TPTP Problem File: SLH0532^1.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 : Universal_Hash_Families/0033_Preliminary_Results/prob_00204_008096__18573174_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1509 ( 633 unt; 237 typ; 0 def)
% Number of atoms : 3242 (1031 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 9996 ( 148 ~; 15 |; 228 &;8383 @)
% ( 0 <=>;1222 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 6 avg)
% Number of types : 30 ( 29 usr)
% Number of type conns : 1059 (1059 >; 0 *; 0 +; 0 <<)
% Number of symbols : 211 ( 208 usr; 21 con; 0-3 aty)
% Number of variables : 3801 ( 540 ^;3192 !; 69 ?;3801 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:46:14.626
%------------------------------------------------------------------------------
% Could-be-implicit typings (29)
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Pr1261947904930325089at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Sum_sum_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__c_J_J_J,type,
set_set_set_c: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
set_set_set_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Set__Oset_Itf__a_J_Mtf__c_J_J,type,
set_set_a_c: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_nat_nat: $tType ).
thf(ty_n_t__Probability____Mass____Function__Opmf_Itf__a_J,type,
probab3364570286911266904_pmf_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Extended____Nonnegative____Real__Oennreal,type,
extend8495563244428889912nnreal: $tType ).
thf(ty_n_t__Set__Oset_It__Extended____Real__Oereal_J,type,
set_Extended_ereal: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mtf__c_J_J,type,
set_nat_c: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__c_J_J,type,
set_set_c: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
set_set_b: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__c_Mtf__c_J_J,type,
set_c_c: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__c_Mtf__a_J_J,type,
set_c_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__c_J_J,type,
set_a_c: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
set_a_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
set_set_o: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Extended____Real__Oereal,type,
extended_ereal: $tType ).
thf(ty_n_t__Set__Oset_Itf__c_J,type,
set_c: $tType ).
thf(ty_n_t__Set__Oset_Itf__b_J,type,
set_b: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Set__Oset_I_Eo_J,type,
set_o: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__c,type,
c: $tType ).
thf(ty_n_tf__b,type,
b: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (208)
thf(sy_c_Complete__Lattices_OInf__class_OInf_001_Eo,type,
complete_Inf_Inf_o: set_o > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_Eo,type,
complete_Sup_Sup_o: set_o > $o ).
thf(sy_c_Finite__Set_OFpow_001t__Nat__Onat,type,
finite_Fpow_nat: set_nat > set_set_nat ).
thf(sy_c_Finite__Set_OFpow_001t__Set__Oset_Itf__a_J,type,
finite_Fpow_set_a: set_set_a > set_set_set_a ).
thf(sy_c_Finite__Set_OFpow_001tf__a,type,
finite_Fpow_a: set_a > set_set_a ).
thf(sy_c_Finite__Set_OFpow_001tf__c,type,
finite_Fpow_c: set_c > set_set_c ).
thf(sy_c_Finite__Set_Ofinite_001_Eo,type,
finite_finite_o: set_o > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
finite6177210948735845034at_nat: set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
finite7209287970140883943_set_a: set_set_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
finite_finite_set_a: set_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__c_J,type,
finite_finite_set_c: set_set_c > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
finite6187706683773761046at_nat: set_Sum_sum_nat_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__b,type,
finite_finite_b: set_b > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__c,type,
finite_finite_c: set_c > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
thf(sy_c_FuncSet_OPi_001t__Nat__Onat_001t__Nat__Onat,type,
pi_nat_nat: set_nat > ( nat > set_nat ) > set_nat_nat ).
thf(sy_c_FuncSet_OPi_001t__Nat__Onat_001tf__c,type,
pi_nat_c: set_nat > ( nat > set_c ) > set_nat_c ).
thf(sy_c_FuncSet_OPi_001t__Set__Oset_Itf__a_J_001tf__c,type,
pi_set_a_c: set_set_a > ( set_a > set_c ) > set_set_a_c ).
thf(sy_c_FuncSet_OPi_001tf__a_001tf__a,type,
pi_a_a: set_a > ( a > set_a ) > set_a_a ).
thf(sy_c_FuncSet_OPi_001tf__a_001tf__c,type,
pi_a_c: set_a > ( a > set_c ) > set_a_c ).
thf(sy_c_FuncSet_OPi_001tf__c_001tf__a,type,
pi_c_a: set_c > ( c > set_a ) > set_c_a ).
thf(sy_c_FuncSet_OPi_001tf__c_001tf__c,type,
pi_c_c: set_c > ( c > set_c ) > set_c_c ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Extended____Nonnegative____Real__Oennreal,type,
minus_8429688780609304081nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > extend8495563244428889912nnreal ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Extended____Real__Oereal,type,
minus_2816186181549245109_ereal: extended_ereal > extended_ereal > extended_ereal ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Nonnegative____Real__Oennreal,type,
zero_z7100319975126383169nnreal: extend8495563244428889912nnreal ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Real__Oereal,type,
zero_z2744965634713055877_ereal: extended_ereal ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_I_Eo_M_Eo_J,type,
inf_inf_o_o: ( $o > $o ) > ( $o > $o ) > $o > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Nat__Onat_M_Eo_J,type,
inf_inf_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Set__Oset_Itf__a_J_M_Eo_J,type,
inf_inf_set_a_o: ( set_a > $o ) > ( set_a > $o ) > set_a > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Set__Oset_Itf__c_J_M_Eo_J,type,
inf_inf_set_c_o: ( set_c > $o ) > ( set_c > $o ) > set_c > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_Itf__a_M_Eo_J,type,
inf_inf_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_Itf__b_M_Eo_J,type,
inf_inf_b_o: ( b > $o ) > ( b > $o ) > b > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_Itf__c_M_Eo_J,type,
inf_inf_c_o: ( c > $o ) > ( c > $o ) > c > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Extended____Nonnegative____Real__Oennreal,type,
inf_in7439215052339218890nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > extend8495563244428889912nnreal ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Extended____Real__Oereal,type,
inf_in2794916579150040252_ereal: extended_ereal > extended_ereal > extended_ereal ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_Eo_J,type,
inf_inf_set_o: set_o > set_o > set_o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
inf_inf_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
inf_in1205276777018777868_set_a: set_set_set_a > set_set_set_a > set_set_set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
inf_inf_set_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_Itf__c_J_J,type,
inf_inf_set_set_c: set_set_c > set_set_c > set_set_c ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__b_J,type,
inf_inf_set_b: set_b > set_b > set_b ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__c_J,type,
inf_inf_set_c: set_c > set_c > set_c ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_Eo_J,type,
ord_less_eq_o_o: ( $o > $o ) > ( $o > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_Itf__a_J_M_Eo_J,type,
ord_less_eq_set_a_o: ( set_a > $o ) > ( set_a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_Itf__c_J_M_Eo_J,type,
ord_less_eq_set_c_o: ( set_c > $o ) > ( set_c > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_M_Eo_J,type,
ord_less_eq_a_o: ( a > $o ) > ( a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__b_M_Eo_J,type,
ord_less_eq_b_o: ( b > $o ) > ( b > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__c_M_Eo_J,type,
ord_less_eq_c_o: ( c > $o ) > ( c > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nonnegative____Real__Oennreal,type,
ord_le3935885782089961368nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Real__Oereal,type,
ord_le1083603963089353582_ereal: extended_ereal > extended_ereal > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
ord_less_eq_set_o: set_o > set_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
ord_le5722252365846178494_set_a: set_set_set_a > set_set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_le3724670747650509150_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__c_J_J,type,
ord_le3866738827743201120_set_c: set_set_c > set_set_c > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__b_J,type,
ord_less_eq_set_b: set_b > set_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__c_J,type,
ord_less_eq_set_c: set_c > set_c > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Extended____Nonnegative____Real__Oennreal,type,
top_to1496364449551166952nnreal: extend8495563244428889912nnreal ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Extended____Real__Oereal,type,
top_to6662034908053899550_ereal: extended_ereal ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
top_top_set_o: set_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
top_to4669805908274784177at_nat: set_Pr1261947904930325089at_nat ).
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__Set__Oset_Itf__a_J_J_J,type,
top_to4027821306633060462_set_a: set_set_set_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
top_top_set_set_a: set_set_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_Itf__c_J_J,type,
top_top_set_set_c: set_set_c ).
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_Itf__a_J,type,
top_top_set_a: set_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__b_J,type,
top_top_set_b: set_b ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__c_J,type,
top_top_set_c: set_c ).
thf(sy_c_Probability__Mass__Function_Oset__pmf_001tf__a,type,
probab49036049091589825_pmf_a: probab3364570286911266904_pmf_a > set_a ).
thf(sy_c_Relation_OPowp_001_Eo,type,
powp_o: ( $o > $o ) > set_o > $o ).
thf(sy_c_Relation_OPowp_001t__Nat__Onat,type,
powp_nat: ( nat > $o ) > set_nat > $o ).
thf(sy_c_Relation_OPowp_001t__Set__Oset_Itf__c_J,type,
powp_set_c: ( set_c > $o ) > set_set_c > $o ).
thf(sy_c_Relation_OPowp_001tf__a,type,
powp_a: ( a > $o ) > set_a > $o ).
thf(sy_c_Relation_OPowp_001tf__b,type,
powp_b: ( b > $o ) > set_b > $o ).
thf(sy_c_Relation_OPowp_001tf__c,type,
powp_c: ( c > $o ) > set_c > $o ).
thf(sy_c_Set_OCollect_001_Eo,type,
collect_o: ( $o > $o ) > set_o ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
collect_set_set_a: ( set_set_a > $o ) > set_set_set_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__c_J,type,
collect_set_c: ( set_c > $o ) > set_set_c ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_OCollect_001tf__b,type,
collect_b: ( b > $o ) > set_b ).
thf(sy_c_Set_OCollect_001tf__c,type,
collect_c: ( c > $o ) > set_c ).
thf(sy_c_Set_OPow_001_Eo,type,
pow_o: set_o > set_set_o ).
thf(sy_c_Set_OPow_001t__Nat__Onat,type,
pow_nat: set_nat > set_set_nat ).
thf(sy_c_Set_OPow_001t__Set__Oset_Itf__a_J,type,
pow_set_a: set_set_a > set_set_set_a ).
thf(sy_c_Set_OPow_001t__Set__Oset_Itf__c_J,type,
pow_set_c: set_set_c > set_set_set_c ).
thf(sy_c_Set_OPow_001tf__a,type,
pow_a: set_a > set_set_a ).
thf(sy_c_Set_OPow_001tf__b,type,
pow_b: set_b > set_set_b ).
thf(sy_c_Set_OPow_001tf__c,type,
pow_c: set_c > set_set_c ).
thf(sy_c_Set_Oimage_001_Eo_001_Eo,type,
image_o_o: ( $o > $o ) > set_o > set_o ).
thf(sy_c_Set_Oimage_001_Eo_001t__Nat__Onat,type,
image_o_nat: ( $o > nat ) > set_o > set_nat ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_Itf__a_J,type,
image_o_set_a: ( $o > set_a ) > set_o > set_set_a ).
thf(sy_c_Set_Oimage_001_Eo_001tf__b,type,
image_o_b: ( $o > b ) > set_o > set_b ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_Eo,type,
image_nat_o: ( nat > $o ) > set_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_Itf__a_J,type,
image_nat_set_a: ( nat > set_a ) > set_nat > set_set_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_Itf__c_J,type,
image_nat_set_c: ( nat > set_c ) > set_nat > set_set_c ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001tf__a,type,
image_nat_a: ( nat > a ) > set_nat > set_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001tf__b,type,
image_nat_b: ( nat > b ) > set_nat > set_b ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001tf__c,type,
image_nat_c: ( nat > c ) > set_nat > set_c ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_Itf__a_J_J_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
image_1042221919965026181_set_a: ( set_set_a > set_set_a ) > set_set_set_a > set_set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001_Eo,type,
image_set_a_o: ( set_a > $o ) > set_set_a > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Nat__Onat,type,
image_set_a_nat: ( set_a > nat ) > set_set_a > set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
image_4955109552351689957_set_a: ( set_a > set_set_a ) > set_set_a > set_set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
image_set_a_set_a: ( set_a > set_a ) > set_set_a > set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__c_J,type,
image_set_a_set_c: ( set_a > set_c ) > set_set_a > set_set_c ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001tf__a,type,
image_set_a_a: ( set_a > a ) > set_set_a > set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001tf__b,type,
image_set_a_b: ( set_a > b ) > set_set_a > set_b ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001tf__c,type,
image_set_a_c: ( set_a > c ) > set_set_a > set_c ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__c_J_001t__Nat__Onat,type,
image_set_c_nat: ( set_c > nat ) > set_set_c > set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__c_J_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
image_8925085205931127395_set_a: ( set_c > set_set_a ) > set_set_c > set_set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__c_J_001t__Set__Oset_Itf__a_J,type,
image_set_c_set_a: ( set_c > set_a ) > set_set_c > set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__c_J_001t__Set__Oset_Itf__c_J,type,
image_set_c_set_c: ( set_c > set_c ) > set_set_c > set_set_c ).
thf(sy_c_Set_Oimage_001tf__a_001t__Nat__Onat,type,
image_a_nat: ( a > nat ) > set_a > set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_Itf__a_J,type,
image_a_set_a: ( a > set_a ) > set_a > set_set_a ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set_Oimage_001tf__a_001tf__c,type,
image_a_c: ( a > c ) > set_a > set_c ).
thf(sy_c_Set_Oimage_001tf__b_001_Eo,type,
image_b_o: ( b > $o ) > set_b > set_o ).
thf(sy_c_Set_Oimage_001tf__b_001t__Nat__Onat,type,
image_b_nat: ( b > nat ) > set_b > set_nat ).
thf(sy_c_Set_Oimage_001tf__b_001t__Set__Oset_Itf__a_J,type,
image_b_set_a: ( b > set_a ) > set_b > set_set_a ).
thf(sy_c_Set_Oimage_001tf__b_001tf__b,type,
image_b_b: ( b > b ) > set_b > set_b ).
thf(sy_c_Set_Oimage_001tf__c_001t__Nat__Onat,type,
image_c_nat: ( c > nat ) > set_c > set_nat ).
thf(sy_c_Set_Oimage_001tf__c_001t__Set__Oset_Itf__a_J,type,
image_c_set_a: ( c > set_a ) > set_c > set_set_a ).
thf(sy_c_Set_Oimage_001tf__c_001t__Set__Oset_Itf__c_J,type,
image_c_set_c: ( c > set_c ) > set_c > set_set_c ).
thf(sy_c_Set_Oimage_001tf__c_001tf__a,type,
image_c_a: ( c > a ) > set_c > set_a ).
thf(sy_c_Set_Oimage_001tf__c_001tf__c,type,
image_c_c: ( c > c ) > set_c > set_c ).
thf(sy_c_Set_Ovimage_001_Eo_001_Eo,type,
vimage_o_o: ( $o > $o ) > set_o > set_o ).
thf(sy_c_Set_Ovimage_001_Eo_001t__Nat__Onat,type,
vimage_o_nat: ( $o > nat ) > set_nat > set_o ).
thf(sy_c_Set_Ovimage_001_Eo_001tf__b,type,
vimage_o_b: ( $o > b ) > set_b > set_o ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001_Eo,type,
vimage_nat_o: ( nat > $o ) > set_o > set_nat ).
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__Set__Oset_Itf__a_J,type,
vimage_nat_set_a: ( nat > set_a ) > set_set_a > set_nat ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001tf__a,type,
vimage_nat_a: ( nat > a ) > set_a > set_nat ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001tf__b,type,
vimage_nat_b: ( nat > b ) > set_b > set_nat ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001tf__c,type,
vimage_nat_c: ( nat > c ) > set_c > set_nat ).
thf(sy_c_Set_Ovimage_001t__Set__Oset_Itf__a_J_001_Eo,type,
vimage_set_a_o: ( set_a > $o ) > set_o > set_set_a ).
thf(sy_c_Set_Ovimage_001t__Set__Oset_Itf__a_J_001t__Nat__Onat,type,
vimage_set_a_nat: ( set_a > nat ) > set_nat > set_set_a ).
thf(sy_c_Set_Ovimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
vimage_set_a_set_a: ( set_a > set_a ) > set_set_a > set_set_a ).
thf(sy_c_Set_Ovimage_001t__Set__Oset_Itf__a_J_001tf__a,type,
vimage_set_a_a: ( set_a > a ) > set_a > set_set_a ).
thf(sy_c_Set_Ovimage_001t__Set__Oset_Itf__a_J_001tf__b,type,
vimage_set_a_b: ( set_a > b ) > set_b > set_set_a ).
thf(sy_c_Set_Ovimage_001t__Set__Oset_Itf__a_J_001tf__c,type,
vimage_set_a_c: ( set_a > c ) > set_c > set_set_a ).
thf(sy_c_Set_Ovimage_001tf__a_001_Eo,type,
vimage_a_o: ( a > $o ) > set_o > set_a ).
thf(sy_c_Set_Ovimage_001tf__a_001t__Nat__Onat,type,
vimage_a_nat: ( a > nat ) > set_nat > set_a ).
thf(sy_c_Set_Ovimage_001tf__a_001t__Set__Oset_Itf__a_J,type,
vimage_a_set_a: ( a > set_a ) > set_set_a > set_a ).
thf(sy_c_Set_Ovimage_001tf__a_001tf__a,type,
vimage_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set_Ovimage_001tf__a_001tf__b,type,
vimage_a_b: ( a > b ) > set_b > set_a ).
thf(sy_c_Set_Ovimage_001tf__a_001tf__c,type,
vimage_a_c: ( a > c ) > set_c > set_a ).
thf(sy_c_Set_Ovimage_001tf__b_001_Eo,type,
vimage_b_o: ( b > $o ) > set_o > set_b ).
thf(sy_c_Set_Ovimage_001tf__b_001t__Nat__Onat,type,
vimage_b_nat: ( b > nat ) > set_nat > set_b ).
thf(sy_c_Set_Ovimage_001tf__b_001tf__b,type,
vimage_b_b: ( b > b ) > set_b > set_b ).
thf(sy_c_Set_Ovimage_001tf__c_001_Eo,type,
vimage_c_o: ( c > $o ) > set_o > set_c ).
thf(sy_c_Set_Ovimage_001tf__c_001t__Nat__Onat,type,
vimage_c_nat: ( c > nat ) > set_nat > set_c ).
thf(sy_c_Set_Ovimage_001tf__c_001t__Set__Oset_Itf__a_J,type,
vimage_c_set_a: ( c > set_a ) > set_set_a > set_c ).
thf(sy_c_Set_Ovimage_001tf__c_001tf__a,type,
vimage_c_a: ( c > a ) > set_a > set_c ).
thf(sy_c_Set_Ovimage_001tf__c_001tf__b,type,
vimage_c_b: ( c > b ) > set_b > set_c ).
thf(sy_c_Set_Ovimage_001tf__c_001tf__c,type,
vimage_c_c: ( c > c ) > set_c > set_c ).
thf(sy_c_Sigma__Algebra_Oclosed__cdi_001tf__a,type,
sigma_closed_cdi_a: set_a > set_set_a > $o ).
thf(sy_c_Sigma__Algebra_Oclosed__cdi_001tf__c,type,
sigma_closed_cdi_c: set_c > set_set_c > $o ).
thf(sy_c_Sigma__Algebra_Omeasure__space_001tf__a,type,
sigma_3179946494550678598pace_a: set_a > set_set_a > ( set_a > extend8495563244428889912nnreal ) > $o ).
thf(sy_c_Sigma__Algebra_Omeasure__space_001tf__c,type,
sigma_3179946494550678600pace_c: set_c > set_set_c > ( set_c > extend8495563244428889912nnreal ) > $o ).
thf(sy_c_Sigma__Algebra_Osigma__sets_001t__Nat__Onat,type,
sigma_sigma_sets_nat: set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Sigma__Algebra_Osigma__sets_001t__Set__Oset_Itf__a_J,type,
sigma_2987359967864564790_set_a: set_set_a > set_set_set_a > set_set_set_a ).
thf(sy_c_Sigma__Algebra_Osigma__sets_001tf__a,type,
sigma_sigma_sets_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Sigma__Algebra_Osigma__sets_001tf__c,type,
sigma_sigma_sets_c: set_c > set_set_c > set_set_c ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
member_nat_nat: ( nat > nat ) > set_nat_nat > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mtf__c_J,type,
member_nat_c: ( nat > c ) > set_nat_c > $o ).
thf(sy_c_member_001_062_It__Set__Oset_Itf__a_J_Mtf__c_J,type,
member_set_a_c: ( set_a > c ) > set_set_a_c > $o ).
thf(sy_c_member_001_062_Itf__a_Mtf__a_J,type,
member_a_a: ( a > a ) > set_a_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mtf__c_J,type,
member_a_c: ( a > c ) > set_a_c > $o ).
thf(sy_c_member_001_062_Itf__c_Mtf__a_J,type,
member_c_a: ( c > a ) > set_c_a > $o ).
thf(sy_c_member_001_062_Itf__c_Mtf__c_J,type,
member_c_c: ( c > c ) > set_c_c > $o ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__Extended____Real__Oereal,type,
member2350847679896131959_ereal: extended_ereal > set_Extended_ereal > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_I_Eo_J,type,
member_set_o: set_o > set_set_o > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
member_set_set_a: set_set_a > set_set_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__c_J_J,type,
member_set_set_c: set_set_c > set_set_set_c > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__b_J,type,
member_set_b: set_b > set_set_b > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__c_J,type,
member_set_c: set_c > set_set_c > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_c_member_001tf__b,type,
member_b: b > set_b > $o ).
thf(sy_c_member_001tf__c,type,
member_c: c > set_c > $o ).
thf(sy_v_F____,type,
f: b > set_set_a ).
thf(sy_v_G____,type,
g: b > set_set_c ).
thf(sy_v_I,type,
i: set_b ).
thf(sy_v_X,type,
x: b > a > c ).
thf(sy_v_i____,type,
i2: b ).
thf(sy_v_p,type,
p: probab3364570286911266904_pmf_a ).
% Relevant facts (1271)
thf(fact_0_a,axiom,
member_b @ i2 @ i ).
% a
thf(fact_1_calculation,axiom,
( ( collect_set_a
@ ^ [U: set_a] :
? [A: set_c] :
( U
= ( inf_inf_set_a @ ( vimage_a_c @ ( x @ i2 ) @ A ) @ ( probab49036049091589825_pmf_a @ p ) ) ) )
= ( collect_set_a
@ ^ [Uu: set_a] :
? [A: set_c] :
( ( Uu
= ( inf_inf_set_a @ ( vimage_a_c @ ( x @ i2 ) @ A ) @ ( probab49036049091589825_pmf_a @ p ) ) )
& ( ord_less_eq_set_c @ A @ ( image_a_c @ ( x @ i2 ) @ ( probab49036049091589825_pmf_a @ p ) ) ) ) ) ) ).
% calculation
thf(fact_2_F__def,axiom,
( f
= ( ^ [I: b] :
( collect_set_a
@ ^ [Uu: set_a] :
? [A2: set_c] :
( ( Uu
= ( inf_inf_set_a @ ( vimage_a_c @ ( x @ I ) @ A2 ) @ ( probab49036049091589825_pmf_a @ p ) ) )
& ( member_set_c @ A2 @ ( g @ I ) ) ) ) ) ) ).
% F_def
thf(fact_3__092_060open_062_092_060And_062A_O_AX_Ai_A_N_096_AA_A_092_060inter_062_Aset__pmf_Ap_A_061_AX_Ai_A_N_096_A_IA_A_092_060inter_062_AX_Ai_A_096_Aset__pmf_Ap_J_A_092_060inter_062_Aset__pmf_Ap_092_060close_062,axiom,
! [A3: set_c] :
( ( inf_inf_set_a @ ( vimage_a_c @ ( x @ i2 ) @ A3 ) @ ( probab49036049091589825_pmf_a @ p ) )
= ( inf_inf_set_a @ ( vimage_a_c @ ( x @ i2 ) @ ( inf_inf_set_c @ A3 @ ( image_a_c @ ( x @ i2 ) @ ( probab49036049091589825_pmf_a @ p ) ) ) ) @ ( probab49036049091589825_pmf_a @ p ) ) ) ).
% \<open>\<And>A. X i -` A \<inter> set_pmf p = X i -` (A \<inter> X i ` set_pmf p) \<inter> set_pmf p\<close>
thf(fact_4_vimage__Int,axiom,
! [F: nat > nat,A3: set_nat,B: set_nat] :
( ( vimage_nat_nat @ F @ ( inf_inf_set_nat @ A3 @ B ) )
= ( inf_inf_set_nat @ ( vimage_nat_nat @ F @ A3 ) @ ( vimage_nat_nat @ F @ B ) ) ) ).
% vimage_Int
thf(fact_5_vimage__Int,axiom,
! [F: a > a,A3: set_a,B: set_a] :
( ( vimage_a_a @ F @ ( inf_inf_set_a @ A3 @ B ) )
= ( inf_inf_set_a @ ( vimage_a_a @ F @ A3 ) @ ( vimage_a_a @ F @ B ) ) ) ).
% vimage_Int
thf(fact_6_vimage__Int,axiom,
! [F: c > a,A3: set_a,B: set_a] :
( ( vimage_c_a @ F @ ( inf_inf_set_a @ A3 @ B ) )
= ( inf_inf_set_c @ ( vimage_c_a @ F @ A3 ) @ ( vimage_c_a @ F @ B ) ) ) ).
% vimage_Int
thf(fact_7_vimage__Int,axiom,
! [F: a > c,A3: set_c,B: set_c] :
( ( vimage_a_c @ F @ ( inf_inf_set_c @ A3 @ B ) )
= ( inf_inf_set_a @ ( vimage_a_c @ F @ A3 ) @ ( vimage_a_c @ F @ B ) ) ) ).
% vimage_Int
thf(fact_8_vimage__Int,axiom,
! [F: c > c,A3: set_c,B: set_c] :
( ( vimage_c_c @ F @ ( inf_inf_set_c @ A3 @ B ) )
= ( inf_inf_set_c @ ( vimage_c_c @ F @ A3 ) @ ( vimage_c_c @ F @ B ) ) ) ).
% vimage_Int
thf(fact_9_vimage__Int,axiom,
! [F: nat > a,A3: set_a,B: set_a] :
( ( vimage_nat_a @ F @ ( inf_inf_set_a @ A3 @ B ) )
= ( inf_inf_set_nat @ ( vimage_nat_a @ F @ A3 ) @ ( vimage_nat_a @ F @ B ) ) ) ).
% vimage_Int
thf(fact_10_vimage__Int,axiom,
! [F: nat > c,A3: set_c,B: set_c] :
( ( vimage_nat_c @ F @ ( inf_inf_set_c @ A3 @ B ) )
= ( inf_inf_set_nat @ ( vimage_nat_c @ F @ A3 ) @ ( vimage_nat_c @ F @ B ) ) ) ).
% vimage_Int
thf(fact_11_vimage__Int,axiom,
! [F: a > nat,A3: set_nat,B: set_nat] :
( ( vimage_a_nat @ F @ ( inf_inf_set_nat @ A3 @ B ) )
= ( inf_inf_set_a @ ( vimage_a_nat @ F @ A3 ) @ ( vimage_a_nat @ F @ B ) ) ) ).
% vimage_Int
thf(fact_12_vimage__Int,axiom,
! [F: c > nat,A3: set_nat,B: set_nat] :
( ( vimage_c_nat @ F @ ( inf_inf_set_nat @ A3 @ B ) )
= ( inf_inf_set_c @ ( vimage_c_nat @ F @ A3 ) @ ( vimage_c_nat @ F @ B ) ) ) ).
% vimage_Int
thf(fact_13_vimage__Int,axiom,
! [F: set_a > a,A3: set_a,B: set_a] :
( ( vimage_set_a_a @ F @ ( inf_inf_set_a @ A3 @ B ) )
= ( inf_inf_set_set_a @ ( vimage_set_a_a @ F @ A3 ) @ ( vimage_set_a_a @ F @ B ) ) ) ).
% vimage_Int
thf(fact_14_Int__subset__iff,axiom,
! [C: set_nat,A3: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C @ ( inf_inf_set_nat @ A3 @ B ) )
= ( ( ord_less_eq_set_nat @ C @ A3 )
& ( ord_less_eq_set_nat @ C @ B ) ) ) ).
% Int_subset_iff
thf(fact_15_Int__subset__iff,axiom,
! [C: set_a,A3: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ ( inf_inf_set_a @ A3 @ B ) )
= ( ( ord_less_eq_set_a @ C @ A3 )
& ( ord_less_eq_set_a @ C @ B ) ) ) ).
% Int_subset_iff
thf(fact_16_Int__subset__iff,axiom,
! [C: set_c,A3: set_c,B: set_c] :
( ( ord_less_eq_set_c @ C @ ( inf_inf_set_c @ A3 @ B ) )
= ( ( ord_less_eq_set_c @ C @ A3 )
& ( ord_less_eq_set_c @ C @ B ) ) ) ).
% Int_subset_iff
thf(fact_17_Int__subset__iff,axiom,
! [C: set_set_a,A3: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C @ ( inf_inf_set_set_a @ A3 @ B ) )
= ( ( ord_le3724670747650509150_set_a @ C @ A3 )
& ( ord_le3724670747650509150_set_a @ C @ B ) ) ) ).
% Int_subset_iff
thf(fact_18_le__inf__iff,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
= ( ( ord_less_eq_set_nat @ X @ Y )
& ( ord_less_eq_set_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_19_le__inf__iff,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( ( ord_less_eq_set_a @ X @ Y )
& ( ord_less_eq_set_a @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_20_le__inf__iff,axiom,
! [X: set_c,Y: set_c,Z: set_c] :
( ( ord_less_eq_set_c @ X @ ( inf_inf_set_c @ Y @ Z ) )
= ( ( ord_less_eq_set_c @ X @ Y )
& ( ord_less_eq_set_c @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_21_le__inf__iff,axiom,
! [X: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ ( inf_inf_set_set_a @ Y @ Z ) )
= ( ( ord_le3724670747650509150_set_a @ X @ Y )
& ( ord_le3724670747650509150_set_a @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_22_le__inf__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_23_le__inf__iff,axiom,
! [X: extended_ereal,Y: extended_ereal,Z: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X @ ( inf_in2794916579150040252_ereal @ Y @ Z ) )
= ( ( ord_le1083603963089353582_ereal @ X @ Y )
& ( ord_le1083603963089353582_ereal @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_24_le__inf__iff,axiom,
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal,Z: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X @ ( inf_in7439215052339218890nnreal @ Y @ Z ) )
= ( ( ord_le3935885782089961368nnreal @ X @ Y )
& ( ord_le3935885782089961368nnreal @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_25_inf_Obounded__iff,axiom,
! [A4: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ ( inf_inf_set_nat @ B2 @ C2 ) )
= ( ( ord_less_eq_set_nat @ A4 @ B2 )
& ( ord_less_eq_set_nat @ A4 @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_26_inf_Obounded__iff,axiom,
! [A4: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A4 @ ( inf_inf_set_a @ B2 @ C2 ) )
= ( ( ord_less_eq_set_a @ A4 @ B2 )
& ( ord_less_eq_set_a @ A4 @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_27_inf_Obounded__iff,axiom,
! [A4: set_c,B2: set_c,C2: set_c] :
( ( ord_less_eq_set_c @ A4 @ ( inf_inf_set_c @ B2 @ C2 ) )
= ( ( ord_less_eq_set_c @ A4 @ B2 )
& ( ord_less_eq_set_c @ A4 @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_28_inf_Obounded__iff,axiom,
! [A4: set_set_a,B2: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ ( inf_inf_set_set_a @ B2 @ C2 ) )
= ( ( ord_le3724670747650509150_set_a @ A4 @ B2 )
& ( ord_le3724670747650509150_set_a @ A4 @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_29_inf_Obounded__iff,axiom,
! [A4: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A4 @ ( inf_inf_nat @ B2 @ C2 ) )
= ( ( ord_less_eq_nat @ A4 @ B2 )
& ( ord_less_eq_nat @ A4 @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_30_inf_Obounded__iff,axiom,
! [A4: extended_ereal,B2: extended_ereal,C2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A4 @ ( inf_in2794916579150040252_ereal @ B2 @ C2 ) )
= ( ( ord_le1083603963089353582_ereal @ A4 @ B2 )
& ( ord_le1083603963089353582_ereal @ A4 @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_31_inf_Obounded__iff,axiom,
! [A4: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A4 @ ( inf_in7439215052339218890nnreal @ B2 @ C2 ) )
= ( ( ord_le3935885782089961368nnreal @ A4 @ B2 )
& ( ord_le3935885782089961368nnreal @ A4 @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_32_vimage__ident,axiom,
! [Y2: set_c] :
( ( vimage_c_c
@ ^ [X2: c] : X2
@ Y2 )
= Y2 ) ).
% vimage_ident
thf(fact_33_vimage__ident,axiom,
! [Y2: set_a] :
( ( vimage_a_a
@ ^ [X2: a] : X2
@ Y2 )
= Y2 ) ).
% vimage_ident
thf(fact_34_vimage__ident,axiom,
! [Y2: set_nat] :
( ( vimage_nat_nat
@ ^ [X2: nat] : X2
@ Y2 )
= Y2 ) ).
% vimage_ident
thf(fact_35_vimage__Collect__eq,axiom,
! [F: a > c,P: c > $o] :
( ( vimage_a_c @ F @ ( collect_c @ P ) )
= ( collect_a
@ ^ [Y3: a] : ( P @ ( F @ Y3 ) ) ) ) ).
% vimage_Collect_eq
thf(fact_36_vimage__Collect__eq,axiom,
! [F: c > c,P: c > $o] :
( ( vimage_c_c @ F @ ( collect_c @ P ) )
= ( collect_c
@ ^ [Y3: c] : ( P @ ( F @ Y3 ) ) ) ) ).
% vimage_Collect_eq
thf(fact_37_vimage__Collect__eq,axiom,
! [F: c > a,P: a > $o] :
( ( vimage_c_a @ F @ ( collect_a @ P ) )
= ( collect_c
@ ^ [Y3: c] : ( P @ ( F @ Y3 ) ) ) ) ).
% vimage_Collect_eq
thf(fact_38_vimage__Collect__eq,axiom,
! [F: a > a,P: a > $o] :
( ( vimage_a_a @ F @ ( collect_a @ P ) )
= ( collect_a
@ ^ [Y3: a] : ( P @ ( F @ Y3 ) ) ) ) ).
% vimage_Collect_eq
thf(fact_39_vimage__Collect__eq,axiom,
! [F: set_a > set_a,P: set_a > $o] :
( ( vimage_set_a_set_a @ F @ ( collect_set_a @ P ) )
= ( collect_set_a
@ ^ [Y3: set_a] : ( P @ ( F @ Y3 ) ) ) ) ).
% vimage_Collect_eq
thf(fact_40_vimage__Collect__eq,axiom,
! [F: nat > set_a,P: set_a > $o] :
( ( vimage_nat_set_a @ F @ ( collect_set_a @ P ) )
= ( collect_nat
@ ^ [Y3: nat] : ( P @ ( F @ Y3 ) ) ) ) ).
% vimage_Collect_eq
thf(fact_41_vimage__Collect__eq,axiom,
! [F: set_a > nat,P: nat > $o] :
( ( vimage_set_a_nat @ F @ ( collect_nat @ P ) )
= ( collect_set_a
@ ^ [Y3: set_a] : ( P @ ( F @ Y3 ) ) ) ) ).
% vimage_Collect_eq
thf(fact_42_vimage__Collect__eq,axiom,
! [F: nat > nat,P: nat > $o] :
( ( vimage_nat_nat @ F @ ( collect_nat @ P ) )
= ( collect_nat
@ ^ [Y3: nat] : ( P @ ( F @ Y3 ) ) ) ) ).
% vimage_Collect_eq
thf(fact_43_sigma__sets__pow,axiom,
! [I2: b] :
( ( member_b @ I2 @ i )
=> ( ( sigma_sigma_sets_c @ ( image_a_c @ ( x @ I2 ) @ ( probab49036049091589825_pmf_a @ p ) ) @ ( g @ I2 ) )
= ( pow_c @ ( image_a_c @ ( x @ I2 ) @ ( probab49036049091589825_pmf_a @ p ) ) ) ) ) ).
% sigma_sets_pow
thf(fact_44_image__vimage__subset,axiom,
! [F: nat > nat,A3: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( vimage_nat_nat @ F @ A3 ) ) @ A3 ) ).
% image_vimage_subset
thf(fact_45_image__vimage__subset,axiom,
! [F: a > c,A3: set_c] : ( ord_less_eq_set_c @ ( image_a_c @ F @ ( vimage_a_c @ F @ A3 ) ) @ A3 ) ).
% image_vimage_subset
thf(fact_46_image__vimage__subset,axiom,
! [F: c > c,A3: set_c] : ( ord_less_eq_set_c @ ( image_c_c @ F @ ( vimage_c_c @ F @ A3 ) ) @ A3 ) ).
% image_vimage_subset
thf(fact_47_image__vimage__subset,axiom,
! [F: c > a,A3: set_a] : ( ord_less_eq_set_a @ ( image_c_a @ F @ ( vimage_c_a @ F @ A3 ) ) @ A3 ) ).
% image_vimage_subset
thf(fact_48_image__vimage__subset,axiom,
! [F: a > a,A3: set_a] : ( ord_less_eq_set_a @ ( image_a_a @ F @ ( vimage_a_a @ F @ A3 ) ) @ A3 ) ).
% image_vimage_subset
thf(fact_49_image__subset__iff__subset__vimage,axiom,
! [F: nat > nat,A3: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A3 ) @ B )
= ( ord_less_eq_set_nat @ A3 @ ( vimage_nat_nat @ F @ B ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_50_image__subset__iff__subset__vimage,axiom,
! [F: c > c,A3: set_c,B: set_c] :
( ( ord_less_eq_set_c @ ( image_c_c @ F @ A3 ) @ B )
= ( ord_less_eq_set_c @ A3 @ ( vimage_c_c @ F @ B ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_51_image__subset__iff__subset__vimage,axiom,
! [F: set_a > c,A3: set_set_a,B: set_c] :
( ( ord_less_eq_set_c @ ( image_set_a_c @ F @ A3 ) @ B )
= ( ord_le3724670747650509150_set_a @ A3 @ ( vimage_set_a_c @ F @ B ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_52_image__subset__iff__subset__vimage,axiom,
! [F: a > c,A3: set_a,B: set_c] :
( ( ord_less_eq_set_c @ ( image_a_c @ F @ A3 ) @ B )
= ( ord_less_eq_set_a @ A3 @ ( vimage_a_c @ F @ B ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_53_image__subset__iff__subset__vimage,axiom,
! [F: c > set_a,A3: set_c,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_c_set_a @ F @ A3 ) @ B )
= ( ord_less_eq_set_c @ A3 @ ( vimage_c_set_a @ F @ B ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_54_image__subset__iff__subset__vimage,axiom,
! [F: set_a > set_a,A3: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ F @ A3 ) @ B )
= ( ord_le3724670747650509150_set_a @ A3 @ ( vimage_set_a_set_a @ F @ B ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_55_image__subset__iff__subset__vimage,axiom,
! [F: a > set_a,A3: set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A3 ) @ B )
= ( ord_less_eq_set_a @ A3 @ ( vimage_a_set_a @ F @ B ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_56_image__subset__iff__subset__vimage,axiom,
! [F: c > a,A3: set_c,B: set_a] :
( ( ord_less_eq_set_a @ ( image_c_a @ F @ A3 ) @ B )
= ( ord_less_eq_set_c @ A3 @ ( vimage_c_a @ F @ B ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_57_image__subset__iff__subset__vimage,axiom,
! [F: set_a > a,A3: set_set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( image_set_a_a @ F @ A3 ) @ B )
= ( ord_le3724670747650509150_set_a @ A3 @ ( vimage_set_a_a @ F @ B ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_58_image__subset__iff__subset__vimage,axiom,
! [F: a > a,A3: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F @ A3 ) @ B )
= ( ord_less_eq_set_a @ A3 @ ( vimage_a_a @ F @ B ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_59_image__Int__subset,axiom,
! [F: a > nat,A3: set_a,B: set_a] : ( ord_less_eq_set_nat @ ( image_a_nat @ F @ ( inf_inf_set_a @ A3 @ B ) ) @ ( inf_inf_set_nat @ ( image_a_nat @ F @ A3 ) @ ( image_a_nat @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_60_image__Int__subset,axiom,
! [F: c > nat,A3: set_c,B: set_c] : ( ord_less_eq_set_nat @ ( image_c_nat @ F @ ( inf_inf_set_c @ A3 @ B ) ) @ ( inf_inf_set_nat @ ( image_c_nat @ F @ A3 ) @ ( image_c_nat @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_61_image__Int__subset,axiom,
! [F: nat > nat,A3: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( inf_inf_set_nat @ A3 @ B ) ) @ ( inf_inf_set_nat @ ( image_nat_nat @ F @ A3 ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_62_image__Int__subset,axiom,
! [F: a > c,A3: set_a,B: set_a] : ( ord_less_eq_set_c @ ( image_a_c @ F @ ( inf_inf_set_a @ A3 @ B ) ) @ ( inf_inf_set_c @ ( image_a_c @ F @ A3 ) @ ( image_a_c @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_63_image__Int__subset,axiom,
! [F: c > c,A3: set_c,B: set_c] : ( ord_less_eq_set_c @ ( image_c_c @ F @ ( inf_inf_set_c @ A3 @ B ) ) @ ( inf_inf_set_c @ ( image_c_c @ F @ A3 ) @ ( image_c_c @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_64_image__Int__subset,axiom,
! [F: nat > c,A3: set_nat,B: set_nat] : ( ord_less_eq_set_c @ ( image_nat_c @ F @ ( inf_inf_set_nat @ A3 @ B ) ) @ ( inf_inf_set_c @ ( image_nat_c @ F @ A3 ) @ ( image_nat_c @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_65_image__Int__subset,axiom,
! [F: a > a,A3: set_a,B: set_a] : ( ord_less_eq_set_a @ ( image_a_a @ F @ ( inf_inf_set_a @ A3 @ B ) ) @ ( inf_inf_set_a @ ( image_a_a @ F @ A3 ) @ ( image_a_a @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_66_image__Int__subset,axiom,
! [F: c > a,A3: set_c,B: set_c] : ( ord_less_eq_set_a @ ( image_c_a @ F @ ( inf_inf_set_c @ A3 @ B ) ) @ ( inf_inf_set_a @ ( image_c_a @ F @ A3 ) @ ( image_c_a @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_67_image__Int__subset,axiom,
! [F: nat > a,A3: set_nat,B: set_nat] : ( ord_less_eq_set_a @ ( image_nat_a @ F @ ( inf_inf_set_nat @ A3 @ B ) ) @ ( inf_inf_set_a @ ( image_nat_a @ F @ A3 ) @ ( image_nat_a @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_68_image__Int__subset,axiom,
! [F: set_a > nat,A3: set_set_a,B: set_set_a] : ( ord_less_eq_set_nat @ ( image_set_a_nat @ F @ ( inf_inf_set_set_a @ A3 @ B ) ) @ ( inf_inf_set_nat @ ( image_set_a_nat @ F @ A3 ) @ ( image_set_a_nat @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_69_image__eqI,axiom,
! [B2: c,F: a > c,X: a,A3: set_a] :
( ( B2
= ( F @ X ) )
=> ( ( member_a @ X @ A3 )
=> ( member_c @ B2 @ ( image_a_c @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_70_image__eqI,axiom,
! [B2: b,F: b > b,X: b,A3: set_b] :
( ( B2
= ( F @ X ) )
=> ( ( member_b @ X @ A3 )
=> ( member_b @ B2 @ ( image_b_b @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_71_image__eqI,axiom,
! [B2: $o,F: b > $o,X: b,A3: set_b] :
( ( B2
= ( F @ X ) )
=> ( ( member_b @ X @ A3 )
=> ( member_o @ B2 @ ( image_b_o @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_72_image__eqI,axiom,
! [B2: nat,F: b > nat,X: b,A3: set_b] :
( ( B2
= ( F @ X ) )
=> ( ( member_b @ X @ A3 )
=> ( member_nat @ B2 @ ( image_b_nat @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_73_image__eqI,axiom,
! [B2: b,F: $o > b,X: $o,A3: set_o] :
( ( B2
= ( F @ X ) )
=> ( ( member_o @ X @ A3 )
=> ( member_b @ B2 @ ( image_o_b @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_74_image__eqI,axiom,
! [B2: $o,F: $o > $o,X: $o,A3: set_o] :
( ( B2
= ( F @ X ) )
=> ( ( member_o @ X @ A3 )
=> ( member_o @ B2 @ ( image_o_o @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_75_image__eqI,axiom,
! [B2: nat,F: $o > nat,X: $o,A3: set_o] :
( ( B2
= ( F @ X ) )
=> ( ( member_o @ X @ A3 )
=> ( member_nat @ B2 @ ( image_o_nat @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_76_image__eqI,axiom,
! [B2: b,F: nat > b,X: nat,A3: set_nat] :
( ( B2
= ( F @ X ) )
=> ( ( member_nat @ X @ A3 )
=> ( member_b @ B2 @ ( image_nat_b @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_77_image__eqI,axiom,
! [B2: $o,F: nat > $o,X: nat,A3: set_nat] :
( ( B2
= ( F @ X ) )
=> ( ( member_nat @ X @ A3 )
=> ( member_o @ B2 @ ( image_nat_o @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_78_image__eqI,axiom,
! [B2: nat,F: nat > nat,X: nat,A3: set_nat] :
( ( B2
= ( F @ X ) )
=> ( ( member_nat @ X @ A3 )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_79_subset__antisym,axiom,
! [A3: set_c,B: set_c] :
( ( ord_less_eq_set_c @ A3 @ B )
=> ( ( ord_less_eq_set_c @ B @ A3 )
=> ( A3 = B ) ) ) ).
% subset_antisym
thf(fact_80_subset__antisym,axiom,
! [A3: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ A3 )
=> ( A3 = B ) ) ) ).
% subset_antisym
thf(fact_81_subset__antisym,axiom,
! [A3: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A3 @ B )
=> ( ( ord_less_eq_set_a @ B @ A3 )
=> ( A3 = B ) ) ) ).
% subset_antisym
thf(fact_82_subsetI,axiom,
! [A3: set_set_c,B: set_set_c] :
( ! [X3: set_c] :
( ( member_set_c @ X3 @ A3 )
=> ( member_set_c @ X3 @ B ) )
=> ( ord_le3866738827743201120_set_c @ A3 @ B ) ) ).
% subsetI
thf(fact_83_subsetI,axiom,
! [A3: set_b,B: set_b] :
( ! [X3: b] :
( ( member_b @ X3 @ A3 )
=> ( member_b @ X3 @ B ) )
=> ( ord_less_eq_set_b @ A3 @ B ) ) ).
% subsetI
thf(fact_84_subsetI,axiom,
! [A3: set_o,B: set_o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A3 )
=> ( member_o @ X3 @ B ) )
=> ( ord_less_eq_set_o @ A3 @ B ) ) ).
% subsetI
thf(fact_85_subsetI,axiom,
! [A3: set_nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( member_nat @ X3 @ B ) )
=> ( ord_less_eq_set_nat @ A3 @ B ) ) ).
% subsetI
thf(fact_86_subsetI,axiom,
! [A3: set_c,B: set_c] :
( ! [X3: c] :
( ( member_c @ X3 @ A3 )
=> ( member_c @ X3 @ B ) )
=> ( ord_less_eq_set_c @ A3 @ B ) ) ).
% subsetI
thf(fact_87_subsetI,axiom,
! [A3: set_set_a,B: set_set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A3 )
=> ( member_set_a @ X3 @ B ) )
=> ( ord_le3724670747650509150_set_a @ A3 @ B ) ) ).
% subsetI
thf(fact_88_subsetI,axiom,
! [A3: set_a,B: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ( member_a @ X3 @ B ) )
=> ( ord_less_eq_set_a @ A3 @ B ) ) ).
% subsetI
thf(fact_89_inf__right__idem,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y )
= ( inf_inf_set_a @ X @ Y ) ) ).
% inf_right_idem
thf(fact_90_inf__right__idem,axiom,
! [X: set_c,Y: set_c] :
( ( inf_inf_set_c @ ( inf_inf_set_c @ X @ Y ) @ Y )
= ( inf_inf_set_c @ X @ Y ) ) ).
% inf_right_idem
thf(fact_91_inf__right__idem,axiom,
! [X: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Y )
= ( inf_inf_set_nat @ X @ Y ) ) ).
% inf_right_idem
thf(fact_92_inf__right__idem,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ X @ Y ) @ Y )
= ( inf_inf_set_set_a @ X @ Y ) ) ).
% inf_right_idem
thf(fact_93_inf__right__idem,axiom,
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( inf_in7439215052339218890nnreal @ ( inf_in7439215052339218890nnreal @ X @ Y ) @ Y )
= ( inf_in7439215052339218890nnreal @ X @ Y ) ) ).
% inf_right_idem
thf(fact_94_inf__right__idem,axiom,
! [X: extended_ereal,Y: extended_ereal] :
( ( inf_in2794916579150040252_ereal @ ( inf_in2794916579150040252_ereal @ X @ Y ) @ Y )
= ( inf_in2794916579150040252_ereal @ X @ Y ) ) ).
% inf_right_idem
thf(fact_95_inf__right__idem,axiom,
! [X: nat,Y: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ X @ Y ) @ Y )
= ( inf_inf_nat @ X @ Y ) ) ).
% inf_right_idem
thf(fact_96_inf_Oright__idem,axiom,
! [A4: set_a,B2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A4 @ B2 ) @ B2 )
= ( inf_inf_set_a @ A4 @ B2 ) ) ).
% inf.right_idem
thf(fact_97_inf_Oright__idem,axiom,
! [A4: set_c,B2: set_c] :
( ( inf_inf_set_c @ ( inf_inf_set_c @ A4 @ B2 ) @ B2 )
= ( inf_inf_set_c @ A4 @ B2 ) ) ).
% inf.right_idem
thf(fact_98_inf_Oright__idem,axiom,
! [A4: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A4 @ B2 ) @ B2 )
= ( inf_inf_set_nat @ A4 @ B2 ) ) ).
% inf.right_idem
thf(fact_99_inf_Oright__idem,axiom,
! [A4: set_set_a,B2: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ A4 @ B2 ) @ B2 )
= ( inf_inf_set_set_a @ A4 @ B2 ) ) ).
% inf.right_idem
thf(fact_100_inf_Oright__idem,axiom,
! [A4: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( inf_in7439215052339218890nnreal @ ( inf_in7439215052339218890nnreal @ A4 @ B2 ) @ B2 )
= ( inf_in7439215052339218890nnreal @ A4 @ B2 ) ) ).
% inf.right_idem
thf(fact_101_inf_Oright__idem,axiom,
! [A4: extended_ereal,B2: extended_ereal] :
( ( inf_in2794916579150040252_ereal @ ( inf_in2794916579150040252_ereal @ A4 @ B2 ) @ B2 )
= ( inf_in2794916579150040252_ereal @ A4 @ B2 ) ) ).
% inf.right_idem
thf(fact_102_inf_Oright__idem,axiom,
! [A4: nat,B2: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ A4 @ B2 ) @ B2 )
= ( inf_inf_nat @ A4 @ B2 ) ) ).
% inf.right_idem
thf(fact_103_inf__left__idem,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
= ( inf_inf_set_a @ X @ Y ) ) ).
% inf_left_idem
thf(fact_104_inf__left__idem,axiom,
! [X: set_c,Y: set_c] :
( ( inf_inf_set_c @ X @ ( inf_inf_set_c @ X @ Y ) )
= ( inf_inf_set_c @ X @ Y ) ) ).
% inf_left_idem
thf(fact_105_inf__left__idem,axiom,
! [X: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ X @ Y ) )
= ( inf_inf_set_nat @ X @ Y ) ) ).
% inf_left_idem
thf(fact_106_inf__left__idem,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( inf_inf_set_set_a @ X @ ( inf_inf_set_set_a @ X @ Y ) )
= ( inf_inf_set_set_a @ X @ Y ) ) ).
% inf_left_idem
thf(fact_107_inf__left__idem,axiom,
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( inf_in7439215052339218890nnreal @ X @ ( inf_in7439215052339218890nnreal @ X @ Y ) )
= ( inf_in7439215052339218890nnreal @ X @ Y ) ) ).
% inf_left_idem
thf(fact_108_inf__left__idem,axiom,
! [X: extended_ereal,Y: extended_ereal] :
( ( inf_in2794916579150040252_ereal @ X @ ( inf_in2794916579150040252_ereal @ X @ Y ) )
= ( inf_in2794916579150040252_ereal @ X @ Y ) ) ).
% inf_left_idem
thf(fact_109_inf__left__idem,axiom,
! [X: nat,Y: nat] :
( ( inf_inf_nat @ X @ ( inf_inf_nat @ X @ Y ) )
= ( inf_inf_nat @ X @ Y ) ) ).
% inf_left_idem
thf(fact_110_inf_Oleft__idem,axiom,
! [A4: set_a,B2: set_a] :
( ( inf_inf_set_a @ A4 @ ( inf_inf_set_a @ A4 @ B2 ) )
= ( inf_inf_set_a @ A4 @ B2 ) ) ).
% inf.left_idem
thf(fact_111_inf_Oleft__idem,axiom,
! [A4: set_c,B2: set_c] :
( ( inf_inf_set_c @ A4 @ ( inf_inf_set_c @ A4 @ B2 ) )
= ( inf_inf_set_c @ A4 @ B2 ) ) ).
% inf.left_idem
thf(fact_112_inf_Oleft__idem,axiom,
! [A4: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ A4 @ ( inf_inf_set_nat @ A4 @ B2 ) )
= ( inf_inf_set_nat @ A4 @ B2 ) ) ).
% inf.left_idem
thf(fact_113_inf_Oleft__idem,axiom,
! [A4: set_set_a,B2: set_set_a] :
( ( inf_inf_set_set_a @ A4 @ ( inf_inf_set_set_a @ A4 @ B2 ) )
= ( inf_inf_set_set_a @ A4 @ B2 ) ) ).
% inf.left_idem
thf(fact_114_inf_Oleft__idem,axiom,
! [A4: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( inf_in7439215052339218890nnreal @ A4 @ ( inf_in7439215052339218890nnreal @ A4 @ B2 ) )
= ( inf_in7439215052339218890nnreal @ A4 @ B2 ) ) ).
% inf.left_idem
thf(fact_115_inf_Oleft__idem,axiom,
! [A4: extended_ereal,B2: extended_ereal] :
( ( inf_in2794916579150040252_ereal @ A4 @ ( inf_in2794916579150040252_ereal @ A4 @ B2 ) )
= ( inf_in2794916579150040252_ereal @ A4 @ B2 ) ) ).
% inf.left_idem
thf(fact_116_inf_Oleft__idem,axiom,
! [A4: nat,B2: nat] :
( ( inf_inf_nat @ A4 @ ( inf_inf_nat @ A4 @ B2 ) )
= ( inf_inf_nat @ A4 @ B2 ) ) ).
% inf.left_idem
thf(fact_117_inf__idem,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ X )
= X ) ).
% inf_idem
thf(fact_118_inf__idem,axiom,
! [X: set_c] :
( ( inf_inf_set_c @ X @ X )
= X ) ).
% inf_idem
thf(fact_119_inf__idem,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ X )
= X ) ).
% inf_idem
thf(fact_120_inf__idem,axiom,
! [X: set_set_a] :
( ( inf_inf_set_set_a @ X @ X )
= X ) ).
% inf_idem
thf(fact_121_inf__idem,axiom,
! [X: extend8495563244428889912nnreal] :
( ( inf_in7439215052339218890nnreal @ X @ X )
= X ) ).
% inf_idem
thf(fact_122_inf__idem,axiom,
! [X: extended_ereal] :
( ( inf_in2794916579150040252_ereal @ X @ X )
= X ) ).
% inf_idem
thf(fact_123_inf__idem,axiom,
! [X: nat] :
( ( inf_inf_nat @ X @ X )
= X ) ).
% inf_idem
thf(fact_124_inf_Oidem,axiom,
! [A4: set_a] :
( ( inf_inf_set_a @ A4 @ A4 )
= A4 ) ).
% inf.idem
thf(fact_125_inf_Oidem,axiom,
! [A4: set_c] :
( ( inf_inf_set_c @ A4 @ A4 )
= A4 ) ).
% inf.idem
thf(fact_126_inf_Oidem,axiom,
! [A4: set_nat] :
( ( inf_inf_set_nat @ A4 @ A4 )
= A4 ) ).
% inf.idem
thf(fact_127_inf_Oidem,axiom,
! [A4: set_set_a] :
( ( inf_inf_set_set_a @ A4 @ A4 )
= A4 ) ).
% inf.idem
thf(fact_128_inf_Oidem,axiom,
! [A4: extend8495563244428889912nnreal] :
( ( inf_in7439215052339218890nnreal @ A4 @ A4 )
= A4 ) ).
% inf.idem
thf(fact_129_inf_Oidem,axiom,
! [A4: extended_ereal] :
( ( inf_in2794916579150040252_ereal @ A4 @ A4 )
= A4 ) ).
% inf.idem
thf(fact_130_inf_Oidem,axiom,
! [A4: nat] :
( ( inf_inf_nat @ A4 @ A4 )
= A4 ) ).
% inf.idem
thf(fact_131_Int__iff,axiom,
! [C2: set_c,A3: set_set_c,B: set_set_c] :
( ( member_set_c @ C2 @ ( inf_inf_set_set_c @ A3 @ B ) )
= ( ( member_set_c @ C2 @ A3 )
& ( member_set_c @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_132_Int__iff,axiom,
! [C2: b,A3: set_b,B: set_b] :
( ( member_b @ C2 @ ( inf_inf_set_b @ A3 @ B ) )
= ( ( member_b @ C2 @ A3 )
& ( member_b @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_133_Int__iff,axiom,
! [C2: $o,A3: set_o,B: set_o] :
( ( member_o @ C2 @ ( inf_inf_set_o @ A3 @ B ) )
= ( ( member_o @ C2 @ A3 )
& ( member_o @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_134_Int__iff,axiom,
! [C2: a,A3: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A3 @ B ) )
= ( ( member_a @ C2 @ A3 )
& ( member_a @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_135_Int__iff,axiom,
! [C2: c,A3: set_c,B: set_c] :
( ( member_c @ C2 @ ( inf_inf_set_c @ A3 @ B ) )
= ( ( member_c @ C2 @ A3 )
& ( member_c @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_136_Int__iff,axiom,
! [C2: nat,A3: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A3 @ B ) )
= ( ( member_nat @ C2 @ A3 )
& ( member_nat @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_137_Int__iff,axiom,
! [C2: set_a,A3: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A3 @ B ) )
= ( ( member_set_a @ C2 @ A3 )
& ( member_set_a @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_138_IntI,axiom,
! [C2: set_c,A3: set_set_c,B: set_set_c] :
( ( member_set_c @ C2 @ A3 )
=> ( ( member_set_c @ C2 @ B )
=> ( member_set_c @ C2 @ ( inf_inf_set_set_c @ A3 @ B ) ) ) ) ).
% IntI
thf(fact_139_IntI,axiom,
! [C2: b,A3: set_b,B: set_b] :
( ( member_b @ C2 @ A3 )
=> ( ( member_b @ C2 @ B )
=> ( member_b @ C2 @ ( inf_inf_set_b @ A3 @ B ) ) ) ) ).
% IntI
thf(fact_140_IntI,axiom,
! [C2: $o,A3: set_o,B: set_o] :
( ( member_o @ C2 @ A3 )
=> ( ( member_o @ C2 @ B )
=> ( member_o @ C2 @ ( inf_inf_set_o @ A3 @ B ) ) ) ) ).
% IntI
thf(fact_141_IntI,axiom,
! [C2: a,A3: set_a,B: set_a] :
( ( member_a @ C2 @ A3 )
=> ( ( member_a @ C2 @ B )
=> ( member_a @ C2 @ ( inf_inf_set_a @ A3 @ B ) ) ) ) ).
% IntI
thf(fact_142_IntI,axiom,
! [C2: c,A3: set_c,B: set_c] :
( ( member_c @ C2 @ A3 )
=> ( ( member_c @ C2 @ B )
=> ( member_c @ C2 @ ( inf_inf_set_c @ A3 @ B ) ) ) ) ).
% IntI
thf(fact_143_IntI,axiom,
! [C2: nat,A3: set_nat,B: set_nat] :
( ( member_nat @ C2 @ A3 )
=> ( ( member_nat @ C2 @ B )
=> ( member_nat @ C2 @ ( inf_inf_set_nat @ A3 @ B ) ) ) ) ).
% IntI
thf(fact_144_IntI,axiom,
! [C2: set_a,A3: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ A3 )
=> ( ( member_set_a @ C2 @ B )
=> ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A3 @ B ) ) ) ) ).
% IntI
thf(fact_145_vimage__eq,axiom,
! [A4: b,F: b > b,B: set_b] :
( ( member_b @ A4 @ ( vimage_b_b @ F @ B ) )
= ( member_b @ ( F @ A4 ) @ B ) ) ).
% vimage_eq
thf(fact_146_vimage__eq,axiom,
! [A4: b,F: b > $o,B: set_o] :
( ( member_b @ A4 @ ( vimage_b_o @ F @ B ) )
= ( member_o @ ( F @ A4 ) @ B ) ) ).
% vimage_eq
thf(fact_147_vimage__eq,axiom,
! [A4: b,F: b > nat,B: set_nat] :
( ( member_b @ A4 @ ( vimage_b_nat @ F @ B ) )
= ( member_nat @ ( F @ A4 ) @ B ) ) ).
% vimage_eq
thf(fact_148_vimage__eq,axiom,
! [A4: $o,F: $o > b,B: set_b] :
( ( member_o @ A4 @ ( vimage_o_b @ F @ B ) )
= ( member_b @ ( F @ A4 ) @ B ) ) ).
% vimage_eq
thf(fact_149_vimage__eq,axiom,
! [A4: $o,F: $o > $o,B: set_o] :
( ( member_o @ A4 @ ( vimage_o_o @ F @ B ) )
= ( member_o @ ( F @ A4 ) @ B ) ) ).
% vimage_eq
thf(fact_150_vimage__eq,axiom,
! [A4: $o,F: $o > nat,B: set_nat] :
( ( member_o @ A4 @ ( vimage_o_nat @ F @ B ) )
= ( member_nat @ ( F @ A4 ) @ B ) ) ).
% vimage_eq
thf(fact_151_vimage__eq,axiom,
! [A4: nat,F: nat > b,B: set_b] :
( ( member_nat @ A4 @ ( vimage_nat_b @ F @ B ) )
= ( member_b @ ( F @ A4 ) @ B ) ) ).
% vimage_eq
thf(fact_152_vimage__eq,axiom,
! [A4: nat,F: nat > $o,B: set_o] :
( ( member_nat @ A4 @ ( vimage_nat_o @ F @ B ) )
= ( member_o @ ( F @ A4 ) @ B ) ) ).
% vimage_eq
thf(fact_153_vimage__eq,axiom,
! [A4: a,F: a > c,B: set_c] :
( ( member_a @ A4 @ ( vimage_a_c @ F @ B ) )
= ( member_c @ ( F @ A4 ) @ B ) ) ).
% vimage_eq
thf(fact_154_vimage__eq,axiom,
! [A4: nat,F: nat > nat,B: set_nat] :
( ( member_nat @ A4 @ ( vimage_nat_nat @ F @ B ) )
= ( member_nat @ ( F @ A4 ) @ B ) ) ).
% vimage_eq
thf(fact_155_vimageI,axiom,
! [F: b > b,A4: b,B2: b,B: set_b] :
( ( ( F @ A4 )
= B2 )
=> ( ( member_b @ B2 @ B )
=> ( member_b @ A4 @ ( vimage_b_b @ F @ B ) ) ) ) ).
% vimageI
thf(fact_156_vimageI,axiom,
! [F: $o > b,A4: $o,B2: b,B: set_b] :
( ( ( F @ A4 )
= B2 )
=> ( ( member_b @ B2 @ B )
=> ( member_o @ A4 @ ( vimage_o_b @ F @ B ) ) ) ) ).
% vimageI
thf(fact_157_vimageI,axiom,
! [F: nat > b,A4: nat,B2: b,B: set_b] :
( ( ( F @ A4 )
= B2 )
=> ( ( member_b @ B2 @ B )
=> ( member_nat @ A4 @ ( vimage_nat_b @ F @ B ) ) ) ) ).
% vimageI
thf(fact_158_vimageI,axiom,
! [F: b > $o,A4: b,B2: $o,B: set_o] :
( ( ( F @ A4 )
= B2 )
=> ( ( member_o @ B2 @ B )
=> ( member_b @ A4 @ ( vimage_b_o @ F @ B ) ) ) ) ).
% vimageI
thf(fact_159_vimageI,axiom,
! [F: $o > $o,A4: $o,B2: $o,B: set_o] :
( ( ( F @ A4 )
= B2 )
=> ( ( member_o @ B2 @ B )
=> ( member_o @ A4 @ ( vimage_o_o @ F @ B ) ) ) ) ).
% vimageI
thf(fact_160_vimageI,axiom,
! [F: nat > $o,A4: nat,B2: $o,B: set_o] :
( ( ( F @ A4 )
= B2 )
=> ( ( member_o @ B2 @ B )
=> ( member_nat @ A4 @ ( vimage_nat_o @ F @ B ) ) ) ) ).
% vimageI
thf(fact_161_vimageI,axiom,
! [F: b > nat,A4: b,B2: nat,B: set_nat] :
( ( ( F @ A4 )
= B2 )
=> ( ( member_nat @ B2 @ B )
=> ( member_b @ A4 @ ( vimage_b_nat @ F @ B ) ) ) ) ).
% vimageI
thf(fact_162_vimageI,axiom,
! [F: $o > nat,A4: $o,B2: nat,B: set_nat] :
( ( ( F @ A4 )
= B2 )
=> ( ( member_nat @ B2 @ B )
=> ( member_o @ A4 @ ( vimage_o_nat @ F @ B ) ) ) ) ).
% vimageI
thf(fact_163_vimageI,axiom,
! [F: a > c,A4: a,B2: c,B: set_c] :
( ( ( F @ A4 )
= B2 )
=> ( ( member_c @ B2 @ B )
=> ( member_a @ A4 @ ( vimage_a_c @ F @ B ) ) ) ) ).
% vimageI
thf(fact_164_vimageI,axiom,
! [F: nat > nat,A4: nat,B2: nat,B: set_nat] :
( ( ( F @ A4 )
= B2 )
=> ( ( member_nat @ B2 @ B )
=> ( member_nat @ A4 @ ( vimage_nat_nat @ F @ B ) ) ) ) ).
% vimageI
thf(fact_165_Pow__iff,axiom,
! [A3: set_c,B: set_c] :
( ( member_set_c @ A3 @ ( pow_c @ B ) )
= ( ord_less_eq_set_c @ A3 @ B ) ) ).
% Pow_iff
thf(fact_166_Pow__iff,axiom,
! [A3: set_set_a,B: set_set_a] :
( ( member_set_set_a @ A3 @ ( pow_set_a @ B ) )
= ( ord_le3724670747650509150_set_a @ A3 @ B ) ) ).
% Pow_iff
thf(fact_167_Pow__iff,axiom,
! [A3: set_a,B: set_a] :
( ( member_set_a @ A3 @ ( pow_a @ B ) )
= ( ord_less_eq_set_a @ A3 @ B ) ) ).
% Pow_iff
thf(fact_168_PowI,axiom,
! [A3: set_c,B: set_c] :
( ( ord_less_eq_set_c @ A3 @ B )
=> ( member_set_c @ A3 @ ( pow_c @ B ) ) ) ).
% PowI
thf(fact_169_PowI,axiom,
! [A3: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B )
=> ( member_set_set_a @ A3 @ ( pow_set_a @ B ) ) ) ).
% PowI
thf(fact_170_PowI,axiom,
! [A3: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A3 @ B )
=> ( member_set_a @ A3 @ ( pow_a @ B ) ) ) ).
% PowI
thf(fact_171_Pow__Int__eq,axiom,
! [A3: set_a,B: set_a] :
( ( pow_a @ ( inf_inf_set_a @ A3 @ B ) )
= ( inf_inf_set_set_a @ ( pow_a @ A3 ) @ ( pow_a @ B ) ) ) ).
% Pow_Int_eq
thf(fact_172_Pow__Int__eq,axiom,
! [A3: set_c,B: set_c] :
( ( pow_c @ ( inf_inf_set_c @ A3 @ B ) )
= ( inf_inf_set_set_c @ ( pow_c @ A3 ) @ ( pow_c @ B ) ) ) ).
% Pow_Int_eq
thf(fact_173_Pow__Int__eq,axiom,
! [A3: set_nat,B: set_nat] :
( ( pow_nat @ ( inf_inf_set_nat @ A3 @ B ) )
= ( inf_inf_set_set_nat @ ( pow_nat @ A3 ) @ ( pow_nat @ B ) ) ) ).
% Pow_Int_eq
thf(fact_174_Pow__Int__eq,axiom,
! [A3: set_set_a,B: set_set_a] :
( ( pow_set_a @ ( inf_inf_set_set_a @ A3 @ B ) )
= ( inf_in1205276777018777868_set_a @ ( pow_set_a @ A3 ) @ ( pow_set_a @ B ) ) ) ).
% Pow_Int_eq
thf(fact_175_Pow__top,axiom,
! [A3: set_a] : ( member_set_a @ A3 @ ( pow_a @ A3 ) ) ).
% Pow_top
thf(fact_176_Pow__top,axiom,
! [A3: set_c] : ( member_set_c @ A3 @ ( pow_c @ A3 ) ) ).
% Pow_top
thf(fact_177_Cantors__paradox,axiom,
! [A3: set_c] :
~ ? [F2: c > set_c] :
( ( image_c_set_c @ F2 @ A3 )
= ( pow_c @ A3 ) ) ).
% Cantors_paradox
thf(fact_178_Cantors__paradox,axiom,
! [A3: set_a] :
~ ? [F2: a > set_a] :
( ( image_a_set_a @ F2 @ A3 )
= ( pow_a @ A3 ) ) ).
% Cantors_paradox
thf(fact_179_image__Pow__surj,axiom,
! [F: c > c,A3: set_c,B: set_c] :
( ( ( image_c_c @ F @ A3 )
= B )
=> ( ( image_set_c_set_c @ ( image_c_c @ F ) @ ( pow_c @ A3 ) )
= ( pow_c @ B ) ) ) ).
% image_Pow_surj
thf(fact_180_image__Pow__surj,axiom,
! [F: c > a,A3: set_c,B: set_a] :
( ( ( image_c_a @ F @ A3 )
= B )
=> ( ( image_set_c_set_a @ ( image_c_a @ F ) @ ( pow_c @ A3 ) )
= ( pow_a @ B ) ) ) ).
% image_Pow_surj
thf(fact_181_image__Pow__surj,axiom,
! [F: a > a,A3: set_a,B: set_a] :
( ( ( image_a_a @ F @ A3 )
= B )
=> ( ( image_set_a_set_a @ ( image_a_a @ F ) @ ( pow_a @ A3 ) )
= ( pow_a @ B ) ) ) ).
% image_Pow_surj
thf(fact_182_image__Pow__surj,axiom,
! [F: a > c,A3: set_a,B: set_c] :
( ( ( image_a_c @ F @ A3 )
= B )
=> ( ( image_set_a_set_c @ ( image_a_c @ F ) @ ( pow_a @ A3 ) )
= ( pow_c @ B ) ) ) ).
% image_Pow_surj
thf(fact_183_Pow__mono,axiom,
! [A3: set_c,B: set_c] :
( ( ord_less_eq_set_c @ A3 @ B )
=> ( ord_le3866738827743201120_set_c @ ( pow_c @ A3 ) @ ( pow_c @ B ) ) ) ).
% Pow_mono
thf(fact_184_Pow__mono,axiom,
! [A3: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B )
=> ( ord_le5722252365846178494_set_a @ ( pow_set_a @ A3 ) @ ( pow_set_a @ B ) ) ) ).
% Pow_mono
thf(fact_185_Pow__mono,axiom,
! [A3: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A3 @ B )
=> ( ord_le3724670747650509150_set_a @ ( pow_a @ A3 ) @ ( pow_a @ B ) ) ) ).
% Pow_mono
thf(fact_186_PowD,axiom,
! [A3: set_c,B: set_c] :
( ( member_set_c @ A3 @ ( pow_c @ B ) )
=> ( ord_less_eq_set_c @ A3 @ B ) ) ).
% PowD
thf(fact_187_PowD,axiom,
! [A3: set_set_a,B: set_set_a] :
( ( member_set_set_a @ A3 @ ( pow_set_a @ B ) )
=> ( ord_le3724670747650509150_set_a @ A3 @ B ) ) ).
% PowD
thf(fact_188_PowD,axiom,
! [A3: set_a,B: set_a] :
( ( member_set_a @ A3 @ ( pow_a @ B ) )
=> ( ord_less_eq_set_a @ A3 @ B ) ) ).
% PowD
thf(fact_189_inf__set__def,axiom,
( inf_inf_set_set_c
= ( ^ [A: set_set_c,B3: set_set_c] :
( collect_set_c
@ ( inf_inf_set_c_o
@ ^ [X2: set_c] : ( member_set_c @ X2 @ A )
@ ^ [X2: set_c] : ( member_set_c @ X2 @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_190_inf__set__def,axiom,
( inf_inf_set_b
= ( ^ [A: set_b,B3: set_b] :
( collect_b
@ ( inf_inf_b_o
@ ^ [X2: b] : ( member_b @ X2 @ A )
@ ^ [X2: b] : ( member_b @ X2 @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_191_inf__set__def,axiom,
( inf_inf_set_o
= ( ^ [A: set_o,B3: set_o] :
( collect_o
@ ( inf_inf_o_o
@ ^ [X2: $o] : ( member_o @ X2 @ A )
@ ^ [X2: $o] : ( member_o @ X2 @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_192_inf__set__def,axiom,
( inf_inf_set_a
= ( ^ [A: set_a,B3: set_a] :
( collect_a
@ ( inf_inf_a_o
@ ^ [X2: a] : ( member_a @ X2 @ A )
@ ^ [X2: a] : ( member_a @ X2 @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_193_inf__set__def,axiom,
( inf_inf_set_c
= ( ^ [A: set_c,B3: set_c] :
( collect_c
@ ( inf_inf_c_o
@ ^ [X2: c] : ( member_c @ X2 @ A )
@ ^ [X2: c] : ( member_c @ X2 @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_194_inf__set__def,axiom,
( inf_inf_set_nat
= ( ^ [A: set_nat,B3: set_nat] :
( collect_nat
@ ( inf_inf_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A )
@ ^ [X2: nat] : ( member_nat @ X2 @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_195_inf__set__def,axiom,
( inf_inf_set_set_a
= ( ^ [A: set_set_a,B3: set_set_a] :
( collect_set_a
@ ( inf_inf_set_a_o
@ ^ [X2: set_a] : ( member_set_a @ X2 @ A )
@ ^ [X2: set_a] : ( member_set_a @ X2 @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_196_Pow__def,axiom,
( pow_c
= ( ^ [A: set_c] :
( collect_set_c
@ ^ [B3: set_c] : ( ord_less_eq_set_c @ B3 @ A ) ) ) ) ).
% Pow_def
thf(fact_197_Pow__def,axiom,
( pow_set_a
= ( ^ [A: set_set_a] :
( collect_set_set_a
@ ^ [B3: set_set_a] : ( ord_le3724670747650509150_set_a @ B3 @ A ) ) ) ) ).
% Pow_def
thf(fact_198_Pow__def,axiom,
( pow_a
= ( ^ [A: set_a] :
( collect_set_a
@ ^ [B3: set_a] : ( ord_less_eq_set_a @ B3 @ A ) ) ) ) ).
% Pow_def
thf(fact_199_less__eq__set__def,axiom,
( ord_le3866738827743201120_set_c
= ( ^ [A: set_set_c,B3: set_set_c] :
( ord_less_eq_set_c_o
@ ^ [X2: set_c] : ( member_set_c @ X2 @ A )
@ ^ [X2: set_c] : ( member_set_c @ X2 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_200_less__eq__set__def,axiom,
( ord_less_eq_set_b
= ( ^ [A: set_b,B3: set_b] :
( ord_less_eq_b_o
@ ^ [X2: b] : ( member_b @ X2 @ A )
@ ^ [X2: b] : ( member_b @ X2 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_201_less__eq__set__def,axiom,
( ord_less_eq_set_o
= ( ^ [A: set_o,B3: set_o] :
( ord_less_eq_o_o
@ ^ [X2: $o] : ( member_o @ X2 @ A )
@ ^ [X2: $o] : ( member_o @ X2 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_202_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A: set_nat,B3: set_nat] :
( ord_less_eq_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A )
@ ^ [X2: nat] : ( member_nat @ X2 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_203_less__eq__set__def,axiom,
( ord_less_eq_set_c
= ( ^ [A: set_c,B3: set_c] :
( ord_less_eq_c_o
@ ^ [X2: c] : ( member_c @ X2 @ A )
@ ^ [X2: c] : ( member_c @ X2 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_204_less__eq__set__def,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A: set_set_a,B3: set_set_a] :
( ord_less_eq_set_a_o
@ ^ [X2: set_a] : ( member_set_a @ X2 @ A )
@ ^ [X2: set_a] : ( member_set_a @ X2 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_205_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A: set_a,B3: set_a] :
( ord_less_eq_a_o
@ ^ [X2: a] : ( member_a @ X2 @ A )
@ ^ [X2: a] : ( member_a @ X2 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_206_image__Pow__mono,axiom,
! [F: c > c,A3: set_c,B: set_c] :
( ( ord_less_eq_set_c @ ( image_c_c @ F @ A3 ) @ B )
=> ( ord_le3866738827743201120_set_c @ ( image_set_c_set_c @ ( image_c_c @ F ) @ ( pow_c @ A3 ) ) @ ( pow_c @ B ) ) ) ).
% image_Pow_mono
thf(fact_207_image__Pow__mono,axiom,
! [F: a > c,A3: set_a,B: set_c] :
( ( ord_less_eq_set_c @ ( image_a_c @ F @ A3 ) @ B )
=> ( ord_le3866738827743201120_set_c @ ( image_set_a_set_c @ ( image_a_c @ F ) @ ( pow_a @ A3 ) ) @ ( pow_c @ B ) ) ) ).
% image_Pow_mono
thf(fact_208_image__Pow__mono,axiom,
! [F: c > set_a,A3: set_c,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_c_set_a @ F @ A3 ) @ B )
=> ( ord_le5722252365846178494_set_a @ ( image_8925085205931127395_set_a @ ( image_c_set_a @ F ) @ ( pow_c @ A3 ) ) @ ( pow_set_a @ B ) ) ) ).
% image_Pow_mono
thf(fact_209_image__Pow__mono,axiom,
! [F: a > set_a,A3: set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A3 ) @ B )
=> ( ord_le5722252365846178494_set_a @ ( image_4955109552351689957_set_a @ ( image_a_set_a @ F ) @ ( pow_a @ A3 ) ) @ ( pow_set_a @ B ) ) ) ).
% image_Pow_mono
thf(fact_210_image__Pow__mono,axiom,
! [F: c > a,A3: set_c,B: set_a] :
( ( ord_less_eq_set_a @ ( image_c_a @ F @ A3 ) @ B )
=> ( ord_le3724670747650509150_set_a @ ( image_set_c_set_a @ ( image_c_a @ F ) @ ( pow_c @ A3 ) ) @ ( pow_a @ B ) ) ) ).
% image_Pow_mono
thf(fact_211_image__Pow__mono,axiom,
! [F: a > a,A3: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F @ A3 ) @ B )
=> ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ ( image_a_a @ F ) @ ( pow_a @ A3 ) ) @ ( pow_a @ B ) ) ) ).
% image_Pow_mono
thf(fact_212_rev__image__eqI,axiom,
! [X: a,A3: set_a,B2: c,F: a > c] :
( ( member_a @ X @ A3 )
=> ( ( B2
= ( F @ X ) )
=> ( member_c @ B2 @ ( image_a_c @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_213_rev__image__eqI,axiom,
! [X: b,A3: set_b,B2: b,F: b > b] :
( ( member_b @ X @ A3 )
=> ( ( B2
= ( F @ X ) )
=> ( member_b @ B2 @ ( image_b_b @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_214_rev__image__eqI,axiom,
! [X: b,A3: set_b,B2: $o,F: b > $o] :
( ( member_b @ X @ A3 )
=> ( ( B2
= ( F @ X ) )
=> ( member_o @ B2 @ ( image_b_o @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_215_rev__image__eqI,axiom,
! [X: b,A3: set_b,B2: nat,F: b > nat] :
( ( member_b @ X @ A3 )
=> ( ( B2
= ( F @ X ) )
=> ( member_nat @ B2 @ ( image_b_nat @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_216_rev__image__eqI,axiom,
! [X: $o,A3: set_o,B2: b,F: $o > b] :
( ( member_o @ X @ A3 )
=> ( ( B2
= ( F @ X ) )
=> ( member_b @ B2 @ ( image_o_b @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_217_rev__image__eqI,axiom,
! [X: $o,A3: set_o,B2: $o,F: $o > $o] :
( ( member_o @ X @ A3 )
=> ( ( B2
= ( F @ X ) )
=> ( member_o @ B2 @ ( image_o_o @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_218_rev__image__eqI,axiom,
! [X: $o,A3: set_o,B2: nat,F: $o > nat] :
( ( member_o @ X @ A3 )
=> ( ( B2
= ( F @ X ) )
=> ( member_nat @ B2 @ ( image_o_nat @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_219_rev__image__eqI,axiom,
! [X: nat,A3: set_nat,B2: b,F: nat > b] :
( ( member_nat @ X @ A3 )
=> ( ( B2
= ( F @ X ) )
=> ( member_b @ B2 @ ( image_nat_b @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_220_rev__image__eqI,axiom,
! [X: nat,A3: set_nat,B2: $o,F: nat > $o] :
( ( member_nat @ X @ A3 )
=> ( ( B2
= ( F @ X ) )
=> ( member_o @ B2 @ ( image_nat_o @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_221_rev__image__eqI,axiom,
! [X: nat,A3: set_nat,B2: nat,F: nat > nat] :
( ( member_nat @ X @ A3 )
=> ( ( B2
= ( F @ X ) )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_222_ball__imageD,axiom,
! [F: a > c,A3: set_a,P: c > $o] :
( ! [X3: c] :
( ( member_c @ X3 @ ( image_a_c @ F @ A3 ) )
=> ( P @ X3 ) )
=> ! [X4: a] :
( ( member_a @ X4 @ A3 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_223_image__cong,axiom,
! [M: set_a,N: set_a,F: a > c,G: a > c] :
( ( M = N )
=> ( ! [X3: a] :
( ( member_a @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_a_c @ F @ M )
= ( image_a_c @ G @ N ) ) ) ) ).
% image_cong
thf(fact_224_mem__Collect__eq,axiom,
! [A4: set_c,P: set_c > $o] :
( ( member_set_c @ A4 @ ( collect_set_c @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_225_mem__Collect__eq,axiom,
! [A4: b,P: b > $o] :
( ( member_b @ A4 @ ( collect_b @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_226_mem__Collect__eq,axiom,
! [A4: $o,P: $o > $o] :
( ( member_o @ A4 @ ( collect_o @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_227_mem__Collect__eq,axiom,
! [A4: set_a,P: set_a > $o] :
( ( member_set_a @ A4 @ ( collect_set_a @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_228_mem__Collect__eq,axiom,
! [A4: nat,P: nat > $o] :
( ( member_nat @ A4 @ ( collect_nat @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_229_Collect__mem__eq,axiom,
! [A3: set_set_c] :
( ( collect_set_c
@ ^ [X2: set_c] : ( member_set_c @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_230_Collect__mem__eq,axiom,
! [A3: set_b] :
( ( collect_b
@ ^ [X2: b] : ( member_b @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_231_Collect__mem__eq,axiom,
! [A3: set_o] :
( ( collect_o
@ ^ [X2: $o] : ( member_o @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_232_Collect__mem__eq,axiom,
! [A3: set_set_a] :
( ( collect_set_a
@ ^ [X2: set_a] : ( member_set_a @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_233_Collect__mem__eq,axiom,
! [A3: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_234_Collect__cong,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X3: set_a] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_set_a @ P )
= ( collect_set_a @ Q ) ) ) ).
% Collect_cong
thf(fact_235_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_236_bex__imageD,axiom,
! [F: a > c,A3: set_a,P: c > $o] :
( ? [X4: c] :
( ( member_c @ X4 @ ( image_a_c @ F @ A3 ) )
& ( P @ X4 ) )
=> ? [X3: a] :
( ( member_a @ X3 @ A3 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_237_image__iff,axiom,
! [Z: c,F: a > c,A3: set_a] :
( ( member_c @ Z @ ( image_a_c @ F @ A3 ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A3 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_238_imageI,axiom,
! [X: a,A3: set_a,F: a > c] :
( ( member_a @ X @ A3 )
=> ( member_c @ ( F @ X ) @ ( image_a_c @ F @ A3 ) ) ) ).
% imageI
thf(fact_239_imageI,axiom,
! [X: b,A3: set_b,F: b > b] :
( ( member_b @ X @ A3 )
=> ( member_b @ ( F @ X ) @ ( image_b_b @ F @ A3 ) ) ) ).
% imageI
thf(fact_240_imageI,axiom,
! [X: b,A3: set_b,F: b > $o] :
( ( member_b @ X @ A3 )
=> ( member_o @ ( F @ X ) @ ( image_b_o @ F @ A3 ) ) ) ).
% imageI
thf(fact_241_imageI,axiom,
! [X: b,A3: set_b,F: b > nat] :
( ( member_b @ X @ A3 )
=> ( member_nat @ ( F @ X ) @ ( image_b_nat @ F @ A3 ) ) ) ).
% imageI
thf(fact_242_imageI,axiom,
! [X: $o,A3: set_o,F: $o > b] :
( ( member_o @ X @ A3 )
=> ( member_b @ ( F @ X ) @ ( image_o_b @ F @ A3 ) ) ) ).
% imageI
thf(fact_243_imageI,axiom,
! [X: $o,A3: set_o,F: $o > $o] :
( ( member_o @ X @ A3 )
=> ( member_o @ ( F @ X ) @ ( image_o_o @ F @ A3 ) ) ) ).
% imageI
thf(fact_244_imageI,axiom,
! [X: $o,A3: set_o,F: $o > nat] :
( ( member_o @ X @ A3 )
=> ( member_nat @ ( F @ X ) @ ( image_o_nat @ F @ A3 ) ) ) ).
% imageI
thf(fact_245_imageI,axiom,
! [X: nat,A3: set_nat,F: nat > b] :
( ( member_nat @ X @ A3 )
=> ( member_b @ ( F @ X ) @ ( image_nat_b @ F @ A3 ) ) ) ).
% imageI
thf(fact_246_imageI,axiom,
! [X: nat,A3: set_nat,F: nat > $o] :
( ( member_nat @ X @ A3 )
=> ( member_o @ ( F @ X ) @ ( image_nat_o @ F @ A3 ) ) ) ).
% imageI
thf(fact_247_imageI,axiom,
! [X: nat,A3: set_nat,F: nat > nat] :
( ( member_nat @ X @ A3 )
=> ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A3 ) ) ) ).
% imageI
thf(fact_248_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X2: nat] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_249_Collect__mono__iff,axiom,
! [P: c > $o,Q: c > $o] :
( ( ord_less_eq_set_c @ ( collect_c @ P ) @ ( collect_c @ Q ) )
= ( ! [X2: c] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_250_Collect__mono__iff,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) )
= ( ! [X2: set_a] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_251_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X2: a] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_252_set__eq__subset,axiom,
( ( ^ [Y4: set_c,Z2: set_c] : ( Y4 = Z2 ) )
= ( ^ [A: set_c,B3: set_c] :
( ( ord_less_eq_set_c @ A @ B3 )
& ( ord_less_eq_set_c @ B3 @ A ) ) ) ) ).
% set_eq_subset
thf(fact_253_set__eq__subset,axiom,
( ( ^ [Y4: set_set_a,Z2: set_set_a] : ( Y4 = Z2 ) )
= ( ^ [A: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B3 )
& ( ord_le3724670747650509150_set_a @ B3 @ A ) ) ) ) ).
% set_eq_subset
thf(fact_254_set__eq__subset,axiom,
( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
= ( ^ [A: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A @ B3 )
& ( ord_less_eq_set_a @ B3 @ A ) ) ) ) ).
% set_eq_subset
thf(fact_255_subset__trans,axiom,
! [A3: set_c,B: set_c,C: set_c] :
( ( ord_less_eq_set_c @ A3 @ B )
=> ( ( ord_less_eq_set_c @ B @ C )
=> ( ord_less_eq_set_c @ A3 @ C ) ) ) ).
% subset_trans
thf(fact_256_subset__trans,axiom,
! [A3: set_set_a,B: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C )
=> ( ord_le3724670747650509150_set_a @ A3 @ C ) ) ) ).
% subset_trans
thf(fact_257_subset__trans,axiom,
! [A3: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A3 @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A3 @ C ) ) ) ).
% subset_trans
thf(fact_258_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_259_Collect__mono,axiom,
! [P: c > $o,Q: c > $o] :
( ! [X3: c] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_c @ ( collect_c @ P ) @ ( collect_c @ Q ) ) ) ).
% Collect_mono
thf(fact_260_Collect__mono,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X3: set_a] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).
% Collect_mono
thf(fact_261_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_262_subset__refl,axiom,
! [A3: set_c] : ( ord_less_eq_set_c @ A3 @ A3 ) ).
% subset_refl
thf(fact_263_subset__refl,axiom,
! [A3: set_set_a] : ( ord_le3724670747650509150_set_a @ A3 @ A3 ) ).
% subset_refl
thf(fact_264_subset__refl,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ A3 @ A3 ) ).
% subset_refl
thf(fact_265_subset__iff,axiom,
( ord_le3866738827743201120_set_c
= ( ^ [A: set_set_c,B3: set_set_c] :
! [T: set_c] :
( ( member_set_c @ T @ A )
=> ( member_set_c @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_266_subset__iff,axiom,
( ord_less_eq_set_b
= ( ^ [A: set_b,B3: set_b] :
! [T: b] :
( ( member_b @ T @ A )
=> ( member_b @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_267_subset__iff,axiom,
( ord_less_eq_set_o
= ( ^ [A: set_o,B3: set_o] :
! [T: $o] :
( ( member_o @ T @ A )
=> ( member_o @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_268_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A: set_nat,B3: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A )
=> ( member_nat @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_269_subset__iff,axiom,
( ord_less_eq_set_c
= ( ^ [A: set_c,B3: set_c] :
! [T: c] :
( ( member_c @ T @ A )
=> ( member_c @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_270_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A: set_set_a,B3: set_set_a] :
! [T: set_a] :
( ( member_set_a @ T @ A )
=> ( member_set_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_271_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A: set_a,B3: set_a] :
! [T: a] :
( ( member_a @ T @ A )
=> ( member_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_272_equalityD2,axiom,
! [A3: set_c,B: set_c] :
( ( A3 = B )
=> ( ord_less_eq_set_c @ B @ A3 ) ) ).
% equalityD2
thf(fact_273_equalityD2,axiom,
! [A3: set_set_a,B: set_set_a] :
( ( A3 = B )
=> ( ord_le3724670747650509150_set_a @ B @ A3 ) ) ).
% equalityD2
thf(fact_274_equalityD2,axiom,
! [A3: set_a,B: set_a] :
( ( A3 = B )
=> ( ord_less_eq_set_a @ B @ A3 ) ) ).
% equalityD2
thf(fact_275_equalityD1,axiom,
! [A3: set_c,B: set_c] :
( ( A3 = B )
=> ( ord_less_eq_set_c @ A3 @ B ) ) ).
% equalityD1
thf(fact_276_equalityD1,axiom,
! [A3: set_set_a,B: set_set_a] :
( ( A3 = B )
=> ( ord_le3724670747650509150_set_a @ A3 @ B ) ) ).
% equalityD1
thf(fact_277_equalityD1,axiom,
! [A3: set_a,B: set_a] :
( ( A3 = B )
=> ( ord_less_eq_set_a @ A3 @ B ) ) ).
% equalityD1
thf(fact_278_subset__eq,axiom,
( ord_le3866738827743201120_set_c
= ( ^ [A: set_set_c,B3: set_set_c] :
! [X2: set_c] :
( ( member_set_c @ X2 @ A )
=> ( member_set_c @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_279_subset__eq,axiom,
( ord_less_eq_set_b
= ( ^ [A: set_b,B3: set_b] :
! [X2: b] :
( ( member_b @ X2 @ A )
=> ( member_b @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_280_subset__eq,axiom,
( ord_less_eq_set_o
= ( ^ [A: set_o,B3: set_o] :
! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( member_o @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_281_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A: set_nat,B3: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_282_subset__eq,axiom,
( ord_less_eq_set_c
= ( ^ [A: set_c,B3: set_c] :
! [X2: c] :
( ( member_c @ X2 @ A )
=> ( member_c @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_283_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A: set_set_a,B3: set_set_a] :
! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( member_set_a @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_284_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A: set_a,B3: set_a] :
! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_a @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_285_equalityE,axiom,
! [A3: set_c,B: set_c] :
( ( A3 = B )
=> ~ ( ( ord_less_eq_set_c @ A3 @ B )
=> ~ ( ord_less_eq_set_c @ B @ A3 ) ) ) ).
% equalityE
thf(fact_286_equalityE,axiom,
! [A3: set_set_a,B: set_set_a] :
( ( A3 = B )
=> ~ ( ( ord_le3724670747650509150_set_a @ A3 @ B )
=> ~ ( ord_le3724670747650509150_set_a @ B @ A3 ) ) ) ).
% equalityE
thf(fact_287_equalityE,axiom,
! [A3: set_a,B: set_a] :
( ( A3 = B )
=> ~ ( ( ord_less_eq_set_a @ A3 @ B )
=> ~ ( ord_less_eq_set_a @ B @ A3 ) ) ) ).
% equalityE
thf(fact_288_subsetD,axiom,
! [A3: set_set_c,B: set_set_c,C2: set_c] :
( ( ord_le3866738827743201120_set_c @ A3 @ B )
=> ( ( member_set_c @ C2 @ A3 )
=> ( member_set_c @ C2 @ B ) ) ) ).
% subsetD
thf(fact_289_subsetD,axiom,
! [A3: set_b,B: set_b,C2: b] :
( ( ord_less_eq_set_b @ A3 @ B )
=> ( ( member_b @ C2 @ A3 )
=> ( member_b @ C2 @ B ) ) ) ).
% subsetD
thf(fact_290_subsetD,axiom,
! [A3: set_o,B: set_o,C2: $o] :
( ( ord_less_eq_set_o @ A3 @ B )
=> ( ( member_o @ C2 @ A3 )
=> ( member_o @ C2 @ B ) ) ) ).
% subsetD
thf(fact_291_subsetD,axiom,
! [A3: set_nat,B: set_nat,C2: nat] :
( ( ord_less_eq_set_nat @ A3 @ B )
=> ( ( member_nat @ C2 @ A3 )
=> ( member_nat @ C2 @ B ) ) ) ).
% subsetD
thf(fact_292_subsetD,axiom,
! [A3: set_c,B: set_c,C2: c] :
( ( ord_less_eq_set_c @ A3 @ B )
=> ( ( member_c @ C2 @ A3 )
=> ( member_c @ C2 @ B ) ) ) ).
% subsetD
thf(fact_293_subsetD,axiom,
! [A3: set_set_a,B: set_set_a,C2: set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B )
=> ( ( member_set_a @ C2 @ A3 )
=> ( member_set_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_294_subsetD,axiom,
! [A3: set_a,B: set_a,C2: a] :
( ( ord_less_eq_set_a @ A3 @ B )
=> ( ( member_a @ C2 @ A3 )
=> ( member_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_295_in__mono,axiom,
! [A3: set_set_c,B: set_set_c,X: set_c] :
( ( ord_le3866738827743201120_set_c @ A3 @ B )
=> ( ( member_set_c @ X @ A3 )
=> ( member_set_c @ X @ B ) ) ) ).
% in_mono
thf(fact_296_in__mono,axiom,
! [A3: set_b,B: set_b,X: b] :
( ( ord_less_eq_set_b @ A3 @ B )
=> ( ( member_b @ X @ A3 )
=> ( member_b @ X @ B ) ) ) ).
% in_mono
thf(fact_297_in__mono,axiom,
! [A3: set_o,B: set_o,X: $o] :
( ( ord_less_eq_set_o @ A3 @ B )
=> ( ( member_o @ X @ A3 )
=> ( member_o @ X @ B ) ) ) ).
% in_mono
thf(fact_298_in__mono,axiom,
! [A3: set_nat,B: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A3 @ B )
=> ( ( member_nat @ X @ A3 )
=> ( member_nat @ X @ B ) ) ) ).
% in_mono
thf(fact_299_in__mono,axiom,
! [A3: set_c,B: set_c,X: c] :
( ( ord_less_eq_set_c @ A3 @ B )
=> ( ( member_c @ X @ A3 )
=> ( member_c @ X @ B ) ) ) ).
% in_mono
thf(fact_300_in__mono,axiom,
! [A3: set_set_a,B: set_set_a,X: set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B )
=> ( ( member_set_a @ X @ A3 )
=> ( member_set_a @ X @ B ) ) ) ).
% in_mono
thf(fact_301_in__mono,axiom,
! [A3: set_a,B: set_a,X: a] :
( ( ord_less_eq_set_a @ A3 @ B )
=> ( ( member_a @ X @ A3 )
=> ( member_a @ X @ B ) ) ) ).
% in_mono
thf(fact_302_inf__left__commute,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_303_inf__left__commute,axiom,
! [X: set_c,Y: set_c,Z: set_c] :
( ( inf_inf_set_c @ X @ ( inf_inf_set_c @ Y @ Z ) )
= ( inf_inf_set_c @ Y @ ( inf_inf_set_c @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_304_inf__left__commute,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
= ( inf_inf_set_nat @ Y @ ( inf_inf_set_nat @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_305_inf__left__commute,axiom,
! [X: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( inf_inf_set_set_a @ X @ ( inf_inf_set_set_a @ Y @ Z ) )
= ( inf_inf_set_set_a @ Y @ ( inf_inf_set_set_a @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_306_inf__left__commute,axiom,
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal,Z: extend8495563244428889912nnreal] :
( ( inf_in7439215052339218890nnreal @ X @ ( inf_in7439215052339218890nnreal @ Y @ Z ) )
= ( inf_in7439215052339218890nnreal @ Y @ ( inf_in7439215052339218890nnreal @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_307_inf__left__commute,axiom,
! [X: extended_ereal,Y: extended_ereal,Z: extended_ereal] :
( ( inf_in2794916579150040252_ereal @ X @ ( inf_in2794916579150040252_ereal @ Y @ Z ) )
= ( inf_in2794916579150040252_ereal @ Y @ ( inf_in2794916579150040252_ereal @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_308_inf__left__commute,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( inf_inf_nat @ Y @ ( inf_inf_nat @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_309_inf_Oleft__commute,axiom,
! [B2: set_a,A4: set_a,C2: set_a] :
( ( inf_inf_set_a @ B2 @ ( inf_inf_set_a @ A4 @ C2 ) )
= ( inf_inf_set_a @ A4 @ ( inf_inf_set_a @ B2 @ C2 ) ) ) ).
% inf.left_commute
thf(fact_310_inf_Oleft__commute,axiom,
! [B2: set_c,A4: set_c,C2: set_c] :
( ( inf_inf_set_c @ B2 @ ( inf_inf_set_c @ A4 @ C2 ) )
= ( inf_inf_set_c @ A4 @ ( inf_inf_set_c @ B2 @ C2 ) ) ) ).
% inf.left_commute
thf(fact_311_inf_Oleft__commute,axiom,
! [B2: set_nat,A4: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ B2 @ ( inf_inf_set_nat @ A4 @ C2 ) )
= ( inf_inf_set_nat @ A4 @ ( inf_inf_set_nat @ B2 @ C2 ) ) ) ).
% inf.left_commute
thf(fact_312_inf_Oleft__commute,axiom,
! [B2: set_set_a,A4: set_set_a,C2: set_set_a] :
( ( inf_inf_set_set_a @ B2 @ ( inf_inf_set_set_a @ A4 @ C2 ) )
= ( inf_inf_set_set_a @ A4 @ ( inf_inf_set_set_a @ B2 @ C2 ) ) ) ).
% inf.left_commute
thf(fact_313_inf_Oleft__commute,axiom,
! [B2: extend8495563244428889912nnreal,A4: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( inf_in7439215052339218890nnreal @ B2 @ ( inf_in7439215052339218890nnreal @ A4 @ C2 ) )
= ( inf_in7439215052339218890nnreal @ A4 @ ( inf_in7439215052339218890nnreal @ B2 @ C2 ) ) ) ).
% inf.left_commute
thf(fact_314_inf_Oleft__commute,axiom,
! [B2: extended_ereal,A4: extended_ereal,C2: extended_ereal] :
( ( inf_in2794916579150040252_ereal @ B2 @ ( inf_in2794916579150040252_ereal @ A4 @ C2 ) )
= ( inf_in2794916579150040252_ereal @ A4 @ ( inf_in2794916579150040252_ereal @ B2 @ C2 ) ) ) ).
% inf.left_commute
thf(fact_315_inf_Oleft__commute,axiom,
! [B2: nat,A4: nat,C2: nat] :
( ( inf_inf_nat @ B2 @ ( inf_inf_nat @ A4 @ C2 ) )
= ( inf_inf_nat @ A4 @ ( inf_inf_nat @ B2 @ C2 ) ) ) ).
% inf.left_commute
thf(fact_316_inf__commute,axiom,
( inf_inf_set_a
= ( ^ [X2: set_a,Y3: set_a] : ( inf_inf_set_a @ Y3 @ X2 ) ) ) ).
% inf_commute
thf(fact_317_inf__commute,axiom,
( inf_inf_set_c
= ( ^ [X2: set_c,Y3: set_c] : ( inf_inf_set_c @ Y3 @ X2 ) ) ) ).
% inf_commute
thf(fact_318_inf__commute,axiom,
( inf_inf_set_nat
= ( ^ [X2: set_nat,Y3: set_nat] : ( inf_inf_set_nat @ Y3 @ X2 ) ) ) ).
% inf_commute
thf(fact_319_inf__commute,axiom,
( inf_inf_set_set_a
= ( ^ [X2: set_set_a,Y3: set_set_a] : ( inf_inf_set_set_a @ Y3 @ X2 ) ) ) ).
% inf_commute
thf(fact_320_inf__commute,axiom,
( inf_in7439215052339218890nnreal
= ( ^ [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] : ( inf_in7439215052339218890nnreal @ Y3 @ X2 ) ) ) ).
% inf_commute
thf(fact_321_inf__commute,axiom,
( inf_in2794916579150040252_ereal
= ( ^ [X2: extended_ereal,Y3: extended_ereal] : ( inf_in2794916579150040252_ereal @ Y3 @ X2 ) ) ) ).
% inf_commute
thf(fact_322_inf__commute,axiom,
( inf_inf_nat
= ( ^ [X2: nat,Y3: nat] : ( inf_inf_nat @ Y3 @ X2 ) ) ) ).
% inf_commute
thf(fact_323_inf_Ocommute,axiom,
( inf_inf_set_a
= ( ^ [A2: set_a,B4: set_a] : ( inf_inf_set_a @ B4 @ A2 ) ) ) ).
% inf.commute
thf(fact_324_inf_Ocommute,axiom,
( inf_inf_set_c
= ( ^ [A2: set_c,B4: set_c] : ( inf_inf_set_c @ B4 @ A2 ) ) ) ).
% inf.commute
thf(fact_325_inf_Ocommute,axiom,
( inf_inf_set_nat
= ( ^ [A2: set_nat,B4: set_nat] : ( inf_inf_set_nat @ B4 @ A2 ) ) ) ).
% inf.commute
thf(fact_326_inf_Ocommute,axiom,
( inf_inf_set_set_a
= ( ^ [A2: set_set_a,B4: set_set_a] : ( inf_inf_set_set_a @ B4 @ A2 ) ) ) ).
% inf.commute
thf(fact_327_inf_Ocommute,axiom,
( inf_in7439215052339218890nnreal
= ( ^ [A2: extend8495563244428889912nnreal,B4: extend8495563244428889912nnreal] : ( inf_in7439215052339218890nnreal @ B4 @ A2 ) ) ) ).
% inf.commute
thf(fact_328_inf_Ocommute,axiom,
( inf_in2794916579150040252_ereal
= ( ^ [A2: extended_ereal,B4: extended_ereal] : ( inf_in2794916579150040252_ereal @ B4 @ A2 ) ) ) ).
% inf.commute
thf(fact_329_inf_Ocommute,axiom,
( inf_inf_nat
= ( ^ [A2: nat,B4: nat] : ( inf_inf_nat @ B4 @ A2 ) ) ) ).
% inf.commute
thf(fact_330_inf__assoc,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_331_inf__assoc,axiom,
! [X: set_c,Y: set_c,Z: set_c] :
( ( inf_inf_set_c @ ( inf_inf_set_c @ X @ Y ) @ Z )
= ( inf_inf_set_c @ X @ ( inf_inf_set_c @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_332_inf__assoc,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Z )
= ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_333_inf__assoc,axiom,
! [X: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ X @ Y ) @ Z )
= ( inf_inf_set_set_a @ X @ ( inf_inf_set_set_a @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_334_inf__assoc,axiom,
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal,Z: extend8495563244428889912nnreal] :
( ( inf_in7439215052339218890nnreal @ ( inf_in7439215052339218890nnreal @ X @ Y ) @ Z )
= ( inf_in7439215052339218890nnreal @ X @ ( inf_in7439215052339218890nnreal @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_335_inf__assoc,axiom,
! [X: extended_ereal,Y: extended_ereal,Z: extended_ereal] :
( ( inf_in2794916579150040252_ereal @ ( inf_in2794916579150040252_ereal @ X @ Y ) @ Z )
= ( inf_in2794916579150040252_ereal @ X @ ( inf_in2794916579150040252_ereal @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_336_inf__assoc,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ X @ Y ) @ Z )
= ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_337_inf_Oassoc,axiom,
! [A4: set_a,B2: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A4 @ B2 ) @ C2 )
= ( inf_inf_set_a @ A4 @ ( inf_inf_set_a @ B2 @ C2 ) ) ) ).
% inf.assoc
thf(fact_338_inf_Oassoc,axiom,
! [A4: set_c,B2: set_c,C2: set_c] :
( ( inf_inf_set_c @ ( inf_inf_set_c @ A4 @ B2 ) @ C2 )
= ( inf_inf_set_c @ A4 @ ( inf_inf_set_c @ B2 @ C2 ) ) ) ).
% inf.assoc
thf(fact_339_inf_Oassoc,axiom,
! [A4: set_nat,B2: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A4 @ B2 ) @ C2 )
= ( inf_inf_set_nat @ A4 @ ( inf_inf_set_nat @ B2 @ C2 ) ) ) ).
% inf.assoc
thf(fact_340_inf_Oassoc,axiom,
! [A4: set_set_a,B2: set_set_a,C2: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ A4 @ B2 ) @ C2 )
= ( inf_inf_set_set_a @ A4 @ ( inf_inf_set_set_a @ B2 @ C2 ) ) ) ).
% inf.assoc
thf(fact_341_inf_Oassoc,axiom,
! [A4: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( inf_in7439215052339218890nnreal @ ( inf_in7439215052339218890nnreal @ A4 @ B2 ) @ C2 )
= ( inf_in7439215052339218890nnreal @ A4 @ ( inf_in7439215052339218890nnreal @ B2 @ C2 ) ) ) ).
% inf.assoc
thf(fact_342_inf_Oassoc,axiom,
! [A4: extended_ereal,B2: extended_ereal,C2: extended_ereal] :
( ( inf_in2794916579150040252_ereal @ ( inf_in2794916579150040252_ereal @ A4 @ B2 ) @ C2 )
= ( inf_in2794916579150040252_ereal @ A4 @ ( inf_in2794916579150040252_ereal @ B2 @ C2 ) ) ) ).
% inf.assoc
thf(fact_343_inf_Oassoc,axiom,
! [A4: nat,B2: nat,C2: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ A4 @ B2 ) @ C2 )
= ( inf_inf_nat @ A4 @ ( inf_inf_nat @ B2 @ C2 ) ) ) ).
% inf.assoc
thf(fact_344_inf__sup__aci_I1_J,axiom,
( inf_inf_set_a
= ( ^ [X2: set_a,Y3: set_a] : ( inf_inf_set_a @ Y3 @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_345_inf__sup__aci_I1_J,axiom,
( inf_inf_set_c
= ( ^ [X2: set_c,Y3: set_c] : ( inf_inf_set_c @ Y3 @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_346_inf__sup__aci_I1_J,axiom,
( inf_inf_set_nat
= ( ^ [X2: set_nat,Y3: set_nat] : ( inf_inf_set_nat @ Y3 @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_347_inf__sup__aci_I1_J,axiom,
( inf_inf_set_set_a
= ( ^ [X2: set_set_a,Y3: set_set_a] : ( inf_inf_set_set_a @ Y3 @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_348_inf__sup__aci_I1_J,axiom,
( inf_in7439215052339218890nnreal
= ( ^ [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] : ( inf_in7439215052339218890nnreal @ Y3 @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_349_inf__sup__aci_I1_J,axiom,
( inf_in2794916579150040252_ereal
= ( ^ [X2: extended_ereal,Y3: extended_ereal] : ( inf_in2794916579150040252_ereal @ Y3 @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_350_inf__sup__aci_I1_J,axiom,
( inf_inf_nat
= ( ^ [X2: nat,Y3: nat] : ( inf_inf_nat @ Y3 @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_351_inf__sup__aci_I2_J,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_352_inf__sup__aci_I2_J,axiom,
! [X: set_c,Y: set_c,Z: set_c] :
( ( inf_inf_set_c @ ( inf_inf_set_c @ X @ Y ) @ Z )
= ( inf_inf_set_c @ X @ ( inf_inf_set_c @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_353_inf__sup__aci_I2_J,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Z )
= ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_354_inf__sup__aci_I2_J,axiom,
! [X: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ X @ Y ) @ Z )
= ( inf_inf_set_set_a @ X @ ( inf_inf_set_set_a @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_355_inf__sup__aci_I2_J,axiom,
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal,Z: extend8495563244428889912nnreal] :
( ( inf_in7439215052339218890nnreal @ ( inf_in7439215052339218890nnreal @ X @ Y ) @ Z )
= ( inf_in7439215052339218890nnreal @ X @ ( inf_in7439215052339218890nnreal @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_356_inf__sup__aci_I2_J,axiom,
! [X: extended_ereal,Y: extended_ereal,Z: extended_ereal] :
( ( inf_in2794916579150040252_ereal @ ( inf_in2794916579150040252_ereal @ X @ Y ) @ Z )
= ( inf_in2794916579150040252_ereal @ X @ ( inf_in2794916579150040252_ereal @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_357_inf__sup__aci_I2_J,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ X @ Y ) @ Z )
= ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_358_inf__sup__aci_I3_J,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_359_inf__sup__aci_I3_J,axiom,
! [X: set_c,Y: set_c,Z: set_c] :
( ( inf_inf_set_c @ X @ ( inf_inf_set_c @ Y @ Z ) )
= ( inf_inf_set_c @ Y @ ( inf_inf_set_c @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_360_inf__sup__aci_I3_J,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
= ( inf_inf_set_nat @ Y @ ( inf_inf_set_nat @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_361_inf__sup__aci_I3_J,axiom,
! [X: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( inf_inf_set_set_a @ X @ ( inf_inf_set_set_a @ Y @ Z ) )
= ( inf_inf_set_set_a @ Y @ ( inf_inf_set_set_a @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_362_inf__sup__aci_I3_J,axiom,
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal,Z: extend8495563244428889912nnreal] :
( ( inf_in7439215052339218890nnreal @ X @ ( inf_in7439215052339218890nnreal @ Y @ Z ) )
= ( inf_in7439215052339218890nnreal @ Y @ ( inf_in7439215052339218890nnreal @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_363_inf__sup__aci_I3_J,axiom,
! [X: extended_ereal,Y: extended_ereal,Z: extended_ereal] :
( ( inf_in2794916579150040252_ereal @ X @ ( inf_in2794916579150040252_ereal @ Y @ Z ) )
= ( inf_in2794916579150040252_ereal @ Y @ ( inf_in2794916579150040252_ereal @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_364_inf__sup__aci_I3_J,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( inf_inf_nat @ Y @ ( inf_inf_nat @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_365_inf__sup__aci_I4_J,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
= ( inf_inf_set_a @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_366_inf__sup__aci_I4_J,axiom,
! [X: set_c,Y: set_c] :
( ( inf_inf_set_c @ X @ ( inf_inf_set_c @ X @ Y ) )
= ( inf_inf_set_c @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_367_inf__sup__aci_I4_J,axiom,
! [X: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ X @ Y ) )
= ( inf_inf_set_nat @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_368_inf__sup__aci_I4_J,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( inf_inf_set_set_a @ X @ ( inf_inf_set_set_a @ X @ Y ) )
= ( inf_inf_set_set_a @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_369_inf__sup__aci_I4_J,axiom,
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( inf_in7439215052339218890nnreal @ X @ ( inf_in7439215052339218890nnreal @ X @ Y ) )
= ( inf_in7439215052339218890nnreal @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_370_inf__sup__aci_I4_J,axiom,
! [X: extended_ereal,Y: extended_ereal] :
( ( inf_in2794916579150040252_ereal @ X @ ( inf_in2794916579150040252_ereal @ X @ Y ) )
= ( inf_in2794916579150040252_ereal @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_371_inf__sup__aci_I4_J,axiom,
! [X: nat,Y: nat] :
( ( inf_inf_nat @ X @ ( inf_inf_nat @ X @ Y ) )
= ( inf_inf_nat @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_372_Int__left__commute,axiom,
! [A3: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ A3 @ ( inf_inf_set_a @ B @ C ) )
= ( inf_inf_set_a @ B @ ( inf_inf_set_a @ A3 @ C ) ) ) ).
% Int_left_commute
thf(fact_373_Int__left__commute,axiom,
! [A3: set_c,B: set_c,C: set_c] :
( ( inf_inf_set_c @ A3 @ ( inf_inf_set_c @ B @ C ) )
= ( inf_inf_set_c @ B @ ( inf_inf_set_c @ A3 @ C ) ) ) ).
% Int_left_commute
thf(fact_374_Int__left__commute,axiom,
! [A3: set_nat,B: set_nat,C: set_nat] :
( ( inf_inf_set_nat @ A3 @ ( inf_inf_set_nat @ B @ C ) )
= ( inf_inf_set_nat @ B @ ( inf_inf_set_nat @ A3 @ C ) ) ) ).
% Int_left_commute
thf(fact_375_Int__left__commute,axiom,
! [A3: set_set_a,B: set_set_a,C: set_set_a] :
( ( inf_inf_set_set_a @ A3 @ ( inf_inf_set_set_a @ B @ C ) )
= ( inf_inf_set_set_a @ B @ ( inf_inf_set_set_a @ A3 @ C ) ) ) ).
% Int_left_commute
thf(fact_376_Int__left__absorb,axiom,
! [A3: set_a,B: set_a] :
( ( inf_inf_set_a @ A3 @ ( inf_inf_set_a @ A3 @ B ) )
= ( inf_inf_set_a @ A3 @ B ) ) ).
% Int_left_absorb
thf(fact_377_Int__left__absorb,axiom,
! [A3: set_c,B: set_c] :
( ( inf_inf_set_c @ A3 @ ( inf_inf_set_c @ A3 @ B ) )
= ( inf_inf_set_c @ A3 @ B ) ) ).
% Int_left_absorb
thf(fact_378_Int__left__absorb,axiom,
! [A3: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ A3 @ ( inf_inf_set_nat @ A3 @ B ) )
= ( inf_inf_set_nat @ A3 @ B ) ) ).
% Int_left_absorb
thf(fact_379_Int__left__absorb,axiom,
! [A3: set_set_a,B: set_set_a] :
( ( inf_inf_set_set_a @ A3 @ ( inf_inf_set_set_a @ A3 @ B ) )
= ( inf_inf_set_set_a @ A3 @ B ) ) ).
% Int_left_absorb
thf(fact_380_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A: set_a,B3: set_a] : ( inf_inf_set_a @ B3 @ A ) ) ) ).
% Int_commute
thf(fact_381_Int__commute,axiom,
( inf_inf_set_c
= ( ^ [A: set_c,B3: set_c] : ( inf_inf_set_c @ B3 @ A ) ) ) ).
% Int_commute
thf(fact_382_Int__commute,axiom,
( inf_inf_set_nat
= ( ^ [A: set_nat,B3: set_nat] : ( inf_inf_set_nat @ B3 @ A ) ) ) ).
% Int_commute
thf(fact_383_Int__commute,axiom,
( inf_inf_set_set_a
= ( ^ [A: set_set_a,B3: set_set_a] : ( inf_inf_set_set_a @ B3 @ A ) ) ) ).
% Int_commute
thf(fact_384_Int__absorb,axiom,
! [A3: set_a] :
( ( inf_inf_set_a @ A3 @ A3 )
= A3 ) ).
% Int_absorb
thf(fact_385_Int__absorb,axiom,
! [A3: set_c] :
( ( inf_inf_set_c @ A3 @ A3 )
= A3 ) ).
% Int_absorb
thf(fact_386_Int__absorb,axiom,
! [A3: set_nat] :
( ( inf_inf_set_nat @ A3 @ A3 )
= A3 ) ).
% Int_absorb
thf(fact_387_Int__absorb,axiom,
! [A3: set_set_a] :
( ( inf_inf_set_set_a @ A3 @ A3 )
= A3 ) ).
% Int_absorb
thf(fact_388_Int__assoc,axiom,
! [A3: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A3 @ B ) @ C )
= ( inf_inf_set_a @ A3 @ ( inf_inf_set_a @ B @ C ) ) ) ).
% Int_assoc
thf(fact_389_Int__assoc,axiom,
! [A3: set_c,B: set_c,C: set_c] :
( ( inf_inf_set_c @ ( inf_inf_set_c @ A3 @ B ) @ C )
= ( inf_inf_set_c @ A3 @ ( inf_inf_set_c @ B @ C ) ) ) ).
% Int_assoc
thf(fact_390_Int__assoc,axiom,
! [A3: set_nat,B: set_nat,C: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A3 @ B ) @ C )
= ( inf_inf_set_nat @ A3 @ ( inf_inf_set_nat @ B @ C ) ) ) ).
% Int_assoc
thf(fact_391_Int__assoc,axiom,
! [A3: set_set_a,B: set_set_a,C: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ A3 @ B ) @ C )
= ( inf_inf_set_set_a @ A3 @ ( inf_inf_set_set_a @ B @ C ) ) ) ).
% Int_assoc
thf(fact_392_IntD2,axiom,
! [C2: set_c,A3: set_set_c,B: set_set_c] :
( ( member_set_c @ C2 @ ( inf_inf_set_set_c @ A3 @ B ) )
=> ( member_set_c @ C2 @ B ) ) ).
% IntD2
thf(fact_393_IntD2,axiom,
! [C2: b,A3: set_b,B: set_b] :
( ( member_b @ C2 @ ( inf_inf_set_b @ A3 @ B ) )
=> ( member_b @ C2 @ B ) ) ).
% IntD2
thf(fact_394_IntD2,axiom,
! [C2: $o,A3: set_o,B: set_o] :
( ( member_o @ C2 @ ( inf_inf_set_o @ A3 @ B ) )
=> ( member_o @ C2 @ B ) ) ).
% IntD2
thf(fact_395_IntD2,axiom,
! [C2: a,A3: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A3 @ B ) )
=> ( member_a @ C2 @ B ) ) ).
% IntD2
thf(fact_396_IntD2,axiom,
! [C2: c,A3: set_c,B: set_c] :
( ( member_c @ C2 @ ( inf_inf_set_c @ A3 @ B ) )
=> ( member_c @ C2 @ B ) ) ).
% IntD2
thf(fact_397_IntD2,axiom,
! [C2: nat,A3: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A3 @ B ) )
=> ( member_nat @ C2 @ B ) ) ).
% IntD2
thf(fact_398_IntD2,axiom,
! [C2: set_a,A3: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A3 @ B ) )
=> ( member_set_a @ C2 @ B ) ) ).
% IntD2
thf(fact_399_IntD1,axiom,
! [C2: set_c,A3: set_set_c,B: set_set_c] :
( ( member_set_c @ C2 @ ( inf_inf_set_set_c @ A3 @ B ) )
=> ( member_set_c @ C2 @ A3 ) ) ).
% IntD1
thf(fact_400_IntD1,axiom,
! [C2: b,A3: set_b,B: set_b] :
( ( member_b @ C2 @ ( inf_inf_set_b @ A3 @ B ) )
=> ( member_b @ C2 @ A3 ) ) ).
% IntD1
thf(fact_401_IntD1,axiom,
! [C2: $o,A3: set_o,B: set_o] :
( ( member_o @ C2 @ ( inf_inf_set_o @ A3 @ B ) )
=> ( member_o @ C2 @ A3 ) ) ).
% IntD1
thf(fact_402_IntD1,axiom,
! [C2: a,A3: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A3 @ B ) )
=> ( member_a @ C2 @ A3 ) ) ).
% IntD1
thf(fact_403_IntD1,axiom,
! [C2: c,A3: set_c,B: set_c] :
( ( member_c @ C2 @ ( inf_inf_set_c @ A3 @ B ) )
=> ( member_c @ C2 @ A3 ) ) ).
% IntD1
thf(fact_404_IntD1,axiom,
! [C2: nat,A3: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A3 @ B ) )
=> ( member_nat @ C2 @ A3 ) ) ).
% IntD1
thf(fact_405_IntD1,axiom,
! [C2: set_a,A3: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A3 @ B ) )
=> ( member_set_a @ C2 @ A3 ) ) ).
% IntD1
thf(fact_406_IntE,axiom,
! [C2: set_c,A3: set_set_c,B: set_set_c] :
( ( member_set_c @ C2 @ ( inf_inf_set_set_c @ A3 @ B ) )
=> ~ ( ( member_set_c @ C2 @ A3 )
=> ~ ( member_set_c @ C2 @ B ) ) ) ).
% IntE
thf(fact_407_IntE,axiom,
! [C2: b,A3: set_b,B: set_b] :
( ( member_b @ C2 @ ( inf_inf_set_b @ A3 @ B ) )
=> ~ ( ( member_b @ C2 @ A3 )
=> ~ ( member_b @ C2 @ B ) ) ) ).
% IntE
thf(fact_408_IntE,axiom,
! [C2: $o,A3: set_o,B: set_o] :
( ( member_o @ C2 @ ( inf_inf_set_o @ A3 @ B ) )
=> ~ ( ( member_o @ C2 @ A3 )
=> ~ ( member_o @ C2 @ B ) ) ) ).
% IntE
thf(fact_409_IntE,axiom,
! [C2: a,A3: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A3 @ B ) )
=> ~ ( ( member_a @ C2 @ A3 )
=> ~ ( member_a @ C2 @ B ) ) ) ).
% IntE
thf(fact_410_IntE,axiom,
! [C2: c,A3: set_c,B: set_c] :
( ( member_c @ C2 @ ( inf_inf_set_c @ A3 @ B ) )
=> ~ ( ( member_c @ C2 @ A3 )
=> ~ ( member_c @ C2 @ B ) ) ) ).
% IntE
thf(fact_411_IntE,axiom,
! [C2: nat,A3: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A3 @ B ) )
=> ~ ( ( member_nat @ C2 @ A3 )
=> ~ ( member_nat @ C2 @ B ) ) ) ).
% IntE
thf(fact_412_IntE,axiom,
! [C2: set_a,A3: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A3 @ B ) )
=> ~ ( ( member_set_a @ C2 @ A3 )
=> ~ ( member_set_a @ C2 @ B ) ) ) ).
% IntE
thf(fact_413_vimage__Collect,axiom,
! [P: c > $o,F: a > c,Q: a > $o] :
( ! [X3: a] :
( ( P @ ( F @ X3 ) )
= ( Q @ X3 ) )
=> ( ( vimage_a_c @ F @ ( collect_c @ P ) )
= ( collect_a @ Q ) ) ) ).
% vimage_Collect
thf(fact_414_vimage__Collect,axiom,
! [P: c > $o,F: c > c,Q: c > $o] :
( ! [X3: c] :
( ( P @ ( F @ X3 ) )
= ( Q @ X3 ) )
=> ( ( vimage_c_c @ F @ ( collect_c @ P ) )
= ( collect_c @ Q ) ) ) ).
% vimage_Collect
thf(fact_415_vimage__Collect,axiom,
! [P: a > $o,F: c > a,Q: c > $o] :
( ! [X3: c] :
( ( P @ ( F @ X3 ) )
= ( Q @ X3 ) )
=> ( ( vimage_c_a @ F @ ( collect_a @ P ) )
= ( collect_c @ Q ) ) ) ).
% vimage_Collect
thf(fact_416_vimage__Collect,axiom,
! [P: a > $o,F: a > a,Q: a > $o] :
( ! [X3: a] :
( ( P @ ( F @ X3 ) )
= ( Q @ X3 ) )
=> ( ( vimage_a_a @ F @ ( collect_a @ P ) )
= ( collect_a @ Q ) ) ) ).
% vimage_Collect
thf(fact_417_vimage__Collect,axiom,
! [P: set_a > $o,F: set_a > set_a,Q: set_a > $o] :
( ! [X3: set_a] :
( ( P @ ( F @ X3 ) )
= ( Q @ X3 ) )
=> ( ( vimage_set_a_set_a @ F @ ( collect_set_a @ P ) )
= ( collect_set_a @ Q ) ) ) ).
% vimage_Collect
thf(fact_418_vimage__Collect,axiom,
! [P: set_a > $o,F: nat > set_a,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ ( F @ X3 ) )
= ( Q @ X3 ) )
=> ( ( vimage_nat_set_a @ F @ ( collect_set_a @ P ) )
= ( collect_nat @ Q ) ) ) ).
% vimage_Collect
thf(fact_419_vimage__Collect,axiom,
! [P: nat > $o,F: set_a > nat,Q: set_a > $o] :
( ! [X3: set_a] :
( ( P @ ( F @ X3 ) )
= ( Q @ X3 ) )
=> ( ( vimage_set_a_nat @ F @ ( collect_nat @ P ) )
= ( collect_set_a @ Q ) ) ) ).
% vimage_Collect
thf(fact_420_vimage__Collect,axiom,
! [P: nat > $o,F: nat > nat,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ ( F @ X3 ) )
= ( Q @ X3 ) )
=> ( ( vimage_nat_nat @ F @ ( collect_nat @ P ) )
= ( collect_nat @ Q ) ) ) ).
% vimage_Collect
thf(fact_421_vimageI2,axiom,
! [F: b > b,A4: b,A3: set_b] :
( ( member_b @ ( F @ A4 ) @ A3 )
=> ( member_b @ A4 @ ( vimage_b_b @ F @ A3 ) ) ) ).
% vimageI2
thf(fact_422_vimageI2,axiom,
! [F: $o > b,A4: $o,A3: set_b] :
( ( member_b @ ( F @ A4 ) @ A3 )
=> ( member_o @ A4 @ ( vimage_o_b @ F @ A3 ) ) ) ).
% vimageI2
thf(fact_423_vimageI2,axiom,
! [F: nat > b,A4: nat,A3: set_b] :
( ( member_b @ ( F @ A4 ) @ A3 )
=> ( member_nat @ A4 @ ( vimage_nat_b @ F @ A3 ) ) ) ).
% vimageI2
thf(fact_424_vimageI2,axiom,
! [F: b > $o,A4: b,A3: set_o] :
( ( member_o @ ( F @ A4 ) @ A3 )
=> ( member_b @ A4 @ ( vimage_b_o @ F @ A3 ) ) ) ).
% vimageI2
thf(fact_425_vimageI2,axiom,
! [F: $o > $o,A4: $o,A3: set_o] :
( ( member_o @ ( F @ A4 ) @ A3 )
=> ( member_o @ A4 @ ( vimage_o_o @ F @ A3 ) ) ) ).
% vimageI2
thf(fact_426_vimageI2,axiom,
! [F: nat > $o,A4: nat,A3: set_o] :
( ( member_o @ ( F @ A4 ) @ A3 )
=> ( member_nat @ A4 @ ( vimage_nat_o @ F @ A3 ) ) ) ).
% vimageI2
thf(fact_427_vimageI2,axiom,
! [F: b > nat,A4: b,A3: set_nat] :
( ( member_nat @ ( F @ A4 ) @ A3 )
=> ( member_b @ A4 @ ( vimage_b_nat @ F @ A3 ) ) ) ).
% vimageI2
thf(fact_428_vimageI2,axiom,
! [F: $o > nat,A4: $o,A3: set_nat] :
( ( member_nat @ ( F @ A4 ) @ A3 )
=> ( member_o @ A4 @ ( vimage_o_nat @ F @ A3 ) ) ) ).
% vimageI2
thf(fact_429_vimageI2,axiom,
! [F: a > c,A4: a,A3: set_c] :
( ( member_c @ ( F @ A4 ) @ A3 )
=> ( member_a @ A4 @ ( vimage_a_c @ F @ A3 ) ) ) ).
% vimageI2
thf(fact_430_vimageI2,axiom,
! [F: nat > nat,A4: nat,A3: set_nat] :
( ( member_nat @ ( F @ A4 ) @ A3 )
=> ( member_nat @ A4 @ ( vimage_nat_nat @ F @ A3 ) ) ) ).
% vimageI2
thf(fact_431_vimageE,axiom,
! [A4: b,F: b > b,B: set_b] :
( ( member_b @ A4 @ ( vimage_b_b @ F @ B ) )
=> ( member_b @ ( F @ A4 ) @ B ) ) ).
% vimageE
thf(fact_432_vimageE,axiom,
! [A4: b,F: b > $o,B: set_o] :
( ( member_b @ A4 @ ( vimage_b_o @ F @ B ) )
=> ( member_o @ ( F @ A4 ) @ B ) ) ).
% vimageE
thf(fact_433_vimageE,axiom,
! [A4: b,F: b > nat,B: set_nat] :
( ( member_b @ A4 @ ( vimage_b_nat @ F @ B ) )
=> ( member_nat @ ( F @ A4 ) @ B ) ) ).
% vimageE
thf(fact_434_vimageE,axiom,
! [A4: $o,F: $o > b,B: set_b] :
( ( member_o @ A4 @ ( vimage_o_b @ F @ B ) )
=> ( member_b @ ( F @ A4 ) @ B ) ) ).
% vimageE
thf(fact_435_vimageE,axiom,
! [A4: $o,F: $o > $o,B: set_o] :
( ( member_o @ A4 @ ( vimage_o_o @ F @ B ) )
=> ( member_o @ ( F @ A4 ) @ B ) ) ).
% vimageE
thf(fact_436_vimageE,axiom,
! [A4: $o,F: $o > nat,B: set_nat] :
( ( member_o @ A4 @ ( vimage_o_nat @ F @ B ) )
=> ( member_nat @ ( F @ A4 ) @ B ) ) ).
% vimageE
thf(fact_437_vimageE,axiom,
! [A4: nat,F: nat > b,B: set_b] :
( ( member_nat @ A4 @ ( vimage_nat_b @ F @ B ) )
=> ( member_b @ ( F @ A4 ) @ B ) ) ).
% vimageE
thf(fact_438_vimageE,axiom,
! [A4: nat,F: nat > $o,B: set_o] :
( ( member_nat @ A4 @ ( vimage_nat_o @ F @ B ) )
=> ( member_o @ ( F @ A4 ) @ B ) ) ).
% vimageE
thf(fact_439_vimageE,axiom,
! [A4: a,F: a > c,B: set_c] :
( ( member_a @ A4 @ ( vimage_a_c @ F @ B ) )
=> ( member_c @ ( F @ A4 ) @ B ) ) ).
% vimageE
thf(fact_440_vimageE,axiom,
! [A4: nat,F: nat > nat,B: set_nat] :
( ( member_nat @ A4 @ ( vimage_nat_nat @ F @ B ) )
=> ( member_nat @ ( F @ A4 ) @ B ) ) ).
% vimageE
thf(fact_441_vimageD,axiom,
! [A4: b,F: b > b,A3: set_b] :
( ( member_b @ A4 @ ( vimage_b_b @ F @ A3 ) )
=> ( member_b @ ( F @ A4 ) @ A3 ) ) ).
% vimageD
thf(fact_442_vimageD,axiom,
! [A4: b,F: b > $o,A3: set_o] :
( ( member_b @ A4 @ ( vimage_b_o @ F @ A3 ) )
=> ( member_o @ ( F @ A4 ) @ A3 ) ) ).
% vimageD
thf(fact_443_vimageD,axiom,
! [A4: b,F: b > nat,A3: set_nat] :
( ( member_b @ A4 @ ( vimage_b_nat @ F @ A3 ) )
=> ( member_nat @ ( F @ A4 ) @ A3 ) ) ).
% vimageD
thf(fact_444_vimageD,axiom,
! [A4: $o,F: $o > b,A3: set_b] :
( ( member_o @ A4 @ ( vimage_o_b @ F @ A3 ) )
=> ( member_b @ ( F @ A4 ) @ A3 ) ) ).
% vimageD
thf(fact_445_vimageD,axiom,
! [A4: $o,F: $o > $o,A3: set_o] :
( ( member_o @ A4 @ ( vimage_o_o @ F @ A3 ) )
=> ( member_o @ ( F @ A4 ) @ A3 ) ) ).
% vimageD
thf(fact_446_vimageD,axiom,
! [A4: $o,F: $o > nat,A3: set_nat] :
( ( member_o @ A4 @ ( vimage_o_nat @ F @ A3 ) )
=> ( member_nat @ ( F @ A4 ) @ A3 ) ) ).
% vimageD
thf(fact_447_vimageD,axiom,
! [A4: nat,F: nat > b,A3: set_b] :
( ( member_nat @ A4 @ ( vimage_nat_b @ F @ A3 ) )
=> ( member_b @ ( F @ A4 ) @ A3 ) ) ).
% vimageD
thf(fact_448_vimageD,axiom,
! [A4: nat,F: nat > $o,A3: set_o] :
( ( member_nat @ A4 @ ( vimage_nat_o @ F @ A3 ) )
=> ( member_o @ ( F @ A4 ) @ A3 ) ) ).
% vimageD
thf(fact_449_vimageD,axiom,
! [A4: a,F: a > c,A3: set_c] :
( ( member_a @ A4 @ ( vimage_a_c @ F @ A3 ) )
=> ( member_c @ ( F @ A4 ) @ A3 ) ) ).
% vimageD
thf(fact_450_vimageD,axiom,
! [A4: nat,F: nat > nat,A3: set_nat] :
( ( member_nat @ A4 @ ( vimage_nat_nat @ F @ A3 ) )
=> ( member_nat @ ( F @ A4 ) @ A3 ) ) ).
% vimageD
thf(fact_451_Compr__image__eq,axiom,
! [F: a > c,A3: set_a,P: c > $o] :
( ( collect_c
@ ^ [X2: c] :
( ( member_c @ X2 @ ( image_a_c @ F @ A3 ) )
& ( P @ X2 ) ) )
= ( image_a_c @ F
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A3 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_452_Compr__image__eq,axiom,
! [F: b > b,A3: set_b,P: b > $o] :
( ( collect_b
@ ^ [X2: b] :
( ( member_b @ X2 @ ( image_b_b @ F @ A3 ) )
& ( P @ X2 ) ) )
= ( image_b_b @ F
@ ( collect_b
@ ^ [X2: b] :
( ( member_b @ X2 @ A3 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_453_Compr__image__eq,axiom,
! [F: $o > b,A3: set_o,P: b > $o] :
( ( collect_b
@ ^ [X2: b] :
( ( member_b @ X2 @ ( image_o_b @ F @ A3 ) )
& ( P @ X2 ) ) )
= ( image_o_b @ F
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A3 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_454_Compr__image__eq,axiom,
! [F: b > $o,A3: set_b,P: $o > $o] :
( ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ ( image_b_o @ F @ A3 ) )
& ( P @ X2 ) ) )
= ( image_b_o @ F
@ ( collect_b
@ ^ [X2: b] :
( ( member_b @ X2 @ A3 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_455_Compr__image__eq,axiom,
! [F: $o > $o,A3: set_o,P: $o > $o] :
( ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ ( image_o_o @ F @ A3 ) )
& ( P @ X2 ) ) )
= ( image_o_o @ F
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A3 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_456_Compr__image__eq,axiom,
! [F: nat > b,A3: set_nat,P: b > $o] :
( ( collect_b
@ ^ [X2: b] :
( ( member_b @ X2 @ ( image_nat_b @ F @ A3 ) )
& ( P @ X2 ) ) )
= ( image_nat_b @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_457_Compr__image__eq,axiom,
! [F: nat > $o,A3: set_nat,P: $o > $o] :
( ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ ( image_nat_o @ F @ A3 ) )
& ( P @ X2 ) ) )
= ( image_nat_o @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_458_Compr__image__eq,axiom,
! [F: b > nat,A3: set_b,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_b_nat @ F @ A3 ) )
& ( P @ X2 ) ) )
= ( image_b_nat @ F
@ ( collect_b
@ ^ [X2: b] :
( ( member_b @ X2 @ A3 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_459_Compr__image__eq,axiom,
! [F: $o > nat,A3: set_o,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_o_nat @ F @ A3 ) )
& ( P @ X2 ) ) )
= ( image_o_nat @ F
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A3 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_460_Compr__image__eq,axiom,
! [F: nat > nat,A3: set_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_nat_nat @ F @ A3 ) )
& ( P @ X2 ) ) )
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_461_image__image,axiom,
! [F: c > c,G: a > c,A3: set_a] :
( ( image_c_c @ F @ ( image_a_c @ G @ A3 ) )
= ( image_a_c
@ ^ [X2: a] : ( F @ ( G @ X2 ) )
@ A3 ) ) ).
% image_image
thf(fact_462_image__image,axiom,
! [F: a > c,G: a > a,A3: set_a] :
( ( image_a_c @ F @ ( image_a_a @ G @ A3 ) )
= ( image_a_c
@ ^ [X2: a] : ( F @ ( G @ X2 ) )
@ A3 ) ) ).
% image_image
thf(fact_463_imageE,axiom,
! [B2: c,F: a > c,A3: set_a] :
( ( member_c @ B2 @ ( image_a_c @ F @ A3 ) )
=> ~ ! [X3: a] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_a @ X3 @ A3 ) ) ) ).
% imageE
thf(fact_464_imageE,axiom,
! [B2: b,F: b > b,A3: set_b] :
( ( member_b @ B2 @ ( image_b_b @ F @ A3 ) )
=> ~ ! [X3: b] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_b @ X3 @ A3 ) ) ) ).
% imageE
thf(fact_465_imageE,axiom,
! [B2: b,F: $o > b,A3: set_o] :
( ( member_b @ B2 @ ( image_o_b @ F @ A3 ) )
=> ~ ! [X3: $o] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_o @ X3 @ A3 ) ) ) ).
% imageE
thf(fact_466_imageE,axiom,
! [B2: b,F: nat > b,A3: set_nat] :
( ( member_b @ B2 @ ( image_nat_b @ F @ A3 ) )
=> ~ ! [X3: nat] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A3 ) ) ) ).
% imageE
thf(fact_467_imageE,axiom,
! [B2: $o,F: b > $o,A3: set_b] :
( ( member_o @ B2 @ ( image_b_o @ F @ A3 ) )
=> ~ ! [X3: b] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_b @ X3 @ A3 ) ) ) ).
% imageE
thf(fact_468_imageE,axiom,
! [B2: $o,F: $o > $o,A3: set_o] :
( ( member_o @ B2 @ ( image_o_o @ F @ A3 ) )
=> ~ ! [X3: $o] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_o @ X3 @ A3 ) ) ) ).
% imageE
thf(fact_469_imageE,axiom,
! [B2: $o,F: nat > $o,A3: set_nat] :
( ( member_o @ B2 @ ( image_nat_o @ F @ A3 ) )
=> ~ ! [X3: nat] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A3 ) ) ) ).
% imageE
thf(fact_470_imageE,axiom,
! [B2: nat,F: b > nat,A3: set_b] :
( ( member_nat @ B2 @ ( image_b_nat @ F @ A3 ) )
=> ~ ! [X3: b] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_b @ X3 @ A3 ) ) ) ).
% imageE
thf(fact_471_imageE,axiom,
! [B2: nat,F: $o > nat,A3: set_o] :
( ( member_nat @ B2 @ ( image_o_nat @ F @ A3 ) )
=> ~ ! [X3: $o] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_o @ X3 @ A3 ) ) ) ).
% imageE
thf(fact_472_imageE,axiom,
! [B2: nat,F: nat > nat,A3: set_nat] :
( ( member_nat @ B2 @ ( image_nat_nat @ F @ A3 ) )
=> ~ ! [X3: nat] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A3 ) ) ) ).
% imageE
thf(fact_473_Collect__subset,axiom,
! [A3: set_set_c,P: set_c > $o] :
( ord_le3866738827743201120_set_c
@ ( collect_set_c
@ ^ [X2: set_c] :
( ( member_set_c @ X2 @ A3 )
& ( P @ X2 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_474_Collect__subset,axiom,
! [A3: set_b,P: b > $o] :
( ord_less_eq_set_b
@ ( collect_b
@ ^ [X2: b] :
( ( member_b @ X2 @ A3 )
& ( P @ X2 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_475_Collect__subset,axiom,
! [A3: set_o,P: $o > $o] :
( ord_less_eq_set_o
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A3 )
& ( P @ X2 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_476_Collect__subset,axiom,
! [A3: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ( P @ X2 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_477_Collect__subset,axiom,
! [A3: set_c,P: c > $o] :
( ord_less_eq_set_c
@ ( collect_c
@ ^ [X2: c] :
( ( member_c @ X2 @ A3 )
& ( P @ X2 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_478_Collect__subset,axiom,
! [A3: set_set_a,P: set_a > $o] :
( ord_le3724670747650509150_set_a
@ ( collect_set_a
@ ^ [X2: set_a] :
( ( member_set_a @ X2 @ A3 )
& ( P @ X2 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_479_Collect__subset,axiom,
! [A3: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A3 )
& ( P @ X2 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_480_Collect__conj__eq,axiom,
! [P: a > $o,Q: a > $o] :
( ( collect_a
@ ^ [X2: a] :
( ( P @ X2 )
& ( Q @ X2 ) ) )
= ( inf_inf_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_481_Collect__conj__eq,axiom,
! [P: c > $o,Q: c > $o] :
( ( collect_c
@ ^ [X2: c] :
( ( P @ X2 )
& ( Q @ X2 ) ) )
= ( inf_inf_set_c @ ( collect_c @ P ) @ ( collect_c @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_482_Collect__conj__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( P @ X2 )
& ( Q @ X2 ) ) )
= ( inf_inf_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_483_Collect__conj__eq,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( collect_set_a
@ ^ [X2: set_a] :
( ( P @ X2 )
& ( Q @ X2 ) ) )
= ( inf_inf_set_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_484_Int__Collect,axiom,
! [X: set_c,A3: set_set_c,P: set_c > $o] :
( ( member_set_c @ X @ ( inf_inf_set_set_c @ A3 @ ( collect_set_c @ P ) ) )
= ( ( member_set_c @ X @ A3 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_485_Int__Collect,axiom,
! [X: b,A3: set_b,P: b > $o] :
( ( member_b @ X @ ( inf_inf_set_b @ A3 @ ( collect_b @ P ) ) )
= ( ( member_b @ X @ A3 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_486_Int__Collect,axiom,
! [X: $o,A3: set_o,P: $o > $o] :
( ( member_o @ X @ ( inf_inf_set_o @ A3 @ ( collect_o @ P ) ) )
= ( ( member_o @ X @ A3 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_487_Int__Collect,axiom,
! [X: a,A3: set_a,P: a > $o] :
( ( member_a @ X @ ( inf_inf_set_a @ A3 @ ( collect_a @ P ) ) )
= ( ( member_a @ X @ A3 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_488_Int__Collect,axiom,
! [X: c,A3: set_c,P: c > $o] :
( ( member_c @ X @ ( inf_inf_set_c @ A3 @ ( collect_c @ P ) ) )
= ( ( member_c @ X @ A3 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_489_Int__Collect,axiom,
! [X: nat,A3: set_nat,P: nat > $o] :
( ( member_nat @ X @ ( inf_inf_set_nat @ A3 @ ( collect_nat @ P ) ) )
= ( ( member_nat @ X @ A3 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_490_Int__Collect,axiom,
! [X: set_a,A3: set_set_a,P: set_a > $o] :
( ( member_set_a @ X @ ( inf_inf_set_set_a @ A3 @ ( collect_set_a @ P ) ) )
= ( ( member_set_a @ X @ A3 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_491_Int__def,axiom,
( inf_inf_set_set_c
= ( ^ [A: set_set_c,B3: set_set_c] :
( collect_set_c
@ ^ [X2: set_c] :
( ( member_set_c @ X2 @ A )
& ( member_set_c @ X2 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_492_Int__def,axiom,
( inf_inf_set_b
= ( ^ [A: set_b,B3: set_b] :
( collect_b
@ ^ [X2: b] :
( ( member_b @ X2 @ A )
& ( member_b @ X2 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_493_Int__def,axiom,
( inf_inf_set_o
= ( ^ [A: set_o,B3: set_o] :
( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A )
& ( member_o @ X2 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_494_Int__def,axiom,
( inf_inf_set_a
= ( ^ [A: set_a,B3: set_a] :
( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A )
& ( member_a @ X2 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_495_Int__def,axiom,
( inf_inf_set_c
= ( ^ [A: set_c,B3: set_c] :
( collect_c
@ ^ [X2: c] :
( ( member_c @ X2 @ A )
& ( member_c @ X2 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_496_Int__def,axiom,
( inf_inf_set_nat
= ( ^ [A: set_nat,B3: set_nat] :
( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A )
& ( member_nat @ X2 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_497_Int__def,axiom,
( inf_inf_set_set_a
= ( ^ [A: set_set_a,B3: set_set_a] :
( collect_set_a
@ ^ [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ( member_set_a @ X2 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_498_vimage__def,axiom,
( vimage_a_c
= ( ^ [F3: a > c,B3: set_c] :
( collect_a
@ ^ [X2: a] : ( member_c @ ( F3 @ X2 ) @ B3 ) ) ) ) ).
% vimage_def
thf(fact_499_vimage__def,axiom,
( vimage_c_c
= ( ^ [F3: c > c,B3: set_c] :
( collect_c
@ ^ [X2: c] : ( member_c @ ( F3 @ X2 ) @ B3 ) ) ) ) ).
% vimage_def
thf(fact_500_vimage__def,axiom,
( vimage_c_a
= ( ^ [F3: c > a,B3: set_a] :
( collect_c
@ ^ [X2: c] : ( member_a @ ( F3 @ X2 ) @ B3 ) ) ) ) ).
% vimage_def
thf(fact_501_vimage__def,axiom,
( vimage_a_a
= ( ^ [F3: a > a,B3: set_a] :
( collect_a
@ ^ [X2: a] : ( member_a @ ( F3 @ X2 ) @ B3 ) ) ) ) ).
% vimage_def
thf(fact_502_vimage__def,axiom,
( vimage_nat_b
= ( ^ [F3: nat > b,B3: set_b] :
( collect_nat
@ ^ [X2: nat] : ( member_b @ ( F3 @ X2 ) @ B3 ) ) ) ) ).
% vimage_def
thf(fact_503_vimage__def,axiom,
( vimage_nat_o
= ( ^ [F3: nat > $o,B3: set_o] :
( collect_nat
@ ^ [X2: nat] : ( member_o @ ( F3 @ X2 ) @ B3 ) ) ) ) ).
% vimage_def
thf(fact_504_vimage__def,axiom,
( vimage_nat_nat
= ( ^ [F3: nat > nat,B3: set_nat] :
( collect_nat
@ ^ [X2: nat] : ( member_nat @ ( F3 @ X2 ) @ B3 ) ) ) ) ).
% vimage_def
thf(fact_505_vimage__def,axiom,
( vimage_set_a_b
= ( ^ [F3: set_a > b,B3: set_b] :
( collect_set_a
@ ^ [X2: set_a] : ( member_b @ ( F3 @ X2 ) @ B3 ) ) ) ) ).
% vimage_def
thf(fact_506_vimage__def,axiom,
( vimage_set_a_o
= ( ^ [F3: set_a > $o,B3: set_o] :
( collect_set_a
@ ^ [X2: set_a] : ( member_o @ ( F3 @ X2 ) @ B3 ) ) ) ) ).
% vimage_def
thf(fact_507_vimage__def,axiom,
( vimage_set_a_nat
= ( ^ [F3: set_a > nat,B3: set_nat] :
( collect_set_a
@ ^ [X2: set_a] : ( member_nat @ ( F3 @ X2 ) @ B3 ) ) ) ) ).
% vimage_def
thf(fact_508_inf_OcoboundedI2,axiom,
! [B2: set_nat,C2: set_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A4 @ B2 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_509_inf_OcoboundedI2,axiom,
! [B2: set_c,C2: set_c,A4: set_c] :
( ( ord_less_eq_set_c @ B2 @ C2 )
=> ( ord_less_eq_set_c @ ( inf_inf_set_c @ A4 @ B2 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_510_inf_OcoboundedI2,axiom,
! [B2: set_set_a,C2: set_set_a,A4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ C2 )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A4 @ B2 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_511_inf_OcoboundedI2,axiom,
! [B2: nat,C2: nat,A4: nat] :
( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A4 @ B2 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_512_inf_OcoboundedI2,axiom,
! [B2: extended_ereal,C2: extended_ereal,A4: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ B2 @ C2 )
=> ( ord_le1083603963089353582_ereal @ ( inf_in2794916579150040252_ereal @ A4 @ B2 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_513_inf_OcoboundedI2,axiom,
! [B2: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal,A4: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B2 @ C2 )
=> ( ord_le3935885782089961368nnreal @ ( inf_in7439215052339218890nnreal @ A4 @ B2 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_514_inf_OcoboundedI2,axiom,
! [B2: set_a,C2: set_a,A4: set_a] :
( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A4 @ B2 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_515_inf_OcoboundedI1,axiom,
! [A4: set_nat,C2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ C2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A4 @ B2 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_516_inf_OcoboundedI1,axiom,
! [A4: set_c,C2: set_c,B2: set_c] :
( ( ord_less_eq_set_c @ A4 @ C2 )
=> ( ord_less_eq_set_c @ ( inf_inf_set_c @ A4 @ B2 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_517_inf_OcoboundedI1,axiom,
! [A4: set_set_a,C2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ C2 )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A4 @ B2 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_518_inf_OcoboundedI1,axiom,
! [A4: nat,C2: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A4 @ B2 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_519_inf_OcoboundedI1,axiom,
! [A4: extended_ereal,C2: extended_ereal,B2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A4 @ C2 )
=> ( ord_le1083603963089353582_ereal @ ( inf_in2794916579150040252_ereal @ A4 @ B2 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_520_inf_OcoboundedI1,axiom,
! [A4: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A4 @ C2 )
=> ( ord_le3935885782089961368nnreal @ ( inf_in7439215052339218890nnreal @ A4 @ B2 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_521_inf_OcoboundedI1,axiom,
! [A4: set_a,C2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A4 @ C2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A4 @ B2 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_522_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [B4: set_nat,A2: set_nat] :
( ( inf_inf_set_nat @ A2 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_523_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_c
= ( ^ [B4: set_c,A2: set_c] :
( ( inf_inf_set_c @ A2 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_524_inf_Oabsorb__iff2,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [B4: set_set_a,A2: set_set_a] :
( ( inf_inf_set_set_a @ A2 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_525_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A2: nat] :
( ( inf_inf_nat @ A2 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_526_inf_Oabsorb__iff2,axiom,
( ord_le1083603963089353582_ereal
= ( ^ [B4: extended_ereal,A2: extended_ereal] :
( ( inf_in2794916579150040252_ereal @ A2 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_527_inf_Oabsorb__iff2,axiom,
( ord_le3935885782089961368nnreal
= ( ^ [B4: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal] :
( ( inf_in7439215052339218890nnreal @ A2 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_528_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A2: set_a] :
( ( inf_inf_set_a @ A2 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_529_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [A2: set_nat,B4: set_nat] :
( ( inf_inf_set_nat @ A2 @ B4 )
= A2 ) ) ) ).
% inf.absorb_iff1
thf(fact_530_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_c
= ( ^ [A2: set_c,B4: set_c] :
( ( inf_inf_set_c @ A2 @ B4 )
= A2 ) ) ) ).
% inf.absorb_iff1
thf(fact_531_inf_Oabsorb__iff1,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A2: set_set_a,B4: set_set_a] :
( ( inf_inf_set_set_a @ A2 @ B4 )
= A2 ) ) ) ).
% inf.absorb_iff1
thf(fact_532_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B4: nat] :
( ( inf_inf_nat @ A2 @ B4 )
= A2 ) ) ) ).
% inf.absorb_iff1
thf(fact_533_inf_Oabsorb__iff1,axiom,
( ord_le1083603963089353582_ereal
= ( ^ [A2: extended_ereal,B4: extended_ereal] :
( ( inf_in2794916579150040252_ereal @ A2 @ B4 )
= A2 ) ) ) ).
% inf.absorb_iff1
thf(fact_534_inf_Oabsorb__iff1,axiom,
( ord_le3935885782089961368nnreal
= ( ^ [A2: extend8495563244428889912nnreal,B4: extend8495563244428889912nnreal] :
( ( inf_in7439215052339218890nnreal @ A2 @ B4 )
= A2 ) ) ) ).
% inf.absorb_iff1
thf(fact_535_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A2: set_a,B4: set_a] :
( ( inf_inf_set_a @ A2 @ B4 )
= A2 ) ) ) ).
% inf.absorb_iff1
thf(fact_536_inf_Ocobounded2,axiom,
! [A4: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A4 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_537_inf_Ocobounded2,axiom,
! [A4: set_c,B2: set_c] : ( ord_less_eq_set_c @ ( inf_inf_set_c @ A4 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_538_inf_Ocobounded2,axiom,
! [A4: set_set_a,B2: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A4 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_539_inf_Ocobounded2,axiom,
! [A4: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A4 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_540_inf_Ocobounded2,axiom,
! [A4: extended_ereal,B2: extended_ereal] : ( ord_le1083603963089353582_ereal @ ( inf_in2794916579150040252_ereal @ A4 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_541_inf_Ocobounded2,axiom,
! [A4: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ ( inf_in7439215052339218890nnreal @ A4 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_542_inf_Ocobounded2,axiom,
! [A4: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A4 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_543_inf_Ocobounded1,axiom,
! [A4: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A4 @ B2 ) @ A4 ) ).
% inf.cobounded1
thf(fact_544_inf_Ocobounded1,axiom,
! [A4: set_c,B2: set_c] : ( ord_less_eq_set_c @ ( inf_inf_set_c @ A4 @ B2 ) @ A4 ) ).
% inf.cobounded1
thf(fact_545_inf_Ocobounded1,axiom,
! [A4: set_set_a,B2: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A4 @ B2 ) @ A4 ) ).
% inf.cobounded1
thf(fact_546_inf_Ocobounded1,axiom,
! [A4: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A4 @ B2 ) @ A4 ) ).
% inf.cobounded1
thf(fact_547_inf_Ocobounded1,axiom,
! [A4: extended_ereal,B2: extended_ereal] : ( ord_le1083603963089353582_ereal @ ( inf_in2794916579150040252_ereal @ A4 @ B2 ) @ A4 ) ).
% inf.cobounded1
thf(fact_548_inf_Ocobounded1,axiom,
! [A4: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ ( inf_in7439215052339218890nnreal @ A4 @ B2 ) @ A4 ) ).
% inf.cobounded1
thf(fact_549_inf_Ocobounded1,axiom,
! [A4: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A4 @ B2 ) @ A4 ) ).
% inf.cobounded1
thf(fact_550_inf_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A2: set_nat,B4: set_nat] :
( A2
= ( inf_inf_set_nat @ A2 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_551_inf_Oorder__iff,axiom,
( ord_less_eq_set_c
= ( ^ [A2: set_c,B4: set_c] :
( A2
= ( inf_inf_set_c @ A2 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_552_inf_Oorder__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A2: set_set_a,B4: set_set_a] :
( A2
= ( inf_inf_set_set_a @ A2 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_553_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B4: nat] :
( A2
= ( inf_inf_nat @ A2 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_554_inf_Oorder__iff,axiom,
( ord_le1083603963089353582_ereal
= ( ^ [A2: extended_ereal,B4: extended_ereal] :
( A2
= ( inf_in2794916579150040252_ereal @ A2 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_555_inf_Oorder__iff,axiom,
( ord_le3935885782089961368nnreal
= ( ^ [A2: extend8495563244428889912nnreal,B4: extend8495563244428889912nnreal] :
( A2
= ( inf_in7439215052339218890nnreal @ A2 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_556_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A2: set_a,B4: set_a] :
( A2
= ( inf_inf_set_a @ A2 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_557_inf__greatest,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ X @ Z )
=> ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_558_inf__greatest,axiom,
! [X: set_c,Y: set_c,Z: set_c] :
( ( ord_less_eq_set_c @ X @ Y )
=> ( ( ord_less_eq_set_c @ X @ Z )
=> ( ord_less_eq_set_c @ X @ ( inf_inf_set_c @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_559_inf__greatest,axiom,
! [X: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y )
=> ( ( ord_le3724670747650509150_set_a @ X @ Z )
=> ( ord_le3724670747650509150_set_a @ X @ ( inf_inf_set_set_a @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_560_inf__greatest,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Z )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_561_inf__greatest,axiom,
! [X: extended_ereal,Y: extended_ereal,Z: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X @ Y )
=> ( ( ord_le1083603963089353582_ereal @ X @ Z )
=> ( ord_le1083603963089353582_ereal @ X @ ( inf_in2794916579150040252_ereal @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_562_inf__greatest,axiom,
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal,Z: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X @ Y )
=> ( ( ord_le3935885782089961368nnreal @ X @ Z )
=> ( ord_le3935885782089961368nnreal @ X @ ( inf_in7439215052339218890nnreal @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_563_inf__greatest,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ X @ Z )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_564_inf_OboundedI,axiom,
! [A4: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B2 )
=> ( ( ord_less_eq_set_nat @ A4 @ C2 )
=> ( ord_less_eq_set_nat @ A4 @ ( inf_inf_set_nat @ B2 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_565_inf_OboundedI,axiom,
! [A4: set_c,B2: set_c,C2: set_c] :
( ( ord_less_eq_set_c @ A4 @ B2 )
=> ( ( ord_less_eq_set_c @ A4 @ C2 )
=> ( ord_less_eq_set_c @ A4 @ ( inf_inf_set_c @ B2 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_566_inf_OboundedI,axiom,
! [A4: set_set_a,B2: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ A4 @ C2 )
=> ( ord_le3724670747650509150_set_a @ A4 @ ( inf_inf_set_set_a @ B2 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_567_inf_OboundedI,axiom,
! [A4: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ord_less_eq_nat @ A4 @ C2 )
=> ( ord_less_eq_nat @ A4 @ ( inf_inf_nat @ B2 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_568_inf_OboundedI,axiom,
! [A4: extended_ereal,B2: extended_ereal,C2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A4 @ B2 )
=> ( ( ord_le1083603963089353582_ereal @ A4 @ C2 )
=> ( ord_le1083603963089353582_ereal @ A4 @ ( inf_in2794916579150040252_ereal @ B2 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_569_inf_OboundedI,axiom,
! [A4: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A4 @ B2 )
=> ( ( ord_le3935885782089961368nnreal @ A4 @ C2 )
=> ( ord_le3935885782089961368nnreal @ A4 @ ( inf_in7439215052339218890nnreal @ B2 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_570_inf_OboundedI,axiom,
! [A4: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A4 @ B2 )
=> ( ( ord_less_eq_set_a @ A4 @ C2 )
=> ( ord_less_eq_set_a @ A4 @ ( inf_inf_set_a @ B2 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_571_inf_OboundedE,axiom,
! [A4: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ ( inf_inf_set_nat @ B2 @ C2 ) )
=> ~ ( ( ord_less_eq_set_nat @ A4 @ B2 )
=> ~ ( ord_less_eq_set_nat @ A4 @ C2 ) ) ) ).
% inf.boundedE
thf(fact_572_inf_OboundedE,axiom,
! [A4: set_c,B2: set_c,C2: set_c] :
( ( ord_less_eq_set_c @ A4 @ ( inf_inf_set_c @ B2 @ C2 ) )
=> ~ ( ( ord_less_eq_set_c @ A4 @ B2 )
=> ~ ( ord_less_eq_set_c @ A4 @ C2 ) ) ) ).
% inf.boundedE
thf(fact_573_inf_OboundedE,axiom,
! [A4: set_set_a,B2: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ ( inf_inf_set_set_a @ B2 @ C2 ) )
=> ~ ( ( ord_le3724670747650509150_set_a @ A4 @ B2 )
=> ~ ( ord_le3724670747650509150_set_a @ A4 @ C2 ) ) ) ).
% inf.boundedE
thf(fact_574_inf_OboundedE,axiom,
! [A4: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A4 @ ( inf_inf_nat @ B2 @ C2 ) )
=> ~ ( ( ord_less_eq_nat @ A4 @ B2 )
=> ~ ( ord_less_eq_nat @ A4 @ C2 ) ) ) ).
% inf.boundedE
thf(fact_575_inf_OboundedE,axiom,
! [A4: extended_ereal,B2: extended_ereal,C2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A4 @ ( inf_in2794916579150040252_ereal @ B2 @ C2 ) )
=> ~ ( ( ord_le1083603963089353582_ereal @ A4 @ B2 )
=> ~ ( ord_le1083603963089353582_ereal @ A4 @ C2 ) ) ) ).
% inf.boundedE
thf(fact_576_inf_OboundedE,axiom,
! [A4: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A4 @ ( inf_in7439215052339218890nnreal @ B2 @ C2 ) )
=> ~ ( ( ord_le3935885782089961368nnreal @ A4 @ B2 )
=> ~ ( ord_le3935885782089961368nnreal @ A4 @ C2 ) ) ) ).
% inf.boundedE
thf(fact_577_inf_OboundedE,axiom,
! [A4: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A4 @ ( inf_inf_set_a @ B2 @ C2 ) )
=> ~ ( ( ord_less_eq_set_a @ A4 @ B2 )
=> ~ ( ord_less_eq_set_a @ A4 @ C2 ) ) ) ).
% inf.boundedE
thf(fact_578_inf__absorb2,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( inf_inf_set_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_579_inf__absorb2,axiom,
! [Y: set_c,X: set_c] :
( ( ord_less_eq_set_c @ Y @ X )
=> ( ( inf_inf_set_c @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_580_inf__absorb2,axiom,
! [Y: set_set_a,X: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y @ X )
=> ( ( inf_inf_set_set_a @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_581_inf__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( inf_inf_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_582_inf__absorb2,axiom,
! [Y: extended_ereal,X: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ Y @ X )
=> ( ( inf_in2794916579150040252_ereal @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_583_inf__absorb2,axiom,
! [Y: extend8495563244428889912nnreal,X: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ Y @ X )
=> ( ( inf_in7439215052339218890nnreal @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_584_inf__absorb2,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( inf_inf_set_a @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_585_inf__absorb1,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( inf_inf_set_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_586_inf__absorb1,axiom,
! [X: set_c,Y: set_c] :
( ( ord_less_eq_set_c @ X @ Y )
=> ( ( inf_inf_set_c @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_587_inf__absorb1,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y )
=> ( ( inf_inf_set_set_a @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_588_inf__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( inf_inf_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_589_inf__absorb1,axiom,
! [X: extended_ereal,Y: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X @ Y )
=> ( ( inf_in2794916579150040252_ereal @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_590_inf__absorb1,axiom,
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X @ Y )
=> ( ( inf_in7439215052339218890nnreal @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_591_inf__absorb1,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( inf_inf_set_a @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_592_inf_Oabsorb2,axiom,
! [B2: set_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A4 )
=> ( ( inf_inf_set_nat @ A4 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_593_inf_Oabsorb2,axiom,
! [B2: set_c,A4: set_c] :
( ( ord_less_eq_set_c @ B2 @ A4 )
=> ( ( inf_inf_set_c @ A4 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_594_inf_Oabsorb2,axiom,
! [B2: set_set_a,A4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A4 )
=> ( ( inf_inf_set_set_a @ A4 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_595_inf_Oabsorb2,axiom,
! [B2: nat,A4: nat] :
( ( ord_less_eq_nat @ B2 @ A4 )
=> ( ( inf_inf_nat @ A4 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_596_inf_Oabsorb2,axiom,
! [B2: extended_ereal,A4: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ B2 @ A4 )
=> ( ( inf_in2794916579150040252_ereal @ A4 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_597_inf_Oabsorb2,axiom,
! [B2: extend8495563244428889912nnreal,A4: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B2 @ A4 )
=> ( ( inf_in7439215052339218890nnreal @ A4 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_598_inf_Oabsorb2,axiom,
! [B2: set_a,A4: set_a] :
( ( ord_less_eq_set_a @ B2 @ A4 )
=> ( ( inf_inf_set_a @ A4 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_599_inf_Oabsorb1,axiom,
! [A4: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B2 )
=> ( ( inf_inf_set_nat @ A4 @ B2 )
= A4 ) ) ).
% inf.absorb1
thf(fact_600_inf_Oabsorb1,axiom,
! [A4: set_c,B2: set_c] :
( ( ord_less_eq_set_c @ A4 @ B2 )
=> ( ( inf_inf_set_c @ A4 @ B2 )
= A4 ) ) ).
% inf.absorb1
thf(fact_601_inf_Oabsorb1,axiom,
! [A4: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ B2 )
=> ( ( inf_inf_set_set_a @ A4 @ B2 )
= A4 ) ) ).
% inf.absorb1
thf(fact_602_inf_Oabsorb1,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( inf_inf_nat @ A4 @ B2 )
= A4 ) ) ).
% inf.absorb1
thf(fact_603_inf_Oabsorb1,axiom,
! [A4: extended_ereal,B2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A4 @ B2 )
=> ( ( inf_in2794916579150040252_ereal @ A4 @ B2 )
= A4 ) ) ).
% inf.absorb1
thf(fact_604_inf_Oabsorb1,axiom,
! [A4: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A4 @ B2 )
=> ( ( inf_in7439215052339218890nnreal @ A4 @ B2 )
= A4 ) ) ).
% inf.absorb1
thf(fact_605_inf_Oabsorb1,axiom,
! [A4: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A4 @ B2 )
=> ( ( inf_inf_set_a @ A4 @ B2 )
= A4 ) ) ).
% inf.absorb1
thf(fact_606_le__iff__inf,axiom,
( ord_less_eq_set_nat
= ( ^ [X2: set_nat,Y3: set_nat] :
( ( inf_inf_set_nat @ X2 @ Y3 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_607_le__iff__inf,axiom,
( ord_less_eq_set_c
= ( ^ [X2: set_c,Y3: set_c] :
( ( inf_inf_set_c @ X2 @ Y3 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_608_le__iff__inf,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [X2: set_set_a,Y3: set_set_a] :
( ( inf_inf_set_set_a @ X2 @ Y3 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_609_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y3: nat] :
( ( inf_inf_nat @ X2 @ Y3 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_610_le__iff__inf,axiom,
( ord_le1083603963089353582_ereal
= ( ^ [X2: extended_ereal,Y3: extended_ereal] :
( ( inf_in2794916579150040252_ereal @ X2 @ Y3 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_611_le__iff__inf,axiom,
( ord_le3935885782089961368nnreal
= ( ^ [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( inf_in7439215052339218890nnreal @ X2 @ Y3 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_612_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X2: set_a,Y3: set_a] :
( ( inf_inf_set_a @ X2 @ Y3 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_613_inf__unique,axiom,
! [F: set_nat > set_nat > set_nat,X: set_nat,Y: set_nat] :
( ! [X3: set_nat,Y5: set_nat] : ( ord_less_eq_set_nat @ ( F @ X3 @ Y5 ) @ X3 )
=> ( ! [X3: set_nat,Y5: set_nat] : ( ord_less_eq_set_nat @ ( F @ X3 @ Y5 ) @ Y5 )
=> ( ! [X3: set_nat,Y5: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y5 )
=> ( ( ord_less_eq_set_nat @ X3 @ Z3 )
=> ( ord_less_eq_set_nat @ X3 @ ( F @ Y5 @ Z3 ) ) ) )
=> ( ( inf_inf_set_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_614_inf__unique,axiom,
! [F: set_c > set_c > set_c,X: set_c,Y: set_c] :
( ! [X3: set_c,Y5: set_c] : ( ord_less_eq_set_c @ ( F @ X3 @ Y5 ) @ X3 )
=> ( ! [X3: set_c,Y5: set_c] : ( ord_less_eq_set_c @ ( F @ X3 @ Y5 ) @ Y5 )
=> ( ! [X3: set_c,Y5: set_c,Z3: set_c] :
( ( ord_less_eq_set_c @ X3 @ Y5 )
=> ( ( ord_less_eq_set_c @ X3 @ Z3 )
=> ( ord_less_eq_set_c @ X3 @ ( F @ Y5 @ Z3 ) ) ) )
=> ( ( inf_inf_set_c @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_615_inf__unique,axiom,
! [F: set_set_a > set_set_a > set_set_a,X: set_set_a,Y: set_set_a] :
( ! [X3: set_set_a,Y5: set_set_a] : ( ord_le3724670747650509150_set_a @ ( F @ X3 @ Y5 ) @ X3 )
=> ( ! [X3: set_set_a,Y5: set_set_a] : ( ord_le3724670747650509150_set_a @ ( F @ X3 @ Y5 ) @ Y5 )
=> ( ! [X3: set_set_a,Y5: set_set_a,Z3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y5 )
=> ( ( ord_le3724670747650509150_set_a @ X3 @ Z3 )
=> ( ord_le3724670747650509150_set_a @ X3 @ ( F @ Y5 @ Z3 ) ) ) )
=> ( ( inf_inf_set_set_a @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_616_inf__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X3: nat,Y5: nat] : ( ord_less_eq_nat @ ( F @ X3 @ Y5 ) @ X3 )
=> ( ! [X3: nat,Y5: nat] : ( ord_less_eq_nat @ ( F @ X3 @ Y5 ) @ Y5 )
=> ( ! [X3: nat,Y5: nat,Z3: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ( ord_less_eq_nat @ X3 @ Z3 )
=> ( ord_less_eq_nat @ X3 @ ( F @ Y5 @ Z3 ) ) ) )
=> ( ( inf_inf_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_617_inf__unique,axiom,
! [F: extended_ereal > extended_ereal > extended_ereal,X: extended_ereal,Y: extended_ereal] :
( ! [X3: extended_ereal,Y5: extended_ereal] : ( ord_le1083603963089353582_ereal @ ( F @ X3 @ Y5 ) @ X3 )
=> ( ! [X3: extended_ereal,Y5: extended_ereal] : ( ord_le1083603963089353582_ereal @ ( F @ X3 @ Y5 ) @ Y5 )
=> ( ! [X3: extended_ereal,Y5: extended_ereal,Z3: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X3 @ Y5 )
=> ( ( ord_le1083603963089353582_ereal @ X3 @ Z3 )
=> ( ord_le1083603963089353582_ereal @ X3 @ ( F @ Y5 @ Z3 ) ) ) )
=> ( ( inf_in2794916579150040252_ereal @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_618_inf__unique,axiom,
! [F: extend8495563244428889912nnreal > extend8495563244428889912nnreal > extend8495563244428889912nnreal,X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ! [X3: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ ( F @ X3 @ Y5 ) @ X3 )
=> ( ! [X3: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ ( F @ X3 @ Y5 ) @ Y5 )
=> ( ! [X3: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal,Z3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y5 )
=> ( ( ord_le3935885782089961368nnreal @ X3 @ Z3 )
=> ( ord_le3935885782089961368nnreal @ X3 @ ( F @ Y5 @ Z3 ) ) ) )
=> ( ( inf_in7439215052339218890nnreal @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_619_inf__unique,axiom,
! [F: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X3: set_a,Y5: set_a] : ( ord_less_eq_set_a @ ( F @ X3 @ Y5 ) @ X3 )
=> ( ! [X3: set_a,Y5: set_a] : ( ord_less_eq_set_a @ ( F @ X3 @ Y5 ) @ Y5 )
=> ( ! [X3: set_a,Y5: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y5 )
=> ( ( ord_less_eq_set_a @ X3 @ Z3 )
=> ( ord_less_eq_set_a @ X3 @ ( F @ Y5 @ Z3 ) ) ) )
=> ( ( inf_inf_set_a @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_620_inf_OorderI,axiom,
! [A4: set_nat,B2: set_nat] :
( ( A4
= ( inf_inf_set_nat @ A4 @ B2 ) )
=> ( ord_less_eq_set_nat @ A4 @ B2 ) ) ).
% inf.orderI
thf(fact_621_inf_OorderI,axiom,
! [A4: set_c,B2: set_c] :
( ( A4
= ( inf_inf_set_c @ A4 @ B2 ) )
=> ( ord_less_eq_set_c @ A4 @ B2 ) ) ).
% inf.orderI
thf(fact_622_inf_OorderI,axiom,
! [A4: set_set_a,B2: set_set_a] :
( ( A4
= ( inf_inf_set_set_a @ A4 @ B2 ) )
=> ( ord_le3724670747650509150_set_a @ A4 @ B2 ) ) ).
% inf.orderI
thf(fact_623_inf_OorderI,axiom,
! [A4: nat,B2: nat] :
( ( A4
= ( inf_inf_nat @ A4 @ B2 ) )
=> ( ord_less_eq_nat @ A4 @ B2 ) ) ).
% inf.orderI
thf(fact_624_inf_OorderI,axiom,
! [A4: extended_ereal,B2: extended_ereal] :
( ( A4
= ( inf_in2794916579150040252_ereal @ A4 @ B2 ) )
=> ( ord_le1083603963089353582_ereal @ A4 @ B2 ) ) ).
% inf.orderI
thf(fact_625_inf_OorderI,axiom,
! [A4: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( A4
= ( inf_in7439215052339218890nnreal @ A4 @ B2 ) )
=> ( ord_le3935885782089961368nnreal @ A4 @ B2 ) ) ).
% inf.orderI
thf(fact_626_inf_OorderI,axiom,
! [A4: set_a,B2: set_a] :
( ( A4
= ( inf_inf_set_a @ A4 @ B2 ) )
=> ( ord_less_eq_set_a @ A4 @ B2 ) ) ).
% inf.orderI
thf(fact_627_inf_OorderE,axiom,
! [A4: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B2 )
=> ( A4
= ( inf_inf_set_nat @ A4 @ B2 ) ) ) ).
% inf.orderE
thf(fact_628_inf_OorderE,axiom,
! [A4: set_c,B2: set_c] :
( ( ord_less_eq_set_c @ A4 @ B2 )
=> ( A4
= ( inf_inf_set_c @ A4 @ B2 ) ) ) ).
% inf.orderE
thf(fact_629_inf_OorderE,axiom,
! [A4: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ B2 )
=> ( A4
= ( inf_inf_set_set_a @ A4 @ B2 ) ) ) ).
% inf.orderE
thf(fact_630_inf_OorderE,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( A4
= ( inf_inf_nat @ A4 @ B2 ) ) ) ).
% inf.orderE
thf(fact_631_inf_OorderE,axiom,
! [A4: extended_ereal,B2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A4 @ B2 )
=> ( A4
= ( inf_in2794916579150040252_ereal @ A4 @ B2 ) ) ) ).
% inf.orderE
thf(fact_632_inf_OorderE,axiom,
! [A4: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A4 @ B2 )
=> ( A4
= ( inf_in7439215052339218890nnreal @ A4 @ B2 ) ) ) ).
% inf.orderE
thf(fact_633_inf_OorderE,axiom,
! [A4: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A4 @ B2 )
=> ( A4
= ( inf_inf_set_a @ A4 @ B2 ) ) ) ).
% inf.orderE
thf(fact_634_le__infI2,axiom,
! [B2: set_nat,X: set_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ X )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A4 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_635_le__infI2,axiom,
! [B2: set_c,X: set_c,A4: set_c] :
( ( ord_less_eq_set_c @ B2 @ X )
=> ( ord_less_eq_set_c @ ( inf_inf_set_c @ A4 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_636_le__infI2,axiom,
! [B2: set_set_a,X: set_set_a,A4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ X )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A4 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_637_le__infI2,axiom,
! [B2: nat,X: nat,A4: nat] :
( ( ord_less_eq_nat @ B2 @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A4 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_638_le__infI2,axiom,
! [B2: extended_ereal,X: extended_ereal,A4: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ B2 @ X )
=> ( ord_le1083603963089353582_ereal @ ( inf_in2794916579150040252_ereal @ A4 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_639_le__infI2,axiom,
! [B2: extend8495563244428889912nnreal,X: extend8495563244428889912nnreal,A4: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B2 @ X )
=> ( ord_le3935885782089961368nnreal @ ( inf_in7439215052339218890nnreal @ A4 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_640_le__infI2,axiom,
! [B2: set_a,X: set_a,A4: set_a] :
( ( ord_less_eq_set_a @ B2 @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A4 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_641_le__infI1,axiom,
! [A4: set_nat,X: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ X )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A4 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_642_le__infI1,axiom,
! [A4: set_c,X: set_c,B2: set_c] :
( ( ord_less_eq_set_c @ A4 @ X )
=> ( ord_less_eq_set_c @ ( inf_inf_set_c @ A4 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_643_le__infI1,axiom,
! [A4: set_set_a,X: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ X )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A4 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_644_le__infI1,axiom,
! [A4: nat,X: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A4 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_645_le__infI1,axiom,
! [A4: extended_ereal,X: extended_ereal,B2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A4 @ X )
=> ( ord_le1083603963089353582_ereal @ ( inf_in2794916579150040252_ereal @ A4 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_646_le__infI1,axiom,
! [A4: extend8495563244428889912nnreal,X: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A4 @ X )
=> ( ord_le3935885782089961368nnreal @ ( inf_in7439215052339218890nnreal @ A4 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_647_le__infI1,axiom,
! [A4: set_a,X: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A4 @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A4 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_648_inf__mono,axiom,
! [A4: set_nat,C2: set_nat,B2: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ D )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A4 @ B2 ) @ ( inf_inf_set_nat @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_649_inf__mono,axiom,
! [A4: set_c,C2: set_c,B2: set_c,D: set_c] :
( ( ord_less_eq_set_c @ A4 @ C2 )
=> ( ( ord_less_eq_set_c @ B2 @ D )
=> ( ord_less_eq_set_c @ ( inf_inf_set_c @ A4 @ B2 ) @ ( inf_inf_set_c @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_650_inf__mono,axiom,
! [A4: set_set_a,C2: set_set_a,B2: set_set_a,D: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ C2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ D )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A4 @ B2 ) @ ( inf_inf_set_set_a @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_651_inf__mono,axiom,
! [A4: nat,C2: nat,B2: nat,D: nat] :
( ( ord_less_eq_nat @ A4 @ C2 )
=> ( ( ord_less_eq_nat @ B2 @ D )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A4 @ B2 ) @ ( inf_inf_nat @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_652_inf__mono,axiom,
! [A4: extended_ereal,C2: extended_ereal,B2: extended_ereal,D: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A4 @ C2 )
=> ( ( ord_le1083603963089353582_ereal @ B2 @ D )
=> ( ord_le1083603963089353582_ereal @ ( inf_in2794916579150040252_ereal @ A4 @ B2 ) @ ( inf_in2794916579150040252_ereal @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_653_inf__mono,axiom,
! [A4: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,D: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A4 @ C2 )
=> ( ( ord_le3935885782089961368nnreal @ B2 @ D )
=> ( ord_le3935885782089961368nnreal @ ( inf_in7439215052339218890nnreal @ A4 @ B2 ) @ ( inf_in7439215052339218890nnreal @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_654_inf__mono,axiom,
! [A4: set_a,C2: set_a,B2: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A4 @ C2 )
=> ( ( ord_less_eq_set_a @ B2 @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A4 @ B2 ) @ ( inf_inf_set_a @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_655_le__infI,axiom,
! [X: set_nat,A4: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ X @ A4 )
=> ( ( ord_less_eq_set_nat @ X @ B2 )
=> ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ A4 @ B2 ) ) ) ) ).
% le_infI
thf(fact_656_le__infI,axiom,
! [X: set_c,A4: set_c,B2: set_c] :
( ( ord_less_eq_set_c @ X @ A4 )
=> ( ( ord_less_eq_set_c @ X @ B2 )
=> ( ord_less_eq_set_c @ X @ ( inf_inf_set_c @ A4 @ B2 ) ) ) ) ).
% le_infI
thf(fact_657_le__infI,axiom,
! [X: set_set_a,A4: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ A4 )
=> ( ( ord_le3724670747650509150_set_a @ X @ B2 )
=> ( ord_le3724670747650509150_set_a @ X @ ( inf_inf_set_set_a @ A4 @ B2 ) ) ) ) ).
% le_infI
thf(fact_658_le__infI,axiom,
! [X: nat,A4: nat,B2: nat] :
( ( ord_less_eq_nat @ X @ A4 )
=> ( ( ord_less_eq_nat @ X @ B2 )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A4 @ B2 ) ) ) ) ).
% le_infI
thf(fact_659_le__infI,axiom,
! [X: extended_ereal,A4: extended_ereal,B2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X @ A4 )
=> ( ( ord_le1083603963089353582_ereal @ X @ B2 )
=> ( ord_le1083603963089353582_ereal @ X @ ( inf_in2794916579150040252_ereal @ A4 @ B2 ) ) ) ) ).
% le_infI
thf(fact_660_le__infI,axiom,
! [X: extend8495563244428889912nnreal,A4: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X @ A4 )
=> ( ( ord_le3935885782089961368nnreal @ X @ B2 )
=> ( ord_le3935885782089961368nnreal @ X @ ( inf_in7439215052339218890nnreal @ A4 @ B2 ) ) ) ) ).
% le_infI
thf(fact_661_le__infI,axiom,
! [X: set_a,A4: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X @ A4 )
=> ( ( ord_less_eq_set_a @ X @ B2 )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A4 @ B2 ) ) ) ) ).
% le_infI
thf(fact_662_le__infE,axiom,
! [X: set_nat,A4: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ A4 @ B2 ) )
=> ~ ( ( ord_less_eq_set_nat @ X @ A4 )
=> ~ ( ord_less_eq_set_nat @ X @ B2 ) ) ) ).
% le_infE
thf(fact_663_le__infE,axiom,
! [X: set_c,A4: set_c,B2: set_c] :
( ( ord_less_eq_set_c @ X @ ( inf_inf_set_c @ A4 @ B2 ) )
=> ~ ( ( ord_less_eq_set_c @ X @ A4 )
=> ~ ( ord_less_eq_set_c @ X @ B2 ) ) ) ).
% le_infE
thf(fact_664_le__infE,axiom,
! [X: set_set_a,A4: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ ( inf_inf_set_set_a @ A4 @ B2 ) )
=> ~ ( ( ord_le3724670747650509150_set_a @ X @ A4 )
=> ~ ( ord_le3724670747650509150_set_a @ X @ B2 ) ) ) ).
% le_infE
thf(fact_665_le__infE,axiom,
! [X: nat,A4: nat,B2: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A4 @ B2 ) )
=> ~ ( ( ord_less_eq_nat @ X @ A4 )
=> ~ ( ord_less_eq_nat @ X @ B2 ) ) ) ).
% le_infE
thf(fact_666_le__infE,axiom,
! [X: extended_ereal,A4: extended_ereal,B2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X @ ( inf_in2794916579150040252_ereal @ A4 @ B2 ) )
=> ~ ( ( ord_le1083603963089353582_ereal @ X @ A4 )
=> ~ ( ord_le1083603963089353582_ereal @ X @ B2 ) ) ) ).
% le_infE
thf(fact_667_le__infE,axiom,
! [X: extend8495563244428889912nnreal,A4: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X @ ( inf_in7439215052339218890nnreal @ A4 @ B2 ) )
=> ~ ( ( ord_le3935885782089961368nnreal @ X @ A4 )
=> ~ ( ord_le3935885782089961368nnreal @ X @ B2 ) ) ) ).
% le_infE
thf(fact_668_le__infE,axiom,
! [X: set_a,A4: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A4 @ B2 ) )
=> ~ ( ( ord_less_eq_set_a @ X @ A4 )
=> ~ ( ord_less_eq_set_a @ X @ B2 ) ) ) ).
% le_infE
thf(fact_669_inf__le2,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_670_inf__le2,axiom,
! [X: set_c,Y: set_c] : ( ord_less_eq_set_c @ ( inf_inf_set_c @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_671_inf__le2,axiom,
! [X: set_set_a,Y: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_672_inf__le2,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_673_inf__le2,axiom,
! [X: extended_ereal,Y: extended_ereal] : ( ord_le1083603963089353582_ereal @ ( inf_in2794916579150040252_ereal @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_674_inf__le2,axiom,
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ ( inf_in7439215052339218890nnreal @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_675_inf__le2,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_676_inf__le1,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_677_inf__le1,axiom,
! [X: set_c,Y: set_c] : ( ord_less_eq_set_c @ ( inf_inf_set_c @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_678_inf__le1,axiom,
! [X: set_set_a,Y: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_679_inf__le1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_680_inf__le1,axiom,
! [X: extended_ereal,Y: extended_ereal] : ( ord_le1083603963089353582_ereal @ ( inf_in2794916579150040252_ereal @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_681_inf__le1,axiom,
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ ( inf_in7439215052339218890nnreal @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_682_inf__le1,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_683_inf__sup__ord_I1_J,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_684_inf__sup__ord_I1_J,axiom,
! [X: set_c,Y: set_c] : ( ord_less_eq_set_c @ ( inf_inf_set_c @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_685_inf__sup__ord_I1_J,axiom,
! [X: set_set_a,Y: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_686_inf__sup__ord_I1_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_687_inf__sup__ord_I1_J,axiom,
! [X: extended_ereal,Y: extended_ereal] : ( ord_le1083603963089353582_ereal @ ( inf_in2794916579150040252_ereal @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_688_inf__sup__ord_I1_J,axiom,
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ ( inf_in7439215052339218890nnreal @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_689_inf__sup__ord_I1_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_690_inf__sup__ord_I2_J,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_691_inf__sup__ord_I2_J,axiom,
! [X: set_c,Y: set_c] : ( ord_less_eq_set_c @ ( inf_inf_set_c @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_692_inf__sup__ord_I2_J,axiom,
! [X: set_set_a,Y: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_693_inf__sup__ord_I2_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_694_inf__sup__ord_I2_J,axiom,
! [X: extended_ereal,Y: extended_ereal] : ( ord_le1083603963089353582_ereal @ ( inf_in2794916579150040252_ereal @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_695_inf__sup__ord_I2_J,axiom,
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ ( inf_in7439215052339218890nnreal @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_696_inf__sup__ord_I2_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_697_subset__image__iff,axiom,
! [B: set_c,F: c > c,A3: set_c] :
( ( ord_less_eq_set_c @ B @ ( image_c_c @ F @ A3 ) )
= ( ? [AA: set_c] :
( ( ord_less_eq_set_c @ AA @ A3 )
& ( B
= ( image_c_c @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_698_subset__image__iff,axiom,
! [B: set_c,F: set_a > c,A3: set_set_a] :
( ( ord_less_eq_set_c @ B @ ( image_set_a_c @ F @ A3 ) )
= ( ? [AA: set_set_a] :
( ( ord_le3724670747650509150_set_a @ AA @ A3 )
& ( B
= ( image_set_a_c @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_699_subset__image__iff,axiom,
! [B: set_c,F: a > c,A3: set_a] :
( ( ord_less_eq_set_c @ B @ ( image_a_c @ F @ A3 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A3 )
& ( B
= ( image_a_c @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_700_subset__image__iff,axiom,
! [B: set_set_a,F: c > set_a,A3: set_c] :
( ( ord_le3724670747650509150_set_a @ B @ ( image_c_set_a @ F @ A3 ) )
= ( ? [AA: set_c] :
( ( ord_less_eq_set_c @ AA @ A3 )
& ( B
= ( image_c_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_701_subset__image__iff,axiom,
! [B: set_set_a,F: set_a > set_a,A3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ ( image_set_a_set_a @ F @ A3 ) )
= ( ? [AA: set_set_a] :
( ( ord_le3724670747650509150_set_a @ AA @ A3 )
& ( B
= ( image_set_a_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_702_subset__image__iff,axiom,
! [B: set_set_a,F: a > set_a,A3: set_a] :
( ( ord_le3724670747650509150_set_a @ B @ ( image_a_set_a @ F @ A3 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A3 )
& ( B
= ( image_a_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_703_subset__image__iff,axiom,
! [B: set_a,F: c > a,A3: set_c] :
( ( ord_less_eq_set_a @ B @ ( image_c_a @ F @ A3 ) )
= ( ? [AA: set_c] :
( ( ord_less_eq_set_c @ AA @ A3 )
& ( B
= ( image_c_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_704_subset__image__iff,axiom,
! [B: set_a,F: set_a > a,A3: set_set_a] :
( ( ord_less_eq_set_a @ B @ ( image_set_a_a @ F @ A3 ) )
= ( ? [AA: set_set_a] :
( ( ord_le3724670747650509150_set_a @ AA @ A3 )
& ( B
= ( image_set_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_705_subset__image__iff,axiom,
! [B: set_a,F: a > a,A3: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A3 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A3 )
& ( B
= ( image_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_706_image__subset__iff,axiom,
! [F: a > c,A3: set_a,B: set_c] :
( ( ord_less_eq_set_c @ ( image_a_c @ F @ A3 ) @ B )
= ( ! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ( member_c @ ( F @ X2 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_707_subset__imageE,axiom,
! [B: set_c,F: c > c,A3: set_c] :
( ( ord_less_eq_set_c @ B @ ( image_c_c @ F @ A3 ) )
=> ~ ! [C3: set_c] :
( ( ord_less_eq_set_c @ C3 @ A3 )
=> ( B
!= ( image_c_c @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_708_subset__imageE,axiom,
! [B: set_c,F: set_a > c,A3: set_set_a] :
( ( ord_less_eq_set_c @ B @ ( image_set_a_c @ F @ A3 ) )
=> ~ ! [C3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C3 @ A3 )
=> ( B
!= ( image_set_a_c @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_709_subset__imageE,axiom,
! [B: set_c,F: a > c,A3: set_a] :
( ( ord_less_eq_set_c @ B @ ( image_a_c @ F @ A3 ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A3 )
=> ( B
!= ( image_a_c @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_710_subset__imageE,axiom,
! [B: set_set_a,F: c > set_a,A3: set_c] :
( ( ord_le3724670747650509150_set_a @ B @ ( image_c_set_a @ F @ A3 ) )
=> ~ ! [C3: set_c] :
( ( ord_less_eq_set_c @ C3 @ A3 )
=> ( B
!= ( image_c_set_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_711_subset__imageE,axiom,
! [B: set_set_a,F: set_a > set_a,A3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ ( image_set_a_set_a @ F @ A3 ) )
=> ~ ! [C3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C3 @ A3 )
=> ( B
!= ( image_set_a_set_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_712_subset__imageE,axiom,
! [B: set_set_a,F: a > set_a,A3: set_a] :
( ( ord_le3724670747650509150_set_a @ B @ ( image_a_set_a @ F @ A3 ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A3 )
=> ( B
!= ( image_a_set_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_713_subset__imageE,axiom,
! [B: set_a,F: c > a,A3: set_c] :
( ( ord_less_eq_set_a @ B @ ( image_c_a @ F @ A3 ) )
=> ~ ! [C3: set_c] :
( ( ord_less_eq_set_c @ C3 @ A3 )
=> ( B
!= ( image_c_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_714_subset__imageE,axiom,
! [B: set_a,F: set_a > a,A3: set_set_a] :
( ( ord_less_eq_set_a @ B @ ( image_set_a_a @ F @ A3 ) )
=> ~ ! [C3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C3 @ A3 )
=> ( B
!= ( image_set_a_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_715_subset__imageE,axiom,
! [B: set_a,F: a > a,A3: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A3 ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A3 )
=> ( B
!= ( image_a_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_716_image__subsetI,axiom,
! [A3: set_b,F: b > b,B: set_b] :
( ! [X3: b] :
( ( member_b @ X3 @ A3 )
=> ( member_b @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_b @ ( image_b_b @ F @ A3 ) @ B ) ) ).
% image_subsetI
thf(fact_717_image__subsetI,axiom,
! [A3: set_b,F: b > $o,B: set_o] :
( ! [X3: b] :
( ( member_b @ X3 @ A3 )
=> ( member_o @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_b_o @ F @ A3 ) @ B ) ) ).
% image_subsetI
thf(fact_718_image__subsetI,axiom,
! [A3: set_b,F: b > nat,B: set_nat] :
( ! [X3: b] :
( ( member_b @ X3 @ A3 )
=> ( member_nat @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_b_nat @ F @ A3 ) @ B ) ) ).
% image_subsetI
thf(fact_719_image__subsetI,axiom,
! [A3: set_o,F: $o > b,B: set_b] :
( ! [X3: $o] :
( ( member_o @ X3 @ A3 )
=> ( member_b @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_b @ ( image_o_b @ F @ A3 ) @ B ) ) ).
% image_subsetI
thf(fact_720_image__subsetI,axiom,
! [A3: set_o,F: $o > $o,B: set_o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A3 )
=> ( member_o @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_o_o @ F @ A3 ) @ B ) ) ).
% image_subsetI
thf(fact_721_image__subsetI,axiom,
! [A3: set_o,F: $o > nat,B: set_nat] :
( ! [X3: $o] :
( ( member_o @ X3 @ A3 )
=> ( member_nat @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_o_nat @ F @ A3 ) @ B ) ) ).
% image_subsetI
thf(fact_722_image__subsetI,axiom,
! [A3: set_nat,F: nat > b,B: set_b] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( member_b @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_b @ ( image_nat_b @ F @ A3 ) @ B ) ) ).
% image_subsetI
thf(fact_723_image__subsetI,axiom,
! [A3: set_nat,F: nat > $o,B: set_o] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( member_o @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_nat_o @ F @ A3 ) @ B ) ) ).
% image_subsetI
thf(fact_724_image__subsetI,axiom,
! [A3: set_nat,F: nat > nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( member_nat @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A3 ) @ B ) ) ).
% image_subsetI
thf(fact_725_image__subsetI,axiom,
! [A3: set_a,F: a > c,B: set_c] :
( ! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ( member_c @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_c @ ( image_a_c @ F @ A3 ) @ B ) ) ).
% image_subsetI
thf(fact_726_image__mono,axiom,
! [A3: set_c,B: set_c,F: c > c] :
( ( ord_less_eq_set_c @ A3 @ B )
=> ( ord_less_eq_set_c @ ( image_c_c @ F @ A3 ) @ ( image_c_c @ F @ B ) ) ) ).
% image_mono
thf(fact_727_image__mono,axiom,
! [A3: set_c,B: set_c,F: c > set_a] :
( ( ord_less_eq_set_c @ A3 @ B )
=> ( ord_le3724670747650509150_set_a @ ( image_c_set_a @ F @ A3 ) @ ( image_c_set_a @ F @ B ) ) ) ).
% image_mono
thf(fact_728_image__mono,axiom,
! [A3: set_c,B: set_c,F: c > a] :
( ( ord_less_eq_set_c @ A3 @ B )
=> ( ord_less_eq_set_a @ ( image_c_a @ F @ A3 ) @ ( image_c_a @ F @ B ) ) ) ).
% image_mono
thf(fact_729_image__mono,axiom,
! [A3: set_set_a,B: set_set_a,F: set_a > c] :
( ( ord_le3724670747650509150_set_a @ A3 @ B )
=> ( ord_less_eq_set_c @ ( image_set_a_c @ F @ A3 ) @ ( image_set_a_c @ F @ B ) ) ) ).
% image_mono
thf(fact_730_image__mono,axiom,
! [A3: set_set_a,B: set_set_a,F: set_a > set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B )
=> ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ F @ A3 ) @ ( image_set_a_set_a @ F @ B ) ) ) ).
% image_mono
thf(fact_731_image__mono,axiom,
! [A3: set_set_a,B: set_set_a,F: set_a > a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B )
=> ( ord_less_eq_set_a @ ( image_set_a_a @ F @ A3 ) @ ( image_set_a_a @ F @ B ) ) ) ).
% image_mono
thf(fact_732_image__mono,axiom,
! [A3: set_a,B: set_a,F: a > c] :
( ( ord_less_eq_set_a @ A3 @ B )
=> ( ord_less_eq_set_c @ ( image_a_c @ F @ A3 ) @ ( image_a_c @ F @ B ) ) ) ).
% image_mono
thf(fact_733_image__mono,axiom,
! [A3: set_a,B: set_a,F: a > set_a] :
( ( ord_less_eq_set_a @ A3 @ B )
=> ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A3 ) @ ( image_a_set_a @ F @ B ) ) ) ).
% image_mono
thf(fact_734_image__mono,axiom,
! [A3: set_a,B: set_a,F: a > a] :
( ( ord_less_eq_set_a @ A3 @ B )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A3 ) @ ( image_a_a @ F @ B ) ) ) ).
% image_mono
thf(fact_735_Int__Collect__mono,axiom,
! [A3: set_set_c,B: set_set_c,P: set_c > $o,Q: set_c > $o] :
( ( ord_le3866738827743201120_set_c @ A3 @ B )
=> ( ! [X3: set_c] :
( ( member_set_c @ X3 @ A3 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le3866738827743201120_set_c @ ( inf_inf_set_set_c @ A3 @ ( collect_set_c @ P ) ) @ ( inf_inf_set_set_c @ B @ ( collect_set_c @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_736_Int__Collect__mono,axiom,
! [A3: set_b,B: set_b,P: b > $o,Q: b > $o] :
( ( ord_less_eq_set_b @ A3 @ B )
=> ( ! [X3: b] :
( ( member_b @ X3 @ A3 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A3 @ ( collect_b @ P ) ) @ ( inf_inf_set_b @ B @ ( collect_b @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_737_Int__Collect__mono,axiom,
! [A3: set_o,B: set_o,P: $o > $o,Q: $o > $o] :
( ( ord_less_eq_set_o @ A3 @ B )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A3 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_o @ ( inf_inf_set_o @ A3 @ ( collect_o @ P ) ) @ ( inf_inf_set_o @ B @ ( collect_o @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_738_Int__Collect__mono,axiom,
! [A3: set_nat,B: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A3 @ B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A3 @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_739_Int__Collect__mono,axiom,
! [A3: set_c,B: set_c,P: c > $o,Q: c > $o] :
( ( ord_less_eq_set_c @ A3 @ B )
=> ( ! [X3: c] :
( ( member_c @ X3 @ A3 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_c @ ( inf_inf_set_c @ A3 @ ( collect_c @ P ) ) @ ( inf_inf_set_c @ B @ ( collect_c @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_740_Int__Collect__mono,axiom,
! [A3: set_set_a,B: set_set_a,P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ A3 @ B )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A3 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A3 @ ( collect_set_a @ P ) ) @ ( inf_inf_set_set_a @ B @ ( collect_set_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_741_Int__Collect__mono,axiom,
! [A3: set_a,B: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A3 @ B )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_742_Int__greatest,axiom,
! [C: set_nat,A3: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C @ A3 )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ ( inf_inf_set_nat @ A3 @ B ) ) ) ) ).
% Int_greatest
thf(fact_743_Int__greatest,axiom,
! [C: set_c,A3: set_c,B: set_c] :
( ( ord_less_eq_set_c @ C @ A3 )
=> ( ( ord_less_eq_set_c @ C @ B )
=> ( ord_less_eq_set_c @ C @ ( inf_inf_set_c @ A3 @ B ) ) ) ) ).
% Int_greatest
thf(fact_744_Int__greatest,axiom,
! [C: set_set_a,A3: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C @ A3 )
=> ( ( ord_le3724670747650509150_set_a @ C @ B )
=> ( ord_le3724670747650509150_set_a @ C @ ( inf_inf_set_set_a @ A3 @ B ) ) ) ) ).
% Int_greatest
thf(fact_745_Int__greatest,axiom,
! [C: set_a,A3: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ A3 )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ ( inf_inf_set_a @ A3 @ B ) ) ) ) ).
% Int_greatest
thf(fact_746_Int__absorb2,axiom,
! [A3: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B )
=> ( ( inf_inf_set_nat @ A3 @ B )
= A3 ) ) ).
% Int_absorb2
thf(fact_747_Int__absorb2,axiom,
! [A3: set_c,B: set_c] :
( ( ord_less_eq_set_c @ A3 @ B )
=> ( ( inf_inf_set_c @ A3 @ B )
= A3 ) ) ).
% Int_absorb2
thf(fact_748_Int__absorb2,axiom,
! [A3: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B )
=> ( ( inf_inf_set_set_a @ A3 @ B )
= A3 ) ) ).
% Int_absorb2
thf(fact_749_Int__absorb2,axiom,
! [A3: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A3 @ B )
=> ( ( inf_inf_set_a @ A3 @ B )
= A3 ) ) ).
% Int_absorb2
thf(fact_750_Int__absorb1,axiom,
! [B: set_nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ B @ A3 )
=> ( ( inf_inf_set_nat @ A3 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_751_Int__absorb1,axiom,
! [B: set_c,A3: set_c] :
( ( ord_less_eq_set_c @ B @ A3 )
=> ( ( inf_inf_set_c @ A3 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_752_Int__absorb1,axiom,
! [B: set_set_a,A3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A3 )
=> ( ( inf_inf_set_set_a @ A3 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_753_Int__absorb1,axiom,
! [B: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B @ A3 )
=> ( ( inf_inf_set_a @ A3 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_754_Int__lower2,axiom,
! [A3: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A3 @ B ) @ B ) ).
% Int_lower2
thf(fact_755_Int__lower2,axiom,
! [A3: set_c,B: set_c] : ( ord_less_eq_set_c @ ( inf_inf_set_c @ A3 @ B ) @ B ) ).
% Int_lower2
thf(fact_756_Int__lower2,axiom,
! [A3: set_set_a,B: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A3 @ B ) @ B ) ).
% Int_lower2
thf(fact_757_Int__lower2,axiom,
! [A3: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B ) @ B ) ).
% Int_lower2
thf(fact_758_Int__lower1,axiom,
! [A3: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A3 @ B ) @ A3 ) ).
% Int_lower1
thf(fact_759_Int__lower1,axiom,
! [A3: set_c,B: set_c] : ( ord_less_eq_set_c @ ( inf_inf_set_c @ A3 @ B ) @ A3 ) ).
% Int_lower1
thf(fact_760_Int__lower1,axiom,
! [A3: set_set_a,B: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A3 @ B ) @ A3 ) ).
% Int_lower1
thf(fact_761_Int__lower1,axiom,
! [A3: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B ) @ A3 ) ).
% Int_lower1
thf(fact_762_Int__mono,axiom,
! [A3: set_nat,C: set_nat,B: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ C )
=> ( ( ord_less_eq_set_nat @ B @ D2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A3 @ B ) @ ( inf_inf_set_nat @ C @ D2 ) ) ) ) ).
% Int_mono
thf(fact_763_Int__mono,axiom,
! [A3: set_c,C: set_c,B: set_c,D2: set_c] :
( ( ord_less_eq_set_c @ A3 @ C )
=> ( ( ord_less_eq_set_c @ B @ D2 )
=> ( ord_less_eq_set_c @ ( inf_inf_set_c @ A3 @ B ) @ ( inf_inf_set_c @ C @ D2 ) ) ) ) ).
% Int_mono
thf(fact_764_Int__mono,axiom,
! [A3: set_set_a,C: set_set_a,B: set_set_a,D2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ C )
=> ( ( ord_le3724670747650509150_set_a @ B @ D2 )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A3 @ B ) @ ( inf_inf_set_set_a @ C @ D2 ) ) ) ) ).
% Int_mono
thf(fact_765_Int__mono,axiom,
! [A3: set_a,C: set_a,B: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A3 @ C )
=> ( ( ord_less_eq_set_a @ B @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B ) @ ( inf_inf_set_a @ C @ D2 ) ) ) ) ).
% Int_mono
thf(fact_766_subset__vimage__iff,axiom,
! [A3: set_nat,F: nat > nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ ( vimage_nat_nat @ F @ B ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A3 )
=> ( member_nat @ ( F @ X2 ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_767_subset__vimage__iff,axiom,
! [A3: set_c,F: c > b,B: set_b] :
( ( ord_less_eq_set_c @ A3 @ ( vimage_c_b @ F @ B ) )
= ( ! [X2: c] :
( ( member_c @ X2 @ A3 )
=> ( member_b @ ( F @ X2 ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_768_subset__vimage__iff,axiom,
! [A3: set_c,F: c > $o,B: set_o] :
( ( ord_less_eq_set_c @ A3 @ ( vimage_c_o @ F @ B ) )
= ( ! [X2: c] :
( ( member_c @ X2 @ A3 )
=> ( member_o @ ( F @ X2 ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_769_subset__vimage__iff,axiom,
! [A3: set_c,F: c > nat,B: set_nat] :
( ( ord_less_eq_set_c @ A3 @ ( vimage_c_nat @ F @ B ) )
= ( ! [X2: c] :
( ( member_c @ X2 @ A3 )
=> ( member_nat @ ( F @ X2 ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_770_subset__vimage__iff,axiom,
! [A3: set_c,F: c > c,B: set_c] :
( ( ord_less_eq_set_c @ A3 @ ( vimage_c_c @ F @ B ) )
= ( ! [X2: c] :
( ( member_c @ X2 @ A3 )
=> ( member_c @ ( F @ X2 ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_771_subset__vimage__iff,axiom,
! [A3: set_c,F: c > a,B: set_a] :
( ( ord_less_eq_set_c @ A3 @ ( vimage_c_a @ F @ B ) )
= ( ! [X2: c] :
( ( member_c @ X2 @ A3 )
=> ( member_a @ ( F @ X2 ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_772_subset__vimage__iff,axiom,
! [A3: set_a,F: a > b,B: set_b] :
( ( ord_less_eq_set_a @ A3 @ ( vimage_a_b @ F @ B ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ( member_b @ ( F @ X2 ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_773_subset__vimage__iff,axiom,
! [A3: set_a,F: a > $o,B: set_o] :
( ( ord_less_eq_set_a @ A3 @ ( vimage_a_o @ F @ B ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ( member_o @ ( F @ X2 ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_774_subset__vimage__iff,axiom,
! [A3: set_a,F: a > nat,B: set_nat] :
( ( ord_less_eq_set_a @ A3 @ ( vimage_a_nat @ F @ B ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ( member_nat @ ( F @ X2 ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_775_subset__vimage__iff,axiom,
! [A3: set_a,F: a > c,B: set_c] :
( ( ord_less_eq_set_a @ A3 @ ( vimage_a_c @ F @ B ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ( member_c @ ( F @ X2 ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_776_vimage__mono,axiom,
! [A3: set_nat,B: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A3 @ B )
=> ( ord_less_eq_set_nat @ ( vimage_nat_nat @ F @ A3 ) @ ( vimage_nat_nat @ F @ B ) ) ) ).
% vimage_mono
thf(fact_777_vimage__mono,axiom,
! [A3: set_c,B: set_c,F: c > c] :
( ( ord_less_eq_set_c @ A3 @ B )
=> ( ord_less_eq_set_c @ ( vimage_c_c @ F @ A3 ) @ ( vimage_c_c @ F @ B ) ) ) ).
% vimage_mono
thf(fact_778_vimage__mono,axiom,
! [A3: set_c,B: set_c,F: set_a > c] :
( ( ord_less_eq_set_c @ A3 @ B )
=> ( ord_le3724670747650509150_set_a @ ( vimage_set_a_c @ F @ A3 ) @ ( vimage_set_a_c @ F @ B ) ) ) ).
% vimage_mono
thf(fact_779_vimage__mono,axiom,
! [A3: set_c,B: set_c,F: a > c] :
( ( ord_less_eq_set_c @ A3 @ B )
=> ( ord_less_eq_set_a @ ( vimage_a_c @ F @ A3 ) @ ( vimage_a_c @ F @ B ) ) ) ).
% vimage_mono
thf(fact_780_vimage__mono,axiom,
! [A3: set_set_a,B: set_set_a,F: c > set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B )
=> ( ord_less_eq_set_c @ ( vimage_c_set_a @ F @ A3 ) @ ( vimage_c_set_a @ F @ B ) ) ) ).
% vimage_mono
thf(fact_781_vimage__mono,axiom,
! [A3: set_set_a,B: set_set_a,F: set_a > set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B )
=> ( ord_le3724670747650509150_set_a @ ( vimage_set_a_set_a @ F @ A3 ) @ ( vimage_set_a_set_a @ F @ B ) ) ) ).
% vimage_mono
thf(fact_782_vimage__mono,axiom,
! [A3: set_set_a,B: set_set_a,F: a > set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B )
=> ( ord_less_eq_set_a @ ( vimage_a_set_a @ F @ A3 ) @ ( vimage_a_set_a @ F @ B ) ) ) ).
% vimage_mono
thf(fact_783_vimage__mono,axiom,
! [A3: set_a,B: set_a,F: c > a] :
( ( ord_less_eq_set_a @ A3 @ B )
=> ( ord_less_eq_set_c @ ( vimage_c_a @ F @ A3 ) @ ( vimage_c_a @ F @ B ) ) ) ).
% vimage_mono
thf(fact_784_vimage__mono,axiom,
! [A3: set_a,B: set_a,F: set_a > a] :
( ( ord_less_eq_set_a @ A3 @ B )
=> ( ord_le3724670747650509150_set_a @ ( vimage_set_a_a @ F @ A3 ) @ ( vimage_set_a_a @ F @ B ) ) ) ).
% vimage_mono
thf(fact_785_vimage__mono,axiom,
! [A3: set_a,B: set_a,F: a > a] :
( ( ord_less_eq_set_a @ A3 @ B )
=> ( ord_less_eq_set_a @ ( vimage_a_a @ F @ A3 ) @ ( vimage_a_a @ F @ B ) ) ) ).
% vimage_mono
thf(fact_786_vimage__inter__cong,axiom,
! [S: set_a,F: a > c,G: a > c,Y: set_c] :
( ! [W: a] :
( ( member_a @ W @ S )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( inf_inf_set_a @ ( vimage_a_c @ F @ Y ) @ S )
= ( inf_inf_set_a @ ( vimage_a_c @ G @ Y ) @ S ) ) ) ).
% vimage_inter_cong
thf(fact_787_vimage__inter__cong,axiom,
! [S: set_a,F: a > a,G: a > a,Y: set_a] :
( ! [W: a] :
( ( member_a @ W @ S )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( inf_inf_set_a @ ( vimage_a_a @ F @ Y ) @ S )
= ( inf_inf_set_a @ ( vimage_a_a @ G @ Y ) @ S ) ) ) ).
% vimage_inter_cong
thf(fact_788_vimage__inter__cong,axiom,
! [S: set_c,F: c > c,G: c > c,Y: set_c] :
( ! [W: c] :
( ( member_c @ W @ S )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( inf_inf_set_c @ ( vimage_c_c @ F @ Y ) @ S )
= ( inf_inf_set_c @ ( vimage_c_c @ G @ Y ) @ S ) ) ) ).
% vimage_inter_cong
thf(fact_789_vimage__inter__cong,axiom,
! [S: set_c,F: c > a,G: c > a,Y: set_a] :
( ! [W: c] :
( ( member_c @ W @ S )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( inf_inf_set_c @ ( vimage_c_a @ F @ Y ) @ S )
= ( inf_inf_set_c @ ( vimage_c_a @ G @ Y ) @ S ) ) ) ).
% vimage_inter_cong
thf(fact_790_vimage__inter__cong,axiom,
! [S: set_nat,F: nat > nat,G: nat > nat,Y: set_nat] :
( ! [W: nat] :
( ( member_nat @ W @ S )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( inf_inf_set_nat @ ( vimage_nat_nat @ F @ Y ) @ S )
= ( inf_inf_set_nat @ ( vimage_nat_nat @ G @ Y ) @ S ) ) ) ).
% vimage_inter_cong
thf(fact_791_setcompr__eq__image,axiom,
! [F: a > c,P: a > $o] :
( ( collect_c
@ ^ [Uu: c] :
? [X2: a] :
( ( Uu
= ( F @ X2 ) )
& ( P @ X2 ) ) )
= ( image_a_c @ F @ ( collect_a @ P ) ) ) ).
% setcompr_eq_image
thf(fact_792_setcompr__eq__image,axiom,
! [F: set_a > set_a,P: set_a > $o] :
( ( collect_set_a
@ ^ [Uu: set_a] :
? [X2: set_a] :
( ( Uu
= ( F @ X2 ) )
& ( P @ X2 ) ) )
= ( image_set_a_set_a @ F @ ( collect_set_a @ P ) ) ) ).
% setcompr_eq_image
thf(fact_793_setcompr__eq__image,axiom,
! [F: nat > set_a,P: nat > $o] :
( ( collect_set_a
@ ^ [Uu: set_a] :
? [X2: nat] :
( ( Uu
= ( F @ X2 ) )
& ( P @ X2 ) ) )
= ( image_nat_set_a @ F @ ( collect_nat @ P ) ) ) ).
% setcompr_eq_image
thf(fact_794_setcompr__eq__image,axiom,
! [F: set_a > nat,P: set_a > $o] :
( ( collect_nat
@ ^ [Uu: nat] :
? [X2: set_a] :
( ( Uu
= ( F @ X2 ) )
& ( P @ X2 ) ) )
= ( image_set_a_nat @ F @ ( collect_set_a @ P ) ) ) ).
% setcompr_eq_image
thf(fact_795_setcompr__eq__image,axiom,
! [F: nat > nat,P: nat > $o] :
( ( collect_nat
@ ^ [Uu: nat] :
? [X2: nat] :
( ( Uu
= ( F @ X2 ) )
& ( P @ X2 ) ) )
= ( image_nat_nat @ F @ ( collect_nat @ P ) ) ) ).
% setcompr_eq_image
thf(fact_796_Setcompr__eq__image,axiom,
! [F: a > c,A3: set_a] :
( ( collect_c
@ ^ [Uu: c] :
? [X2: a] :
( ( Uu
= ( F @ X2 ) )
& ( member_a @ X2 @ A3 ) ) )
= ( image_a_c @ F @ A3 ) ) ).
% Setcompr_eq_image
thf(fact_797_Setcompr__eq__image,axiom,
! [F: set_c > set_a,A3: set_set_c] :
( ( collect_set_a
@ ^ [Uu: set_a] :
? [X2: set_c] :
( ( Uu
= ( F @ X2 ) )
& ( member_set_c @ X2 @ A3 ) ) )
= ( image_set_c_set_a @ F @ A3 ) ) ).
% Setcompr_eq_image
thf(fact_798_Setcompr__eq__image,axiom,
! [F: b > set_a,A3: set_b] :
( ( collect_set_a
@ ^ [Uu: set_a] :
? [X2: b] :
( ( Uu
= ( F @ X2 ) )
& ( member_b @ X2 @ A3 ) ) )
= ( image_b_set_a @ F @ A3 ) ) ).
% Setcompr_eq_image
thf(fact_799_Setcompr__eq__image,axiom,
! [F: $o > set_a,A3: set_o] :
( ( collect_set_a
@ ^ [Uu: set_a] :
? [X2: $o] :
( ( Uu
= ( F @ X2 ) )
& ( member_o @ X2 @ A3 ) ) )
= ( image_o_set_a @ F @ A3 ) ) ).
% Setcompr_eq_image
thf(fact_800_Setcompr__eq__image,axiom,
! [F: nat > set_a,A3: set_nat] :
( ( collect_set_a
@ ^ [Uu: set_a] :
? [X2: nat] :
( ( Uu
= ( F @ X2 ) )
& ( member_nat @ X2 @ A3 ) ) )
= ( image_nat_set_a @ F @ A3 ) ) ).
% Setcompr_eq_image
thf(fact_801_Setcompr__eq__image,axiom,
! [F: set_c > nat,A3: set_set_c] :
( ( collect_nat
@ ^ [Uu: nat] :
? [X2: set_c] :
( ( Uu
= ( F @ X2 ) )
& ( member_set_c @ X2 @ A3 ) ) )
= ( image_set_c_nat @ F @ A3 ) ) ).
% Setcompr_eq_image
thf(fact_802_Setcompr__eq__image,axiom,
! [F: b > nat,A3: set_b] :
( ( collect_nat
@ ^ [Uu: nat] :
? [X2: b] :
( ( Uu
= ( F @ X2 ) )
& ( member_b @ X2 @ A3 ) ) )
= ( image_b_nat @ F @ A3 ) ) ).
% Setcompr_eq_image
thf(fact_803_Setcompr__eq__image,axiom,
! [F: $o > nat,A3: set_o] :
( ( collect_nat
@ ^ [Uu: nat] :
? [X2: $o] :
( ( Uu
= ( F @ X2 ) )
& ( member_o @ X2 @ A3 ) ) )
= ( image_o_nat @ F @ A3 ) ) ).
% Setcompr_eq_image
thf(fact_804_Setcompr__eq__image,axiom,
! [F: nat > nat,A3: set_nat] :
( ( collect_nat
@ ^ [Uu: nat] :
? [X2: nat] :
( ( Uu
= ( F @ X2 ) )
& ( member_nat @ X2 @ A3 ) ) )
= ( image_nat_nat @ F @ A3 ) ) ).
% Setcompr_eq_image
thf(fact_805_F__in__events,axiom,
! [I2: b] :
( ( member_b @ I2 @ i )
=> ( ord_le3724670747650509150_set_a @ ( f @ I2 ) @ ( pow_a @ ( probab49036049091589825_pmf_a @ p ) ) ) ) ).
% F_in_events
thf(fact_806_restricted__sigma,axiom,
! [S: set_nat,Omega: set_nat,M: set_set_nat] :
( ( member_set_nat @ S @ ( sigma_sigma_sets_nat @ Omega @ M ) )
=> ( ( ord_le6893508408891458716et_nat @ M @ ( pow_nat @ Omega ) )
=> ( ( image_7916887816326733075et_nat @ ( inf_inf_set_nat @ S ) @ ( sigma_sigma_sets_nat @ Omega @ M ) )
= ( sigma_sigma_sets_nat @ S @ ( image_7916887816326733075et_nat @ ( inf_inf_set_nat @ S ) @ M ) ) ) ) ) ).
% restricted_sigma
thf(fact_807_restricted__sigma,axiom,
! [S: set_set_a,Omega: set_set_a,M: set_set_set_a] :
( ( member_set_set_a @ S @ ( sigma_2987359967864564790_set_a @ Omega @ M ) )
=> ( ( ord_le5722252365846178494_set_a @ M @ ( pow_set_a @ Omega ) )
=> ( ( image_1042221919965026181_set_a @ ( inf_inf_set_set_a @ S ) @ ( sigma_2987359967864564790_set_a @ Omega @ M ) )
= ( sigma_2987359967864564790_set_a @ S @ ( image_1042221919965026181_set_a @ ( inf_inf_set_set_a @ S ) @ M ) ) ) ) ) ).
% restricted_sigma
thf(fact_808_restricted__sigma,axiom,
! [S: set_c,Omega: set_c,M: set_set_c] :
( ( member_set_c @ S @ ( sigma_sigma_sets_c @ Omega @ M ) )
=> ( ( ord_le3866738827743201120_set_c @ M @ ( pow_c @ Omega ) )
=> ( ( image_set_c_set_c @ ( inf_inf_set_c @ S ) @ ( sigma_sigma_sets_c @ Omega @ M ) )
= ( sigma_sigma_sets_c @ S @ ( image_set_c_set_c @ ( inf_inf_set_c @ S ) @ M ) ) ) ) ) ).
% restricted_sigma
thf(fact_809_restricted__sigma,axiom,
! [S: set_a,Omega: set_a,M: set_set_a] :
( ( member_set_a @ S @ ( sigma_sigma_sets_a @ Omega @ M ) )
=> ( ( ord_le3724670747650509150_set_a @ M @ ( pow_a @ Omega ) )
=> ( ( image_set_a_set_a @ ( inf_inf_set_a @ S ) @ ( sigma_sigma_sets_a @ Omega @ M ) )
= ( sigma_sigma_sets_a @ S @ ( image_set_a_set_a @ ( inf_inf_set_a @ S ) @ M ) ) ) ) ) ).
% restricted_sigma
thf(fact_810_sigma__sets__into__sp,axiom,
! [A3: set_set_c,Sp: set_c,X: set_c] :
( ( ord_le3866738827743201120_set_c @ A3 @ ( pow_c @ Sp ) )
=> ( ( member_set_c @ X @ ( sigma_sigma_sets_c @ Sp @ A3 ) )
=> ( ord_less_eq_set_c @ X @ Sp ) ) ) ).
% sigma_sets_into_sp
thf(fact_811_sigma__sets__into__sp,axiom,
! [A3: set_set_set_a,Sp: set_set_a,X: set_set_a] :
( ( ord_le5722252365846178494_set_a @ A3 @ ( pow_set_a @ Sp ) )
=> ( ( member_set_set_a @ X @ ( sigma_2987359967864564790_set_a @ Sp @ A3 ) )
=> ( ord_le3724670747650509150_set_a @ X @ Sp ) ) ) ).
% sigma_sets_into_sp
thf(fact_812_sigma__sets__into__sp,axiom,
! [A3: set_set_a,Sp: set_a,X: set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ ( pow_a @ Sp ) )
=> ( ( member_set_a @ X @ ( sigma_sigma_sets_a @ Sp @ A3 ) )
=> ( ord_less_eq_set_a @ X @ Sp ) ) ) ).
% sigma_sets_into_sp
thf(fact_813_sigma__sets__Int,axiom,
! [A3: set_nat,Sp: set_nat,St: set_set_nat] :
( ( member_set_nat @ A3 @ ( sigma_sigma_sets_nat @ Sp @ St ) )
=> ( ( ord_less_eq_set_nat @ A3 @ Sp )
=> ( ( image_7916887816326733075et_nat @ ( inf_inf_set_nat @ A3 ) @ ( sigma_sigma_sets_nat @ Sp @ St ) )
= ( sigma_sigma_sets_nat @ A3 @ ( image_7916887816326733075et_nat @ ( inf_inf_set_nat @ A3 ) @ St ) ) ) ) ) ).
% sigma_sets_Int
thf(fact_814_sigma__sets__Int,axiom,
! [A3: set_c,Sp: set_c,St: set_set_c] :
( ( member_set_c @ A3 @ ( sigma_sigma_sets_c @ Sp @ St ) )
=> ( ( ord_less_eq_set_c @ A3 @ Sp )
=> ( ( image_set_c_set_c @ ( inf_inf_set_c @ A3 ) @ ( sigma_sigma_sets_c @ Sp @ St ) )
= ( sigma_sigma_sets_c @ A3 @ ( image_set_c_set_c @ ( inf_inf_set_c @ A3 ) @ St ) ) ) ) ) ).
% sigma_sets_Int
thf(fact_815_sigma__sets__Int,axiom,
! [A3: set_set_a,Sp: set_set_a,St: set_set_set_a] :
( ( member_set_set_a @ A3 @ ( sigma_2987359967864564790_set_a @ Sp @ St ) )
=> ( ( ord_le3724670747650509150_set_a @ A3 @ Sp )
=> ( ( image_1042221919965026181_set_a @ ( inf_inf_set_set_a @ A3 ) @ ( sigma_2987359967864564790_set_a @ Sp @ St ) )
= ( sigma_2987359967864564790_set_a @ A3 @ ( image_1042221919965026181_set_a @ ( inf_inf_set_set_a @ A3 ) @ St ) ) ) ) ) ).
% sigma_sets_Int
thf(fact_816_sigma__sets__Int,axiom,
! [A3: set_a,Sp: set_a,St: set_set_a] :
( ( member_set_a @ A3 @ ( sigma_sigma_sets_a @ Sp @ St ) )
=> ( ( ord_less_eq_set_a @ A3 @ Sp )
=> ( ( image_set_a_set_a @ ( inf_inf_set_a @ A3 ) @ ( sigma_sigma_sets_a @ Sp @ St ) )
= ( sigma_sigma_sets_a @ A3 @ ( image_set_a_set_a @ ( inf_inf_set_a @ A3 ) @ St ) ) ) ) ) ).
% sigma_sets_Int
thf(fact_817_dual__order_Orefl,axiom,
! [A4: set_c] : ( ord_less_eq_set_c @ A4 @ A4 ) ).
% dual_order.refl
thf(fact_818_dual__order_Orefl,axiom,
! [A4: set_set_a] : ( ord_le3724670747650509150_set_a @ A4 @ A4 ) ).
% dual_order.refl
thf(fact_819_dual__order_Orefl,axiom,
! [A4: nat] : ( ord_less_eq_nat @ A4 @ A4 ) ).
% dual_order.refl
thf(fact_820_dual__order_Orefl,axiom,
! [A4: extended_ereal] : ( ord_le1083603963089353582_ereal @ A4 @ A4 ) ).
% dual_order.refl
thf(fact_821_dual__order_Orefl,axiom,
! [A4: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ A4 @ A4 ) ).
% dual_order.refl
thf(fact_822_dual__order_Orefl,axiom,
! [A4: set_a] : ( ord_less_eq_set_a @ A4 @ A4 ) ).
% dual_order.refl
thf(fact_823_order__refl,axiom,
! [X: set_c] : ( ord_less_eq_set_c @ X @ X ) ).
% order_refl
thf(fact_824_order__refl,axiom,
! [X: set_set_a] : ( ord_le3724670747650509150_set_a @ X @ X ) ).
% order_refl
thf(fact_825_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_826_order__refl,axiom,
! [X: extended_ereal] : ( ord_le1083603963089353582_ereal @ X @ X ) ).
% order_refl
thf(fact_827_order__refl,axiom,
! [X: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ X @ X ) ).
% order_refl
thf(fact_828_order__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% order_refl
thf(fact_829_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > b,B: set_b] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( member_b @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_b @ ( image_nat_b @ F @ ( collect_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_830_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > $o,B: set_o] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( member_o @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_nat_o @ F @ ( collect_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_831_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > nat,B: set_nat] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( member_nat @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( collect_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_832_image__Collect__subsetI,axiom,
! [P: a > $o,F: a > c,B: set_c] :
( ! [X3: a] :
( ( P @ X3 )
=> ( member_c @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_c @ ( image_a_c @ F @ ( collect_a @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_833_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > c,B: set_c] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( member_c @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_c @ ( image_nat_c @ F @ ( collect_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_834_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > a,B: set_a] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( member_a @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_nat_a @ F @ ( collect_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_835_image__Collect__subsetI,axiom,
! [P: set_a > $o,F: set_a > b,B: set_b] :
( ! [X3: set_a] :
( ( P @ X3 )
=> ( member_b @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_b @ ( image_set_a_b @ F @ ( collect_set_a @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_836_image__Collect__subsetI,axiom,
! [P: set_a > $o,F: set_a > $o,B: set_o] :
( ! [X3: set_a] :
( ( P @ X3 )
=> ( member_o @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_set_a_o @ F @ ( collect_set_a @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_837_image__Collect__subsetI,axiom,
! [P: set_a > $o,F: set_a > nat,B: set_nat] :
( ! [X3: set_a] :
( ( P @ X3 )
=> ( member_nat @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_set_a_nat @ F @ ( collect_set_a @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_838_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > set_c,B: set_set_c] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( member_set_c @ ( F @ X3 ) @ B ) )
=> ( ord_le3866738827743201120_set_c @ ( image_nat_set_c @ F @ ( collect_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_839_sigma__sets__mono_H_H,axiom,
! [A3: set_c,C: set_c,D2: set_set_c,B: set_set_c] :
( ( member_set_c @ A3 @ ( sigma_sigma_sets_c @ C @ D2 ) )
=> ( ( ord_le3866738827743201120_set_c @ B @ D2 )
=> ( ( ord_le3866738827743201120_set_c @ D2 @ ( pow_c @ C ) )
=> ( ord_le3866738827743201120_set_c @ ( sigma_sigma_sets_c @ A3 @ B ) @ ( sigma_sigma_sets_c @ C @ D2 ) ) ) ) ) ).
% sigma_sets_mono''
thf(fact_840_sigma__sets__mono_H_H,axiom,
! [A3: set_a,C: set_a,D2: set_set_a,B: set_set_a] :
( ( member_set_a @ A3 @ ( sigma_sigma_sets_a @ C @ D2 ) )
=> ( ( ord_le3724670747650509150_set_a @ B @ D2 )
=> ( ( ord_le3724670747650509150_set_a @ D2 @ ( pow_a @ C ) )
=> ( ord_le3724670747650509150_set_a @ ( sigma_sigma_sets_a @ A3 @ B ) @ ( sigma_sigma_sets_a @ C @ D2 ) ) ) ) ) ).
% sigma_sets_mono''
thf(fact_841_sigma__sets__sigma__sets__eq,axiom,
! [M: set_set_c,S: set_c] :
( ( ord_le3866738827743201120_set_c @ M @ ( pow_c @ S ) )
=> ( ( sigma_sigma_sets_c @ S @ ( sigma_sigma_sets_c @ S @ M ) )
= ( sigma_sigma_sets_c @ S @ M ) ) ) ).
% sigma_sets_sigma_sets_eq
thf(fact_842_sigma__sets__sigma__sets__eq,axiom,
! [M: set_set_a,S: set_a] :
( ( ord_le3724670747650509150_set_a @ M @ ( pow_a @ S ) )
=> ( ( sigma_sigma_sets_a @ S @ ( sigma_sigma_sets_a @ S @ M ) )
= ( sigma_sigma_sets_a @ S @ M ) ) ) ).
% sigma_sets_sigma_sets_eq
thf(fact_843_all__subset__image,axiom,
! [F: c > c,A3: set_c,P: set_c > $o] :
( ( ! [B3: set_c] :
( ( ord_less_eq_set_c @ B3 @ ( image_c_c @ F @ A3 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_c] :
( ( ord_less_eq_set_c @ B3 @ A3 )
=> ( P @ ( image_c_c @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_844_all__subset__image,axiom,
! [F: set_a > c,A3: set_set_a,P: set_c > $o] :
( ( ! [B3: set_c] :
( ( ord_less_eq_set_c @ B3 @ ( image_set_a_c @ F @ A3 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ A3 )
=> ( P @ ( image_set_a_c @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_845_all__subset__image,axiom,
! [F: a > c,A3: set_a,P: set_c > $o] :
( ( ! [B3: set_c] :
( ( ord_less_eq_set_c @ B3 @ ( image_a_c @ F @ A3 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
=> ( P @ ( image_a_c @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_846_all__subset__image,axiom,
! [F: c > set_a,A3: set_c,P: set_set_a > $o] :
( ( ! [B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ ( image_c_set_a @ F @ A3 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_c] :
( ( ord_less_eq_set_c @ B3 @ A3 )
=> ( P @ ( image_c_set_a @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_847_all__subset__image,axiom,
! [F: set_a > set_a,A3: set_set_a,P: set_set_a > $o] :
( ( ! [B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ ( image_set_a_set_a @ F @ A3 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ A3 )
=> ( P @ ( image_set_a_set_a @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_848_all__subset__image,axiom,
! [F: a > set_a,A3: set_a,P: set_set_a > $o] :
( ( ! [B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ ( image_a_set_a @ F @ A3 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
=> ( P @ ( image_a_set_a @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_849_all__subset__image,axiom,
! [F: c > a,A3: set_c,P: set_a > $o] :
( ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ ( image_c_a @ F @ A3 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_c] :
( ( ord_less_eq_set_c @ B3 @ A3 )
=> ( P @ ( image_c_a @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_850_all__subset__image,axiom,
! [F: set_a > a,A3: set_set_a,P: set_a > $o] :
( ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ ( image_set_a_a @ F @ A3 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ A3 )
=> ( P @ ( image_set_a_a @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_851_all__subset__image,axiom,
! [F: a > a,A3: set_a,P: set_a > $o] :
( ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ ( image_a_a @ F @ A3 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
=> ( P @ ( image_a_a @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_852_inf__Int__eq,axiom,
! [R: set_set_c,S: set_set_c] :
( ( inf_inf_set_c_o
@ ^ [X2: set_c] : ( member_set_c @ X2 @ R )
@ ^ [X2: set_c] : ( member_set_c @ X2 @ S ) )
= ( ^ [X2: set_c] : ( member_set_c @ X2 @ ( inf_inf_set_set_c @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_853_inf__Int__eq,axiom,
! [R: set_b,S: set_b] :
( ( inf_inf_b_o
@ ^ [X2: b] : ( member_b @ X2 @ R )
@ ^ [X2: b] : ( member_b @ X2 @ S ) )
= ( ^ [X2: b] : ( member_b @ X2 @ ( inf_inf_set_b @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_854_inf__Int__eq,axiom,
! [R: set_o,S: set_o] :
( ( inf_inf_o_o
@ ^ [X2: $o] : ( member_o @ X2 @ R )
@ ^ [X2: $o] : ( member_o @ X2 @ S ) )
= ( ^ [X2: $o] : ( member_o @ X2 @ ( inf_inf_set_o @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_855_inf__Int__eq,axiom,
! [R: set_a,S: set_a] :
( ( inf_inf_a_o
@ ^ [X2: a] : ( member_a @ X2 @ R )
@ ^ [X2: a] : ( member_a @ X2 @ S ) )
= ( ^ [X2: a] : ( member_a @ X2 @ ( inf_inf_set_a @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_856_inf__Int__eq,axiom,
! [R: set_c,S: set_c] :
( ( inf_inf_c_o
@ ^ [X2: c] : ( member_c @ X2 @ R )
@ ^ [X2: c] : ( member_c @ X2 @ S ) )
= ( ^ [X2: c] : ( member_c @ X2 @ ( inf_inf_set_c @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_857_inf__Int__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( inf_inf_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ R )
@ ^ [X2: nat] : ( member_nat @ X2 @ S ) )
= ( ^ [X2: nat] : ( member_nat @ X2 @ ( inf_inf_set_nat @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_858_inf__Int__eq,axiom,
! [R: set_set_a,S: set_set_a] :
( ( inf_inf_set_a_o
@ ^ [X2: set_a] : ( member_set_a @ X2 @ R )
@ ^ [X2: set_a] : ( member_set_a @ X2 @ S ) )
= ( ^ [X2: set_a] : ( member_set_a @ X2 @ ( inf_inf_set_set_a @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_859_pred__subset__eq,axiom,
! [R: set_set_c,S: set_set_c] :
( ( ord_less_eq_set_c_o
@ ^ [X2: set_c] : ( member_set_c @ X2 @ R )
@ ^ [X2: set_c] : ( member_set_c @ X2 @ S ) )
= ( ord_le3866738827743201120_set_c @ R @ S ) ) ).
% pred_subset_eq
thf(fact_860_pred__subset__eq,axiom,
! [R: set_b,S: set_b] :
( ( ord_less_eq_b_o
@ ^ [X2: b] : ( member_b @ X2 @ R )
@ ^ [X2: b] : ( member_b @ X2 @ S ) )
= ( ord_less_eq_set_b @ R @ S ) ) ).
% pred_subset_eq
thf(fact_861_pred__subset__eq,axiom,
! [R: set_o,S: set_o] :
( ( ord_less_eq_o_o
@ ^ [X2: $o] : ( member_o @ X2 @ R )
@ ^ [X2: $o] : ( member_o @ X2 @ S ) )
= ( ord_less_eq_set_o @ R @ S ) ) ).
% pred_subset_eq
thf(fact_862_pred__subset__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ R )
@ ^ [X2: nat] : ( member_nat @ X2 @ S ) )
= ( ord_less_eq_set_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_863_pred__subset__eq,axiom,
! [R: set_c,S: set_c] :
( ( ord_less_eq_c_o
@ ^ [X2: c] : ( member_c @ X2 @ R )
@ ^ [X2: c] : ( member_c @ X2 @ S ) )
= ( ord_less_eq_set_c @ R @ S ) ) ).
% pred_subset_eq
thf(fact_864_pred__subset__eq,axiom,
! [R: set_set_a,S: set_set_a] :
( ( ord_less_eq_set_a_o
@ ^ [X2: set_a] : ( member_set_a @ X2 @ R )
@ ^ [X2: set_a] : ( member_set_a @ X2 @ S ) )
= ( ord_le3724670747650509150_set_a @ R @ S ) ) ).
% pred_subset_eq
thf(fact_865_pred__subset__eq,axiom,
! [R: set_a,S: set_a] :
( ( ord_less_eq_a_o
@ ^ [X2: a] : ( member_a @ X2 @ R )
@ ^ [X2: a] : ( member_a @ X2 @ S ) )
= ( ord_less_eq_set_a @ R @ S ) ) ).
% pred_subset_eq
thf(fact_866_nle__le,axiom,
! [A4: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A4 @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A4 )
& ( B2 != A4 ) ) ) ).
% nle_le
thf(fact_867_nle__le,axiom,
! [A4: extended_ereal,B2: extended_ereal] :
( ( ~ ( ord_le1083603963089353582_ereal @ A4 @ B2 ) )
= ( ( ord_le1083603963089353582_ereal @ B2 @ A4 )
& ( B2 != A4 ) ) ) ).
% nle_le
thf(fact_868_nle__le,axiom,
! [A4: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( ~ ( ord_le3935885782089961368nnreal @ A4 @ B2 ) )
= ( ( ord_le3935885782089961368nnreal @ B2 @ A4 )
& ( B2 != A4 ) ) ) ).
% nle_le
thf(fact_869_le__cases3,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_870_le__cases3,axiom,
! [X: extended_ereal,Y: extended_ereal,Z: extended_ereal] :
( ( ( ord_le1083603963089353582_ereal @ X @ Y )
=> ~ ( ord_le1083603963089353582_ereal @ Y @ Z ) )
=> ( ( ( ord_le1083603963089353582_ereal @ Y @ X )
=> ~ ( ord_le1083603963089353582_ereal @ X @ Z ) )
=> ( ( ( ord_le1083603963089353582_ereal @ X @ Z )
=> ~ ( ord_le1083603963089353582_ereal @ Z @ Y ) )
=> ( ( ( ord_le1083603963089353582_ereal @ Z @ Y )
=> ~ ( ord_le1083603963089353582_ereal @ Y @ X ) )
=> ( ( ( ord_le1083603963089353582_ereal @ Y @ Z )
=> ~ ( ord_le1083603963089353582_ereal @ Z @ X ) )
=> ~ ( ( ord_le1083603963089353582_ereal @ Z @ X )
=> ~ ( ord_le1083603963089353582_ereal @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_871_le__cases3,axiom,
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal,Z: extend8495563244428889912nnreal] :
( ( ( ord_le3935885782089961368nnreal @ X @ Y )
=> ~ ( ord_le3935885782089961368nnreal @ Y @ Z ) )
=> ( ( ( ord_le3935885782089961368nnreal @ Y @ X )
=> ~ ( ord_le3935885782089961368nnreal @ X @ Z ) )
=> ( ( ( ord_le3935885782089961368nnreal @ X @ Z )
=> ~ ( ord_le3935885782089961368nnreal @ Z @ Y ) )
=> ( ( ( ord_le3935885782089961368nnreal @ Z @ Y )
=> ~ ( ord_le3935885782089961368nnreal @ Y @ X ) )
=> ( ( ( ord_le3935885782089961368nnreal @ Y @ Z )
=> ~ ( ord_le3935885782089961368nnreal @ Z @ X ) )
=> ~ ( ( ord_le3935885782089961368nnreal @ Z @ X )
=> ~ ( ord_le3935885782089961368nnreal @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_872_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_c,Z2: set_c] : ( Y4 = Z2 ) )
= ( ^ [X2: set_c,Y3: set_c] :
( ( ord_less_eq_set_c @ X2 @ Y3 )
& ( ord_less_eq_set_c @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_873_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_set_a,Z2: set_set_a] : ( Y4 = Z2 ) )
= ( ^ [X2: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y3 )
& ( ord_le3724670747650509150_set_a @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_874_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_875_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: extended_ereal,Z2: extended_ereal] : ( Y4 = Z2 ) )
= ( ^ [X2: extended_ereal,Y3: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X2 @ Y3 )
& ( ord_le1083603963089353582_ereal @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_876_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: extend8495563244428889912nnreal,Z2: extend8495563244428889912nnreal] : ( Y4 = Z2 ) )
= ( ^ [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X2 @ Y3 )
& ( ord_le3935885782089961368nnreal @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_877_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
= ( ^ [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
& ( ord_less_eq_set_a @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_878_ord__eq__le__trans,axiom,
! [A4: set_c,B2: set_c,C2: set_c] :
( ( A4 = B2 )
=> ( ( ord_less_eq_set_c @ B2 @ C2 )
=> ( ord_less_eq_set_c @ A4 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_879_ord__eq__le__trans,axiom,
! [A4: set_set_a,B2: set_set_a,C2: set_set_a] :
( ( A4 = B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C2 )
=> ( ord_le3724670747650509150_set_a @ A4 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_880_ord__eq__le__trans,axiom,
! [A4: nat,B2: nat,C2: nat] :
( ( A4 = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_eq_nat @ A4 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_881_ord__eq__le__trans,axiom,
! [A4: extended_ereal,B2: extended_ereal,C2: extended_ereal] :
( ( A4 = B2 )
=> ( ( ord_le1083603963089353582_ereal @ B2 @ C2 )
=> ( ord_le1083603963089353582_ereal @ A4 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_882_ord__eq__le__trans,axiom,
! [A4: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( A4 = B2 )
=> ( ( ord_le3935885782089961368nnreal @ B2 @ C2 )
=> ( ord_le3935885782089961368nnreal @ A4 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_883_ord__eq__le__trans,axiom,
! [A4: set_a,B2: set_a,C2: set_a] :
( ( A4 = B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ A4 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_884_ord__le__eq__trans,axiom,
! [A4: set_c,B2: set_c,C2: set_c] :
( ( ord_less_eq_set_c @ A4 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_set_c @ A4 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_885_ord__le__eq__trans,axiom,
! [A4: set_set_a,B2: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_le3724670747650509150_set_a @ A4 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_886_ord__le__eq__trans,axiom,
! [A4: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_nat @ A4 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_887_ord__le__eq__trans,axiom,
! [A4: extended_ereal,B2: extended_ereal,C2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A4 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_le1083603963089353582_ereal @ A4 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_888_ord__le__eq__trans,axiom,
! [A4: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A4 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_le3935885782089961368nnreal @ A4 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_889_ord__le__eq__trans,axiom,
! [A4: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A4 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_set_a @ A4 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_890_order__antisym,axiom,
! [X: set_c,Y: set_c] :
( ( ord_less_eq_set_c @ X @ Y )
=> ( ( ord_less_eq_set_c @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_891_order__antisym,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y )
=> ( ( ord_le3724670747650509150_set_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_892_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_893_order__antisym,axiom,
! [X: extended_ereal,Y: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X @ Y )
=> ( ( ord_le1083603963089353582_ereal @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_894_order__antisym,axiom,
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X @ Y )
=> ( ( ord_le3935885782089961368nnreal @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_895_order__antisym,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_896_order_Otrans,axiom,
! [A4: set_c,B2: set_c,C2: set_c] :
( ( ord_less_eq_set_c @ A4 @ B2 )
=> ( ( ord_less_eq_set_c @ B2 @ C2 )
=> ( ord_less_eq_set_c @ A4 @ C2 ) ) ) ).
% order.trans
thf(fact_897_order_Otrans,axiom,
! [A4: set_set_a,B2: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C2 )
=> ( ord_le3724670747650509150_set_a @ A4 @ C2 ) ) ) ).
% order.trans
thf(fact_898_order_Otrans,axiom,
! [A4: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_eq_nat @ A4 @ C2 ) ) ) ).
% order.trans
thf(fact_899_order_Otrans,axiom,
! [A4: extended_ereal,B2: extended_ereal,C2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A4 @ B2 )
=> ( ( ord_le1083603963089353582_ereal @ B2 @ C2 )
=> ( ord_le1083603963089353582_ereal @ A4 @ C2 ) ) ) ).
% order.trans
thf(fact_900_order_Otrans,axiom,
! [A4: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A4 @ B2 )
=> ( ( ord_le3935885782089961368nnreal @ B2 @ C2 )
=> ( ord_le3935885782089961368nnreal @ A4 @ C2 ) ) ) ).
% order.trans
thf(fact_901_order_Otrans,axiom,
! [A4: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A4 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ A4 @ C2 ) ) ) ).
% order.trans
thf(fact_902_order__trans,axiom,
! [X: set_c,Y: set_c,Z: set_c] :
( ( ord_less_eq_set_c @ X @ Y )
=> ( ( ord_less_eq_set_c @ Y @ Z )
=> ( ord_less_eq_set_c @ X @ Z ) ) ) ).
% order_trans
thf(fact_903_order__trans,axiom,
! [X: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y )
=> ( ( ord_le3724670747650509150_set_a @ Y @ Z )
=> ( ord_le3724670747650509150_set_a @ X @ Z ) ) ) ).
% order_trans
thf(fact_904_order__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_905_order__trans,axiom,
! [X: extended_ereal,Y: extended_ereal,Z: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X @ Y )
=> ( ( ord_le1083603963089353582_ereal @ Y @ Z )
=> ( ord_le1083603963089353582_ereal @ X @ Z ) ) ) ).
% order_trans
thf(fact_906_order__trans,axiom,
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal,Z: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X @ Y )
=> ( ( ord_le3935885782089961368nnreal @ Y @ Z )
=> ( ord_le3935885782089961368nnreal @ X @ Z ) ) ) ).
% order_trans
thf(fact_907_order__trans,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z )
=> ( ord_less_eq_set_a @ X @ Z ) ) ) ).
% order_trans
thf(fact_908_linorder__wlog,axiom,
! [P: nat > nat > $o,A4: nat,B2: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: nat,B5: nat] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A4 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_909_linorder__wlog,axiom,
! [P: extended_ereal > extended_ereal > $o,A4: extended_ereal,B2: extended_ereal] :
( ! [A5: extended_ereal,B5: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: extended_ereal,B5: extended_ereal] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A4 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_910_linorder__wlog,axiom,
! [P: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o,A4: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ! [A5: extend8495563244428889912nnreal,B5: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: extend8495563244428889912nnreal,B5: extend8495563244428889912nnreal] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A4 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_911_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_c,Z2: set_c] : ( Y4 = Z2 ) )
= ( ^ [A2: set_c,B4: set_c] :
( ( ord_less_eq_set_c @ B4 @ A2 )
& ( ord_less_eq_set_c @ A2 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_912_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_set_a,Z2: set_set_a] : ( Y4 = Z2 ) )
= ( ^ [A2: set_set_a,B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B4 @ A2 )
& ( ord_le3724670747650509150_set_a @ A2 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_913_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A2: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A2 )
& ( ord_less_eq_nat @ A2 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_914_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: extended_ereal,Z2: extended_ereal] : ( Y4 = Z2 ) )
= ( ^ [A2: extended_ereal,B4: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ B4 @ A2 )
& ( ord_le1083603963089353582_ereal @ A2 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_915_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: extend8495563244428889912nnreal,Z2: extend8495563244428889912nnreal] : ( Y4 = Z2 ) )
= ( ^ [A2: extend8495563244428889912nnreal,B4: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B4 @ A2 )
& ( ord_le3935885782089961368nnreal @ A2 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_916_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
= ( ^ [A2: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A2 )
& ( ord_less_eq_set_a @ A2 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_917_dual__order_Oantisym,axiom,
! [B2: set_c,A4: set_c] :
( ( ord_less_eq_set_c @ B2 @ A4 )
=> ( ( ord_less_eq_set_c @ A4 @ B2 )
=> ( A4 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_918_dual__order_Oantisym,axiom,
! [B2: set_set_a,A4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A4 )
=> ( ( ord_le3724670747650509150_set_a @ A4 @ B2 )
=> ( A4 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_919_dual__order_Oantisym,axiom,
! [B2: nat,A4: nat] :
( ( ord_less_eq_nat @ B2 @ A4 )
=> ( ( ord_less_eq_nat @ A4 @ B2 )
=> ( A4 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_920_dual__order_Oantisym,axiom,
! [B2: extended_ereal,A4: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ B2 @ A4 )
=> ( ( ord_le1083603963089353582_ereal @ A4 @ B2 )
=> ( A4 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_921_dual__order_Oantisym,axiom,
! [B2: extend8495563244428889912nnreal,A4: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B2 @ A4 )
=> ( ( ord_le3935885782089961368nnreal @ A4 @ B2 )
=> ( A4 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_922_dual__order_Oantisym,axiom,
! [B2: set_a,A4: set_a] :
( ( ord_less_eq_set_a @ B2 @ A4 )
=> ( ( ord_less_eq_set_a @ A4 @ B2 )
=> ( A4 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_923_dual__order_Otrans,axiom,
! [B2: set_c,A4: set_c,C2: set_c] :
( ( ord_less_eq_set_c @ B2 @ A4 )
=> ( ( ord_less_eq_set_c @ C2 @ B2 )
=> ( ord_less_eq_set_c @ C2 @ A4 ) ) ) ).
% dual_order.trans
thf(fact_924_dual__order_Otrans,axiom,
! [B2: set_set_a,A4: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A4 )
=> ( ( ord_le3724670747650509150_set_a @ C2 @ B2 )
=> ( ord_le3724670747650509150_set_a @ C2 @ A4 ) ) ) ).
% dual_order.trans
thf(fact_925_dual__order_Otrans,axiom,
! [B2: nat,A4: nat,C2: nat] :
( ( ord_less_eq_nat @ B2 @ A4 )
=> ( ( ord_less_eq_nat @ C2 @ B2 )
=> ( ord_less_eq_nat @ C2 @ A4 ) ) ) ).
% dual_order.trans
thf(fact_926_dual__order_Otrans,axiom,
! [B2: extended_ereal,A4: extended_ereal,C2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ B2 @ A4 )
=> ( ( ord_le1083603963089353582_ereal @ C2 @ B2 )
=> ( ord_le1083603963089353582_ereal @ C2 @ A4 ) ) ) ).
% dual_order.trans
thf(fact_927_dual__order_Otrans,axiom,
! [B2: extend8495563244428889912nnreal,A4: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B2 @ A4 )
=> ( ( ord_le3935885782089961368nnreal @ C2 @ B2 )
=> ( ord_le3935885782089961368nnreal @ C2 @ A4 ) ) ) ).
% dual_order.trans
thf(fact_928_dual__order_Otrans,axiom,
! [B2: set_a,A4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A4 )
=> ( ( ord_less_eq_set_a @ C2 @ B2 )
=> ( ord_less_eq_set_a @ C2 @ A4 ) ) ) ).
% dual_order.trans
thf(fact_929_antisym,axiom,
! [A4: set_c,B2: set_c] :
( ( ord_less_eq_set_c @ A4 @ B2 )
=> ( ( ord_less_eq_set_c @ B2 @ A4 )
=> ( A4 = B2 ) ) ) ).
% antisym
thf(fact_930_antisym,axiom,
! [A4: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ A4 )
=> ( A4 = B2 ) ) ) ).
% antisym
thf(fact_931_antisym,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A4 )
=> ( A4 = B2 ) ) ) ).
% antisym
thf(fact_932_antisym,axiom,
! [A4: extended_ereal,B2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A4 @ B2 )
=> ( ( ord_le1083603963089353582_ereal @ B2 @ A4 )
=> ( A4 = B2 ) ) ) ).
% antisym
thf(fact_933_antisym,axiom,
! [A4: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A4 @ B2 )
=> ( ( ord_le3935885782089961368nnreal @ B2 @ A4 )
=> ( A4 = B2 ) ) ) ).
% antisym
thf(fact_934_antisym,axiom,
! [A4: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A4 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A4 )
=> ( A4 = B2 ) ) ) ).
% antisym
thf(fact_935_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_c,Z2: set_c] : ( Y4 = Z2 ) )
= ( ^ [A2: set_c,B4: set_c] :
( ( ord_less_eq_set_c @ A2 @ B4 )
& ( ord_less_eq_set_c @ B4 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_936_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_set_a,Z2: set_set_a] : ( Y4 = Z2 ) )
= ( ^ [A2: set_set_a,B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B4 )
& ( ord_le3724670747650509150_set_a @ B4 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_937_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A2: nat,B4: nat] :
( ( ord_less_eq_nat @ A2 @ B4 )
& ( ord_less_eq_nat @ B4 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_938_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: extended_ereal,Z2: extended_ereal] : ( Y4 = Z2 ) )
= ( ^ [A2: extended_ereal,B4: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A2 @ B4 )
& ( ord_le1083603963089353582_ereal @ B4 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_939_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: extend8495563244428889912nnreal,Z2: extend8495563244428889912nnreal] : ( Y4 = Z2 ) )
= ( ^ [A2: extend8495563244428889912nnreal,B4: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A2 @ B4 )
& ( ord_le3935885782089961368nnreal @ B4 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_940_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
= ( ^ [A2: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A2 @ B4 )
& ( ord_less_eq_set_a @ B4 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_941_order__subst1,axiom,
! [A4: nat,F: nat > nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_942_order__subst1,axiom,
! [A4: nat,F: extended_ereal > nat,B2: extended_ereal,C2: extended_ereal] :
( ( ord_less_eq_nat @ A4 @ ( F @ B2 ) )
=> ( ( ord_le1083603963089353582_ereal @ B2 @ C2 )
=> ( ! [X3: extended_ereal,Y5: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_943_order__subst1,axiom,
! [A4: nat,F: extend8495563244428889912nnreal > nat,B2: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_less_eq_nat @ A4 @ ( F @ B2 ) )
=> ( ( ord_le3935885782089961368nnreal @ B2 @ C2 )
=> ( ! [X3: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_944_order__subst1,axiom,
! [A4: extended_ereal,F: nat > extended_ereal,B2: nat,C2: nat] :
( ( ord_le1083603963089353582_ereal @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_le1083603963089353582_ereal @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le1083603963089353582_ereal @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_945_order__subst1,axiom,
! [A4: extended_ereal,F: extended_ereal > extended_ereal,B2: extended_ereal,C2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A4 @ ( F @ B2 ) )
=> ( ( ord_le1083603963089353582_ereal @ B2 @ C2 )
=> ( ! [X3: extended_ereal,Y5: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X3 @ Y5 )
=> ( ord_le1083603963089353582_ereal @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le1083603963089353582_ereal @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_946_order__subst1,axiom,
! [A4: extended_ereal,F: extend8495563244428889912nnreal > extended_ereal,B2: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le1083603963089353582_ereal @ A4 @ ( F @ B2 ) )
=> ( ( ord_le3935885782089961368nnreal @ B2 @ C2 )
=> ( ! [X3: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y5 )
=> ( ord_le1083603963089353582_ereal @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le1083603963089353582_ereal @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_947_order__subst1,axiom,
! [A4: extend8495563244428889912nnreal,F: nat > extend8495563244428889912nnreal,B2: nat,C2: nat] :
( ( ord_le3935885782089961368nnreal @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le3935885782089961368nnreal @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_948_order__subst1,axiom,
! [A4: extend8495563244428889912nnreal,F: extended_ereal > extend8495563244428889912nnreal,B2: extended_ereal,C2: extended_ereal] :
( ( ord_le3935885782089961368nnreal @ A4 @ ( F @ B2 ) )
=> ( ( ord_le1083603963089353582_ereal @ B2 @ C2 )
=> ( ! [X3: extended_ereal,Y5: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X3 @ Y5 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le3935885782089961368nnreal @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_949_order__subst1,axiom,
! [A4: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A4 @ ( F @ B2 ) )
=> ( ( ord_le3935885782089961368nnreal @ B2 @ C2 )
=> ( ! [X3: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y5 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le3935885782089961368nnreal @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_950_order__subst1,axiom,
! [A4: set_c,F: nat > set_c,B2: nat,C2: nat] :
( ( ord_less_eq_set_c @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_less_eq_set_c @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_c @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_951_order__subst2,axiom,
! [A4: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_952_order__subst2,axiom,
! [A4: nat,B2: nat,F: nat > extended_ereal,C2: extended_ereal] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ord_le1083603963089353582_ereal @ ( F @ B2 ) @ C2 )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_le1083603963089353582_ereal @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le1083603963089353582_ereal @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_953_order__subst2,axiom,
! [A4: nat,B2: nat,F: nat > extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ord_le3935885782089961368nnreal @ ( F @ B2 ) @ C2 )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le3935885782089961368nnreal @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_954_order__subst2,axiom,
! [A4: extended_ereal,B2: extended_ereal,F: extended_ereal > nat,C2: nat] :
( ( ord_le1083603963089353582_ereal @ A4 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X3: extended_ereal,Y5: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_955_order__subst2,axiom,
! [A4: extended_ereal,B2: extended_ereal,F: extended_ereal > extended_ereal,C2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A4 @ B2 )
=> ( ( ord_le1083603963089353582_ereal @ ( F @ B2 ) @ C2 )
=> ( ! [X3: extended_ereal,Y5: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X3 @ Y5 )
=> ( ord_le1083603963089353582_ereal @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le1083603963089353582_ereal @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_956_order__subst2,axiom,
! [A4: extended_ereal,B2: extended_ereal,F: extended_ereal > extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le1083603963089353582_ereal @ A4 @ B2 )
=> ( ( ord_le3935885782089961368nnreal @ ( F @ B2 ) @ C2 )
=> ( ! [X3: extended_ereal,Y5: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X3 @ Y5 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le3935885782089961368nnreal @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_957_order__subst2,axiom,
! [A4: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > nat,C2: nat] :
( ( ord_le3935885782089961368nnreal @ A4 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X3: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_958_order__subst2,axiom,
! [A4: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extended_ereal,C2: extended_ereal] :
( ( ord_le3935885782089961368nnreal @ A4 @ B2 )
=> ( ( ord_le1083603963089353582_ereal @ ( F @ B2 ) @ C2 )
=> ( ! [X3: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y5 )
=> ( ord_le1083603963089353582_ereal @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le1083603963089353582_ereal @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_959_order__subst2,axiom,
! [A4: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A4 @ B2 )
=> ( ( ord_le3935885782089961368nnreal @ ( F @ B2 ) @ C2 )
=> ( ! [X3: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y5 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le3935885782089961368nnreal @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_960_order__subst2,axiom,
! [A4: set_c,B2: set_c,F: set_c > nat,C2: nat] :
( ( ord_less_eq_set_c @ A4 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X3: set_c,Y5: set_c] :
( ( ord_less_eq_set_c @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_961_order__eq__refl,axiom,
! [X: set_c,Y: set_c] :
( ( X = Y )
=> ( ord_less_eq_set_c @ X @ Y ) ) ).
% order_eq_refl
thf(fact_962_order__eq__refl,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( X = Y )
=> ( ord_le3724670747650509150_set_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_963_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_964_order__eq__refl,axiom,
! [X: extended_ereal,Y: extended_ereal] :
( ( X = Y )
=> ( ord_le1083603963089353582_ereal @ X @ Y ) ) ).
% order_eq_refl
thf(fact_965_order__eq__refl,axiom,
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( X = Y )
=> ( ord_le3935885782089961368nnreal @ X @ Y ) ) ).
% order_eq_refl
thf(fact_966_order__eq__refl,axiom,
! [X: set_a,Y: set_a] :
( ( X = Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_967_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_968_linorder__linear,axiom,
! [X: extended_ereal,Y: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X @ Y )
| ( ord_le1083603963089353582_ereal @ Y @ X ) ) ).
% linorder_linear
thf(fact_969_linorder__linear,axiom,
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X @ Y )
| ( ord_le3935885782089961368nnreal @ Y @ X ) ) ).
% linorder_linear
thf(fact_970_ord__eq__le__subst,axiom,
! [A4: nat,F: nat > nat,B2: nat,C2: nat] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_971_ord__eq__le__subst,axiom,
! [A4: extended_ereal,F: nat > extended_ereal,B2: nat,C2: nat] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_le1083603963089353582_ereal @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le1083603963089353582_ereal @ A4 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_972_ord__eq__le__subst,axiom,
! [A4: extend8495563244428889912nnreal,F: nat > extend8495563244428889912nnreal,B2: nat,C2: nat] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le3935885782089961368nnreal @ A4 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_973_ord__eq__le__subst,axiom,
! [A4: nat,F: extended_ereal > nat,B2: extended_ereal,C2: extended_ereal] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_le1083603963089353582_ereal @ B2 @ C2 )
=> ( ! [X3: extended_ereal,Y5: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_974_ord__eq__le__subst,axiom,
! [A4: extended_ereal,F: extended_ereal > extended_ereal,B2: extended_ereal,C2: extended_ereal] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_le1083603963089353582_ereal @ B2 @ C2 )
=> ( ! [X3: extended_ereal,Y5: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X3 @ Y5 )
=> ( ord_le1083603963089353582_ereal @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le1083603963089353582_ereal @ A4 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_975_ord__eq__le__subst,axiom,
! [A4: extend8495563244428889912nnreal,F: extended_ereal > extend8495563244428889912nnreal,B2: extended_ereal,C2: extended_ereal] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_le1083603963089353582_ereal @ B2 @ C2 )
=> ( ! [X3: extended_ereal,Y5: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X3 @ Y5 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le3935885782089961368nnreal @ A4 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_976_ord__eq__le__subst,axiom,
! [A4: nat,F: extend8495563244428889912nnreal > nat,B2: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_le3935885782089961368nnreal @ B2 @ C2 )
=> ( ! [X3: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_977_ord__eq__le__subst,axiom,
! [A4: extended_ereal,F: extend8495563244428889912nnreal > extended_ereal,B2: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_le3935885782089961368nnreal @ B2 @ C2 )
=> ( ! [X3: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y5 )
=> ( ord_le1083603963089353582_ereal @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le1083603963089353582_ereal @ A4 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_978_ord__eq__le__subst,axiom,
! [A4: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_le3935885782089961368nnreal @ B2 @ C2 )
=> ( ! [X3: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y5 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le3935885782089961368nnreal @ A4 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_979_ord__eq__le__subst,axiom,
! [A4: nat,F: set_c > nat,B2: set_c,C2: set_c] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_c @ B2 @ C2 )
=> ( ! [X3: set_c,Y5: set_c] :
( ( ord_less_eq_set_c @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_980_ord__le__eq__subst,axiom,
! [A4: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A4 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_981_ord__le__eq__subst,axiom,
! [A4: nat,B2: nat,F: nat > extended_ereal,C2: extended_ereal] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_le1083603963089353582_ereal @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le1083603963089353582_ereal @ ( F @ A4 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_982_ord__le__eq__subst,axiom,
! [A4: nat,B2: nat,F: nat > extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le3935885782089961368nnreal @ ( F @ A4 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_983_ord__le__eq__subst,axiom,
! [A4: extended_ereal,B2: extended_ereal,F: extended_ereal > nat,C2: nat] :
( ( ord_le1083603963089353582_ereal @ A4 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X3: extended_ereal,Y5: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A4 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_984_ord__le__eq__subst,axiom,
! [A4: extended_ereal,B2: extended_ereal,F: extended_ereal > extended_ereal,C2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A4 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X3: extended_ereal,Y5: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X3 @ Y5 )
=> ( ord_le1083603963089353582_ereal @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le1083603963089353582_ereal @ ( F @ A4 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_985_ord__le__eq__subst,axiom,
! [A4: extended_ereal,B2: extended_ereal,F: extended_ereal > extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le1083603963089353582_ereal @ A4 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X3: extended_ereal,Y5: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X3 @ Y5 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le3935885782089961368nnreal @ ( F @ A4 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_986_ord__le__eq__subst,axiom,
! [A4: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > nat,C2: nat] :
( ( ord_le3935885782089961368nnreal @ A4 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X3: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A4 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_987_ord__le__eq__subst,axiom,
! [A4: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extended_ereal,C2: extended_ereal] :
( ( ord_le3935885782089961368nnreal @ A4 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X3: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y5 )
=> ( ord_le1083603963089353582_ereal @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le1083603963089353582_ereal @ ( F @ A4 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_988_ord__le__eq__subst,axiom,
! [A4: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A4 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X3: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y5 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le3935885782089961368nnreal @ ( F @ A4 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_989_ord__le__eq__subst,axiom,
! [A4: set_c,B2: set_c,F: set_c > nat,C2: nat] :
( ( ord_less_eq_set_c @ A4 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X3: set_c,Y5: set_c] :
( ( ord_less_eq_set_c @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A4 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_990_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_991_linorder__le__cases,axiom,
! [X: extended_ereal,Y: extended_ereal] :
( ~ ( ord_le1083603963089353582_ereal @ X @ Y )
=> ( ord_le1083603963089353582_ereal @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_992_linorder__le__cases,axiom,
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ~ ( ord_le3935885782089961368nnreal @ X @ Y )
=> ( ord_le3935885782089961368nnreal @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_993_order__antisym__conv,axiom,
! [Y: set_c,X: set_c] :
( ( ord_less_eq_set_c @ Y @ X )
=> ( ( ord_less_eq_set_c @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_994_order__antisym__conv,axiom,
! [Y: set_set_a,X: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y @ X )
=> ( ( ord_le3724670747650509150_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_995_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_996_order__antisym__conv,axiom,
! [Y: extended_ereal,X: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ Y @ X )
=> ( ( ord_le1083603963089353582_ereal @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_997_order__antisym__conv,axiom,
! [Y: extend8495563244428889912nnreal,X: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ Y @ X )
=> ( ( ord_le3935885782089961368nnreal @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_998_order__antisym__conv,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_999_sigma__sets_OBasic,axiom,
! [A4: set_c,A3: set_set_c,Sp: set_c] :
( ( member_set_c @ A4 @ A3 )
=> ( member_set_c @ A4 @ ( sigma_sigma_sets_c @ Sp @ A3 ) ) ) ).
% sigma_sets.Basic
thf(fact_1000_sigma__sets__eqI,axiom,
! [A3: set_set_c,M: set_c,B: set_set_c] :
( ! [A5: set_c] :
( ( member_set_c @ A5 @ A3 )
=> ( member_set_c @ A5 @ ( sigma_sigma_sets_c @ M @ B ) ) )
=> ( ! [B5: set_c] :
( ( member_set_c @ B5 @ B )
=> ( member_set_c @ B5 @ ( sigma_sigma_sets_c @ M @ A3 ) ) )
=> ( ( sigma_sigma_sets_c @ M @ A3 )
= ( sigma_sigma_sets_c @ M @ B ) ) ) ) ).
% sigma_sets_eqI
thf(fact_1001_sigma__sets__top,axiom,
! [Sp: set_c,A3: set_set_c] : ( member_set_c @ Sp @ ( sigma_sigma_sets_c @ Sp @ A3 ) ) ).
% sigma_sets_top
thf(fact_1002_sigma__sets__mono,axiom,
! [A3: set_set_c,X5: set_c,B: set_set_c] :
( ( ord_le3866738827743201120_set_c @ A3 @ ( sigma_sigma_sets_c @ X5 @ B ) )
=> ( ord_le3866738827743201120_set_c @ ( sigma_sigma_sets_c @ X5 @ A3 ) @ ( sigma_sigma_sets_c @ X5 @ B ) ) ) ).
% sigma_sets_mono
thf(fact_1003_sigma__sets__mono,axiom,
! [A3: set_set_a,X5: set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ ( sigma_sigma_sets_a @ X5 @ B ) )
=> ( ord_le3724670747650509150_set_a @ ( sigma_sigma_sets_a @ X5 @ A3 ) @ ( sigma_sigma_sets_a @ X5 @ B ) ) ) ).
% sigma_sets_mono
thf(fact_1004_sigma__sets__subseteq,axiom,
! [A3: set_set_c,B: set_set_c,X5: set_c] :
( ( ord_le3866738827743201120_set_c @ A3 @ B )
=> ( ord_le3866738827743201120_set_c @ ( sigma_sigma_sets_c @ X5 @ A3 ) @ ( sigma_sigma_sets_c @ X5 @ B ) ) ) ).
% sigma_sets_subseteq
thf(fact_1005_sigma__sets__subseteq,axiom,
! [A3: set_set_a,B: set_set_a,X5: set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B )
=> ( ord_le3724670747650509150_set_a @ ( sigma_sigma_sets_a @ X5 @ A3 ) @ ( sigma_sigma_sets_a @ X5 @ B ) ) ) ).
% sigma_sets_subseteq
thf(fact_1006_sigma__sets__superset__generator,axiom,
! [A3: set_set_c,X5: set_c] : ( ord_le3866738827743201120_set_c @ A3 @ ( sigma_sigma_sets_c @ X5 @ A3 ) ) ).
% sigma_sets_superset_generator
thf(fact_1007_sigma__sets__superset__generator,axiom,
! [A3: set_set_a,X5: set_a] : ( ord_le3724670747650509150_set_a @ A3 @ ( sigma_sigma_sets_a @ X5 @ A3 ) ) ).
% sigma_sets_superset_generator
thf(fact_1008_prop__restrict,axiom,
! [X: set_c,Z4: set_set_c,X5: set_set_c,P: set_c > $o] :
( ( member_set_c @ X @ Z4 )
=> ( ( ord_le3866738827743201120_set_c @ Z4
@ ( collect_set_c
@ ^ [X2: set_c] :
( ( member_set_c @ X2 @ X5 )
& ( P @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_1009_prop__restrict,axiom,
! [X: b,Z4: set_b,X5: set_b,P: b > $o] :
( ( member_b @ X @ Z4 )
=> ( ( ord_less_eq_set_b @ Z4
@ ( collect_b
@ ^ [X2: b] :
( ( member_b @ X2 @ X5 )
& ( P @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_1010_prop__restrict,axiom,
! [X: $o,Z4: set_o,X5: set_o,P: $o > $o] :
( ( member_o @ X @ Z4 )
=> ( ( ord_less_eq_set_o @ Z4
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ X5 )
& ( P @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_1011_prop__restrict,axiom,
! [X: nat,Z4: set_nat,X5: set_nat,P: nat > $o] :
( ( member_nat @ X @ Z4 )
=> ( ( ord_less_eq_set_nat @ Z4
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ X5 )
& ( P @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_1012_prop__restrict,axiom,
! [X: c,Z4: set_c,X5: set_c,P: c > $o] :
( ( member_c @ X @ Z4 )
=> ( ( ord_less_eq_set_c @ Z4
@ ( collect_c
@ ^ [X2: c] :
( ( member_c @ X2 @ X5 )
& ( P @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_1013_prop__restrict,axiom,
! [X: set_a,Z4: set_set_a,X5: set_set_a,P: set_a > $o] :
( ( member_set_a @ X @ Z4 )
=> ( ( ord_le3724670747650509150_set_a @ Z4
@ ( collect_set_a
@ ^ [X2: set_a] :
( ( member_set_a @ X2 @ X5 )
& ( P @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_1014_prop__restrict,axiom,
! [X: a,Z4: set_a,X5: set_a,P: a > $o] :
( ( member_a @ X @ Z4 )
=> ( ( ord_less_eq_set_a @ Z4
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ X5 )
& ( P @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_1015_Collect__restrict,axiom,
! [X5: set_set_c,P: set_c > $o] :
( ord_le3866738827743201120_set_c
@ ( collect_set_c
@ ^ [X2: set_c] :
( ( member_set_c @ X2 @ X5 )
& ( P @ X2 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_1016_Collect__restrict,axiom,
! [X5: set_b,P: b > $o] :
( ord_less_eq_set_b
@ ( collect_b
@ ^ [X2: b] :
( ( member_b @ X2 @ X5 )
& ( P @ X2 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_1017_Collect__restrict,axiom,
! [X5: set_o,P: $o > $o] :
( ord_less_eq_set_o
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ X5 )
& ( P @ X2 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_1018_Collect__restrict,axiom,
! [X5: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ X5 )
& ( P @ X2 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_1019_Collect__restrict,axiom,
! [X5: set_c,P: c > $o] :
( ord_less_eq_set_c
@ ( collect_c
@ ^ [X2: c] :
( ( member_c @ X2 @ X5 )
& ( P @ X2 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_1020_Collect__restrict,axiom,
! [X5: set_set_a,P: set_a > $o] :
( ord_le3724670747650509150_set_a
@ ( collect_set_a
@ ^ [X2: set_a] :
( ( member_set_a @ X2 @ X5 )
& ( P @ X2 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_1021_Collect__restrict,axiom,
! [X5: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ X5 )
& ( P @ X2 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_1022_image__Fpow__mono,axiom,
! [F: a > c,A3: set_a,B: set_c] :
( ( ord_less_eq_set_c @ ( image_a_c @ F @ A3 ) @ B )
=> ( ord_le3866738827743201120_set_c @ ( image_set_a_set_c @ ( image_a_c @ F ) @ ( finite_Fpow_a @ A3 ) ) @ ( finite_Fpow_c @ B ) ) ) ).
% image_Fpow_mono
thf(fact_1023_conj__subset__def,axiom,
! [A3: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A3
@ ( collect_nat
@ ^ [X2: nat] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) )
= ( ( ord_less_eq_set_nat @ A3 @ ( collect_nat @ P ) )
& ( ord_less_eq_set_nat @ A3 @ ( collect_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_1024_conj__subset__def,axiom,
! [A3: set_c,P: c > $o,Q: c > $o] :
( ( ord_less_eq_set_c @ A3
@ ( collect_c
@ ^ [X2: c] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) )
= ( ( ord_less_eq_set_c @ A3 @ ( collect_c @ P ) )
& ( ord_less_eq_set_c @ A3 @ ( collect_c @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_1025_conj__subset__def,axiom,
! [A3: set_set_a,P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ A3
@ ( collect_set_a
@ ^ [X2: set_a] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) )
= ( ( ord_le3724670747650509150_set_a @ A3 @ ( collect_set_a @ P ) )
& ( ord_le3724670747650509150_set_a @ A3 @ ( collect_set_a @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_1026_conj__subset__def,axiom,
! [A3: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A3
@ ( collect_a
@ ^ [X2: a] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) )
= ( ( ord_less_eq_set_a @ A3 @ ( collect_a @ P ) )
& ( ord_less_eq_set_a @ A3 @ ( collect_a @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_1027_subset__CollectI,axiom,
! [B: set_set_c,A3: set_set_c,Q: set_c > $o,P: set_c > $o] :
( ( ord_le3866738827743201120_set_c @ B @ A3 )
=> ( ! [X3: set_c] :
( ( member_set_c @ X3 @ B )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_le3866738827743201120_set_c
@ ( collect_set_c
@ ^ [X2: set_c] :
( ( member_set_c @ X2 @ B )
& ( Q @ X2 ) ) )
@ ( collect_set_c
@ ^ [X2: set_c] :
( ( member_set_c @ X2 @ A3 )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_1028_subset__CollectI,axiom,
! [B: set_b,A3: set_b,Q: b > $o,P: b > $o] :
( ( ord_less_eq_set_b @ B @ A3 )
=> ( ! [X3: b] :
( ( member_b @ X3 @ B )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_less_eq_set_b
@ ( collect_b
@ ^ [X2: b] :
( ( member_b @ X2 @ B )
& ( Q @ X2 ) ) )
@ ( collect_b
@ ^ [X2: b] :
( ( member_b @ X2 @ A3 )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_1029_subset__CollectI,axiom,
! [B: set_o,A3: set_o,Q: $o > $o,P: $o > $o] :
( ( ord_less_eq_set_o @ B @ A3 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_less_eq_set_o
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ B )
& ( Q @ X2 ) ) )
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A3 )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_1030_subset__CollectI,axiom,
! [B: set_nat,A3: set_nat,Q: nat > $o,P: nat > $o] :
( ( ord_less_eq_set_nat @ B @ A3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ B )
& ( Q @ X2 ) ) )
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_1031_subset__CollectI,axiom,
! [B: set_c,A3: set_c,Q: c > $o,P: c > $o] :
( ( ord_less_eq_set_c @ B @ A3 )
=> ( ! [X3: c] :
( ( member_c @ X3 @ B )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_less_eq_set_c
@ ( collect_c
@ ^ [X2: c] :
( ( member_c @ X2 @ B )
& ( Q @ X2 ) ) )
@ ( collect_c
@ ^ [X2: c] :
( ( member_c @ X2 @ A3 )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_1032_subset__CollectI,axiom,
! [B: set_set_a,A3: set_set_a,Q: set_a > $o,P: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ B @ A3 )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ B )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_le3724670747650509150_set_a
@ ( collect_set_a
@ ^ [X2: set_a] :
( ( member_set_a @ X2 @ B )
& ( Q @ X2 ) ) )
@ ( collect_set_a
@ ^ [X2: set_a] :
( ( member_set_a @ X2 @ A3 )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_1033_subset__CollectI,axiom,
! [B: set_a,A3: set_a,Q: a > $o,P: a > $o] :
( ( ord_less_eq_set_a @ B @ A3 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_less_eq_set_a
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ B )
& ( Q @ X2 ) ) )
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A3 )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_1034_subset__Collect__iff,axiom,
! [B: set_set_c,A3: set_set_c,P: set_c > $o] :
( ( ord_le3866738827743201120_set_c @ B @ A3 )
=> ( ( ord_le3866738827743201120_set_c @ B
@ ( collect_set_c
@ ^ [X2: set_c] :
( ( member_set_c @ X2 @ A3 )
& ( P @ X2 ) ) ) )
= ( ! [X2: set_c] :
( ( member_set_c @ X2 @ B )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_1035_subset__Collect__iff,axiom,
! [B: set_b,A3: set_b,P: b > $o] :
( ( ord_less_eq_set_b @ B @ A3 )
=> ( ( ord_less_eq_set_b @ B
@ ( collect_b
@ ^ [X2: b] :
( ( member_b @ X2 @ A3 )
& ( P @ X2 ) ) ) )
= ( ! [X2: b] :
( ( member_b @ X2 @ B )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_1036_subset__Collect__iff,axiom,
! [B: set_o,A3: set_o,P: $o > $o] :
( ( ord_less_eq_set_o @ B @ A3 )
=> ( ( ord_less_eq_set_o @ B
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A3 )
& ( P @ X2 ) ) ) )
= ( ! [X2: $o] :
( ( member_o @ X2 @ B )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_1037_subset__Collect__iff,axiom,
! [B: set_nat,A3: set_nat,P: nat > $o] :
( ( ord_less_eq_set_nat @ B @ A3 )
=> ( ( ord_less_eq_set_nat @ B
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ( P @ X2 ) ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ B )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_1038_subset__Collect__iff,axiom,
! [B: set_c,A3: set_c,P: c > $o] :
( ( ord_less_eq_set_c @ B @ A3 )
=> ( ( ord_less_eq_set_c @ B
@ ( collect_c
@ ^ [X2: c] :
( ( member_c @ X2 @ A3 )
& ( P @ X2 ) ) ) )
= ( ! [X2: c] :
( ( member_c @ X2 @ B )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_1039_subset__Collect__iff,axiom,
! [B: set_set_a,A3: set_set_a,P: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ B @ A3 )
=> ( ( ord_le3724670747650509150_set_a @ B
@ ( collect_set_a
@ ^ [X2: set_a] :
( ( member_set_a @ X2 @ A3 )
& ( P @ X2 ) ) ) )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ B )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_1040_subset__Collect__iff,axiom,
! [B: set_a,A3: set_a,P: a > $o] :
( ( ord_less_eq_set_a @ B @ A3 )
=> ( ( ord_less_eq_set_a @ B
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A3 )
& ( P @ X2 ) ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_1041_Sup_OSUP__cong,axiom,
! [A3: set_a,B: set_a,C: a > c,D2: a > c,Sup: set_c > c] :
( ( A3 = B )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B )
=> ( ( C @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( Sup @ ( image_a_c @ C @ A3 ) )
= ( Sup @ ( image_a_c @ D2 @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_1042_Inf_OINF__cong,axiom,
! [A3: set_a,B: set_a,C: a > c,D2: a > c,Inf: set_c > c] :
( ( A3 = B )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B )
=> ( ( C @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( Inf @ ( image_a_c @ C @ A3 ) )
= ( Inf @ ( image_a_c @ D2 @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_1043_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_a,K: set_a,B2: set_a,A4: set_a] :
( ( B
= ( inf_inf_set_a @ K @ B2 ) )
=> ( ( inf_inf_set_a @ A4 @ B )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A4 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_1044_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_c,K: set_c,B2: set_c,A4: set_c] :
( ( B
= ( inf_inf_set_c @ K @ B2 ) )
=> ( ( inf_inf_set_c @ A4 @ B )
= ( inf_inf_set_c @ K @ ( inf_inf_set_c @ A4 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_1045_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_nat,K: set_nat,B2: set_nat,A4: set_nat] :
( ( B
= ( inf_inf_set_nat @ K @ B2 ) )
=> ( ( inf_inf_set_nat @ A4 @ B )
= ( inf_inf_set_nat @ K @ ( inf_inf_set_nat @ A4 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_1046_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_set_a,K: set_set_a,B2: set_set_a,A4: set_set_a] :
( ( B
= ( inf_inf_set_set_a @ K @ B2 ) )
=> ( ( inf_inf_set_set_a @ A4 @ B )
= ( inf_inf_set_set_a @ K @ ( inf_inf_set_set_a @ A4 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_1047_boolean__algebra__cancel_Oinf2,axiom,
! [B: extend8495563244428889912nnreal,K: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,A4: extend8495563244428889912nnreal] :
( ( B
= ( inf_in7439215052339218890nnreal @ K @ B2 ) )
=> ( ( inf_in7439215052339218890nnreal @ A4 @ B )
= ( inf_in7439215052339218890nnreal @ K @ ( inf_in7439215052339218890nnreal @ A4 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_1048_boolean__algebra__cancel_Oinf2,axiom,
! [B: extended_ereal,K: extended_ereal,B2: extended_ereal,A4: extended_ereal] :
( ( B
= ( inf_in2794916579150040252_ereal @ K @ B2 ) )
=> ( ( inf_in2794916579150040252_ereal @ A4 @ B )
= ( inf_in2794916579150040252_ereal @ K @ ( inf_in2794916579150040252_ereal @ A4 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_1049_boolean__algebra__cancel_Oinf2,axiom,
! [B: nat,K: nat,B2: nat,A4: nat] :
( ( B
= ( inf_inf_nat @ K @ B2 ) )
=> ( ( inf_inf_nat @ A4 @ B )
= ( inf_inf_nat @ K @ ( inf_inf_nat @ A4 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_1050_boolean__algebra__cancel_Oinf1,axiom,
! [A3: set_a,K: set_a,A4: set_a,B2: set_a] :
( ( A3
= ( inf_inf_set_a @ K @ A4 ) )
=> ( ( inf_inf_set_a @ A3 @ B2 )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A4 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_1051_boolean__algebra__cancel_Oinf1,axiom,
! [A3: set_c,K: set_c,A4: set_c,B2: set_c] :
( ( A3
= ( inf_inf_set_c @ K @ A4 ) )
=> ( ( inf_inf_set_c @ A3 @ B2 )
= ( inf_inf_set_c @ K @ ( inf_inf_set_c @ A4 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_1052_boolean__algebra__cancel_Oinf1,axiom,
! [A3: set_nat,K: set_nat,A4: set_nat,B2: set_nat] :
( ( A3
= ( inf_inf_set_nat @ K @ A4 ) )
=> ( ( inf_inf_set_nat @ A3 @ B2 )
= ( inf_inf_set_nat @ K @ ( inf_inf_set_nat @ A4 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_1053_boolean__algebra__cancel_Oinf1,axiom,
! [A3: set_set_a,K: set_set_a,A4: set_set_a,B2: set_set_a] :
( ( A3
= ( inf_inf_set_set_a @ K @ A4 ) )
=> ( ( inf_inf_set_set_a @ A3 @ B2 )
= ( inf_inf_set_set_a @ K @ ( inf_inf_set_set_a @ A4 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_1054_boolean__algebra__cancel_Oinf1,axiom,
! [A3: extend8495563244428889912nnreal,K: extend8495563244428889912nnreal,A4: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( A3
= ( inf_in7439215052339218890nnreal @ K @ A4 ) )
=> ( ( inf_in7439215052339218890nnreal @ A3 @ B2 )
= ( inf_in7439215052339218890nnreal @ K @ ( inf_in7439215052339218890nnreal @ A4 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_1055_boolean__algebra__cancel_Oinf1,axiom,
! [A3: extended_ereal,K: extended_ereal,A4: extended_ereal,B2: extended_ereal] :
( ( A3
= ( inf_in2794916579150040252_ereal @ K @ A4 ) )
=> ( ( inf_in2794916579150040252_ereal @ A3 @ B2 )
= ( inf_in2794916579150040252_ereal @ K @ ( inf_in2794916579150040252_ereal @ A4 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_1056_boolean__algebra__cancel_Oinf1,axiom,
! [A3: nat,K: nat,A4: nat,B2: nat] :
( ( A3
= ( inf_inf_nat @ K @ A4 ) )
=> ( ( inf_inf_nat @ A3 @ B2 )
= ( inf_inf_nat @ K @ ( inf_inf_nat @ A4 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_1057_Fpow__mono,axiom,
! [A3: set_c,B: set_c] :
( ( ord_less_eq_set_c @ A3 @ B )
=> ( ord_le3866738827743201120_set_c @ ( finite_Fpow_c @ A3 ) @ ( finite_Fpow_c @ B ) ) ) ).
% Fpow_mono
thf(fact_1058_Fpow__mono,axiom,
! [A3: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B )
=> ( ord_le5722252365846178494_set_a @ ( finite_Fpow_set_a @ A3 ) @ ( finite_Fpow_set_a @ B ) ) ) ).
% Fpow_mono
thf(fact_1059_Fpow__mono,axiom,
! [A3: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A3 @ B )
=> ( ord_le3724670747650509150_set_a @ ( finite_Fpow_a @ A3 ) @ ( finite_Fpow_a @ B ) ) ) ).
% Fpow_mono
thf(fact_1060_Fpow__subset__Pow,axiom,
! [A3: set_c] : ( ord_le3866738827743201120_set_c @ ( finite_Fpow_c @ A3 ) @ ( pow_c @ A3 ) ) ).
% Fpow_subset_Pow
thf(fact_1061_Fpow__subset__Pow,axiom,
! [A3: set_a] : ( ord_le3724670747650509150_set_a @ ( finite_Fpow_a @ A3 ) @ ( pow_a @ A3 ) ) ).
% Fpow_subset_Pow
thf(fact_1062_Fpow__Pow__finite,axiom,
( finite_Fpow_c
= ( ^ [A: set_c] : ( inf_inf_set_set_c @ ( pow_c @ A ) @ ( collect_set_c @ finite_finite_c ) ) ) ) ).
% Fpow_Pow_finite
thf(fact_1063_Fpow__Pow__finite,axiom,
( finite_Fpow_nat
= ( ^ [A: set_nat] : ( inf_inf_set_set_nat @ ( pow_nat @ A ) @ ( collect_set_nat @ finite_finite_nat ) ) ) ) ).
% Fpow_Pow_finite
thf(fact_1064_Fpow__Pow__finite,axiom,
( finite_Fpow_a
= ( ^ [A: set_a] : ( inf_inf_set_set_a @ ( pow_a @ A ) @ ( collect_set_a @ finite_finite_a ) ) ) ) ).
% Fpow_Pow_finite
thf(fact_1065_Powp__Pow__eq,axiom,
! [A3: set_a] :
( ( powp_a
@ ^ [X2: a] : ( member_a @ X2 @ A3 ) )
= ( ^ [X2: set_a] : ( member_set_a @ X2 @ ( pow_a @ A3 ) ) ) ) ).
% Powp_Pow_eq
thf(fact_1066_Powp__Pow__eq,axiom,
! [A3: set_c] :
( ( powp_c
@ ^ [X2: c] : ( member_c @ X2 @ A3 ) )
= ( ^ [X2: set_c] : ( member_set_c @ X2 @ ( pow_c @ A3 ) ) ) ) ).
% Powp_Pow_eq
thf(fact_1067_Powp__Pow__eq,axiom,
! [A3: set_set_c] :
( ( powp_set_c
@ ^ [X2: set_c] : ( member_set_c @ X2 @ A3 ) )
= ( ^ [X2: set_set_c] : ( member_set_set_c @ X2 @ ( pow_set_c @ A3 ) ) ) ) ).
% Powp_Pow_eq
thf(fact_1068_Powp__Pow__eq,axiom,
! [A3: set_b] :
( ( powp_b
@ ^ [X2: b] : ( member_b @ X2 @ A3 ) )
= ( ^ [X2: set_b] : ( member_set_b @ X2 @ ( pow_b @ A3 ) ) ) ) ).
% Powp_Pow_eq
thf(fact_1069_Powp__Pow__eq,axiom,
! [A3: set_o] :
( ( powp_o
@ ^ [X2: $o] : ( member_o @ X2 @ A3 ) )
= ( ^ [X2: set_o] : ( member_set_o @ X2 @ ( pow_o @ A3 ) ) ) ) ).
% Powp_Pow_eq
thf(fact_1070_Powp__Pow__eq,axiom,
! [A3: set_nat] :
( ( powp_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A3 ) )
= ( ^ [X2: set_nat] : ( member_set_nat @ X2 @ ( pow_nat @ A3 ) ) ) ) ).
% Powp_Pow_eq
thf(fact_1071_image__vimage__eq,axiom,
! [F: c > a,A3: set_a] :
( ( image_c_a @ F @ ( vimage_c_a @ F @ A3 ) )
= ( inf_inf_set_a @ A3 @ ( image_c_a @ F @ top_top_set_c ) ) ) ).
% image_vimage_eq
thf(fact_1072_image__vimage__eq,axiom,
! [F: a > a,A3: set_a] :
( ( image_a_a @ F @ ( vimage_a_a @ F @ A3 ) )
= ( inf_inf_set_a @ A3 @ ( image_a_a @ F @ top_top_set_a ) ) ) ).
% image_vimage_eq
thf(fact_1073_image__vimage__eq,axiom,
! [F: a > c,A3: set_c] :
( ( image_a_c @ F @ ( vimage_a_c @ F @ A3 ) )
= ( inf_inf_set_c @ A3 @ ( image_a_c @ F @ top_top_set_a ) ) ) ).
% image_vimage_eq
thf(fact_1074_image__vimage__eq,axiom,
! [F: c > c,A3: set_c] :
( ( image_c_c @ F @ ( vimage_c_c @ F @ A3 ) )
= ( inf_inf_set_c @ A3 @ ( image_c_c @ F @ top_top_set_c ) ) ) ).
% image_vimage_eq
thf(fact_1075_image__vimage__eq,axiom,
! [F: nat > a,A3: set_a] :
( ( image_nat_a @ F @ ( vimage_nat_a @ F @ A3 ) )
= ( inf_inf_set_a @ A3 @ ( image_nat_a @ F @ top_top_set_nat ) ) ) ).
% image_vimage_eq
thf(fact_1076_image__vimage__eq,axiom,
! [F: nat > c,A3: set_c] :
( ( image_nat_c @ F @ ( vimage_nat_c @ F @ A3 ) )
= ( inf_inf_set_c @ A3 @ ( image_nat_c @ F @ top_top_set_nat ) ) ) ).
% image_vimage_eq
thf(fact_1077_image__vimage__eq,axiom,
! [F: nat > nat,A3: set_nat] :
( ( image_nat_nat @ F @ ( vimage_nat_nat @ F @ A3 ) )
= ( inf_inf_set_nat @ A3 @ ( image_nat_nat @ F @ top_top_set_nat ) ) ) ).
% image_vimage_eq
thf(fact_1078_image__vimage__eq,axiom,
! [F: nat > set_a,A3: set_set_a] :
( ( image_nat_set_a @ F @ ( vimage_nat_set_a @ F @ A3 ) )
= ( inf_inf_set_set_a @ A3 @ ( image_nat_set_a @ F @ top_top_set_nat ) ) ) ).
% image_vimage_eq
thf(fact_1079_sigma__sets__vimage__commute,axiom,
! [X5: a > a,Omega: set_a,Omega2: set_a,M2: set_set_a] :
( ( member_a_a @ X5
@ ( pi_a_a @ Omega
@ ^ [Uu: a] : Omega2 ) )
=> ( ( collect_set_a
@ ^ [Uu: set_a] :
? [A: set_a] :
( ( Uu
= ( inf_inf_set_a @ ( vimage_a_a @ X5 @ A ) @ Omega ) )
& ( member_set_a @ A @ ( sigma_sigma_sets_a @ Omega2 @ M2 ) ) ) )
= ( sigma_sigma_sets_a @ Omega
@ ( collect_set_a
@ ^ [Uu: set_a] :
? [A: set_a] :
( ( Uu
= ( inf_inf_set_a @ ( vimage_a_a @ X5 @ A ) @ Omega ) )
& ( member_set_a @ A @ M2 ) ) ) ) ) ) ).
% sigma_sets_vimage_commute
thf(fact_1080_sigma__sets__vimage__commute,axiom,
! [X5: nat > nat,Omega: set_nat,Omega2: set_nat,M2: set_set_nat] :
( ( member_nat_nat @ X5
@ ( pi_nat_nat @ Omega
@ ^ [Uu: nat] : Omega2 ) )
=> ( ( collect_set_nat
@ ^ [Uu: set_nat] :
? [A: set_nat] :
( ( Uu
= ( inf_inf_set_nat @ ( vimage_nat_nat @ X5 @ A ) @ Omega ) )
& ( member_set_nat @ A @ ( sigma_sigma_sets_nat @ Omega2 @ M2 ) ) ) )
= ( sigma_sigma_sets_nat @ Omega
@ ( collect_set_nat
@ ^ [Uu: set_nat] :
? [A: set_nat] :
( ( Uu
= ( inf_inf_set_nat @ ( vimage_nat_nat @ X5 @ A ) @ Omega ) )
& ( member_set_nat @ A @ M2 ) ) ) ) ) ) ).
% sigma_sets_vimage_commute
thf(fact_1081_sigma__sets__vimage__commute,axiom,
! [X5: c > a,Omega: set_c,Omega2: set_a,M2: set_set_a] :
( ( member_c_a @ X5
@ ( pi_c_a @ Omega
@ ^ [Uu: c] : Omega2 ) )
=> ( ( collect_set_c
@ ^ [Uu: set_c] :
? [A: set_a] :
( ( Uu
= ( inf_inf_set_c @ ( vimage_c_a @ X5 @ A ) @ Omega ) )
& ( member_set_a @ A @ ( sigma_sigma_sets_a @ Omega2 @ M2 ) ) ) )
= ( sigma_sigma_sets_c @ Omega
@ ( collect_set_c
@ ^ [Uu: set_c] :
? [A: set_a] :
( ( Uu
= ( inf_inf_set_c @ ( vimage_c_a @ X5 @ A ) @ Omega ) )
& ( member_set_a @ A @ M2 ) ) ) ) ) ) ).
% sigma_sets_vimage_commute
thf(fact_1082_sigma__sets__vimage__commute,axiom,
! [X5: a > c,Omega: set_a,Omega2: set_c,M2: set_set_c] :
( ( member_a_c @ X5
@ ( pi_a_c @ Omega
@ ^ [Uu: a] : Omega2 ) )
=> ( ( collect_set_a
@ ^ [Uu: set_a] :
? [A: set_c] :
( ( Uu
= ( inf_inf_set_a @ ( vimage_a_c @ X5 @ A ) @ Omega ) )
& ( member_set_c @ A @ ( sigma_sigma_sets_c @ Omega2 @ M2 ) ) ) )
= ( sigma_sigma_sets_a @ Omega
@ ( collect_set_a
@ ^ [Uu: set_a] :
? [A: set_c] :
( ( Uu
= ( inf_inf_set_a @ ( vimage_a_c @ X5 @ A ) @ Omega ) )
& ( member_set_c @ A @ M2 ) ) ) ) ) ) ).
% sigma_sets_vimage_commute
thf(fact_1083_sigma__sets__vimage__commute,axiom,
! [X5: nat > c,Omega: set_nat,Omega2: set_c,M2: set_set_c] :
( ( member_nat_c @ X5
@ ( pi_nat_c @ Omega
@ ^ [Uu: nat] : Omega2 ) )
=> ( ( collect_set_nat
@ ^ [Uu: set_nat] :
? [A: set_c] :
( ( Uu
= ( inf_inf_set_nat @ ( vimage_nat_c @ X5 @ A ) @ Omega ) )
& ( member_set_c @ A @ ( sigma_sigma_sets_c @ Omega2 @ M2 ) ) ) )
= ( sigma_sigma_sets_nat @ Omega
@ ( collect_set_nat
@ ^ [Uu: set_nat] :
? [A: set_c] :
( ( Uu
= ( inf_inf_set_nat @ ( vimage_nat_c @ X5 @ A ) @ Omega ) )
& ( member_set_c @ A @ M2 ) ) ) ) ) ) ).
% sigma_sets_vimage_commute
thf(fact_1084_sigma__sets__vimage__commute,axiom,
! [X5: set_a > c,Omega: set_set_a,Omega2: set_c,M2: set_set_c] :
( ( member_set_a_c @ X5
@ ( pi_set_a_c @ Omega
@ ^ [Uu: set_a] : Omega2 ) )
=> ( ( collect_set_set_a
@ ^ [Uu: set_set_a] :
? [A: set_c] :
( ( Uu
= ( inf_inf_set_set_a @ ( vimage_set_a_c @ X5 @ A ) @ Omega ) )
& ( member_set_c @ A @ ( sigma_sigma_sets_c @ Omega2 @ M2 ) ) ) )
= ( sigma_2987359967864564790_set_a @ Omega
@ ( collect_set_set_a
@ ^ [Uu: set_set_a] :
? [A: set_c] :
( ( Uu
= ( inf_inf_set_set_a @ ( vimage_set_a_c @ X5 @ A ) @ Omega ) )
& ( member_set_c @ A @ M2 ) ) ) ) ) ) ).
% sigma_sets_vimage_commute
thf(fact_1085_sigma__sets__vimage__commute,axiom,
! [X5: c > c,Omega: set_c,Omega2: set_c,M2: set_set_c] :
( ( member_c_c @ X5
@ ( pi_c_c @ Omega
@ ^ [Uu: c] : Omega2 ) )
=> ( ( collect_set_c
@ ^ [Uu: set_c] :
? [A: set_c] :
( ( Uu
= ( inf_inf_set_c @ ( vimage_c_c @ X5 @ A ) @ Omega ) )
& ( member_set_c @ A @ ( sigma_sigma_sets_c @ Omega2 @ M2 ) ) ) )
= ( sigma_sigma_sets_c @ Omega
@ ( collect_set_c
@ ^ [Uu: set_c] :
? [A: set_c] :
( ( Uu
= ( inf_inf_set_c @ ( vimage_c_c @ X5 @ A ) @ Omega ) )
& ( member_set_c @ A @ M2 ) ) ) ) ) ) ).
% sigma_sets_vimage_commute
thf(fact_1086_closed__cdi__subset,axiom,
! [Omega: set_c,M: set_set_c] :
( ( sigma_closed_cdi_c @ Omega @ M )
=> ( ord_le3866738827743201120_set_c @ M @ ( pow_c @ Omega ) ) ) ).
% closed_cdi_subset
thf(fact_1087_closed__cdi__subset,axiom,
! [Omega: set_a,M: set_set_a] :
( ( sigma_closed_cdi_a @ Omega @ M )
=> ( ord_le3724670747650509150_set_a @ M @ ( pow_a @ Omega ) ) ) ).
% closed_cdi_subset
thf(fact_1088_measure__space__eq,axiom,
! [A3: set_set_c,Omega: set_c,Mu: set_c > extend8495563244428889912nnreal,Mu2: set_c > extend8495563244428889912nnreal] :
( ( ord_le3866738827743201120_set_c @ A3 @ ( pow_c @ Omega ) )
=> ( ! [A5: set_c] :
( ( member_set_c @ A5 @ ( sigma_sigma_sets_c @ Omega @ A3 ) )
=> ( ( Mu @ A5 )
= ( Mu2 @ A5 ) ) )
=> ( ( sigma_3179946494550678600pace_c @ Omega @ ( sigma_sigma_sets_c @ Omega @ A3 ) @ Mu )
= ( sigma_3179946494550678600pace_c @ Omega @ ( sigma_sigma_sets_c @ Omega @ A3 ) @ Mu2 ) ) ) ) ).
% measure_space_eq
thf(fact_1089_measure__space__eq,axiom,
! [A3: set_set_a,Omega: set_a,Mu: set_a > extend8495563244428889912nnreal,Mu2: set_a > extend8495563244428889912nnreal] :
( ( ord_le3724670747650509150_set_a @ A3 @ ( pow_a @ Omega ) )
=> ( ! [A5: set_a] :
( ( member_set_a @ A5 @ ( sigma_sigma_sets_a @ Omega @ A3 ) )
=> ( ( Mu @ A5 )
= ( Mu2 @ A5 ) ) )
=> ( ( sigma_3179946494550678598pace_a @ Omega @ ( sigma_sigma_sets_a @ Omega @ A3 ) @ Mu )
= ( sigma_3179946494550678598pace_a @ Omega @ ( sigma_sigma_sets_a @ Omega @ A3 ) @ Mu2 ) ) ) ) ).
% measure_space_eq
thf(fact_1090_range__inter,axiom,
! [F4: set_a] :
( ( image_set_a_set_a @ ( inf_inf_set_a @ F4 ) @ top_top_set_set_a )
= ( pow_a @ F4 ) ) ).
% range_inter
thf(fact_1091_range__inter,axiom,
! [F4: set_c] :
( ( image_set_c_set_c @ ( inf_inf_set_c @ F4 ) @ top_top_set_set_c )
= ( pow_c @ F4 ) ) ).
% range_inter
thf(fact_1092_range__inter,axiom,
! [F4: set_nat] :
( ( image_7916887816326733075et_nat @ ( inf_inf_set_nat @ F4 ) @ top_top_set_set_nat )
= ( pow_nat @ F4 ) ) ).
% range_inter
thf(fact_1093_range__inter,axiom,
! [F4: set_set_a] :
( ( image_1042221919965026181_set_a @ ( inf_inf_set_set_a @ F4 ) @ top_to4027821306633060462_set_a )
= ( pow_set_a @ F4 ) ) ).
% range_inter
thf(fact_1094_UNIV__I,axiom,
! [X: set_c] : ( member_set_c @ X @ top_top_set_set_c ) ).
% UNIV_I
thf(fact_1095_UNIV__I,axiom,
! [X: b] : ( member_b @ X @ top_top_set_b ) ).
% UNIV_I
thf(fact_1096_UNIV__I,axiom,
! [X: $o] : ( member_o @ X @ top_top_set_o ) ).
% UNIV_I
thf(fact_1097_UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% UNIV_I
thf(fact_1098_finite__Collect__disjI,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( finite_finite_set_a
@ ( collect_set_a
@ ^ [X2: set_a] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite_finite_set_a @ ( collect_set_a @ P ) )
& ( finite_finite_set_a @ ( collect_set_a @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_1099_finite__Collect__disjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_1100_finite__Collect__conjI,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( ( finite_finite_set_a @ ( collect_set_a @ P ) )
| ( finite_finite_set_a @ ( collect_set_a @ Q ) ) )
=> ( finite_finite_set_a
@ ( collect_set_a
@ ^ [X2: set_a] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_1101_finite__Collect__conjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P ) )
| ( finite_finite_nat @ ( collect_nat @ Q ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_1102_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_1103_finite__imageI,axiom,
! [F4: set_a,H: a > c] :
( ( finite_finite_a @ F4 )
=> ( finite_finite_c @ ( image_a_c @ H @ F4 ) ) ) ).
% finite_imageI
thf(fact_1104_finite__imageI,axiom,
! [F4: set_nat,H: nat > nat] :
( ( finite_finite_nat @ F4 )
=> ( finite_finite_nat @ ( image_nat_nat @ H @ F4 ) ) ) ).
% finite_imageI
thf(fact_1105_inf__top__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ top_top_set_a @ X )
= X ) ).
% inf_top_left
thf(fact_1106_inf__top__left,axiom,
! [X: set_c] :
( ( inf_inf_set_c @ top_top_set_c @ X )
= X ) ).
% inf_top_left
thf(fact_1107_inf__top__left,axiom,
! [X: set_set_a] :
( ( inf_inf_set_set_a @ top_top_set_set_a @ X )
= X ) ).
% inf_top_left
thf(fact_1108_inf__top__left,axiom,
! [X: extended_ereal] :
( ( inf_in2794916579150040252_ereal @ top_to6662034908053899550_ereal @ X )
= X ) ).
% inf_top_left
thf(fact_1109_inf__top__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ X )
= X ) ).
% inf_top_left
thf(fact_1110_inf__top__left,axiom,
! [X: extend8495563244428889912nnreal] :
( ( inf_in7439215052339218890nnreal @ top_to1496364449551166952nnreal @ X )
= X ) ).
% inf_top_left
thf(fact_1111_inf__top__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ top_top_set_a )
= X ) ).
% inf_top_right
thf(fact_1112_inf__top__right,axiom,
! [X: set_c] :
( ( inf_inf_set_c @ X @ top_top_set_c )
= X ) ).
% inf_top_right
thf(fact_1113_inf__top__right,axiom,
! [X: set_set_a] :
( ( inf_inf_set_set_a @ X @ top_top_set_set_a )
= X ) ).
% inf_top_right
thf(fact_1114_inf__top__right,axiom,
! [X: extended_ereal] :
( ( inf_in2794916579150040252_ereal @ X @ top_to6662034908053899550_ereal )
= X ) ).
% inf_top_right
thf(fact_1115_inf__top__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ top_top_set_nat )
= X ) ).
% inf_top_right
thf(fact_1116_inf__top__right,axiom,
! [X: extend8495563244428889912nnreal] :
( ( inf_in7439215052339218890nnreal @ X @ top_to1496364449551166952nnreal )
= X ) ).
% inf_top_right
thf(fact_1117_inf__eq__top__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ( inf_inf_set_a @ X @ Y )
= top_top_set_a )
= ( ( X = top_top_set_a )
& ( Y = top_top_set_a ) ) ) ).
% inf_eq_top_iff
thf(fact_1118_inf__eq__top__iff,axiom,
! [X: set_c,Y: set_c] :
( ( ( inf_inf_set_c @ X @ Y )
= top_top_set_c )
= ( ( X = top_top_set_c )
& ( Y = top_top_set_c ) ) ) ).
% inf_eq_top_iff
thf(fact_1119_inf__eq__top__iff,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( ( inf_inf_set_set_a @ X @ Y )
= top_top_set_set_a )
= ( ( X = top_top_set_set_a )
& ( Y = top_top_set_set_a ) ) ) ).
% inf_eq_top_iff
thf(fact_1120_inf__eq__top__iff,axiom,
! [X: extended_ereal,Y: extended_ereal] :
( ( ( inf_in2794916579150040252_ereal @ X @ Y )
= top_to6662034908053899550_ereal )
= ( ( X = top_to6662034908053899550_ereal )
& ( Y = top_to6662034908053899550_ereal ) ) ) ).
% inf_eq_top_iff
thf(fact_1121_inf__eq__top__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( inf_inf_set_nat @ X @ Y )
= top_top_set_nat )
= ( ( X = top_top_set_nat )
& ( Y = top_top_set_nat ) ) ) ).
% inf_eq_top_iff
thf(fact_1122_inf__eq__top__iff,axiom,
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( ( inf_in7439215052339218890nnreal @ X @ Y )
= top_to1496364449551166952nnreal )
= ( ( X = top_to1496364449551166952nnreal )
& ( Y = top_to1496364449551166952nnreal ) ) ) ).
% inf_eq_top_iff
thf(fact_1123_top__eq__inf__iff,axiom,
! [X: set_a,Y: set_a] :
( ( top_top_set_a
= ( inf_inf_set_a @ X @ Y ) )
= ( ( X = top_top_set_a )
& ( Y = top_top_set_a ) ) ) ).
% top_eq_inf_iff
thf(fact_1124_top__eq__inf__iff,axiom,
! [X: set_c,Y: set_c] :
( ( top_top_set_c
= ( inf_inf_set_c @ X @ Y ) )
= ( ( X = top_top_set_c )
& ( Y = top_top_set_c ) ) ) ).
% top_eq_inf_iff
thf(fact_1125_top__eq__inf__iff,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( top_top_set_set_a
= ( inf_inf_set_set_a @ X @ Y ) )
= ( ( X = top_top_set_set_a )
& ( Y = top_top_set_set_a ) ) ) ).
% top_eq_inf_iff
thf(fact_1126_top__eq__inf__iff,axiom,
! [X: extended_ereal,Y: extended_ereal] :
( ( top_to6662034908053899550_ereal
= ( inf_in2794916579150040252_ereal @ X @ Y ) )
= ( ( X = top_to6662034908053899550_ereal )
& ( Y = top_to6662034908053899550_ereal ) ) ) ).
% top_eq_inf_iff
thf(fact_1127_top__eq__inf__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( top_top_set_nat
= ( inf_inf_set_nat @ X @ Y ) )
= ( ( X = top_top_set_nat )
& ( Y = top_top_set_nat ) ) ) ).
% top_eq_inf_iff
thf(fact_1128_top__eq__inf__iff,axiom,
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( top_to1496364449551166952nnreal
= ( inf_in7439215052339218890nnreal @ X @ Y ) )
= ( ( X = top_to1496364449551166952nnreal )
& ( Y = top_to1496364449551166952nnreal ) ) ) ).
% top_eq_inf_iff
thf(fact_1129_inf__top_Oeq__neutr__iff,axiom,
! [A4: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ A4 @ B2 )
= top_top_set_a )
= ( ( A4 = top_top_set_a )
& ( B2 = top_top_set_a ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_1130_inf__top_Oeq__neutr__iff,axiom,
! [A4: set_c,B2: set_c] :
( ( ( inf_inf_set_c @ A4 @ B2 )
= top_top_set_c )
= ( ( A4 = top_top_set_c )
& ( B2 = top_top_set_c ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_1131_inf__top_Oeq__neutr__iff,axiom,
! [A4: set_set_a,B2: set_set_a] :
( ( ( inf_inf_set_set_a @ A4 @ B2 )
= top_top_set_set_a )
= ( ( A4 = top_top_set_set_a )
& ( B2 = top_top_set_set_a ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_1132_inf__top_Oeq__neutr__iff,axiom,
! [A4: extended_ereal,B2: extended_ereal] :
( ( ( inf_in2794916579150040252_ereal @ A4 @ B2 )
= top_to6662034908053899550_ereal )
= ( ( A4 = top_to6662034908053899550_ereal )
& ( B2 = top_to6662034908053899550_ereal ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_1133_inf__top_Oeq__neutr__iff,axiom,
! [A4: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A4 @ B2 )
= top_top_set_nat )
= ( ( A4 = top_top_set_nat )
& ( B2 = top_top_set_nat ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_1134_inf__top_Oeq__neutr__iff,axiom,
! [A4: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( ( inf_in7439215052339218890nnreal @ A4 @ B2 )
= top_to1496364449551166952nnreal )
= ( ( A4 = top_to1496364449551166952nnreal )
& ( B2 = top_to1496364449551166952nnreal ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_1135_inf__top_Oleft__neutral,axiom,
! [A4: set_a] :
( ( inf_inf_set_a @ top_top_set_a @ A4 )
= A4 ) ).
% inf_top.left_neutral
thf(fact_1136_inf__top_Oleft__neutral,axiom,
! [A4: set_c] :
( ( inf_inf_set_c @ top_top_set_c @ A4 )
= A4 ) ).
% inf_top.left_neutral
thf(fact_1137_inf__top_Oleft__neutral,axiom,
! [A4: set_set_a] :
( ( inf_inf_set_set_a @ top_top_set_set_a @ A4 )
= A4 ) ).
% inf_top.left_neutral
thf(fact_1138_inf__top_Oleft__neutral,axiom,
! [A4: extended_ereal] :
( ( inf_in2794916579150040252_ereal @ top_to6662034908053899550_ereal @ A4 )
= A4 ) ).
% inf_top.left_neutral
thf(fact_1139_inf__top_Oleft__neutral,axiom,
! [A4: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ A4 )
= A4 ) ).
% inf_top.left_neutral
thf(fact_1140_inf__top_Oleft__neutral,axiom,
! [A4: extend8495563244428889912nnreal] :
( ( inf_in7439215052339218890nnreal @ top_to1496364449551166952nnreal @ A4 )
= A4 ) ).
% inf_top.left_neutral
thf(fact_1141_inf__top_Oneutr__eq__iff,axiom,
! [A4: set_a,B2: set_a] :
( ( top_top_set_a
= ( inf_inf_set_a @ A4 @ B2 ) )
= ( ( A4 = top_top_set_a )
& ( B2 = top_top_set_a ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_1142_inf__top_Oneutr__eq__iff,axiom,
! [A4: set_c,B2: set_c] :
( ( top_top_set_c
= ( inf_inf_set_c @ A4 @ B2 ) )
= ( ( A4 = top_top_set_c )
& ( B2 = top_top_set_c ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_1143_inf__top_Oneutr__eq__iff,axiom,
! [A4: set_set_a,B2: set_set_a] :
( ( top_top_set_set_a
= ( inf_inf_set_set_a @ A4 @ B2 ) )
= ( ( A4 = top_top_set_set_a )
& ( B2 = top_top_set_set_a ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_1144_inf__top_Oneutr__eq__iff,axiom,
! [A4: extended_ereal,B2: extended_ereal] :
( ( top_to6662034908053899550_ereal
= ( inf_in2794916579150040252_ereal @ A4 @ B2 ) )
= ( ( A4 = top_to6662034908053899550_ereal )
& ( B2 = top_to6662034908053899550_ereal ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_1145_inf__top_Oneutr__eq__iff,axiom,
! [A4: set_nat,B2: set_nat] :
( ( top_top_set_nat
= ( inf_inf_set_nat @ A4 @ B2 ) )
= ( ( A4 = top_top_set_nat )
& ( B2 = top_top_set_nat ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_1146_inf__top_Oneutr__eq__iff,axiom,
! [A4: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( top_to1496364449551166952nnreal
= ( inf_in7439215052339218890nnreal @ A4 @ B2 ) )
= ( ( A4 = top_to1496364449551166952nnreal )
& ( B2 = top_to1496364449551166952nnreal ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_1147_inf__top_Oright__neutral,axiom,
! [A4: set_a] :
( ( inf_inf_set_a @ A4 @ top_top_set_a )
= A4 ) ).
% inf_top.right_neutral
thf(fact_1148_inf__top_Oright__neutral,axiom,
! [A4: set_c] :
( ( inf_inf_set_c @ A4 @ top_top_set_c )
= A4 ) ).
% inf_top.right_neutral
thf(fact_1149_inf__top_Oright__neutral,axiom,
! [A4: set_set_a] :
( ( inf_inf_set_set_a @ A4 @ top_top_set_set_a )
= A4 ) ).
% inf_top.right_neutral
thf(fact_1150_inf__top_Oright__neutral,axiom,
! [A4: extended_ereal] :
( ( inf_in2794916579150040252_ereal @ A4 @ top_to6662034908053899550_ereal )
= A4 ) ).
% inf_top.right_neutral
thf(fact_1151_inf__top_Oright__neutral,axiom,
! [A4: set_nat] :
( ( inf_inf_set_nat @ A4 @ top_top_set_nat )
= A4 ) ).
% inf_top.right_neutral
thf(fact_1152_inf__top_Oright__neutral,axiom,
! [A4: extend8495563244428889912nnreal] :
( ( inf_in7439215052339218890nnreal @ A4 @ top_to1496364449551166952nnreal )
= A4 ) ).
% inf_top.right_neutral
thf(fact_1153_Int__UNIV,axiom,
! [A3: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A3 @ B )
= top_top_set_a )
= ( ( A3 = top_top_set_a )
& ( B = top_top_set_a ) ) ) ).
% Int_UNIV
thf(fact_1154_Int__UNIV,axiom,
! [A3: set_c,B: set_c] :
( ( ( inf_inf_set_c @ A3 @ B )
= top_top_set_c )
= ( ( A3 = top_top_set_c )
& ( B = top_top_set_c ) ) ) ).
% Int_UNIV
thf(fact_1155_Int__UNIV,axiom,
! [A3: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ A3 @ B )
= top_top_set_set_a )
= ( ( A3 = top_top_set_set_a )
& ( B = top_top_set_set_a ) ) ) ).
% Int_UNIV
thf(fact_1156_Int__UNIV,axiom,
! [A3: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A3 @ B )
= top_top_set_nat )
= ( ( A3 = top_top_set_nat )
& ( B = top_top_set_nat ) ) ) ).
% Int_UNIV
thf(fact_1157_finite__Int,axiom,
! [F4: set_a,G2: set_a] :
( ( ( finite_finite_a @ F4 )
| ( finite_finite_a @ G2 ) )
=> ( finite_finite_a @ ( inf_inf_set_a @ F4 @ G2 ) ) ) ).
% finite_Int
thf(fact_1158_finite__Int,axiom,
! [F4: set_c,G2: set_c] :
( ( ( finite_finite_c @ F4 )
| ( finite_finite_c @ G2 ) )
=> ( finite_finite_c @ ( inf_inf_set_c @ F4 @ G2 ) ) ) ).
% finite_Int
thf(fact_1159_finite__Int,axiom,
! [F4: set_nat,G2: set_nat] :
( ( ( finite_finite_nat @ F4 )
| ( finite_finite_nat @ G2 ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F4 @ G2 ) ) ) ).
% finite_Int
thf(fact_1160_finite__Int,axiom,
! [F4: set_set_a,G2: set_set_a] :
( ( ( finite_finite_set_a @ F4 )
| ( finite_finite_set_a @ G2 ) )
=> ( finite_finite_set_a @ ( inf_inf_set_set_a @ F4 @ G2 ) ) ) ).
% finite_Int
thf(fact_1161_vimage__UNIV,axiom,
! [F: a > c] :
( ( vimage_a_c @ F @ top_top_set_c )
= top_top_set_a ) ).
% vimage_UNIV
thf(fact_1162_vimage__UNIV,axiom,
! [F: c > c] :
( ( vimage_c_c @ F @ top_top_set_c )
= top_top_set_c ) ).
% vimage_UNIV
thf(fact_1163_vimage__UNIV,axiom,
! [F: c > a] :
( ( vimage_c_a @ F @ top_top_set_a )
= top_top_set_c ) ).
% vimage_UNIV
thf(fact_1164_vimage__UNIV,axiom,
! [F: a > a] :
( ( vimage_a_a @ F @ top_top_set_a )
= top_top_set_a ) ).
% vimage_UNIV
thf(fact_1165_vimage__UNIV,axiom,
! [F: nat > nat] :
( ( vimage_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat ) ).
% vimage_UNIV
thf(fact_1166_finite__Pow__iff,axiom,
! [A3: set_c] :
( ( finite_finite_set_c @ ( pow_c @ A3 ) )
= ( finite_finite_c @ A3 ) ) ).
% finite_Pow_iff
thf(fact_1167_finite__Pow__iff,axiom,
! [A3: set_a] :
( ( finite_finite_set_a @ ( pow_a @ A3 ) )
= ( finite_finite_a @ A3 ) ) ).
% finite_Pow_iff
thf(fact_1168_finite__Pow__iff,axiom,
! [A3: set_nat] :
( ( finite1152437895449049373et_nat @ ( pow_nat @ A3 ) )
= ( finite_finite_nat @ A3 ) ) ).
% finite_Pow_iff
thf(fact_1169_finite__Collect__not,axiom,
! [P: set_a > $o] :
( ( finite_finite_set_a @ ( collect_set_a @ P ) )
=> ( ( finite_finite_set_a
@ ( collect_set_a
@ ^ [X2: set_a] :
~ ( P @ X2 ) ) )
= ( finite_finite_set_a @ top_top_set_set_a ) ) ) ).
% finite_Collect_not
thf(fact_1170_finite__Collect__not,axiom,
! [P: nat > $o] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
~ ( P @ X2 ) ) )
= ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Collect_not
thf(fact_1171_finite__Collect__subsets,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B3: set_nat] : ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_1172_finite__Collect__subsets,axiom,
! [A3: set_c] :
( ( finite_finite_c @ A3 )
=> ( finite_finite_set_c
@ ( collect_set_c
@ ^ [B3: set_c] : ( ord_less_eq_set_c @ B3 @ A3 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_1173_finite__Collect__subsets,axiom,
! [A3: set_set_a] :
( ( finite_finite_set_a @ A3 )
=> ( finite7209287970140883943_set_a
@ ( collect_set_set_a
@ ^ [B3: set_set_a] : ( ord_le3724670747650509150_set_a @ B3 @ A3 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_1174_finite__Collect__subsets,axiom,
! [A3: set_a] :
( ( finite_finite_a @ A3 )
=> ( finite_finite_set_a
@ ( collect_set_a
@ ^ [B3: set_a] : ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_1175_finite__Collect__bounded__ex,axiom,
! [P: set_a > $o,Q: set_a > set_a > $o] :
( ( finite_finite_set_a @ ( collect_set_a @ P ) )
=> ( ( finite_finite_set_a
@ ( collect_set_a
@ ^ [X2: set_a] :
? [Y3: set_a] :
( ( P @ Y3 )
& ( Q @ X2 @ Y3 ) ) ) )
= ( ! [Y3: set_a] :
( ( P @ Y3 )
=> ( finite_finite_set_a
@ ( collect_set_a
@ ^ [X2: set_a] : ( Q @ X2 @ Y3 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_1176_finite__Collect__bounded__ex,axiom,
! [P: set_a > $o,Q: nat > set_a > $o] :
( ( finite_finite_set_a @ ( collect_set_a @ P ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
? [Y3: set_a] :
( ( P @ Y3 )
& ( Q @ X2 @ Y3 ) ) ) )
= ( ! [Y3: set_a] :
( ( P @ Y3 )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] : ( Q @ X2 @ Y3 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_1177_finite__Collect__bounded__ex,axiom,
! [P: nat > $o,Q: set_a > nat > $o] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite_finite_set_a
@ ( collect_set_a
@ ^ [X2: set_a] :
? [Y3: nat] :
( ( P @ Y3 )
& ( Q @ X2 @ Y3 ) ) ) )
= ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( finite_finite_set_a
@ ( collect_set_a
@ ^ [X2: set_a] : ( Q @ X2 @ Y3 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_1178_finite__Collect__bounded__ex,axiom,
! [P: nat > $o,Q: nat > nat > $o] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
? [Y3: nat] :
( ( P @ Y3 )
& ( Q @ X2 @ Y3 ) ) ) )
= ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] : ( Q @ X2 @ Y3 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_1179_Pow__UNIV,axiom,
( ( pow_c @ top_top_set_c )
= top_top_set_set_c ) ).
% Pow_UNIV
thf(fact_1180_Pow__UNIV,axiom,
( ( pow_a @ top_top_set_a )
= top_top_set_set_a ) ).
% Pow_UNIV
thf(fact_1181_Pow__UNIV,axiom,
( ( pow_nat @ top_top_set_nat )
= top_top_set_set_nat ) ).
% Pow_UNIV
thf(fact_1182_UNIV__def,axiom,
( top_top_set_set_a
= ( collect_set_a
@ ^ [X2: set_a] : $true ) ) ).
% UNIV_def
thf(fact_1183_UNIV__def,axiom,
( top_top_set_nat
= ( collect_nat
@ ^ [X2: nat] : $true ) ) ).
% UNIV_def
thf(fact_1184_finite__range__imageI,axiom,
! [G: a > c,F: c > c] :
( ( finite_finite_c @ ( image_a_c @ G @ top_top_set_a ) )
=> ( finite_finite_c
@ ( image_a_c
@ ^ [X2: a] : ( F @ ( G @ X2 ) )
@ top_top_set_a ) ) ) ).
% finite_range_imageI
thf(fact_1185_finite__range__imageI,axiom,
! [G: a > c,F: c > nat] :
( ( finite_finite_c @ ( image_a_c @ G @ top_top_set_a ) )
=> ( finite_finite_nat
@ ( image_a_nat
@ ^ [X2: a] : ( F @ ( G @ X2 ) )
@ top_top_set_a ) ) ) ).
% finite_range_imageI
thf(fact_1186_finite__range__imageI,axiom,
! [G: a > nat,F: nat > c] :
( ( finite_finite_nat @ ( image_a_nat @ G @ top_top_set_a ) )
=> ( finite_finite_c
@ ( image_a_c
@ ^ [X2: a] : ( F @ ( G @ X2 ) )
@ top_top_set_a ) ) ) ).
% finite_range_imageI
thf(fact_1187_finite__range__imageI,axiom,
! [G: nat > nat,F: nat > nat] :
( ( finite_finite_nat @ ( image_nat_nat @ G @ top_top_set_nat ) )
=> ( finite_finite_nat
@ ( image_nat_nat
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ top_top_set_nat ) ) ) ).
% finite_range_imageI
thf(fact_1188_UNIV__eq__I,axiom,
! [A3: set_set_c] :
( ! [X3: set_c] : ( member_set_c @ X3 @ A3 )
=> ( top_top_set_set_c = A3 ) ) ).
% UNIV_eq_I
thf(fact_1189_UNIV__eq__I,axiom,
! [A3: set_b] :
( ! [X3: b] : ( member_b @ X3 @ A3 )
=> ( top_top_set_b = A3 ) ) ).
% UNIV_eq_I
thf(fact_1190_UNIV__eq__I,axiom,
! [A3: set_o] :
( ! [X3: $o] : ( member_o @ X3 @ A3 )
=> ( top_top_set_o = A3 ) ) ).
% UNIV_eq_I
thf(fact_1191_UNIV__eq__I,axiom,
! [A3: set_nat] :
( ! [X3: nat] : ( member_nat @ X3 @ A3 )
=> ( top_top_set_nat = A3 ) ) ).
% UNIV_eq_I
thf(fact_1192_UNIV__witness,axiom,
? [X3: set_c] : ( member_set_c @ X3 @ top_top_set_set_c ) ).
% UNIV_witness
thf(fact_1193_UNIV__witness,axiom,
? [X3: b] : ( member_b @ X3 @ top_top_set_b ) ).
% UNIV_witness
thf(fact_1194_UNIV__witness,axiom,
? [X3: $o] : ( member_o @ X3 @ top_top_set_o ) ).
% UNIV_witness
thf(fact_1195_UNIV__witness,axiom,
? [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_1196_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_1197_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_1198_ex__new__if__finite,axiom,
! [A3: set_set_c] :
( ~ ( finite_finite_set_c @ top_top_set_set_c )
=> ( ( finite_finite_set_c @ A3 )
=> ? [A5: set_c] :
~ ( member_set_c @ A5 @ A3 ) ) ) ).
% ex_new_if_finite
thf(fact_1199_ex__new__if__finite,axiom,
! [A3: set_b] :
( ~ ( finite_finite_b @ top_top_set_b )
=> ( ( finite_finite_b @ A3 )
=> ? [A5: b] :
~ ( member_b @ A5 @ A3 ) ) ) ).
% ex_new_if_finite
thf(fact_1200_ex__new__if__finite,axiom,
! [A3: set_o] :
( ~ ( finite_finite_o @ top_top_set_o )
=> ( ( finite_finite_o @ A3 )
=> ? [A5: $o] :
~ ( member_o @ A5 @ A3 ) ) ) ).
% ex_new_if_finite
thf(fact_1201_ex__new__if__finite,axiom,
! [A3: set_nat] :
( ~ ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_nat @ A3 )
=> ? [A5: nat] :
~ ( member_nat @ A5 @ A3 ) ) ) ).
% ex_new_if_finite
thf(fact_1202_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_1203_not__finite__existsD,axiom,
! [P: set_a > $o] :
( ~ ( finite_finite_set_a @ ( collect_set_a @ P ) )
=> ? [X_1: set_a] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_1204_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_1: nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_1205_infinite__UNIV__char__0,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_char_0
thf(fact_1206_pigeonhole__infinite__rel,axiom,
! [A3: set_nat,B: set_nat,R: nat > nat > $o] :
( ~ ( finite_finite_nat @ A3 )
=> ( ( finite_finite_nat @ B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A2: nat] :
( ( member_nat @ A2 @ A3 )
& ( R @ A2 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_1207_infinite__UNIV__nat,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_nat
thf(fact_1208_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_1209_Inf__bool__def,axiom,
( complete_Inf_Inf_o
= ( ^ [A: set_o] :
~ ( member_o @ $false @ A ) ) ) ).
% Inf_bool_def
thf(fact_1210_infinite__nat__iff__unbounded__le,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M3: nat] :
? [N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
& ( member_nat @ N2 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_1211_nat__not__finite,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% nat_not_finite
thf(fact_1212_finite__less__ub,axiom,
! [F: nat > nat,U2: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( F @ N3 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ U2 ) ) ) ) ).
% finite_less_ub
thf(fact_1213_bot__nat__0_Oextremum,axiom,
! [A4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A4 ) ).
% bot_nat_0.extremum
thf(fact_1214_le0,axiom,
! [N4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N4 ) ).
% le0
thf(fact_1215_le__0__eq,axiom,
! [N4: nat] :
( ( ord_less_eq_nat @ N4 @ zero_zero_nat )
= ( N4 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1216_bot__nat__0_Oextremum__uniqueI,axiom,
! [A4: nat] :
( ( ord_less_eq_nat @ A4 @ zero_zero_nat )
=> ( A4 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1217_bot__nat__0_Oextremum__unique,axiom,
! [A4: nat] :
( ( ord_less_eq_nat @ A4 @ zero_zero_nat )
= ( A4 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1218_less__eq__nat_Osimps_I1_J,axiom,
! [N4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N4 ) ).
% less_eq_nat.simps(1)
thf(fact_1219_le__refl,axiom,
! [N4: nat] : ( ord_less_eq_nat @ N4 @ N4 ) ).
% le_refl
thf(fact_1220_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_1221_eq__imp__le,axiom,
! [M4: nat,N4: nat] :
( ( M4 = N4 )
=> ( ord_less_eq_nat @ M4 @ N4 ) ) ).
% eq_imp_le
thf(fact_1222_le__antisym,axiom,
! [M4: nat,N4: nat] :
( ( ord_less_eq_nat @ M4 @ N4 )
=> ( ( ord_less_eq_nat @ N4 @ M4 )
=> ( M4 = N4 ) ) ) ).
% le_antisym
thf(fact_1223_nat__le__linear,axiom,
! [M4: nat,N4: nat] :
( ( ord_less_eq_nat @ M4 @ N4 )
| ( ord_less_eq_nat @ N4 @ M4 ) ) ).
% nat_le_linear
thf(fact_1224_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ B2 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1225_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M ) )
=> ~ ! [M5: nat] :
( ( P @ M5 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M5 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1226_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M3: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N5 )
=> ( ord_less_eq_nat @ X2 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1227_Sup__bool__def,axiom,
( complete_Sup_Sup_o
= ( member_o @ $true ) ) ).
% Sup_bool_def
thf(fact_1228_ereal__complete__Inf,axiom,
! [S: set_Extended_ereal] :
? [X3: extended_ereal] :
( ! [Xa: extended_ereal] :
( ( member2350847679896131959_ereal @ Xa @ S )
=> ( ord_le1083603963089353582_ereal @ X3 @ Xa ) )
& ! [Z5: extended_ereal] :
( ! [Xa2: extended_ereal] :
( ( member2350847679896131959_ereal @ Xa2 @ S )
=> ( ord_le1083603963089353582_ereal @ Z5 @ Xa2 ) )
=> ( ord_le1083603963089353582_ereal @ Z5 @ X3 ) ) ) ).
% ereal_complete_Inf
thf(fact_1229_ereal__complete__Sup,axiom,
! [S: set_Extended_ereal] :
? [X3: extended_ereal] :
( ! [Xa: extended_ereal] :
( ( member2350847679896131959_ereal @ Xa @ S )
=> ( ord_le1083603963089353582_ereal @ Xa @ X3 ) )
& ! [Z5: extended_ereal] :
( ! [Xa2: extended_ereal] :
( ( member2350847679896131959_ereal @ Xa2 @ S )
=> ( ord_le1083603963089353582_ereal @ Xa2 @ Z5 ) )
=> ( ord_le1083603963089353582_ereal @ X3 @ Z5 ) ) ) ).
% ereal_complete_Sup
thf(fact_1230_Suc__le__mono,axiom,
! [N4: nat,M4: nat] :
( ( ord_less_eq_nat @ ( suc @ N4 ) @ ( suc @ M4 ) )
= ( ord_less_eq_nat @ N4 @ M4 ) ) ).
% Suc_le_mono
thf(fact_1231_Suc__leD,axiom,
! [M4: nat,N4: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N4 )
=> ( ord_less_eq_nat @ M4 @ N4 ) ) ).
% Suc_leD
thf(fact_1232_le__SucE,axiom,
! [M4: nat,N4: nat] :
( ( ord_less_eq_nat @ M4 @ ( suc @ N4 ) )
=> ( ~ ( ord_less_eq_nat @ M4 @ N4 )
=> ( M4
= ( suc @ N4 ) ) ) ) ).
% le_SucE
thf(fact_1233_le__SucI,axiom,
! [M4: nat,N4: nat] :
( ( ord_less_eq_nat @ M4 @ N4 )
=> ( ord_less_eq_nat @ M4 @ ( suc @ N4 ) ) ) ).
% le_SucI
thf(fact_1234_Suc__le__D,axiom,
! [N4: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N4 ) @ M6 )
=> ? [M5: nat] :
( M6
= ( suc @ M5 ) ) ) ).
% Suc_le_D
thf(fact_1235_le__Suc__eq,axiom,
! [M4: nat,N4: nat] :
( ( ord_less_eq_nat @ M4 @ ( suc @ N4 ) )
= ( ( ord_less_eq_nat @ M4 @ N4 )
| ( M4
= ( suc @ N4 ) ) ) ) ).
% le_Suc_eq
thf(fact_1236_Suc__n__not__le__n,axiom,
! [N4: nat] :
~ ( ord_less_eq_nat @ ( suc @ N4 ) @ N4 ) ).
% Suc_n_not_le_n
thf(fact_1237_not__less__eq__eq,axiom,
! [M4: nat,N4: nat] :
( ( ~ ( ord_less_eq_nat @ M4 @ N4 ) )
= ( ord_less_eq_nat @ ( suc @ N4 ) @ M4 ) ) ).
% not_less_eq_eq
thf(fact_1238_full__nat__induct,axiom,
! [P: nat > $o,N4: nat] :
( ! [N3: nat] :
( ! [M7: nat] :
( ( ord_less_eq_nat @ ( suc @ M7 ) @ N3 )
=> ( P @ M7 ) )
=> ( P @ N3 ) )
=> ( P @ N4 ) ) ).
% full_nat_induct
thf(fact_1239_nat__induct__at__least,axiom,
! [M4: nat,N4: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M4 @ N4 )
=> ( ( P @ M4 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M4 @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N4 ) ) ) ) ).
% nat_induct_at_least
thf(fact_1240_transitive__stepwise__le,axiom,
! [M4: nat,N4: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M4 @ N4 )
=> ( ! [X3: nat] : ( R @ X3 @ X3 )
=> ( ! [X3: nat,Y5: nat,Z3: nat] :
( ( R @ X3 @ Y5 )
=> ( ( R @ Y5 @ Z3 )
=> ( R @ X3 @ Z3 ) ) )
=> ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
=> ( R @ M4 @ N4 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1241_diff__diff__cancel,axiom,
! [I2: nat,N4: nat] :
( ( ord_less_eq_nat @ I2 @ N4 )
=> ( ( minus_minus_nat @ N4 @ ( minus_minus_nat @ N4 @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_1242_diff__is__0__eq_H,axiom,
! [M4: nat,N4: nat] :
( ( ord_less_eq_nat @ M4 @ N4 )
=> ( ( minus_minus_nat @ M4 @ N4 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1243_diff__is__0__eq,axiom,
! [M4: nat,N4: nat] :
( ( ( minus_minus_nat @ M4 @ N4 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M4 @ N4 ) ) ).
% diff_is_0_eq
thf(fact_1244_Suc__diff__le,axiom,
! [N4: nat,M4: nat] :
( ( ord_less_eq_nat @ N4 @ M4 )
=> ( ( minus_minus_nat @ ( suc @ M4 ) @ N4 )
= ( suc @ ( minus_minus_nat @ M4 @ N4 ) ) ) ) ).
% Suc_diff_le
thf(fact_1245_inj__on__diff__nat,axiom,
! [N: set_nat,K: nat] :
( ! [N3: nat] :
( ( member_nat @ N3 @ N )
=> ( ord_less_eq_nat @ K @ N3 ) )
=> ( inj_on_nat_nat
@ ^ [N2: nat] : ( minus_minus_nat @ N2 @ K )
@ N ) ) ).
% inj_on_diff_nat
thf(fact_1246_diff__le__mono2,axiom,
! [M4: nat,N4: nat,L: nat] :
( ( ord_less_eq_nat @ M4 @ N4 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N4 ) @ ( minus_minus_nat @ L @ M4 ) ) ) ).
% diff_le_mono2
thf(fact_1247_le__diff__iff_H,axiom,
! [A4: nat,C2: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ C2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A4 ) @ ( minus_minus_nat @ C2 @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A4 ) ) ) ) ).
% le_diff_iff'
thf(fact_1248_diff__le__self,axiom,
! [M4: nat,N4: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M4 @ N4 ) @ M4 ) ).
% diff_le_self
thf(fact_1249_diff__le__mono,axiom,
! [M4: nat,N4: nat,L: nat] :
( ( ord_less_eq_nat @ M4 @ N4 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M4 @ L ) @ ( minus_minus_nat @ N4 @ L ) ) ) ).
% diff_le_mono
thf(fact_1250_Nat_Odiff__diff__eq,axiom,
! [K: nat,M4: nat,N4: nat] :
( ( ord_less_eq_nat @ K @ M4 )
=> ( ( ord_less_eq_nat @ K @ N4 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M4 @ K ) @ ( minus_minus_nat @ N4 @ K ) )
= ( minus_minus_nat @ M4 @ N4 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1251_le__diff__iff,axiom,
! [K: nat,M4: nat,N4: nat] :
( ( ord_less_eq_nat @ K @ M4 )
=> ( ( ord_less_eq_nat @ K @ N4 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M4 @ K ) @ ( minus_minus_nat @ N4 @ K ) )
= ( ord_less_eq_nat @ M4 @ N4 ) ) ) ) ).
% le_diff_iff
thf(fact_1252_eq__diff__iff,axiom,
! [K: nat,M4: nat,N4: nat] :
( ( ord_less_eq_nat @ K @ M4 )
=> ( ( ord_less_eq_nat @ K @ N4 )
=> ( ( ( minus_minus_nat @ M4 @ K )
= ( minus_minus_nat @ N4 @ K ) )
= ( M4 = N4 ) ) ) ) ).
% eq_diff_iff
thf(fact_1253_ereal__minus__mono,axiom,
! [A3: extended_ereal,B: extended_ereal,D2: extended_ereal,C: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A3 @ B )
=> ( ( ord_le1083603963089353582_ereal @ D2 @ C )
=> ( ord_le1083603963089353582_ereal @ ( minus_2816186181549245109_ereal @ A3 @ C ) @ ( minus_2816186181549245109_ereal @ B @ D2 ) ) ) ) ).
% ereal_minus_mono
thf(fact_1254_ereal__diff__le__mono__left,axiom,
! [X: extended_ereal,Z: extended_ereal,Y: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X @ Z )
=> ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ Y )
=> ( ord_le1083603963089353582_ereal @ ( minus_2816186181549245109_ereal @ X @ Y ) @ Z ) ) ) ).
% ereal_diff_le_mono_left
thf(fact_1255_ereal__diff__positive,axiom,
! [A4: extended_ereal,B2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A4 @ B2 )
=> ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( minus_2816186181549245109_ereal @ B2 @ A4 ) ) ) ).
% ereal_diff_positive
thf(fact_1256_ereal__diff__le__self,axiom,
! [Y: extended_ereal,X: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ Y )
=> ( ord_le1083603963089353582_ereal @ ( minus_2816186181549245109_ereal @ X @ Y ) @ X ) ) ).
% ereal_diff_le_self
thf(fact_1257_finite__vimage__Suc__iff,axiom,
! [F4: set_nat] :
( ( finite_finite_nat @ ( vimage_nat_nat @ suc @ F4 ) )
= ( finite_finite_nat @ F4 ) ) ).
% finite_vimage_Suc_iff
thf(fact_1258_ennreal__diff__self,axiom,
! [A4: extend8495563244428889912nnreal] :
( ( A4 != top_to1496364449551166952nnreal )
=> ( ( minus_8429688780609304081nnreal @ A4 @ A4 )
= zero_z7100319975126383169nnreal ) ) ).
% ennreal_diff_self
thf(fact_1259_ennreal__top__minus,axiom,
! [X: extend8495563244428889912nnreal] :
( ( minus_8429688780609304081nnreal @ top_to1496364449551166952nnreal @ X )
= top_to1496364449551166952nnreal ) ).
% ennreal_top_minus
thf(fact_1260_ennreal__minus__eq__top,axiom,
! [A4: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( ( minus_8429688780609304081nnreal @ A4 @ B2 )
= top_to1496364449551166952nnreal )
= ( A4 = top_to1496364449551166952nnreal ) ) ).
% ennreal_minus_eq_top
thf(fact_1261_neq__top__trans,axiom,
! [Y: extend8495563244428889912nnreal,X: extend8495563244428889912nnreal] :
( ( Y != top_to1496364449551166952nnreal )
=> ( ( ord_le3935885782089961368nnreal @ X @ Y )
=> ( X != top_to1496364449551166952nnreal ) ) ) ).
% neq_top_trans
thf(fact_1262_ennreal__diff__le__mono__left,axiom,
! [A4: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A4 @ B2 )
=> ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A4 @ C2 ) @ B2 ) ) ).
% ennreal_diff_le_mono_left
thf(fact_1263_diff__le__self__ennreal,axiom,
! [A4: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A4 @ B2 ) @ A4 ) ).
% diff_le_self_ennreal
thf(fact_1264_ennreal__mono__minus,axiom,
! [C2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,A4: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ C2 @ B2 )
=> ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A4 @ B2 ) @ ( minus_8429688780609304081nnreal @ A4 @ C2 ) ) ) ).
% ennreal_mono_minus
thf(fact_1265_ennreal__minus__mono,axiom,
! [A4: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal,D: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A4 @ C2 )
=> ( ( ord_le3935885782089961368nnreal @ D @ B2 )
=> ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A4 @ B2 ) @ ( minus_8429688780609304081nnreal @ C2 @ D ) ) ) ) ).
% ennreal_minus_mono
thf(fact_1266_ennreal__minus__cancel,axiom,
! [C2: extend8495563244428889912nnreal,A4: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( C2 != top_to1496364449551166952nnreal )
=> ( ( ord_le3935885782089961368nnreal @ A4 @ C2 )
=> ( ( ord_le3935885782089961368nnreal @ B2 @ C2 )
=> ( ( ( minus_8429688780609304081nnreal @ C2 @ A4 )
= ( minus_8429688780609304081nnreal @ C2 @ B2 ) )
=> ( A4 = B2 ) ) ) ) ) ).
% ennreal_minus_cancel
thf(fact_1267_ennreal__minus__cancel__iff,axiom,
! [A4: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ( minus_8429688780609304081nnreal @ A4 @ B2 )
= ( minus_8429688780609304081nnreal @ A4 @ C2 ) )
= ( ( B2 = C2 )
| ( ( ord_le3935885782089961368nnreal @ A4 @ B2 )
& ( ord_le3935885782089961368nnreal @ A4 @ C2 ) )
| ( A4 = top_to1496364449551166952nnreal ) ) ) ).
% ennreal_minus_cancel_iff
thf(fact_1268_ennreal__zero__neq__top,axiom,
zero_z7100319975126383169nnreal != top_to1496364449551166952nnreal ).
% ennreal_zero_neq_top
thf(fact_1269_ennreal__minus__eq__0,axiom,
! [A4: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( ( minus_8429688780609304081nnreal @ A4 @ B2 )
= zero_z7100319975126383169nnreal )
=> ( ord_le3935885782089961368nnreal @ A4 @ B2 ) ) ).
% ennreal_minus_eq_0
thf(fact_1270_minus__top__ennreal,axiom,
! [X: extend8495563244428889912nnreal] :
( ( ( X = top_to1496364449551166952nnreal )
=> ( ( minus_8429688780609304081nnreal @ X @ top_to1496364449551166952nnreal )
= top_to1496364449551166952nnreal ) )
& ( ( X != top_to1496364449551166952nnreal )
=> ( ( minus_8429688780609304081nnreal @ X @ top_to1496364449551166952nnreal )
= zero_z7100319975126383169nnreal ) ) ) ).
% minus_top_ennreal
% Conjectures (1)
thf(conj_0,conjecture,
( ( collect_set_a
@ ^ [Uu: set_a] :
? [A: set_c] :
( ( Uu
= ( inf_inf_set_a @ ( vimage_a_c @ ( x @ i2 ) @ A ) @ ( probab49036049091589825_pmf_a @ p ) ) )
& ( ord_less_eq_set_c @ A @ ( image_a_c @ ( x @ i2 ) @ ( probab49036049091589825_pmf_a @ p ) ) ) ) )
= ( collect_set_a
@ ^ [Uu: set_a] :
? [A: set_c] :
( ( Uu
= ( inf_inf_set_a @ ( vimage_a_c @ ( x @ i2 ) @ A ) @ ( probab49036049091589825_pmf_a @ p ) ) )
& ( member_set_c @ A @ ( sigma_sigma_sets_c @ ( image_a_c @ ( x @ i2 ) @ ( probab49036049091589825_pmf_a @ p ) ) @ ( g @ i2 ) ) ) ) ) ) ).
%------------------------------------------------------------------------------