A2 != top_top_set_nat )
= ( ord_less_set_nat @ A2 @ top_top_set_nat ) ) ).
% top.not_eq_extremum
thf(fact_788_Finite__Set_Ofinite__set,axiom,
( ( finite1152437895449049373et_nat @ top_top_set_set_nat )
= ( finite_finite_nat @ top_top_set_nat ) ) ).
% Finite_Set.finite_set
thf(fact_789_Finite__Set_Ofinite__set,axiom,
( ( finite3465821965481346669l_num0 @ top_to3433798952193014772l_num0 )
= ( finite1111429032697314573l_num0 @ top_to3689904424835650196l_num0 ) ) ).
% Finite_Set.finite_set
thf(fact_790_Finite__Set_Ofinite__set,axiom,
( ( finite2869373537460367197iteral @ top_to5694933271948605156iteral )
= ( finite5847741373460823677iteral @ top_top_set_literal ) ) ).
% Finite_Set.finite_set
thf(fact_791_Finite__Set_Ofinite__set,axiom,
( ( finite3465821969784575470l_num1 @ top_to3504832992239360757l_num1 )
= ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 ) ) ).
% Finite_Set.finite_set
thf(fact_792_Finite__Set_Ofinite__set,axiom,
( ( finite1772178364199683094t_unit @ top_to1767297665138865437t_unit )
= ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ).
% Finite_Set.finite_set
thf(fact_793_finite__prod,axiom,
( ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_prod
thf(fact_794_finite__prod,axiom,
( ( finite6806726112283354912l_num0 @ top_to987872205429544815l_num0 )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite1111429032697314573l_num0 @ top_to3689904424835650196l_num0 ) ) ) ).
% finite_prod
thf(fact_795_finite__prod,axiom,
( ( finite211349803975347344iteral @ top_to6658620532179778271iteral )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite5847741373460823677iteral @ top_top_set_literal ) ) ) ).
% finite_prod
thf(fact_796_finite__prod,axiom,
( ( finite6806726116586583713l_num1 @ top_to1058906245475890800l_num1 )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 ) ) ) ).
% finite_prod
thf(fact_797_finite__prod,axiom,
( ( finite5113082511001691337t_unit @ top_to8544742955230171288t_unit )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_prod
thf(fact_798_finite__prod,axiom,
( ( finite3886154922020948986m0_nat @ top_to9085566620183473993m0_nat )
= ( ( finite1111429032697314573l_num0 @ top_to3689904424835650196l_num0 )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_prod
thf(fact_799_finite__prod,axiom,
( ( finite4894210157890413008l_num0 @ top_to2553066108271082967l_num0 )
= ( ( finite1111429032697314573l_num0 @ top_to3689904424835650196l_num0 )
& ( finite1111429032697314573l_num0 @ top_to3689904424835650196l_num0 ) ) ) ).
% finite_prod
thf(fact_800_finite__prod,axiom,
( ( finite3746811924666371904iteral @ top_to7655489177542131015iteral )
= ( ( finite1111429032697314573l_num0 @ top_to3689904424835650196l_num0 )
& ( finite5847741373460823677iteral @ top_top_set_literal ) ) ) ).
% finite_prod
thf(fact_801_finite__prod,axiom,
( ( finite4894210162193641809l_num1 @ top_to2624100148317428952l_num1 )
= ( ( finite1111429032697314573l_num0 @ top_to3689904424835650196l_num0 )
& ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 ) ) ) ).
% finite_prod
thf(fact_802_finite__prod,axiom,
( ( finite3200566556608749433t_unit @ top_to886564821216933632t_unit )
= ( ( finite1111429032697314573l_num0 @ top_to3689904424835650196l_num0 )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_prod
thf(fact_803_finite__Prod__UNIV,axiom,
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_nat @ top_top_set_nat )
=> ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat ) ) ) ).
% finite_Prod_UNIV
thf(fact_804_finite__Prod__UNIV,axiom,
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite1111429032697314573l_num0 @ top_to3689904424835650196l_num0 )
=> ( finite6806726112283354912l_num0 @ top_to987872205429544815l_num0 ) ) ) ).
% finite_Prod_UNIV
thf(fact_805_finite__Prod__UNIV,axiom,
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite5847741373460823677iteral @ top_top_set_literal )
=> ( finite211349803975347344iteral @ top_to6658620532179778271iteral ) ) ) ).
% finite_Prod_UNIV
thf(fact_806_finite__Prod__UNIV,axiom,
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 )
=> ( finite6806726116586583713l_num1 @ top_to1058906245475890800l_num1 ) ) ) ).
% finite_Prod_UNIV
thf(fact_807_finite__Prod__UNIV,axiom,
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( finite5113082511001691337t_unit @ top_to8544742955230171288t_unit ) ) ) ).
% finite_Prod_UNIV
thf(fact_808_finite__Prod__UNIV,axiom,
( ( finite1111429032697314573l_num0 @ top_to3689904424835650196l_num0 )
=> ( ( finite_finite_nat @ top_top_set_nat )
=> ( finite3886154922020948986m0_nat @ top_to9085566620183473993m0_nat ) ) ) ).
% finite_Prod_UNIV
thf(fact_809_finite__Prod__UNIV,axiom,
( ( finite1111429032697314573l_num0 @ top_to3689904424835650196l_num0 )
=> ( ( finite1111429032697314573l_num0 @ top_to3689904424835650196l_num0 )
=> ( finite4894210157890413008l_num0 @ top_to2553066108271082967l_num0 ) ) ) ).
% finite_Prod_UNIV
thf(fact_810_finite__Prod__UNIV,axiom,
( ( finite1111429032697314573l_num0 @ top_to3689904424835650196l_num0 )
=> ( ( finite5847741373460823677iteral @ top_top_set_literal )
=> ( finite3746811924666371904iteral @ top_to7655489177542131015iteral ) ) ) ).
% finite_Prod_UNIV
thf(fact_811_finite__Prod__UNIV,axiom,
( ( finite1111429032697314573l_num0 @ top_to3689904424835650196l_num0 )
=> ( ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 )
=> ( finite4894210162193641809l_num1 @ top_to2624100148317428952l_num1 ) ) ) ).
% finite_Prod_UNIV
thf(fact_812_finite__Prod__UNIV,axiom,
( ( finite1111429032697314573l_num0 @ top_to3689904424835650196l_num0 )
=> ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( finite3200566556608749433t_unit @ top_to886564821216933632t_unit ) ) ) ).
% finite_Prod_UNIV
thf(fact_813_UNIV__eq__I,axiom,
! [A: set_nat] :
( ! [X: nat] : ( member_nat @ X @ A )
=> ( top_top_set_nat = A ) ) ).
% UNIV_eq_I
thf(fact_814_UNIV__eq__I,axiom,
! [A: set_Numeral_num0] :
( ! [X: numeral_num0] : ( member_Numeral_num0 @ X @ A )
=> ( top_to3689904424835650196l_num0 = A ) ) ).
% UNIV_eq_I
thf(fact_815_UNIV__eq__I,axiom,
! [A: set_literal] :
( ! [X: literal] : ( member_literal @ X @ A )
=> ( top_top_set_literal = A ) ) ).
% UNIV_eq_I
thf(fact_816_UNIV__eq__I,axiom,
! [A: set_Numeral_num1] :
( ! [X: numeral_num1] : ( member_Numeral_num1 @ X @ A )
=> ( top_to3689904429138878997l_num1 = A ) ) ).
% UNIV_eq_I
thf(fact_817_UNIV__eq__I,axiom,
! [A: set_Product_unit] :
( ! [X: product_unit] : ( member_Product_unit @ X @ A )
=> ( top_to1996260823553986621t_unit = A ) ) ).
% UNIV_eq_I
thf(fact_818_UNIV__witness,axiom,
? [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_819_UNIV__witness,axiom,
? [X: numeral_num0] : ( member_Numeral_num0 @ X @ top_to3689904424835650196l_num0 ) ).
% UNIV_witness
thf(fact_820_UNIV__witness,axiom,
? [X: literal] : ( member_literal @ X @ top_top_set_literal ) ).
% UNIV_witness
thf(fact_821_UNIV__witness,axiom,
? [X: numeral_num1] : ( member_Numeral_num1 @ X @ top_to3689904429138878997l_num1 ) ).
% UNIV_witness
thf(fact_822_UNIV__witness,axiom,
? [X: product_unit] : ( member_Product_unit @ X @ top_to1996260823553986621t_unit ) ).
% UNIV_witness
thf(fact_823_PowD,axiom,
! [A: set_nat,B: set_nat] :
( ( member_set_nat @ A @ ( pow_nat @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% PowD
thf(fact_824_rangeI,axiom,
! [F: nat > nat,X2: nat] : ( member_nat @ ( F @ X2 ) @ ( image_nat_nat @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_825_rangeI,axiom,
! [F: numeral_num0 > nat,X2: numeral_num0] : ( member_nat @ ( F @ X2 ) @ ( image_8797574156932312687m0_nat @ F @ top_to3689904424835650196l_num0 ) ) ).
% rangeI
thf(fact_826_rangeI,axiom,
! [F: literal > nat,X2: literal] : ( member_nat @ ( F @ X2 ) @ ( image_literal_nat @ F @ top_top_set_literal ) ) ).
% rangeI
thf(fact_827_rangeI,axiom,
! [F: numeral_num1 > nat,X2: numeral_num1] : ( member_nat @ ( F @ X2 ) @ ( image_809646449033931376m1_nat @ F @ top_to3689904429138878997l_num1 ) ) ).
% rangeI
thf(fact_828_rangeI,axiom,
! [F: product_unit > nat,X2: product_unit] : ( member_nat @ ( F @ X2 ) @ ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit ) ) ).
% rangeI
thf(fact_829_range__eqI,axiom,
! [B2: nat,F: nat > nat,X2: nat] :
( ( B2
= ( F @ X2 ) )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_830_range__eqI,axiom,
! [B2: nat,F: numeral_num0 > nat,X2: numeral_num0] :
( ( B2
= ( F @ X2 ) )
=> ( member_nat @ B2 @ ( image_8797574156932312687m0_nat @ F @ top_to3689904424835650196l_num0 ) ) ) ).
% range_eqI
thf(fact_831_range__eqI,axiom,
! [B2: nat,F: literal > nat,X2: literal] :
( ( B2
= ( F @ X2 ) )
=> ( member_nat @ B2 @ ( image_literal_nat @ F @ top_top_set_literal ) ) ) ).
% range_eqI
thf(fact_832_range__eqI,axiom,
! [B2: nat,F: numeral_num1 > nat,X2: numeral_num1] :
( ( B2
= ( F @ X2 ) )
=> ( member_nat @ B2 @ ( image_809646449033931376m1_nat @ F @ top_to3689904429138878997l_num1 ) ) ) ).
% range_eqI
thf(fact_833_range__eqI,axiom,
! [B2: nat,F: product_unit > nat,X2: product_unit] :
( ( B2
= ( F @ X2 ) )
=> ( member_nat @ B2 @ ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit ) ) ) ).
% range_eqI
thf(fact_834_infinite__UNIV__char__0,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_char_0
thf(fact_835_ex__new__if__finite,axiom,
! [A: set_nat] :
( ~ ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_nat @ A )
=> ? [A3: nat] :
~ ( member_nat @ A3 @ A ) ) ) ).
% ex_new_if_finite
thf(fact_836_ex__new__if__finite,axiom,
! [A: set_Numeral_num0] :
( ~ ( finite1111429032697314573l_num0 @ top_to3689904424835650196l_num0 )
=> ( ( finite1111429032697314573l_num0 @ A )
=> ? [A3: numeral_num0] :
~ ( member_Numeral_num0 @ A3 @ A ) ) ) ).
% ex_new_if_finite
thf(fact_837_ex__new__if__finite,axiom,
! [A: set_literal] :
( ~ ( finite5847741373460823677iteral @ top_top_set_literal )
=> ( ( finite5847741373460823677iteral @ A )
=> ? [A3: literal] :
~ ( member_literal @ A3 @ A ) ) ) ).
% ex_new_if_finite
thf(fact_838_ex__new__if__finite,axiom,
! [A: set_Numeral_num1] :
( ~ ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 )
=> ( ( finite1111429032697314574l_num1 @ A )
=> ? [A3: numeral_num1] :
~ ( member_Numeral_num1 @ A3 @ A ) ) ) ).
% ex_new_if_finite
thf(fact_839_ex__new__if__finite,axiom,
! [A: set_Product_unit] :
( ~ ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ( finite4290736615968046902t_unit @ A )
=> ? [A3: product_unit] :
~ ( member_Product_unit @ A3 @ A ) ) ) ).
% ex_new_if_finite
thf(fact_840_finite__class_Ofinite__UNIV,axiom,
finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 ).
% finite_class.finite_UNIV
thf(fact_841_finite__class_Ofinite__UNIV,axiom,
finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ).
% finite_class.finite_UNIV
thf(fact_842_infinite__UNIV,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV
thf(fact_843_subset__UNIV,axiom,
! [A: set_Numeral_num0] : ( ord_le5200684351691877604l_num0 @ A @ top_to3689904424835650196l_num0 ) ).
% subset_UNIV
thf(fact_844_subset__UNIV,axiom,
! [A: set_literal] : ( ord_le7307670543136651348iteral @ A @ top_top_set_literal ) ).
% subset_UNIV
thf(fact_845_subset__UNIV,axiom,
! [A: set_Numeral_num1] : ( ord_le5200684355995106405l_num1 @ A @ top_to3689904429138878997l_num1 ) ).
% subset_UNIV
thf(fact_846_subset__UNIV,axiom,
! [A: set_Product_unit] : ( ord_le3507040750410214029t_unit @ A @ top_to1996260823553986621t_unit ) ).
% subset_UNIV
thf(fact_847_subset__UNIV,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% subset_UNIV
thf(fact_848_Int__UNIV__left,axiom,
! [B: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ B )
= B ) ).
% Int_UNIV_left
thf(fact_849_Int__UNIV__left,axiom,
! [B: set_Numeral_num0] :
( ( inf_in6354261966906920242l_num0 @ top_to3689904424835650196l_num0 @ B )
= B ) ).
% Int_UNIV_left
thf(fact_850_Int__UNIV__left,axiom,
! [B: set_literal] :
( ( inf_inf_set_literal @ top_top_set_literal @ B )
= B ) ).
% Int_UNIV_left
thf(fact_851_Int__UNIV__left,axiom,
! [B: set_Numeral_num1] :
( ( inf_in6354261971210149043l_num1 @ top_to3689904429138878997l_num1 @ B )
= B ) ).
% Int_UNIV_left
thf(fact_852_Int__UNIV__left,axiom,
! [B: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ top_to1996260823553986621t_unit @ B )
= B ) ).
% Int_UNIV_left
thf(fact_853_Int__UNIV__right,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ top_top_set_nat )
= A ) ).
% Int_UNIV_right
thf(fact_854_Int__UNIV__right,axiom,
! [A: set_Numeral_num0] :
( ( inf_in6354261966906920242l_num0 @ A @ top_to3689904424835650196l_num0 )
= A ) ).
% Int_UNIV_right
thf(fact_855_Int__UNIV__right,axiom,
! [A: set_literal] :
( ( inf_inf_set_literal @ A @ top_top_set_literal )
= A ) ).
% Int_UNIV_right
thf(fact_856_Int__UNIV__right,axiom,
! [A: set_Numeral_num1] :
( ( inf_in6354261971210149043l_num1 @ A @ top_to3689904429138878997l_num1 )
= A ) ).
% Int_UNIV_right
thf(fact_857_Int__UNIV__right,axiom,
! [A: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ A @ top_to1996260823553986621t_unit )
= A ) ).
% Int_UNIV_right
thf(fact_858_infinite__UNIV__nat,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_nat
thf(fact_859_range__subsetD,axiom,
! [F: nat > nat,B: set_nat,I2: nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ top_top_set_nat ) @ B )
=> ( member_nat @ ( F @ I2 ) @ B ) ) ).
% range_subsetD
thf(fact_860_range__subsetD,axiom,
! [F: numeral_num0 > nat,B: set_nat,I2: numeral_num0] :
( ( ord_less_eq_set_nat @ ( image_8797574156932312687m0_nat @ F @ top_to3689904424835650196l_num0 ) @ B )
=> ( member_nat @ ( F @ I2 ) @ B ) ) ).
% range_subsetD
thf(fact_861_range__subsetD,axiom,
! [F: literal > nat,B: set_nat,I2: literal] :
( ( ord_less_eq_set_nat @ ( image_literal_nat @ F @ top_top_set_literal ) @ B )
=> ( member_nat @ ( F @ I2 ) @ B ) ) ).
% range_subsetD
thf(fact_862_range__subsetD,axiom,
! [F: numeral_num1 > nat,B: set_nat,I2: numeral_num1] :
( ( ord_less_eq_set_nat @ ( image_809646449033931376m1_nat @ F @ top_to3689904429138878997l_num1 ) @ B )
=> ( member_nat @ ( F @ I2 ) @ B ) ) ).
% range_subsetD
thf(fact_863_range__subsetD,axiom,
! [F: product_unit > nat,B: set_nat,I2: product_unit] :
( ( ord_less_eq_set_nat @ ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit ) @ B )
=> ( member_nat @ ( F @ I2 ) @ B ) ) ).
% range_subsetD
thf(fact_864_infinite__iff__countable__subset,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ? [F3: nat > nat] :
( ( inj_on_nat_nat @ F3 @ top_top_set_nat )
& ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ top_top_set_nat ) @ S ) ) ) ) ).
% infinite_iff_countable_subset
thf(fact_865_infinite__countable__subset,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ? [F4: nat > nat] :
( ( inj_on_nat_nat @ F4 @ top_top_set_nat )
& ( ord_less_eq_set_nat @ ( image_nat_nat @ F4 @ top_top_set_nat ) @ S ) ) ) ).
% infinite_countable_subset
thf(fact_866_linorder__injI,axiom,
! [F: nat > nat] :
( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ( F @ X )
!= ( F @ Y3 ) ) )
=> ( inj_on_nat_nat @ F @ top_top_set_nat ) ) ).
% linorder_injI
thf(fact_867_card__eq__UNIV__imp__eq__UNIV,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( ( finite_card_nat @ A )
= ( finite_card_nat @ top_top_set_nat ) )
=> ( A = top_top_set_nat ) ) ) ).
% card_eq_UNIV_imp_eq_UNIV
thf(fact_868_card__eq__UNIV__imp__eq__UNIV,axiom,
! [A: set_Numeral_num0] :
( ( finite1111429032697314573l_num0 @ top_to3689904424835650196l_num0 )
=> ( ( ( finite6454714172617411596l_num0 @ A )
= ( finite6454714172617411596l_num0 @ top_to3689904424835650196l_num0 ) )
=> ( A = top_to3689904424835650196l_num0 ) ) ) ).
% card_eq_UNIV_imp_eq_UNIV
thf(fact_869_card__eq__UNIV__imp__eq__UNIV,axiom,
! [A: set_literal] :
( ( finite5847741373460823677iteral @ top_top_set_literal )
=> ( ( ( finite_card_literal @ A )
= ( finite_card_literal @ top_top_set_literal ) )
=> ( A = top_top_set_literal ) ) ) ).
% card_eq_UNIV_imp_eq_UNIV
thf(fact_870_card__eq__UNIV__imp__eq__UNIV,axiom,
! [A: set_Numeral_num1] :
( ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 )
=> ( ( ( finite6454714172617411597l_num1 @ A )
= ( finite6454714172617411597l_num1 @ top_to3689904429138878997l_num1 ) )
=> ( A = top_to3689904429138878997l_num1 ) ) ) ).
% card_eq_UNIV_imp_eq_UNIV
thf(fact_871_card__eq__UNIV__imp__eq__UNIV,axiom,
! [A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ( ( finite410649719033368117t_unit @ A )
= ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit ) )
=> ( A = top_to1996260823553986621t_unit ) ) ) ).
% card_eq_UNIV_imp_eq_UNIV
thf(fact_872_Pow__mono,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_le6893508408891458716et_nat @ ( pow_nat @ A ) @ ( pow_nat @ B ) ) ) ).
% Pow_mono
thf(fact_873_image__Pow__surj,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( ( image_nat_nat @ F @ A )
= B )
=> ( ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ ( pow_nat @ A ) )
= ( pow_nat @ B ) ) ) ).
% image_Pow_surj
thf(fact_874_finite__UNIV__inj__surj,axiom,
! [F: nat > nat] :
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat ) ) ) ).
% finite_UNIV_inj_surj
thf(fact_875_finite__UNIV__inj__surj,axiom,
! [F: numeral_num0 > numeral_num0] :
( ( finite1111429032697314573l_num0 @ top_to3689904424835650196l_num0 )
=> ( ( inj_on1355912882866956657l_num0 @ F @ top_to3689904424835650196l_num0 )
=> ( ( image_2832974300507296261l_num0 @ F @ top_to3689904424835650196l_num0 )
= top_to3689904424835650196l_num0 ) ) ) ).
% finite_UNIV_inj_surj
thf(fact_876_finite__UNIV__inj__surj,axiom,
! [F: literal > literal] :
( ( finite5847741373460823677iteral @ top_top_set_literal )
=> ( ( inj_on602069361295035377iteral @ F @ top_top_set_literal )
=> ( ( image_8195128725298311301iteral @ F @ top_top_set_literal )
= top_top_set_literal ) ) ) ).
% finite_UNIV_inj_surj
thf(fact_877_finite__UNIV__inj__surj,axiom,
! [F: numeral_num1 > numeral_num1] :
( ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 )
=> ( ( inj_on5306804518485545137l_num1 @ F @ top_to3689904429138878997l_num1 )
=> ( ( image_6783865936125884741l_num1 @ F @ top_to3689904429138878997l_num1 )
= top_to3689904429138878997l_num1 ) ) ) ).
% finite_UNIV_inj_surj
thf(fact_878_finite__UNIV__inj__surj,axiom,
! [F: product_unit > product_unit] :
( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ( inj_on8151373323710067377t_unit @ F @ top_to1996260823553986621t_unit )
=> ( ( image_405062704495631173t_unit @ F @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit ) ) ) ).
% finite_UNIV_inj_surj
thf(fact_879_finite__UNIV__surj__inj,axiom,
! [F: nat > nat] :
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( inj_on_nat_nat @ F @ top_top_set_nat ) ) ) ).
% finite_UNIV_surj_inj
thf(fact_880_finite__UNIV__surj__inj,axiom,
! [F: numeral_num0 > numeral_num0] :
( ( finite1111429032697314573l_num0 @ top_to3689904424835650196l_num0 )
=> ( ( ( image_2832974300507296261l_num0 @ F @ top_to3689904424835650196l_num0 )
= top_to3689904424835650196l_num0 )
=> ( inj_on1355912882866956657l_num0 @ F @ top_to3689904424835650196l_num0 ) ) ) ).
% finite_UNIV_surj_inj
thf(fact_881_finite__UNIV__surj__inj,axiom,
! [F: literal > literal] :
( ( finite5847741373460823677iteral @ top_top_set_literal )
=> ( ( ( image_8195128725298311301iteral @ F @ top_top_set_literal )
= top_top_set_literal )
=> ( inj_on602069361295035377iteral @ F @ top_top_set_literal ) ) ) ).
% finite_UNIV_surj_inj
thf(fact_882_finite__UNIV__surj__inj,axiom,
! [F: numeral_num1 > numeral_num1] :
( ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 )
=> ( ( ( image_6783865936125884741l_num1 @ F @ top_to3689904429138878997l_num1 )
= top_to3689904429138878997l_num1 )
=> ( inj_on5306804518485545137l_num1 @ F @ top_to3689904429138878997l_num1 ) ) ) ).
% finite_UNIV_surj_inj
thf(fact_883_finite__UNIV__surj__inj,axiom,
! [F: product_unit > product_unit] :
( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ( ( image_405062704495631173t_unit @ F @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit )
=> ( inj_on8151373323710067377t_unit @ F @ top_to1996260823553986621t_unit ) ) ) ).
% finite_UNIV_surj_inj
thf(fact_884_inj__image__subset__iff,axiom,
! [F: numeral_num0 > nat,A: set_Numeral_num0,B: set_Numeral_num0] :
( ( inj_on7129071197303617539m0_nat @ F @ top_to3689904424835650196l_num0 )
=> ( ( ord_less_eq_set_nat @ ( image_8797574156932312687m0_nat @ F @ A ) @ ( image_8797574156932312687m0_nat @ F @ B ) )
= ( ord_le5200684351691877604l_num0 @ A @ B ) ) ) ).
% inj_image_subset_iff
thf(fact_885_inj__image__subset__iff,axiom,
! [F: literal > nat,A: set_literal,B: set_literal] :
( ( inj_on_literal_nat @ F @ top_top_set_literal )
=> ( ( ord_less_eq_set_nat @ ( image_literal_nat @ F @ A ) @ ( image_literal_nat @ F @ B ) )
= ( ord_le7307670543136651348iteral @ A @ B ) ) ) ).
% inj_image_subset_iff
thf(fact_886_inj__image__subset__iff,axiom,
! [F: numeral_num1 > nat,A: set_Numeral_num1,B: set_Numeral_num1] :
( ( inj_on8364515526260012036m1_nat @ F @ top_to3689904429138878997l_num1 )
=> ( ( ord_less_eq_set_nat @ ( image_809646449033931376m1_nat @ F @ A ) @ ( image_809646449033931376m1_nat @ F @ B ) )
= ( ord_le5200684355995106405l_num1 @ A @ B ) ) ) ).
% inj_image_subset_iff
thf(fact_887_inj__image__subset__iff,axiom,
! [F: product_unit > nat,A: set_Product_unit,B: set_Product_unit] :
( ( inj_on8430439091780834860it_nat @ F @ top_to1996260823553986621t_unit )
=> ( ( ord_less_eq_set_nat @ ( image_875570014554754200it_nat @ F @ A ) @ ( image_875570014554754200it_nat @ F @ B ) )
= ( ord_le3507040750410214029t_unit @ A @ B ) ) ) ).
% inj_image_subset_iff
thf(fact_888_inj__image__subset__iff,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) )
= ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% inj_image_subset_iff
thf(fact_889_image__set__diff,axiom,
! [F: numeral_num0 > nat,A: set_Numeral_num0,B: set_Numeral_num0] :
( ( inj_on7129071197303617539m0_nat @ F @ top_to3689904424835650196l_num0 )
=> ( ( image_8797574156932312687m0_nat @ F @ ( minus_8146479927826647979l_num0 @ A @ B ) )
= ( minus_minus_set_nat @ ( image_8797574156932312687m0_nat @ F @ A ) @ ( image_8797574156932312687m0_nat @ F @ B ) ) ) ) ).
% image_set_diff
thf(fact_890_image__set__diff,axiom,
! [F: literal > nat,A: set_literal,B: set_literal] :
( ( inj_on_literal_nat @ F @ top_top_set_literal )
=> ( ( image_literal_nat @ F @ ( minus_7832829386415567259iteral @ A @ B ) )
= ( minus_minus_set_nat @ ( image_literal_nat @ F @ A ) @ ( image_literal_nat @ F @ B ) ) ) ) ).
% image_set_diff
thf(fact_891_image__set__diff,axiom,
! [F: numeral_num1 > nat,A: set_Numeral_num1,B: set_Numeral_num1] :
( ( inj_on8364515526260012036m1_nat @ F @ top_to3689904429138878997l_num1 )
=> ( ( image_809646449033931376m1_nat @ F @ ( minus_8146479932129876780l_num1 @ A @ B ) )
= ( minus_minus_set_nat @ ( image_809646449033931376m1_nat @ F @ A ) @ ( image_809646449033931376m1_nat @ F @ B ) ) ) ) ).
% image_set_diff
thf(fact_892_image__set__diff,axiom,
! [F: product_unit > nat,A: set_Product_unit,B: set_Product_unit] :
( ( inj_on8430439091780834860it_nat @ F @ top_to1996260823553986621t_unit )
=> ( ( image_875570014554754200it_nat @ F @ ( minus_6452836326544984404t_unit @ A @ B ) )
= ( minus_minus_set_nat @ ( image_875570014554754200it_nat @ F @ A ) @ ( image_875570014554754200it_nat @ F @ B ) ) ) ) ).
% image_set_diff
thf(fact_893_image__set__diff,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( image_nat_nat @ F @ ( minus_minus_set_nat @ A @ B ) )
= ( minus_minus_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ) ).
% image_set_diff
thf(fact_894_finite__vimageD,axiom,
! [H: nat > nat,F2: set_nat] :
( ( finite_finite_nat @ ( vimage_nat_nat @ H @ F2 ) )
=> ( ( ( image_nat_nat @ H @ top_top_set_nat )
= top_top_set_nat )
=> ( finite_finite_nat @ F2 ) ) ) ).
% finite_vimageD
thf(fact_895_finite__vimageD,axiom,
! [H: nat > numeral_num0,F2: set_Numeral_num0] :
( ( finite_finite_nat @ ( vimage1705183035017847703l_num0 @ H @ F2 ) )
=> ( ( ( image_5550796612950789325l_num0 @ H @ top_top_set_nat )
= top_to3689904424835650196l_num0 )
=> ( finite1111429032697314573l_num0 @ F2 ) ) ) ).
% finite_vimageD
thf(fact_896_finite__vimageD,axiom,
! [H: nat > literal,F2: set_literal] :
( ( finite_finite_nat @ ( vimage_nat_literal @ H @ F2 ) )
=> ( ( ( image_nat_literal @ H @ top_top_set_nat )
= top_top_set_literal )
=> ( finite5847741373460823677iteral @ F2 ) ) ) ).
% finite_vimageD
thf(fact_897_finite__vimageD,axiom,
! [H: nat > numeral_num1,F2: set_Numeral_num1] :
( ( finite_finite_nat @ ( vimage1705183035017847704l_num1 @ H @ F2 ) )
=> ( ( ( image_5550796612950789326l_num1 @ H @ top_top_set_nat )
= top_to3689904429138878997l_num1 )
=> ( finite1111429032697314574l_num1 @ F2 ) ) ) ).
% finite_vimageD
thf(fact_898_finite__vimageD,axiom,
! [H: nat > product_unit,F2: set_Product_unit] :
( ( finite_finite_nat @ ( vimage4884490618288580032t_unit @ H @ F2 ) )
=> ( ( ( image_8730104196221521654t_unit @ H @ top_top_set_nat )
= top_to1996260823553986621t_unit )
=> ( finite4290736615968046902t_unit @ F2 ) ) ) ).
% finite_vimageD
thf(fact_899_finite__vimageD,axiom,
! [H: numeral_num0 > nat,F2: set_nat] :
( ( finite1111429032697314573l_num0 @ ( vimage4951960578999371065m0_nat @ H @ F2 ) )
=> ( ( ( image_8797574156932312687m0_nat @ H @ top_to3689904424835650196l_num0 )
= top_top_set_nat )
=> ( finite_finite_nat @ F2 ) ) ) ).
% finite_vimageD
thf(fact_900_finite__vimageD,axiom,
! [H: numeral_num0 > numeral_num0,F2: set_Numeral_num0] :
( ( finite1111429032697314573l_num0 @ ( vimage1199591675108543419l_num0 @ H @ F2 ) )
=> ( ( ( image_2832974300507296261l_num0 @ H @ top_to3689904424835650196l_num0 )
= top_to3689904424835650196l_num0 )
=> ( finite1111429032697314573l_num0 @ F2 ) ) ) ).
% finite_vimageD
thf(fact_901_finite__vimageD,axiom,
! [H: numeral_num0 > literal,F2: set_literal] :
( ( finite1111429032697314573l_num0 @ ( vimage7997836657330795051iteral @ H @ F2 ) )
=> ( ( ( image_8737817577461598069iteral @ H @ top_to3689904424835650196l_num0 )
= top_top_set_literal )
=> ( finite5847741373460823677iteral @ F2 ) ) ) ).
% finite_vimageD
thf(fact_902_finite__vimageD,axiom,
! [H: numeral_num0 > numeral_num1,F2: set_Numeral_num1] :
( ( finite1111429032697314573l_num0 @ ( vimage1199591675108543420l_num1 @ H @ F2 ) )
=> ( ( ( image_2832974300507296262l_num1 @ H @ top_to3689904424835650196l_num0 )
= top_to3689904429138878997l_num1 )
=> ( finite1111429032697314574l_num1 @ F2 ) ) ) ).
% finite_vimageD
thf(fact_903_finite__vimageD,axiom,
! [H: numeral_num0 > product_unit,F2: set_Product_unit] :
( ( finite1111429032697314573l_num0 @ ( vimage4378899258379275748t_unit @ H @ F2 ) )
=> ( ( ( image_6012281883778028590t_unit @ H @ top_to3689904424835650196l_num0 )
= top_to1996260823553986621t_unit )
=> ( finite4290736615968046902t_unit @ F2 ) ) ) ).
% finite_vimageD
thf(fact_904_vimage__subsetD,axiom,
! [F: numeral_num0 > numeral_num0,B: set_Numeral_num0,A: set_Numeral_num0] :
( ( ( image_2832974300507296261l_num0 @ F @ top_to3689904424835650196l_num0 )
= top_to3689904424835650196l_num0 )
=> ( ( ord_le5200684351691877604l_num0 @ ( vimage1199591675108543419l_num0 @ F @ B ) @ A )
=> ( ord_le5200684351691877604l_num0 @ B @ ( image_2832974300507296261l_num0 @ F @ A ) ) ) ) ).
% vimage_subsetD
thf(fact_905_vimage__subsetD,axiom,
! [F: numeral_num0 > literal,B: set_literal,A: set_Numeral_num0] :
( ( ( image_8737817577461598069iteral @ F @ top_to3689904424835650196l_num0 )
= top_top_set_literal )
=> ( ( ord_le5200684351691877604l_num0 @ ( vimage7997836657330795051iteral @ F @ B ) @ A )
=> ( ord_le7307670543136651348iteral @ B @ ( image_8737817577461598069iteral @ F @ A ) ) ) ) ).
% vimage_subsetD
thf(fact_906_vimage__subsetD,axiom,
! [F: numeral_num0 > numeral_num1,B: set_Numeral_num1,A: set_Numeral_num0] :
( ( ( image_2832974300507296262l_num1 @ F @ top_to3689904424835650196l_num0 )
= top_to3689904429138878997l_num1 )
=> ( ( ord_le5200684351691877604l_num0 @ ( vimage1199591675108543420l_num1 @ F @ B ) @ A )
=> ( ord_le5200684355995106405l_num1 @ B @ ( image_2832974300507296262l_num1 @ F @ A ) ) ) ) ).
% vimage_subsetD
thf(fact_907_vimage__subsetD,axiom,
! [F: numeral_num0 > product_unit,B: set_Product_unit,A: set_Numeral_num0] :
( ( ( image_6012281883778028590t_unit @ F @ top_to3689904424835650196l_num0 )
= top_to1996260823553986621t_unit )
=> ( ( ord_le5200684351691877604l_num0 @ ( vimage4378899258379275748t_unit @ F @ B ) @ A )
=> ( ord_le3507040750410214029t_unit @ B @ ( image_6012281883778028590t_unit @ F @ A ) ) ) ) ).
% vimage_subsetD
thf(fact_908_vimage__subsetD,axiom,
! [F: literal > numeral_num0,B: set_Numeral_num0,A: set_literal] :
( ( ( image_3608546570274595605l_num0 @ F @ top_top_set_literal )
= top_to3689904424835650196l_num0 )
=> ( ( ord_le7307670543136651348iteral @ ( vimage2868565650143792587l_num0 @ F @ B ) @ A )
=> ( ord_le5200684351691877604l_num0 @ B @ ( image_3608546570274595605l_num0 @ F @ A ) ) ) ) ).
% vimage_subsetD
thf(fact_909_vimage__subsetD,axiom,
! [F: literal > literal,B: set_literal,A: set_literal] :
( ( ( image_8195128725298311301iteral @ F @ top_top_set_literal )
= top_top_set_literal )
=> ( ( ord_le7307670543136651348iteral @ ( vimage8238609917233974331iteral @ F @ B ) @ A )
=> ( ord_le7307670543136651348iteral @ B @ ( image_8195128725298311301iteral @ F @ A ) ) ) ) ).
% vimage_subsetD
thf(fact_910_vimage__subsetD,axiom,
! [F: literal > numeral_num1,B: set_Numeral_num1,A: set_literal] :
( ( ( image_3608546570274595606l_num1 @ F @ top_top_set_literal )
= top_to3689904429138878997l_num1 )
=> ( ( ord_le7307670543136651348iteral @ ( vimage2868565650143792588l_num1 @ F @ B ) @ A )
=> ( ord_le5200684355995106405l_num1 @ B @ ( image_3608546570274595606l_num1 @ F @ A ) ) ) ) ).
% vimage_subsetD
thf(fact_911_vimage__subsetD,axiom,
! [F: literal > product_unit,B: set_Product_unit,A: set_literal] :
( ( ( image_6787854153545327934t_unit @ F @ top_top_set_literal )
= top_to1996260823553986621t_unit )
=> ( ( ord_le7307670543136651348iteral @ ( vimage6047873233414524916t_unit @ F @ B ) @ A )
=> ( ord_le3507040750410214029t_unit @ B @ ( image_6787854153545327934t_unit @ F @ A ) ) ) ) ).
% vimage_subsetD
thf(fact_912_vimage__subsetD,axiom,
! [F: numeral_num1 > numeral_num0,B: set_Numeral_num0,A: set_Numeral_num1] :
( ( ( image_6783865936125884740l_num0 @ F @ top_to3689904429138878997l_num1 )
= top_to3689904424835650196l_num0 )
=> ( ( ord_le5200684355995106405l_num1 @ ( vimage5150483310727131898l_num0 @ F @ B ) @ A )
=> ( ord_le5200684351691877604l_num0 @ B @ ( image_6783865936125884740l_num0 @ F @ A ) ) ) ) ).
% vimage_subsetD
thf(fact_913_vimage__subsetD,axiom,
! [F: numeral_num1 > literal,B: set_literal,A: set_Numeral_num1] :
( ( ( image_5852747068178070836iteral @ F @ top_to3689904429138878997l_num1 )
= top_top_set_literal )
=> ( ( ord_le5200684355995106405l_num1 @ ( vimage5112766148047267818iteral @ F @ B ) @ A )
=> ( ord_le7307670543136651348iteral @ B @ ( image_5852747068178070836iteral @ F @ A ) ) ) ) ).
% vimage_subsetD
thf(fact_914_finite__vimageI,axiom,
! [F2: set_nat,H: nat > nat] :
( ( finite_finite_nat @ F2 )
=> ( ( inj_on_nat_nat @ H @ top_top_set_nat )
=> ( finite_finite_nat @ ( vimage_nat_nat @ H @ F2 ) ) ) ) ).
% finite_vimageI
thf(fact_915_finite__vimageI,axiom,
! [F2: set_nat,H: numeral_num0 > nat] :
( ( finite_finite_nat @ F2 )
=> ( ( inj_on7129071197303617539m0_nat @ H @ top_to3689904424835650196l_num0 )
=> ( finite1111429032697314573l_num0 @ ( vimage4951960578999371065m0_nat @ H @ F2 ) ) ) ) ).
% finite_vimageI
thf(fact_916_finite__vimageI,axiom,
! [F2: set_nat,H: literal > nat] :
( ( finite_finite_nat @ F2 )
=> ( ( inj_on_literal_nat @ H @ top_top_set_literal )
=> ( finite5847741373460823677iteral @ ( vimage_literal_nat @ H @ F2 ) ) ) ) ).
% finite_vimageI
thf(fact_917_finite__vimageI,axiom,
! [F2: set_nat,H: numeral_num1 > nat] :
( ( finite_finite_nat @ F2 )
=> ( ( inj_on8364515526260012036m1_nat @ H @ top_to3689904429138878997l_num1 )
=> ( finite1111429032697314574l_num1 @ ( vimage6187404907955765562m1_nat @ H @ F2 ) ) ) ) ).
% finite_vimageI
thf(fact_918_finite__vimageI,axiom,
! [F2: set_nat,H: product_unit > nat] :
( ( finite_finite_nat @ F2 )
=> ( ( inj_on8430439091780834860it_nat @ H @ top_to1996260823553986621t_unit )
=> ( finite4290736615968046902t_unit @ ( vimage6253328473476588386it_nat @ H @ F2 ) ) ) ) ).
% finite_vimageI
thf(fact_919_finite__vimageD_H,axiom,
! [F: nat > nat,A: set_nat] :
( ( finite_finite_nat @ ( vimage_nat_nat @ F @ A ) )
=> ( ( ord_less_eq_set_nat @ A @ ( image_nat_nat @ F @ top_top_set_nat ) )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_vimageD'
thf(fact_920_finite__vimageD_H,axiom,
! [F: numeral_num0 > nat,A: set_nat] :
( ( finite1111429032697314573l_num0 @ ( vimage4951960578999371065m0_nat @ F @ A ) )
=> ( ( ord_less_eq_set_nat @ A @ ( image_8797574156932312687m0_nat @ F @ top_to3689904424835650196l_num0 ) )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_vimageD'
thf(fact_921_finite__vimageD_H,axiom,
! [F: literal > nat,A: set_nat] :
( ( finite5847741373460823677iteral @ ( vimage_literal_nat @ F @ A ) )
=> ( ( ord_less_eq_set_nat @ A @ ( image_literal_nat @ F @ top_top_set_literal ) )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_vimageD'
thf(fact_922_finite__vimageD_H,axiom,
! [F: numeral_num1 > nat,A: set_nat] :
( ( finite1111429032697314574l_num1 @ ( vimage6187404907955765562m1_nat @ F @ A ) )
=> ( ( ord_less_eq_set_nat @ A @ ( image_809646449033931376m1_nat @ F @ top_to3689904429138878997l_num1 ) )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_vimageD'
thf(fact_923_finite__vimageD_H,axiom,
! [F: product_unit > nat,A: set_nat] :
( ( finite4290736615968046902t_unit @ ( vimage6253328473476588386it_nat @ F @ A ) )
=> ( ( ord_less_eq_set_nat @ A @ ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit ) )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_vimageD'
thf(fact_924_finite__option__UNIV,axiom,
( ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat )
= ( finite_finite_nat @ top_top_set_nat ) ) ).
% finite_option_UNIV
thf(fact_925_finite__option__UNIV,axiom,
( ( finite3139260970856576979l_num0 @ top_to4357361496606412890l_num0 )
= ( finite1111429032697314573l_num0 @ top_to3689904424835650196l_num0 ) ) ).
% finite_option_UNIV
thf(fact_926_finite__option__UNIV,axiom,
( ( finite5071707688241699267iteral @ top_to8248435444729185354iteral )
= ( finite5847741373460823677iteral @ top_top_set_literal ) ) ).
% finite_option_UNIV
thf(fact_927_finite__option__UNIV,axiom,
( ( finite3139260975159805780l_num1 @ top_to4428395536652758875l_num1 )
= ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 ) ) ).
% finite_option_UNIV
thf(fact_928_finite__option__UNIV,axiom,
( ( finite1445617369574913404t_unit @ top_to2690860209552263555t_unit )
= ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ).
% finite_option_UNIV
thf(fact_929_inf__top_Osemilattice__neutr__order__axioms,axiom,
semila6117336903898717938l_num0 @ inf_in6354261966906920242l_num0 @ top_to3689904424835650196l_num0 @ ord_le5200684351691877604l_num0 @ ord_le526730871819019248l_num0 ).
% inf_top.semilattice_neutr_order_axioms
thf(fact_930_inf__top_Osemilattice__neutr__order__axioms,axiom,
semila836322847636774754iteral @ inf_inf_set_literal @ top_top_set_literal @ ord_le7307670543136651348iteral @ ord_less_set_literal ).
% inf_top.semilattice_neutr_order_axioms
thf(fact_931_inf__top_Osemilattice__neutr__order__axioms,axiom,
semila6117336908201946739l_num1 @ inf_in6354261971210149043l_num1 @ top_to3689904429138878997l_num1 @ ord_le5200684355995106405l_num1 @ ord_le526730876122248049l_num1 ).
% inf_top.semilattice_neutr_order_axioms
thf(fact_932_inf__top_Osemilattice__neutr__order__axioms,axiom,
semila4423693302617054363t_unit @ inf_in4660618365625256667t_unit @ top_to1996260823553986621t_unit @ ord_le3507040750410214029t_unit @ ord_le8056459307392131481t_unit ).
% inf_top.semilattice_neutr_order_axioms
thf(fact_933_inf__top_Osemilattice__neutr__order__axioms,axiom,
semila1667268886620078168et_nat @ inf_inf_set_nat @ top_top_set_nat @ ord_less_eq_set_nat @ ord_less_set_nat ).
% inf_top.semilattice_neutr_order_axioms
thf(fact_934_card__range__greater__zero,axiom,
! [F: nat > numeral_num0] :
( ( finite1111429032697314573l_num0 @ ( image_5550796612950789325l_num0 @ F @ top_top_set_nat ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite6454714172617411596l_num0 @ ( image_5550796612950789325l_num0 @ F @ top_top_set_nat ) ) ) ) ).
% card_range_greater_zero
thf(fact_935_card__range__greater__zero,axiom,
! [F: nat > literal] :
( ( finite5847741373460823677iteral @ ( image_nat_literal @ F @ top_top_set_nat ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_literal @ ( image_nat_literal @ F @ top_top_set_nat ) ) ) ) ).
% card_range_greater_zero
thf(fact_936_card__range__greater__zero,axiom,
! [F: nat > numeral_num1] :
( ( finite1111429032697314574l_num1 @ ( image_5550796612950789326l_num1 @ F @ top_top_set_nat ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite6454714172617411597l_num1 @ ( image_5550796612950789326l_num1 @ F @ top_top_set_nat ) ) ) ) ).
% card_range_greater_zero
thf(fact_937_card__range__greater__zero,axiom,
! [F: nat > product_unit] :
( ( finite4290736615968046902t_unit @ ( image_8730104196221521654t_unit @ F @ top_top_set_nat ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite410649719033368117t_unit @ ( image_8730104196221521654t_unit @ F @ top_top_set_nat ) ) ) ) ).
% card_range_greater_zero
thf(fact_938_card__range__greater__zero,axiom,
! [F: nat > nat] :
( ( finite_finite_nat @ ( image_nat_nat @ F @ top_top_set_nat ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ ( image_nat_nat @ F @ top_top_set_nat ) ) ) ) ).
% card_range_greater_zero
thf(fact_939_card__range__greater__zero,axiom,
! [F: numeral_num0 > numeral_num0] :
( ( finite1111429032697314573l_num0 @ ( image_2832974300507296261l_num0 @ F @ top_to3689904424835650196l_num0 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite6454714172617411596l_num0 @ ( image_2832974300507296261l_num0 @ F @ top_to3689904424835650196l_num0 ) ) ) ) ).
% card_range_greater_zero
thf(fact_940_card__range__greater__zero,axiom,
! [F: numeral_num0 > literal] :
( ( finite5847741373460823677iteral @ ( image_8737817577461598069iteral @ F @ top_to3689904424835650196l_num0 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_literal @ ( image_8737817577461598069iteral @ F @ top_to3689904424835650196l_num0 ) ) ) ) ).
% card_range_greater_zero
thf(fact_941_card__range__greater__zero,axiom,
! [F: numeral_num0 > numeral_num1] :
( ( finite1111429032697314574l_num1 @ ( image_2832974300507296262l_num1 @ F @ top_to3689904424835650196l_num0 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite6454714172617411597l_num1 @ ( image_2832974300507296262l_num1 @ F @ top_to3689904424835650196l_num0 ) ) ) ) ).
% card_range_greater_zero
thf(fact_942_card__range__greater__zero,axiom,
! [F: numeral_num0 > product_unit] :
( ( finite4290736615968046902t_unit @ ( image_6012281883778028590t_unit @ F @ top_to3689904424835650196l_num0 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite410649719033368117t_unit @ ( image_6012281883778028590t_unit @ F @ top_to3689904424835650196l_num0 ) ) ) ) ).
% card_range_greater_zero
thf(fact_943_card__range__greater__zero,axiom,
! [F: numeral_num0 > nat] :
( ( finite_finite_nat @ ( image_8797574156932312687m0_nat @ F @ top_to3689904424835650196l_num0 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ ( image_8797574156932312687m0_nat @ F @ top_to3689904424835650196l_num0 ) ) ) ) ).
% card_range_greater_zero
thf(fact_944_top_Oordering__top__axioms,axiom,
orderi4629348754929624565l_num0 @ ord_le5200684351691877604l_num0 @ ord_le526730871819019248l_num0 @ top_to3689904424835650196l_num0 ).
% top.ordering_top_axioms
thf(fact_945_top_Oordering__top__axioms,axiom,
orderi829509329793325797iteral @ ord_le7307670543136651348iteral @ ord_less_set_literal @ top_top_set_literal ).
% top.ordering_top_axioms
thf(fact_946_top_Oordering__top__axioms,axiom,
orderi4629348759232853366l_num1 @ ord_le5200684355995106405l_num1 @ ord_le526730876122248049l_num1 @ top_to3689904429138878997l_num1 ).
% top.ordering_top_axioms
thf(fact_947_top_Oordering__top__axioms,axiom,
orderi2935705153647960990t_unit @ ord_le3507040750410214029t_unit @ ord_le8056459307392131481t_unit @ top_to1996260823553986621t_unit ).
% top.ordering_top_axioms
thf(fact_948_top_Oordering__top__axioms,axiom,
ordering_top_set_nat @ ord_less_eq_set_nat @ ord_less_set_nat @ top_top_set_nat ).
% top.ordering_top_axioms
thf(fact_949_le__zero__eq,axiom,
! [N5: nat] :
( ( ord_less_eq_nat @ N5 @ zero_zero_nat )
= ( N5 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_950_not__gr__zero,axiom,
! [N5: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N5 ) )
= ( N5 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_951_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ A2 )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_952_diff__zero,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% diff_zero
thf(fact_953_zero__diff,axiom,
! [A2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_954_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_955_neq0__conv,axiom,
! [N5: nat] :
( ( N5 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N5 ) ) ).
% neq0_conv
thf(fact_956_less__nat__zero__code,axiom,
! [N5: nat] :
~ ( ord_less_nat @ N5 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_957_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_958_le0,axiom,
! [N5: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N5 ) ).
% le0
thf(fact_959_diff__self__eq__0,axiom,
! [M4: nat] :
( ( minus_minus_nat @ M4 @ M4 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_960_diff__0__eq__0,axiom,
! [N5: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N5 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_961_zero__less__diff,axiom,
! [N5: nat,M4: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N5 @ M4 ) )
= ( ord_less_nat @ M4 @ N5 ) ) ).
% zero_less_diff
thf(fact_962_diff__is__0__eq,axiom,
! [M4: nat,N5: nat] :
( ( ( minus_minus_nat @ M4 @ N5 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M4 @ N5 ) ) ).
% diff_is_0_eq
thf(fact_963_diff__is__0__eq_H,axiom,
! [M4: nat,N5: nat] :
( ( ord_less_eq_nat @ M4 @ N5 )
=> ( ( minus_minus_nat @ M4 @ N5 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_964_card_Oinfinite,axiom,
! [A: set_Numeral_num0] :
( ~ ( finite1111429032697314573l_num0 @ A )
=> ( ( finite6454714172617411596l_num0 @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_965_card_Oinfinite,axiom,
! [A: set_literal] :
( ~ ( finite5847741373460823677iteral @ A )
=> ( ( finite_card_literal @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_966_card_Oinfinite,axiom,
! [A: set_Numeral_num1] :
( ~ ( finite1111429032697314574l_num1 @ A )
=> ( ( finite6454714172617411597l_num1 @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_967_card_Oinfinite,axiom,
! [A: set_Product_unit] :
( ~ ( finite4290736615968046902t_unit @ A )
=> ( ( finite410649719033368117t_unit @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_968_card_Oinfinite,axiom,
! [A: set_nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_card_nat @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_969_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_970_top__set__def,axiom,
( top_to3689904424835650196l_num0
= ( collect_Numeral_num0 @ top_to4648304687082283337num0_o ) ) ).
% top_set_def
thf(fact_971_top__set__def,axiom,
( top_top_set_literal
= ( collect_literal @ top_top_literal_o ) ) ).
% top_set_def
thf(fact_972_top__set__def,axiom,
( top_to3689904429138878997l_num1
= ( collect_Numeral_num1 @ top_to1749082287617889032num1_o ) ) ).
% top_set_def
thf(fact_973_top__set__def,axiom,
( top_to1996260823553986621t_unit
= ( collect_Product_unit @ top_to2465898995584390880unit_o ) ) ).
% top_set_def
thf(fact_974_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_975_zero__less__iff__neq__zero,axiom,
! [N5: nat] :
( ( ord_less_nat @ zero_zero_nat @ N5 )
= ( N5 != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_976_gr__implies__not__zero,axiom,
! [M4: nat,N5: nat] :
( ( ord_less_nat @ M4 @ N5 )
=> ( N5 != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_977_not__less__zero,axiom,
! [N5: nat] :
~ ( ord_less_nat @ N5 @ zero_zero_nat ) ).
% not_less_zero
thf(fact_978_gr__zeroI,axiom,
! [N5: nat] :
( ( N5 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N5 ) ) ).
% gr_zeroI
thf(fact_979_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_980_infinite__descent0,axiom,
! [P: nat > $o,N5: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ~ ( P @ N )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N )
& ~ ( P @ M5 ) ) ) )
=> ( P @ N5 ) ) ) ).
% infinite_descent0
thf(fact_981_gr__implies__not0,axiom,
! [M4: nat,N5: nat] :
( ( ord_less_nat @ M4 @ N5 )
=> ( N5 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_982_less__zeroE,axiom,
! [N5: nat] :
~ ( ord_less_nat @ N5 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_983_not__less0,axiom,
! [N5: nat] :
~ ( ord_less_nat @ N5 @ zero_zero_nat ) ).
% not_less0
thf(fact_984_not__gr0,axiom,
! [N5: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N5 ) )
= ( N5 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_985_gr0I,axiom,
! [N5: nat] :
( ( N5 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N5 ) ) ).
% gr0I
thf(fact_986_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_987_le__0__eq,axiom,
! [N5: nat] :
( ( ord_less_eq_nat @ N5 @ zero_zero_nat )
= ( N5 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_988_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_989_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_990_less__eq__nat_Osimps_I1_J,axiom,
! [N5: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N5 ) ).
% less_eq_nat.simps(1)
thf(fact_991_diffs0__imp__equal,axiom,
! [M4: nat,N5: nat] :
( ( ( minus_minus_nat @ M4 @ N5 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N5 @ M4 )
= zero_zero_nat )
=> ( M4 = N5 ) ) ) ).
% diffs0_imp_equal
thf(fact_992_minus__nat_Odiff__0,axiom,
! [M4: nat] :
( ( minus_minus_nat @ M4 @ zero_zero_nat )
= M4 ) ).
% minus_nat.diff_0
thf(fact_993_ex__least__nat__le,axiom,
! [P: nat > $o,N5: nat] :
( ( P @ N5 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N5 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_994_diff__less,axiom,
! [N5: nat,M4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N5 )
=> ( ( ord_less_nat @ zero_zero_nat @ M4 )
=> ( ord_less_nat @ ( minus_minus_nat @ M4 @ N5 ) @ M4 ) ) ) ).
% diff_less
thf(fact_995_card__ge__0__finite,axiom,
! [A: set_Numeral_num0] :
( ( ord_less_nat @ zero_zero_nat @ ( finite6454714172617411596l_num0 @ A ) )
=> ( finite1111429032697314573l_num0 @ A ) ) ).
% card_ge_0_finite
thf(fact_996_card__ge__0__finite,axiom,
! [A: set_literal] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_literal @ A ) )
=> ( finite5847741373460823677iteral @ A ) ) ).
% card_ge_0_finite
thf(fact_997_card__ge__0__finite,axiom,
! [A: set_Numeral_num1] :
( ( ord_less_nat @ zero_zero_nat @ ( finite6454714172617411597l_num1 @ A ) )
=> ( finite1111429032697314574l_num1 @ A ) ) ).
% card_ge_0_finite
thf(fact_998_card__ge__0__finite,axiom,
! [A: set_Product_unit] :
( ( ord_less_nat @ zero_zero_nat @ ( finite410649719033368117t_unit @ A ) )
=> ( finite4290736615968046902t_unit @ A ) ) ).
% card_ge_0_finite
thf(fact_999_card__ge__0__finite,axiom,
! [A: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A ) )
=> ( finite_finite_nat @ A ) ) ).
% card_ge_0_finite
thf(fact_1000_finite__UNIV__card__ge__0,axiom,
( ( finite_finite_nat @ top_top_set_nat )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ top_top_set_nat ) ) ) ).
% finite_UNIV_card_ge_0
thf(fact_1001_finite__UNIV__card__ge__0,axiom,
( ( finite1111429032697314573l_num0 @ top_to3689904424835650196l_num0 )
=> ( ord_less_nat @ zero_zero_nat @ ( finite6454714172617411596l_num0 @ top_to3689904424835650196l_num0 ) ) ) ).
% finite_UNIV_card_ge_0
thf(fact_1002_finite__UNIV__card__ge__0,axiom,
( ( finite5847741373460823677iteral @ top_top_set_literal )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_literal @ top_top_set_literal ) ) ) ).
% finite_UNIV_card_ge_0
thf(fact_1003_finite__UNIV__card__ge__0,axiom,
( ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 )
=> ( ord_less_nat @ zero_zero_nat @ ( finite6454714172617411597l_num1 @ top_to3689904429138878997l_num1 ) ) ) ).
% finite_UNIV_card_ge_0
thf(fact_1004_finite__UNIV__card__ge__0,axiom,
( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ord_less_nat @ zero_zero_nat @ ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_UNIV_card_ge_0
thf(fact_1005_card__nat,axiom,
( ( finite_card_nat @ top_top_set_nat )
= zero_zero_nat ) ).
% card_nat
thf(fact_1006_zero__less__card__finite,axiom,
ord_less_nat @ zero_zero_nat @ ( finite6454714172617411597l_num1 @ top_to3689904429138878997l_num1 ) ).
% zero_less_card_finite
thf(fact_1007_zero__less__card__finite,axiom,
ord_less_nat @ zero_zero_nat @ ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit ) ).
% zero_less_card_finite
thf(fact_1008_card__num0,axiom,
( ( finite6454714172617411596l_num0 @ top_to3689904424835650196l_num0 )
= zero_zero_nat ) ).
% card_num0
thf(fact_1009_finite__fun__UNIVD1,axiom,
( ( finite2115694454571419734at_nat @ top_top_set_nat_nat )
=> ( ( ( finite_card_nat @ top_top_set_nat )
!= ( suc @ zero_zero_nat ) )
=> ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_fun_UNIVD1
thf(fact_1010_finite__fun__UNIVD1,axiom,
( ( finite3030762105040614734m0_nat @ top_to8982932540221636693m0_nat )
=> ( ( ( finite_card_nat @ top_top_set_nat )
!= ( suc @ zero_zero_nat ) )
=> ( finite1111429032697314573l_num0 @ top_to3689904424835650196l_num0 ) ) ) ).
% finite_fun_UNIVD1
thf(fact_1011_finite__fun__UNIVD1,axiom,
( ( finite4607804209690727486al_nat @ top_to8927312373334552901al_nat )
=> ( ( ( finite_card_nat @ top_top_set_nat )
!= ( suc @ zero_zero_nat ) )
=> ( finite5847741373460823677iteral @ top_top_set_literal ) ) ) ).
% finite_fun_UNIVD1
thf(fact_1012_finite__fun__UNIVD1,axiom,
( ( finite4266206433997009231m1_nat @ top_to5355779118554991574m1_nat )
=> ( ( ( finite_card_nat @ top_top_set_nat )
!= ( suc @ zero_zero_nat ) )
=> ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 ) ) ) ).
% finite_fun_UNIVD1
thf(fact_1013_finite__fun__UNIVD1,axiom,
( ( finite4332129999517832055it_nat @ top_to5871476398150932990it_nat )
=> ( ( ( finite_card_nat @ top_top_set_nat )
!= ( suc @ zero_zero_nat ) )
=> ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_fun_UNIVD1
thf(fact_1014_finite__fun__UNIVD1,axiom,
( ( finite5951333295303020660l_num0 @ top_to885238125467707515l_num0 )
=> ( ( ( finite6454714172617411596l_num0 @ top_to3689904424835650196l_num0 )
!= ( suc @ zero_zero_nat ) )
=> ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_fun_UNIVD1
thf(fact_1015_finite__fun__UNIVD1,axiom,
( ( finite4742813036018141564l_num0 @ top_to1375916195660207051l_num0 )
=> ( ( ( finite6454714172617411596l_num0 @ top_to3689904424835650196l_num0 )
!= ( suc @ zero_zero_nat ) )
=> ( finite1111429032697314573l_num0 @ top_to3689904424835650196l_num0 ) ) ) ).
% finite_fun_UNIVD1
thf(fact_1016_finite__fun__UNIVD1,axiom,
( ( finite3175997883609872012l_num0 @ top_to1450039601738329819l_num0 )
=> ( ( ( finite6454714172617411596l_num0 @ top_to3689904424835650196l_num0 )
!= ( suc @ zero_zero_nat ) )
=> ( finite5847741373460823677iteral @ top_top_set_literal ) ) ) ).
% finite_fun_UNIVD1
thf(fact_1017_finite__fun__UNIVD1,axiom,
( ( finite8693704671636730043l_num0 @ top_to7513531990524988298l_num0 )
=> ( ( ( finite6454714172617411596l_num0 @ top_to3689904424835650196l_num0 )
!= ( suc @ zero_zero_nat ) )
=> ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 ) ) ) ).
% finite_fun_UNIVD1
thf(fact_1018_finite__fun__UNIVD1,axiom,
( ( finite8358965893590519955l_num0 @ top_to2325158523423817570l_num0 )
=> ( ( ( finite6454714172617411596l_num0 @ top_to3689904424835650196l_num0 )
!= ( suc @ zero_zero_nat ) )
=> ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_fun_UNIVD1
thf(fact_1019_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_1020_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1021_Suc__less__eq,axiom,
! [M4: nat,N5: nat] :
( ( ord_less_nat @ ( suc @ M4 ) @ ( suc @ N5 ) )
= ( ord_less_nat @ M4 @ N5 ) ) ).
% Suc_less_eq
thf(fact_1022_Suc__mono,axiom,
! [M4: nat,N5: nat] :
( ( ord_less_nat @ M4 @ N5 )
=> ( ord_less_nat @ ( suc @ M4 ) @ ( suc @ N5 ) ) ) ).
% Suc_mono
thf(fact_1023_lessI,axiom,
! [N5: nat] : ( ord_less_nat @ N5 @ ( suc @ N5 ) ) ).
% lessI
thf(fact_1024_Suc__le__mono,axiom,
! [N5: nat,M4: nat] :
( ( ord_less_eq_nat @ ( suc @ N5 ) @ ( suc @ M4 ) )
= ( ord_less_eq_nat @ N5 @ M4 ) ) ).
% Suc_le_mono
thf(fact_1025_diff__Suc__Suc,axiom,
! [M4: nat,N5: nat] :
( ( minus_minus_nat @ ( suc @ M4 ) @ ( suc @ N5 ) )
= ( minus_minus_nat @ M4 @ N5 ) ) ).
% diff_Suc_Suc
thf(fact_1026_Suc__diff__diff,axiom,
! [M4: nat,N5: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M4 ) @ N5 ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M4 @ N5 ) @ K ) ) ).
% Suc_diff_diff
thf(fact_1027_zero__less__Suc,axiom,
! [N5: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N5 ) ) ).
% zero_less_Suc
thf(fact_1028_less__Suc0,axiom,
! [N5: nat] :
( ( ord_less_nat @ N5 @ ( suc @ zero_zero_nat ) )
= ( N5 = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1029_Suc__pred,axiom,
! [N5: nat] :
( ( ord_less_nat @ zero_zero_nat @ N5 )
=> ( ( suc @ ( minus_minus_nat @ N5 @ ( suc @ zero_zero_nat ) ) )
= N5 ) ) ).
% Suc_pred
thf(fact_1030_card__option,axiom,
( ( finite6050419269668842451l_num1 @ top_to4428395536652758875l_num1 )
= ( suc @ ( finite6454714172617411597l_num1 @ top_to3689904429138878997l_num1 ) ) ) ).
% card_option
thf(fact_1031_card__option,axiom,
( ( finite4356775664083950075t_unit @ top_to2690860209552263555t_unit )
= ( suc @ ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% card_option
thf(fact_1032_not0__implies__Suc,axiom,
! [N5: nat] :
( ( N5 != zero_zero_nat )
=> ? [M3: nat] :
( N5
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_1033_Zero__not__Suc,axiom,
! [M4: nat] :
( zero_zero_nat
!= ( suc @ M4 ) ) ).
% Zero_not_Suc
thf(fact_1034_Zero__neq__Suc,axiom,
! [M4: nat] :
( zero_zero_nat
!= ( suc @ M4 ) ) ).
% Zero_neq_Suc
thf(fact_1035_Suc__neq__Zero,axiom,
! [M4: nat] :
( ( suc @ M4 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1036_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N: nat] :
( ( P @ ( suc @ N ) )
=> ( P @ N ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1037_diff__induct,axiom,
! [P: nat > nat > $o,M4: nat,N5: nat] :
( ! [X: nat] : ( P @ X @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X: nat,Y3: nat] :
( ( P @ X @ Y3 )
=> ( P @ ( suc @ X ) @ ( suc @ Y3 ) ) )
=> ( P @ M4 @ N5 ) ) ) ) ).
% diff_induct
thf(fact_1038_nat__induct,axiom,
! [P: nat > $o,N5: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N: nat] :
( ( P @ N )
=> ( P @ ( suc @ N ) ) )
=> ( P @ N5 ) ) ) ).
% nat_induct
thf(fact_1039_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1040_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1041_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1042_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1043_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_1044_inj__Suc,axiom,
! [N4: set_nat] : ( inj_on_nat_nat @ suc @ N4 ) ).
% inj_Suc
thf(fact_1045_zero__notin__Suc__image,axiom,
! [A: set_nat] :
~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A ) ) ).
% zero_notin_Suc_image
thf(fact_1046_not__less__less__Suc__eq,axiom,
! [N5: nat,M4: nat] :
( ~ ( ord_less_nat @ N5 @ M4 )
=> ( ( ord_less_nat @ N5 @ ( suc @ M4 ) )
= ( N5 = M4 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1047_strict__inc__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I2 ) ) ) ) ).
% strict_inc_induct
thf(fact_1048_less__Suc__induct,axiom,
! [I2: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I3 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I3 @ K2 ) ) ) ) )
=> ( P @ I2 @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1049_less__trans__Suc,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I2 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_1050_Suc__less__SucD,axiom,
! [M4: nat,N5: nat] :
( ( ord_less_nat @ ( suc @ M4 ) @ ( suc @ N5 ) )
=> ( ord_less_nat @ M4 @ N5 ) ) ).
% Suc_less_SucD
thf(fact_1051_less__antisym,axiom,
! [N5: nat,M4: nat] :
( ~ ( ord_less_nat @ N5 @ M4 )
=> ( ( ord_less_nat @ N5 @ ( suc @ M4 ) )
=> ( M4 = N5 ) ) ) ).
% less_antisym
thf(fact_1052_Suc__less__eq2,axiom,
! [N5: nat,M4: nat] :
( ( ord_less_nat @ ( suc @ N5 ) @ M4 )
= ( ? [M6: nat] :
( ( M4
= ( suc @ M6 ) )
& ( ord_less_nat @ N5 @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1053_Nat_OAll__less__Suc,axiom,
! [N5: nat,P: nat > $o] :
( ( ! [I6: nat] :
( ( ord_less_nat @ I6 @ ( suc @ N5 ) )
=> ( P @ I6 ) ) )
= ( ( P @ N5 )
& ! [I6: nat] :
( ( ord_less_nat @ I6 @ N5 )
=> ( P @ I6 ) ) ) ) ).
% Nat.All_less_Suc
thf(fact_1054_not__less__eq,axiom,
! [M4: nat,N5: nat] :
( ( ~ ( ord_less_nat @ M4 @ N5 ) )
= ( ord_less_nat @ N5 @ ( suc @ M4 ) ) ) ).
% not_less_eq
thf(fact_1055_less__Suc__eq,axiom,
! [M4: nat,N5: nat] :
( ( ord_less_nat @ M4 @ ( suc @ N5 ) )
= ( ( ord_less_nat @ M4 @ N5 )
| ( M4 = N5 ) ) ) ).
% less_Suc_eq
thf(fact_1056_Ex__less__Suc,axiom,
! [N5: nat,P: nat > $o] :
( ( ? [I6: nat] :
( ( ord_less_nat @ I6 @ ( suc @ N5 ) )
& ( P @ I6 ) ) )
= ( ( P @ N5 )
| ? [I6: nat] :
( ( ord_less_nat @ I6 @ N5 )
& ( P @ I6 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1057_less__SucI,axiom,
! [M4: nat,N5: nat] :
( ( ord_less_nat @ M4 @ N5 )
=> ( ord_less_nat @ M4 @ ( suc @ N5 ) ) ) ).
% less_SucI
thf(fact_1058_less__SucE,axiom,
! [M4: nat,N5: nat] :
( ( ord_less_nat @ M4 @ ( suc @ N5 ) )
=> ( ~ ( ord_less_nat @ M4 @ N5 )
=> ( M4 = N5 ) ) ) ).
% less_SucE
thf(fact_1059_Suc__lessI,axiom,
! [M4: nat,N5: nat] :
( ( ord_less_nat @ M4 @ N5 )
=> ( ( ( suc @ M4 )
!= N5 )
=> ( ord_less_nat @ ( suc @ M4 ) @ N5 ) ) ) ).
% Suc_lessI
thf(fact_1060_Suc__lessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_1061_Suc__lessD,axiom,
! [M4: nat,N5: nat] :
( ( ord_less_nat @ ( suc @ M4 ) @ N5 )
=> ( ord_less_nat @ M4 @ N5 ) ) ).
% Suc_lessD
thf(fact_1062_Nat_OlessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( K
!= ( suc @ I2 ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1063_transitive__stepwise__le,axiom,
! [M4: nat,N5: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M4 @ N5 )
=> ( ! [X: nat] : ( R2 @ X @ X )
=> ( ! [X: nat,Y3: nat,Z3: nat] :
( ( R2 @ X @ Y3 )
=> ( ( R2 @ Y3 @ Z3 )
=> ( R2 @ X @ Z3 ) ) )
=> ( ! [N: nat] : ( R2 @ N @ ( suc @ N ) )
=> ( R2 @ M4 @ N5 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1064_nat__induct__at__least,axiom,
! [M4: nat,N5: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M4 @ N5 )
=> ( ( P @ M4 )
=> ( ! [N: nat] :
( ( ord_less_eq_nat @ M4 @ N )
=> ( ( P @ N )
=> ( P @ ( suc @ N ) ) ) )
=> ( P @ N5 ) ) ) ) ).
% nat_induct_at_least
thf(fact_1065_full__nat__induct,axiom,
! [P: nat > $o,N5: nat] :
( ! [N: nat] :
( ! [M5: nat] :
( ( ord_less_eq_nat @ ( suc @ M5 ) @ N )
=> ( P @ M5 ) )
=> ( P @ N ) )
=> ( P @ N5 ) ) ).
% full_nat_induct
thf(fact_1066_not__less__eq__eq,axiom,
! [M4: nat,N5: nat] :
( ( ~ ( ord_less_eq_nat @ M4 @ N5 ) )
= ( ord_less_eq_nat @ ( suc @ N5 ) @ M4 ) ) ).
% not_less_eq_eq
thf(fact_1067_Suc__n__not__le__n,axiom,
! [N5: nat] :
~ ( ord_less_eq_nat @ ( suc @ N5 ) @ N5 ) ).
% Suc_n_not_le_n
thf(fact_1068_le__Suc__eq,axiom,
! [M4: nat,N5: nat] :
( ( ord_less_eq_nat @ M4 @ ( suc @ N5 ) )
= ( ( ord_less_eq_nat @ M4 @ N5 )
| ( M4
= ( suc @ N5 ) ) ) ) ).
% le_Suc_eq
thf(fact_1069_Suc__le__D,axiom,
! [N5: nat,M7: nat] :
( ( ord_less_eq_nat @ ( suc @ N5 ) @ M7 )
=> ? [M3: nat] :
( M7
= ( suc @ M3 ) ) ) ).
% Suc_le_D
thf(fact_1070_le__SucI,axiom,
! [M4: nat,N5: nat] :
( ( ord_less_eq_nat @ M4 @ N5 )
=> ( ord_less_eq_nat @ M4 @ ( suc @ N5 ) ) ) ).
% le_SucI
thf(fact_1071_le__SucE,axiom,
! [M4: nat,N5: nat] :
( ( ord_less_eq_nat @ M4 @ ( suc @ N5 ) )
=> ( ~ ( ord_less_eq_nat @ M4 @ N5 )
=> ( M4
= ( suc @ N5 ) ) ) ) ).
% le_SucE
thf(fact_1072_Suc__leD,axiom,
! [M4: nat,N5: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N5 )
=> ( ord_less_eq_nat @ M4 @ N5 ) ) ).
% Suc_leD
thf(fact_1073_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I2: nat] :
( ( P @ K )
=> ( ! [N: nat] :
( ( P @ ( suc @ N ) )
=> ( P @ N ) )
=> ( P @ ( minus_minus_nat @ K @ I2 ) ) ) ) ).
% zero_induct_lemma
thf(fact_1074_Suc__inject,axiom,
! [X2: nat,Y: nat] :
( ( ( suc @ X2 )
= ( suc @ Y ) )
=> ( X2 = Y ) ) ).
% Suc_inject
thf(fact_1075_n__not__Suc__n,axiom,
! [N5: nat] :
( N5
!= ( suc @ N5 ) ) ).
% n_not_Suc_n
thf(fact_1076_lift__Suc__mono__less,axiom,
! [F: nat > set_nat,N5: nat,N6: nat] :
( ! [N: nat] : ( ord_less_set_nat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
=> ( ( ord_less_nat @ N5 @ N6 )
=> ( ord_less_set_nat @ ( F @ N5 ) @ ( F @ N6 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_1077_lift__Suc__mono__less,axiom,
! [F: nat > nat,N5: nat,N6: nat] :
( ! [N: nat] : ( ord_less_nat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
=> ( ( ord_less_nat @ N5 @ N6 )
=> ( ord_less_nat @ ( F @ N5 ) @ ( F @ N6 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_1078_lift__Suc__mono__less__iff,axiom,
! [F: nat > set_nat,N5: nat,M4: nat] :
( ! [N: nat] : ( ord_less_set_nat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
=> ( ( ord_less_set_nat @ ( F @ N5 ) @ ( F @ M4 ) )
= ( ord_less_nat @ N5 @ M4 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_1079_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N5: nat,M4: nat] :
( ! [N: nat] : ( ord_less_nat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
=> ( ( ord_less_nat @ ( F @ N5 ) @ ( F @ M4 ) )
= ( ord_less_nat @ N5 @ M4 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_1080_lift__Suc__antimono__le,axiom,
! [F: nat > set_nat,N5: nat,N6: nat] :
( ! [N: nat] : ( ord_less_eq_set_nat @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
=> ( ( ord_less_eq_nat @ N5 @ N6 )
=> ( ord_less_eq_set_nat @ ( F @ N6 ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1081_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N5: nat,N6: nat] :
( ! [N: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
=> ( ( ord_less_eq_nat @ N5 @ N6 )
=> ( ord_less_eq_nat @ ( F @ N6 ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1082_lift__Suc__mono__le,axiom,
! [F: nat > set_nat,N5: nat,N6: nat] :
( ! [N: nat] : ( ord_less_eq_set_nat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ N5 @ N6 )
=> ( ord_less_eq_set_nat @ ( F @ N5 ) @ ( F @ N6 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1083_lift__Suc__mono__le,axiom,
! [F: nat > nat,N5: nat,N6: nat] :
( ! [N: nat] : ( ord_less_eq_nat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ N5 @ N6 )
=> ( ord_less_eq_nat @ ( F @ N5 ) @ ( F @ N6 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1084_less__Suc__eq__0__disj,axiom,
! [M4: nat,N5: nat] :
( ( ord_less_nat @ M4 @ ( suc @ N5 ) )
= ( ( M4 = zero_zero_nat )
| ? [J3: nat] :
( ( M4
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N5 ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1085_gr0__implies__Suc,axiom,
! [N5: nat] :
( ( ord_less_nat @ zero_zero_nat @ N5 )
=> ? [M3: nat] :
( N5
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_1086_All__less__Suc2,axiom,
! [N5: nat,P: nat > $o] :
( ( ! [I6: nat] :
( ( ord_less_nat @ I6 @ ( suc @ N5 ) )
=> ( P @ I6 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I6: nat] :
( ( ord_less_nat @ I6 @ N5 )
=> ( P @ ( suc @ I6 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1087_gr0__conv__Suc,axiom,
! [N5: nat] :
( ( ord_less_nat @ zero_zero_nat @ N5 )
= ( ? [M: nat] :
( N5
= ( suc @ M ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1088_Ex__less__Suc2,axiom,
! [N5: nat,P: nat > $o] :
( ( ? [I6: nat] :
( ( ord_less_nat @ I6 @ ( suc @ N5 ) )
& ( P @ I6 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I6: nat] :
( ( ord_less_nat @ I6 @ N5 )
& ( P @ ( suc @ I6 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1089_le__imp__less__Suc,axiom,
! [M4: nat,N5: nat] :
( ( ord_less_eq_nat @ M4 @ N5 )
=> ( ord_less_nat @ M4 @ ( suc @ N5 ) ) ) ).
% le_imp_less_Suc
thf(fact_1090_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1091_less__Suc__eq__le,axiom,
! [M4: nat,N5: nat] :
( ( ord_less_nat @ M4 @ ( suc @ N5 ) )
= ( ord_less_eq_nat @ M4 @ N5 ) ) ).
% less_Suc_eq_le
thf(fact_1092_le__less__Suc__eq,axiom,
! [M4: nat,N5: nat] :
( ( ord_less_eq_nat @ M4 @ N5 )
=> ( ( ord_less_nat @ N5 @ ( suc @ M4 ) )
= ( N5 = M4 ) ) ) ).
% le_less_Suc_eq
thf(fact_1093_Suc__le__lessD,axiom,
! [M4: nat,N5: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N5 )
=> ( ord_less_nat @ M4 @ N5 ) ) ).
% Suc_le_lessD
thf(fact_1094_inc__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( P @ J )
=> ( ! [N: nat] :
( ( ord_less_eq_nat @ I2 @ N )
=> ( ( ord_less_nat @ N @ J )
=> ( ( P @ ( suc @ N ) )
=> ( P @ N ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% inc_induct
thf(fact_1095_dec__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( P @ I2 )
=> ( ! [N: nat] :
( ( ord_less_eq_nat @ I2 @ N )
=> ( ( ord_less_nat @ N @ J )
=> ( ( P @ N )
=> ( P @ ( suc @ N ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1096_Suc__le__eq,axiom,
! [M4: nat,N5: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N5 )
= ( ord_less_nat @ M4 @ N5 ) ) ).
% Suc_le_eq
thf(fact_1097_Suc__leI,axiom,
! [M4: nat,N5: nat] :
( ( ord_less_nat @ M4 @ N5 )
=> ( ord_less_eq_nat @ ( suc @ M4 ) @ N5 ) ) ).
% Suc_leI
thf(fact_1098_Suc__diff__Suc,axiom,
! [N5: nat,M4: nat] :
( ( ord_less_nat @ N5 @ M4 )
=> ( ( suc @ ( minus_minus_nat @ M4 @ ( suc @ N5 ) ) )
= ( minus_minus_nat @ M4 @ N5 ) ) ) ).
% Suc_diff_Suc
thf(fact_1099_diff__less__Suc,axiom,
! [M4: nat,N5: nat] : ( ord_less_nat @ ( minus_minus_nat @ M4 @ N5 ) @ ( suc @ M4 ) ) ).
% diff_less_Suc
thf(fact_1100_Suc__diff__le,axiom,
! [N5: nat,M4: nat] :
( ( ord_less_eq_nat @ N5 @ M4 )
=> ( ( minus_minus_nat @ ( suc @ M4 ) @ N5 )
= ( suc @ ( minus_minus_nat @ M4 @ N5 ) ) ) ) ).
% Suc_diff_le
thf(fact_1101_one__le__card__finite,axiom,
ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( finite6454714172617411597l_num1 @ top_to3689904429138878997l_num1 ) ).
% one_le_card_finite
thf(fact_1102_one__le__card__finite,axiom,
ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit ) ).
% one_le_card_finite
thf(fact_1103_ex__least__nat__less,axiom,
! [P: nat > $o,N5: nat] :
( ( P @ N5 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N5 )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1104_diff__Suc__less,axiom,
! [N5: nat,I2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N5 )
=> ( ord_less_nat @ ( minus_minus_nat @ N5 @ ( suc @ I2 ) ) @ N5 ) ) ).
% diff_Suc_less
thf(fact_1105_card__literal,axiom,
( ( finite_card_literal @ top_top_set_literal )
= zero_zero_nat ) ).
% card_literal
thf(fact_1106_card__le__Suc0__iff__eq,axiom,
! [A: set_Numeral_num0] :
( ( finite1111429032697314573l_num0 @ A )
=> ( ( ord_less_eq_nat @ ( finite6454714172617411596l_num0 @ A ) @ ( suc @ zero_zero_nat ) )
= ( ! [X4: numeral_num0] :
( ( member_Numeral_num0 @ X4 @ A )
=> ! [Y4: numeral_num0] :
( ( member_Numeral_num0 @ Y4 @ A )
=> ( X4 = Y4 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_1107_card__le__Suc0__iff__eq,axiom,
! [A: set_literal] :
( ( finite5847741373460823677iteral @ A )
=> ( ( ord_less_eq_nat @ ( finite_card_literal @ A ) @ ( suc @ zero_zero_nat ) )
= ( ! [X4: literal] :
( ( member_literal @ X4 @ A )
=> ! [Y4: literal] :
( ( member_literal @ Y4 @ A )
=> ( X4 = Y4 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_1108_card__le__Suc0__iff__eq,axiom,
! [A: set_Numeral_num1] :
( ( finite1111429032697314574l_num1 @ A )
=> ( ( ord_less_eq_nat @ ( finite6454714172617411597l_num1 @ A ) @ ( suc @ zero_zero_nat ) )
= ( ! [X4: numeral_num1] :
( ( member_Numeral_num1 @ X4 @ A )
=> ! [Y4: numeral_num1] :
( ( member_Numeral_num1 @ Y4 @ A )
=> ( X4 = Y4 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_1109_card__le__Suc0__iff__eq,axiom,
! [A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ ( suc @ zero_zero_nat ) )
= ( ! [X4: product_unit] :
( ( member_Product_unit @ X4 @ A )
=> ! [Y4: product_unit] :
( ( member_Product_unit @ Y4 @ A )
=> ( X4 = Y4 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_1110_card__le__Suc0__iff__eq,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( suc @ zero_zero_nat ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ! [Y4: nat] :
( ( member_nat @ Y4 @ A )
=> ( X4 = Y4 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_1111_Comparator__Generator_OAll__less__Suc,axiom,
! [X2: nat,P: nat > $o] :
( ( ! [I6: nat] :
( ( ord_less_nat @ I6 @ ( suc @ X2 ) )
=> ( P @ I6 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I6: nat] :
( ( ord_less_nat @ I6 @ X2 )
=> ( P @ ( suc @ I6 ) ) ) ) ) ).
% Comparator_Generator.All_less_Suc
thf(fact_1112_forall__finite_I2_J,axiom,
! [P: nat > $o] :
( ( ! [I6: nat] :
( ( ord_less_nat @ I6 @ ( suc @ zero_zero_nat ) )
=> ( P @ I6 ) ) )
= ( P @ zero_zero_nat ) ) ).
% forall_finite(2)
thf(fact_1113_forall__finite_I3_J,axiom,
! [X2: nat,P: nat > $o] :
( ( ! [I6: nat] :
( ( ord_less_nat @ I6 @ ( suc @ ( suc @ X2 ) ) )
=> ( P @ I6 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I6: nat] :
( ( ord_less_nat @ I6 @ ( suc @ X2 ) )
=> ( P @ ( suc @ I6 ) ) ) ) ) ).
% forall_finite(3)
thf(fact_1114_forall__finite_I1_J,axiom,
! [P: nat > $o,I4: nat] :
( ( ord_less_nat @ I4 @ zero_zero_nat )
=> ( P @ I4 ) ) ).
% forall_finite(1)
thf(fact_1115_finite__vimage__Suc__iff,axiom,
! [F2: set_nat] :
( ( finite_finite_nat @ ( vimage_nat_nat @ suc @ F2 ) )
= ( finite_finite_nat @ F2 ) ) ).
% finite_vimage_Suc_iff
thf(fact_1116_finite__mono__remains__stable__implies__strict__prefix,axiom,
! [F: nat > set_nat] :
( ( finite1152437895449049373et_nat @ ( image_nat_set_nat @ F @ top_top_set_nat ) )
=> ( ( monoto6489329683466618047et_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_set_nat @ F )
=> ( ! [N: nat] :
( ( ( F @ N )
= ( F @ ( suc @ N ) ) )
=> ( ( F @ ( suc @ N ) )
= ( F @ ( suc @ ( suc @ N ) ) ) ) )
=> ? [N7: nat] :
( ! [N8: nat] :
( ( ord_less_eq_nat @ N8 @ N7 )
=> ! [M5: nat] :
( ( ord_less_eq_nat @ M5 @ N7 )
=> ( ( ord_less_nat @ M5 @ N8 )
=> ( ord_less_set_nat @ ( F @ M5 ) @ ( F @ N8 ) ) ) ) )
& ! [N8: nat] :
( ( ord_less_eq_nat @ N7 @ N8 )
=> ( ( F @ N7 )
= ( F @ N8 ) ) ) ) ) ) ) ).
% finite_mono_remains_stable_implies_strict_prefix
thf(fact_1117_finite__mono__remains__stable__implies__strict__prefix,axiom,
! [F: nat > nat] :
( ( finite_finite_nat @ ( image_nat_nat @ F @ top_top_set_nat ) )
=> ( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ! [N: nat] :
( ( ( F @ N )
= ( F @ ( suc @ N ) ) )
=> ( ( F @ ( suc @ N ) )
= ( F @ ( suc @ ( suc @ N ) ) ) ) )
=> ? [N7: nat] :
( ! [N8: nat] :
( ( ord_less_eq_nat @ N8 @ N7 )
=> ! [M5: nat] :
( ( ord_less_eq_nat @ M5 @ N7 )
=> ( ( ord_less_nat @ M5 @ N8 )
=> ( ord_less_nat @ ( F @ M5 ) @ ( F @ N8 ) ) ) ) )
& ! [N8: nat] :
( ( ord_less_eq_nat @ N7 @ N8 )
=> ( ( F @ N7 )
= ( F @ N8 ) ) ) ) ) ) ) ).
% finite_mono_remains_stable_implies_strict_prefix
thf(fact_1118_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_1119_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_1120_mono__on__greaterD,axiom,
! [A: set_nat,G3: nat > nat,X2: nat,Y: nat] :
( ( monotone_on_nat_nat @ A @ ord_less_eq_nat @ ord_less_eq_nat @ G3 )
=> ( ( member_nat @ X2 @ A )
=> ( ( member_nat @ Y @ A )
=> ( ( ord_less_nat @ ( G3 @ Y ) @ ( G3 @ X2 ) )
=> ( ord_less_nat @ Y @ X2 ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_1121_strict__mono__on__leD,axiom,
! [A: set_nat,F: nat > set_nat,X2: nat,Y: nat] :
( ( monoto6489329683466618047et_nat @ A @ ord_less_nat @ ord_less_set_nat @ F )
=> ( ( member_nat @ X2 @ A )
=> ( ( member_nat @ Y @ A )
=> ( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_1122_strict__mono__on__leD,axiom,
! [A: set_nat,F: nat > nat,X2: nat,Y: nat] :
( ( monotone_on_nat_nat @ A @ ord_less_nat @ ord_less_nat @ F )
=> ( ( member_nat @ X2 @ A )
=> ( ( member_nat @ Y @ A )
=> ( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_1123_strict__mono__on__imp__mono__on,axiom,
! [A: set_nat,F: nat > set_nat] :
( ( monoto6489329683466618047et_nat @ A @ ord_less_nat @ ord_less_set_nat @ F )
=> ( monoto6489329683466618047et_nat @ A @ ord_less_eq_nat @ ord_less_eq_set_nat @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_1124_strict__mono__on__imp__mono__on,axiom,
! [A: set_nat,F: nat > nat] :
( ( monotone_on_nat_nat @ A @ ord_less_nat @ ord_less_nat @ F )
=> ( monotone_on_nat_nat @ A @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_1125_mono__imp__mono__on,axiom,
! [F: literal > set_nat,A: set_literal] :
( ( monoto994342333799602483et_nat @ top_top_set_literal @ ord_less_eq_literal @ ord_less_eq_set_nat @ F )
=> ( monoto994342333799602483et_nat @ A @ ord_less_eq_literal @ ord_less_eq_set_nat @ F ) ) ).
% mono_imp_mono_on
thf(fact_1126_mono__imp__mono__on,axiom,
! [F: numeral_num1 > set_nat,A: set_Numeral_num1] :
( ( monoto1869804516012902724et_nat @ top_to3689904429138878997l_num1 @ ord_le41944437919317893l_num1 @ ord_less_eq_set_nat @ F )
=> ( monoto1869804516012902724et_nat @ A @ ord_le41944437919317893l_num1 @ ord_less_eq_set_nat @ F ) ) ).
% mono_imp_mono_on
thf(fact_1127_mono__imp__mono__on,axiom,
! [F: product_unit > set_nat,A: set_Product_unit] :
( ( monoto1410268124396407660et_nat @ top_to1996260823553986621t_unit @ ord_le3221252021190050221t_unit @ ord_less_eq_set_nat @ F )
=> ( monoto1410268124396407660et_nat @ A @ ord_le3221252021190050221t_unit @ ord_less_eq_set_nat @ F ) ) ).
% mono_imp_mono_on
thf(fact_1128_mono__imp__mono__on,axiom,
! [F: literal > nat,A: set_literal] :
( ( monoto6092665527236862333al_nat @ top_top_set_literal @ ord_less_eq_literal @ ord_less_eq_nat @ F )
=> ( monoto6092665527236862333al_nat @ A @ ord_less_eq_literal @ ord_less_eq_nat @ F ) ) ).
% mono_imp_mono_on
thf(fact_1129_mono__imp__mono__on,axiom,
! [F: numeral_num1 > nat,A: set_Numeral_num1] :
( ( monoto7167505876871681422m1_nat @ top_to3689904429138878997l_num1 @ ord_le41944437919317893l_num1 @ ord_less_eq_nat @ F )
=> ( monoto7167505876871681422m1_nat @ A @ ord_le41944437919317893l_num1 @ ord_less_eq_nat @ F ) ) ).
% mono_imp_mono_on
thf(fact_1130_mono__imp__mono__on,axiom,
! [F: product_unit > nat,A: set_Product_unit] :
( ( monoto7233429442392504246it_nat @ top_to1996260823553986621t_unit @ ord_le3221252021190050221t_unit @ ord_less_eq_nat @ F )
=> ( monoto7233429442392504246it_nat @ A @ ord_le3221252021190050221t_unit @ ord_less_eq_nat @ F ) ) ).
% mono_imp_mono_on
thf(fact_1131_mono__imp__mono__on,axiom,
! [F: set_nat > set_nat,A: set_set_nat] :
( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( monoto1748750089227133045et_nat @ A @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F ) ) ).
% mono_imp_mono_on
thf(fact_1132_mono__imp__mono__on,axiom,
! [F: set_nat > nat,A: set_set_nat] :
( ( monoto2923694778811248831at_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_nat @ F )
=> ( monoto2923694778811248831at_nat @ A @ ord_less_eq_set_nat @ ord_less_eq_nat @ F ) ) ).
% mono_imp_mono_on
thf(fact_1133_mono__imp__mono__on,axiom,
! [F: nat > set_nat,A: set_nat] :
( ( monoto6489329683466618047et_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_set_nat @ F )
=> ( monoto6489329683466618047et_nat @ A @ ord_less_eq_nat @ ord_less_eq_set_nat @ F ) ) ).
% mono_imp_mono_on
thf(fact_1134_mono__imp__mono__on,axiom,
! [F: nat > nat,A: set_nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( monotone_on_nat_nat @ A @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% mono_imp_mono_on
thf(fact_1135_monoI,axiom,
! [F: literal > set_nat] :
( ! [X: literal,Y3: literal] :
( ( ord_less_eq_literal @ X @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( monoto994342333799602483et_nat @ top_top_set_literal @ ord_less_eq_literal @ ord_less_eq_set_nat @ F ) ) ).
% monoI
thf(fact_1136_monoI,axiom,
! [F: numeral_num1 > set_nat] :
( ! [X: numeral_num1,Y3: numeral_num1] :
( ( ord_le41944437919317893l_num1 @ X @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( monoto1869804516012902724et_nat @ top_to3689904429138878997l_num1 @ ord_le41944437919317893l_num1 @ ord_less_eq_set_nat @ F ) ) ).
% monoI
thf(fact_1137_monoI,axiom,
! [F: product_unit > set_nat] :
( ! [X: product_unit,Y3: product_unit] :
( ( ord_le3221252021190050221t_unit @ X @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( monoto1410268124396407660et_nat @ top_to1996260823553986621t_unit @ ord_le3221252021190050221t_unit @ ord_less_eq_set_nat @ F ) ) ).
% monoI
thf(fact_1138_monoI,axiom,
! [F: literal > nat] :
( ! [X: literal,Y3: literal] :
( ( ord_less_eq_literal @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( monoto6092665527236862333al_nat @ top_top_set_literal @ ord_less_eq_literal @ ord_less_eq_nat @ F ) ) ).
% monoI
thf(fact_1139_monoI,axiom,
! [F: numeral_num1 > nat] :
( ! [X: numeral_num1,Y3: numeral_num1] :
( ( ord_le41944437919317893l_num1 @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( monoto7167505876871681422m1_nat @ top_to3689904429138878997l_num1 @ ord_le41944437919317893l_num1 @ ord_less_eq_nat @ F ) ) ).
% monoI
thf(fact_1140_monoI,axiom,
! [F: product_unit > nat] :
( ! [X: product_unit,Y3: product_unit] :
( ( ord_le3221252021190050221t_unit @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( monoto7233429442392504246it_nat @ top_to1996260823553986621t_unit @ ord_le3221252021190050221t_unit @ ord_less_eq_nat @ F ) ) ).
% monoI
thf(fact_1141_monoI,axiom,
! [F: set_nat > set_nat] :
( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F ) ) ).
% monoI
thf(fact_1142_monoI,axiom,
! [F: set_nat > nat] :
( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( monoto2923694778811248831at_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_nat @ F ) ) ).
% monoI
thf(fact_1143_monoI,axiom,
! [F: nat > set_nat] :
( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( monoto6489329683466618047et_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_set_nat @ F ) ) ).
% monoI
thf(fact_1144_monoI,axiom,
! [F: nat > nat] :
( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% monoI
thf(fact_1145_monoE,axiom,
! [F: literal > set_nat,X2: literal,Y: literal] :
( ( monoto994342333799602483et_nat @ top_top_set_literal @ ord_less_eq_literal @ ord_less_eq_set_nat @ F )
=> ( ( ord_less_eq_literal @ X2 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_1146_monoE,axiom,
! [F: numeral_num1 > set_nat,X2: numeral_num1,Y: numeral_num1] :
( ( monoto1869804516012902724et_nat @ top_to3689904429138878997l_num1 @ ord_le41944437919317893l_num1 @ ord_less_eq_set_nat @ F )
=> ( ( ord_le41944437919317893l_num1 @ X2 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_1147_monoE,axiom,
! [F: product_unit > set_nat,X2: product_unit,Y: product_unit] :
( ( monoto1410268124396407660et_nat @ top_to1996260823553986621t_unit @ ord_le3221252021190050221t_unit @ ord_less_eq_set_nat @ F )
=> ( ( ord_le3221252021190050221t_unit @ X2 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_1148_monoE,axiom,
! [F: literal > nat,X2: literal,Y: literal] :
( ( monoto6092665527236862333al_nat @ top_top_set_literal @ ord_less_eq_literal @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_literal @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_1149_monoE,axiom,
! [F: numeral_num1 > nat,X2: numeral_num1,Y: numeral_num1] :
( ( monoto7167505876871681422m1_nat @ top_to3689904429138878997l_num1 @ ord_le41944437919317893l_num1 @ ord_less_eq_nat @ F )
=> ( ( ord_le41944437919317893l_num1 @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_1150_monoE,axiom,
! [F: product_unit > nat,X2: product_unit,Y: product_unit] :
( ( monoto7233429442392504246it_nat @ top_to1996260823553986621t_unit @ ord_le3221252021190050221t_unit @ ord_less_eq_nat @ F )
=> ( ( ord_le3221252021190050221t_unit @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_1151_monoE,axiom,
! [F: set_nat > set_nat,X2: set_nat,Y: set_nat] :
( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_1152_monoE,axiom,
! [F: set_nat > nat,X2: set_nat,Y: set_nat] :
( ( monoto2923694778811248831at_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_1153_monoE,axiom,
! [F: nat > set_nat,X2: nat,Y: nat] :
( ( monoto6489329683466618047et_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_set_nat @ F )
=> ( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_1154_monoE,axiom,
! [F: nat > nat,X2: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_1155_monoD,axiom,
! [F: literal > set_nat,X2: literal,Y: literal] :
( ( monoto994342333799602483et_nat @ top_top_set_literal @ ord_less_eq_literal @ ord_less_eq_set_nat @ F )
=> ( ( ord_less_eq_literal @ X2 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_1156_monoD,axiom,
! [F: numeral_num1 > set_nat,X2: numeral_num1,Y: numeral_num1] :
( ( monoto1869804516012902724et_nat @ top_to3689904429138878997l_num1 @ ord_le41944437919317893l_num1 @ ord_less_eq_set_nat @ F )
=> ( ( ord_le41944437919317893l_num1 @ X2 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_1157_monoD,axiom,
! [F: product_unit > set_nat,X2: product_unit,Y: product_unit] :
( ( monoto1410268124396407660et_nat @ top_to1996260823553986621t_unit @ ord_le3221252021190050221t_unit @ ord_less_eq_set_nat @ F )
=> ( ( ord_le3221252021190050221t_unit @ X2 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_1158_monoD,axiom,
! [F: literal > nat,X2: literal,Y: literal] :
( ( monoto6092665527236862333al_nat @ top_top_set_literal @ ord_less_eq_literal @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_literal @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_1159_monoD,axiom,
! [F: numeral_num1 > nat,X2: numeral_num1,Y: numeral_num1] :
( ( monoto7167505876871681422m1_nat @ top_to3689904429138878997l_num1 @ ord_le41944437919317893l_num1 @ ord_less_eq_nat @ F )
=> ( ( ord_le41944437919317893l_num1 @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_1160_monoD,axiom,
! [F: product_unit > nat,X2: product_unit,Y: product_unit] :
( ( monoto7233429442392504246it_nat @ top_to1996260823553986621t_unit @ ord_le3221252021190050221t_unit @ ord_less_eq_nat @ F )
=> ( ( ord_le3221252021190050221t_unit @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_1161_monoD,axiom,
! [F: set_nat > set_nat,X2: set_nat,Y: set_nat] :
( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_1162_monoD,axiom,
! [F: set_nat > nat,X2: set_nat,Y: set_nat] :
( ( monoto2923694778811248831at_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_1163_monoD,axiom,
! [F: nat > set_nat,X2: nat,Y: nat] :
( ( monoto6489329683466618047et_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_set_nat @ F )
=> ( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_1164_monoD,axiom,
! [F: nat > nat,X2: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_1165_strict__monoD,axiom,
! [F: literal > nat,X2: literal,Y: literal] :
( ( monoto6092665527236862333al_nat @ top_top_set_literal @ ord_less_literal @ ord_less_nat @ F )
=> ( ( ord_less_literal @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ).
% strict_monoD
thf(fact_1166_strict__monoD,axiom,
! [F: numeral_num1 > nat,X2: numeral_num1,Y: numeral_num1] :
( ( monoto7167505876871681422m1_nat @ top_to3689904429138878997l_num1 @ ord_le6405328735288452753l_num1 @ ord_less_nat @ F )
=> ( ( ord_le6405328735288452753l_num1 @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ).
% strict_monoD
thf(fact_1167_strict__monoD,axiom,
! [F: product_unit > nat,X2: product_unit,Y: product_unit] :
( ( monoto7233429442392504246it_nat @ top_to1996260823553986621t_unit @ ord_le361264281704409273t_unit @ ord_less_nat @ F )
=> ( ( ord_le361264281704409273t_unit @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ).
% strict_monoD
thf(fact_1168_strict__monoD,axiom,
! [F: set_nat > set_nat,X2: set_nat,Y: set_nat] :
( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_set_nat @ ord_less_set_nat @ F )
=> ( ( ord_less_set_nat @ X2 @ Y )
=> ( ord_less_set_nat @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ).
% strict_monoD
thf(fact_1169_strict__monoD,axiom,
! [F: set_nat > nat,X2: set_nat,Y: set_nat] :
( ( monoto2923694778811248831at_nat @ top_top_set_set_nat @ ord_less_set_nat @ ord_less_nat @ F )
=> ( ( ord_less_set_nat @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ).
% strict_monoD
thf(fact_1170_strict__monoD,axiom,
! [F: nat > set_nat,X2: nat,Y: nat] :
( ( monoto6489329683466618047et_nat @ top_top_set_nat @ ord_less_nat @ ord_less_set_nat @ F )
=> ( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_set_nat @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ).
% strict_monoD
thf(fact_1171_strict__monoD,axiom,
! [F: nat > nat,X2: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ).
% strict_monoD
thf(fact_1172_mono__Suc,axiom,
monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ suc ).
% mono_Suc
thf(fact_1173_strict__mono__imp__increasing,axiom,
! [F: nat > nat,N5: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( ord_less_eq_nat @ N5 @ ( F @ N5 ) ) ) ).
% strict_mono_imp_increasing
thf(fact_1174_Suc__diff__1,axiom,
! [N5: nat] :
( ( ord_less_nat @ zero_zero_nat @ N5 )
=> ( ( suc @ ( minus_minus_nat @ N5 @ one_one_nat ) )
= N5 ) ) ).
% Suc_diff_1
thf(fact_1175_greaterThan__0,axiom,
( ( set_or1210151606488870762an_nat @ zero_zero_nat )
= ( image_nat_nat @ suc @ top_top_set_nat ) ) ).
% greaterThan_0
thf(fact_1176_card__num1,axiom,
( ( finite6454714172617411597l_num1 @ top_to3689904429138878997l_num1 )
= one_one_nat ) ).
% card_num1
thf(fact_1177_card__UNIV__unit,axiom,
( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
= one_one_nat ) ).
% card_UNIV_unit
thf(fact_1178_less__one,axiom,
! [N5: nat] :
( ( ord_less_nat @ N5 @ one_one_nat )
= ( N5 = zero_zero_nat ) ) ).
% less_one
thf(fact_1179_diff__Suc__1,axiom,
! [N5: nat] :
( ( minus_minus_nat @ ( suc @ N5 ) @ one_one_nat )
= N5 ) ).
% diff_Suc_1
thf(fact_1180_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1181_diff__Suc__eq__diff__pred,axiom,
! [M4: nat,N5: nat] :
( ( minus_minus_nat @ M4 @ ( suc @ N5 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M4 @ one_one_nat ) @ N5 ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1182_nat__induct__non__zero,axiom,
! [N5: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N5 )
=> ( ( P @ one_one_nat )
=> ( ! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ N )
=> ( P @ ( suc @ N ) ) ) )
=> ( P @ N5 ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1183_Suc__pred_H,axiom,
! [N5: nat] :
( ( ord_less_nat @ zero_zero_nat @ N5 )
=> ( N5
= ( suc @ ( minus_minus_nat @ N5 @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1184_Suc__diff__eq__diff__pred,axiom,
! [N5: nat,M4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N5 )
=> ( ( minus_minus_nat @ ( suc @ M4 ) @ N5 )
= ( minus_minus_nat @ M4 @ ( minus_minus_nat @ N5 @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1185_mono__times__nat,axiom,
! [N5: nat] :
( ( ord_less_nat @ zero_zero_nat @ N5 )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ ( times_times_nat @ N5 ) ) ) ).
% mono_times_nat
thf(fact_1186_add__is__0,axiom,
! [M4: nat,N5: nat] :
( ( ( plus_plus_nat @ M4 @ N5 )
= zero_zero_nat )
= ( ( M4 = zero_zero_nat )
& ( N5 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1187_Nat_Oadd__0__right,axiom,
! [M4: nat] :
( ( plus_plus_nat @ M4 @ zero_zero_nat )
= M4 ) ).
% Nat.add_0_right
thf(fact_1188_add__Suc__right,axiom,
! [M4: nat,N5: nat] :
( ( plus_plus_nat @ M4 @ ( suc @ N5 ) )
= ( suc @ ( plus_plus_nat @ M4 @ N5 ) ) ) ).
% add_Suc_right
thf(fact_1189_nat__add__left__cancel__less,axiom,
! [K: nat,M4: nat,N5: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M4 ) @ ( plus_plus_nat @ K @ N5 ) )
= ( ord_less_nat @ M4 @ N5 ) ) ).
% nat_add_left_cancel_less
thf(fact_1190_nat__add__left__cancel__le,axiom,
! [K: nat,M4: nat,N5: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M4 ) @ ( plus_plus_nat @ K @ N5 ) )
= ( ord_less_eq_nat @ M4 @ N5 ) ) ).
% nat_add_left_cancel_le
thf(fact_1191_mult__is__0,axiom,
! [M4: nat,N5: nat] :
( ( ( times_times_nat @ M4 @ N5 )
= zero_zero_nat )
= ( ( M4 = zero_zero_nat )
| ( N5 = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1192_mult__0__right,axiom,
! [M4: nat] :
( ( times_times_nat @ M4 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1193_mult__cancel1,axiom,
! [K: nat,M4: nat,N5: nat] :
( ( ( times_times_nat @ K @ M4 )
= ( times_times_nat @ K @ N5 ) )
= ( ( M4 = N5 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1194_mult__cancel2,axiom,
! [M4: nat,K: nat,N5: nat] :
( ( ( times_times_nat @ M4 @ K )
= ( times_times_nat @ N5 @ K ) )
= ( ( M4 = N5 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1195_diff__diff__left,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
= ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1196_nat__mult__eq__1__iff,axiom,
! [M4: nat,N5: nat] :
( ( ( times_times_nat @ M4 @ N5 )
= one_one_nat )
= ( ( M4 = one_one_nat )
& ( N5 = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1197_nat__1__eq__mult__iff,axiom,
! [M4: nat,N5: nat] :
( ( one_one_nat
= ( times_times_nat @ M4 @ N5 ) )
= ( ( M4 = one_one_nat )
& ( N5 = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1198_add__gr__0,axiom,
! [M4: nat,N5: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M4 @ N5 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M4 )
| ( ord_less_nat @ zero_zero_nat @ N5 ) ) ) ).
% add_gr_0
thf(fact_1199_mult__eq__1__iff,axiom,
! [M4: nat,N5: nat] :
( ( ( times_times_nat @ M4 @ N5 )
= ( suc @ zero_zero_nat ) )
= ( ( M4
= ( suc @ zero_zero_nat ) )
& ( N5
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1200_one__eq__mult__iff,axiom,
! [M4: nat,N5: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M4 @ N5 ) )
= ( ( M4
= ( suc @ zero_zero_nat ) )
& ( N5
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1201_mult__less__cancel2,axiom,
! [M4: nat,K: nat,N5: nat] :
( ( ord_less_nat @ ( times_times_nat @ M4 @ K ) @ ( times_times_nat @ N5 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M4 @ N5 ) ) ) ).
% mult_less_cancel2
thf(fact_1202_nat__0__less__mult__iff,axiom,
! [M4: nat,N5: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M4 @ N5 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M4 )
& ( ord_less_nat @ zero_zero_nat @ N5 ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1203_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1204_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1205_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1206_mult__Suc__right,axiom,
! [M4: nat,N5: nat] :
( ( times_times_nat @ M4 @ ( suc @ N5 ) )
= ( plus_plus_nat @ M4 @ ( times_times_nat @ M4 @ N5 ) ) ) ).
% mult_Suc_right
thf(fact_1207_one__le__mult__iff,axiom,
! [M4: nat,N5: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M4 @ N5 ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M4 )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N5 ) ) ) ).
% one_le_mult_iff
thf(fact_1208_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I2 )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1209_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I2 @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1210_mult__le__cancel2,axiom,
! [M4: nat,K: nat,N5: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M4 @ K ) @ ( times_times_nat @ N5 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M4 @ N5 ) ) ) ).
% mult_le_cancel2
thf(fact_1211_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M: nat,N2: nat] : ( if_nat @ ( M = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N2 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1212_nat__mult__1__right,axiom,
! [N5: nat] :
( ( times_times_nat @ N5 @ one_one_nat )
= N5 ) ).
% nat_mult_1_right
thf(fact_1213_nat__mult__1,axiom,
! [N5: nat] :
( ( times_times_nat @ one_one_nat @ N5 )
= N5 ) ).
% nat_mult_1
thf(fact_1214_one__is__add,axiom,
! [M4: nat,N5: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M4 @ N5 ) )
= ( ( ( M4
= ( suc @ zero_zero_nat ) )
& ( N5 = zero_zero_nat ) )
| ( ( M4 = zero_zero_nat )
& ( N5
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1215_add__is__1,axiom,
! [M4: nat,N5: nat] :
( ( ( plus_plus_nat @ M4 @ N5 )
= ( suc @ zero_zero_nat ) )
= ( ( ( M4
= ( suc @ zero_zero_nat ) )
& ( N5 = zero_zero_nat ) )
| ( ( M4 = zero_zero_nat )
& ( N5
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1216_less__imp__add__positive,axiom,
! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I2 @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1217_less__imp__Suc__add,axiom,
! [M4: nat,N5: nat] :
( ( ord_less_nat @ M4 @ N5 )
=> ? [K2: nat] :
( N5
= ( suc @ ( plus_plus_nat @ M4 @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1218_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M: nat,N2: nat] :
? [K3: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1219_less__add__Suc2,axiom,
! [I2: nat,M4: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ M4 @ I2 ) ) ) ).
% less_add_Suc2
thf(fact_1220_less__add__Suc1,axiom,
! [I2: nat,M4: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ I2 @ M4 ) ) ) ).
% less_add_Suc1
thf(fact_1221_less__natE,axiom,
! [M4: nat,N5: nat] :
( ( ord_less_nat @ M4 @ N5 )
=> ~ ! [Q4: nat] :
( N5
!= ( suc @ ( plus_plus_nat @ M4 @ Q4 ) ) ) ) ).
% less_natE
thf(fact_1222_mult__less__mono2,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1223_mult__less__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1224_Suc__mult__less__cancel1,axiom,
! [K: nat,M4: nat,N5: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M4 ) @ ( times_times_nat @ ( suc @ K ) @ N5 ) )
= ( ord_less_nat @ M4 @ N5 ) ) ).
% Suc_mult_less_cancel1
thf(fact_1225_diff__add__0,axiom,
! [N5: nat,M4: nat] :
( ( minus_minus_nat @ N5 @ ( plus_plus_nat @ N5 @ M4 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1226_add__eq__self__zero,axiom,
! [M4: nat,N5: nat] :
( ( ( plus_plus_nat @ M4 @ N5 )
= M4 )
=> ( N5 = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1227_mult__0,axiom,
! [N5: nat] :
( ( times_times_nat @ zero_zero_nat @ N5 )
= zero_zero_nat ) ).
% mult_0
thf(fact_1228_plus__nat_Oadd__0,axiom,
! [N5: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N5 )
= N5 ) ).
% plus_nat.add_0
thf(fact_1229_nat__arith_Osuc1,axiom,
! [A: nat,K: nat,A2: nat] :
( ( A
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( suc @ A )
= ( plus_plus_nat @ K @ ( suc @ A2 ) ) ) ) ).
% nat_arith.suc1
thf(fact_1230_add__Suc,axiom,
! [M4: nat,N5: nat] :
( ( plus_plus_nat @ ( suc @ M4 ) @ N5 )
= ( suc @ ( plus_plus_nat @ M4 @ N5 ) ) ) ).
% add_Suc
thf(fact_1231_mult__Suc,axiom,
! [M4: nat,N5: nat] :
( ( times_times_nat @ ( suc @ M4 ) @ N5 )
= ( plus_plus_nat @ N5 @ ( times_times_nat @ M4 @ N5 ) ) ) ).
% mult_Suc
thf(fact_1232_add__Suc__shift,axiom,
! [M4: nat,N5: nat] :
( ( plus_plus_nat @ ( suc @ M4 ) @ N5 )
= ( plus_plus_nat @ M4 @ ( suc @ N5 ) ) ) ).
% add_Suc_shift
thf(fact_1233_Suc__mult__cancel1,axiom,
! [K: nat,M4: nat,N5: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M4 )
= ( times_times_nat @ ( suc @ K ) @ N5 ) )
= ( M4 = N5 ) ) ).
% Suc_mult_cancel1
thf(fact_1234_add__mult__distrib2,axiom,
! [K: nat,M4: nat,N5: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M4 @ N5 ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M4 ) @ ( times_times_nat @ K @ N5 ) ) ) ).
% add_mult_distrib2
thf(fact_1235_add__mult__distrib,axiom,
! [M4: nat,N5: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M4 @ N5 ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M4 @ K ) @ ( times_times_nat @ N5 @ K ) ) ) ).
% add_mult_distrib
thf(fact_1236_diff__mult__distrib,axiom,
! [M4: nat,N5: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M4 @ N5 ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M4 @ K ) @ ( times_times_nat @ N5 @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1237_diff__mult__distrib2,axiom,
! [K: nat,M4: nat,N5: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M4 @ N5 ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M4 ) @ ( times_times_nat @ K @ N5 ) ) ) ).
% diff_mult_distrib2
thf(fact_1238_le__cube,axiom,
! [M4: nat] : ( ord_less_eq_nat @ M4 @ ( times_times_nat @ M4 @ ( times_times_nat @ M4 @ M4 ) ) ) ).
% le_cube
thf(fact_1239_le__square,axiom,
! [M4: nat] : ( ord_less_eq_nat @ M4 @ ( times_times_nat @ M4 @ M4 ) ) ).
% le_square
thf(fact_1240_mult__le__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1241_mult__le__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_1242_mult__le__mono2,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_1243_Nat_Odiff__cancel,axiom,
! [K: nat,M4: nat,N5: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M4 ) @ ( plus_plus_nat @ K @ N5 ) )
= ( minus_minus_nat @ M4 @ N5 ) ) ).
% Nat.diff_cancel
thf(fact_1244_diff__cancel2,axiom,
! [M4: nat,K: nat,N5: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M4 @ K ) @ ( plus_plus_nat @ N5 @ K ) )
= ( minus_minus_nat @ M4 @ N5 ) ) ).
% diff_cancel2
thf(fact_1245_diff__add__inverse,axiom,
! [N5: nat,M4: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N5 @ M4 ) @ N5 )
= M4 ) ).
% diff_add_inverse
thf(fact_1246_diff__add__inverse2,axiom,
! [M4: nat,N5: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M4 @ N5 ) @ N5 )
= M4 ) ).
% diff_add_inverse2
thf(fact_1247_add__leE,axiom,
! [M4: nat,K: nat,N5: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M4 @ K ) @ N5 )
=> ~ ( ( ord_less_eq_nat @ M4 @ N5 )
=> ~ ( ord_less_eq_nat @ K @ N5 ) ) ) ).
% add_leE
thf(fact_1248_le__add1,axiom,
! [N5: nat,M4: nat] : ( ord_less_eq_nat @ N5 @ ( plus_plus_nat @ N5 @ M4 ) ) ).
% le_add1
thf(fact_1249_le__add2,axiom,
! [N5: nat,M4: nat] : ( ord_less_eq_nat @ N5 @ ( plus_plus_nat @ M4 @ N5 ) ) ).
% le_add2
thf(fact_1250_add__leD1,axiom,
! [M4: nat,K: nat,N5: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M4 @ K ) @ N5 )
=> ( ord_less_eq_nat @ M4 @ N5 ) ) ).
% add_leD1
thf(fact_1251_add__leD2,axiom,
! [M4: nat,K: nat,N5: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M4 @ K ) @ N5 )
=> ( ord_less_eq_nat @ K @ N5 ) ) ).
% add_leD2
thf(fact_1252_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N: nat] :
( L
= ( plus_plus_nat @ K @ N ) ) ) ).
% le_Suc_ex
thf(fact_1253_add__le__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1254_add__le__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1255_trans__le__add1,axiom,
! [I2: nat,J: nat,M4: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J @ M4 ) ) ) ).
% trans_le_add1
thf(fact_1256_trans__le__add2,axiom,
! [I2: nat,J: nat,M4: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M4 @ J ) ) ) ).
% trans_le_add2
thf(fact_1257_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N2: nat] :
? [K3: nat] :
( N2
= ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1258_add__lessD1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
=> ( ord_less_nat @ I2 @ K ) ) ).
% add_lessD1
thf(fact_1259_add__less__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1260_not__add__less1,axiom,
! [I2: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ I2 ) ).
% not_add_less1
thf(fact_1261_not__add__less2,axiom,
! [J: nat,I2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I2 ) @ I2 ) ).
% not_add_less2
thf(fact_1262_add__less__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1263_trans__less__add1,axiom,
! [I2: nat,J: nat,M4: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ J @ M4 ) ) ) ).
% trans_less_add1
thf(fact_1264_trans__less__add2,axiom,
! [I2: nat,J: nat,M4: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ M4 @ J ) ) ) ).
% trans_less_add2
thf(fact_1265_less__add__eq__less,axiom,
! [K: nat,L: nat,M4: nat,N5: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M4 @ L )
= ( plus_plus_nat @ K @ N5 ) )
=> ( ord_less_nat @ M4 @ N5 ) ) ) ).
% less_add_eq_less
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y: nat] :
( ( if_nat @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y: nat] :
( ( if_nat @ $true @ X2 @ Y )
= X2 ) ).
% Conjectures (2)
thf(conj_0,hypothesis,
! [X5: nat,Sigma5: nat > a] :
( ( member_nat @ X5 @ ( minus_minus_set_nat @ x @ ( relational_fv_a_b @ q ) ) )
=> ( ( ord_less_eq_set_nat @ ( relational_fv_a_b @ q ) @ x )
=> ( ( relational_sat_a_b @ q @ i @ Sigma5 )
=> thesis ) ) ) ).
thf(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------