TPTP Problem File: SLH0608^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Safe_Range_RC/0021_Relational_Calculus/prob_01645_062133__17500178_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1468 ( 401 unt; 187 typ; 0 def)
% Number of atoms : 4366 (1051 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 13226 ( 355 ~; 20 |; 302 &;10424 @)
% ( 0 <=>;2125 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 8 avg)
% Number of types : 16 ( 15 usr)
% Number of type conns : 1270 (1270 >; 0 *; 0 +; 0 <<)
% Number of symbols : 175 ( 172 usr; 14 con; 0-4 aty)
% Number of variables : 4269 ( 371 ^;3728 !; 170 ?;4269 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:26:21.400
%------------------------------------------------------------------------------
% Could-be-implicit typings (15)
thf(ty_n_t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
set_Re381260168593705685la_a_b: $tType ).
thf(ty_n_t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
relational_fmla_a_b: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J,type,
product_prod_b_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_Eo_J_J,type,
set_nat_o: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
set_list_a: $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_It__Set__Oset_I_Eo_J_J,type,
set_set_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_M_Eo_J_J,type,
set_a_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_Eo_M_Eo_J_J,type,
set_o_o: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $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__a,type,
a: $tType ).
% Explicit typings (172)
thf(sy_c_Complete__Lattices_OInf__class_OInf_001_062_I_Eo_M_Eo_J,type,
complete_Inf_Inf_o_o: set_o_o > $o > $o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001_062_It__Nat__Onat_M_Eo_J,type,
comple6214475593288795910_nat_o: set_nat_o > nat > $o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001_062_Itf__a_M_Eo_J,type,
complete_Inf_Inf_a_o: set_a_o > a > $o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001_Eo,type,
complete_Inf_Inf_o: set_o > $o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Nat__Onat,type,
complete_Inf_Inf_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_I_Eo_J,type,
comple3063163877087187839_set_o: set_set_o > set_o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Nat__Onat_J,type,
comple7806235888213564991et_nat: set_set_nat > set_nat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_Itf__a_J,type,
comple6135023378680113637_set_a: set_set_a > set_a ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_I_Eo_M_Eo_J,type,
complete_Sup_Sup_o_o: set_o_o > $o > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Nat__Onat_M_Eo_J,type,
comple8317665133742190828_nat_o: set_nat_o > nat > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_Itf__a_M_Eo_J,type,
complete_Sup_Sup_a_o: set_a_o > a > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_Eo,type,
complete_Sup_Sup_o: set_o > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
complete_Sup_Sup_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_I_Eo_J,type,
comple90263536869209701_set_o: set_set_o > set_o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
comple7399068483239264473et_nat: set_set_nat > set_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_Itf__a_J,type,
comple2307003609928055243_set_a: set_set_a > set_a ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001_Eo,type,
condit5488710616941104124bove_o: set_o > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Nat__Onat,type,
condit2214826472909112428ve_nat: set_nat > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Set__Oset_It__Nat__Onat_J,type,
condit5477540289124974626et_nat: set_set_nat > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001tf__a,type,
condit5209368051240477026bove_a: set_a > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below_001_Eo,type,
condit5413489452508810728elow_o: set_o > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below_001t__Nat__Onat,type,
condit1738341127787009408ow_nat: set_nat > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below_001t__Set__Oset_It__Nat__Onat_J,type,
condit68592940725977398et_nat: set_set_nat > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below_001tf__a,type,
condit5901475214736682318elow_a: set_a > $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_001tf__a,type,
finite_Fpow_a: set_a > set_set_a ).
thf(sy_c_Finite__Set_Ocard_001_Eo,type,
finite_card_o: set_o > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001tf__a,type,
finite_card_a: set_a > nat ).
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__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
finite5600759454172676150la_a_b: set_Re381260168593705685la_a_b > $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_Itf__a_J,type,
finite_finite_set_a: set_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001t__Nat__Onat,type,
fun_upd_nat_nat: ( nat > nat ) > nat > nat > nat > nat ).
thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001tf__a,type,
fun_upd_nat_a: ( nat > a ) > nat > a > nat > a ).
thf(sy_c_Fun_Ofun__upd_001tf__a_001t__Nat__Onat,type,
fun_upd_a_nat: ( a > nat ) > a > nat > a > nat ).
thf(sy_c_Fun_Ofun__upd_001tf__a_001tf__a,type,
fun_upd_a_a: ( a > a ) > a > a > a > a ).
thf(sy_c_Fun_Oinj__on_001_Eo_001_Eo,type,
inj_on_o_o: ( $o > $o ) > set_o > $o ).
thf(sy_c_Fun_Oinj__on_001_Eo_001t__Nat__Onat,type,
inj_on_o_nat: ( $o > nat ) > set_o > $o ).
thf(sy_c_Fun_Oinj__on_001_Eo_001tf__a,type,
inj_on_o_a: ( $o > a ) > set_o > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001_Eo,type,
inj_on_nat_o: ( nat > $o ) > set_nat > $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_Fun_Oinj__on_001t__Nat__Onat_001tf__a,type,
inj_on_nat_a: ( nat > a ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
inj_on4604407203859583615et_nat: ( set_nat > set_nat ) > set_set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_Itf__a_J,type,
inj_on_set_nat_set_a: ( set_nat > set_a ) > set_set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_Itf__a_J_001t__Set__Oset_It__Nat__Onat_J,type,
inj_on_set_a_set_nat: ( set_a > set_nat ) > set_set_a > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
inj_on_set_a_set_a: ( set_a > set_a ) > set_set_a > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001_Eo,type,
inj_on_a_o: ( a > $o ) > set_a > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001t__Nat__Onat,type,
inj_on_a_nat: ( a > nat ) > set_a > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001tf__a,type,
inj_on_a_a: ( a > a ) > set_a > $o ).
thf(sy_c_Fun_Othe__inv__into_001_Eo_001_Eo,type,
the_inv_into_o_o: set_o > ( $o > $o ) > $o > $o ).
thf(sy_c_Fun_Othe__inv__into_001_Eo_001t__Nat__Onat,type,
the_inv_into_o_nat: set_o > ( $o > nat ) > nat > $o ).
thf(sy_c_Fun_Othe__inv__into_001_Eo_001tf__a,type,
the_inv_into_o_a: set_o > ( $o > a ) > a > $o ).
thf(sy_c_Fun_Othe__inv__into_001t__Nat__Onat_001_Eo,type,
the_inv_into_nat_o: set_nat > ( nat > $o ) > $o > nat ).
thf(sy_c_Fun_Othe__inv__into_001t__Nat__Onat_001t__Nat__Onat,type,
the_inv_into_nat_nat: set_nat > ( nat > nat ) > nat > nat ).
thf(sy_c_Fun_Othe__inv__into_001t__Nat__Onat_001tf__a,type,
the_inv_into_nat_a: set_nat > ( nat > a ) > a > nat ).
thf(sy_c_Fun_Othe__inv__into_001tf__a_001_Eo,type,
the_inv_into_a_o: set_a > ( a > $o ) > $o > a ).
thf(sy_c_Fun_Othe__inv__into_001tf__a_001t__Nat__Onat,type,
the_inv_into_a_nat: set_a > ( a > nat ) > nat > a ).
thf(sy_c_Fun_Othe__inv__into_001tf__a_001tf__a,type,
the_inv_into_a_a: set_a > ( a > a ) > a > a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_I_Eo_M_Eo_J,type,
uminus_uminus_o_o: ( $o > $o ) > $o > $o ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Nat__Onat_M_Eo_J,type,
uminus_uminus_nat_o: ( nat > $o ) > nat > $o ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_Itf__a_M_Eo_J,type,
uminus_uminus_a_o: ( a > $o ) > a > $o ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001_Eo,type,
uminus_uminus_o: $o > $o ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_I_Eo_J,type,
uminus_uminus_set_o: set_o > set_o ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Nat__Onat_J,type,
uminus5710092332889474511et_nat: set_nat > set_nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_Itf__a_J,type,
uminus_uminus_set_a: set_a > set_a ).
thf(sy_c_If_001tf__a,type,
if_a: $o > a > a > a ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_I_Eo_M_Eo_J,type,
sup_sup_o_o: ( $o > $o ) > ( $o > $o ) > $o > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Nat__Onat_M_Eo_J,type,
sup_sup_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_Itf__a_M_Eo_J,type,
sup_sup_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_Eo,type,
sup_sup_o: $o > $o > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_Eo_J,type,
sup_sup_set_o: set_o > set_o > set_o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
sup_sup_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
sup_sup_set_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_I_Eo_M_Eo_J,type,
bot_bot_o_o: $o > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
bot_bot_a_o: a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_Eo,type,
bot_bot_o: $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_Eo_J,type,
bot_bot_set_o: set_o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
bot_bo4495933725496725865la_a_b: set_Re381260168593705685la_a_b ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bot_bot_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
bot_bot_set_set_a: set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
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_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_Eo,type,
ord_less_eq_o: $o > $o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_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_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_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001tf__a,type,
ord_less_eq_a: a > a > $o ).
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_001tf__a,type,
powp_a: ( a > $o ) > set_a > $o ).
thf(sy_c_Relational__Calculus_Oadom_001tf__b_001tf__a,type,
relational_adom_b_a: ( product_prod_b_nat > set_list_a ) > set_a ).
thf(sy_c_Relational__Calculus_Oap_001tf__a_001tf__b,type,
relational_ap_a_b: relational_fmla_a_b > $o ).
thf(sy_c_Relational__Calculus_Ocov_001tf__a_001tf__b,type,
relational_cov_a_b: nat > relational_fmla_a_b > set_Re381260168593705685la_a_b > $o ).
thf(sy_c_Relational__Calculus_Ocsts_001tf__a_001tf__b,type,
relational_csts_a_b: relational_fmla_a_b > set_a ).
thf(sy_c_Relational__Calculus_Oequiv_001tf__a_001tf__b,type,
relational_equiv_a_b: relational_fmla_a_b > relational_fmla_a_b > $o ).
thf(sy_c_Relational__Calculus_Oerase_001tf__a_001tf__b,type,
relational_erase_a_b: relational_fmla_a_b > nat > relational_fmla_a_b ).
thf(sy_c_Relational__Calculus_Oeval_001tf__a_001tf__b,type,
relational_eval_a_b: relational_fmla_a_b > ( product_prod_b_nat > set_list_a ) > set_list_a ).
thf(sy_c_Relational__Calculus_Oeval__on_001tf__a_001tf__b,type,
relati8814510239606734169on_a_b: set_nat > relational_fmla_a_b > ( product_prod_b_nat > set_list_a ) > set_list_a ).
thf(sy_c_Relational__Calculus_Ofresh__val_001tf__a_001tf__b,type,
relati2318939533276802993al_a_b: relational_fmla_a_b > ( product_prod_b_nat > set_list_a ) > set_a > a ).
thf(sy_c_Relational__Calculus_Ofv_001tf__a_001tf__b,type,
relational_fv_a_b: relational_fmla_a_b > set_nat ).
thf(sy_c_Relational__Calculus_Oqp_001tf__a_001tf__b,type,
relational_qp_a_b: relational_fmla_a_b > $o ).
thf(sy_c_Relational__Calculus_Osat_001tf__a_001tf__b,type,
relational_sat_a_b: relational_fmla_a_b > ( product_prod_b_nat > set_list_a ) > ( nat > a ) > $o ).
thf(sy_c_Set_OCollect_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_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
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_001tf__a,type,
pow_a: set_a > set_set_a ).
thf(sy_c_Set_Oimage_001_062_I_Eo_M_Eo_J_001t__Set__Oset_I_Eo_J,type,
image_o_o_set_o: ( ( $o > $o ) > set_o ) > set_o_o > set_set_o ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_M_Eo_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_o_set_nat: ( ( nat > $o ) > set_nat ) > set_nat_o > set_set_nat ).
thf(sy_c_Set_Oimage_001_062_Itf__a_M_Eo_J_001t__Set__Oset_Itf__a_J,type,
image_a_o_set_a: ( ( a > $o ) > set_a ) > set_a_o > set_set_a ).
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_I_Eo_J,type,
image_o_set_o: ( $o > set_o ) > set_o > set_set_o ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_It__Nat__Onat_J,type,
image_o_set_nat: ( $o > set_nat ) > set_o > set_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__a,type,
image_o_a: ( $o > a ) > set_o > set_a ).
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_I_Eo_J,type,
image_nat_set_o: ( nat > set_o ) > set_nat > set_set_o ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_Itf__a_J,type,
image_nat_set_a: ( nat > set_a ) > set_nat > set_set_a ).
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__Set__Oset_I_Eo_J_001_062_I_Eo_M_Eo_J,type,
image_set_o_o_o: ( set_o > $o > $o ) > set_set_o > set_o_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_Eo_J_001_Eo,type,
image_set_o_o: ( set_o > $o ) > set_set_o > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001_062_It__Nat__Onat_M_Eo_J,type,
image_set_nat_nat_o: ( set_nat > nat > $o ) > set_set_nat > set_nat_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001_Eo,type,
image_set_nat_o: ( set_nat > $o ) > set_set_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_Itf__a_J,type,
image_set_nat_set_a: ( set_nat > set_a ) > set_set_nat > set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001_062_Itf__a_M_Eo_J,type,
image_set_a_a_o: ( set_a > a > $o ) > set_set_a > set_a_o ).
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__Set__Oset_It__Nat__Onat_J,type,
image_set_a_set_nat: ( set_a > set_nat ) > set_set_a > set_set_nat ).
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_001tf__a_001_Eo,type,
image_a_o: ( a > $o ) > set_a > set_o ).
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_I_Eo_J,type,
image_a_set_o: ( a > set_o ) > set_a > set_set_o ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_It__Nat__Onat_J,type,
image_a_set_nat: ( a > set_nat ) > set_a > set_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_fChoice_001_Eo,type,
fChoice_o: ( $o > $o ) > $o ).
thf(sy_c_fChoice_001t__Nat__Onat,type,
fChoice_nat: ( nat > $o ) > nat ).
thf(sy_c_fChoice_001tf__a,type,
fChoice_a: ( a > $o ) > a ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
member4680049679412964150la_a_b: relational_fmla_a_b > set_Re381260168593705685la_a_b > $o ).
thf(sy_c_member_001t__Set__Oset_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_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_G,type,
g: set_Re381260168593705685la_a_b ).
thf(sy_v_I,type,
i: product_prod_b_nat > set_list_a ).
thf(sy_v_Q,type,
q: relational_fmla_a_b ).
thf(sy_v_Q3____,type,
q3: relational_fmla_a_b ).
thf(sy_v__092_060sigma_062,type,
sigma: nat > a ).
thf(sy_v_x,type,
x: nat ).
% Relevant facts (1274)
thf(fact_0_assms_I3_J,axiom,
finite_finite_a @ ( relational_adom_b_a @ i ) ).
% assms(3)
thf(fact_1_assms_I2_J,axiom,
member_nat @ x @ ( relational_fv_a_b @ q ) ).
% assms(2)
thf(fact_2_finite__csts,axiom,
! [T: relational_fmla_a_b] : ( finite_finite_a @ ( relational_csts_a_b @ T ) ) ).
% finite_csts
thf(fact_3_ex__fresh__val,axiom,
! [I: product_prod_b_nat > set_list_a,A: set_a,Q: relational_fmla_a_b] :
( ( finite_finite_a @ ( relational_adom_b_a @ I ) )
=> ( ( finite_finite_a @ A )
=> ? [X: a] :
( ~ ( member_a @ X @ ( relational_adom_b_a @ I ) )
& ~ ( member_a @ X @ ( relational_csts_a_b @ Q ) )
& ~ ( member_a @ X @ A ) ) ) ) ).
% ex_fresh_val
thf(fact_4_finite__Un,axiom,
! [F: set_nat,G: set_nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ F @ G ) )
= ( ( finite_finite_nat @ F )
& ( finite_finite_nat @ G ) ) ) ).
% finite_Un
thf(fact_5_finite__Un,axiom,
! [F: set_a,G: set_a] :
( ( finite_finite_a @ ( sup_sup_set_a @ F @ G ) )
= ( ( finite_finite_a @ F )
& ( finite_finite_a @ G ) ) ) ).
% finite_Un
thf(fact_6_finite__imageI,axiom,
! [F: set_a,H: a > a] :
( ( finite_finite_a @ F )
=> ( finite_finite_a @ ( image_a_a @ H @ F ) ) ) ).
% finite_imageI
thf(fact_7_finite__imageI,axiom,
! [F: set_a,H: a > nat] :
( ( finite_finite_a @ F )
=> ( finite_finite_nat @ ( image_a_nat @ H @ F ) ) ) ).
% finite_imageI
thf(fact_8_finite__imageI,axiom,
! [F: set_nat,H: nat > a] :
( ( finite_finite_nat @ F )
=> ( finite_finite_a @ ( image_nat_a @ H @ F ) ) ) ).
% finite_imageI
thf(fact_9_finite__imageI,axiom,
! [F: set_nat,H: nat > nat] :
( ( finite_finite_nat @ F )
=> ( finite_finite_nat @ ( image_nat_nat @ H @ F ) ) ) ).
% finite_imageI
thf(fact_10_finite__Collect__conjI,axiom,
! [P: a > $o,Q: a > $o] :
( ( ( finite_finite_a @ ( collect_a @ P ) )
| ( finite_finite_a @ ( collect_a @ Q ) ) )
=> ( finite_finite_a
@ ( collect_a
@ ^ [X2: a] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_11_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_12_finite__Collect__disjI,axiom,
! [P: a > $o,Q: a > $o] :
( ( finite_finite_a
@ ( collect_a
@ ^ [X2: a] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite_finite_a @ ( collect_a @ P ) )
& ( finite_finite_a @ ( collect_a @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_13_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_14_image__ident,axiom,
! [Y: set_nat] :
( ( image_nat_nat
@ ^ [X2: nat] : X2
@ Y )
= Y ) ).
% image_ident
thf(fact_15_image__ident,axiom,
! [Y: set_a] :
( ( image_a_a
@ ^ [X2: a] : X2
@ Y )
= Y ) ).
% image_ident
thf(fact_16_fresh__val_I2_J,axiom,
! [I: product_prod_b_nat > set_list_a,A: set_a,Q: relational_fmla_a_b] :
( ( finite_finite_a @ ( relational_adom_b_a @ I ) )
=> ( ( finite_finite_a @ A )
=> ~ ( member_a @ ( relati2318939533276802993al_a_b @ Q @ I @ A ) @ ( relational_csts_a_b @ Q ) ) ) ) ).
% fresh_val(2)
thf(fact_17_UnCI,axiom,
! [C: $o,B: set_o,A: set_o] :
( ( ~ ( member_o @ C @ B )
=> ( member_o @ C @ A ) )
=> ( member_o @ C @ ( sup_sup_set_o @ A @ B ) ) ) ).
% UnCI
thf(fact_18_UnCI,axiom,
! [C: a,B: set_a,A: set_a] :
( ( ~ ( member_a @ C @ B )
=> ( member_a @ C @ A ) )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnCI
thf(fact_19_UnCI,axiom,
! [C: nat,B: set_nat,A: set_nat] :
( ( ~ ( member_nat @ C @ B )
=> ( member_nat @ C @ A ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_20_Un__iff,axiom,
! [C: $o,A: set_o,B: set_o] :
( ( member_o @ C @ ( sup_sup_set_o @ A @ B ) )
= ( ( member_o @ C @ A )
| ( member_o @ C @ B ) ) ) ).
% Un_iff
thf(fact_21_Un__iff,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A @ B ) )
= ( ( member_a @ C @ A )
| ( member_a @ C @ B ) ) ) ).
% Un_iff
thf(fact_22_Un__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) )
= ( ( member_nat @ C @ A )
| ( member_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_23_sup_Oidem,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_24_sup_Oidem,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_25_sup__idem,axiom,
! [X3: set_a] :
( ( sup_sup_set_a @ X3 @ X3 )
= X3 ) ).
% sup_idem
thf(fact_26_sup__idem,axiom,
! [X3: set_nat] :
( ( sup_sup_set_nat @ X3 @ X3 )
= X3 ) ).
% sup_idem
thf(fact_27_sup_Oleft__idem,axiom,
! [A2: set_a,B2: set_a] :
( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B2 ) )
= ( sup_sup_set_a @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_28_sup_Oleft__idem,axiom,
! [A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_29_sup__left__idem,axiom,
! [X3: set_a,Y2: set_a] :
( ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ X3 @ Y2 ) )
= ( sup_sup_set_a @ X3 @ Y2 ) ) ).
% sup_left_idem
thf(fact_30_sup__left__idem,axiom,
! [X3: set_nat,Y2: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( sup_sup_set_nat @ X3 @ Y2 ) )
= ( sup_sup_set_nat @ X3 @ Y2 ) ) ).
% sup_left_idem
thf(fact_31_image__eqI,axiom,
! [B2: a,F2: a > a,X3: a,A: set_a] :
( ( B2
= ( F2 @ X3 ) )
=> ( ( member_a @ X3 @ A )
=> ( member_a @ B2 @ ( image_a_a @ F2 @ A ) ) ) ) ).
% image_eqI
thf(fact_32_image__eqI,axiom,
! [B2: nat,F2: a > nat,X3: a,A: set_a] :
( ( B2
= ( F2 @ X3 ) )
=> ( ( member_a @ X3 @ A )
=> ( member_nat @ B2 @ ( image_a_nat @ F2 @ A ) ) ) ) ).
% image_eqI
thf(fact_33_image__eqI,axiom,
! [B2: $o,F2: a > $o,X3: a,A: set_a] :
( ( B2
= ( F2 @ X3 ) )
=> ( ( member_a @ X3 @ A )
=> ( member_o @ B2 @ ( image_a_o @ F2 @ A ) ) ) ) ).
% image_eqI
thf(fact_34_image__eqI,axiom,
! [B2: a,F2: nat > a,X3: nat,A: set_nat] :
( ( B2
= ( F2 @ X3 ) )
=> ( ( member_nat @ X3 @ A )
=> ( member_a @ B2 @ ( image_nat_a @ F2 @ A ) ) ) ) ).
% image_eqI
thf(fact_35_image__eqI,axiom,
! [B2: nat,F2: nat > nat,X3: nat,A: set_nat] :
( ( B2
= ( F2 @ X3 ) )
=> ( ( member_nat @ X3 @ A )
=> ( member_nat @ B2 @ ( image_nat_nat @ F2 @ A ) ) ) ) ).
% image_eqI
thf(fact_36_image__eqI,axiom,
! [B2: $o,F2: nat > $o,X3: nat,A: set_nat] :
( ( B2
= ( F2 @ X3 ) )
=> ( ( member_nat @ X3 @ A )
=> ( member_o @ B2 @ ( image_nat_o @ F2 @ A ) ) ) ) ).
% image_eqI
thf(fact_37_image__eqI,axiom,
! [B2: a,F2: $o > a,X3: $o,A: set_o] :
( ( B2
= ( F2 @ X3 ) )
=> ( ( member_o @ X3 @ A )
=> ( member_a @ B2 @ ( image_o_a @ F2 @ A ) ) ) ) ).
% image_eqI
thf(fact_38_image__eqI,axiom,
! [B2: nat,F2: $o > nat,X3: $o,A: set_o] :
( ( B2
= ( F2 @ X3 ) )
=> ( ( member_o @ X3 @ A )
=> ( member_nat @ B2 @ ( image_o_nat @ F2 @ A ) ) ) ) ).
% image_eqI
thf(fact_39_image__eqI,axiom,
! [B2: $o,F2: $o > $o,X3: $o,A: set_o] :
( ( B2
= ( F2 @ X3 ) )
=> ( ( member_o @ X3 @ A )
=> ( member_o @ B2 @ ( image_o_o @ F2 @ A ) ) ) ) ).
% image_eqI
thf(fact_40_sup_Oright__idem,axiom,
! [A2: set_a,B2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ B2 )
= ( sup_sup_set_a @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_41_sup_Oright__idem,axiom,
! [A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ B2 )
= ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_42_sup__set__def,axiom,
( sup_sup_set_o
= ( ^ [A3: set_o,B3: set_o] :
( collect_o
@ ( sup_sup_o_o
@ ^ [X2: $o] : ( member_o @ X2 @ A3 )
@ ^ [X2: $o] : ( member_o @ X2 @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_43_sup__set__def,axiom,
( sup_sup_set_a
= ( ^ [A3: set_a,B3: set_a] :
( collect_a
@ ( sup_sup_a_o
@ ^ [X2: a] : ( member_a @ X2 @ A3 )
@ ^ [X2: a] : ( member_a @ X2 @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_44_sup__set__def,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( collect_nat
@ ( sup_sup_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A3 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_45_finite__fv,axiom,
! [Phi: relational_fmla_a_b] : ( finite_finite_nat @ ( relational_fv_a_b @ Phi ) ) ).
% finite_fv
thf(fact_46_rev__image__eqI,axiom,
! [X3: a,A: set_a,B2: a,F2: a > a] :
( ( member_a @ X3 @ A )
=> ( ( B2
= ( F2 @ X3 ) )
=> ( member_a @ B2 @ ( image_a_a @ F2 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_47_rev__image__eqI,axiom,
! [X3: a,A: set_a,B2: nat,F2: a > nat] :
( ( member_a @ X3 @ A )
=> ( ( B2
= ( F2 @ X3 ) )
=> ( member_nat @ B2 @ ( image_a_nat @ F2 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_48_rev__image__eqI,axiom,
! [X3: a,A: set_a,B2: $o,F2: a > $o] :
( ( member_a @ X3 @ A )
=> ( ( B2
= ( F2 @ X3 ) )
=> ( member_o @ B2 @ ( image_a_o @ F2 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_49_rev__image__eqI,axiom,
! [X3: nat,A: set_nat,B2: a,F2: nat > a] :
( ( member_nat @ X3 @ A )
=> ( ( B2
= ( F2 @ X3 ) )
=> ( member_a @ B2 @ ( image_nat_a @ F2 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_50_rev__image__eqI,axiom,
! [X3: nat,A: set_nat,B2: nat,F2: nat > nat] :
( ( member_nat @ X3 @ A )
=> ( ( B2
= ( F2 @ X3 ) )
=> ( member_nat @ B2 @ ( image_nat_nat @ F2 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_51_rev__image__eqI,axiom,
! [X3: nat,A: set_nat,B2: $o,F2: nat > $o] :
( ( member_nat @ X3 @ A )
=> ( ( B2
= ( F2 @ X3 ) )
=> ( member_o @ B2 @ ( image_nat_o @ F2 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_52_rev__image__eqI,axiom,
! [X3: $o,A: set_o,B2: a,F2: $o > a] :
( ( member_o @ X3 @ A )
=> ( ( B2
= ( F2 @ X3 ) )
=> ( member_a @ B2 @ ( image_o_a @ F2 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_53_rev__image__eqI,axiom,
! [X3: $o,A: set_o,B2: nat,F2: $o > nat] :
( ( member_o @ X3 @ A )
=> ( ( B2
= ( F2 @ X3 ) )
=> ( member_nat @ B2 @ ( image_o_nat @ F2 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_54_rev__image__eqI,axiom,
! [X3: $o,A: set_o,B2: $o,F2: $o > $o] :
( ( member_o @ X3 @ A )
=> ( ( B2
= ( F2 @ X3 ) )
=> ( member_o @ B2 @ ( image_o_o @ F2 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_55_ball__imageD,axiom,
! [F2: nat > a,A: set_nat,P: a > $o] :
( ! [X: a] :
( ( member_a @ X @ ( image_nat_a @ F2 @ A ) )
=> ( P @ X ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( P @ ( F2 @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_56_ball__imageD,axiom,
! [F2: nat > nat,A: set_nat,P: nat > $o] :
( ! [X: nat] :
( ( member_nat @ X @ ( image_nat_nat @ F2 @ A ) )
=> ( P @ X ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( P @ ( F2 @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_57_ball__imageD,axiom,
! [F2: a > nat,A: set_a,P: nat > $o] :
( ! [X: nat] :
( ( member_nat @ X @ ( image_a_nat @ F2 @ A ) )
=> ( P @ X ) )
=> ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( P @ ( F2 @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_58_ball__imageD,axiom,
! [F2: a > a,A: set_a,P: a > $o] :
( ! [X: a] :
( ( member_a @ X @ ( image_a_a @ F2 @ A ) )
=> ( P @ X ) )
=> ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( P @ ( F2 @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_59_image__cong,axiom,
! [M: set_a,N: set_a,F2: a > nat,G2: a > nat] :
( ( M = N )
=> ( ! [X: a] :
( ( member_a @ X @ N )
=> ( ( F2 @ X )
= ( G2 @ X ) ) )
=> ( ( image_a_nat @ F2 @ M )
= ( image_a_nat @ G2 @ N ) ) ) ) ).
% image_cong
thf(fact_60_image__cong,axiom,
! [M: set_a,N: set_a,F2: a > a,G2: a > a] :
( ( M = N )
=> ( ! [X: a] :
( ( member_a @ X @ N )
=> ( ( F2 @ X )
= ( G2 @ X ) ) )
=> ( ( image_a_a @ F2 @ M )
= ( image_a_a @ G2 @ N ) ) ) ) ).
% image_cong
thf(fact_61_image__cong,axiom,
! [M: set_nat,N: set_nat,F2: nat > a,G2: nat > a] :
( ( M = N )
=> ( ! [X: nat] :
( ( member_nat @ X @ N )
=> ( ( F2 @ X )
= ( G2 @ X ) ) )
=> ( ( image_nat_a @ F2 @ M )
= ( image_nat_a @ G2 @ N ) ) ) ) ).
% image_cong
thf(fact_62_image__cong,axiom,
! [M: set_nat,N: set_nat,F2: nat > nat,G2: nat > nat] :
( ( M = N )
=> ( ! [X: nat] :
( ( member_nat @ X @ N )
=> ( ( F2 @ X )
= ( G2 @ X ) ) )
=> ( ( image_nat_nat @ F2 @ M )
= ( image_nat_nat @ G2 @ N ) ) ) ) ).
% image_cong
thf(fact_63_bex__imageD,axiom,
! [F2: nat > a,A: set_nat,P: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( image_nat_a @ F2 @ A ) )
& ( P @ X4 ) )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ( P @ ( F2 @ X ) ) ) ) ).
% bex_imageD
thf(fact_64_bex__imageD,axiom,
! [F2: nat > nat,A: set_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( image_nat_nat @ F2 @ A ) )
& ( P @ X4 ) )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ( P @ ( F2 @ X ) ) ) ) ).
% bex_imageD
thf(fact_65_bex__imageD,axiom,
! [F2: a > nat,A: set_a,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( image_a_nat @ F2 @ A ) )
& ( P @ X4 ) )
=> ? [X: a] :
( ( member_a @ X @ A )
& ( P @ ( F2 @ X ) ) ) ) ).
% bex_imageD
thf(fact_66_bex__imageD,axiom,
! [F2: a > a,A: set_a,P: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( image_a_a @ F2 @ A ) )
& ( P @ X4 ) )
=> ? [X: a] :
( ( member_a @ X @ A )
& ( P @ ( F2 @ X ) ) ) ) ).
% bex_imageD
thf(fact_67_image__iff,axiom,
! [Z: a,F2: nat > a,A: set_nat] :
( ( member_a @ Z @ ( image_nat_a @ F2 @ A ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( Z
= ( F2 @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_68_image__iff,axiom,
! [Z: a,F2: a > a,A: set_a] :
( ( member_a @ Z @ ( image_a_a @ F2 @ A ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A )
& ( Z
= ( F2 @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_69_image__iff,axiom,
! [Z: nat,F2: nat > nat,A: set_nat] :
( ( member_nat @ Z @ ( image_nat_nat @ F2 @ A ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( Z
= ( F2 @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_70_image__iff,axiom,
! [Z: nat,F2: a > nat,A: set_a] :
( ( member_nat @ Z @ ( image_a_nat @ F2 @ A ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A )
& ( Z
= ( F2 @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_71_imageI,axiom,
! [X3: a,A: set_a,F2: a > a] :
( ( member_a @ X3 @ A )
=> ( member_a @ ( F2 @ X3 ) @ ( image_a_a @ F2 @ A ) ) ) ).
% imageI
thf(fact_72_imageI,axiom,
! [X3: a,A: set_a,F2: a > nat] :
( ( member_a @ X3 @ A )
=> ( member_nat @ ( F2 @ X3 ) @ ( image_a_nat @ F2 @ A ) ) ) ).
% imageI
thf(fact_73_imageI,axiom,
! [X3: a,A: set_a,F2: a > $o] :
( ( member_a @ X3 @ A )
=> ( member_o @ ( F2 @ X3 ) @ ( image_a_o @ F2 @ A ) ) ) ).
% imageI
thf(fact_74_imageI,axiom,
! [X3: nat,A: set_nat,F2: nat > a] :
( ( member_nat @ X3 @ A )
=> ( member_a @ ( F2 @ X3 ) @ ( image_nat_a @ F2 @ A ) ) ) ).
% imageI
thf(fact_75_imageI,axiom,
! [X3: nat,A: set_nat,F2: nat > nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat @ ( F2 @ X3 ) @ ( image_nat_nat @ F2 @ A ) ) ) ).
% imageI
thf(fact_76_imageI,axiom,
! [X3: nat,A: set_nat,F2: nat > $o] :
( ( member_nat @ X3 @ A )
=> ( member_o @ ( F2 @ X3 ) @ ( image_nat_o @ F2 @ A ) ) ) ).
% imageI
thf(fact_77_imageI,axiom,
! [X3: $o,A: set_o,F2: $o > a] :
( ( member_o @ X3 @ A )
=> ( member_a @ ( F2 @ X3 ) @ ( image_o_a @ F2 @ A ) ) ) ).
% imageI
thf(fact_78_imageI,axiom,
! [X3: $o,A: set_o,F2: $o > nat] :
( ( member_o @ X3 @ A )
=> ( member_nat @ ( F2 @ X3 ) @ ( image_o_nat @ F2 @ A ) ) ) ).
% imageI
thf(fact_79_imageI,axiom,
! [X3: $o,A: set_o,F2: $o > $o] :
( ( member_o @ X3 @ A )
=> ( member_o @ ( F2 @ X3 ) @ ( image_o_o @ F2 @ A ) ) ) ).
% imageI
thf(fact_80_sup__left__commute,axiom,
! [X3: set_a,Y2: set_a,Z: set_a] :
( ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ Y2 @ Z ) )
= ( sup_sup_set_a @ Y2 @ ( sup_sup_set_a @ X3 @ Z ) ) ) ).
% sup_left_commute
thf(fact_81_sup__left__commute,axiom,
! [X3: set_nat,Y2: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( sup_sup_set_nat @ Y2 @ Z ) )
= ( sup_sup_set_nat @ Y2 @ ( sup_sup_set_nat @ X3 @ Z ) ) ) ).
% sup_left_commute
thf(fact_82_sup_Oleft__commute,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( sup_sup_set_a @ B2 @ ( sup_sup_set_a @ A2 @ C ) )
= ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_83_sup_Oleft__commute,axiom,
! [B2: set_nat,A2: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ B2 @ ( sup_sup_set_nat @ A2 @ C ) )
= ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_84_sup__commute,axiom,
( sup_sup_set_a
= ( ^ [X2: set_a,Y3: set_a] : ( sup_sup_set_a @ Y3 @ X2 ) ) ) ).
% sup_commute
thf(fact_85_sup__commute,axiom,
( sup_sup_set_nat
= ( ^ [X2: set_nat,Y3: set_nat] : ( sup_sup_set_nat @ Y3 @ X2 ) ) ) ).
% sup_commute
thf(fact_86_sup_Ocommute,axiom,
( sup_sup_set_a
= ( ^ [A4: set_a,B4: set_a] : ( sup_sup_set_a @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_87_sup_Ocommute,axiom,
( sup_sup_set_nat
= ( ^ [A4: set_nat,B4: set_nat] : ( sup_sup_set_nat @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_88_sup__assoc,axiom,
! [X3: set_a,Y2: set_a,Z: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X3 @ Y2 ) @ Z )
= ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ Y2 @ Z ) ) ) ).
% sup_assoc
thf(fact_89_sup__assoc,axiom,
! [X3: set_nat,Y2: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X3 @ Y2 ) @ Z )
= ( sup_sup_set_nat @ X3 @ ( sup_sup_set_nat @ Y2 @ Z ) ) ) ).
% sup_assoc
thf(fact_90_sup_Oassoc,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ C )
= ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_91_sup_Oassoc,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C )
= ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_92_inf__sup__aci_I5_J,axiom,
( sup_sup_set_a
= ( ^ [X2: set_a,Y3: set_a] : ( sup_sup_set_a @ Y3 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_93_inf__sup__aci_I5_J,axiom,
( sup_sup_set_nat
= ( ^ [X2: set_nat,Y3: set_nat] : ( sup_sup_set_nat @ Y3 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_94_inf__sup__aci_I6_J,axiom,
! [X3: set_a,Y2: set_a,Z: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X3 @ Y2 ) @ Z )
= ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ Y2 @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_95_inf__sup__aci_I6_J,axiom,
! [X3: set_nat,Y2: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X3 @ Y2 ) @ Z )
= ( sup_sup_set_nat @ X3 @ ( sup_sup_set_nat @ Y2 @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_96_inf__sup__aci_I7_J,axiom,
! [X3: set_a,Y2: set_a,Z: set_a] :
( ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ Y2 @ Z ) )
= ( sup_sup_set_a @ Y2 @ ( sup_sup_set_a @ X3 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_97_inf__sup__aci_I7_J,axiom,
! [X3: set_nat,Y2: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( sup_sup_set_nat @ Y2 @ Z ) )
= ( sup_sup_set_nat @ Y2 @ ( sup_sup_set_nat @ X3 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_98_inf__sup__aci_I8_J,axiom,
! [X3: set_a,Y2: set_a] :
( ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ X3 @ Y2 ) )
= ( sup_sup_set_a @ X3 @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_99_inf__sup__aci_I8_J,axiom,
! [X3: set_nat,Y2: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( sup_sup_set_nat @ X3 @ Y2 ) )
= ( sup_sup_set_nat @ X3 @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_100_Un__left__commute,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B @ C2 ) )
= ( sup_sup_set_a @ B @ ( sup_sup_set_a @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_101_Un__left__commute,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) )
= ( sup_sup_set_nat @ B @ ( sup_sup_set_nat @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_102_Un__left__absorb,axiom,
! [A: set_a,B: set_a] :
( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ A @ B ) )
= ( sup_sup_set_a @ A @ B ) ) ).
% Un_left_absorb
thf(fact_103_Un__left__absorb,axiom,
! [A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_set_nat @ A @ B ) ) ).
% Un_left_absorb
thf(fact_104_Un__commute,axiom,
( sup_sup_set_a
= ( ^ [A3: set_a,B3: set_a] : ( sup_sup_set_a @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_105_Un__commute,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B3: set_nat] : ( sup_sup_set_nat @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_106_Un__absorb,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ A )
= A ) ).
% Un_absorb
thf(fact_107_Un__absorb,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_108_mem__Collect__eq,axiom,
! [A2: $o,P: $o > $o] :
( ( member_o @ A2 @ ( collect_o @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_109_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_110_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_111_Collect__mem__eq,axiom,
! [A: set_o] :
( ( collect_o
@ ^ [X2: $o] : ( member_o @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_112_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_113_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_114_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X: a] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_115_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X: nat] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_116_Un__assoc,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B ) @ C2 )
= ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_117_Un__assoc,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C2 )
= ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_118_ball__Un,axiom,
! [A: set_a,B: set_a,P: a > $o] :
( ( ! [X2: a] :
( ( member_a @ X2 @ ( sup_sup_set_a @ A @ B ) )
=> ( P @ X2 ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( P @ X2 ) )
& ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_119_ball__Un,axiom,
! [A: set_nat,B: set_nat,P: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat @ X2 @ ( sup_sup_set_nat @ A @ B ) )
=> ( P @ X2 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( P @ X2 ) )
& ! [X2: nat] :
( ( member_nat @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_120_bex__Un,axiom,
! [A: set_a,B: set_a,P: a > $o] :
( ( ? [X2: a] :
( ( member_a @ X2 @ ( sup_sup_set_a @ A @ B ) )
& ( P @ X2 ) ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A )
& ( P @ X2 ) )
| ? [X2: a] :
( ( member_a @ X2 @ B )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_121_bex__Un,axiom,
! [A: set_nat,B: set_nat,P: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat @ X2 @ ( sup_sup_set_nat @ A @ B ) )
& ( P @ X2 ) ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ X2 ) )
| ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_122_UnI2,axiom,
! [C: $o,B: set_o,A: set_o] :
( ( member_o @ C @ B )
=> ( member_o @ C @ ( sup_sup_set_o @ A @ B ) ) ) ).
% UnI2
thf(fact_123_UnI2,axiom,
! [C: a,B: set_a,A: set_a] :
( ( member_a @ C @ B )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnI2
thf(fact_124_UnI2,axiom,
! [C: nat,B: set_nat,A: set_nat] :
( ( member_nat @ C @ B )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_125_UnI1,axiom,
! [C: $o,A: set_o,B: set_o] :
( ( member_o @ C @ A )
=> ( member_o @ C @ ( sup_sup_set_o @ A @ B ) ) ) ).
% UnI1
thf(fact_126_UnI1,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ A )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnI1
thf(fact_127_UnI1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ A )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_128_UnE,axiom,
! [C: $o,A: set_o,B: set_o] :
( ( member_o @ C @ ( sup_sup_set_o @ A @ B ) )
=> ( ~ ( member_o @ C @ A )
=> ( member_o @ C @ B ) ) ) ).
% UnE
thf(fact_129_UnE,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A @ B ) )
=> ( ~ ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% UnE
thf(fact_130_UnE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) )
=> ( ~ ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% UnE
thf(fact_131_Compr__image__eq,axiom,
! [F2: $o > $o,A: set_o,P: $o > $o] :
( ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ ( image_o_o @ F2 @ A ) )
& ( P @ X2 ) ) )
= ( image_o_o @ F2
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A )
& ( P @ ( F2 @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_132_Compr__image__eq,axiom,
! [F2: a > $o,A: set_a,P: $o > $o] :
( ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ ( image_a_o @ F2 @ A ) )
& ( P @ X2 ) ) )
= ( image_a_o @ F2
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A )
& ( P @ ( F2 @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_133_Compr__image__eq,axiom,
! [F2: nat > $o,A: set_nat,P: $o > $o] :
( ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ ( image_nat_o @ F2 @ A ) )
& ( P @ X2 ) ) )
= ( image_nat_o @ F2
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ ( F2 @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_134_Compr__image__eq,axiom,
! [F2: $o > a,A: set_o,P: a > $o] :
( ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ ( image_o_a @ F2 @ A ) )
& ( P @ X2 ) ) )
= ( image_o_a @ F2
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A )
& ( P @ ( F2 @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_135_Compr__image__eq,axiom,
! [F2: a > a,A: set_a,P: a > $o] :
( ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ ( image_a_a @ F2 @ A ) )
& ( P @ X2 ) ) )
= ( image_a_a @ F2
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A )
& ( P @ ( F2 @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_136_Compr__image__eq,axiom,
! [F2: nat > a,A: set_nat,P: a > $o] :
( ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ ( image_nat_a @ F2 @ A ) )
& ( P @ X2 ) ) )
= ( image_nat_a @ F2
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ ( F2 @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_137_Compr__image__eq,axiom,
! [F2: $o > nat,A: set_o,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_o_nat @ F2 @ A ) )
& ( P @ X2 ) ) )
= ( image_o_nat @ F2
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A )
& ( P @ ( F2 @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_138_Compr__image__eq,axiom,
! [F2: a > nat,A: set_a,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_a_nat @ F2 @ A ) )
& ( P @ X2 ) ) )
= ( image_a_nat @ F2
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A )
& ( P @ ( F2 @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_139_Compr__image__eq,axiom,
! [F2: nat > nat,A: set_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_nat_nat @ F2 @ A ) )
& ( P @ X2 ) ) )
= ( image_nat_nat @ F2
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ ( F2 @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_140_image__image,axiom,
! [F2: nat > a,G2: nat > nat,A: set_nat] :
( ( image_nat_a @ F2 @ ( image_nat_nat @ G2 @ A ) )
= ( image_nat_a
@ ^ [X2: nat] : ( F2 @ ( G2 @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_141_image__image,axiom,
! [F2: nat > a,G2: a > nat,A: set_a] :
( ( image_nat_a @ F2 @ ( image_a_nat @ G2 @ A ) )
= ( image_a_a
@ ^ [X2: a] : ( F2 @ ( G2 @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_142_image__image,axiom,
! [F2: nat > nat,G2: nat > nat,A: set_nat] :
( ( image_nat_nat @ F2 @ ( image_nat_nat @ G2 @ A ) )
= ( image_nat_nat
@ ^ [X2: nat] : ( F2 @ ( G2 @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_143_image__image,axiom,
! [F2: nat > nat,G2: a > nat,A: set_a] :
( ( image_nat_nat @ F2 @ ( image_a_nat @ G2 @ A ) )
= ( image_a_nat
@ ^ [X2: a] : ( F2 @ ( G2 @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_144_image__image,axiom,
! [F2: a > nat,G2: nat > a,A: set_nat] :
( ( image_a_nat @ F2 @ ( image_nat_a @ G2 @ A ) )
= ( image_nat_nat
@ ^ [X2: nat] : ( F2 @ ( G2 @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_145_image__image,axiom,
! [F2: a > nat,G2: a > a,A: set_a] :
( ( image_a_nat @ F2 @ ( image_a_a @ G2 @ A ) )
= ( image_a_nat
@ ^ [X2: a] : ( F2 @ ( G2 @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_146_image__image,axiom,
! [F2: a > a,G2: nat > a,A: set_nat] :
( ( image_a_a @ F2 @ ( image_nat_a @ G2 @ A ) )
= ( image_nat_a
@ ^ [X2: nat] : ( F2 @ ( G2 @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_147_image__image,axiom,
! [F2: a > a,G2: a > a,A: set_a] :
( ( image_a_a @ F2 @ ( image_a_a @ G2 @ A ) )
= ( image_a_a
@ ^ [X2: a] : ( F2 @ ( G2 @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_148_imageE,axiom,
! [B2: a,F2: a > a,A: set_a] :
( ( member_a @ B2 @ ( image_a_a @ F2 @ A ) )
=> ~ ! [X: a] :
( ( B2
= ( F2 @ X ) )
=> ~ ( member_a @ X @ A ) ) ) ).
% imageE
thf(fact_149_imageE,axiom,
! [B2: a,F2: nat > a,A: set_nat] :
( ( member_a @ B2 @ ( image_nat_a @ F2 @ A ) )
=> ~ ! [X: nat] :
( ( B2
= ( F2 @ X ) )
=> ~ ( member_nat @ X @ A ) ) ) ).
% imageE
thf(fact_150_imageE,axiom,
! [B2: a,F2: $o > a,A: set_o] :
( ( member_a @ B2 @ ( image_o_a @ F2 @ A ) )
=> ~ ! [X: $o] :
( ( B2
= ( F2 @ X ) )
=> ~ ( member_o @ X @ A ) ) ) ).
% imageE
thf(fact_151_imageE,axiom,
! [B2: nat,F2: a > nat,A: set_a] :
( ( member_nat @ B2 @ ( image_a_nat @ F2 @ A ) )
=> ~ ! [X: a] :
( ( B2
= ( F2 @ X ) )
=> ~ ( member_a @ X @ A ) ) ) ).
% imageE
thf(fact_152_imageE,axiom,
! [B2: nat,F2: nat > nat,A: set_nat] :
( ( member_nat @ B2 @ ( image_nat_nat @ F2 @ A ) )
=> ~ ! [X: nat] :
( ( B2
= ( F2 @ X ) )
=> ~ ( member_nat @ X @ A ) ) ) ).
% imageE
thf(fact_153_imageE,axiom,
! [B2: nat,F2: $o > nat,A: set_o] :
( ( member_nat @ B2 @ ( image_o_nat @ F2 @ A ) )
=> ~ ! [X: $o] :
( ( B2
= ( F2 @ X ) )
=> ~ ( member_o @ X @ A ) ) ) ).
% imageE
thf(fact_154_imageE,axiom,
! [B2: $o,F2: a > $o,A: set_a] :
( ( member_o @ B2 @ ( image_a_o @ F2 @ A ) )
=> ~ ! [X: a] :
( ( B2
= ( F2 @ X ) )
=> ~ ( member_a @ X @ A ) ) ) ).
% imageE
thf(fact_155_imageE,axiom,
! [B2: $o,F2: nat > $o,A: set_nat] :
( ( member_o @ B2 @ ( image_nat_o @ F2 @ A ) )
=> ~ ! [X: nat] :
( ( B2
= ( F2 @ X ) )
=> ~ ( member_nat @ X @ A ) ) ) ).
% imageE
thf(fact_156_imageE,axiom,
! [B2: $o,F2: $o > $o,A: set_o] :
( ( member_o @ B2 @ ( image_o_o @ F2 @ A ) )
=> ~ ! [X: $o] :
( ( B2
= ( F2 @ X ) )
=> ~ ( member_o @ X @ A ) ) ) ).
% imageE
thf(fact_157_pigeonhole__infinite__rel,axiom,
! [A: set_o,B: set_a,R: $o > a > $o] :
( ~ ( finite_finite_o @ A )
=> ( ( finite_finite_a @ B )
=> ( ! [X: $o] :
( ( member_o @ X @ A )
=> ? [Xa: a] :
( ( member_a @ Xa @ B )
& ( R @ X @ Xa ) ) )
=> ? [X: a] :
( ( member_a @ X @ B )
& ~ ( finite_finite_o
@ ( collect_o
@ ^ [A4: $o] :
( ( member_o @ A4 @ A )
& ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_158_pigeonhole__infinite__rel,axiom,
! [A: set_o,B: set_nat,R: $o > nat > $o] :
( ~ ( finite_finite_o @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X: $o] :
( ( member_o @ X @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X @ Xa ) ) )
=> ? [X: nat] :
( ( member_nat @ X @ B )
& ~ ( finite_finite_o
@ ( collect_o
@ ^ [A4: $o] :
( ( member_o @ A4 @ A )
& ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_159_pigeonhole__infinite__rel,axiom,
! [A: set_a,B: set_a,R: a > a > $o] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ? [Xa: a] :
( ( member_a @ Xa @ B )
& ( R @ X @ Xa ) ) )
=> ? [X: a] :
( ( member_a @ X @ B )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A )
& ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_160_pigeonhole__infinite__rel,axiom,
! [A: set_a,B: set_nat,R: a > nat > $o] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X @ Xa ) ) )
=> ? [X: nat] :
( ( member_nat @ X @ B )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A )
& ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_161_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B: set_a,R: nat > a > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_a @ B )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ? [Xa: a] :
( ( member_a @ Xa @ B )
& ( R @ X @ Xa ) ) )
=> ? [X: a] :
( ( member_a @ X @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_162_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B: set_nat,R: nat > nat > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X @ Xa ) ) )
=> ? [X: nat] :
( ( member_nat @ X @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_163_not__finite__existsD,axiom,
! [P: a > $o] :
( ~ ( finite_finite_a @ ( collect_a @ P ) )
=> ? [X_1: a] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_164_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_1: nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_165_Collect__disj__eq,axiom,
! [P: a > $o,Q: a > $o] :
( ( collect_a
@ ^ [X2: a] :
( ( P @ X2 )
| ( Q @ X2 ) ) )
= ( sup_sup_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_166_Collect__disj__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( P @ X2 )
| ( Q @ X2 ) ) )
= ( sup_sup_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_167_Un__def,axiom,
( sup_sup_set_o
= ( ^ [A3: set_o,B3: set_o] :
( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A3 )
| ( member_o @ X2 @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_168_Un__def,axiom,
( sup_sup_set_a
= ( ^ [A3: set_a,B3: set_a] :
( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A3 )
| ( member_a @ X2 @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_169_Un__def,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A3 )
| ( member_nat @ X2 @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_170_fresh__val_I1_J,axiom,
! [I: product_prod_b_nat > set_list_a,A: set_a,Q: relational_fmla_a_b] :
( ( finite_finite_a @ ( relational_adom_b_a @ I ) )
=> ( ( finite_finite_a @ A )
=> ~ ( member_a @ ( relati2318939533276802993al_a_b @ Q @ I @ A ) @ ( relational_adom_b_a @ I ) ) ) ) ).
% fresh_val(1)
thf(fact_171_fresh__val_I3_J,axiom,
! [I: product_prod_b_nat > set_list_a,A: set_a,Q: relational_fmla_a_b] :
( ( finite_finite_a @ ( relational_adom_b_a @ I ) )
=> ( ( finite_finite_a @ A )
=> ~ ( member_a @ ( relati2318939533276802993al_a_b @ Q @ I @ A ) @ A ) ) ) ).
% fresh_val(3)
thf(fact_172_image__Un,axiom,
! [F2: a > a,A: set_a,B: set_a] :
( ( image_a_a @ F2 @ ( sup_sup_set_a @ A @ B ) )
= ( sup_sup_set_a @ ( image_a_a @ F2 @ A ) @ ( image_a_a @ F2 @ B ) ) ) ).
% image_Un
thf(fact_173_image__Un,axiom,
! [F2: a > nat,A: set_a,B: set_a] :
( ( image_a_nat @ F2 @ ( sup_sup_set_a @ A @ B ) )
= ( sup_sup_set_nat @ ( image_a_nat @ F2 @ A ) @ ( image_a_nat @ F2 @ B ) ) ) ).
% image_Un
thf(fact_174_image__Un,axiom,
! [F2: nat > a,A: set_nat,B: set_nat] :
( ( image_nat_a @ F2 @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_set_a @ ( image_nat_a @ F2 @ A ) @ ( image_nat_a @ F2 @ B ) ) ) ).
% image_Un
thf(fact_175_image__Un,axiom,
! [F2: nat > nat,A: set_nat,B: set_nat] :
( ( image_nat_nat @ F2 @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_set_nat @ ( image_nat_nat @ F2 @ A ) @ ( image_nat_nat @ F2 @ B ) ) ) ).
% image_Un
thf(fact_176_infinite__Un,axiom,
! [S: set_a,T2: set_a] :
( ( ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T2 ) ) )
= ( ~ ( finite_finite_a @ S )
| ~ ( finite_finite_a @ T2 ) ) ) ).
% infinite_Un
thf(fact_177_infinite__Un,axiom,
! [S: set_nat,T2: set_nat] :
( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T2 ) ) )
= ( ~ ( finite_finite_nat @ S )
| ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_Un
thf(fact_178_Un__infinite,axiom,
! [S: set_a,T2: set_a] :
( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T2 ) ) ) ).
% Un_infinite
thf(fact_179_Un__infinite,axiom,
! [S: set_nat,T2: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T2 ) ) ) ).
% Un_infinite
thf(fact_180_finite__UnI,axiom,
! [F: set_a,G: set_a] :
( ( finite_finite_a @ F )
=> ( ( finite_finite_a @ G )
=> ( finite_finite_a @ ( sup_sup_set_a @ F @ G ) ) ) ) ).
% finite_UnI
thf(fact_181_finite__UnI,axiom,
! [F: set_nat,G: set_nat] :
( ( finite_finite_nat @ F )
=> ( ( finite_finite_nat @ G )
=> ( finite_finite_nat @ ( sup_sup_set_nat @ F @ G ) ) ) ) ).
% finite_UnI
thf(fact_182_pigeonhole__infinite,axiom,
! [A: set_o,F2: $o > a] :
( ~ ( finite_finite_o @ A )
=> ( ( finite_finite_a @ ( image_o_a @ F2 @ A ) )
=> ? [X: $o] :
( ( member_o @ X @ A )
& ~ ( finite_finite_o
@ ( collect_o
@ ^ [A4: $o] :
( ( member_o @ A4 @ A )
& ( ( F2 @ A4 )
= ( F2 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_183_pigeonhole__infinite,axiom,
! [A: set_o,F2: $o > nat] :
( ~ ( finite_finite_o @ A )
=> ( ( finite_finite_nat @ ( image_o_nat @ F2 @ A ) )
=> ? [X: $o] :
( ( member_o @ X @ A )
& ~ ( finite_finite_o
@ ( collect_o
@ ^ [A4: $o] :
( ( member_o @ A4 @ A )
& ( ( F2 @ A4 )
= ( F2 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_184_pigeonhole__infinite,axiom,
! [A: set_a,F2: a > a] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_finite_a @ ( image_a_a @ F2 @ A ) )
=> ? [X: a] :
( ( member_a @ X @ A )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A )
& ( ( F2 @ A4 )
= ( F2 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_185_pigeonhole__infinite,axiom,
! [A: set_a,F2: a > nat] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_finite_nat @ ( image_a_nat @ F2 @ A ) )
=> ? [X: a] :
( ( member_a @ X @ A )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A )
& ( ( F2 @ A4 )
= ( F2 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_186_pigeonhole__infinite,axiom,
! [A: set_nat,F2: nat > a] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_a @ ( image_nat_a @ F2 @ A ) )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( ( F2 @ A4 )
= ( F2 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_187_pigeonhole__infinite,axiom,
! [A: set_nat,F2: nat > nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ ( image_nat_nat @ F2 @ A ) )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( ( F2 @ A4 )
= ( F2 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_188_fin__fv,axiom,
finite_finite_nat @ ( relational_fv_a_b @ q3 ) ).
% fin_fv
thf(fact_189_sat,axiom,
! [D: a] :
( ~ ( member_a @ D @ ( sup_sup_set_a @ ( sup_sup_set_a @ ( relational_adom_b_a @ i ) @ ( relational_csts_a_b @ q ) ) @ ( image_nat_a @ sigma @ ( relational_fv_a_b @ q ) ) ) )
=> ( relational_sat_a_b @ q3 @ i @ ( fun_upd_nat_a @ sigma @ x @ D ) ) ) ).
% sat
thf(fact_190_assms_I4_J,axiom,
relational_sat_a_b @ ( relational_erase_a_b @ q @ x ) @ i @ sigma ).
% assms(4)
thf(fact_191_assms_I1_J,axiom,
relational_cov_a_b @ x @ q @ g ).
% assms(1)
thf(fact_192_fresh__val__def,axiom,
( relati2318939533276802993al_a_b
= ( ^ [Q2: relational_fmla_a_b,I2: product_prod_b_nat > set_list_a,A3: set_a] :
( fChoice_a
@ ^ [X2: a] :
( ~ ( member_a @ X2 @ ( relational_adom_b_a @ I2 ) )
& ~ ( member_a @ X2 @ ( relational_csts_a_b @ Q2 ) )
& ~ ( member_a @ X2 @ A3 ) ) ) ) ) ).
% fresh_val_def
thf(fact_193_sup__Un__eq,axiom,
! [R: set_o,S: set_o] :
( ( sup_sup_o_o
@ ^ [X2: $o] : ( member_o @ X2 @ R )
@ ^ [X2: $o] : ( member_o @ X2 @ S ) )
= ( ^ [X2: $o] : ( member_o @ X2 @ ( sup_sup_set_o @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_194_sup__Un__eq,axiom,
! [R: set_a,S: set_a] :
( ( sup_sup_a_o
@ ^ [X2: a] : ( member_a @ X2 @ R )
@ ^ [X2: a] : ( member_a @ X2 @ S ) )
= ( ^ [X2: a] : ( member_a @ X2 @ ( sup_sup_set_a @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_195_sup__Un__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( sup_sup_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ R )
@ ^ [X2: nat] : ( member_nat @ X2 @ S ) )
= ( ^ [X2: nat] : ( member_nat @ X2 @ ( sup_sup_set_nat @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_196_Sup_OSUP__identity__eq,axiom,
! [Sup: set_nat > nat,A: set_nat] :
( ( Sup
@ ( image_nat_nat
@ ^ [X2: nat] : X2
@ A ) )
= ( Sup @ A ) ) ).
% Sup.SUP_identity_eq
thf(fact_197_Sup_OSUP__identity__eq,axiom,
! [Sup: set_a > a,A: set_a] :
( ( Sup
@ ( image_a_a
@ ^ [X2: a] : X2
@ A ) )
= ( Sup @ A ) ) ).
% Sup.SUP_identity_eq
thf(fact_198_Inf_OINF__identity__eq,axiom,
! [Inf: set_nat > nat,A: set_nat] :
( ( Inf
@ ( image_nat_nat
@ ^ [X2: nat] : X2
@ A ) )
= ( Inf @ A ) ) ).
% Inf.INF_identity_eq
thf(fact_199_Inf_OINF__identity__eq,axiom,
! [Inf: set_a > a,A: set_a] :
( ( Inf
@ ( image_a_a
@ ^ [X2: a] : X2
@ A ) )
= ( Inf @ A ) ) ).
% Inf.INF_identity_eq
thf(fact_200_fv123_I3_J,axiom,
ord_less_eq_set_nat @ ( relational_fv_a_b @ q3 ) @ ( relational_fv_a_b @ q ) ).
% fv123(3)
thf(fact_201_boolean__algebra__cancel_Osup2,axiom,
! [B: set_a,K: set_a,B2: set_a,A2: set_a] :
( ( B
= ( sup_sup_set_a @ K @ B2 ) )
=> ( ( sup_sup_set_a @ A2 @ B )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_202_boolean__algebra__cancel_Osup2,axiom,
! [B: set_nat,K: set_nat,B2: set_nat,A2: set_nat] :
( ( B
= ( sup_sup_set_nat @ K @ B2 ) )
=> ( ( sup_sup_set_nat @ A2 @ B )
= ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_203_boolean__algebra__cancel_Osup1,axiom,
! [A: set_a,K: set_a,A2: set_a,B2: set_a] :
( ( A
= ( sup_sup_set_a @ K @ A2 ) )
=> ( ( sup_sup_set_a @ A @ B2 )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_204_boolean__algebra__cancel_Osup1,axiom,
! [A: set_nat,K: set_nat,A2: set_nat,B2: set_nat] :
( ( A
= ( sup_sup_set_nat @ K @ A2 ) )
=> ( ( sup_sup_set_nat @ A @ B2 )
= ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_205_Sup_OSUP__cong,axiom,
! [A: set_a,B: set_a,C2: a > nat,D2: a > nat,Sup: set_nat > nat] :
( ( A = B )
=> ( ! [X: a] :
( ( member_a @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( Sup @ ( image_a_nat @ C2 @ A ) )
= ( Sup @ ( image_a_nat @ D2 @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_206_Sup_OSUP__cong,axiom,
! [A: set_a,B: set_a,C2: a > a,D2: a > a,Sup: set_a > a] :
( ( A = B )
=> ( ! [X: a] :
( ( member_a @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( Sup @ ( image_a_a @ C2 @ A ) )
= ( Sup @ ( image_a_a @ D2 @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_207_Sup_OSUP__cong,axiom,
! [A: set_nat,B: set_nat,C2: nat > a,D2: nat > a,Sup: set_a > a] :
( ( A = B )
=> ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( Sup @ ( image_nat_a @ C2 @ A ) )
= ( Sup @ ( image_nat_a @ D2 @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_208_Sup_OSUP__cong,axiom,
! [A: set_nat,B: set_nat,C2: nat > nat,D2: nat > nat,Sup: set_nat > nat] :
( ( A = B )
=> ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( Sup @ ( image_nat_nat @ C2 @ A ) )
= ( Sup @ ( image_nat_nat @ D2 @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_209_subsetI,axiom,
! [A: set_a,B: set_a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_a @ X @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% subsetI
thf(fact_210_subsetI,axiom,
! [A: set_o,B: set_o] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( member_o @ X @ B ) )
=> ( ord_less_eq_set_o @ A @ B ) ) ).
% subsetI
thf(fact_211_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_nat @ X @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_212_subset__antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_213_le__sup__iff,axiom,
! [X3: set_a,Y2: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X3 @ Y2 ) @ Z )
= ( ( ord_less_eq_set_a @ X3 @ Z )
& ( ord_less_eq_set_a @ Y2 @ Z ) ) ) ).
% le_sup_iff
thf(fact_214_le__sup__iff,axiom,
! [X3: set_nat,Y2: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X3 @ Y2 ) @ Z )
= ( ( ord_less_eq_set_nat @ X3 @ Z )
& ( ord_less_eq_set_nat @ Y2 @ Z ) ) ) ).
% le_sup_iff
thf(fact_215_le__sup__iff,axiom,
! [X3: nat,Y2: nat,Z: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X3 @ Y2 ) @ Z )
= ( ( ord_less_eq_nat @ X3 @ Z )
& ( ord_less_eq_nat @ Y2 @ Z ) ) ) ).
% le_sup_iff
thf(fact_216_sup_Obounded__iff,axiom,
! [B2: set_a,C: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A2 )
= ( ( ord_less_eq_set_a @ B2 @ A2 )
& ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_217_sup_Obounded__iff,axiom,
! [B2: set_nat,C: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A2 )
= ( ( ord_less_eq_set_nat @ B2 @ A2 )
& ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_218_sup_Obounded__iff,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_219_Un__subset__iff,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ C2 )
= ( ( ord_less_eq_set_a @ A @ C2 )
& ( ord_less_eq_set_a @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_220_Un__subset__iff,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C2 )
= ( ( ord_less_eq_set_nat @ A @ C2 )
& ( ord_less_eq_set_nat @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_221_finite__Collect__subsets,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( finite_finite_set_a
@ ( collect_set_a
@ ^ [B3: set_a] : ( ord_less_eq_set_a @ B3 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_222_finite__Collect__subsets,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B3: set_nat] : ( ord_less_eq_set_nat @ B3 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_223_in__mono,axiom,
! [A: set_a,B: set_a,X3: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ X3 @ A )
=> ( member_a @ X3 @ B ) ) ) ).
% in_mono
thf(fact_224_in__mono,axiom,
! [A: set_o,B: set_o,X3: $o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( member_o @ X3 @ A )
=> ( member_o @ X3 @ B ) ) ) ).
% in_mono
thf(fact_225_in__mono,axiom,
! [A: set_nat,B: set_nat,X3: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X3 @ A )
=> ( member_nat @ X3 @ B ) ) ) ).
% in_mono
thf(fact_226_subsetD,axiom,
! [A: set_a,B: set_a,C: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% subsetD
thf(fact_227_subsetD,axiom,
! [A: set_o,B: set_o,C: $o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( member_o @ C @ A )
=> ( member_o @ C @ B ) ) ) ).
% subsetD
thf(fact_228_subsetD,axiom,
! [A: set_nat,B: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_229_equalityE,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_nat @ A @ B )
=> ~ ( ord_less_eq_set_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_230_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ( member_a @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_231_subset__eq,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B3: set_o] :
! [X2: $o] :
( ( member_o @ X2 @ A3 )
=> ( member_o @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_232_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A3 )
=> ( member_nat @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_233_equalityD1,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% equalityD1
thf(fact_234_Set_OequalityD2,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% Set.equalityD2
thf(fact_235_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
! [T3: a] :
( ( member_a @ T3 @ A3 )
=> ( member_a @ T3 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_236_subset__iff,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B3: set_o] :
! [T3: $o] :
( ( member_o @ T3 @ A3 )
=> ( member_o @ T3 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_237_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [T3: nat] :
( ( member_nat @ T3 @ A3 )
=> ( member_nat @ T3 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_238_subset__refl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% subset_refl
thf(fact_239_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X: a] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_240_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X: nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_241_subset__trans,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_242_set__eq__subset,axiom,
( ( ^ [Y4: set_nat,Z2: set_nat] : ( Y4 = Z2 ) )
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_243_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_244_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_245_cov__finite,axiom,
! [X3: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_cov_a_b @ X3 @ Q @ G )
=> ( finite5600759454172676150la_a_b @ G ) ) ).
% cov_finite
thf(fact_246_sat__erase,axiom,
! [Q: relational_fmla_a_b,X3: nat,I: product_prod_b_nat > set_list_a,Sigma: nat > a,Z: a] :
( ( relational_sat_a_b @ ( relational_erase_a_b @ Q @ X3 ) @ I @ ( fun_upd_nat_a @ Sigma @ X3 @ Z ) )
= ( relational_sat_a_b @ ( relational_erase_a_b @ Q @ X3 ) @ I @ Sigma ) ) ).
% sat_erase
thf(fact_247_sat__fun__upd,axiom,
! [N2: nat,Q: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > a,Z: a] :
( ~ ( member_nat @ N2 @ ( relational_fv_a_b @ Q ) )
=> ( ( relational_sat_a_b @ Q @ I @ ( fun_upd_nat_a @ Sigma @ N2 @ Z ) )
= ( relational_sat_a_b @ Q @ I @ Sigma ) ) ) ).
% sat_fun_upd
thf(fact_248_cov__fv,axiom,
! [X3: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Qqp: relational_fmla_a_b] :
( ( relational_cov_a_b @ X3 @ Q @ G )
=> ( ( member_nat @ X3 @ ( relational_fv_a_b @ Q ) )
=> ( ( member4680049679412964150la_a_b @ Qqp @ G )
=> ( ( member_nat @ X3 @ ( relational_fv_a_b @ Qqp ) )
& ( ord_less_eq_set_nat @ ( relational_fv_a_b @ Qqp ) @ ( relational_fv_a_b @ Q ) ) ) ) ) ) ).
% cov_fv
thf(fact_249_cov__csts,axiom,
! [X3: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Qqp: relational_fmla_a_b] :
( ( relational_cov_a_b @ X3 @ Q @ G )
=> ( ( member4680049679412964150la_a_b @ Qqp @ G )
=> ( ord_less_eq_set_a @ ( relational_csts_a_b @ Qqp ) @ ( relational_csts_a_b @ Q ) ) ) ) ).
% cov_csts
thf(fact_250_Collect__subset,axiom,
! [A: set_o,P: $o > $o] :
( ord_less_eq_set_o
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_251_Collect__subset,axiom,
! [A: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_252_Collect__subset,axiom,
! [A: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_253_sat__fv__cong,axiom,
! [Phi: relational_fmla_a_b,Sigma: nat > a,Sigma2: nat > a,I: product_prod_b_nat > set_list_a] :
( ! [N3: nat] :
( ( member_nat @ N3 @ ( relational_fv_a_b @ Phi ) )
=> ( ( Sigma @ N3 )
= ( Sigma2 @ N3 ) ) )
=> ( ( relational_sat_a_b @ Phi @ I @ Sigma )
= ( relational_sat_a_b @ Phi @ I @ Sigma2 ) ) ) ).
% sat_fv_cong
thf(fact_254_finite__has__maximal2,axiom,
! [A: set_o,A2: $o] :
( ( finite_finite_o @ A )
=> ( ( member_o @ A2 @ A )
=> ? [X: $o] :
( ( member_o @ X @ A )
& ( ord_less_eq_o @ A2 @ X )
& ! [Xa: $o] :
( ( member_o @ Xa @ A )
=> ( ( ord_less_eq_o @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_255_finite__has__maximal2,axiom,
! [A: set_a,A2: a] :
( ( finite_finite_a @ A )
=> ( ( member_a @ A2 @ A )
=> ? [X: a] :
( ( member_a @ X @ A )
& ( ord_less_eq_a @ A2 @ X )
& ! [Xa: a] :
( ( member_a @ Xa @ A )
=> ( ( ord_less_eq_a @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_256_finite__has__maximal2,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_set_nat @ A2 @ A )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A )
& ( ord_less_eq_set_nat @ A2 @ X )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_257_finite__has__maximal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ( ord_less_eq_nat @ A2 @ X )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_258_finite__has__minimal2,axiom,
! [A: set_o,A2: $o] :
( ( finite_finite_o @ A )
=> ( ( member_o @ A2 @ A )
=> ? [X: $o] :
( ( member_o @ X @ A )
& ( ord_less_eq_o @ X @ A2 )
& ! [Xa: $o] :
( ( member_o @ Xa @ A )
=> ( ( ord_less_eq_o @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_259_finite__has__minimal2,axiom,
! [A: set_a,A2: a] :
( ( finite_finite_a @ A )
=> ( ( member_a @ A2 @ A )
=> ? [X: a] :
( ( member_a @ X @ A )
& ( ord_less_eq_a @ X @ A2 )
& ! [Xa: a] :
( ( member_a @ Xa @ A )
=> ( ( ord_less_eq_a @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_260_finite__has__minimal2,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_set_nat @ A2 @ A )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A )
& ( ord_less_eq_set_nat @ X @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_261_finite__has__minimal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ( ord_less_eq_nat @ X @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_262_image__mono,axiom,
! [A: set_a,B: set_a,F2: a > a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ ( image_a_a @ F2 @ A ) @ ( image_a_a @ F2 @ B ) ) ) ).
% image_mono
thf(fact_263_image__mono,axiom,
! [A: set_a,B: set_a,F2: a > nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_nat @ ( image_a_nat @ F2 @ A ) @ ( image_a_nat @ F2 @ B ) ) ) ).
% image_mono
thf(fact_264_image__mono,axiom,
! [A: set_nat,B: set_nat,F2: nat > a] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_a @ ( image_nat_a @ F2 @ A ) @ ( image_nat_a @ F2 @ B ) ) ) ).
% image_mono
thf(fact_265_image__mono,axiom,
! [A: set_nat,B: set_nat,F2: nat > nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A ) @ ( image_nat_nat @ F2 @ B ) ) ) ).
% image_mono
thf(fact_266_image__subsetI,axiom,
! [A: set_a,F2: a > a,B: set_a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_a @ ( F2 @ X ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F2 @ A ) @ B ) ) ).
% image_subsetI
thf(fact_267_image__subsetI,axiom,
! [A: set_a,F2: a > $o,B: set_o] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_o @ ( F2 @ X ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_a_o @ F2 @ A ) @ B ) ) ).
% image_subsetI
thf(fact_268_image__subsetI,axiom,
! [A: set_nat,F2: nat > a,B: set_a] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_a @ ( F2 @ X ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_nat_a @ F2 @ A ) @ B ) ) ).
% image_subsetI
thf(fact_269_image__subsetI,axiom,
! [A: set_nat,F2: nat > $o,B: set_o] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_o @ ( F2 @ X ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_nat_o @ F2 @ A ) @ B ) ) ).
% image_subsetI
thf(fact_270_image__subsetI,axiom,
! [A: set_o,F2: $o > a,B: set_a] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( member_a @ ( F2 @ X ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_o_a @ F2 @ A ) @ B ) ) ).
% image_subsetI
thf(fact_271_image__subsetI,axiom,
! [A: set_o,F2: $o > $o,B: set_o] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( member_o @ ( F2 @ X ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_o_o @ F2 @ A ) @ B ) ) ).
% image_subsetI
thf(fact_272_image__subsetI,axiom,
! [A: set_a,F2: a > nat,B: set_nat] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_nat @ ( F2 @ X ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_a_nat @ F2 @ A ) @ B ) ) ).
% image_subsetI
thf(fact_273_image__subsetI,axiom,
! [A: set_nat,F2: nat > nat,B: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_nat @ ( F2 @ X ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A ) @ B ) ) ).
% image_subsetI
thf(fact_274_image__subsetI,axiom,
! [A: set_o,F2: $o > nat,B: set_nat] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( member_nat @ ( F2 @ X ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_o_nat @ F2 @ A ) @ B ) ) ).
% image_subsetI
thf(fact_275_subset__imageE,axiom,
! [B: set_a,F2: a > a,A: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F2 @ A ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A )
=> ( B
!= ( image_a_a @ F2 @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_276_subset__imageE,axiom,
! [B: set_a,F2: nat > a,A: set_nat] :
( ( ord_less_eq_set_a @ B @ ( image_nat_a @ F2 @ A ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
=> ( B
!= ( image_nat_a @ F2 @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_277_subset__imageE,axiom,
! [B: set_nat,F2: a > nat,A: set_a] :
( ( ord_less_eq_set_nat @ B @ ( image_a_nat @ F2 @ A ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A )
=> ( B
!= ( image_a_nat @ F2 @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_278_subset__imageE,axiom,
! [B: set_nat,F2: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F2 @ A ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
=> ( B
!= ( image_nat_nat @ F2 @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_279_image__subset__iff,axiom,
! [F2: nat > a,A: set_nat,B: set_a] :
( ( ord_less_eq_set_a @ ( image_nat_a @ F2 @ A ) @ B )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_a @ ( F2 @ X2 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_280_image__subset__iff,axiom,
! [F2: a > a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F2 @ A ) @ B )
= ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_a @ ( F2 @ X2 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_281_image__subset__iff,axiom,
! [F2: nat > nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A ) @ B )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ ( F2 @ X2 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_282_image__subset__iff,axiom,
! [F2: a > nat,A: set_a,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_a_nat @ F2 @ A ) @ B )
= ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_nat @ ( F2 @ X2 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_283_subset__image__iff,axiom,
! [B: set_a,F2: a > a,A: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F2 @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B
= ( image_a_a @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_284_subset__image__iff,axiom,
! [B: set_a,F2: nat > a,A: set_nat] :
( ( ord_less_eq_set_a @ B @ ( image_nat_a @ F2 @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B
= ( image_nat_a @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_285_subset__image__iff,axiom,
! [B: set_nat,F2: a > nat,A: set_a] :
( ( ord_less_eq_set_nat @ B @ ( image_a_nat @ F2 @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B
= ( image_a_nat @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_286_subset__image__iff,axiom,
! [B: set_nat,F2: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F2 @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B
= ( image_nat_nat @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_287_all__subset__image,axiom,
! [F2: a > a,A: set_a,P: set_a > $o] :
( ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ ( image_a_a @ F2 @ A ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A )
=> ( P @ ( image_a_a @ F2 @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_288_all__subset__image,axiom,
! [F2: nat > a,A: set_nat,P: set_a > $o] :
( ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ ( image_nat_a @ F2 @ A ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A )
=> ( P @ ( image_nat_a @ F2 @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_289_all__subset__image,axiom,
! [F2: a > nat,A: set_a,P: set_nat > $o] :
( ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ ( image_a_nat @ F2 @ A ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A )
=> ( P @ ( image_a_nat @ F2 @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_290_all__subset__image,axiom,
! [F2: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F2 @ A ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A )
=> ( P @ ( image_nat_nat @ F2 @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_291_finite__subset,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ A ) ) ) ).
% finite_subset
thf(fact_292_finite__subset,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_subset
thf(fact_293_infinite__super,axiom,
! [S: set_a,T2: set_a] :
( ( ord_less_eq_set_a @ S @ T2 )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ T2 ) ) ) ).
% infinite_super
thf(fact_294_infinite__super,axiom,
! [S: set_nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S @ T2 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_295_rev__finite__subset,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( finite_finite_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_296_rev__finite__subset,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_297_inf__sup__ord_I4_J,axiom,
! [Y2: set_a,X3: set_a] : ( ord_less_eq_set_a @ Y2 @ ( sup_sup_set_a @ X3 @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_298_inf__sup__ord_I4_J,axiom,
! [Y2: set_nat,X3: set_nat] : ( ord_less_eq_set_nat @ Y2 @ ( sup_sup_set_nat @ X3 @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_299_inf__sup__ord_I4_J,axiom,
! [Y2: nat,X3: nat] : ( ord_less_eq_nat @ Y2 @ ( sup_sup_nat @ X3 @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_300_inf__sup__ord_I3_J,axiom,
! [X3: set_a,Y2: set_a] : ( ord_less_eq_set_a @ X3 @ ( sup_sup_set_a @ X3 @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_301_inf__sup__ord_I3_J,axiom,
! [X3: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ X3 @ ( sup_sup_set_nat @ X3 @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_302_inf__sup__ord_I3_J,axiom,
! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ X3 @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_303_le__supE,axiom,
! [A2: set_a,B2: set_a,X3: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ X3 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ X3 )
=> ~ ( ord_less_eq_set_a @ B2 @ X3 ) ) ) ).
% le_supE
thf(fact_304_le__supE,axiom,
! [A2: set_nat,B2: set_nat,X3: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ X3 )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ X3 )
=> ~ ( ord_less_eq_set_nat @ B2 @ X3 ) ) ) ).
% le_supE
thf(fact_305_le__supE,axiom,
! [A2: nat,B2: nat,X3: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X3 )
=> ~ ( ( ord_less_eq_nat @ A2 @ X3 )
=> ~ ( ord_less_eq_nat @ B2 @ X3 ) ) ) ).
% le_supE
thf(fact_306_le__supI,axiom,
! [A2: set_a,X3: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ X3 )
=> ( ( ord_less_eq_set_a @ B2 @ X3 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ X3 ) ) ) ).
% le_supI
thf(fact_307_le__supI,axiom,
! [A2: set_nat,X3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ X3 )
=> ( ( ord_less_eq_set_nat @ B2 @ X3 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ X3 ) ) ) ).
% le_supI
thf(fact_308_le__supI,axiom,
! [A2: nat,X3: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ X3 )
=> ( ( ord_less_eq_nat @ B2 @ X3 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X3 ) ) ) ).
% le_supI
thf(fact_309_sup__ge1,axiom,
! [X3: set_a,Y2: set_a] : ( ord_less_eq_set_a @ X3 @ ( sup_sup_set_a @ X3 @ Y2 ) ) ).
% sup_ge1
thf(fact_310_sup__ge1,axiom,
! [X3: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ X3 @ ( sup_sup_set_nat @ X3 @ Y2 ) ) ).
% sup_ge1
thf(fact_311_sup__ge1,axiom,
! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ X3 @ Y2 ) ) ).
% sup_ge1
thf(fact_312_sup__ge2,axiom,
! [Y2: set_a,X3: set_a] : ( ord_less_eq_set_a @ Y2 @ ( sup_sup_set_a @ X3 @ Y2 ) ) ).
% sup_ge2
thf(fact_313_sup__ge2,axiom,
! [Y2: set_nat,X3: set_nat] : ( ord_less_eq_set_nat @ Y2 @ ( sup_sup_set_nat @ X3 @ Y2 ) ) ).
% sup_ge2
thf(fact_314_sup__ge2,axiom,
! [Y2: nat,X3: nat] : ( ord_less_eq_nat @ Y2 @ ( sup_sup_nat @ X3 @ Y2 ) ) ).
% sup_ge2
thf(fact_315_le__supI1,axiom,
! [X3: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X3 @ A2 )
=> ( ord_less_eq_set_a @ X3 @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_316_le__supI1,axiom,
! [X3: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ X3 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_317_le__supI1,axiom,
! [X3: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_318_le__supI2,axiom,
! [X3: set_a,B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ X3 @ B2 )
=> ( ord_less_eq_set_a @ X3 @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_319_le__supI2,axiom,
! [X3: set_nat,B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ B2 )
=> ( ord_less_eq_set_nat @ X3 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_320_le__supI2,axiom,
! [X3: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ X3 @ B2 )
=> ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_321_sup_Omono,axiom,
! [C: set_a,A2: set_a,D: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C @ A2 )
=> ( ( ord_less_eq_set_a @ D @ B2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ C @ D ) @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_322_sup_Omono,axiom,
! [C: set_nat,A2: set_nat,D: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C @ A2 )
=> ( ( ord_less_eq_set_nat @ D @ B2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C @ D ) @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_323_sup_Omono,axiom,
! [C: nat,A2: nat,D: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ( ord_less_eq_nat @ D @ B2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D ) @ ( sup_sup_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_324_sup__mono,axiom,
! [A2: set_a,C: set_a,B2: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B2 @ D )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ ( sup_sup_set_a @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_325_sup__mono,axiom,
! [A2: set_nat,C: set_nat,B2: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( ord_less_eq_set_nat @ B2 @ D )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ ( sup_sup_set_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_326_sup__mono,axiom,
! [A2: nat,C: nat,B2: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ ( sup_sup_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_327_sup__least,axiom,
! [Y2: set_a,X3: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X3 )
=> ( ( ord_less_eq_set_a @ Z @ X3 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y2 @ Z ) @ X3 ) ) ) ).
% sup_least
thf(fact_328_sup__least,axiom,
! [Y2: set_nat,X3: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X3 )
=> ( ( ord_less_eq_set_nat @ Z @ X3 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ Y2 @ Z ) @ X3 ) ) ) ).
% sup_least
thf(fact_329_sup__least,axiom,
! [Y2: nat,X3: nat,Z: nat] :
( ( ord_less_eq_nat @ Y2 @ X3 )
=> ( ( ord_less_eq_nat @ Z @ X3 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y2 @ Z ) @ X3 ) ) ) ).
% sup_least
thf(fact_330_le__iff__sup,axiom,
( ord_less_eq_set_a
= ( ^ [X2: set_a,Y3: set_a] :
( ( sup_sup_set_a @ X2 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_331_le__iff__sup,axiom,
( ord_less_eq_set_nat
= ( ^ [X2: set_nat,Y3: set_nat] :
( ( sup_sup_set_nat @ X2 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_332_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y3: nat] :
( ( sup_sup_nat @ X2 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_333_sup_OorderE,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2
= ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_334_sup_OorderE,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( A2
= ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_335_sup_OorderE,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2
= ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_336_sup_OorderI,axiom,
! [A2: set_a,B2: set_a] :
( ( A2
= ( sup_sup_set_a @ A2 @ B2 ) )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_337_sup_OorderI,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2
= ( sup_sup_set_nat @ A2 @ B2 ) )
=> ( ord_less_eq_set_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_338_sup_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( sup_sup_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_339_sup__unique,axiom,
! [F2: set_a > set_a > set_a,X3: set_a,Y2: set_a] :
( ! [X: set_a,Y5: set_a] : ( ord_less_eq_set_a @ X @ ( F2 @ X @ Y5 ) )
=> ( ! [X: set_a,Y5: set_a] : ( ord_less_eq_set_a @ Y5 @ ( F2 @ X @ Y5 ) )
=> ( ! [X: set_a,Y5: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ Y5 @ X )
=> ( ( ord_less_eq_set_a @ Z3 @ X )
=> ( ord_less_eq_set_a @ ( F2 @ Y5 @ Z3 ) @ X ) ) )
=> ( ( sup_sup_set_a @ X3 @ Y2 )
= ( F2 @ X3 @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_340_sup__unique,axiom,
! [F2: set_nat > set_nat > set_nat,X3: set_nat,Y2: set_nat] :
( ! [X: set_nat,Y5: set_nat] : ( ord_less_eq_set_nat @ X @ ( F2 @ X @ Y5 ) )
=> ( ! [X: set_nat,Y5: set_nat] : ( ord_less_eq_set_nat @ Y5 @ ( F2 @ X @ Y5 ) )
=> ( ! [X: set_nat,Y5: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ Y5 @ X )
=> ( ( ord_less_eq_set_nat @ Z3 @ X )
=> ( ord_less_eq_set_nat @ ( F2 @ Y5 @ Z3 ) @ X ) ) )
=> ( ( sup_sup_set_nat @ X3 @ Y2 )
= ( F2 @ X3 @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_341_sup__unique,axiom,
! [F2: nat > nat > nat,X3: nat,Y2: nat] :
( ! [X: nat,Y5: nat] : ( ord_less_eq_nat @ X @ ( F2 @ X @ Y5 ) )
=> ( ! [X: nat,Y5: nat] : ( ord_less_eq_nat @ Y5 @ ( F2 @ X @ Y5 ) )
=> ( ! [X: nat,Y5: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y5 @ X )
=> ( ( ord_less_eq_nat @ Z3 @ X )
=> ( ord_less_eq_nat @ ( F2 @ Y5 @ Z3 ) @ X ) ) )
=> ( ( sup_sup_nat @ X3 @ Y2 )
= ( F2 @ X3 @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_342_sup_Oabsorb1,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_343_sup_Oabsorb1,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( sup_sup_set_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_344_sup_Oabsorb1,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_345_sup_Oabsorb2,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_346_sup_Oabsorb2,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( sup_sup_set_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_347_sup_Oabsorb2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_348_sup__absorb1,axiom,
! [Y2: set_a,X3: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X3 )
=> ( ( sup_sup_set_a @ X3 @ Y2 )
= X3 ) ) ).
% sup_absorb1
thf(fact_349_sup__absorb1,axiom,
! [Y2: set_nat,X3: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X3 )
=> ( ( sup_sup_set_nat @ X3 @ Y2 )
= X3 ) ) ).
% sup_absorb1
thf(fact_350_sup__absorb1,axiom,
! [Y2: nat,X3: nat] :
( ( ord_less_eq_nat @ Y2 @ X3 )
=> ( ( sup_sup_nat @ X3 @ Y2 )
= X3 ) ) ).
% sup_absorb1
thf(fact_351_sup__absorb2,axiom,
! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ( sup_sup_set_a @ X3 @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_352_sup__absorb2,axiom,
! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ( sup_sup_set_nat @ X3 @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_353_sup__absorb2,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( sup_sup_nat @ X3 @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_354_sup_OboundedE,axiom,
! [B2: set_a,C: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ~ ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_355_sup_OboundedE,axiom,
! [B2: set_nat,C: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ~ ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_356_sup_OboundedE,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_nat @ B2 @ A2 )
=> ~ ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_357_sup_OboundedI,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ C @ A2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_358_sup_OboundedI,axiom,
! [B2: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ A2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_359_sup_OboundedI,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_360_sup_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A4: set_a] :
( A4
= ( sup_sup_set_a @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_361_sup_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [B4: set_nat,A4: set_nat] :
( A4
= ( sup_sup_set_nat @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_362_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( A4
= ( sup_sup_nat @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_363_sup_Ocobounded1,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_364_sup_Ocobounded1,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_365_sup_Ocobounded1,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_366_sup_Ocobounded2,axiom,
! [B2: set_a,A2: set_a] : ( ord_less_eq_set_a @ B2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_367_sup_Ocobounded2,axiom,
! [B2: set_nat,A2: set_nat] : ( ord_less_eq_set_nat @ B2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_368_sup_Ocobounded2,axiom,
! [B2: nat,A2: nat] : ( ord_less_eq_nat @ B2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_369_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A4: set_a] :
( ( sup_sup_set_a @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_370_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [B4: set_nat,A4: set_nat] :
( ( sup_sup_set_nat @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_371_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( sup_sup_nat @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_372_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( sup_sup_set_a @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_373_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( sup_sup_set_nat @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_374_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( sup_sup_nat @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_375_sup_OcoboundedI1,axiom,
! [C: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C @ A2 )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_376_sup_OcoboundedI1,axiom,
! [C: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C @ A2 )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_377_sup_OcoboundedI1,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_378_sup_OcoboundedI2,axiom,
! [C: set_a,B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ C @ B2 )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_379_sup_OcoboundedI2,axiom,
! [C: set_nat,B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ C @ B2 )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_380_sup_OcoboundedI2,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_381_Un__mono,axiom,
! [A: set_a,C2: set_a,B: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B @ D2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_382_Un__mono,axiom,
! [A: set_nat,C2: set_nat,B: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ D2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_383_Un__least,axiom,
! [A: set_a,C2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_384_Un__least,axiom,
! [A: set_nat,C2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_385_Un__upper1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ A @ B ) ) ).
% Un_upper1
thf(fact_386_Un__upper1,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) ) ).
% Un_upper1
thf(fact_387_Un__upper2,axiom,
! [B: set_a,A: set_a] : ( ord_less_eq_set_a @ B @ ( sup_sup_set_a @ A @ B ) ) ).
% Un_upper2
thf(fact_388_Un__upper2,axiom,
! [B: set_nat,A: set_nat] : ( ord_less_eq_set_nat @ B @ ( sup_sup_set_nat @ A @ B ) ) ).
% Un_upper2
thf(fact_389_Un__absorb1,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( sup_sup_set_a @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_390_Un__absorb1,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( sup_sup_set_nat @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_391_Un__absorb2,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( sup_sup_set_a @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_392_Un__absorb2,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( sup_sup_set_nat @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_393_subset__UnE,axiom,
! [C2: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A @ B ) )
=> ~ ! [A5: set_a] :
( ( ord_less_eq_set_a @ A5 @ A )
=> ! [B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ B )
=> ( C2
!= ( sup_sup_set_a @ A5 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_394_subset__UnE,axiom,
! [C2: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) )
=> ~ ! [A5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ A )
=> ! [B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ B )
=> ( C2
!= ( sup_sup_set_nat @ A5 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_395_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( sup_sup_set_a @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_396_subset__Un__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( sup_sup_set_nat @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_397_finite__update__induct,axiom,
! [F2: nat > a,C: a,P: ( nat > a ) > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( F2 @ A4 )
!= C ) ) )
=> ( ( P
@ ^ [A4: nat] : C )
=> ( ! [A6: nat,B6: a,F3: nat > a] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [C4: nat] :
( ( F3 @ C4 )
!= C ) ) )
=> ( ( ( F3 @ A6 )
= C )
=> ( ( B6 != C )
=> ( ( P @ F3 )
=> ( P @ ( fun_upd_nat_a @ F3 @ A6 @ B6 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_update_induct
thf(fact_398_finite__surj,axiom,
! [A: set_a,B: set_a,F2: a > a] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_a @ B @ ( image_a_a @ F2 @ A ) )
=> ( finite_finite_a @ B ) ) ) ).
% finite_surj
thf(fact_399_finite__surj,axiom,
! [A: set_nat,B: set_a,F2: nat > a] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_a @ B @ ( image_nat_a @ F2 @ A ) )
=> ( finite_finite_a @ B ) ) ) ).
% finite_surj
thf(fact_400_finite__surj,axiom,
! [A: set_a,B: set_nat,F2: a > nat] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_nat @ B @ ( image_a_nat @ F2 @ A ) )
=> ( finite_finite_nat @ B ) ) ) ).
% finite_surj
thf(fact_401_finite__surj,axiom,
! [A: set_nat,B: set_nat,F2: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F2 @ A ) )
=> ( finite_finite_nat @ B ) ) ) ).
% finite_surj
thf(fact_402_finite__subset__image,axiom,
! [B: set_a,F2: a > a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ B @ ( image_a_a @ F2 @ A ) )
=> ? [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A )
& ( finite_finite_a @ C3 )
& ( B
= ( image_a_a @ F2 @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_403_finite__subset__image,axiom,
! [B: set_a,F2: nat > a,A: set_nat] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ B @ ( image_nat_a @ F2 @ A ) )
=> ? [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
& ( finite_finite_nat @ C3 )
& ( B
= ( image_nat_a @ F2 @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_404_finite__subset__image,axiom,
! [B: set_nat,F2: a > nat,A: set_a] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_a_nat @ F2 @ A ) )
=> ? [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A )
& ( finite_finite_a @ C3 )
& ( B
= ( image_a_nat @ F2 @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_405_finite__subset__image,axiom,
! [B: set_nat,F2: nat > nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F2 @ A ) )
=> ? [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
& ( finite_finite_nat @ C3 )
& ( B
= ( image_nat_nat @ F2 @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_406_ex__finite__subset__image,axiom,
! [F2: a > a,A: set_a,P: set_a > $o] :
( ( ? [B3: set_a] :
( ( finite_finite_a @ B3 )
& ( ord_less_eq_set_a @ B3 @ ( image_a_a @ F2 @ A ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_a] :
( ( finite_finite_a @ B3 )
& ( ord_less_eq_set_a @ B3 @ A )
& ( P @ ( image_a_a @ F2 @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_407_ex__finite__subset__image,axiom,
! [F2: nat > a,A: set_nat,P: set_a > $o] :
( ( ? [B3: set_a] :
( ( finite_finite_a @ B3 )
& ( ord_less_eq_set_a @ B3 @ ( image_nat_a @ F2 @ A ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A )
& ( P @ ( image_nat_a @ F2 @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_408_ex__finite__subset__image,axiom,
! [F2: a > nat,A: set_a,P: set_nat > $o] :
( ( ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ ( image_a_nat @ F2 @ A ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_a] :
( ( finite_finite_a @ B3 )
& ( ord_less_eq_set_a @ B3 @ A )
& ( P @ ( image_a_nat @ F2 @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_409_ex__finite__subset__image,axiom,
! [F2: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F2 @ A ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A )
& ( P @ ( image_nat_nat @ F2 @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_410_all__finite__subset__image,axiom,
! [F2: a > a,A: set_a,P: set_a > $o] :
( ( ! [B3: set_a] :
( ( ( finite_finite_a @ B3 )
& ( ord_less_eq_set_a @ B3 @ ( image_a_a @ F2 @ A ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_a] :
( ( ( finite_finite_a @ B3 )
& ( ord_less_eq_set_a @ B3 @ A ) )
=> ( P @ ( image_a_a @ F2 @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_411_all__finite__subset__image,axiom,
! [F2: nat > a,A: set_nat,P: set_a > $o] :
( ( ! [B3: set_a] :
( ( ( finite_finite_a @ B3 )
& ( ord_less_eq_set_a @ B3 @ ( image_nat_a @ F2 @ A ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A ) )
=> ( P @ ( image_nat_a @ F2 @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_412_all__finite__subset__image,axiom,
! [F2: a > nat,A: set_a,P: set_nat > $o] :
( ( ! [B3: set_nat] :
( ( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ ( image_a_nat @ F2 @ A ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_a] :
( ( ( finite_finite_a @ B3 )
& ( ord_less_eq_set_a @ B3 @ A ) )
=> ( P @ ( image_a_nat @ F2 @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_413_all__finite__subset__image,axiom,
! [F2: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ! [B3: set_nat] :
( ( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F2 @ A ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A ) )
=> ( P @ ( image_nat_nat @ F2 @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_414_Inf_OINF__cong,axiom,
! [A: set_a,B: set_a,C2: a > nat,D2: a > nat,Inf: set_nat > nat] :
( ( A = B )
=> ( ! [X: a] :
( ( member_a @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( Inf @ ( image_a_nat @ C2 @ A ) )
= ( Inf @ ( image_a_nat @ D2 @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_415_Inf_OINF__cong,axiom,
! [A: set_a,B: set_a,C2: a > a,D2: a > a,Inf: set_a > a] :
( ( A = B )
=> ( ! [X: a] :
( ( member_a @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( Inf @ ( image_a_a @ C2 @ A ) )
= ( Inf @ ( image_a_a @ D2 @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_416_Inf_OINF__cong,axiom,
! [A: set_nat,B: set_nat,C2: nat > a,D2: nat > a,Inf: set_a > a] :
( ( A = B )
=> ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( Inf @ ( image_nat_a @ C2 @ A ) )
= ( Inf @ ( image_nat_a @ D2 @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_417_Inf_OINF__cong,axiom,
! [A: set_nat,B: set_nat,C2: nat > nat,D2: nat > nat,Inf: set_nat > nat] :
( ( A = B )
=> ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( Inf @ ( image_nat_nat @ C2 @ A ) )
= ( Inf @ ( image_nat_nat @ D2 @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_418_infinite__surj,axiom,
! [A: set_a,F2: a > a,B: set_a] :
( ~ ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_a @ A @ ( image_a_a @ F2 @ B ) )
=> ~ ( finite_finite_a @ B ) ) ) ).
% infinite_surj
thf(fact_419_infinite__surj,axiom,
! [A: set_a,F2: nat > a,B: set_nat] :
( ~ ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_a @ A @ ( image_nat_a @ F2 @ B ) )
=> ~ ( finite_finite_nat @ B ) ) ) ).
% infinite_surj
thf(fact_420_infinite__surj,axiom,
! [A: set_nat,F2: a > nat,B: set_a] :
( ~ ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( image_a_nat @ F2 @ B ) )
=> ~ ( finite_finite_a @ B ) ) ) ).
% infinite_surj
thf(fact_421_infinite__surj,axiom,
! [A: set_nat,F2: nat > nat,B: set_nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( image_nat_nat @ F2 @ B ) )
=> ~ ( finite_finite_nat @ B ) ) ) ).
% infinite_surj
thf(fact_422_fun__upd__apply,axiom,
( fun_upd_nat_a
= ( ^ [F4: nat > a,X2: nat,Y3: a,Z4: nat] : ( if_a @ ( Z4 = X2 ) @ Y3 @ ( F4 @ Z4 ) ) ) ) ).
% fun_upd_apply
thf(fact_423_fun__upd__triv,axiom,
! [F2: nat > a,X3: nat] :
( ( fun_upd_nat_a @ F2 @ X3 @ ( F2 @ X3 ) )
= F2 ) ).
% fun_upd_triv
thf(fact_424_fun__upd__upd,axiom,
! [F2: nat > a,X3: nat,Y2: a,Z: a] :
( ( fun_upd_nat_a @ ( fun_upd_nat_a @ F2 @ X3 @ Y2 ) @ X3 @ Z )
= ( fun_upd_nat_a @ F2 @ X3 @ Z ) ) ).
% fun_upd_upd
thf(fact_425_dual__order_Orefl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_426_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_427_order__refl,axiom,
! [X3: set_nat] : ( ord_less_eq_set_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_428_order__refl,axiom,
! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_429_image__Collect__subsetI,axiom,
! [P: a > $o,F2: a > a,B: set_a] :
( ! [X: a] :
( ( P @ X )
=> ( member_a @ ( F2 @ X ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F2 @ ( collect_a @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_430_image__Collect__subsetI,axiom,
! [P: a > $o,F2: a > $o,B: set_o] :
( ! [X: a] :
( ( P @ X )
=> ( member_o @ ( F2 @ X ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_a_o @ F2 @ ( collect_a @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_431_image__Collect__subsetI,axiom,
! [P: nat > $o,F2: nat > a,B: set_a] :
( ! [X: nat] :
( ( P @ X )
=> ( member_a @ ( F2 @ X ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_nat_a @ F2 @ ( collect_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_432_image__Collect__subsetI,axiom,
! [P: nat > $o,F2: nat > $o,B: set_o] :
( ! [X: nat] :
( ( P @ X )
=> ( member_o @ ( F2 @ X ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_nat_o @ F2 @ ( collect_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_433_image__Collect__subsetI,axiom,
! [P: a > $o,F2: a > nat,B: set_nat] :
( ! [X: a] :
( ( P @ X )
=> ( member_nat @ ( F2 @ X ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_a_nat @ F2 @ ( collect_a @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_434_image__Collect__subsetI,axiom,
! [P: nat > $o,F2: nat > nat,B: set_nat] :
( ! [X: nat] :
( ( P @ X )
=> ( member_nat @ ( F2 @ X ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ ( collect_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_435_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_eq_nat @ N4 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_436_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_437_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_438_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_439_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ord_less_eq_a_o
@ ^ [X2: a] : ( member_a @ X2 @ A3 )
@ ^ [X2: a] : ( member_a @ X2 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_440_less__eq__set__def,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B3: set_o] :
( ord_less_eq_o_o
@ ^ [X2: $o] : ( member_o @ X2 @ A3 )
@ ^ [X2: $o] : ( member_o @ X2 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_441_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ord_less_eq_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A3 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_442_nle__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_443_le__cases3,axiom,
! [X3: nat,Y2: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X3 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Z ) )
=> ( ( ( ord_less_eq_nat @ X3 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X3 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X3 ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_444_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat,Z2: set_nat] : ( Y4 = Z2 ) )
= ( ^ [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
& ( ord_less_eq_set_nat @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_445_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_446_ord__eq__le__trans,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_447_ord__eq__le__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_448_ord__le__eq__trans,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_449_ord__le__eq__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_450_order__antisym,axiom,
! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_set_nat @ Y2 @ X3 )
=> ( X3 = Y2 ) ) ) ).
% order_antisym
thf(fact_451_order__antisym,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X3 )
=> ( X3 = Y2 ) ) ) ).
% order_antisym
thf(fact_452_order_Otrans,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_453_order_Otrans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_454_order__trans,axiom,
! [X3: set_nat,Y2: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_set_nat @ Y2 @ Z )
=> ( ord_less_eq_set_nat @ X3 @ Z ) ) ) ).
% order_trans
thf(fact_455_order__trans,axiom,
! [X3: nat,Y2: nat,Z: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z )
=> ( ord_less_eq_nat @ X3 @ Z ) ) ) ).
% order_trans
thf(fact_456_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A6: nat,B6: nat] :
( ( ord_less_eq_nat @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: nat,B6: nat] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_457_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_nat,Z2: set_nat] : ( Y4 = Z2 ) )
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A4 )
& ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_458_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_459_dual__order_Oantisym,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_460_dual__order_Oantisym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_461_dual__order_Otrans,axiom,
! [B2: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ B2 )
=> ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_462_dual__order_Otrans,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_463_antisym,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_464_antisym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_465_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat,Z2: set_nat] : ( Y4 = Z2 ) )
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_466_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_467_order__subst1,axiom,
! [A2: set_nat,F2: set_nat > set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y5 )
=> ( ord_less_eq_set_nat @ ( F2 @ X ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_468_order__subst1,axiom,
! [A2: set_nat,F2: nat > set_nat,B2: nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X: nat,Y5: nat] :
( ( ord_less_eq_nat @ X @ Y5 )
=> ( ord_less_eq_set_nat @ ( F2 @ X ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_469_order__subst1,axiom,
! [A2: nat,F2: set_nat > nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y5 )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_470_order__subst1,axiom,
! [A2: nat,F2: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X: nat,Y5: nat] :
( ( ord_less_eq_nat @ X @ Y5 )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_471_order__subst2,axiom,
! [A2: set_nat,B2: set_nat,F2: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ ( F2 @ B2 ) @ C )
=> ( ! [X: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y5 )
=> ( ord_less_eq_set_nat @ ( F2 @ X ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_eq_set_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_472_order__subst2,axiom,
! [A2: set_nat,B2: set_nat,F2: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F2 @ B2 ) @ C )
=> ( ! [X: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y5 )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_473_order__subst2,axiom,
! [A2: nat,B2: nat,F2: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ ( F2 @ B2 ) @ C )
=> ( ! [X: nat,Y5: nat] :
( ( ord_less_eq_nat @ X @ Y5 )
=> ( ord_less_eq_set_nat @ ( F2 @ X ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_eq_set_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_474_order__subst2,axiom,
! [A2: nat,B2: nat,F2: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F2 @ B2 ) @ C )
=> ( ! [X: nat,Y5: nat] :
( ( ord_less_eq_nat @ X @ Y5 )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_475_order__eq__refl,axiom,
! [X3: set_nat,Y2: set_nat] :
( ( X3 = Y2 )
=> ( ord_less_eq_set_nat @ X3 @ Y2 ) ) ).
% order_eq_refl
thf(fact_476_order__eq__refl,axiom,
! [X3: nat,Y2: nat] :
( ( X3 = Y2 )
=> ( ord_less_eq_nat @ X3 @ Y2 ) ) ).
% order_eq_refl
thf(fact_477_linorder__linear,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X3 ) ) ).
% linorder_linear
thf(fact_478_ord__eq__le__subst,axiom,
! [A2: set_nat,F2: set_nat > set_nat,B2: set_nat,C: set_nat] :
( ( A2
= ( F2 @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y5 )
=> ( ord_less_eq_set_nat @ ( F2 @ X ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_479_ord__eq__le__subst,axiom,
! [A2: nat,F2: set_nat > nat,B2: set_nat,C: set_nat] :
( ( A2
= ( F2 @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y5 )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_480_ord__eq__le__subst,axiom,
! [A2: set_nat,F2: nat > set_nat,B2: nat,C: nat] :
( ( A2
= ( F2 @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X: nat,Y5: nat] :
( ( ord_less_eq_nat @ X @ Y5 )
=> ( ord_less_eq_set_nat @ ( F2 @ X ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_481_ord__eq__le__subst,axiom,
! [A2: nat,F2: nat > nat,B2: nat,C: nat] :
( ( A2
= ( F2 @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X: nat,Y5: nat] :
( ( ord_less_eq_nat @ X @ Y5 )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_482_ord__le__eq__subst,axiom,
! [A2: set_nat,B2: set_nat,F2: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ( F2 @ B2 )
= C )
=> ( ! [X: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y5 )
=> ( ord_less_eq_set_nat @ ( F2 @ X ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_eq_set_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_483_ord__le__eq__subst,axiom,
! [A2: set_nat,B2: set_nat,F2: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ( F2 @ B2 )
= C )
=> ( ! [X: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y5 )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_484_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F2: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F2 @ B2 )
= C )
=> ( ! [X: nat,Y5: nat] :
( ( ord_less_eq_nat @ X @ Y5 )
=> ( ord_less_eq_set_nat @ ( F2 @ X ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_eq_set_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_485_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F2: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F2 @ B2 )
= C )
=> ( ! [X: nat,Y5: nat] :
( ( ord_less_eq_nat @ X @ Y5 )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_486_linorder__le__cases,axiom,
! [X3: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X3 ) ) ).
% linorder_le_cases
thf(fact_487_order__antisym__conv,axiom,
! [Y2: set_nat,X3: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X3 )
=> ( ( ord_less_eq_set_nat @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_488_order__antisym__conv,axiom,
! [Y2: nat,X3: nat] :
( ( ord_less_eq_nat @ Y2 @ X3 )
=> ( ( ord_less_eq_nat @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_489_fun__upd__def,axiom,
( fun_upd_nat_a
= ( ^ [F4: nat > a,A4: nat,B4: a,X2: nat] : ( if_a @ ( X2 = A4 ) @ B4 @ ( F4 @ X2 ) ) ) ) ).
% fun_upd_def
thf(fact_490_fun__upd__eqD,axiom,
! [F2: nat > a,X3: nat,Y2: a,G2: nat > a,Z: a] :
( ( ( fun_upd_nat_a @ F2 @ X3 @ Y2 )
= ( fun_upd_nat_a @ G2 @ X3 @ Z ) )
=> ( Y2 = Z ) ) ).
% fun_upd_eqD
thf(fact_491_fun__upd__idem,axiom,
! [F2: nat > a,X3: nat,Y2: a] :
( ( ( F2 @ X3 )
= Y2 )
=> ( ( fun_upd_nat_a @ F2 @ X3 @ Y2 )
= F2 ) ) ).
% fun_upd_idem
thf(fact_492_fun__upd__same,axiom,
! [F2: nat > a,X3: nat,Y2: a] :
( ( fun_upd_nat_a @ F2 @ X3 @ Y2 @ X3 )
= Y2 ) ).
% fun_upd_same
thf(fact_493_fun__upd__other,axiom,
! [Z: nat,X3: nat,F2: nat > a,Y2: a] :
( ( Z != X3 )
=> ( ( fun_upd_nat_a @ F2 @ X3 @ Y2 @ Z )
= ( F2 @ Z ) ) ) ).
% fun_upd_other
thf(fact_494_fun__upd__twist,axiom,
! [A2: nat,C: nat,M2: nat > a,B2: a,D: a] :
( ( A2 != C )
=> ( ( fun_upd_nat_a @ ( fun_upd_nat_a @ M2 @ A2 @ B2 ) @ C @ D )
= ( fun_upd_nat_a @ ( fun_upd_nat_a @ M2 @ C @ D ) @ A2 @ B2 ) ) ) ).
% fun_upd_twist
thf(fact_495_fun__upd__idem__iff,axiom,
! [F2: nat > a,X3: nat,Y2: a] :
( ( ( fun_upd_nat_a @ F2 @ X3 @ Y2 )
= F2 )
= ( ( F2 @ X3 )
= Y2 ) ) ).
% fun_upd_idem_iff
thf(fact_496_prop__restrict,axiom,
! [X3: $o,Z5: set_o,X5: set_o,P: $o > $o] :
( ( member_o @ X3 @ Z5 )
=> ( ( ord_less_eq_set_o @ Z5
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ X5 )
& ( P @ X2 ) ) ) )
=> ( P @ X3 ) ) ) ).
% prop_restrict
thf(fact_497_prop__restrict,axiom,
! [X3: a,Z5: set_a,X5: set_a,P: a > $o] :
( ( member_a @ X3 @ Z5 )
=> ( ( ord_less_eq_set_a @ Z5
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ X5 )
& ( P @ X2 ) ) ) )
=> ( P @ X3 ) ) ) ).
% prop_restrict
thf(fact_498_prop__restrict,axiom,
! [X3: nat,Z5: set_nat,X5: set_nat,P: nat > $o] :
( ( member_nat @ X3 @ Z5 )
=> ( ( ord_less_eq_set_nat @ Z5
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ X5 )
& ( P @ X2 ) ) ) )
=> ( P @ X3 ) ) ) ).
% prop_restrict
thf(fact_499_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_500_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_501_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_502_someI2__bex,axiom,
! [A: set_a,P: a > $o,Q: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ A )
& ( P @ X4 ) )
=> ( ! [X: a] :
( ( ( member_a @ X @ A )
& ( P @ X ) )
=> ( Q @ X ) )
=> ( Q
@ ( fChoice_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A )
& ( P @ X2 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_503_someI2__bex,axiom,
! [A: set_nat,P: nat > $o,Q: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( P @ X4 ) )
=> ( ! [X: nat] :
( ( ( member_nat @ X @ A )
& ( P @ X ) )
=> ( Q @ X ) )
=> ( Q
@ ( fChoice_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ X2 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_504_someI2__bex,axiom,
! [A: set_o,P: $o > $o,Q: $o > $o] :
( ? [X4: $o] :
( ( member_o @ X4 @ A )
& ( P @ X4 ) )
=> ( ! [X: $o] :
( ( ( member_o @ X @ A )
& ( P @ X ) )
=> ( Q @ X ) )
=> ( Q
@ ( fChoice_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A )
& ( P @ X2 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_505_ap__fresh__val,axiom,
! [Q: relational_fmla_a_b,Sigma: nat > a,X3: nat,I: product_prod_b_nat > set_list_a] :
( ( relational_ap_a_b @ Q )
=> ( ~ ( member_a @ ( Sigma @ X3 ) @ ( relational_adom_b_a @ I ) )
=> ( ~ ( member_a @ ( Sigma @ X3 ) @ ( relational_csts_a_b @ Q ) )
=> ( ( relational_sat_a_b @ Q @ I @ Sigma )
=> ~ ( member_nat @ X3 @ ( relational_fv_a_b @ Q ) ) ) ) ) ) ).
% ap_fresh_val
thf(fact_506_finite__less__ub,axiom,
! [F2: nat > nat,U: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( F2 @ N3 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_eq_nat @ ( F2 @ N4 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_507_bounded__Max__nat,axiom,
! [P: nat > $o,X3: nat,M: nat] :
( ( P @ X3 )
=> ( ! [X: nat] :
( ( P @ X )
=> ( ord_less_eq_nat @ X @ M ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_508_verit__la__disequality,axiom,
! [A2: nat,B2: nat] :
( ( A2 = B2 )
| ~ ( ord_less_eq_nat @ A2 @ B2 )
| ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% verit_la_disequality
thf(fact_509_verit__comp__simplify1_I2_J,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_510_verit__comp__simplify1_I2_J,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_511_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M4: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N5 )
=> ( ord_less_eq_nat @ X2 @ M4 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_512_qp__fresh__val,axiom,
! [Q: relational_fmla_a_b,Sigma: nat > a,X3: nat,I: product_prod_b_nat > set_list_a] :
( ( relational_qp_a_b @ Q )
=> ( ~ ( member_a @ ( Sigma @ X3 ) @ ( relational_adom_b_a @ I ) )
=> ( ~ ( member_a @ ( Sigma @ X3 ) @ ( relational_csts_a_b @ Q ) )
=> ( ( relational_sat_a_b @ Q @ I @ Sigma )
=> ~ ( member_nat @ X3 @ ( relational_fv_a_b @ Q ) ) ) ) ) ) ).
% qp_fresh_val
thf(fact_513_subset__Collect__iff,axiom,
! [B: set_o,A: set_o,P: $o > $o] :
( ( ord_less_eq_set_o @ B @ A )
=> ( ( ord_less_eq_set_o @ B
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A )
& ( P @ X2 ) ) ) )
= ( ! [X2: $o] :
( ( member_o @ X2 @ B )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_514_subset__Collect__iff,axiom,
! [B: set_a,A: set_a,P: a > $o] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ B
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A )
& ( P @ X2 ) ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_515_subset__Collect__iff,axiom,
! [B: set_nat,A: set_nat,P: nat > $o] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ B
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ X2 ) ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ B )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_516_subset__CollectI,axiom,
! [B: set_o,A: set_o,Q: $o > $o,P: $o > $o] :
( ( ord_less_eq_set_o @ B @ A )
=> ( ! [X: $o] :
( ( member_o @ X @ B )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_less_eq_set_o
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ B )
& ( Q @ X2 ) ) )
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_517_subset__CollectI,axiom,
! [B: set_a,A: set_a,Q: a > $o,P: a > $o] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ! [X: a] :
( ( member_a @ X @ B )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_less_eq_set_a
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ B )
& ( Q @ X2 ) ) )
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_518_subset__CollectI,axiom,
! [B: set_nat,A: set_nat,Q: nat > $o,P: nat > $o] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ B )
& ( Q @ X2 ) ) )
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_519_conj__subset__def,axiom,
! [A: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A
@ ( collect_a
@ ^ [X2: a] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) )
= ( ( ord_less_eq_set_a @ A @ ( collect_a @ P ) )
& ( ord_less_eq_set_a @ A @ ( collect_a @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_520_conj__subset__def,axiom,
! [A: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A
@ ( collect_nat
@ ^ [X2: nat] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) )
= ( ( ord_less_eq_set_nat @ A @ ( collect_nat @ P ) )
& ( ord_less_eq_set_nat @ A @ ( collect_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_521_Fpow__def,axiom,
( finite_Fpow_a
= ( ^ [A3: set_a] :
( collect_set_a
@ ^ [X6: set_a] :
( ( ord_less_eq_set_a @ X6 @ A3 )
& ( finite_finite_a @ X6 ) ) ) ) ) ).
% Fpow_def
thf(fact_522_Fpow__def,axiom,
( finite_Fpow_nat
= ( ^ [A3: set_nat] :
( collect_set_nat
@ ^ [X6: set_nat] :
( ( ord_less_eq_set_nat @ X6 @ A3 )
& ( finite_finite_nat @ X6 ) ) ) ) ) ).
% Fpow_def
thf(fact_523_surj__card__le,axiom,
! [A: set_a,B: set_a,F2: a > a] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_a @ B @ ( image_a_a @ F2 @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ B ) @ ( finite_card_a @ A ) ) ) ) ).
% surj_card_le
thf(fact_524_surj__card__le,axiom,
! [A: set_nat,B: set_a,F2: nat > a] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_a @ B @ ( image_nat_a @ F2 @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ B ) @ ( finite_card_nat @ A ) ) ) ) ).
% surj_card_le
thf(fact_525_surj__card__le,axiom,
! [A: set_a,B: set_nat,F2: a > nat] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_nat @ B @ ( image_a_nat @ F2 @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_a @ A ) ) ) ) ).
% surj_card_le
thf(fact_526_surj__card__le,axiom,
! [A: set_nat,B: set_nat,F2: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F2 @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_nat @ A ) ) ) ) ).
% surj_card_le
thf(fact_527_card__subset__eq,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( finite_card_a @ A )
= ( finite_card_a @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_528_card__subset__eq,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( finite_card_nat @ A )
= ( finite_card_nat @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_529_infinite__arbitrarily__large,axiom,
! [A: set_a,N2: nat] :
( ~ ( finite_finite_a @ A )
=> ? [B7: set_a] :
( ( finite_finite_a @ B7 )
& ( ( finite_card_a @ B7 )
= N2 )
& ( ord_less_eq_set_a @ B7 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_530_infinite__arbitrarily__large,axiom,
! [A: set_nat,N2: nat] :
( ~ ( finite_finite_nat @ A )
=> ? [B7: set_nat] :
( ( finite_finite_nat @ B7 )
& ( ( finite_card_nat @ B7 )
= N2 )
& ( ord_less_eq_set_nat @ B7 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_531_card__le__if__inj__on__rel,axiom,
! [B: set_o,A: set_a,R2: a > $o > $o] :
( ( finite_finite_o @ B )
=> ( ! [A6: a] :
( ( member_a @ A6 @ A )
=> ? [B8: $o] :
( ( member_o @ B8 @ B )
& ( R2 @ A6 @ B8 ) ) )
=> ( ! [A1: a,A22: a,B6: $o] :
( ( member_a @ A1 @ A )
=> ( ( member_a @ A22 @ A )
=> ( ( member_o @ B6 @ B )
=> ( ( R2 @ A1 @ B6 )
=> ( ( R2 @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_o @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_532_card__le__if__inj__on__rel,axiom,
! [B: set_o,A: set_nat,R2: nat > $o > $o] :
( ( finite_finite_o @ B )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ A )
=> ? [B8: $o] :
( ( member_o @ B8 @ B )
& ( R2 @ A6 @ B8 ) ) )
=> ( ! [A1: nat,A22: nat,B6: $o] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_o @ B6 @ B )
=> ( ( R2 @ A1 @ B6 )
=> ( ( R2 @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_o @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_533_card__le__if__inj__on__rel,axiom,
! [B: set_o,A: set_o,R2: $o > $o > $o] :
( ( finite_finite_o @ B )
=> ( ! [A6: $o] :
( ( member_o @ A6 @ A )
=> ? [B8: $o] :
( ( member_o @ B8 @ B )
& ( R2 @ A6 @ B8 ) ) )
=> ( ! [A1: $o,A22: $o,B6: $o] :
( ( member_o @ A1 @ A )
=> ( ( member_o @ A22 @ A )
=> ( ( member_o @ B6 @ B )
=> ( ( R2 @ A1 @ B6 )
=> ( ( R2 @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_o @ A ) @ ( finite_card_o @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_534_card__le__if__inj__on__rel,axiom,
! [B: set_a,A: set_a,R2: a > a > $o] :
( ( finite_finite_a @ B )
=> ( ! [A6: a] :
( ( member_a @ A6 @ A )
=> ? [B8: a] :
( ( member_a @ B8 @ B )
& ( R2 @ A6 @ B8 ) ) )
=> ( ! [A1: a,A22: a,B6: a] :
( ( member_a @ A1 @ A )
=> ( ( member_a @ A22 @ A )
=> ( ( member_a @ B6 @ B )
=> ( ( R2 @ A1 @ B6 )
=> ( ( R2 @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_535_card__le__if__inj__on__rel,axiom,
! [B: set_a,A: set_nat,R2: nat > a > $o] :
( ( finite_finite_a @ B )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ A )
=> ? [B8: a] :
( ( member_a @ B8 @ B )
& ( R2 @ A6 @ B8 ) ) )
=> ( ! [A1: nat,A22: nat,B6: a] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_a @ B6 @ B )
=> ( ( R2 @ A1 @ B6 )
=> ( ( R2 @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_a @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_536_card__le__if__inj__on__rel,axiom,
! [B: set_a,A: set_o,R2: $o > a > $o] :
( ( finite_finite_a @ B )
=> ( ! [A6: $o] :
( ( member_o @ A6 @ A )
=> ? [B8: a] :
( ( member_a @ B8 @ B )
& ( R2 @ A6 @ B8 ) ) )
=> ( ! [A1: $o,A22: $o,B6: a] :
( ( member_o @ A1 @ A )
=> ( ( member_o @ A22 @ A )
=> ( ( member_a @ B6 @ B )
=> ( ( R2 @ A1 @ B6 )
=> ( ( R2 @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_o @ A ) @ ( finite_card_a @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_537_card__le__if__inj__on__rel,axiom,
! [B: set_nat,A: set_a,R2: a > nat > $o] :
( ( finite_finite_nat @ B )
=> ( ! [A6: a] :
( ( member_a @ A6 @ A )
=> ? [B8: nat] :
( ( member_nat @ B8 @ B )
& ( R2 @ A6 @ B8 ) ) )
=> ( ! [A1: a,A22: a,B6: nat] :
( ( member_a @ A1 @ A )
=> ( ( member_a @ A22 @ A )
=> ( ( member_nat @ B6 @ B )
=> ( ( R2 @ A1 @ B6 )
=> ( ( R2 @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_538_card__le__if__inj__on__rel,axiom,
! [B: set_nat,A: set_nat,R2: nat > nat > $o] :
( ( finite_finite_nat @ B )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ A )
=> ? [B8: nat] :
( ( member_nat @ B8 @ B )
& ( R2 @ A6 @ B8 ) ) )
=> ( ! [A1: nat,A22: nat,B6: nat] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_nat @ B6 @ B )
=> ( ( R2 @ A1 @ B6 )
=> ( ( R2 @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_539_card__le__if__inj__on__rel,axiom,
! [B: set_nat,A: set_o,R2: $o > nat > $o] :
( ( finite_finite_nat @ B )
=> ( ! [A6: $o] :
( ( member_o @ A6 @ A )
=> ? [B8: nat] :
( ( member_nat @ B8 @ B )
& ( R2 @ A6 @ B8 ) ) )
=> ( ! [A1: $o,A22: $o,B6: nat] :
( ( member_o @ A1 @ A )
=> ( ( member_o @ A22 @ A )
=> ( ( member_nat @ B6 @ B )
=> ( ( R2 @ A1 @ B6 )
=> ( ( R2 @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_o @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_540_Fpow__mono,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_le6893508408891458716et_nat @ ( finite_Fpow_nat @ A ) @ ( finite_Fpow_nat @ B ) ) ) ).
% Fpow_mono
thf(fact_541_card__image__le,axiom,
! [A: set_a,F2: a > nat] :
( ( finite_finite_a @ A )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_a_nat @ F2 @ A ) ) @ ( finite_card_a @ A ) ) ) ).
% card_image_le
thf(fact_542_card__image__le,axiom,
! [A: set_a,F2: a > a] :
( ( finite_finite_a @ A )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( image_a_a @ F2 @ A ) ) @ ( finite_card_a @ A ) ) ) ).
% card_image_le
thf(fact_543_card__image__le,axiom,
! [A: set_nat,F2: nat > a] :
( ( finite_finite_nat @ A )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( image_nat_a @ F2 @ A ) ) @ ( finite_card_nat @ A ) ) ) ).
% card_image_le
thf(fact_544_card__image__le,axiom,
! [A: set_nat,F2: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_nat_nat @ F2 @ A ) ) @ ( finite_card_nat @ A ) ) ) ).
% card_image_le
thf(fact_545_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_a,C2: nat] :
( ! [G3: set_a] :
( ( ord_less_eq_set_a @ G3 @ F )
=> ( ( finite_finite_a @ G3 )
=> ( ord_less_eq_nat @ ( finite_card_a @ G3 ) @ C2 ) ) )
=> ( ( finite_finite_a @ F )
& ( ord_less_eq_nat @ ( finite_card_a @ F ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_546_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_nat,C2: nat] :
( ! [G3: set_nat] :
( ( ord_less_eq_set_nat @ G3 @ F )
=> ( ( finite_finite_nat @ G3 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G3 ) @ C2 ) ) )
=> ( ( finite_finite_nat @ F )
& ( ord_less_eq_nat @ ( finite_card_nat @ F ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_547_obtain__subset__with__card__n,axiom,
! [N2: nat,S: set_a] :
( ( ord_less_eq_nat @ N2 @ ( finite_card_a @ S ) )
=> ~ ! [T4: set_a] :
( ( ord_less_eq_set_a @ T4 @ S )
=> ( ( ( finite_card_a @ T4 )
= N2 )
=> ~ ( finite_finite_a @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_548_obtain__subset__with__card__n,axiom,
! [N2: nat,S: set_nat] :
( ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ S ) )
=> ~ ! [T4: set_nat] :
( ( ord_less_eq_set_nat @ T4 @ S )
=> ( ( ( finite_card_nat @ T4 )
= N2 )
=> ~ ( finite_finite_nat @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_549_exists__subset__between,axiom,
! [A: set_a,N2: nat,C2: set_a] :
( ( ord_less_eq_nat @ ( finite_card_a @ A ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( finite_card_a @ C2 ) )
=> ( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( finite_finite_a @ C2 )
=> ? [B7: set_a] :
( ( ord_less_eq_set_a @ A @ B7 )
& ( ord_less_eq_set_a @ B7 @ C2 )
& ( ( finite_card_a @ B7 )
= N2 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_550_exists__subset__between,axiom,
! [A: set_nat,N2: nat,C2: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ C2 ) )
=> ( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( finite_finite_nat @ C2 )
=> ? [B7: set_nat] :
( ( ord_less_eq_set_nat @ A @ B7 )
& ( ord_less_eq_set_nat @ B7 @ C2 )
& ( ( finite_card_nat @ B7 )
= N2 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_551_card__seteq,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ B ) @ ( finite_card_a @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_552_card__seteq,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_nat @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_553_card__mono,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B ) ) ) ) ).
% card_mono
thf(fact_554_card__mono,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ).
% card_mono
thf(fact_555_image__Fpow__mono,axiom,
! [F2: nat > a,A: set_nat,B: set_a] :
( ( ord_less_eq_set_a @ ( image_nat_a @ F2 @ A ) @ B )
=> ( ord_le3724670747650509150_set_a @ ( image_set_nat_set_a @ ( image_nat_a @ F2 ) @ ( finite_Fpow_nat @ A ) ) @ ( finite_Fpow_a @ B ) ) ) ).
% image_Fpow_mono
thf(fact_556_image__Fpow__mono,axiom,
! [F2: a > a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F2 @ A ) @ B )
=> ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ ( image_a_a @ F2 ) @ ( finite_Fpow_a @ A ) ) @ ( finite_Fpow_a @ B ) ) ) ).
% image_Fpow_mono
thf(fact_557_image__Fpow__mono,axiom,
! [F2: nat > nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A ) @ B )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F2 ) @ ( finite_Fpow_nat @ A ) ) @ ( finite_Fpow_nat @ B ) ) ) ).
% image_Fpow_mono
thf(fact_558_image__Fpow__mono,axiom,
! [F2: a > nat,A: set_a,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_a_nat @ F2 @ A ) @ B )
=> ( ord_le6893508408891458716et_nat @ ( image_set_a_set_nat @ ( image_a_nat @ F2 ) @ ( finite_Fpow_a @ A ) ) @ ( finite_Fpow_nat @ B ) ) ) ).
% image_Fpow_mono
thf(fact_559_inj__on__iff__card__le,axiom,
! [A: set_a,B: set_a] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B )
=> ( ( ? [F4: a > a] :
( ( inj_on_a_a @ F4 @ A )
& ( ord_less_eq_set_a @ ( image_a_a @ F4 @ A ) @ B ) ) )
= ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_560_inj__on__iff__card__le,axiom,
! [A: set_nat,B: set_a] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_a @ B )
=> ( ( ? [F4: nat > a] :
( ( inj_on_nat_a @ F4 @ A )
& ( ord_less_eq_set_a @ ( image_nat_a @ F4 @ A ) @ B ) ) )
= ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_a @ B ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_561_inj__on__iff__card__le,axiom,
! [A: set_a,B: set_nat] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( ? [F4: a > nat] :
( ( inj_on_a_nat @ F4 @ A )
& ( ord_less_eq_set_nat @ ( image_a_nat @ F4 @ A ) @ B ) ) )
= ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_562_inj__on__iff__card__le,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( ? [F4: nat > nat] :
( ( inj_on_nat_nat @ F4 @ A )
& ( ord_less_eq_set_nat @ ( image_nat_nat @ F4 @ A ) @ B ) ) )
= ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_563_card__inj__on__le,axiom,
! [F2: nat > a,A: set_nat,B: set_a] :
( ( inj_on_nat_a @ F2 @ A )
=> ( ( ord_less_eq_set_a @ ( image_nat_a @ F2 @ A ) @ B )
=> ( ( finite_finite_a @ B )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_a @ B ) ) ) ) ) ).
% card_inj_on_le
thf(fact_564_card__inj__on__le,axiom,
! [F2: a > a,A: set_a,B: set_a] :
( ( inj_on_a_a @ F2 @ A )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ F2 @ A ) @ B )
=> ( ( finite_finite_a @ B )
=> ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B ) ) ) ) ) ).
% card_inj_on_le
thf(fact_565_card__inj__on__le,axiom,
! [F2: nat > nat,A: set_nat,B: set_nat] :
( ( inj_on_nat_nat @ F2 @ A )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A ) @ B )
=> ( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_inj_on_le
thf(fact_566_card__inj__on__le,axiom,
! [F2: a > nat,A: set_a,B: set_nat] :
( ( inj_on_a_nat @ F2 @ A )
=> ( ( ord_less_eq_set_nat @ ( image_a_nat @ F2 @ A ) @ B )
=> ( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_inj_on_le
thf(fact_567_card__le__inj,axiom,
! [A: set_a,B: set_a] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B ) )
=> ? [F3: a > a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F3 @ A ) @ B )
& ( inj_on_a_a @ F3 @ A ) ) ) ) ) ).
% card_le_inj
thf(fact_568_card__le__inj,axiom,
! [A: set_nat,B: set_a] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_a @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_a @ B ) )
=> ? [F3: nat > a] :
( ( ord_less_eq_set_a @ ( image_nat_a @ F3 @ A ) @ B )
& ( inj_on_nat_a @ F3 @ A ) ) ) ) ) ).
% card_le_inj
thf(fact_569_card__le__inj,axiom,
! [A: set_a,B: set_nat] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_nat @ B ) )
=> ? [F3: a > nat] :
( ( ord_less_eq_set_nat @ ( image_a_nat @ F3 @ A ) @ B )
& ( inj_on_a_nat @ F3 @ A ) ) ) ) ) ).
% card_le_inj
thf(fact_570_card__le__inj,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) )
=> ? [F3: nat > nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ A ) @ B )
& ( inj_on_nat_nat @ F3 @ A ) ) ) ) ) ).
% card_le_inj
thf(fact_571_Relational__Calculus_Oequiv__def,axiom,
( relational_equiv_a_b
= ( ^ [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
! [I2: product_prod_b_nat > set_list_a,Sigma3: nat > a] :
( ( finite_finite_a @ ( relational_adom_b_a @ I2 ) )
=> ( ( relational_sat_a_b @ Q1 @ I2 @ Sigma3 )
= ( relational_sat_a_b @ Q22 @ I2 @ Sigma3 ) ) ) ) ) ).
% Relational_Calculus.equiv_def
thf(fact_572_surjective__iff__injective__gen,axiom,
! [S: set_a,T2: set_a,F2: a > a] :
( ( finite_finite_a @ S )
=> ( ( finite_finite_a @ T2 )
=> ( ( ( finite_card_a @ S )
= ( finite_card_a @ T2 ) )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ F2 @ S ) @ T2 )
=> ( ( ! [X2: a] :
( ( member_a @ X2 @ T2 )
=> ? [Y3: a] :
( ( member_a @ Y3 @ S )
& ( ( F2 @ Y3 )
= X2 ) ) ) )
= ( inj_on_a_a @ F2 @ S ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_573_surjective__iff__injective__gen,axiom,
! [S: set_nat,T2: set_a,F2: nat > a] :
( ( finite_finite_nat @ S )
=> ( ( finite_finite_a @ T2 )
=> ( ( ( finite_card_nat @ S )
= ( finite_card_a @ T2 ) )
=> ( ( ord_less_eq_set_a @ ( image_nat_a @ F2 @ S ) @ T2 )
=> ( ( ! [X2: a] :
( ( member_a @ X2 @ T2 )
=> ? [Y3: nat] :
( ( member_nat @ Y3 @ S )
& ( ( F2 @ Y3 )
= X2 ) ) ) )
= ( inj_on_nat_a @ F2 @ S ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_574_surjective__iff__injective__gen,axiom,
! [S: set_a,T2: set_nat,F2: a > nat] :
( ( finite_finite_a @ S )
=> ( ( finite_finite_nat @ T2 )
=> ( ( ( finite_card_a @ S )
= ( finite_card_nat @ T2 ) )
=> ( ( ord_less_eq_set_nat @ ( image_a_nat @ F2 @ S ) @ T2 )
=> ( ( ! [X2: nat] :
( ( member_nat @ X2 @ T2 )
=> ? [Y3: a] :
( ( member_a @ Y3 @ S )
& ( ( F2 @ Y3 )
= X2 ) ) ) )
= ( inj_on_a_nat @ F2 @ S ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_575_surjective__iff__injective__gen,axiom,
! [S: set_nat,T2: set_nat,F2: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( finite_finite_nat @ T2 )
=> ( ( ( finite_card_nat @ S )
= ( finite_card_nat @ T2 ) )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ S ) @ T2 )
=> ( ( ! [X2: nat] :
( ( member_nat @ X2 @ T2 )
=> ? [Y3: nat] :
( ( member_nat @ Y3 @ S )
& ( ( F2 @ Y3 )
= X2 ) ) ) )
= ( inj_on_nat_nat @ F2 @ S ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_576_card__bij__eq,axiom,
! [F2: a > a,A: set_a,B: set_a,G2: a > a] :
( ( inj_on_a_a @ F2 @ A )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ F2 @ A ) @ B )
=> ( ( inj_on_a_a @ G2 @ B )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ G2 @ B ) @ A )
=> ( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B )
=> ( ( finite_card_a @ A )
= ( finite_card_a @ B ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_577_card__bij__eq,axiom,
! [F2: nat > a,A: set_nat,B: set_a,G2: a > nat] :
( ( inj_on_nat_a @ F2 @ A )
=> ( ( ord_less_eq_set_a @ ( image_nat_a @ F2 @ A ) @ B )
=> ( ( inj_on_a_nat @ G2 @ B )
=> ( ( ord_less_eq_set_nat @ ( image_a_nat @ G2 @ B ) @ A )
=> ( ( finite_finite_nat @ A )
=> ( ( finite_finite_a @ B )
=> ( ( finite_card_nat @ A )
= ( finite_card_a @ B ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_578_card__bij__eq,axiom,
! [F2: a > nat,A: set_a,B: set_nat,G2: nat > a] :
( ( inj_on_a_nat @ F2 @ A )
=> ( ( ord_less_eq_set_nat @ ( image_a_nat @ F2 @ A ) @ B )
=> ( ( inj_on_nat_a @ G2 @ B )
=> ( ( ord_less_eq_set_a @ ( image_nat_a @ G2 @ B ) @ A )
=> ( ( finite_finite_a @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( finite_card_a @ A )
= ( finite_card_nat @ B ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_579_card__bij__eq,axiom,
! [F2: nat > nat,A: set_nat,B: set_nat,G2: nat > nat] :
( ( inj_on_nat_nat @ F2 @ A )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A ) @ B )
=> ( ( inj_on_nat_nat @ G2 @ B )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ G2 @ B ) @ A )
=> ( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( finite_card_nat @ A )
= ( finite_card_nat @ B ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_580_inj__on__image__Fpow,axiom,
! [F2: nat > a,A: set_nat] :
( ( inj_on_nat_a @ F2 @ A )
=> ( inj_on_set_nat_set_a @ ( image_nat_a @ F2 ) @ ( finite_Fpow_nat @ A ) ) ) ).
% inj_on_image_Fpow
thf(fact_581_inj__on__image__Fpow,axiom,
! [F2: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F2 @ A )
=> ( inj_on4604407203859583615et_nat @ ( image_nat_nat @ F2 ) @ ( finite_Fpow_nat @ A ) ) ) ).
% inj_on_image_Fpow
thf(fact_582_inj__on__image__Fpow,axiom,
! [F2: a > nat,A: set_a] :
( ( inj_on_a_nat @ F2 @ A )
=> ( inj_on_set_a_set_nat @ ( image_a_nat @ F2 ) @ ( finite_Fpow_a @ A ) ) ) ).
% inj_on_image_Fpow
thf(fact_583_inj__on__image__Fpow,axiom,
! [F2: a > a,A: set_a] :
( ( inj_on_a_a @ F2 @ A )
=> ( inj_on_set_a_set_a @ ( image_a_a @ F2 ) @ ( finite_Fpow_a @ A ) ) ) ).
% inj_on_image_Fpow
thf(fact_584_finite__inverse__image__gen,axiom,
! [A: set_o,F2: $o > $o,D2: set_o] :
( ( finite_finite_o @ A )
=> ( ( inj_on_o_o @ F2 @ D2 )
=> ( finite_finite_o
@ ( collect_o
@ ^ [J: $o] :
( ( member_o @ J @ D2 )
& ( member_o @ ( F2 @ J ) @ A ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_585_finite__inverse__image__gen,axiom,
! [A: set_o,F2: a > $o,D2: set_a] :
( ( finite_finite_o @ A )
=> ( ( inj_on_a_o @ F2 @ D2 )
=> ( finite_finite_a
@ ( collect_a
@ ^ [J: a] :
( ( member_a @ J @ D2 )
& ( member_o @ ( F2 @ J ) @ A ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_586_finite__inverse__image__gen,axiom,
! [A: set_o,F2: nat > $o,D2: set_nat] :
( ( finite_finite_o @ A )
=> ( ( inj_on_nat_o @ F2 @ D2 )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [J: nat] :
( ( member_nat @ J @ D2 )
& ( member_o @ ( F2 @ J ) @ A ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_587_finite__inverse__image__gen,axiom,
! [A: set_a,F2: $o > a,D2: set_o] :
( ( finite_finite_a @ A )
=> ( ( inj_on_o_a @ F2 @ D2 )
=> ( finite_finite_o
@ ( collect_o
@ ^ [J: $o] :
( ( member_o @ J @ D2 )
& ( member_a @ ( F2 @ J ) @ A ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_588_finite__inverse__image__gen,axiom,
! [A: set_a,F2: a > a,D2: set_a] :
( ( finite_finite_a @ A )
=> ( ( inj_on_a_a @ F2 @ D2 )
=> ( finite_finite_a
@ ( collect_a
@ ^ [J: a] :
( ( member_a @ J @ D2 )
& ( member_a @ ( F2 @ J ) @ A ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_589_finite__inverse__image__gen,axiom,
! [A: set_a,F2: nat > a,D2: set_nat] :
( ( finite_finite_a @ A )
=> ( ( inj_on_nat_a @ F2 @ D2 )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [J: nat] :
( ( member_nat @ J @ D2 )
& ( member_a @ ( F2 @ J ) @ A ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_590_finite__inverse__image__gen,axiom,
! [A: set_nat,F2: $o > nat,D2: set_o] :
( ( finite_finite_nat @ A )
=> ( ( inj_on_o_nat @ F2 @ D2 )
=> ( finite_finite_o
@ ( collect_o
@ ^ [J: $o] :
( ( member_o @ J @ D2 )
& ( member_nat @ ( F2 @ J ) @ A ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_591_finite__inverse__image__gen,axiom,
! [A: set_nat,F2: a > nat,D2: set_a] :
( ( finite_finite_nat @ A )
=> ( ( inj_on_a_nat @ F2 @ D2 )
=> ( finite_finite_a
@ ( collect_a
@ ^ [J: a] :
( ( member_a @ J @ D2 )
& ( member_nat @ ( F2 @ J ) @ A ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_592_finite__inverse__image__gen,axiom,
! [A: set_nat,F2: nat > nat,D2: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( inj_on_nat_nat @ F2 @ D2 )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [J: nat] :
( ( member_nat @ J @ D2 )
& ( member_nat @ ( F2 @ J ) @ A ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_593_finite__imageD,axiom,
! [F2: a > a,A: set_a] :
( ( finite_finite_a @ ( image_a_a @ F2 @ A ) )
=> ( ( inj_on_a_a @ F2 @ A )
=> ( finite_finite_a @ A ) ) ) ).
% finite_imageD
thf(fact_594_finite__imageD,axiom,
! [F2: nat > a,A: set_nat] :
( ( finite_finite_a @ ( image_nat_a @ F2 @ A ) )
=> ( ( inj_on_nat_a @ F2 @ A )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_imageD
thf(fact_595_finite__imageD,axiom,
! [F2: a > nat,A: set_a] :
( ( finite_finite_nat @ ( image_a_nat @ F2 @ A ) )
=> ( ( inj_on_a_nat @ F2 @ A )
=> ( finite_finite_a @ A ) ) ) ).
% finite_imageD
thf(fact_596_finite__imageD,axiom,
! [F2: nat > nat,A: set_nat] :
( ( finite_finite_nat @ ( image_nat_nat @ F2 @ A ) )
=> ( ( inj_on_nat_nat @ F2 @ A )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_imageD
thf(fact_597_finite__image__iff,axiom,
! [F2: a > a,A: set_a] :
( ( inj_on_a_a @ F2 @ A )
=> ( ( finite_finite_a @ ( image_a_a @ F2 @ A ) )
= ( finite_finite_a @ A ) ) ) ).
% finite_image_iff
thf(fact_598_finite__image__iff,axiom,
! [F2: nat > a,A: set_nat] :
( ( inj_on_nat_a @ F2 @ A )
=> ( ( finite_finite_a @ ( image_nat_a @ F2 @ A ) )
= ( finite_finite_nat @ A ) ) ) ).
% finite_image_iff
thf(fact_599_finite__image__iff,axiom,
! [F2: a > nat,A: set_a] :
( ( inj_on_a_nat @ F2 @ A )
=> ( ( finite_finite_nat @ ( image_a_nat @ F2 @ A ) )
= ( finite_finite_a @ A ) ) ) ).
% finite_image_iff
thf(fact_600_finite__image__iff,axiom,
! [F2: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F2 @ A )
=> ( ( finite_finite_nat @ ( image_nat_nat @ F2 @ A ) )
= ( finite_finite_nat @ A ) ) ) ).
% finite_image_iff
thf(fact_601_inj__on__image__eq__iff,axiom,
! [F2: a > nat,C2: set_a,A: set_a,B: set_a] :
( ( inj_on_a_nat @ F2 @ C2 )
=> ( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ( ( image_a_nat @ F2 @ A )
= ( image_a_nat @ F2 @ B ) )
= ( A = B ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_602_inj__on__image__eq__iff,axiom,
! [F2: a > a,C2: set_a,A: set_a,B: set_a] :
( ( inj_on_a_a @ F2 @ C2 )
=> ( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ( ( image_a_a @ F2 @ A )
= ( image_a_a @ F2 @ B ) )
= ( A = B ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_603_inj__on__image__eq__iff,axiom,
! [F2: nat > a,C2: set_nat,A: set_nat,B: set_nat] :
( ( inj_on_nat_a @ F2 @ C2 )
=> ( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ( ( image_nat_a @ F2 @ A )
= ( image_nat_a @ F2 @ B ) )
= ( A = B ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_604_inj__on__image__eq__iff,axiom,
! [F2: nat > nat,C2: set_nat,A: set_nat,B: set_nat] :
( ( inj_on_nat_nat @ F2 @ C2 )
=> ( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ( ( image_nat_nat @ F2 @ A )
= ( image_nat_nat @ F2 @ B ) )
= ( A = B ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_605_inj__on__image__mem__iff,axiom,
! [F2: a > a,B: set_a,A2: a,A: set_a] :
( ( inj_on_a_a @ F2 @ B )
=> ( ( member_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ ( F2 @ A2 ) @ ( image_a_a @ F2 @ A ) )
= ( member_a @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_606_inj__on__image__mem__iff,axiom,
! [F2: a > nat,B: set_a,A2: a,A: set_a] :
( ( inj_on_a_nat @ F2 @ B )
=> ( ( member_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_nat @ ( F2 @ A2 ) @ ( image_a_nat @ F2 @ A ) )
= ( member_a @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_607_inj__on__image__mem__iff,axiom,
! [F2: a > $o,B: set_a,A2: a,A: set_a] :
( ( inj_on_a_o @ F2 @ B )
=> ( ( member_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_o @ ( F2 @ A2 ) @ ( image_a_o @ F2 @ A ) )
= ( member_a @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_608_inj__on__image__mem__iff,axiom,
! [F2: $o > a,B: set_o,A2: $o,A: set_o] :
( ( inj_on_o_a @ F2 @ B )
=> ( ( member_o @ A2 @ B )
=> ( ( ord_less_eq_set_o @ A @ B )
=> ( ( member_a @ ( F2 @ A2 ) @ ( image_o_a @ F2 @ A ) )
= ( member_o @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_609_inj__on__image__mem__iff,axiom,
! [F2: $o > nat,B: set_o,A2: $o,A: set_o] :
( ( inj_on_o_nat @ F2 @ B )
=> ( ( member_o @ A2 @ B )
=> ( ( ord_less_eq_set_o @ A @ B )
=> ( ( member_nat @ ( F2 @ A2 ) @ ( image_o_nat @ F2 @ A ) )
= ( member_o @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_610_inj__on__image__mem__iff,axiom,
! [F2: $o > $o,B: set_o,A2: $o,A: set_o] :
( ( inj_on_o_o @ F2 @ B )
=> ( ( member_o @ A2 @ B )
=> ( ( ord_less_eq_set_o @ A @ B )
=> ( ( member_o @ ( F2 @ A2 ) @ ( image_o_o @ F2 @ A ) )
= ( member_o @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_611_inj__on__image__mem__iff,axiom,
! [F2: nat > a,B: set_nat,A2: nat,A: set_nat] :
( ( inj_on_nat_a @ F2 @ B )
=> ( ( member_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_a @ ( F2 @ A2 ) @ ( image_nat_a @ F2 @ A ) )
= ( member_nat @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_612_inj__on__image__mem__iff,axiom,
! [F2: nat > nat,B: set_nat,A2: nat,A: set_nat] :
( ( inj_on_nat_nat @ F2 @ B )
=> ( ( member_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ ( F2 @ A2 ) @ ( image_nat_nat @ F2 @ A ) )
= ( member_nat @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_613_inj__on__image__mem__iff,axiom,
! [F2: nat > $o,B: set_nat,A2: nat,A: set_nat] :
( ( inj_on_nat_o @ F2 @ B )
=> ( ( member_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_o @ ( F2 @ A2 ) @ ( image_nat_o @ F2 @ A ) )
= ( member_nat @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_614_subset__image__inj,axiom,
! [S: set_a,F2: a > a,T2: set_a] :
( ( ord_less_eq_set_a @ S @ ( image_a_a @ F2 @ T2 ) )
= ( ? [U2: set_a] :
( ( ord_less_eq_set_a @ U2 @ T2 )
& ( inj_on_a_a @ F2 @ U2 )
& ( S
= ( image_a_a @ F2 @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_615_subset__image__inj,axiom,
! [S: set_a,F2: nat > a,T2: set_nat] :
( ( ord_less_eq_set_a @ S @ ( image_nat_a @ F2 @ T2 ) )
= ( ? [U2: set_nat] :
( ( ord_less_eq_set_nat @ U2 @ T2 )
& ( inj_on_nat_a @ F2 @ U2 )
& ( S
= ( image_nat_a @ F2 @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_616_subset__image__inj,axiom,
! [S: set_nat,F2: a > nat,T2: set_a] :
( ( ord_less_eq_set_nat @ S @ ( image_a_nat @ F2 @ T2 ) )
= ( ? [U2: set_a] :
( ( ord_less_eq_set_a @ U2 @ T2 )
& ( inj_on_a_nat @ F2 @ U2 )
& ( S
= ( image_a_nat @ F2 @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_617_subset__image__inj,axiom,
! [S: set_nat,F2: nat > nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S @ ( image_nat_nat @ F2 @ T2 ) )
= ( ? [U2: set_nat] :
( ( ord_less_eq_set_nat @ U2 @ T2 )
& ( inj_on_nat_nat @ F2 @ U2 )
& ( S
= ( image_nat_nat @ F2 @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_618_inj__on__Un__image__eq__iff,axiom,
! [F2: a > nat,A: set_a,B: set_a] :
( ( inj_on_a_nat @ F2 @ ( sup_sup_set_a @ A @ B ) )
=> ( ( ( image_a_nat @ F2 @ A )
= ( image_a_nat @ F2 @ B ) )
= ( A = B ) ) ) ).
% inj_on_Un_image_eq_iff
thf(fact_619_inj__on__Un__image__eq__iff,axiom,
! [F2: a > a,A: set_a,B: set_a] :
( ( inj_on_a_a @ F2 @ ( sup_sup_set_a @ A @ B ) )
=> ( ( ( image_a_a @ F2 @ A )
= ( image_a_a @ F2 @ B ) )
= ( A = B ) ) ) ).
% inj_on_Un_image_eq_iff
thf(fact_620_inj__on__Un__image__eq__iff,axiom,
! [F2: nat > a,A: set_nat,B: set_nat] :
( ( inj_on_nat_a @ F2 @ ( sup_sup_set_nat @ A @ B ) )
=> ( ( ( image_nat_a @ F2 @ A )
= ( image_nat_a @ F2 @ B ) )
= ( A = B ) ) ) ).
% inj_on_Un_image_eq_iff
thf(fact_621_inj__on__Un__image__eq__iff,axiom,
! [F2: nat > nat,A: set_nat,B: set_nat] :
( ( inj_on_nat_nat @ F2 @ ( sup_sup_set_nat @ A @ B ) )
=> ( ( ( image_nat_nat @ F2 @ A )
= ( image_nat_nat @ F2 @ B ) )
= ( A = B ) ) ) ).
% inj_on_Un_image_eq_iff
thf(fact_622_card__image,axiom,
! [F2: nat > a,A: set_nat] :
( ( inj_on_nat_a @ F2 @ A )
=> ( ( finite_card_a @ ( image_nat_a @ F2 @ A ) )
= ( finite_card_nat @ A ) ) ) ).
% card_image
thf(fact_623_card__image,axiom,
! [F2: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F2 @ A )
=> ( ( finite_card_nat @ ( image_nat_nat @ F2 @ A ) )
= ( finite_card_nat @ A ) ) ) ).
% card_image
thf(fact_624_card__image,axiom,
! [F2: a > nat,A: set_a] :
( ( inj_on_a_nat @ F2 @ A )
=> ( ( finite_card_nat @ ( image_a_nat @ F2 @ A ) )
= ( finite_card_a @ A ) ) ) ).
% card_image
thf(fact_625_card__image,axiom,
! [F2: a > a,A: set_a] :
( ( inj_on_a_a @ F2 @ A )
=> ( ( finite_card_a @ ( image_a_a @ F2 @ A ) )
= ( finite_card_a @ A ) ) ) ).
% card_image
thf(fact_626_inj__on__fun__updI,axiom,
! [F2: a > a,A: set_a,Y2: a,X3: a] :
( ( inj_on_a_a @ F2 @ A )
=> ( ~ ( member_a @ Y2 @ ( image_a_a @ F2 @ A ) )
=> ( inj_on_a_a @ ( fun_upd_a_a @ F2 @ X3 @ Y2 ) @ A ) ) ) ).
% inj_on_fun_updI
thf(fact_627_inj__on__fun__updI,axiom,
! [F2: nat > nat,A: set_nat,Y2: nat,X3: nat] :
( ( inj_on_nat_nat @ F2 @ A )
=> ( ~ ( member_nat @ Y2 @ ( image_nat_nat @ F2 @ A ) )
=> ( inj_on_nat_nat @ ( fun_upd_nat_nat @ F2 @ X3 @ Y2 ) @ A ) ) ) ).
% inj_on_fun_updI
thf(fact_628_inj__on__fun__updI,axiom,
! [F2: a > nat,A: set_a,Y2: nat,X3: a] :
( ( inj_on_a_nat @ F2 @ A )
=> ( ~ ( member_nat @ Y2 @ ( image_a_nat @ F2 @ A ) )
=> ( inj_on_a_nat @ ( fun_upd_a_nat @ F2 @ X3 @ Y2 ) @ A ) ) ) ).
% inj_on_fun_updI
thf(fact_629_inj__on__fun__updI,axiom,
! [F2: nat > a,A: set_nat,Y2: a,X3: nat] :
( ( inj_on_nat_a @ F2 @ A )
=> ( ~ ( member_a @ Y2 @ ( image_nat_a @ F2 @ A ) )
=> ( inj_on_nat_a @ ( fun_upd_nat_a @ F2 @ X3 @ Y2 ) @ A ) ) ) ).
% inj_on_fun_updI
thf(fact_630_endo__inj__surj,axiom,
! [A: set_a,F2: a > a] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ F2 @ A ) @ A )
=> ( ( inj_on_a_a @ F2 @ A )
=> ( ( image_a_a @ F2 @ A )
= A ) ) ) ) ).
% endo_inj_surj
thf(fact_631_endo__inj__surj,axiom,
! [A: set_nat,F2: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A ) @ A )
=> ( ( inj_on_nat_nat @ F2 @ A )
=> ( ( image_nat_nat @ F2 @ A )
= A ) ) ) ) ).
% endo_inj_surj
thf(fact_632_inj__on__finite,axiom,
! [F2: a > a,A: set_a,B: set_a] :
( ( inj_on_a_a @ F2 @ A )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ F2 @ A ) @ B )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ A ) ) ) ) ).
% inj_on_finite
thf(fact_633_inj__on__finite,axiom,
! [F2: nat > a,A: set_nat,B: set_a] :
( ( inj_on_nat_a @ F2 @ A )
=> ( ( ord_less_eq_set_a @ ( image_nat_a @ F2 @ A ) @ B )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_nat @ A ) ) ) ) ).
% inj_on_finite
thf(fact_634_inj__on__finite,axiom,
! [F2: a > nat,A: set_a,B: set_nat] :
( ( inj_on_a_nat @ F2 @ A )
=> ( ( ord_less_eq_set_nat @ ( image_a_nat @ F2 @ A ) @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_a @ A ) ) ) ) ).
% inj_on_finite
thf(fact_635_inj__on__finite,axiom,
! [F2: nat > nat,A: set_nat,B: set_nat] :
( ( inj_on_nat_nat @ F2 @ A )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A ) @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A ) ) ) ) ).
% inj_on_finite
thf(fact_636_finite__surj__inj,axiom,
! [A: set_a,F2: a > a] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_a @ A @ ( image_a_a @ F2 @ A ) )
=> ( inj_on_a_a @ F2 @ A ) ) ) ).
% finite_surj_inj
thf(fact_637_finite__surj__inj,axiom,
! [A: set_nat,F2: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( image_nat_nat @ F2 @ A ) )
=> ( inj_on_nat_nat @ F2 @ A ) ) ) ).
% finite_surj_inj
thf(fact_638_eq__card__imp__inj__on,axiom,
! [A: set_a,F2: a > nat] :
( ( finite_finite_a @ A )
=> ( ( ( finite_card_nat @ ( image_a_nat @ F2 @ A ) )
= ( finite_card_a @ A ) )
=> ( inj_on_a_nat @ F2 @ A ) ) ) ).
% eq_card_imp_inj_on
thf(fact_639_eq__card__imp__inj__on,axiom,
! [A: set_a,F2: a > a] :
( ( finite_finite_a @ A )
=> ( ( ( finite_card_a @ ( image_a_a @ F2 @ A ) )
= ( finite_card_a @ A ) )
=> ( inj_on_a_a @ F2 @ A ) ) ) ).
% eq_card_imp_inj_on
thf(fact_640_eq__card__imp__inj__on,axiom,
! [A: set_nat,F2: nat > a] :
( ( finite_finite_nat @ A )
=> ( ( ( finite_card_a @ ( image_nat_a @ F2 @ A ) )
= ( finite_card_nat @ A ) )
=> ( inj_on_nat_a @ F2 @ A ) ) ) ).
% eq_card_imp_inj_on
thf(fact_641_eq__card__imp__inj__on,axiom,
! [A: set_nat,F2: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ( finite_card_nat @ ( image_nat_nat @ F2 @ A ) )
= ( finite_card_nat @ A ) )
=> ( inj_on_nat_nat @ F2 @ A ) ) ) ).
% eq_card_imp_inj_on
thf(fact_642_inj__on__iff__eq__card,axiom,
! [A: set_a,F2: a > nat] :
( ( finite_finite_a @ A )
=> ( ( inj_on_a_nat @ F2 @ A )
= ( ( finite_card_nat @ ( image_a_nat @ F2 @ A ) )
= ( finite_card_a @ A ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_643_inj__on__iff__eq__card,axiom,
! [A: set_a,F2: a > a] :
( ( finite_finite_a @ A )
=> ( ( inj_on_a_a @ F2 @ A )
= ( ( finite_card_a @ ( image_a_a @ F2 @ A ) )
= ( finite_card_a @ A ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_644_inj__on__iff__eq__card,axiom,
! [A: set_nat,F2: nat > a] :
( ( finite_finite_nat @ A )
=> ( ( inj_on_nat_a @ F2 @ A )
= ( ( finite_card_a @ ( image_nat_a @ F2 @ A ) )
= ( finite_card_nat @ A ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_645_inj__on__iff__eq__card,axiom,
! [A: set_nat,F2: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( inj_on_nat_nat @ F2 @ A )
= ( ( finite_card_nat @ ( image_nat_nat @ F2 @ A ) )
= ( finite_card_nat @ A ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_646_equiv__eval__eqI,axiom,
! [I: product_prod_b_nat > set_list_a,Q: relational_fmla_a_b,Q3: relational_fmla_a_b] :
( ( finite_finite_a @ ( relational_adom_b_a @ I ) )
=> ( ( ( relational_fv_a_b @ Q )
= ( relational_fv_a_b @ Q3 ) )
=> ( ( relational_equiv_a_b @ Q @ Q3 )
=> ( ( relational_eval_a_b @ Q @ I )
= ( relational_eval_a_b @ Q3 @ I ) ) ) ) ) ).
% equiv_eval_eqI
thf(fact_647_equiv__eval__on__eqI,axiom,
! [I: product_prod_b_nat > set_list_a,Q: relational_fmla_a_b,Q3: relational_fmla_a_b,X5: set_nat] :
( ( finite_finite_a @ ( relational_adom_b_a @ I ) )
=> ( ( relational_equiv_a_b @ Q @ Q3 )
=> ( ( relati8814510239606734169on_a_b @ X5 @ Q @ I )
= ( relati8814510239606734169on_a_b @ X5 @ Q3 @ I ) ) ) ) ).
% equiv_eval_on_eqI
thf(fact_648_equiv__eval__on__eval__eqI,axiom,
! [I: product_prod_b_nat > set_list_a,Q: relational_fmla_a_b,Q3: relational_fmla_a_b] :
( ( finite_finite_a @ ( relational_adom_b_a @ I ) )
=> ( ( ord_less_eq_set_nat @ ( relational_fv_a_b @ Q ) @ ( relational_fv_a_b @ Q3 ) )
=> ( ( relational_equiv_a_b @ Q @ Q3 )
=> ( ( relati8814510239606734169on_a_b @ ( relational_fv_a_b @ Q3 ) @ Q @ I )
= ( relational_eval_a_b @ Q3 @ I ) ) ) ) ) ).
% equiv_eval_on_eval_eqI
thf(fact_649_image__Pow__mono,axiom,
! [F2: nat > a,A: set_nat,B: set_a] :
( ( ord_less_eq_set_a @ ( image_nat_a @ F2 @ A ) @ B )
=> ( ord_le3724670747650509150_set_a @ ( image_set_nat_set_a @ ( image_nat_a @ F2 ) @ ( pow_nat @ A ) ) @ ( pow_a @ B ) ) ) ).
% image_Pow_mono
thf(fact_650_image__Pow__mono,axiom,
! [F2: a > a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F2 @ A ) @ B )
=> ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ ( image_a_a @ F2 ) @ ( pow_a @ A ) ) @ ( pow_a @ B ) ) ) ).
% image_Pow_mono
thf(fact_651_image__Pow__mono,axiom,
! [F2: nat > nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A ) @ B )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F2 ) @ ( pow_nat @ A ) ) @ ( pow_nat @ B ) ) ) ).
% image_Pow_mono
thf(fact_652_image__Pow__mono,axiom,
! [F2: a > nat,A: set_a,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_a_nat @ F2 @ A ) @ B )
=> ( ord_le6893508408891458716et_nat @ ( image_set_a_set_nat @ ( image_a_nat @ F2 ) @ ( pow_a @ A ) ) @ ( pow_nat @ B ) ) ) ).
% image_Pow_mono
thf(fact_653_the__inv__into__into,axiom,
! [F2: a > a,A: set_a,X3: a,B: set_a] :
( ( inj_on_a_a @ F2 @ A )
=> ( ( member_a @ X3 @ ( image_a_a @ F2 @ A ) )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( member_a @ ( the_inv_into_a_a @ A @ F2 @ X3 ) @ B ) ) ) ) ).
% the_inv_into_into
thf(fact_654_the__inv__into__into,axiom,
! [F2: $o > a,A: set_o,X3: a,B: set_o] :
( ( inj_on_o_a @ F2 @ A )
=> ( ( member_a @ X3 @ ( image_o_a @ F2 @ A ) )
=> ( ( ord_less_eq_set_o @ A @ B )
=> ( member_o @ ( the_inv_into_o_a @ A @ F2 @ X3 ) @ B ) ) ) ) ).
% the_inv_into_into
thf(fact_655_the__inv__into__into,axiom,
! [F2: a > nat,A: set_a,X3: nat,B: set_a] :
( ( inj_on_a_nat @ F2 @ A )
=> ( ( member_nat @ X3 @ ( image_a_nat @ F2 @ A ) )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( member_a @ ( the_inv_into_a_nat @ A @ F2 @ X3 ) @ B ) ) ) ) ).
% the_inv_into_into
thf(fact_656_the__inv__into__into,axiom,
! [F2: $o > nat,A: set_o,X3: nat,B: set_o] :
( ( inj_on_o_nat @ F2 @ A )
=> ( ( member_nat @ X3 @ ( image_o_nat @ F2 @ A ) )
=> ( ( ord_less_eq_set_o @ A @ B )
=> ( member_o @ ( the_inv_into_o_nat @ A @ F2 @ X3 ) @ B ) ) ) ) ).
% the_inv_into_into
thf(fact_657_the__inv__into__into,axiom,
! [F2: a > $o,A: set_a,X3: $o,B: set_a] :
( ( inj_on_a_o @ F2 @ A )
=> ( ( member_o @ X3 @ ( image_a_o @ F2 @ A ) )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( member_a @ ( the_inv_into_a_o @ A @ F2 @ X3 ) @ B ) ) ) ) ).
% the_inv_into_into
thf(fact_658_the__inv__into__into,axiom,
! [F2: $o > $o,A: set_o,X3: $o,B: set_o] :
( ( inj_on_o_o @ F2 @ A )
=> ( ( member_o @ X3 @ ( image_o_o @ F2 @ A ) )
=> ( ( ord_less_eq_set_o @ A @ B )
=> ( member_o @ ( the_inv_into_o_o @ A @ F2 @ X3 ) @ B ) ) ) ) ).
% the_inv_into_into
thf(fact_659_the__inv__into__into,axiom,
! [F2: nat > a,A: set_nat,X3: a,B: set_nat] :
( ( inj_on_nat_a @ F2 @ A )
=> ( ( member_a @ X3 @ ( image_nat_a @ F2 @ A ) )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( member_nat @ ( the_inv_into_nat_a @ A @ F2 @ X3 ) @ B ) ) ) ) ).
% the_inv_into_into
thf(fact_660_the__inv__into__into,axiom,
! [F2: nat > nat,A: set_nat,X3: nat,B: set_nat] :
( ( inj_on_nat_nat @ F2 @ A )
=> ( ( member_nat @ X3 @ ( image_nat_nat @ F2 @ A ) )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( member_nat @ ( the_inv_into_nat_nat @ A @ F2 @ X3 ) @ B ) ) ) ) ).
% the_inv_into_into
thf(fact_661_the__inv__into__into,axiom,
! [F2: nat > $o,A: set_nat,X3: $o,B: set_nat] :
( ( inj_on_nat_o @ F2 @ A )
=> ( ( member_o @ X3 @ ( image_nat_o @ F2 @ A ) )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( member_nat @ ( the_inv_into_nat_o @ A @ F2 @ X3 ) @ B ) ) ) ) ).
% the_inv_into_into
thf(fact_662_finite__subset__Union,axiom,
! [A: set_a,B9: set_set_a] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_a @ A @ ( comple2307003609928055243_set_a @ B9 ) )
=> ~ ! [F5: set_set_a] :
( ( finite_finite_set_a @ F5 )
=> ( ( ord_le3724670747650509150_set_a @ F5 @ B9 )
=> ~ ( ord_less_eq_set_a @ A @ ( comple2307003609928055243_set_a @ F5 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_663_finite__subset__Union,axiom,
! [A: set_nat,B9: set_set_nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( comple7399068483239264473et_nat @ B9 ) )
=> ~ ! [F5: set_set_nat] :
( ( finite1152437895449049373et_nat @ F5 )
=> ( ( ord_le6893508408891458716et_nat @ F5 @ B9 )
=> ~ ( ord_less_eq_set_nat @ A @ ( comple7399068483239264473et_nat @ F5 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_664_image__INT,axiom,
! [F2: a > nat,C2: set_a,A: set_a,B: a > set_a,J2: a] :
( ( inj_on_a_nat @ F2 @ C2 )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_set_a @ ( B @ X ) @ C2 ) )
=> ( ( member_a @ J2 @ A )
=> ( ( image_a_nat @ F2 @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ B @ A ) ) )
= ( comple7806235888213564991et_nat
@ ( image_a_set_nat
@ ^ [X2: a] : ( image_a_nat @ F2 @ ( B @ X2 ) )
@ A ) ) ) ) ) ) ).
% image_INT
thf(fact_665_image__INT,axiom,
! [F2: a > a,C2: set_a,A: set_a,B: a > set_a,J2: a] :
( ( inj_on_a_a @ F2 @ C2 )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_set_a @ ( B @ X ) @ C2 ) )
=> ( ( member_a @ J2 @ A )
=> ( ( image_a_a @ F2 @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ B @ A ) ) )
= ( comple6135023378680113637_set_a
@ ( image_a_set_a
@ ^ [X2: a] : ( image_a_a @ F2 @ ( B @ X2 ) )
@ A ) ) ) ) ) ) ).
% image_INT
thf(fact_666_image__INT,axiom,
! [F2: a > nat,C2: set_a,A: set_nat,B: nat > set_a,J2: nat] :
( ( inj_on_a_nat @ F2 @ C2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_set_a @ ( B @ X ) @ C2 ) )
=> ( ( member_nat @ J2 @ A )
=> ( ( image_a_nat @ F2 @ ( comple6135023378680113637_set_a @ ( image_nat_set_a @ B @ A ) ) )
= ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( image_a_nat @ F2 @ ( B @ X2 ) )
@ A ) ) ) ) ) ) ).
% image_INT
thf(fact_667_image__INT,axiom,
! [F2: a > a,C2: set_a,A: set_nat,B: nat > set_a,J2: nat] :
( ( inj_on_a_a @ F2 @ C2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_set_a @ ( B @ X ) @ C2 ) )
=> ( ( member_nat @ J2 @ A )
=> ( ( image_a_a @ F2 @ ( comple6135023378680113637_set_a @ ( image_nat_set_a @ B @ A ) ) )
= ( comple6135023378680113637_set_a
@ ( image_nat_set_a
@ ^ [X2: nat] : ( image_a_a @ F2 @ ( B @ X2 ) )
@ A ) ) ) ) ) ) ).
% image_INT
thf(fact_668_image__INT,axiom,
! [F2: a > nat,C2: set_a,A: set_o,B: $o > set_a,J2: $o] :
( ( inj_on_a_nat @ F2 @ C2 )
=> ( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_set_a @ ( B @ X ) @ C2 ) )
=> ( ( member_o @ J2 @ A )
=> ( ( image_a_nat @ F2 @ ( comple6135023378680113637_set_a @ ( image_o_set_a @ B @ A ) ) )
= ( comple7806235888213564991et_nat
@ ( image_o_set_nat
@ ^ [X2: $o] : ( image_a_nat @ F2 @ ( B @ X2 ) )
@ A ) ) ) ) ) ) ).
% image_INT
thf(fact_669_image__INT,axiom,
! [F2: a > a,C2: set_a,A: set_o,B: $o > set_a,J2: $o] :
( ( inj_on_a_a @ F2 @ C2 )
=> ( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_set_a @ ( B @ X ) @ C2 ) )
=> ( ( member_o @ J2 @ A )
=> ( ( image_a_a @ F2 @ ( comple6135023378680113637_set_a @ ( image_o_set_a @ B @ A ) ) )
= ( comple6135023378680113637_set_a
@ ( image_o_set_a
@ ^ [X2: $o] : ( image_a_a @ F2 @ ( B @ X2 ) )
@ A ) ) ) ) ) ) ).
% image_INT
thf(fact_670_image__INT,axiom,
! [F2: nat > a,C2: set_nat,A: set_a,B: a > set_nat,J2: a] :
( ( inj_on_nat_a @ F2 @ C2 )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_set_nat @ ( B @ X ) @ C2 ) )
=> ( ( member_a @ J2 @ A )
=> ( ( image_nat_a @ F2 @ ( comple7806235888213564991et_nat @ ( image_a_set_nat @ B @ A ) ) )
= ( comple6135023378680113637_set_a
@ ( image_a_set_a
@ ^ [X2: a] : ( image_nat_a @ F2 @ ( B @ X2 ) )
@ A ) ) ) ) ) ) ).
% image_INT
thf(fact_671_image__INT,axiom,
! [F2: nat > nat,C2: set_nat,A: set_a,B: a > set_nat,J2: a] :
( ( inj_on_nat_nat @ F2 @ C2 )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_set_nat @ ( B @ X ) @ C2 ) )
=> ( ( member_a @ J2 @ A )
=> ( ( image_nat_nat @ F2 @ ( comple7806235888213564991et_nat @ ( image_a_set_nat @ B @ A ) ) )
= ( comple7806235888213564991et_nat
@ ( image_a_set_nat
@ ^ [X2: a] : ( image_nat_nat @ F2 @ ( B @ X2 ) )
@ A ) ) ) ) ) ) ).
% image_INT
thf(fact_672_image__INT,axiom,
! [F2: nat > a,C2: set_nat,A: set_nat,B: nat > set_nat,J2: nat] :
( ( inj_on_nat_a @ F2 @ C2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_set_nat @ ( B @ X ) @ C2 ) )
=> ( ( member_nat @ J2 @ A )
=> ( ( image_nat_a @ F2 @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B @ A ) ) )
= ( comple6135023378680113637_set_a
@ ( image_nat_set_a
@ ^ [X2: nat] : ( image_nat_a @ F2 @ ( B @ X2 ) )
@ A ) ) ) ) ) ) ).
% image_INT
thf(fact_673_image__INT,axiom,
! [F2: nat > nat,C2: set_nat,A: set_nat,B: nat > set_nat,J2: nat] :
( ( inj_on_nat_nat @ F2 @ C2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_set_nat @ ( B @ X ) @ C2 ) )
=> ( ( member_nat @ J2 @ A )
=> ( ( image_nat_nat @ F2 @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B @ A ) ) )
= ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( image_nat_nat @ F2 @ ( B @ X2 ) )
@ A ) ) ) ) ) ) ).
% image_INT
thf(fact_674_Union__iff,axiom,
! [A: a,C2: set_set_a] :
( ( member_a @ A @ ( comple2307003609928055243_set_a @ C2 ) )
= ( ? [X2: set_a] :
( ( member_set_a @ X2 @ C2 )
& ( member_a @ A @ X2 ) ) ) ) ).
% Union_iff
thf(fact_675_Union__iff,axiom,
! [A: nat,C2: set_set_nat] :
( ( member_nat @ A @ ( comple7399068483239264473et_nat @ C2 ) )
= ( ? [X2: set_nat] :
( ( member_set_nat @ X2 @ C2 )
& ( member_nat @ A @ X2 ) ) ) ) ).
% Union_iff
thf(fact_676_Union__iff,axiom,
! [A: $o,C2: set_set_o] :
( ( member_o @ A @ ( comple90263536869209701_set_o @ C2 ) )
= ( ? [X2: set_o] :
( ( member_set_o @ X2 @ C2 )
& ( member_o @ A @ X2 ) ) ) ) ).
% Union_iff
thf(fact_677_UnionI,axiom,
! [X5: set_a,C2: set_set_a,A: a] :
( ( member_set_a @ X5 @ C2 )
=> ( ( member_a @ A @ X5 )
=> ( member_a @ A @ ( comple2307003609928055243_set_a @ C2 ) ) ) ) ).
% UnionI
thf(fact_678_UnionI,axiom,
! [X5: set_nat,C2: set_set_nat,A: nat] :
( ( member_set_nat @ X5 @ C2 )
=> ( ( member_nat @ A @ X5 )
=> ( member_nat @ A @ ( comple7399068483239264473et_nat @ C2 ) ) ) ) ).
% UnionI
thf(fact_679_UnionI,axiom,
! [X5: set_o,C2: set_set_o,A: $o] :
( ( member_set_o @ X5 @ C2 )
=> ( ( member_o @ A @ X5 )
=> ( member_o @ A @ ( comple90263536869209701_set_o @ C2 ) ) ) ) ).
% UnionI
thf(fact_680_Inter__iff,axiom,
! [A: a,C2: set_set_a] :
( ( member_a @ A @ ( comple6135023378680113637_set_a @ C2 ) )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ C2 )
=> ( member_a @ A @ X2 ) ) ) ) ).
% Inter_iff
thf(fact_681_Inter__iff,axiom,
! [A: nat,C2: set_set_nat] :
( ( member_nat @ A @ ( comple7806235888213564991et_nat @ C2 ) )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ C2 )
=> ( member_nat @ A @ X2 ) ) ) ) ).
% Inter_iff
thf(fact_682_Inter__iff,axiom,
! [A: $o,C2: set_set_o] :
( ( member_o @ A @ ( comple3063163877087187839_set_o @ C2 ) )
= ( ! [X2: set_o] :
( ( member_set_o @ X2 @ C2 )
=> ( member_o @ A @ X2 ) ) ) ) ).
% Inter_iff
thf(fact_683_InterI,axiom,
! [C2: set_set_a,A: a] :
( ! [X7: set_a] :
( ( member_set_a @ X7 @ C2 )
=> ( member_a @ A @ X7 ) )
=> ( member_a @ A @ ( comple6135023378680113637_set_a @ C2 ) ) ) ).
% InterI
thf(fact_684_InterI,axiom,
! [C2: set_set_nat,A: nat] :
( ! [X7: set_nat] :
( ( member_set_nat @ X7 @ C2 )
=> ( member_nat @ A @ X7 ) )
=> ( member_nat @ A @ ( comple7806235888213564991et_nat @ C2 ) ) ) ).
% InterI
thf(fact_685_InterI,axiom,
! [C2: set_set_o,A: $o] :
( ! [X7: set_o] :
( ( member_set_o @ X7 @ C2 )
=> ( member_o @ A @ X7 ) )
=> ( member_o @ A @ ( comple3063163877087187839_set_o @ C2 ) ) ) ).
% InterI
thf(fact_686_Union__Un__distrib,axiom,
! [A: set_set_a,B: set_set_a] :
( ( comple2307003609928055243_set_a @ ( sup_sup_set_set_a @ A @ B ) )
= ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ A ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Union_Un_distrib
thf(fact_687_Union__Un__distrib,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( comple7399068483239264473et_nat @ ( sup_sup_set_set_nat @ A @ B ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Union_Un_distrib
thf(fact_688_finite__Inter,axiom,
! [M: set_set_a] :
( ? [X4: set_a] :
( ( member_set_a @ X4 @ M )
& ( finite_finite_a @ X4 ) )
=> ( finite_finite_a @ ( comple6135023378680113637_set_a @ M ) ) ) ).
% finite_Inter
thf(fact_689_finite__Inter,axiom,
! [M: set_set_nat] :
( ? [X4: set_nat] :
( ( member_set_nat @ X4 @ M )
& ( finite_finite_nat @ X4 ) )
=> ( finite_finite_nat @ ( comple7806235888213564991et_nat @ M ) ) ) ).
% finite_Inter
thf(fact_690_Pow__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( member_set_nat @ A @ ( pow_nat @ B ) )
= ( ord_less_eq_set_nat @ A @ B ) ) ).
% Pow_iff
thf(fact_691_PowI,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( member_set_nat @ A @ ( pow_nat @ B ) ) ) ).
% PowI
thf(fact_692_SUP__identity__eq,axiom,
! [A: set_nat] :
( ( complete_Sup_Sup_nat
@ ( image_nat_nat
@ ^ [X2: nat] : X2
@ A ) )
= ( complete_Sup_Sup_nat @ A ) ) ).
% SUP_identity_eq
thf(fact_693_SUP__identity__eq,axiom,
! [A: set_o] :
( ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [X2: $o] : X2
@ A ) )
= ( complete_Sup_Sup_o @ A ) ) ).
% SUP_identity_eq
thf(fact_694_INF__identity__eq,axiom,
! [A: set_nat] :
( ( complete_Inf_Inf_nat
@ ( image_nat_nat
@ ^ [X2: nat] : X2
@ A ) )
= ( complete_Inf_Inf_nat @ A ) ) ).
% INF_identity_eq
thf(fact_695_INF__identity__eq,axiom,
! [A: set_o] :
( ( complete_Inf_Inf_o
@ ( image_o_o
@ ^ [X2: $o] : X2
@ A ) )
= ( complete_Inf_Inf_o @ A ) ) ).
% INF_identity_eq
thf(fact_696_UN__I,axiom,
! [A2: a,A: set_a,B2: a,B: a > set_a] :
( ( member_a @ A2 @ A )
=> ( ( member_a @ B2 @ ( B @ A2 ) )
=> ( member_a @ B2 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_697_UN__I,axiom,
! [A2: a,A: set_a,B2: nat,B: a > set_nat] :
( ( member_a @ A2 @ A )
=> ( ( member_nat @ B2 @ ( B @ A2 ) )
=> ( member_nat @ B2 @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_698_UN__I,axiom,
! [A2: a,A: set_a,B2: $o,B: a > set_o] :
( ( member_a @ A2 @ A )
=> ( ( member_o @ B2 @ ( B @ A2 ) )
=> ( member_o @ B2 @ ( comple90263536869209701_set_o @ ( image_a_set_o @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_699_UN__I,axiom,
! [A2: nat,A: set_nat,B2: a,B: nat > set_a] :
( ( member_nat @ A2 @ A )
=> ( ( member_a @ B2 @ ( B @ A2 ) )
=> ( member_a @ B2 @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_700_UN__I,axiom,
! [A2: nat,A: set_nat,B2: nat,B: nat > set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( member_nat @ B2 @ ( B @ A2 ) )
=> ( member_nat @ B2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_701_UN__I,axiom,
! [A2: nat,A: set_nat,B2: $o,B: nat > set_o] :
( ( member_nat @ A2 @ A )
=> ( ( member_o @ B2 @ ( B @ A2 ) )
=> ( member_o @ B2 @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_702_UN__I,axiom,
! [A2: $o,A: set_o,B2: a,B: $o > set_a] :
( ( member_o @ A2 @ A )
=> ( ( member_a @ B2 @ ( B @ A2 ) )
=> ( member_a @ B2 @ ( comple2307003609928055243_set_a @ ( image_o_set_a @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_703_UN__I,axiom,
! [A2: $o,A: set_o,B2: nat,B: $o > set_nat] :
( ( member_o @ A2 @ A )
=> ( ( member_nat @ B2 @ ( B @ A2 ) )
=> ( member_nat @ B2 @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_704_UN__I,axiom,
! [A2: $o,A: set_o,B2: $o,B: $o > set_o] :
( ( member_o @ A2 @ A )
=> ( ( member_o @ B2 @ ( B @ A2 ) )
=> ( member_o @ B2 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_705_INT__I,axiom,
! [A: set_a,B2: a,B: a > set_a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_a @ B2 @ ( B @ X ) ) )
=> ( member_a @ B2 @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ B @ A ) ) ) ) ).
% INT_I
thf(fact_706_INT__I,axiom,
! [A: set_a,B2: nat,B: a > set_nat] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_nat @ B2 @ ( B @ X ) ) )
=> ( member_nat @ B2 @ ( comple7806235888213564991et_nat @ ( image_a_set_nat @ B @ A ) ) ) ) ).
% INT_I
thf(fact_707_INT__I,axiom,
! [A: set_a,B2: $o,B: a > set_o] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_o @ B2 @ ( B @ X ) ) )
=> ( member_o @ B2 @ ( comple3063163877087187839_set_o @ ( image_a_set_o @ B @ A ) ) ) ) ).
% INT_I
thf(fact_708_INT__I,axiom,
! [A: set_nat,B2: a,B: nat > set_a] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_a @ B2 @ ( B @ X ) ) )
=> ( member_a @ B2 @ ( comple6135023378680113637_set_a @ ( image_nat_set_a @ B @ A ) ) ) ) ).
% INT_I
thf(fact_709_INT__I,axiom,
! [A: set_nat,B2: nat,B: nat > set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_nat @ B2 @ ( B @ X ) ) )
=> ( member_nat @ B2 @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B @ A ) ) ) ) ).
% INT_I
thf(fact_710_INT__I,axiom,
! [A: set_nat,B2: $o,B: nat > set_o] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_o @ B2 @ ( B @ X ) ) )
=> ( member_o @ B2 @ ( comple3063163877087187839_set_o @ ( image_nat_set_o @ B @ A ) ) ) ) ).
% INT_I
thf(fact_711_INT__I,axiom,
! [A: set_o,B2: a,B: $o > set_a] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( member_a @ B2 @ ( B @ X ) ) )
=> ( member_a @ B2 @ ( comple6135023378680113637_set_a @ ( image_o_set_a @ B @ A ) ) ) ) ).
% INT_I
thf(fact_712_INT__I,axiom,
! [A: set_o,B2: nat,B: $o > set_nat] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( member_nat @ B2 @ ( B @ X ) ) )
=> ( member_nat @ B2 @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ B @ A ) ) ) ) ).
% INT_I
thf(fact_713_INT__I,axiom,
! [A: set_o,B2: $o,B: $o > set_o] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( member_o @ B2 @ ( B @ X ) ) )
=> ( member_o @ B2 @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ B @ A ) ) ) ) ).
% INT_I
thf(fact_714_finite__UN,axiom,
! [A: set_a,B: a > set_a] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( finite_finite_a @ ( B @ X2 ) ) ) ) ) ) ).
% finite_UN
thf(fact_715_finite__UN,axiom,
! [A: set_a,B: a > set_nat] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ B @ A ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( finite_finite_nat @ ( B @ X2 ) ) ) ) ) ) ).
% finite_UN
thf(fact_716_finite__UN,axiom,
! [A: set_nat,B: nat > set_a] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_a @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ B @ A ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( finite_finite_a @ ( B @ X2 ) ) ) ) ) ) ).
% finite_UN
thf(fact_717_finite__UN,axiom,
! [A: set_nat,B: nat > set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( finite_finite_nat @ ( B @ X2 ) ) ) ) ) ) ).
% finite_UN
thf(fact_718_finite__Union,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ! [M5: set_a] :
( ( member_set_a @ M5 @ A )
=> ( finite_finite_a @ M5 ) )
=> ( finite_finite_a @ ( comple2307003609928055243_set_a @ A ) ) ) ) ).
% finite_Union
thf(fact_719_finite__Union,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ! [M5: set_nat] :
( ( member_set_nat @ M5 @ A )
=> ( finite_finite_nat @ M5 ) )
=> ( finite_finite_nat @ ( comple7399068483239264473et_nat @ A ) ) ) ) ).
% finite_Union
thf(fact_720_the__inv__into__onto,axiom,
! [F2: a > nat,A: set_a] :
( ( inj_on_a_nat @ F2 @ A )
=> ( ( image_nat_a @ ( the_inv_into_a_nat @ A @ F2 ) @ ( image_a_nat @ F2 @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_721_the__inv__into__onto,axiom,
! [F2: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F2 @ A )
=> ( ( image_nat_nat @ ( the_inv_into_nat_nat @ A @ F2 ) @ ( image_nat_nat @ F2 @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_722_the__inv__into__onto,axiom,
! [F2: nat > a,A: set_nat] :
( ( inj_on_nat_a @ F2 @ A )
=> ( ( image_a_nat @ ( the_inv_into_nat_a @ A @ F2 ) @ ( image_nat_a @ F2 @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_723_the__inv__into__onto,axiom,
! [F2: a > a,A: set_a] :
( ( inj_on_a_a @ F2 @ A )
=> ( ( image_a_a @ ( the_inv_into_a_a @ A @ F2 ) @ ( image_a_a @ F2 @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_724_finite__Pow__iff,axiom,
! [A: set_a] :
( ( finite_finite_set_a @ ( pow_a @ A ) )
= ( finite_finite_a @ A ) ) ).
% finite_Pow_iff
thf(fact_725_finite__Pow__iff,axiom,
! [A: set_nat] :
( ( finite1152437895449049373et_nat @ ( pow_nat @ A ) )
= ( finite_finite_nat @ A ) ) ).
% finite_Pow_iff
thf(fact_726_finite__UN__I,axiom,
! [A: set_o,B: $o > set_a] :
( ( finite_finite_o @ A )
=> ( ! [A6: $o] :
( ( member_o @ A6 @ A )
=> ( finite_finite_a @ ( B @ A6 ) ) )
=> ( finite_finite_a @ ( comple2307003609928055243_set_a @ ( image_o_set_a @ B @ A ) ) ) ) ) ).
% finite_UN_I
thf(fact_727_finite__UN__I,axiom,
! [A: set_o,B: $o > set_nat] :
( ( finite_finite_o @ A )
=> ( ! [A6: $o] :
( ( member_o @ A6 @ A )
=> ( finite_finite_nat @ ( B @ A6 ) ) )
=> ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B @ A ) ) ) ) ) ).
% finite_UN_I
thf(fact_728_finite__UN__I,axiom,
! [A: set_a,B: a > set_a] :
( ( finite_finite_a @ A )
=> ( ! [A6: a] :
( ( member_a @ A6 @ A )
=> ( finite_finite_a @ ( B @ A6 ) ) )
=> ( finite_finite_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A ) ) ) ) ) ).
% finite_UN_I
thf(fact_729_finite__UN__I,axiom,
! [A: set_a,B: a > set_nat] :
( ( finite_finite_a @ A )
=> ( ! [A6: a] :
( ( member_a @ A6 @ A )
=> ( finite_finite_nat @ ( B @ A6 ) ) )
=> ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ B @ A ) ) ) ) ) ).
% finite_UN_I
thf(fact_730_finite__UN__I,axiom,
! [A: set_nat,B: nat > set_a] :
( ( finite_finite_nat @ A )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ A )
=> ( finite_finite_a @ ( B @ A6 ) ) )
=> ( finite_finite_a @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ B @ A ) ) ) ) ) ).
% finite_UN_I
thf(fact_731_finite__UN__I,axiom,
! [A: set_nat,B: nat > set_nat] :
( ( finite_finite_nat @ A )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ A )
=> ( finite_finite_nat @ ( B @ A6 ) ) )
=> ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) ) ) ) ).
% finite_UN_I
thf(fact_732_UN__Un,axiom,
! [M: a > set_a,A: set_a,B: set_a] :
( ( comple2307003609928055243_set_a @ ( image_a_set_a @ M @ ( sup_sup_set_a @ A @ B ) ) )
= ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ M @ A ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ M @ B ) ) ) ) ).
% UN_Un
thf(fact_733_UN__Un,axiom,
! [M: a > set_nat,A: set_a,B: set_a] :
( ( comple7399068483239264473et_nat @ ( image_a_set_nat @ M @ ( sup_sup_set_a @ A @ B ) ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ M @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ M @ B ) ) ) ) ).
% UN_Un
thf(fact_734_UN__Un,axiom,
! [M: nat > set_a,A: set_nat,B: set_nat] :
( ( comple2307003609928055243_set_a @ ( image_nat_set_a @ M @ ( sup_sup_set_nat @ A @ B ) ) )
= ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ M @ A ) ) @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ M @ B ) ) ) ) ).
% UN_Un
thf(fact_735_UN__Un,axiom,
! [M: nat > set_nat,A: set_nat,B: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M @ ( sup_sup_set_nat @ A @ B ) ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M @ B ) ) ) ) ).
% UN_Un
thf(fact_736_UnionE,axiom,
! [A: a,C2: set_set_a] :
( ( member_a @ A @ ( comple2307003609928055243_set_a @ C2 ) )
=> ~ ! [X7: set_a] :
( ( member_a @ A @ X7 )
=> ~ ( member_set_a @ X7 @ C2 ) ) ) ).
% UnionE
thf(fact_737_UnionE,axiom,
! [A: nat,C2: set_set_nat] :
( ( member_nat @ A @ ( comple7399068483239264473et_nat @ C2 ) )
=> ~ ! [X7: set_nat] :
( ( member_nat @ A @ X7 )
=> ~ ( member_set_nat @ X7 @ C2 ) ) ) ).
% UnionE
thf(fact_738_UnionE,axiom,
! [A: $o,C2: set_set_o] :
( ( member_o @ A @ ( comple90263536869209701_set_o @ C2 ) )
=> ~ ! [X7: set_o] :
( ( member_o @ A @ X7 )
=> ~ ( member_set_o @ X7 @ C2 ) ) ) ).
% UnionE
thf(fact_739_InterE,axiom,
! [A: a,C2: set_set_a,X5: set_a] :
( ( member_a @ A @ ( comple6135023378680113637_set_a @ C2 ) )
=> ( ( member_set_a @ X5 @ C2 )
=> ( member_a @ A @ X5 ) ) ) ).
% InterE
thf(fact_740_InterE,axiom,
! [A: nat,C2: set_set_nat,X5: set_nat] :
( ( member_nat @ A @ ( comple7806235888213564991et_nat @ C2 ) )
=> ( ( member_set_nat @ X5 @ C2 )
=> ( member_nat @ A @ X5 ) ) ) ).
% InterE
thf(fact_741_InterE,axiom,
! [A: $o,C2: set_set_o,X5: set_o] :
( ( member_o @ A @ ( comple3063163877087187839_set_o @ C2 ) )
=> ( ( member_set_o @ X5 @ C2 )
=> ( member_o @ A @ X5 ) ) ) ).
% InterE
thf(fact_742_InterD,axiom,
! [A: a,C2: set_set_a,X5: set_a] :
( ( member_a @ A @ ( comple6135023378680113637_set_a @ C2 ) )
=> ( ( member_set_a @ X5 @ C2 )
=> ( member_a @ A @ X5 ) ) ) ).
% InterD
thf(fact_743_InterD,axiom,
! [A: nat,C2: set_set_nat,X5: set_nat] :
( ( member_nat @ A @ ( comple7806235888213564991et_nat @ C2 ) )
=> ( ( member_set_nat @ X5 @ C2 )
=> ( member_nat @ A @ X5 ) ) ) ).
% InterD
thf(fact_744_InterD,axiom,
! [A: $o,C2: set_set_o,X5: set_o] :
( ( member_o @ A @ ( comple3063163877087187839_set_o @ C2 ) )
=> ( ( member_set_o @ X5 @ C2 )
=> ( member_o @ A @ X5 ) ) ) ).
% InterD
thf(fact_745_Relational__Calculus_Oeval__def,axiom,
( relational_eval_a_b
= ( ^ [Q2: relational_fmla_a_b] : ( relati8814510239606734169on_a_b @ ( relational_fv_a_b @ Q2 ) @ Q2 ) ) ) ).
% Relational_Calculus.eval_def
thf(fact_746_Inf__eqI,axiom,
! [A: set_set_nat,X3: set_nat] :
( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ A )
=> ( ord_less_eq_set_nat @ X3 @ I3 ) )
=> ( ! [Y5: set_nat] :
( ! [I4: set_nat] :
( ( member_set_nat @ I4 @ A )
=> ( ord_less_eq_set_nat @ Y5 @ I4 ) )
=> ( ord_less_eq_set_nat @ Y5 @ X3 ) )
=> ( ( comple7806235888213564991et_nat @ A )
= X3 ) ) ) ).
% Inf_eqI
thf(fact_747_Inf__eqI,axiom,
! [A: set_o,X3: $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ( ord_less_eq_o @ X3 @ I3 ) )
=> ( ! [Y5: $o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A )
=> ( ord_less_eq_o @ Y5 @ I4 ) )
=> ( ord_less_eq_o @ Y5 @ X3 ) )
=> ( ( complete_Inf_Inf_o @ A )
= X3 ) ) ) ).
% Inf_eqI
thf(fact_748_Inf__mono,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ! [B6: set_nat] :
( ( member_set_nat @ B6 @ B )
=> ? [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
& ( ord_less_eq_set_nat @ X4 @ B6 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A ) @ ( comple7806235888213564991et_nat @ B ) ) ) ).
% Inf_mono
thf(fact_749_Inf__mono,axiom,
! [B: set_o,A: set_o] :
( ! [B6: $o] :
( ( member_o @ B6 @ B )
=> ? [X4: $o] :
( ( member_o @ X4 @ A )
& ( ord_less_eq_o @ X4 @ B6 ) ) )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A ) @ ( complete_Inf_Inf_o @ B ) ) ) ).
% Inf_mono
thf(fact_750_Inf__lower,axiom,
! [X3: set_nat,A: set_set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A ) @ X3 ) ) ).
% Inf_lower
thf(fact_751_Inf__lower,axiom,
! [X3: $o,A: set_o] :
( ( member_o @ X3 @ A )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A ) @ X3 ) ) ).
% Inf_lower
thf(fact_752_Inf__lower2,axiom,
! [U: set_nat,A: set_set_nat,V: set_nat] :
( ( member_set_nat @ U @ A )
=> ( ( ord_less_eq_set_nat @ U @ V )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A ) @ V ) ) ) ).
% Inf_lower2
thf(fact_753_Inf__lower2,axiom,
! [U: $o,A: set_o,V: $o] :
( ( member_o @ U @ A )
=> ( ( ord_less_eq_o @ U @ V )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A ) @ V ) ) ) ).
% Inf_lower2
thf(fact_754_le__Inf__iff,axiom,
! [B2: set_nat,A: set_set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( comple7806235888213564991et_nat @ A ) )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( ord_less_eq_set_nat @ B2 @ X2 ) ) ) ) ).
% le_Inf_iff
thf(fact_755_le__Inf__iff,axiom,
! [B2: $o,A: set_o] :
( ( ord_less_eq_o @ B2 @ ( complete_Inf_Inf_o @ A ) )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( ord_less_eq_o @ B2 @ X2 ) ) ) ) ).
% le_Inf_iff
thf(fact_756_Inf__greatest,axiom,
! [A: set_set_nat,Z: set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ( ord_less_eq_set_nat @ Z @ X ) )
=> ( ord_less_eq_set_nat @ Z @ ( comple7806235888213564991et_nat @ A ) ) ) ).
% Inf_greatest
thf(fact_757_Inf__greatest,axiom,
! [A: set_o,Z: $o] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_o @ Z @ X ) )
=> ( ord_less_eq_o @ Z @ ( complete_Inf_Inf_o @ A ) ) ) ).
% Inf_greatest
thf(fact_758_Sup__eqI,axiom,
! [A: set_set_nat,X3: set_nat] :
( ! [Y5: set_nat] :
( ( member_set_nat @ Y5 @ A )
=> ( ord_less_eq_set_nat @ Y5 @ X3 ) )
=> ( ! [Y5: set_nat] :
( ! [Z6: set_nat] :
( ( member_set_nat @ Z6 @ A )
=> ( ord_less_eq_set_nat @ Z6 @ Y5 ) )
=> ( ord_less_eq_set_nat @ X3 @ Y5 ) )
=> ( ( comple7399068483239264473et_nat @ A )
= X3 ) ) ) ).
% Sup_eqI
thf(fact_759_Sup__eqI,axiom,
! [A: set_o,X3: $o] :
( ! [Y5: $o] :
( ( member_o @ Y5 @ A )
=> ( ord_less_eq_o @ Y5 @ X3 ) )
=> ( ! [Y5: $o] :
( ! [Z6: $o] :
( ( member_o @ Z6 @ A )
=> ( ord_less_eq_o @ Z6 @ Y5 ) )
=> ( ord_less_eq_o @ X3 @ Y5 ) )
=> ( ( complete_Sup_Sup_o @ A )
= X3 ) ) ) ).
% Sup_eqI
thf(fact_760_Sup__mono,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ! [A6: set_nat] :
( ( member_set_nat @ A6 @ A )
=> ? [X4: set_nat] :
( ( member_set_nat @ X4 @ B )
& ( ord_less_eq_set_nat @ A6 @ X4 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Sup_mono
thf(fact_761_Sup__mono,axiom,
! [A: set_o,B: set_o] :
( ! [A6: $o] :
( ( member_o @ A6 @ A )
=> ? [X4: $o] :
( ( member_o @ X4 @ B )
& ( ord_less_eq_o @ A6 @ X4 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ ( complete_Sup_Sup_o @ B ) ) ) ).
% Sup_mono
thf(fact_762_Sup__least,axiom,
! [A: set_set_nat,Z: set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ( ord_less_eq_set_nat @ X @ Z ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ Z ) ) ).
% Sup_least
thf(fact_763_Sup__least,axiom,
! [A: set_o,Z: $o] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_o @ X @ Z ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ Z ) ) ).
% Sup_least
thf(fact_764_Sup__upper,axiom,
! [X3: set_nat,A: set_set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ X3 @ ( comple7399068483239264473et_nat @ A ) ) ) ).
% Sup_upper
thf(fact_765_Sup__upper,axiom,
! [X3: $o,A: set_o] :
( ( member_o @ X3 @ A )
=> ( ord_less_eq_o @ X3 @ ( complete_Sup_Sup_o @ A ) ) ) ).
% Sup_upper
thf(fact_766_Sup__le__iff,axiom,
! [A: set_set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ B2 )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( ord_less_eq_set_nat @ X2 @ B2 ) ) ) ) ).
% Sup_le_iff
thf(fact_767_Sup__le__iff,axiom,
! [A: set_o,B2: $o] :
( ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ B2 )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( ord_less_eq_o @ X2 @ B2 ) ) ) ) ).
% Sup_le_iff
thf(fact_768_Sup__upper2,axiom,
! [U: set_nat,A: set_set_nat,V: set_nat] :
( ( member_set_nat @ U @ A )
=> ( ( ord_less_eq_set_nat @ V @ U )
=> ( ord_less_eq_set_nat @ V @ ( comple7399068483239264473et_nat @ A ) ) ) ) ).
% Sup_upper2
thf(fact_769_Sup__upper2,axiom,
! [U: $o,A: set_o,V: $o] :
( ( member_o @ U @ A )
=> ( ( ord_less_eq_o @ V @ U )
=> ( ord_less_eq_o @ V @ ( complete_Sup_Sup_o @ A ) ) ) ) ).
% Sup_upper2
thf(fact_770_INF__cong,axiom,
! [A: set_a,B: set_a,C2: a > nat,D2: a > nat] :
( ( A = B )
=> ( ! [X: a] :
( ( member_a @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( complete_Inf_Inf_nat @ ( image_a_nat @ C2 @ A ) )
= ( complete_Inf_Inf_nat @ ( image_a_nat @ D2 @ B ) ) ) ) ) ).
% INF_cong
thf(fact_771_INF__cong,axiom,
! [A: set_nat,B: set_nat,C2: nat > nat,D2: nat > nat] :
( ( A = B )
=> ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( complete_Inf_Inf_nat @ ( image_nat_nat @ C2 @ A ) )
= ( complete_Inf_Inf_nat @ ( image_nat_nat @ D2 @ B ) ) ) ) ) ).
% INF_cong
thf(fact_772_INF__cong,axiom,
! [A: set_a,B: set_a,C2: a > $o,D2: a > $o] :
( ( A = B )
=> ( ! [X: a] :
( ( member_a @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_a_o @ C2 @ A ) )
= ( complete_Inf_Inf_o @ ( image_a_o @ D2 @ B ) ) ) ) ) ).
% INF_cong
thf(fact_773_INF__cong,axiom,
! [A: set_nat,B: set_nat,C2: nat > $o,D2: nat > $o] :
( ( A = B )
=> ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_nat_o @ C2 @ A ) )
= ( complete_Inf_Inf_o @ ( image_nat_o @ D2 @ B ) ) ) ) ) ).
% INF_cong
thf(fact_774_INF__cong,axiom,
! [A: set_o,B: set_o,C2: $o > $o,D2: $o > $o] :
( ( A = B )
=> ( ! [X: $o] :
( ( member_o @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ C2 @ A ) )
= ( complete_Inf_Inf_o @ ( image_o_o @ D2 @ B ) ) ) ) ) ).
% INF_cong
thf(fact_775_SUP__cong,axiom,
! [A: set_a,B: set_a,C2: a > nat,D2: a > nat] :
( ( A = B )
=> ( ! [X: a] :
( ( member_a @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_a_nat @ C2 @ A ) )
= ( complete_Sup_Sup_nat @ ( image_a_nat @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_776_SUP__cong,axiom,
! [A: set_nat,B: set_nat,C2: nat > nat,D2: nat > nat] :
( ( A = B )
=> ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat @ C2 @ A ) )
= ( complete_Sup_Sup_nat @ ( image_nat_nat @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_777_SUP__cong,axiom,
! [A: set_a,B: set_a,C2: a > $o,D2: a > $o] :
( ( A = B )
=> ( ! [X: a] :
( ( member_a @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_a_o @ C2 @ A ) )
= ( complete_Sup_Sup_o @ ( image_a_o @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_778_SUP__cong,axiom,
! [A: set_nat,B: set_nat,C2: nat > $o,D2: nat > $o] :
( ( A = B )
=> ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ C2 @ A ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_779_SUP__cong,axiom,
! [A: set_o,B: set_o,C2: $o > $o,D2: $o > $o] :
( ( A = B )
=> ( ! [X: $o] :
( ( member_o @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ C2 @ A ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_780_Inter__lower,axiom,
! [B: set_nat,A: set_set_nat] :
( ( member_set_nat @ B @ A )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A ) @ B ) ) ).
% Inter_lower
thf(fact_781_Inter__greatest,axiom,
! [A: set_set_nat,C2: set_nat] :
( ! [X7: set_nat] :
( ( member_set_nat @ X7 @ A )
=> ( ord_less_eq_set_nat @ C2 @ X7 ) )
=> ( ord_less_eq_set_nat @ C2 @ ( comple7806235888213564991et_nat @ A ) ) ) ).
% Inter_greatest
thf(fact_782_Union__least,axiom,
! [A: set_set_nat,C2: set_nat] :
( ! [X7: set_nat] :
( ( member_set_nat @ X7 @ A )
=> ( ord_less_eq_set_nat @ X7 @ C2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ C2 ) ) ).
% Union_least
thf(fact_783_Union__upper,axiom,
! [B: set_nat,A: set_set_nat] :
( ( member_set_nat @ B @ A )
=> ( ord_less_eq_set_nat @ B @ ( comple7399068483239264473et_nat @ A ) ) ) ).
% Union_upper
thf(fact_784_Union__subsetI,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ? [Y6: set_nat] :
( ( member_set_nat @ Y6 @ B )
& ( ord_less_eq_set_nat @ X @ Y6 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Union_subsetI
thf(fact_785_PowD,axiom,
! [A: set_nat,B: set_nat] :
( ( member_set_nat @ A @ ( pow_nat @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% PowD
thf(fact_786_INT__D,axiom,
! [B2: a,B: a > set_a,A: set_a,A2: a] :
( ( member_a @ B2 @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ B @ A ) ) )
=> ( ( member_a @ A2 @ A )
=> ( member_a @ B2 @ ( B @ A2 ) ) ) ) ).
% INT_D
thf(fact_787_INT__D,axiom,
! [B2: a,B: nat > set_a,A: set_nat,A2: nat] :
( ( member_a @ B2 @ ( comple6135023378680113637_set_a @ ( image_nat_set_a @ B @ A ) ) )
=> ( ( member_nat @ A2 @ A )
=> ( member_a @ B2 @ ( B @ A2 ) ) ) ) ).
% INT_D
thf(fact_788_INT__D,axiom,
! [B2: a,B: $o > set_a,A: set_o,A2: $o] :
( ( member_a @ B2 @ ( comple6135023378680113637_set_a @ ( image_o_set_a @ B @ A ) ) )
=> ( ( member_o @ A2 @ A )
=> ( member_a @ B2 @ ( B @ A2 ) ) ) ) ).
% INT_D
thf(fact_789_INT__D,axiom,
! [B2: nat,B: a > set_nat,A: set_a,A2: a] :
( ( member_nat @ B2 @ ( comple7806235888213564991et_nat @ ( image_a_set_nat @ B @ A ) ) )
=> ( ( member_a @ A2 @ A )
=> ( member_nat @ B2 @ ( B @ A2 ) ) ) ) ).
% INT_D
thf(fact_790_INT__D,axiom,
! [B2: nat,B: nat > set_nat,A: set_nat,A2: nat] :
( ( member_nat @ B2 @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B @ A ) ) )
=> ( ( member_nat @ A2 @ A )
=> ( member_nat @ B2 @ ( B @ A2 ) ) ) ) ).
% INT_D
thf(fact_791_INT__D,axiom,
! [B2: nat,B: $o > set_nat,A: set_o,A2: $o] :
( ( member_nat @ B2 @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ B @ A ) ) )
=> ( ( member_o @ A2 @ A )
=> ( member_nat @ B2 @ ( B @ A2 ) ) ) ) ).
% INT_D
thf(fact_792_INT__D,axiom,
! [B2: $o,B: a > set_o,A: set_a,A2: a] :
( ( member_o @ B2 @ ( comple3063163877087187839_set_o @ ( image_a_set_o @ B @ A ) ) )
=> ( ( member_a @ A2 @ A )
=> ( member_o @ B2 @ ( B @ A2 ) ) ) ) ).
% INT_D
thf(fact_793_INT__D,axiom,
! [B2: $o,B: nat > set_o,A: set_nat,A2: nat] :
( ( member_o @ B2 @ ( comple3063163877087187839_set_o @ ( image_nat_set_o @ B @ A ) ) )
=> ( ( member_nat @ A2 @ A )
=> ( member_o @ B2 @ ( B @ A2 ) ) ) ) ).
% INT_D
thf(fact_794_INT__D,axiom,
! [B2: $o,B: $o > set_o,A: set_o,A2: $o] :
( ( member_o @ B2 @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ B @ A ) ) )
=> ( ( member_o @ A2 @ A )
=> ( member_o @ B2 @ ( B @ A2 ) ) ) ) ).
% INT_D
thf(fact_795_INT__E,axiom,
! [B2: a,B: a > set_a,A: set_a,A2: a] :
( ( member_a @ B2 @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ B @ A ) ) )
=> ( ~ ( member_a @ B2 @ ( B @ A2 ) )
=> ~ ( member_a @ A2 @ A ) ) ) ).
% INT_E
thf(fact_796_INT__E,axiom,
! [B2: a,B: nat > set_a,A: set_nat,A2: nat] :
( ( member_a @ B2 @ ( comple6135023378680113637_set_a @ ( image_nat_set_a @ B @ A ) ) )
=> ( ~ ( member_a @ B2 @ ( B @ A2 ) )
=> ~ ( member_nat @ A2 @ A ) ) ) ).
% INT_E
thf(fact_797_INT__E,axiom,
! [B2: a,B: $o > set_a,A: set_o,A2: $o] :
( ( member_a @ B2 @ ( comple6135023378680113637_set_a @ ( image_o_set_a @ B @ A ) ) )
=> ( ~ ( member_a @ B2 @ ( B @ A2 ) )
=> ~ ( member_o @ A2 @ A ) ) ) ).
% INT_E
thf(fact_798_INT__E,axiom,
! [B2: nat,B: a > set_nat,A: set_a,A2: a] :
( ( member_nat @ B2 @ ( comple7806235888213564991et_nat @ ( image_a_set_nat @ B @ A ) ) )
=> ( ~ ( member_nat @ B2 @ ( B @ A2 ) )
=> ~ ( member_a @ A2 @ A ) ) ) ).
% INT_E
thf(fact_799_INT__E,axiom,
! [B2: nat,B: nat > set_nat,A: set_nat,A2: nat] :
( ( member_nat @ B2 @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B @ A ) ) )
=> ( ~ ( member_nat @ B2 @ ( B @ A2 ) )
=> ~ ( member_nat @ A2 @ A ) ) ) ).
% INT_E
thf(fact_800_INT__E,axiom,
! [B2: nat,B: $o > set_nat,A: set_o,A2: $o] :
( ( member_nat @ B2 @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ B @ A ) ) )
=> ( ~ ( member_nat @ B2 @ ( B @ A2 ) )
=> ~ ( member_o @ A2 @ A ) ) ) ).
% INT_E
thf(fact_801_INT__E,axiom,
! [B2: $o,B: a > set_o,A: set_a,A2: a] :
( ( member_o @ B2 @ ( comple3063163877087187839_set_o @ ( image_a_set_o @ B @ A ) ) )
=> ( ~ ( member_o @ B2 @ ( B @ A2 ) )
=> ~ ( member_a @ A2 @ A ) ) ) ).
% INT_E
thf(fact_802_INT__E,axiom,
! [B2: $o,B: nat > set_o,A: set_nat,A2: nat] :
( ( member_o @ B2 @ ( comple3063163877087187839_set_o @ ( image_nat_set_o @ B @ A ) ) )
=> ( ~ ( member_o @ B2 @ ( B @ A2 ) )
=> ~ ( member_nat @ A2 @ A ) ) ) ).
% INT_E
thf(fact_803_INT__E,axiom,
! [B2: $o,B: $o > set_o,A: set_o,A2: $o] :
( ( member_o @ B2 @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ B @ A ) ) )
=> ( ~ ( member_o @ B2 @ ( B @ A2 ) )
=> ~ ( member_o @ A2 @ A ) ) ) ).
% INT_E
thf(fact_804_UN__E,axiom,
! [B2: a,B: a > set_a,A: set_a] :
( ( member_a @ B2 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A ) ) )
=> ~ ! [X: a] :
( ( member_a @ X @ A )
=> ~ ( member_a @ B2 @ ( B @ X ) ) ) ) ).
% UN_E
thf(fact_805_UN__E,axiom,
! [B2: a,B: nat > set_a,A: set_nat] :
( ( member_a @ B2 @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ B @ A ) ) )
=> ~ ! [X: nat] :
( ( member_nat @ X @ A )
=> ~ ( member_a @ B2 @ ( B @ X ) ) ) ) ).
% UN_E
thf(fact_806_UN__E,axiom,
! [B2: a,B: $o > set_a,A: set_o] :
( ( member_a @ B2 @ ( comple2307003609928055243_set_a @ ( image_o_set_a @ B @ A ) ) )
=> ~ ! [X: $o] :
( ( member_o @ X @ A )
=> ~ ( member_a @ B2 @ ( B @ X ) ) ) ) ).
% UN_E
thf(fact_807_UN__E,axiom,
! [B2: nat,B: a > set_nat,A: set_a] :
( ( member_nat @ B2 @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ B @ A ) ) )
=> ~ ! [X: a] :
( ( member_a @ X @ A )
=> ~ ( member_nat @ B2 @ ( B @ X ) ) ) ) ).
% UN_E
thf(fact_808_UN__E,axiom,
! [B2: nat,B: nat > set_nat,A: set_nat] :
( ( member_nat @ B2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) )
=> ~ ! [X: nat] :
( ( member_nat @ X @ A )
=> ~ ( member_nat @ B2 @ ( B @ X ) ) ) ) ).
% UN_E
thf(fact_809_UN__E,axiom,
! [B2: nat,B: $o > set_nat,A: set_o] :
( ( member_nat @ B2 @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B @ A ) ) )
=> ~ ! [X: $o] :
( ( member_o @ X @ A )
=> ~ ( member_nat @ B2 @ ( B @ X ) ) ) ) ).
% UN_E
thf(fact_810_UN__E,axiom,
! [B2: $o,B: a > set_o,A: set_a] :
( ( member_o @ B2 @ ( comple90263536869209701_set_o @ ( image_a_set_o @ B @ A ) ) )
=> ~ ! [X: a] :
( ( member_a @ X @ A )
=> ~ ( member_o @ B2 @ ( B @ X ) ) ) ) ).
% UN_E
thf(fact_811_UN__E,axiom,
! [B2: $o,B: nat > set_o,A: set_nat] :
( ( member_o @ B2 @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B @ A ) ) )
=> ~ ! [X: nat] :
( ( member_nat @ X @ A )
=> ~ ( member_o @ B2 @ ( B @ X ) ) ) ) ).
% UN_E
thf(fact_812_UN__E,axiom,
! [B2: $o,B: $o > set_o,A: set_o] :
( ( member_o @ B2 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ A ) ) )
=> ~ ! [X: $o] :
( ( member_o @ X @ A )
=> ~ ( member_o @ B2 @ ( B @ X ) ) ) ) ).
% UN_E
thf(fact_813_Pow__def,axiom,
( pow_nat
= ( ^ [A3: set_nat] :
( collect_set_nat
@ ^ [B3: set_nat] : ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).
% Pow_def
thf(fact_814_INF__eq,axiom,
! [A: set_a,B: set_a,G2: a > $o,F2: a > $o] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ? [X4: a] :
( ( member_a @ X4 @ B )
& ( ord_less_eq_o @ ( G2 @ X4 ) @ ( F2 @ I3 ) ) ) )
=> ( ! [J3: a] :
( ( member_a @ J3 @ B )
=> ? [X4: a] :
( ( member_a @ X4 @ A )
& ( ord_less_eq_o @ ( F2 @ X4 ) @ ( G2 @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_a_o @ F2 @ A ) )
= ( complete_Inf_Inf_o @ ( image_a_o @ G2 @ B ) ) ) ) ) ).
% INF_eq
thf(fact_815_INF__eq,axiom,
! [A: set_a,B: set_nat,G2: nat > $o,F2: a > $o] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B )
& ( ord_less_eq_o @ ( G2 @ X4 ) @ ( F2 @ I3 ) ) ) )
=> ( ! [J3: nat] :
( ( member_nat @ J3 @ B )
=> ? [X4: a] :
( ( member_a @ X4 @ A )
& ( ord_less_eq_o @ ( F2 @ X4 ) @ ( G2 @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_a_o @ F2 @ A ) )
= ( complete_Inf_Inf_o @ ( image_nat_o @ G2 @ B ) ) ) ) ) ).
% INF_eq
thf(fact_816_INF__eq,axiom,
! [A: set_a,B: set_o,G2: $o > $o,F2: a > $o] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ? [X4: $o] :
( ( member_o @ X4 @ B )
& ( ord_less_eq_o @ ( G2 @ X4 ) @ ( F2 @ I3 ) ) ) )
=> ( ! [J3: $o] :
( ( member_o @ J3 @ B )
=> ? [X4: a] :
( ( member_a @ X4 @ A )
& ( ord_less_eq_o @ ( F2 @ X4 ) @ ( G2 @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_a_o @ F2 @ A ) )
= ( complete_Inf_Inf_o @ ( image_o_o @ G2 @ B ) ) ) ) ) ).
% INF_eq
thf(fact_817_INF__eq,axiom,
! [A: set_nat,B: set_a,G2: a > $o,F2: nat > $o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ? [X4: a] :
( ( member_a @ X4 @ B )
& ( ord_less_eq_o @ ( G2 @ X4 ) @ ( F2 @ I3 ) ) ) )
=> ( ! [J3: a] :
( ( member_a @ J3 @ B )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ord_less_eq_o @ ( F2 @ X4 ) @ ( G2 @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_nat_o @ F2 @ A ) )
= ( complete_Inf_Inf_o @ ( image_a_o @ G2 @ B ) ) ) ) ) ).
% INF_eq
thf(fact_818_INF__eq,axiom,
! [A: set_nat,B: set_nat,G2: nat > $o,F2: nat > $o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B )
& ( ord_less_eq_o @ ( G2 @ X4 ) @ ( F2 @ I3 ) ) ) )
=> ( ! [J3: nat] :
( ( member_nat @ J3 @ B )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ord_less_eq_o @ ( F2 @ X4 ) @ ( G2 @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_nat_o @ F2 @ A ) )
= ( complete_Inf_Inf_o @ ( image_nat_o @ G2 @ B ) ) ) ) ) ).
% INF_eq
thf(fact_819_INF__eq,axiom,
! [A: set_nat,B: set_o,G2: $o > $o,F2: nat > $o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ? [X4: $o] :
( ( member_o @ X4 @ B )
& ( ord_less_eq_o @ ( G2 @ X4 ) @ ( F2 @ I3 ) ) ) )
=> ( ! [J3: $o] :
( ( member_o @ J3 @ B )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ord_less_eq_o @ ( F2 @ X4 ) @ ( G2 @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_nat_o @ F2 @ A ) )
= ( complete_Inf_Inf_o @ ( image_o_o @ G2 @ B ) ) ) ) ) ).
% INF_eq
thf(fact_820_INF__eq,axiom,
! [A: set_o,B: set_a,G2: a > $o,F2: $o > $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ? [X4: a] :
( ( member_a @ X4 @ B )
& ( ord_less_eq_o @ ( G2 @ X4 ) @ ( F2 @ I3 ) ) ) )
=> ( ! [J3: a] :
( ( member_a @ J3 @ B )
=> ? [X4: $o] :
( ( member_o @ X4 @ A )
& ( ord_less_eq_o @ ( F2 @ X4 ) @ ( G2 @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ F2 @ A ) )
= ( complete_Inf_Inf_o @ ( image_a_o @ G2 @ B ) ) ) ) ) ).
% INF_eq
thf(fact_821_INF__eq,axiom,
! [A: set_o,B: set_nat,G2: nat > $o,F2: $o > $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B )
& ( ord_less_eq_o @ ( G2 @ X4 ) @ ( F2 @ I3 ) ) ) )
=> ( ! [J3: nat] :
( ( member_nat @ J3 @ B )
=> ? [X4: $o] :
( ( member_o @ X4 @ A )
& ( ord_less_eq_o @ ( F2 @ X4 ) @ ( G2 @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ F2 @ A ) )
= ( complete_Inf_Inf_o @ ( image_nat_o @ G2 @ B ) ) ) ) ) ).
% INF_eq
thf(fact_822_INF__eq,axiom,
! [A: set_o,B: set_o,G2: $o > $o,F2: $o > $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ? [X4: $o] :
( ( member_o @ X4 @ B )
& ( ord_less_eq_o @ ( G2 @ X4 ) @ ( F2 @ I3 ) ) ) )
=> ( ! [J3: $o] :
( ( member_o @ J3 @ B )
=> ? [X4: $o] :
( ( member_o @ X4 @ A )
& ( ord_less_eq_o @ ( F2 @ X4 ) @ ( G2 @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ F2 @ A ) )
= ( complete_Inf_Inf_o @ ( image_o_o @ G2 @ B ) ) ) ) ) ).
% INF_eq
thf(fact_823_INF__eq,axiom,
! [A: set_a,B: set_a,G2: a > set_nat,F2: a > set_nat] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ? [X4: a] :
( ( member_a @ X4 @ B )
& ( ord_less_eq_set_nat @ ( G2 @ X4 ) @ ( F2 @ I3 ) ) ) )
=> ( ! [J3: a] :
( ( member_a @ J3 @ B )
=> ? [X4: a] :
( ( member_a @ X4 @ A )
& ( ord_less_eq_set_nat @ ( F2 @ X4 ) @ ( G2 @ J3 ) ) ) )
=> ( ( comple7806235888213564991et_nat @ ( image_a_set_nat @ F2 @ A ) )
= ( comple7806235888213564991et_nat @ ( image_a_set_nat @ G2 @ B ) ) ) ) ) ).
% INF_eq
thf(fact_824_SUP__eq,axiom,
! [A: set_a,B: set_a,F2: a > $o,G2: a > $o] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ? [X4: a] :
( ( member_a @ X4 @ B )
& ( ord_less_eq_o @ ( F2 @ I3 ) @ ( G2 @ X4 ) ) ) )
=> ( ! [J3: a] :
( ( member_a @ J3 @ B )
=> ? [X4: a] :
( ( member_a @ X4 @ A )
& ( ord_less_eq_o @ ( G2 @ J3 ) @ ( F2 @ X4 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_a_o @ F2 @ A ) )
= ( complete_Sup_Sup_o @ ( image_a_o @ G2 @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_825_SUP__eq,axiom,
! [A: set_a,B: set_nat,F2: a > $o,G2: nat > $o] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B )
& ( ord_less_eq_o @ ( F2 @ I3 ) @ ( G2 @ X4 ) ) ) )
=> ( ! [J3: nat] :
( ( member_nat @ J3 @ B )
=> ? [X4: a] :
( ( member_a @ X4 @ A )
& ( ord_less_eq_o @ ( G2 @ J3 ) @ ( F2 @ X4 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_a_o @ F2 @ A ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ G2 @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_826_SUP__eq,axiom,
! [A: set_a,B: set_o,F2: a > $o,G2: $o > $o] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ? [X4: $o] :
( ( member_o @ X4 @ B )
& ( ord_less_eq_o @ ( F2 @ I3 ) @ ( G2 @ X4 ) ) ) )
=> ( ! [J3: $o] :
( ( member_o @ J3 @ B )
=> ? [X4: a] :
( ( member_a @ X4 @ A )
& ( ord_less_eq_o @ ( G2 @ J3 ) @ ( F2 @ X4 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_a_o @ F2 @ A ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ G2 @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_827_SUP__eq,axiom,
! [A: set_nat,B: set_a,F2: nat > $o,G2: a > $o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ? [X4: a] :
( ( member_a @ X4 @ B )
& ( ord_less_eq_o @ ( F2 @ I3 ) @ ( G2 @ X4 ) ) ) )
=> ( ! [J3: a] :
( ( member_a @ J3 @ B )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ord_less_eq_o @ ( G2 @ J3 ) @ ( F2 @ X4 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F2 @ A ) )
= ( complete_Sup_Sup_o @ ( image_a_o @ G2 @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_828_SUP__eq,axiom,
! [A: set_nat,B: set_nat,F2: nat > $o,G2: nat > $o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B )
& ( ord_less_eq_o @ ( F2 @ I3 ) @ ( G2 @ X4 ) ) ) )
=> ( ! [J3: nat] :
( ( member_nat @ J3 @ B )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ord_less_eq_o @ ( G2 @ J3 ) @ ( F2 @ X4 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F2 @ A ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ G2 @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_829_SUP__eq,axiom,
! [A: set_nat,B: set_o,F2: nat > $o,G2: $o > $o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ? [X4: $o] :
( ( member_o @ X4 @ B )
& ( ord_less_eq_o @ ( F2 @ I3 ) @ ( G2 @ X4 ) ) ) )
=> ( ! [J3: $o] :
( ( member_o @ J3 @ B )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ord_less_eq_o @ ( G2 @ J3 ) @ ( F2 @ X4 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F2 @ A ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ G2 @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_830_SUP__eq,axiom,
! [A: set_o,B: set_a,F2: $o > $o,G2: a > $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ? [X4: a] :
( ( member_a @ X4 @ B )
& ( ord_less_eq_o @ ( F2 @ I3 ) @ ( G2 @ X4 ) ) ) )
=> ( ! [J3: a] :
( ( member_a @ J3 @ B )
=> ? [X4: $o] :
( ( member_o @ X4 @ A )
& ( ord_less_eq_o @ ( G2 @ J3 ) @ ( F2 @ X4 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F2 @ A ) )
= ( complete_Sup_Sup_o @ ( image_a_o @ G2 @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_831_SUP__eq,axiom,
! [A: set_o,B: set_nat,F2: $o > $o,G2: nat > $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B )
& ( ord_less_eq_o @ ( F2 @ I3 ) @ ( G2 @ X4 ) ) ) )
=> ( ! [J3: nat] :
( ( member_nat @ J3 @ B )
=> ? [X4: $o] :
( ( member_o @ X4 @ A )
& ( ord_less_eq_o @ ( G2 @ J3 ) @ ( F2 @ X4 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F2 @ A ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ G2 @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_832_SUP__eq,axiom,
! [A: set_o,B: set_o,F2: $o > $o,G2: $o > $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ? [X4: $o] :
( ( member_o @ X4 @ B )
& ( ord_less_eq_o @ ( F2 @ I3 ) @ ( G2 @ X4 ) ) ) )
=> ( ! [J3: $o] :
( ( member_o @ J3 @ B )
=> ? [X4: $o] :
( ( member_o @ X4 @ A )
& ( ord_less_eq_o @ ( G2 @ J3 ) @ ( F2 @ X4 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F2 @ A ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ G2 @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_833_SUP__eq,axiom,
! [A: set_a,B: set_a,F2: a > set_nat,G2: a > set_nat] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ? [X4: a] :
( ( member_a @ X4 @ B )
& ( ord_less_eq_set_nat @ ( F2 @ I3 ) @ ( G2 @ X4 ) ) ) )
=> ( ! [J3: a] :
( ( member_a @ J3 @ B )
=> ? [X4: a] :
( ( member_a @ X4 @ A )
& ( ord_less_eq_set_nat @ ( G2 @ J3 ) @ ( F2 @ X4 ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_a_set_nat @ F2 @ A ) )
= ( comple7399068483239264473et_nat @ ( image_a_set_nat @ G2 @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_834_Inf__superset__mono,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A ) @ ( comple7806235888213564991et_nat @ B ) ) ) ).
% Inf_superset_mono
thf(fact_835_Inf__superset__mono,axiom,
! [B: set_o,A: set_o] :
( ( ord_less_eq_set_o @ B @ A )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A ) @ ( complete_Inf_Inf_o @ B ) ) ) ).
% Inf_superset_mono
thf(fact_836_Sup__subset__mono,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Sup_subset_mono
thf(fact_837_Sup__subset__mono,axiom,
! [A: set_o,B: set_o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ ( complete_Sup_Sup_o @ B ) ) ) ).
% Sup_subset_mono
thf(fact_838_sup__Inf,axiom,
! [A2: set_a,B: set_set_a] :
( ( sup_sup_set_a @ A2 @ ( comple6135023378680113637_set_a @ B ) )
= ( comple6135023378680113637_set_a @ ( image_set_a_set_a @ ( sup_sup_set_a @ A2 ) @ B ) ) ) ).
% sup_Inf
thf(fact_839_sup__Inf,axiom,
! [A2: set_nat,B: set_set_nat] :
( ( sup_sup_set_nat @ A2 @ ( comple7806235888213564991et_nat @ B ) )
= ( comple7806235888213564991et_nat @ ( image_7916887816326733075et_nat @ ( sup_sup_set_nat @ A2 ) @ B ) ) ) ).
% sup_Inf
thf(fact_840_sup__Inf,axiom,
! [A2: $o,B: set_o] :
( ( sup_sup_o @ A2 @ ( complete_Inf_Inf_o @ B ) )
= ( complete_Inf_Inf_o @ ( image_o_o @ ( sup_sup_o @ A2 ) @ B ) ) ) ).
% sup_Inf
thf(fact_841_Sup__union__distrib,axiom,
! [A: set_set_a,B: set_set_a] :
( ( comple2307003609928055243_set_a @ ( sup_sup_set_set_a @ A @ B ) )
= ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ A ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Sup_union_distrib
thf(fact_842_Sup__union__distrib,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( comple7399068483239264473et_nat @ ( sup_sup_set_set_nat @ A @ B ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Sup_union_distrib
thf(fact_843_Sup__union__distrib,axiom,
! [A: set_o,B: set_o] :
( ( complete_Sup_Sup_o @ ( sup_sup_set_o @ A @ B ) )
= ( sup_sup_o @ ( complete_Sup_Sup_o @ A ) @ ( complete_Sup_Sup_o @ B ) ) ) ).
% Sup_union_distrib
thf(fact_844_inj__on__image,axiom,
! [F2: nat > a,A: set_set_nat] :
( ( inj_on_nat_a @ F2 @ ( comple7399068483239264473et_nat @ A ) )
=> ( inj_on_set_nat_set_a @ ( image_nat_a @ F2 ) @ A ) ) ).
% inj_on_image
thf(fact_845_inj__on__image,axiom,
! [F2: nat > nat,A: set_set_nat] :
( ( inj_on_nat_nat @ F2 @ ( comple7399068483239264473et_nat @ A ) )
=> ( inj_on4604407203859583615et_nat @ ( image_nat_nat @ F2 ) @ A ) ) ).
% inj_on_image
thf(fact_846_inj__on__image,axiom,
! [F2: a > nat,A: set_set_a] :
( ( inj_on_a_nat @ F2 @ ( comple2307003609928055243_set_a @ A ) )
=> ( inj_on_set_a_set_nat @ ( image_a_nat @ F2 ) @ A ) ) ).
% inj_on_image
thf(fact_847_inj__on__image,axiom,
! [F2: a > a,A: set_set_a] :
( ( inj_on_a_a @ F2 @ ( comple2307003609928055243_set_a @ A ) )
=> ( inj_on_set_a_set_a @ ( image_a_a @ F2 ) @ A ) ) ).
% inj_on_image
thf(fact_848_Inter__anti__mono,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A ) @ ( comple7806235888213564991et_nat @ B ) ) ) ).
% Inter_anti_mono
thf(fact_849_Union__mono,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Union_mono
thf(fact_850_inj__on__image__Pow,axiom,
! [F2: nat > a,A: set_nat] :
( ( inj_on_nat_a @ F2 @ A )
=> ( inj_on_set_nat_set_a @ ( image_nat_a @ F2 ) @ ( pow_nat @ A ) ) ) ).
% inj_on_image_Pow
thf(fact_851_inj__on__image__Pow,axiom,
! [F2: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F2 @ A )
=> ( inj_on4604407203859583615et_nat @ ( image_nat_nat @ F2 ) @ ( pow_nat @ A ) ) ) ).
% inj_on_image_Pow
thf(fact_852_inj__on__image__Pow,axiom,
! [F2: a > nat,A: set_a] :
( ( inj_on_a_nat @ F2 @ A )
=> ( inj_on_set_a_set_nat @ ( image_a_nat @ F2 ) @ ( pow_a @ A ) ) ) ).
% inj_on_image_Pow
thf(fact_853_inj__on__image__Pow,axiom,
! [F2: a > a,A: set_a] :
( ( inj_on_a_a @ F2 @ A )
=> ( inj_on_set_a_set_a @ ( image_a_a @ F2 ) @ ( pow_a @ A ) ) ) ).
% inj_on_image_Pow
thf(fact_854_INF__eqI,axiom,
! [A: set_a,X3: set_nat,F2: a > set_nat] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ( ord_less_eq_set_nat @ X3 @ ( F2 @ I3 ) ) )
=> ( ! [Y5: set_nat] :
( ! [I4: a] :
( ( member_a @ I4 @ A )
=> ( ord_less_eq_set_nat @ Y5 @ ( F2 @ I4 ) ) )
=> ( ord_less_eq_set_nat @ Y5 @ X3 ) )
=> ( ( comple7806235888213564991et_nat @ ( image_a_set_nat @ F2 @ A ) )
= X3 ) ) ) ).
% INF_eqI
thf(fact_855_INF__eqI,axiom,
! [A: set_nat,X3: set_nat,F2: nat > set_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ( ord_less_eq_set_nat @ X3 @ ( F2 @ I3 ) ) )
=> ( ! [Y5: set_nat] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A )
=> ( ord_less_eq_set_nat @ Y5 @ ( F2 @ I4 ) ) )
=> ( ord_less_eq_set_nat @ Y5 @ X3 ) )
=> ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F2 @ A ) )
= X3 ) ) ) ).
% INF_eqI
thf(fact_856_INF__eqI,axiom,
! [A: set_o,X3: set_nat,F2: $o > set_nat] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ( ord_less_eq_set_nat @ X3 @ ( F2 @ I3 ) ) )
=> ( ! [Y5: set_nat] :
( ! [I4: $o] :
( ( member_o @ I4 @ A )
=> ( ord_less_eq_set_nat @ Y5 @ ( F2 @ I4 ) ) )
=> ( ord_less_eq_set_nat @ Y5 @ X3 ) )
=> ( ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F2 @ A ) )
= X3 ) ) ) ).
% INF_eqI
thf(fact_857_INF__eqI,axiom,
! [A: set_a,X3: $o,F2: a > $o] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ( ord_less_eq_o @ X3 @ ( F2 @ I3 ) ) )
=> ( ! [Y5: $o] :
( ! [I4: a] :
( ( member_a @ I4 @ A )
=> ( ord_less_eq_o @ Y5 @ ( F2 @ I4 ) ) )
=> ( ord_less_eq_o @ Y5 @ X3 ) )
=> ( ( complete_Inf_Inf_o @ ( image_a_o @ F2 @ A ) )
= X3 ) ) ) ).
% INF_eqI
thf(fact_858_INF__eqI,axiom,
! [A: set_nat,X3: $o,F2: nat > $o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ( ord_less_eq_o @ X3 @ ( F2 @ I3 ) ) )
=> ( ! [Y5: $o] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A )
=> ( ord_less_eq_o @ Y5 @ ( F2 @ I4 ) ) )
=> ( ord_less_eq_o @ Y5 @ X3 ) )
=> ( ( complete_Inf_Inf_o @ ( image_nat_o @ F2 @ A ) )
= X3 ) ) ) ).
% INF_eqI
thf(fact_859_INF__eqI,axiom,
! [A: set_o,X3: $o,F2: $o > $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ( ord_less_eq_o @ X3 @ ( F2 @ I3 ) ) )
=> ( ! [Y5: $o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A )
=> ( ord_less_eq_o @ Y5 @ ( F2 @ I4 ) ) )
=> ( ord_less_eq_o @ Y5 @ X3 ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ F2 @ A ) )
= X3 ) ) ) ).
% INF_eqI
thf(fact_860_INF__lower,axiom,
! [I5: a,A: set_a,F2: a > set_nat] :
( ( member_a @ I5 @ A )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_a_set_nat @ F2 @ A ) ) @ ( F2 @ I5 ) ) ) ).
% INF_lower
thf(fact_861_INF__lower,axiom,
! [I5: nat,A: set_nat,F2: nat > set_nat] :
( ( member_nat @ I5 @ A )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F2 @ A ) ) @ ( F2 @ I5 ) ) ) ).
% INF_lower
thf(fact_862_INF__lower,axiom,
! [I5: $o,A: set_o,F2: $o > set_nat] :
( ( member_o @ I5 @ A )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F2 @ A ) ) @ ( F2 @ I5 ) ) ) ).
% INF_lower
thf(fact_863_INF__lower,axiom,
! [I5: a,A: set_a,F2: a > $o] :
( ( member_a @ I5 @ A )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_a_o @ F2 @ A ) ) @ ( F2 @ I5 ) ) ) ).
% INF_lower
thf(fact_864_INF__lower,axiom,
! [I5: nat,A: set_nat,F2: nat > $o] :
( ( member_nat @ I5 @ A )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_nat_o @ F2 @ A ) ) @ ( F2 @ I5 ) ) ) ).
% INF_lower
thf(fact_865_INF__lower,axiom,
! [I5: $o,A: set_o,F2: $o > $o] :
( ( member_o @ I5 @ A )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_o_o @ F2 @ A ) ) @ ( F2 @ I5 ) ) ) ).
% INF_lower
thf(fact_866_INF__lower2,axiom,
! [I5: a,A: set_a,F2: a > set_nat,U: set_nat] :
( ( member_a @ I5 @ A )
=> ( ( ord_less_eq_set_nat @ ( F2 @ I5 ) @ U )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_a_set_nat @ F2 @ A ) ) @ U ) ) ) ).
% INF_lower2
thf(fact_867_INF__lower2,axiom,
! [I5: nat,A: set_nat,F2: nat > set_nat,U: set_nat] :
( ( member_nat @ I5 @ A )
=> ( ( ord_less_eq_set_nat @ ( F2 @ I5 ) @ U )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F2 @ A ) ) @ U ) ) ) ).
% INF_lower2
thf(fact_868_INF__lower2,axiom,
! [I5: $o,A: set_o,F2: $o > set_nat,U: set_nat] :
( ( member_o @ I5 @ A )
=> ( ( ord_less_eq_set_nat @ ( F2 @ I5 ) @ U )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F2 @ A ) ) @ U ) ) ) ).
% INF_lower2
thf(fact_869_INF__lower2,axiom,
! [I5: a,A: set_a,F2: a > $o,U: $o] :
( ( member_a @ I5 @ A )
=> ( ( ord_less_eq_o @ ( F2 @ I5 ) @ U )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_a_o @ F2 @ A ) ) @ U ) ) ) ).
% INF_lower2
thf(fact_870_INF__lower2,axiom,
! [I5: nat,A: set_nat,F2: nat > $o,U: $o] :
( ( member_nat @ I5 @ A )
=> ( ( ord_less_eq_o @ ( F2 @ I5 ) @ U )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_nat_o @ F2 @ A ) ) @ U ) ) ) ).
% INF_lower2
thf(fact_871_INF__lower2,axiom,
! [I5: $o,A: set_o,F2: $o > $o,U: $o] :
( ( member_o @ I5 @ A )
=> ( ( ord_less_eq_o @ ( F2 @ I5 ) @ U )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_o_o @ F2 @ A ) ) @ U ) ) ) ).
% INF_lower2
thf(fact_872_INF__greatest,axiom,
! [A: set_a,U: set_nat,F2: a > set_nat] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ( ord_less_eq_set_nat @ U @ ( F2 @ I3 ) ) )
=> ( ord_less_eq_set_nat @ U @ ( comple7806235888213564991et_nat @ ( image_a_set_nat @ F2 @ A ) ) ) ) ).
% INF_greatest
thf(fact_873_INF__greatest,axiom,
! [A: set_nat,U: set_nat,F2: nat > set_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ( ord_less_eq_set_nat @ U @ ( F2 @ I3 ) ) )
=> ( ord_less_eq_set_nat @ U @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F2 @ A ) ) ) ) ).
% INF_greatest
thf(fact_874_INF__greatest,axiom,
! [A: set_o,U: set_nat,F2: $o > set_nat] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ( ord_less_eq_set_nat @ U @ ( F2 @ I3 ) ) )
=> ( ord_less_eq_set_nat @ U @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F2 @ A ) ) ) ) ).
% INF_greatest
thf(fact_875_INF__greatest,axiom,
! [A: set_a,U: $o,F2: a > $o] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ( ord_less_eq_o @ U @ ( F2 @ I3 ) ) )
=> ( ord_less_eq_o @ U @ ( complete_Inf_Inf_o @ ( image_a_o @ F2 @ A ) ) ) ) ).
% INF_greatest
thf(fact_876_INF__greatest,axiom,
! [A: set_nat,U: $o,F2: nat > $o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ( ord_less_eq_o @ U @ ( F2 @ I3 ) ) )
=> ( ord_less_eq_o @ U @ ( complete_Inf_Inf_o @ ( image_nat_o @ F2 @ A ) ) ) ) ).
% INF_greatest
thf(fact_877_INF__greatest,axiom,
! [A: set_o,U: $o,F2: $o > $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ( ord_less_eq_o @ U @ ( F2 @ I3 ) ) )
=> ( ord_less_eq_o @ U @ ( complete_Inf_Inf_o @ ( image_o_o @ F2 @ A ) ) ) ) ).
% INF_greatest
thf(fact_878_SUP__eqI,axiom,
! [A: set_a,F2: a > set_nat,X3: set_nat] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ( ord_less_eq_set_nat @ ( F2 @ I3 ) @ X3 ) )
=> ( ! [Y5: set_nat] :
( ! [I4: a] :
( ( member_a @ I4 @ A )
=> ( ord_less_eq_set_nat @ ( F2 @ I4 ) @ Y5 ) )
=> ( ord_less_eq_set_nat @ X3 @ Y5 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_a_set_nat @ F2 @ A ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_879_SUP__eqI,axiom,
! [A: set_nat,F2: nat > set_nat,X3: set_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ( ord_less_eq_set_nat @ ( F2 @ I3 ) @ X3 ) )
=> ( ! [Y5: set_nat] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A )
=> ( ord_less_eq_set_nat @ ( F2 @ I4 ) @ Y5 ) )
=> ( ord_less_eq_set_nat @ X3 @ Y5 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ A ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_880_SUP__eqI,axiom,
! [A: set_o,F2: $o > set_nat,X3: set_nat] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ( ord_less_eq_set_nat @ ( F2 @ I3 ) @ X3 ) )
=> ( ! [Y5: set_nat] :
( ! [I4: $o] :
( ( member_o @ I4 @ A )
=> ( ord_less_eq_set_nat @ ( F2 @ I4 ) @ Y5 ) )
=> ( ord_less_eq_set_nat @ X3 @ Y5 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F2 @ A ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_881_SUP__eqI,axiom,
! [A: set_a,F2: a > $o,X3: $o] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ( ord_less_eq_o @ ( F2 @ I3 ) @ X3 ) )
=> ( ! [Y5: $o] :
( ! [I4: a] :
( ( member_a @ I4 @ A )
=> ( ord_less_eq_o @ ( F2 @ I4 ) @ Y5 ) )
=> ( ord_less_eq_o @ X3 @ Y5 ) )
=> ( ( complete_Sup_Sup_o @ ( image_a_o @ F2 @ A ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_882_SUP__eqI,axiom,
! [A: set_nat,F2: nat > $o,X3: $o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ( ord_less_eq_o @ ( F2 @ I3 ) @ X3 ) )
=> ( ! [Y5: $o] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A )
=> ( ord_less_eq_o @ ( F2 @ I4 ) @ Y5 ) )
=> ( ord_less_eq_o @ X3 @ Y5 ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F2 @ A ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_883_SUP__eqI,axiom,
! [A: set_o,F2: $o > $o,X3: $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ( ord_less_eq_o @ ( F2 @ I3 ) @ X3 ) )
=> ( ! [Y5: $o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A )
=> ( ord_less_eq_o @ ( F2 @ I4 ) @ Y5 ) )
=> ( ord_less_eq_o @ X3 @ Y5 ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F2 @ A ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_884_SUP__least,axiom,
! [A: set_a,F2: a > set_nat,U: set_nat] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ( ord_less_eq_set_nat @ ( F2 @ I3 ) @ U ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ F2 @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_885_SUP__least,axiom,
! [A: set_nat,F2: nat > set_nat,U: set_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ( ord_less_eq_set_nat @ ( F2 @ I3 ) @ U ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_886_SUP__least,axiom,
! [A: set_o,F2: $o > set_nat,U: set_nat] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ( ord_less_eq_set_nat @ ( F2 @ I3 ) @ U ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F2 @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_887_SUP__least,axiom,
! [A: set_a,F2: a > $o,U: $o] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ( ord_less_eq_o @ ( F2 @ I3 ) @ U ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_a_o @ F2 @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_888_SUP__least,axiom,
! [A: set_nat,F2: nat > $o,U: $o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ( ord_less_eq_o @ ( F2 @ I3 ) @ U ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_nat_o @ F2 @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_889_SUP__least,axiom,
! [A: set_o,F2: $o > $o,U: $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ( ord_less_eq_o @ ( F2 @ I3 ) @ U ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F2 @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_890_SUP__upper,axiom,
! [I5: a,A: set_a,F2: a > set_nat] :
( ( member_a @ I5 @ A )
=> ( ord_less_eq_set_nat @ ( F2 @ I5 ) @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ F2 @ A ) ) ) ) ).
% SUP_upper
thf(fact_891_SUP__upper,axiom,
! [I5: nat,A: set_nat,F2: nat > set_nat] :
( ( member_nat @ I5 @ A )
=> ( ord_less_eq_set_nat @ ( F2 @ I5 ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ A ) ) ) ) ).
% SUP_upper
thf(fact_892_SUP__upper,axiom,
! [I5: $o,A: set_o,F2: $o > set_nat] :
( ( member_o @ I5 @ A )
=> ( ord_less_eq_set_nat @ ( F2 @ I5 ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F2 @ A ) ) ) ) ).
% SUP_upper
thf(fact_893_SUP__upper,axiom,
! [I5: a,A: set_a,F2: a > $o] :
( ( member_a @ I5 @ A )
=> ( ord_less_eq_o @ ( F2 @ I5 ) @ ( complete_Sup_Sup_o @ ( image_a_o @ F2 @ A ) ) ) ) ).
% SUP_upper
thf(fact_894_SUP__upper,axiom,
! [I5: nat,A: set_nat,F2: nat > $o] :
( ( member_nat @ I5 @ A )
=> ( ord_less_eq_o @ ( F2 @ I5 ) @ ( complete_Sup_Sup_o @ ( image_nat_o @ F2 @ A ) ) ) ) ).
% SUP_upper
thf(fact_895_SUP__upper,axiom,
! [I5: $o,A: set_o,F2: $o > $o] :
( ( member_o @ I5 @ A )
=> ( ord_less_eq_o @ ( F2 @ I5 ) @ ( complete_Sup_Sup_o @ ( image_o_o @ F2 @ A ) ) ) ) ).
% SUP_upper
thf(fact_896_SUP__upper2,axiom,
! [I5: a,A: set_a,U: set_nat,F2: a > set_nat] :
( ( member_a @ I5 @ A )
=> ( ( ord_less_eq_set_nat @ U @ ( F2 @ I5 ) )
=> ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ F2 @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_897_SUP__upper2,axiom,
! [I5: nat,A: set_nat,U: set_nat,F2: nat > set_nat] :
( ( member_nat @ I5 @ A )
=> ( ( ord_less_eq_set_nat @ U @ ( F2 @ I5 ) )
=> ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_898_SUP__upper2,axiom,
! [I5: $o,A: set_o,U: set_nat,F2: $o > set_nat] :
( ( member_o @ I5 @ A )
=> ( ( ord_less_eq_set_nat @ U @ ( F2 @ I5 ) )
=> ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F2 @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_899_SUP__upper2,axiom,
! [I5: a,A: set_a,U: $o,F2: a > $o] :
( ( member_a @ I5 @ A )
=> ( ( ord_less_eq_o @ U @ ( F2 @ I5 ) )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ ( image_a_o @ F2 @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_900_SUP__upper2,axiom,
! [I5: nat,A: set_nat,U: $o,F2: nat > $o] :
( ( member_nat @ I5 @ A )
=> ( ( ord_less_eq_o @ U @ ( F2 @ I5 ) )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ ( image_nat_o @ F2 @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_901_SUP__upper2,axiom,
! [I5: $o,A: set_o,U: $o,F2: $o > $o] :
( ( member_o @ I5 @ A )
=> ( ( ord_less_eq_o @ U @ ( F2 @ I5 ) )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ ( image_o_o @ F2 @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_902_Inf__sup,axiom,
! [B: set_set_a,A2: set_a] :
( ( sup_sup_set_a @ ( comple6135023378680113637_set_a @ B ) @ A2 )
= ( comple6135023378680113637_set_a
@ ( image_set_a_set_a
@ ^ [B4: set_a] : ( sup_sup_set_a @ B4 @ A2 )
@ B ) ) ) ).
% Inf_sup
thf(fact_903_Inf__sup,axiom,
! [B: set_set_nat,A2: set_nat] :
( ( sup_sup_set_nat @ ( comple7806235888213564991et_nat @ B ) @ A2 )
= ( comple7806235888213564991et_nat
@ ( image_7916887816326733075et_nat
@ ^ [B4: set_nat] : ( sup_sup_set_nat @ B4 @ A2 )
@ B ) ) ) ).
% Inf_sup
thf(fact_904_Inf__sup,axiom,
! [B: set_o,A2: $o] :
( ( sup_sup_o @ ( complete_Inf_Inf_o @ B ) @ A2 )
= ( complete_Inf_Inf_o
@ ( image_o_o
@ ^ [B4: $o] : ( sup_sup_o @ B4 @ A2 )
@ B ) ) ) ).
% Inf_sup
thf(fact_905_SUP__absorb,axiom,
! [K: a,I: set_a,A: a > set_a] :
( ( member_a @ K @ I )
=> ( ( sup_sup_set_a @ ( A @ K ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ A @ I ) ) )
= ( comple2307003609928055243_set_a @ ( image_a_set_a @ A @ I ) ) ) ) ).
% SUP_absorb
thf(fact_906_SUP__absorb,axiom,
! [K: nat,I: set_nat,A: nat > set_a] :
( ( member_nat @ K @ I )
=> ( ( sup_sup_set_a @ ( A @ K ) @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ A @ I ) ) )
= ( comple2307003609928055243_set_a @ ( image_nat_set_a @ A @ I ) ) ) ) ).
% SUP_absorb
thf(fact_907_SUP__absorb,axiom,
! [K: $o,I: set_o,A: $o > set_a] :
( ( member_o @ K @ I )
=> ( ( sup_sup_set_a @ ( A @ K ) @ ( comple2307003609928055243_set_a @ ( image_o_set_a @ A @ I ) ) )
= ( comple2307003609928055243_set_a @ ( image_o_set_a @ A @ I ) ) ) ) ).
% SUP_absorb
thf(fact_908_SUP__absorb,axiom,
! [K: a,I: set_a,A: a > set_nat] :
( ( member_a @ K @ I )
=> ( ( sup_sup_set_nat @ ( A @ K ) @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ A @ I ) ) )
= ( comple7399068483239264473et_nat @ ( image_a_set_nat @ A @ I ) ) ) ) ).
% SUP_absorb
thf(fact_909_SUP__absorb,axiom,
! [K: nat,I: set_nat,A: nat > set_nat] :
( ( member_nat @ K @ I )
=> ( ( sup_sup_set_nat @ ( A @ K ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A @ I ) ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A @ I ) ) ) ) ).
% SUP_absorb
thf(fact_910_SUP__absorb,axiom,
! [K: $o,I: set_o,A: $o > set_nat] :
( ( member_o @ K @ I )
=> ( ( sup_sup_set_nat @ ( A @ K ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A @ I ) ) )
= ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A @ I ) ) ) ) ).
% SUP_absorb
thf(fact_911_SUP__absorb,axiom,
! [K: a,I: set_a,A: a > $o] :
( ( member_a @ K @ I )
=> ( ( sup_sup_o @ ( A @ K ) @ ( complete_Sup_Sup_o @ ( image_a_o @ A @ I ) ) )
= ( complete_Sup_Sup_o @ ( image_a_o @ A @ I ) ) ) ) ).
% SUP_absorb
thf(fact_912_SUP__absorb,axiom,
! [K: nat,I: set_nat,A: nat > $o] :
( ( member_nat @ K @ I )
=> ( ( sup_sup_o @ ( A @ K ) @ ( complete_Sup_Sup_o @ ( image_nat_o @ A @ I ) ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ A @ I ) ) ) ) ).
% SUP_absorb
thf(fact_913_SUP__absorb,axiom,
! [K: $o,I: set_o,A: $o > $o] :
( ( member_o @ K @ I )
=> ( ( sup_sup_o @ ( A @ K ) @ ( complete_Sup_Sup_o @ ( image_o_o @ A @ I ) ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ A @ I ) ) ) ) ).
% SUP_absorb
thf(fact_914_finite__UnionD,axiom,
! [A: set_set_a] :
( ( finite_finite_a @ ( comple2307003609928055243_set_a @ A ) )
=> ( finite_finite_set_a @ A ) ) ).
% finite_UnionD
thf(fact_915_finite__UnionD,axiom,
! [A: set_set_nat] :
( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ A ) )
=> ( finite1152437895449049373et_nat @ A ) ) ).
% finite_UnionD
thf(fact_916_Pow__mono,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_le6893508408891458716et_nat @ ( pow_nat @ A ) @ ( pow_nat @ B ) ) ) ).
% Pow_mono
thf(fact_917_Un__Pow__subset,axiom,
! [A: set_a,B: set_a] : ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ ( pow_a @ A ) @ ( pow_a @ B ) ) @ ( pow_a @ ( sup_sup_set_a @ A @ B ) ) ) ).
% Un_Pow_subset
thf(fact_918_Un__Pow__subset,axiom,
! [A: set_nat,B: set_nat] : ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ ( pow_nat @ A ) @ ( pow_nat @ B ) ) @ ( pow_nat @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% Un_Pow_subset
thf(fact_919_INT__anti__mono,axiom,
! [A: set_a,B: set_a,F2: a > set_nat,G2: a > set_nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_set_nat @ ( F2 @ X ) @ ( G2 @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_a_set_nat @ F2 @ B ) ) @ ( comple7806235888213564991et_nat @ ( image_a_set_nat @ G2 @ A ) ) ) ) ) ).
% INT_anti_mono
thf(fact_920_INT__anti__mono,axiom,
! [A: set_o,B: set_o,F2: $o > set_nat,G2: $o > set_nat] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_set_nat @ ( F2 @ X ) @ ( G2 @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F2 @ B ) ) @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ G2 @ A ) ) ) ) ) ).
% INT_anti_mono
thf(fact_921_INT__anti__mono,axiom,
! [A: set_nat,B: set_nat,F2: nat > set_nat,G2: nat > set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_set_nat @ ( F2 @ X ) @ ( G2 @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F2 @ B ) ) @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ G2 @ A ) ) ) ) ) ).
% INT_anti_mono
thf(fact_922_INT__greatest,axiom,
! [A: set_a,C2: set_nat,B: a > set_nat] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_set_nat @ C2 @ ( B @ X ) ) )
=> ( ord_less_eq_set_nat @ C2 @ ( comple7806235888213564991et_nat @ ( image_a_set_nat @ B @ A ) ) ) ) ).
% INT_greatest
thf(fact_923_INT__greatest,axiom,
! [A: set_nat,C2: set_nat,B: nat > set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_set_nat @ C2 @ ( B @ X ) ) )
=> ( ord_less_eq_set_nat @ C2 @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B @ A ) ) ) ) ).
% INT_greatest
thf(fact_924_INT__greatest,axiom,
! [A: set_o,C2: set_nat,B: $o > set_nat] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_set_nat @ C2 @ ( B @ X ) ) )
=> ( ord_less_eq_set_nat @ C2 @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ B @ A ) ) ) ) ).
% INT_greatest
thf(fact_925_INT__lower,axiom,
! [A2: a,A: set_a,B: a > set_nat] :
( ( member_a @ A2 @ A )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_a_set_nat @ B @ A ) ) @ ( B @ A2 ) ) ) ).
% INT_lower
thf(fact_926_INT__lower,axiom,
! [A2: nat,A: set_nat,B: nat > set_nat] :
( ( member_nat @ A2 @ A )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B @ A ) ) @ ( B @ A2 ) ) ) ).
% INT_lower
thf(fact_927_INT__lower,axiom,
! [A2: $o,A: set_o,B: $o > set_nat] :
( ( member_o @ A2 @ A )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ B @ A ) ) @ ( B @ A2 ) ) ) ).
% INT_lower
thf(fact_928_image__Pow__surj,axiom,
! [F2: nat > a,A: set_nat,B: set_a] :
( ( ( image_nat_a @ F2 @ A )
= B )
=> ( ( image_set_nat_set_a @ ( image_nat_a @ F2 ) @ ( pow_nat @ A ) )
= ( pow_a @ B ) ) ) ).
% image_Pow_surj
thf(fact_929_image__Pow__surj,axiom,
! [F2: nat > nat,A: set_nat,B: set_nat] :
( ( ( image_nat_nat @ F2 @ A )
= B )
=> ( ( image_7916887816326733075et_nat @ ( image_nat_nat @ F2 ) @ ( pow_nat @ A ) )
= ( pow_nat @ B ) ) ) ).
% image_Pow_surj
thf(fact_930_image__Pow__surj,axiom,
! [F2: a > nat,A: set_a,B: set_nat] :
( ( ( image_a_nat @ F2 @ A )
= B )
=> ( ( image_set_a_set_nat @ ( image_a_nat @ F2 ) @ ( pow_a @ A ) )
= ( pow_nat @ B ) ) ) ).
% image_Pow_surj
thf(fact_931_image__Pow__surj,axiom,
! [F2: a > a,A: set_a,B: set_a] :
( ( ( image_a_a @ F2 @ A )
= B )
=> ( ( image_set_a_set_a @ ( image_a_a @ F2 ) @ ( pow_a @ A ) )
= ( pow_a @ B ) ) ) ).
% image_Pow_surj
thf(fact_932_UN__upper,axiom,
! [A2: a,A: set_a,B: a > set_nat] :
( ( member_a @ A2 @ A )
=> ( ord_less_eq_set_nat @ ( B @ A2 ) @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ B @ A ) ) ) ) ).
% UN_upper
thf(fact_933_UN__upper,axiom,
! [A2: nat,A: set_nat,B: nat > set_nat] :
( ( member_nat @ A2 @ A )
=> ( ord_less_eq_set_nat @ ( B @ A2 ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) ) ) ).
% UN_upper
thf(fact_934_UN__upper,axiom,
! [A2: $o,A: set_o,B: $o > set_nat] :
( ( member_o @ A2 @ A )
=> ( ord_less_eq_set_nat @ ( B @ A2 ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B @ A ) ) ) ) ).
% UN_upper
thf(fact_935_UN__least,axiom,
! [A: set_a,B: a > set_nat,C2: set_nat] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_set_nat @ ( B @ X ) @ C2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ B @ A ) ) @ C2 ) ) ).
% UN_least
thf(fact_936_UN__least,axiom,
! [A: set_nat,B: nat > set_nat,C2: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_set_nat @ ( B @ X ) @ C2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) @ C2 ) ) ).
% UN_least
thf(fact_937_UN__least,axiom,
! [A: set_o,B: $o > set_nat,C2: set_nat] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_set_nat @ ( B @ X ) @ C2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B @ A ) ) @ C2 ) ) ).
% UN_least
thf(fact_938_UN__mono,axiom,
! [A: set_a,B: set_a,F2: a > set_nat,G2: a > set_nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_set_nat @ ( F2 @ X ) @ ( G2 @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ F2 @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ G2 @ B ) ) ) ) ) ).
% UN_mono
thf(fact_939_UN__mono,axiom,
! [A: set_o,B: set_o,F2: $o > set_nat,G2: $o > set_nat] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_set_nat @ ( F2 @ X ) @ ( G2 @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F2 @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ G2 @ B ) ) ) ) ) ).
% UN_mono
thf(fact_940_UN__mono,axiom,
! [A: set_nat,B: set_nat,F2: nat > set_nat,G2: nat > set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_set_nat @ ( F2 @ X ) @ ( G2 @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G2 @ B ) ) ) ) ) ).
% UN_mono
thf(fact_941_UN__absorb,axiom,
! [K: a,I: set_a,A: a > set_a] :
( ( member_a @ K @ I )
=> ( ( sup_sup_set_a @ ( A @ K ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ A @ I ) ) )
= ( comple2307003609928055243_set_a @ ( image_a_set_a @ A @ I ) ) ) ) ).
% UN_absorb
thf(fact_942_UN__absorb,axiom,
! [K: nat,I: set_nat,A: nat > set_a] :
( ( member_nat @ K @ I )
=> ( ( sup_sup_set_a @ ( A @ K ) @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ A @ I ) ) )
= ( comple2307003609928055243_set_a @ ( image_nat_set_a @ A @ I ) ) ) ) ).
% UN_absorb
thf(fact_943_UN__absorb,axiom,
! [K: $o,I: set_o,A: $o > set_a] :
( ( member_o @ K @ I )
=> ( ( sup_sup_set_a @ ( A @ K ) @ ( comple2307003609928055243_set_a @ ( image_o_set_a @ A @ I ) ) )
= ( comple2307003609928055243_set_a @ ( image_o_set_a @ A @ I ) ) ) ) ).
% UN_absorb
thf(fact_944_UN__absorb,axiom,
! [K: a,I: set_a,A: a > set_nat] :
( ( member_a @ K @ I )
=> ( ( sup_sup_set_nat @ ( A @ K ) @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ A @ I ) ) )
= ( comple7399068483239264473et_nat @ ( image_a_set_nat @ A @ I ) ) ) ) ).
% UN_absorb
thf(fact_945_UN__absorb,axiom,
! [K: nat,I: set_nat,A: nat > set_nat] :
( ( member_nat @ K @ I )
=> ( ( sup_sup_set_nat @ ( A @ K ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A @ I ) ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A @ I ) ) ) ) ).
% UN_absorb
thf(fact_946_UN__absorb,axiom,
! [K: $o,I: set_o,A: $o > set_nat] :
( ( member_o @ K @ I )
=> ( ( sup_sup_set_nat @ ( A @ K ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A @ I ) ) )
= ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A @ I ) ) ) ) ).
% UN_absorb
thf(fact_947_Un__Inter,axiom,
! [A: set_a,B: set_set_a] :
( ( sup_sup_set_a @ A @ ( comple6135023378680113637_set_a @ B ) )
= ( comple6135023378680113637_set_a @ ( image_set_a_set_a @ ( sup_sup_set_a @ A ) @ B ) ) ) ).
% Un_Inter
thf(fact_948_Un__Inter,axiom,
! [A: set_nat,B: set_set_nat] :
( ( sup_sup_set_nat @ A @ ( comple7806235888213564991et_nat @ B ) )
= ( comple7806235888213564991et_nat @ ( image_7916887816326733075et_nat @ ( sup_sup_set_nat @ A ) @ B ) ) ) ).
% Un_Inter
thf(fact_949_image__Union,axiom,
! [F2: nat > a,S: set_set_nat] :
( ( image_nat_a @ F2 @ ( comple7399068483239264473et_nat @ S ) )
= ( comple2307003609928055243_set_a @ ( image_set_nat_set_a @ ( image_nat_a @ F2 ) @ S ) ) ) ).
% image_Union
thf(fact_950_image__Union,axiom,
! [F2: nat > nat,S: set_set_nat] :
( ( image_nat_nat @ F2 @ ( comple7399068483239264473et_nat @ S ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F2 ) @ S ) ) ) ).
% image_Union
thf(fact_951_image__Union,axiom,
! [F2: a > nat,S: set_set_a] :
( ( image_a_nat @ F2 @ ( comple2307003609928055243_set_a @ S ) )
= ( comple7399068483239264473et_nat @ ( image_set_a_set_nat @ ( image_a_nat @ F2 ) @ S ) ) ) ).
% image_Union
thf(fact_952_image__Union,axiom,
! [F2: a > a,S: set_set_a] :
( ( image_a_a @ F2 @ ( comple2307003609928055243_set_a @ S ) )
= ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ ( image_a_a @ F2 ) @ S ) ) ) ).
% image_Union
thf(fact_953_f__the__inv__into__f,axiom,
! [F2: nat > a,A: set_nat,Y2: a] :
( ( inj_on_nat_a @ F2 @ A )
=> ( ( member_a @ Y2 @ ( image_nat_a @ F2 @ A ) )
=> ( ( F2 @ ( the_inv_into_nat_a @ A @ F2 @ Y2 ) )
= Y2 ) ) ) ).
% f_the_inv_into_f
thf(fact_954_f__the__inv__into__f,axiom,
! [F2: a > a,A: set_a,Y2: a] :
( ( inj_on_a_a @ F2 @ A )
=> ( ( member_a @ Y2 @ ( image_a_a @ F2 @ A ) )
=> ( ( F2 @ ( the_inv_into_a_a @ A @ F2 @ Y2 ) )
= Y2 ) ) ) ).
% f_the_inv_into_f
thf(fact_955_f__the__inv__into__f,axiom,
! [F2: nat > nat,A: set_nat,Y2: nat] :
( ( inj_on_nat_nat @ F2 @ A )
=> ( ( member_nat @ Y2 @ ( image_nat_nat @ F2 @ A ) )
=> ( ( F2 @ ( the_inv_into_nat_nat @ A @ F2 @ Y2 ) )
= Y2 ) ) ) ).
% f_the_inv_into_f
thf(fact_956_f__the__inv__into__f,axiom,
! [F2: a > nat,A: set_a,Y2: nat] :
( ( inj_on_a_nat @ F2 @ A )
=> ( ( member_nat @ Y2 @ ( image_a_nat @ F2 @ A ) )
=> ( ( F2 @ ( the_inv_into_a_nat @ A @ F2 @ Y2 ) )
= Y2 ) ) ) ).
% f_the_inv_into_f
thf(fact_957_inj__on__the__inv__into,axiom,
! [F2: nat > a,A: set_nat] :
( ( inj_on_nat_a @ F2 @ A )
=> ( inj_on_a_nat @ ( the_inv_into_nat_a @ A @ F2 ) @ ( image_nat_a @ F2 @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_958_inj__on__the__inv__into,axiom,
! [F2: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F2 @ A )
=> ( inj_on_nat_nat @ ( the_inv_into_nat_nat @ A @ F2 ) @ ( image_nat_nat @ F2 @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_959_inj__on__the__inv__into,axiom,
! [F2: a > nat,A: set_a] :
( ( inj_on_a_nat @ F2 @ A )
=> ( inj_on_nat_a @ ( the_inv_into_a_nat @ A @ F2 ) @ ( image_a_nat @ F2 @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_960_inj__on__the__inv__into,axiom,
! [F2: a > a,A: set_a] :
( ( inj_on_a_a @ F2 @ A )
=> ( inj_on_a_a @ ( the_inv_into_a_a @ A @ F2 ) @ ( image_a_a @ F2 @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_961_INF__superset__mono,axiom,
! [B: set_a,A: set_a,F2: a > set_nat,G2: a > set_nat] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ! [X: a] :
( ( member_a @ X @ B )
=> ( ord_less_eq_set_nat @ ( F2 @ X ) @ ( G2 @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_a_set_nat @ F2 @ A ) ) @ ( comple7806235888213564991et_nat @ ( image_a_set_nat @ G2 @ B ) ) ) ) ) ).
% INF_superset_mono
thf(fact_962_INF__superset__mono,axiom,
! [B: set_o,A: set_o,F2: $o > set_nat,G2: $o > set_nat] :
( ( ord_less_eq_set_o @ B @ A )
=> ( ! [X: $o] :
( ( member_o @ X @ B )
=> ( ord_less_eq_set_nat @ ( F2 @ X ) @ ( G2 @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F2 @ A ) ) @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ G2 @ B ) ) ) ) ) ).
% INF_superset_mono
thf(fact_963_INF__superset__mono,axiom,
! [B: set_nat,A: set_nat,F2: nat > set_nat,G2: nat > set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( ord_less_eq_set_nat @ ( F2 @ X ) @ ( G2 @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F2 @ A ) ) @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ G2 @ B ) ) ) ) ) ).
% INF_superset_mono
thf(fact_964_INF__superset__mono,axiom,
! [B: set_a,A: set_a,F2: a > $o,G2: a > $o] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ! [X: a] :
( ( member_a @ X @ B )
=> ( ord_less_eq_o @ ( F2 @ X ) @ ( G2 @ X ) ) )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_a_o @ F2 @ A ) ) @ ( complete_Inf_Inf_o @ ( image_a_o @ G2 @ B ) ) ) ) ) ).
% INF_superset_mono
thf(fact_965_INF__superset__mono,axiom,
! [B: set_o,A: set_o,F2: $o > $o,G2: $o > $o] :
( ( ord_less_eq_set_o @ B @ A )
=> ( ! [X: $o] :
( ( member_o @ X @ B )
=> ( ord_less_eq_o @ ( F2 @ X ) @ ( G2 @ X ) ) )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_o_o @ F2 @ A ) ) @ ( complete_Inf_Inf_o @ ( image_o_o @ G2 @ B ) ) ) ) ) ).
% INF_superset_mono
thf(fact_966_INF__superset__mono,axiom,
! [B: set_nat,A: set_nat,F2: nat > $o,G2: nat > $o] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( ord_less_eq_o @ ( F2 @ X ) @ ( G2 @ X ) ) )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_nat_o @ F2 @ A ) ) @ ( complete_Inf_Inf_o @ ( image_nat_o @ G2 @ B ) ) ) ) ) ).
% INF_superset_mono
thf(fact_967_SUP__subset__mono,axiom,
! [A: set_a,B: set_a,F2: a > set_nat,G2: a > set_nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_set_nat @ ( F2 @ X ) @ ( G2 @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ F2 @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ G2 @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_968_SUP__subset__mono,axiom,
! [A: set_o,B: set_o,F2: $o > set_nat,G2: $o > set_nat] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_set_nat @ ( F2 @ X ) @ ( G2 @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F2 @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ G2 @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_969_SUP__subset__mono,axiom,
! [A: set_nat,B: set_nat,F2: nat > set_nat,G2: nat > set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_set_nat @ ( F2 @ X ) @ ( G2 @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G2 @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_970_SUP__subset__mono,axiom,
! [A: set_a,B: set_a,F2: a > $o,G2: a > $o] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_o @ ( F2 @ X ) @ ( G2 @ X ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_a_o @ F2 @ A ) ) @ ( complete_Sup_Sup_o @ ( image_a_o @ G2 @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_971_SUP__subset__mono,axiom,
! [A: set_o,B: set_o,F2: $o > $o,G2: $o > $o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_o @ ( F2 @ X ) @ ( G2 @ X ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F2 @ A ) ) @ ( complete_Sup_Sup_o @ ( image_o_o @ G2 @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_972_SUP__subset__mono,axiom,
! [A: set_nat,B: set_nat,F2: nat > $o,G2: nat > $o] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_o @ ( F2 @ X ) @ ( G2 @ X ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_nat_o @ F2 @ A ) ) @ ( complete_Sup_Sup_o @ ( image_nat_o @ G2 @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_973_SUP__union,axiom,
! [M: a > set_a,A: set_a,B: set_a] :
( ( comple2307003609928055243_set_a @ ( image_a_set_a @ M @ ( sup_sup_set_a @ A @ B ) ) )
= ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ M @ A ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ M @ B ) ) ) ) ).
% SUP_union
thf(fact_974_SUP__union,axiom,
! [M: a > set_nat,A: set_a,B: set_a] :
( ( comple7399068483239264473et_nat @ ( image_a_set_nat @ M @ ( sup_sup_set_a @ A @ B ) ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ M @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ M @ B ) ) ) ) ).
% SUP_union
thf(fact_975_SUP__union,axiom,
! [M: nat > set_a,A: set_nat,B: set_nat] :
( ( comple2307003609928055243_set_a @ ( image_nat_set_a @ M @ ( sup_sup_set_nat @ A @ B ) ) )
= ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ M @ A ) ) @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ M @ B ) ) ) ) ).
% SUP_union
thf(fact_976_SUP__union,axiom,
! [M: nat > set_nat,A: set_nat,B: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M @ ( sup_sup_set_nat @ A @ B ) ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M @ B ) ) ) ) ).
% SUP_union
thf(fact_977_SUP__union,axiom,
! [M: a > $o,A: set_a,B: set_a] :
( ( complete_Sup_Sup_o @ ( image_a_o @ M @ ( sup_sup_set_a @ A @ B ) ) )
= ( sup_sup_o @ ( complete_Sup_Sup_o @ ( image_a_o @ M @ A ) ) @ ( complete_Sup_Sup_o @ ( image_a_o @ M @ B ) ) ) ) ).
% SUP_union
thf(fact_978_SUP__union,axiom,
! [M: nat > $o,A: set_nat,B: set_nat] :
( ( complete_Sup_Sup_o @ ( image_nat_o @ M @ ( sup_sup_set_nat @ A @ B ) ) )
= ( sup_sup_o @ ( complete_Sup_Sup_o @ ( image_nat_o @ M @ A ) ) @ ( complete_Sup_Sup_o @ ( image_nat_o @ M @ B ) ) ) ) ).
% SUP_union
thf(fact_979_cInf__le__finite,axiom,
! [X5: set_set_nat,X3: set_nat] :
( ( finite1152437895449049373et_nat @ X5 )
=> ( ( member_set_nat @ X3 @ X5 )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ X5 ) @ X3 ) ) ) ).
% cInf_le_finite
thf(fact_980_cInf__le__finite,axiom,
! [X5: set_nat,X3: nat] :
( ( finite_finite_nat @ X5 )
=> ( ( member_nat @ X3 @ X5 )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ X5 ) @ X3 ) ) ) ).
% cInf_le_finite
thf(fact_981_cInf__le__finite,axiom,
! [X5: set_o,X3: $o] :
( ( finite_finite_o @ X5 )
=> ( ( member_o @ X3 @ X5 )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ X5 ) @ X3 ) ) ) ).
% cInf_le_finite
thf(fact_982_le__cSup__finite,axiom,
! [X5: set_set_nat,X3: set_nat] :
( ( finite1152437895449049373et_nat @ X5 )
=> ( ( member_set_nat @ X3 @ X5 )
=> ( ord_less_eq_set_nat @ X3 @ ( comple7399068483239264473et_nat @ X5 ) ) ) ) ).
% le_cSup_finite
thf(fact_983_le__cSup__finite,axiom,
! [X5: set_nat,X3: nat] :
( ( finite_finite_nat @ X5 )
=> ( ( member_nat @ X3 @ X5 )
=> ( ord_less_eq_nat @ X3 @ ( complete_Sup_Sup_nat @ X5 ) ) ) ) ).
% le_cSup_finite
thf(fact_984_le__cSup__finite,axiom,
! [X5: set_o,X3: $o] :
( ( finite_finite_o @ X5 )
=> ( ( member_o @ X3 @ X5 )
=> ( ord_less_eq_o @ X3 @ ( complete_Sup_Sup_o @ X5 ) ) ) ) ).
% le_cSup_finite
thf(fact_985_wellorder__Inf__le1,axiom,
! [K: nat,A: set_nat] :
( ( member_nat @ K @ A )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ A ) @ K ) ) ).
% wellorder_Inf_le1
thf(fact_986_cInf__eq,axiom,
! [X5: set_nat,A2: nat] :
( ! [X: nat] :
( ( member_nat @ X @ X5 )
=> ( ord_less_eq_nat @ A2 @ X ) )
=> ( ! [Y5: nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ X5 )
=> ( ord_less_eq_nat @ Y5 @ X4 ) )
=> ( ord_less_eq_nat @ Y5 @ A2 ) )
=> ( ( complete_Inf_Inf_nat @ X5 )
= A2 ) ) ) ).
% cInf_eq
thf(fact_987_cInf__eq__minimum,axiom,
! [Z: set_nat,X5: set_set_nat] :
( ( member_set_nat @ Z @ X5 )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ X5 )
=> ( ord_less_eq_set_nat @ Z @ X ) )
=> ( ( comple7806235888213564991et_nat @ X5 )
= Z ) ) ) ).
% cInf_eq_minimum
thf(fact_988_cInf__eq__minimum,axiom,
! [Z: nat,X5: set_nat] :
( ( member_nat @ Z @ X5 )
=> ( ! [X: nat] :
( ( member_nat @ X @ X5 )
=> ( ord_less_eq_nat @ Z @ X ) )
=> ( ( complete_Inf_Inf_nat @ X5 )
= Z ) ) ) ).
% cInf_eq_minimum
thf(fact_989_cInf__eq__minimum,axiom,
! [Z: $o,X5: set_o] :
( ( member_o @ Z @ X5 )
=> ( ! [X: $o] :
( ( member_o @ X @ X5 )
=> ( ord_less_eq_o @ Z @ X ) )
=> ( ( complete_Inf_Inf_o @ X5 )
= Z ) ) ) ).
% cInf_eq_minimum
thf(fact_990_Sup__SUP__eq,axiom,
( complete_Sup_Sup_o_o
= ( ^ [S2: set_o_o,X2: $o] : ( member_o @ X2 @ ( comple90263536869209701_set_o @ ( image_o_o_set_o @ collect_o @ S2 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_991_Sup__SUP__eq,axiom,
( complete_Sup_Sup_a_o
= ( ^ [S2: set_a_o,X2: a] : ( member_a @ X2 @ ( comple2307003609928055243_set_a @ ( image_a_o_set_a @ collect_a @ S2 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_992_Sup__SUP__eq,axiom,
( comple8317665133742190828_nat_o
= ( ^ [S2: set_nat_o,X2: nat] : ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ ( image_nat_o_set_nat @ collect_nat @ S2 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_993_Inf__INT__eq,axiom,
( complete_Inf_Inf_o_o
= ( ^ [S2: set_o_o,X2: $o] : ( member_o @ X2 @ ( comple3063163877087187839_set_o @ ( image_o_o_set_o @ collect_o @ S2 ) ) ) ) ) ).
% Inf_INT_eq
thf(fact_994_Inf__INT__eq,axiom,
( complete_Inf_Inf_a_o
= ( ^ [S2: set_a_o,X2: a] : ( member_a @ X2 @ ( comple6135023378680113637_set_a @ ( image_a_o_set_a @ collect_a @ S2 ) ) ) ) ) ).
% Inf_INT_eq
thf(fact_995_Inf__INT__eq,axiom,
( comple6214475593288795910_nat_o
= ( ^ [S2: set_nat_o,X2: nat] : ( member_nat @ X2 @ ( comple7806235888213564991et_nat @ ( image_nat_o_set_nat @ collect_nat @ S2 ) ) ) ) ) ).
% Inf_INT_eq
thf(fact_996_SUP__Sup__eq,axiom,
! [S: set_set_a] :
( ( complete_Sup_Sup_a_o
@ ( image_set_a_a_o
@ ^ [I6: set_a,X2: a] : ( member_a @ X2 @ I6 )
@ S ) )
= ( ^ [X2: a] : ( member_a @ X2 @ ( comple2307003609928055243_set_a @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_997_SUP__Sup__eq,axiom,
! [S: set_set_nat] :
( ( comple8317665133742190828_nat_o
@ ( image_set_nat_nat_o
@ ^ [I6: set_nat,X2: nat] : ( member_nat @ X2 @ I6 )
@ S ) )
= ( ^ [X2: nat] : ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_998_SUP__Sup__eq,axiom,
! [S: set_set_o] :
( ( complete_Sup_Sup_o_o
@ ( image_set_o_o_o
@ ^ [I6: set_o,X2: $o] : ( member_o @ X2 @ I6 )
@ S ) )
= ( ^ [X2: $o] : ( member_o @ X2 @ ( comple90263536869209701_set_o @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_999_INF__Int__eq,axiom,
! [S: set_set_a] :
( ( complete_Inf_Inf_a_o
@ ( image_set_a_a_o
@ ^ [I6: set_a,X2: a] : ( member_a @ X2 @ I6 )
@ S ) )
= ( ^ [X2: a] : ( member_a @ X2 @ ( comple6135023378680113637_set_a @ S ) ) ) ) ).
% INF_Int_eq
thf(fact_1000_INF__Int__eq,axiom,
! [S: set_set_nat] :
( ( comple6214475593288795910_nat_o
@ ( image_set_nat_nat_o
@ ^ [I6: set_nat,X2: nat] : ( member_nat @ X2 @ I6 )
@ S ) )
= ( ^ [X2: nat] : ( member_nat @ X2 @ ( comple7806235888213564991et_nat @ S ) ) ) ) ).
% INF_Int_eq
thf(fact_1001_INF__Int__eq,axiom,
! [S: set_set_o] :
( ( complete_Inf_Inf_o_o
@ ( image_set_o_o_o
@ ^ [I6: set_o,X2: $o] : ( member_o @ X2 @ I6 )
@ S ) )
= ( ^ [X2: $o] : ( member_o @ X2 @ ( comple3063163877087187839_set_o @ S ) ) ) ) ).
% INF_Int_eq
thf(fact_1002_Sup__set__def,axiom,
( comple90263536869209701_set_o
= ( ^ [A3: set_set_o] :
( collect_o
@ ^ [X2: $o] : ( complete_Sup_Sup_o @ ( image_set_o_o @ ( member_o @ X2 ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_1003_Sup__set__def,axiom,
( comple2307003609928055243_set_a
= ( ^ [A3: set_set_a] :
( collect_a
@ ^ [X2: a] : ( complete_Sup_Sup_o @ ( image_set_a_o @ ( member_a @ X2 ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_1004_Sup__set__def,axiom,
( comple7399068483239264473et_nat
= ( ^ [A3: set_set_nat] :
( collect_nat
@ ^ [X2: nat] : ( complete_Sup_Sup_o @ ( image_set_nat_o @ ( member_nat @ X2 ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_1005_Inf__set__def,axiom,
( comple3063163877087187839_set_o
= ( ^ [A3: set_set_o] :
( collect_o
@ ^ [X2: $o] : ( complete_Inf_Inf_o @ ( image_set_o_o @ ( member_o @ X2 ) @ A3 ) ) ) ) ) ).
% Inf_set_def
thf(fact_1006_Inf__set__def,axiom,
( comple6135023378680113637_set_a
= ( ^ [A3: set_set_a] :
( collect_a
@ ^ [X2: a] : ( complete_Inf_Inf_o @ ( image_set_a_o @ ( member_a @ X2 ) @ A3 ) ) ) ) ) ).
% Inf_set_def
thf(fact_1007_Inf__set__def,axiom,
( comple7806235888213564991et_nat
= ( ^ [A3: set_set_nat] :
( collect_nat
@ ^ [X2: nat] : ( complete_Inf_Inf_o @ ( image_set_nat_o @ ( member_nat @ X2 ) @ A3 ) ) ) ) ) ).
% Inf_set_def
thf(fact_1008_cSup__eq__maximum,axiom,
! [Z: set_nat,X5: set_set_nat] :
( ( member_set_nat @ Z @ X5 )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ X5 )
=> ( ord_less_eq_set_nat @ X @ Z ) )
=> ( ( comple7399068483239264473et_nat @ X5 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_1009_cSup__eq__maximum,axiom,
! [Z: nat,X5: set_nat] :
( ( member_nat @ Z @ X5 )
=> ( ! [X: nat] :
( ( member_nat @ X @ X5 )
=> ( ord_less_eq_nat @ X @ Z ) )
=> ( ( complete_Sup_Sup_nat @ X5 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_1010_cSup__eq__maximum,axiom,
! [Z: $o,X5: set_o] :
( ( member_o @ Z @ X5 )
=> ( ! [X: $o] :
( ( member_o @ X @ X5 )
=> ( ord_less_eq_o @ X @ Z ) )
=> ( ( complete_Sup_Sup_o @ X5 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_1011_Powp__Pow__eq,axiom,
! [A: set_a] :
( ( powp_a
@ ^ [X2: a] : ( member_a @ X2 @ A ) )
= ( ^ [X2: set_a] : ( member_set_a @ X2 @ ( pow_a @ A ) ) ) ) ).
% Powp_Pow_eq
thf(fact_1012_Powp__Pow__eq,axiom,
! [A: set_nat] :
( ( powp_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A ) )
= ( ^ [X2: set_nat] : ( member_set_nat @ X2 @ ( pow_nat @ A ) ) ) ) ).
% Powp_Pow_eq
thf(fact_1013_Powp__Pow__eq,axiom,
! [A: set_o] :
( ( powp_o
@ ^ [X2: $o] : ( member_o @ X2 @ A ) )
= ( ^ [X2: set_o] : ( member_set_o @ X2 @ ( pow_o @ A ) ) ) ) ).
% Powp_Pow_eq
thf(fact_1014_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_1015_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_1016_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_1017_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_1018_all__not__in__conv,axiom,
! [A: set_a] :
( ( ! [X2: a] :
~ ( member_a @ X2 @ A ) )
= ( A = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_1019_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X2: nat] :
~ ( member_nat @ X2 @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_1020_all__not__in__conv,axiom,
! [A: set_o] :
( ( ! [X2: $o] :
~ ( member_o @ X2 @ A ) )
= ( A = bot_bot_set_o ) ) ).
% all_not_in_conv
thf(fact_1021_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_1022_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_1023_empty__iff,axiom,
! [C: $o] :
~ ( member_o @ C @ bot_bot_set_o ) ).
% empty_iff
thf(fact_1024_Compl__iff,axiom,
! [C: a,A: set_a] :
( ( member_a @ C @ ( uminus_uminus_set_a @ A ) )
= ( ~ ( member_a @ C @ A ) ) ) ).
% Compl_iff
thf(fact_1025_Compl__iff,axiom,
! [C: nat,A: set_nat] :
( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A ) )
= ( ~ ( member_nat @ C @ A ) ) ) ).
% Compl_iff
thf(fact_1026_Compl__iff,axiom,
! [C: $o,A: set_o] :
( ( member_o @ C @ ( uminus_uminus_set_o @ A ) )
= ( ~ ( member_o @ C @ A ) ) ) ).
% Compl_iff
thf(fact_1027_ComplI,axiom,
! [C: a,A: set_a] :
( ~ ( member_a @ C @ A )
=> ( member_a @ C @ ( uminus_uminus_set_a @ A ) ) ) ).
% ComplI
thf(fact_1028_ComplI,axiom,
! [C: nat,A: set_nat] :
( ~ ( member_nat @ C @ A )
=> ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A ) ) ) ).
% ComplI
thf(fact_1029_ComplI,axiom,
! [C: $o,A: set_o] :
( ~ ( member_o @ C @ A )
=> ( member_o @ C @ ( uminus_uminus_set_o @ A ) ) ) ).
% ComplI
thf(fact_1030_compl__le__compl__iff,axiom,
! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X3 ) @ ( uminus5710092332889474511et_nat @ Y2 ) )
= ( ord_less_eq_set_nat @ Y2 @ X3 ) ) ).
% compl_le_compl_iff
thf(fact_1031_image__is__empty,axiom,
! [F2: nat > a,A: set_nat] :
( ( ( image_nat_a @ F2 @ A )
= bot_bot_set_a )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_1032_image__is__empty,axiom,
! [F2: nat > nat,A: set_nat] :
( ( ( image_nat_nat @ F2 @ A )
= bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_1033_image__is__empty,axiom,
! [F2: a > nat,A: set_a] :
( ( ( image_a_nat @ F2 @ A )
= bot_bot_set_nat )
= ( A = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_1034_image__is__empty,axiom,
! [F2: a > a,A: set_a] :
( ( ( image_a_a @ F2 @ A )
= bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_1035_empty__is__image,axiom,
! [F2: nat > a,A: set_nat] :
( ( bot_bot_set_a
= ( image_nat_a @ F2 @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_1036_empty__is__image,axiom,
! [F2: nat > nat,A: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat @ F2 @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_1037_empty__is__image,axiom,
! [F2: a > nat,A: set_a] :
( ( bot_bot_set_nat
= ( image_a_nat @ F2 @ A ) )
= ( A = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_1038_empty__is__image,axiom,
! [F2: a > a,A: set_a] :
( ( bot_bot_set_a
= ( image_a_a @ F2 @ A ) )
= ( A = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_1039_image__empty,axiom,
! [F2: nat > a] :
( ( image_nat_a @ F2 @ bot_bot_set_nat )
= bot_bot_set_a ) ).
% image_empty
thf(fact_1040_image__empty,axiom,
! [F2: nat > nat] :
( ( image_nat_nat @ F2 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_1041_image__empty,axiom,
! [F2: a > nat] :
( ( image_a_nat @ F2 @ bot_bot_set_a )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_1042_image__empty,axiom,
! [F2: a > a] :
( ( image_a_a @ F2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% image_empty
thf(fact_1043_empty__subsetI,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% empty_subsetI
thf(fact_1044_subset__empty,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_1045_sup__bot_Oright__neutral,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_1046_sup__bot_Oright__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_1047_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_a )
& ( B2 = bot_bot_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_1048_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_1049_sup__bot_Oleft__neutral,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_1050_sup__bot_Oleft__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_1051_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ( sup_sup_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ( A2 = bot_bot_set_a )
& ( B2 = bot_bot_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_1052_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( sup_sup_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ( A2 = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_1053_sup__eq__bot__iff,axiom,
! [X3: set_a,Y2: set_a] :
( ( ( sup_sup_set_a @ X3 @ Y2 )
= bot_bot_set_a )
= ( ( X3 = bot_bot_set_a )
& ( Y2 = bot_bot_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_1054_sup__eq__bot__iff,axiom,
! [X3: set_nat,Y2: set_nat] :
( ( ( sup_sup_set_nat @ X3 @ Y2 )
= bot_bot_set_nat )
= ( ( X3 = bot_bot_set_nat )
& ( Y2 = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_1055_bot__eq__sup__iff,axiom,
! [X3: set_a,Y2: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ X3 @ Y2 ) )
= ( ( X3 = bot_bot_set_a )
& ( Y2 = bot_bot_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_1056_bot__eq__sup__iff,axiom,
! [X3: set_nat,Y2: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X3 @ Y2 ) )
= ( ( X3 = bot_bot_set_nat )
& ( Y2 = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_1057_sup__bot__right,axiom,
! [X3: set_a] :
( ( sup_sup_set_a @ X3 @ bot_bot_set_a )
= X3 ) ).
% sup_bot_right
thf(fact_1058_sup__bot__right,axiom,
! [X3: set_nat] :
( ( sup_sup_set_nat @ X3 @ bot_bot_set_nat )
= X3 ) ).
% sup_bot_right
thf(fact_1059_sup__bot__left,axiom,
! [X3: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_1060_sup__bot__left,axiom,
! [X3: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_1061_Sup__bot__conv_I1_J,axiom,
! [A: set_o] :
( ( ( complete_Sup_Sup_o @ A )
= bot_bot_o )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( X2 = bot_bot_o ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_1062_Sup__bot__conv_I2_J,axiom,
! [A: set_o] :
( ( bot_bot_o
= ( complete_Sup_Sup_o @ A ) )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( X2 = bot_bot_o ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_1063_Un__empty,axiom,
! [A: set_a,B: set_a] :
( ( ( sup_sup_set_a @ A @ B )
= bot_bot_set_a )
= ( ( A = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% Un_empty
thf(fact_1064_Un__empty,axiom,
! [A: set_nat,B: set_nat] :
( ( ( sup_sup_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ( A = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_1065_Compl__subset__Compl__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ ( uminus5710092332889474511et_nat @ B ) )
= ( ord_less_eq_set_nat @ B @ A ) ) ).
% Compl_subset_Compl_iff
thf(fact_1066_Compl__anti__mono,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ B ) @ ( uminus5710092332889474511et_nat @ A ) ) ) ).
% Compl_anti_mono
thf(fact_1067_Sup__empty,axiom,
( ( complete_Sup_Sup_o @ bot_bot_set_o )
= bot_bot_o ) ).
% Sup_empty
thf(fact_1068_cSUP__const,axiom,
! [A: set_nat,C: nat] :
( ( A != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_nat
@ ( image_nat_nat
@ ^ [X2: nat] : C
@ A ) )
= C ) ) ).
% cSUP_const
thf(fact_1069_cSUP__const,axiom,
! [A: set_a,C: nat] :
( ( A != bot_bot_set_a )
=> ( ( complete_Sup_Sup_nat
@ ( image_a_nat
@ ^ [X2: a] : C
@ A ) )
= C ) ) ).
% cSUP_const
thf(fact_1070_cINF__const,axiom,
! [A: set_nat,C: nat] :
( ( A != bot_bot_set_nat )
=> ( ( complete_Inf_Inf_nat
@ ( image_nat_nat
@ ^ [X2: nat] : C
@ A ) )
= C ) ) ).
% cINF_const
thf(fact_1071_cINF__const,axiom,
! [A: set_a,C: nat] :
( ( A != bot_bot_set_a )
=> ( ( complete_Inf_Inf_nat
@ ( image_a_nat
@ ^ [X2: a] : C
@ A ) )
= C ) ) ).
% cINF_const
thf(fact_1072_Sup__bool__def,axiom,
( complete_Sup_Sup_o
= ( member_o @ $true ) ) ).
% Sup_bool_def
thf(fact_1073_Inf__bool__def,axiom,
( complete_Inf_Inf_o
= ( ^ [A3: set_o] :
~ ( member_o @ $false @ A3 ) ) ) ).
% Inf_bool_def
thf(fact_1074_Compl__eq,axiom,
( uminus_uminus_set_o
= ( ^ [A3: set_o] :
( collect_o
@ ^ [X2: $o] :
~ ( member_o @ X2 @ A3 ) ) ) ) ).
% Compl_eq
thf(fact_1075_Compl__eq,axiom,
( uminus_uminus_set_a
= ( ^ [A3: set_a] :
( collect_a
@ ^ [X2: a] :
~ ( member_a @ X2 @ A3 ) ) ) ) ).
% Compl_eq
thf(fact_1076_Compl__eq,axiom,
( uminus5710092332889474511et_nat
= ( ^ [A3: set_nat] :
( collect_nat
@ ^ [X2: nat] :
~ ( member_nat @ X2 @ A3 ) ) ) ) ).
% Compl_eq
thf(fact_1077_Collect__neg__eq,axiom,
! [P: a > $o] :
( ( collect_a
@ ^ [X2: a] :
~ ( P @ X2 ) )
= ( uminus_uminus_set_a @ ( collect_a @ P ) ) ) ).
% Collect_neg_eq
thf(fact_1078_Collect__neg__eq,axiom,
! [P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
~ ( P @ X2 ) )
= ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) ) ).
% Collect_neg_eq
thf(fact_1079_empty__def,axiom,
( bot_bot_set_a
= ( collect_a
@ ^ [X2: a] : $false ) ) ).
% empty_def
thf(fact_1080_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X2: nat] : $false ) ) ).
% empty_def
thf(fact_1081_bot_Oextremum__uniqueI,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
=> ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_1082_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_1083_bot_Oextremum__unique,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_1084_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_1085_bot_Oextremum,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% bot.extremum
thf(fact_1086_bot_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot.extremum
thf(fact_1087_boolean__algebra_Odisj__zero__right,axiom,
! [X3: set_a] :
( ( sup_sup_set_a @ X3 @ bot_bot_set_a )
= X3 ) ).
% boolean_algebra.disj_zero_right
thf(fact_1088_boolean__algebra_Odisj__zero__right,axiom,
! [X3: set_nat] :
( ( sup_sup_set_nat @ X3 @ bot_bot_set_nat )
= X3 ) ).
% boolean_algebra.disj_zero_right
thf(fact_1089_compl__le__swap2,axiom,
! [Y2: set_nat,X3: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y2 ) @ X3 )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X3 ) @ Y2 ) ) ).
% compl_le_swap2
thf(fact_1090_compl__le__swap1,axiom,
! [Y2: set_nat,X3: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ ( uminus5710092332889474511et_nat @ X3 ) )
=> ( ord_less_eq_set_nat @ X3 @ ( uminus5710092332889474511et_nat @ Y2 ) ) ) ).
% compl_le_swap1
thf(fact_1091_compl__mono,axiom,
! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y2 ) @ ( uminus5710092332889474511et_nat @ X3 ) ) ) ).
% compl_mono
thf(fact_1092_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_1093_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_1094_infinite__imp__nonempty,axiom,
! [S: set_a] :
( ~ ( finite_finite_a @ S )
=> ( S != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_1095_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_1096_subset__emptyI,axiom,
! [A: set_a] :
( ! [X: a] :
~ ( member_a @ X @ A )
=> ( ord_less_eq_set_a @ A @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_1097_subset__emptyI,axiom,
! [A: set_o] :
( ! [X: $o] :
~ ( member_o @ X @ A )
=> ( ord_less_eq_set_o @ A @ bot_bot_set_o ) ) ).
% subset_emptyI
thf(fact_1098_subset__emptyI,axiom,
! [A: set_nat] :
( ! [X: nat] :
~ ( member_nat @ X @ A )
=> ( ord_less_eq_set_nat @ A @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_1099_Un__empty__left,axiom,
! [B: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ B )
= B ) ).
% Un_empty_left
thf(fact_1100_Un__empty__left,axiom,
! [B: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_1101_Un__empty__right,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ bot_bot_set_a )
= A ) ).
% Un_empty_right
thf(fact_1102_Un__empty__right,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Un_empty_right
thf(fact_1103_subset__Compl__self__eq,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( uminus5710092332889474511et_nat @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% subset_Compl_self_eq
thf(fact_1104_ex__in__conv,axiom,
! [A: set_a] :
( ( ? [X2: a] : ( member_a @ X2 @ A ) )
= ( A != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_1105_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X2: nat] : ( member_nat @ X2 @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_1106_ex__in__conv,axiom,
! [A: set_o] :
( ( ? [X2: $o] : ( member_o @ X2 @ A ) )
= ( A != bot_bot_set_o ) ) ).
% ex_in_conv
thf(fact_1107_equals0I,axiom,
! [A: set_a] :
( ! [Y5: a] :
~ ( member_a @ Y5 @ A )
=> ( A = bot_bot_set_a ) ) ).
% equals0I
thf(fact_1108_equals0I,axiom,
! [A: set_nat] :
( ! [Y5: nat] :
~ ( member_nat @ Y5 @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_1109_equals0I,axiom,
! [A: set_o] :
( ! [Y5: $o] :
~ ( member_o @ Y5 @ A )
=> ( A = bot_bot_set_o ) ) ).
% equals0I
thf(fact_1110_equals0D,axiom,
! [A: set_a,A2: a] :
( ( A = bot_bot_set_a )
=> ~ ( member_a @ A2 @ A ) ) ).
% equals0D
thf(fact_1111_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_1112_equals0D,axiom,
! [A: set_o,A2: $o] :
( ( A = bot_bot_set_o )
=> ~ ( member_o @ A2 @ A ) ) ).
% equals0D
thf(fact_1113_emptyE,axiom,
! [A2: a] :
~ ( member_a @ A2 @ bot_bot_set_a ) ).
% emptyE
thf(fact_1114_emptyE,axiom,
! [A2: nat] :
~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_1115_emptyE,axiom,
! [A2: $o] :
~ ( member_o @ A2 @ bot_bot_set_o ) ).
% emptyE
thf(fact_1116_ComplD,axiom,
! [C: a,A: set_a] :
( ( member_a @ C @ ( uminus_uminus_set_a @ A ) )
=> ~ ( member_a @ C @ A ) ) ).
% ComplD
thf(fact_1117_ComplD,axiom,
! [C: nat,A: set_nat] :
( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A ) )
=> ~ ( member_nat @ C @ A ) ) ).
% ComplD
thf(fact_1118_ComplD,axiom,
! [C: $o,A: set_o] :
( ( member_o @ C @ ( uminus_uminus_set_o @ A ) )
=> ~ ( member_o @ C @ A ) ) ).
% ComplD
thf(fact_1119_Collect__imp__eq,axiom,
! [P: a > $o,Q: a > $o] :
( ( collect_a
@ ^ [X2: a] :
( ( P @ X2 )
=> ( Q @ X2 ) ) )
= ( sup_sup_set_a @ ( uminus_uminus_set_a @ ( collect_a @ P ) ) @ ( collect_a @ Q ) ) ) ).
% Collect_imp_eq
thf(fact_1120_Collect__imp__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( P @ X2 )
=> ( Q @ X2 ) ) )
= ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) @ ( collect_nat @ Q ) ) ) ).
% Collect_imp_eq
thf(fact_1121_some__in__eq,axiom,
! [A: set_a] :
( ( member_a
@ ( fChoice_a
@ ^ [X2: a] : ( member_a @ X2 @ A ) )
@ A )
= ( A != bot_bot_set_a ) ) ).
% some_in_eq
thf(fact_1122_some__in__eq,axiom,
! [A: set_nat] :
( ( member_nat
@ ( fChoice_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A ) )
@ A )
= ( A != bot_bot_set_nat ) ) ).
% some_in_eq
thf(fact_1123_some__in__eq,axiom,
! [A: set_o] :
( ( member_o
@ ( fChoice_o
@ ^ [X2: $o] : ( member_o @ X2 @ A ) )
@ A )
= ( A != bot_bot_set_o ) ) ).
% some_in_eq
thf(fact_1124_finite__has__maximal,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( ( A != bot_bot_set_a )
=> ? [X: a] :
( ( member_a @ X @ A )
& ! [Xa: a] :
( ( member_a @ Xa @ A )
=> ( ( ord_less_eq_a @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1125_finite__has__maximal,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1126_finite__has__maximal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1127_finite__has__minimal,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( ( A != bot_bot_set_a )
=> ? [X: a] :
( ( member_a @ X @ A )
& ! [Xa: a] :
( ( member_a @ Xa @ A )
=> ( ( ord_less_eq_a @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1128_finite__has__minimal,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1129_finite__has__minimal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1130_less__eq__Sup,axiom,
! [A: set_set_nat,U: set_nat] :
( ! [V2: set_nat] :
( ( member_set_nat @ V2 @ A )
=> ( ord_less_eq_set_nat @ U @ V2 ) )
=> ( ( A != bot_bot_set_set_nat )
=> ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ A ) ) ) ) ).
% less_eq_Sup
thf(fact_1131_less__eq__Sup,axiom,
! [A: set_o,U: $o] :
( ! [V2: $o] :
( ( member_o @ V2 @ A )
=> ( ord_less_eq_o @ U @ V2 ) )
=> ( ( A != bot_bot_set_o )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ A ) ) ) ) ).
% less_eq_Sup
thf(fact_1132_cSup__least,axiom,
! [X5: set_set_nat,Z: set_nat] :
( ( X5 != bot_bot_set_set_nat )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ X5 )
=> ( ord_less_eq_set_nat @ X @ Z ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ X5 ) @ Z ) ) ) ).
% cSup_least
thf(fact_1133_cSup__least,axiom,
! [X5: set_nat,Z: nat] :
( ( X5 != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ X5 )
=> ( ord_less_eq_nat @ X @ Z ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ X5 ) @ Z ) ) ) ).
% cSup_least
thf(fact_1134_cSup__least,axiom,
! [X5: set_o,Z: $o] :
( ( X5 != bot_bot_set_o )
=> ( ! [X: $o] :
( ( member_o @ X @ X5 )
=> ( ord_less_eq_o @ X @ Z ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ X5 ) @ Z ) ) ) ).
% cSup_least
thf(fact_1135_cSup__eq__non__empty,axiom,
! [X5: set_set_nat,A2: set_nat] :
( ( X5 != bot_bot_set_set_nat )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ X5 )
=> ( ord_less_eq_set_nat @ X @ A2 ) )
=> ( ! [Y5: set_nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ X5 )
=> ( ord_less_eq_set_nat @ X4 @ Y5 ) )
=> ( ord_less_eq_set_nat @ A2 @ Y5 ) )
=> ( ( comple7399068483239264473et_nat @ X5 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_1136_cSup__eq__non__empty,axiom,
! [X5: set_nat,A2: nat] :
( ( X5 != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ X5 )
=> ( ord_less_eq_nat @ X @ A2 ) )
=> ( ! [Y5: nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ X5 )
=> ( ord_less_eq_nat @ X4 @ Y5 ) )
=> ( ord_less_eq_nat @ A2 @ Y5 ) )
=> ( ( complete_Sup_Sup_nat @ X5 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_1137_cSup__eq__non__empty,axiom,
! [X5: set_o,A2: $o] :
( ( X5 != bot_bot_set_o )
=> ( ! [X: $o] :
( ( member_o @ X @ X5 )
=> ( ord_less_eq_o @ X @ A2 ) )
=> ( ! [Y5: $o] :
( ! [X4: $o] :
( ( member_o @ X4 @ X5 )
=> ( ord_less_eq_o @ X4 @ Y5 ) )
=> ( ord_less_eq_o @ A2 @ Y5 ) )
=> ( ( complete_Sup_Sup_o @ X5 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_1138_Inf__less__eq,axiom,
! [A: set_set_nat,U: set_nat] :
( ! [V2: set_nat] :
( ( member_set_nat @ V2 @ A )
=> ( ord_less_eq_set_nat @ V2 @ U ) )
=> ( ( A != bot_bot_set_set_nat )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A ) @ U ) ) ) ).
% Inf_less_eq
thf(fact_1139_Inf__less__eq,axiom,
! [A: set_o,U: $o] :
( ! [V2: $o] :
( ( member_o @ V2 @ A )
=> ( ord_less_eq_o @ V2 @ U ) )
=> ( ( A != bot_bot_set_o )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A ) @ U ) ) ) ).
% Inf_less_eq
thf(fact_1140_cInf__greatest,axiom,
! [X5: set_set_nat,Z: set_nat] :
( ( X5 != bot_bot_set_set_nat )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ X5 )
=> ( ord_less_eq_set_nat @ Z @ X ) )
=> ( ord_less_eq_set_nat @ Z @ ( comple7806235888213564991et_nat @ X5 ) ) ) ) ).
% cInf_greatest
thf(fact_1141_cInf__greatest,axiom,
! [X5: set_nat,Z: nat] :
( ( X5 != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ X5 )
=> ( ord_less_eq_nat @ Z @ X ) )
=> ( ord_less_eq_nat @ Z @ ( complete_Inf_Inf_nat @ X5 ) ) ) ) ).
% cInf_greatest
thf(fact_1142_cInf__greatest,axiom,
! [X5: set_o,Z: $o] :
( ( X5 != bot_bot_set_o )
=> ( ! [X: $o] :
( ( member_o @ X @ X5 )
=> ( ord_less_eq_o @ Z @ X ) )
=> ( ord_less_eq_o @ Z @ ( complete_Inf_Inf_o @ X5 ) ) ) ) ).
% cInf_greatest
thf(fact_1143_cInf__eq__non__empty,axiom,
! [X5: set_set_nat,A2: set_nat] :
( ( X5 != bot_bot_set_set_nat )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ X5 )
=> ( ord_less_eq_set_nat @ A2 @ X ) )
=> ( ! [Y5: set_nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ X5 )
=> ( ord_less_eq_set_nat @ Y5 @ X4 ) )
=> ( ord_less_eq_set_nat @ Y5 @ A2 ) )
=> ( ( comple7806235888213564991et_nat @ X5 )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_1144_cInf__eq__non__empty,axiom,
! [X5: set_nat,A2: nat] :
( ( X5 != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ X5 )
=> ( ord_less_eq_nat @ A2 @ X ) )
=> ( ! [Y5: nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ X5 )
=> ( ord_less_eq_nat @ Y5 @ X4 ) )
=> ( ord_less_eq_nat @ Y5 @ A2 ) )
=> ( ( complete_Inf_Inf_nat @ X5 )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_1145_cInf__eq__non__empty,axiom,
! [X5: set_o,A2: $o] :
( ( X5 != bot_bot_set_o )
=> ( ! [X: $o] :
( ( member_o @ X @ X5 )
=> ( ord_less_eq_o @ A2 @ X ) )
=> ( ! [Y5: $o] :
( ! [X4: $o] :
( ( member_o @ X4 @ X5 )
=> ( ord_less_eq_o @ Y5 @ X4 ) )
=> ( ord_less_eq_o @ Y5 @ A2 ) )
=> ( ( complete_Inf_Inf_o @ X5 )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_1146_SUP__eq__const,axiom,
! [I: set_a,F2: a > $o,X3: $o] :
( ( I != bot_bot_set_a )
=> ( ! [I3: a] :
( ( member_a @ I3 @ I )
=> ( ( F2 @ I3 )
= X3 ) )
=> ( ( complete_Sup_Sup_o @ ( image_a_o @ F2 @ I ) )
= X3 ) ) ) ).
% SUP_eq_const
thf(fact_1147_SUP__eq__const,axiom,
! [I: set_nat,F2: nat > $o,X3: $o] :
( ( I != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I )
=> ( ( F2 @ I3 )
= X3 ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F2 @ I ) )
= X3 ) ) ) ).
% SUP_eq_const
thf(fact_1148_SUP__eq__const,axiom,
! [I: set_o,F2: $o > $o,X3: $o] :
( ( I != bot_bot_set_o )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I )
=> ( ( F2 @ I3 )
= X3 ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F2 @ I ) )
= X3 ) ) ) ).
% SUP_eq_const
thf(fact_1149_INF__eq__const,axiom,
! [I: set_a,F2: a > $o,X3: $o] :
( ( I != bot_bot_set_a )
=> ( ! [I3: a] :
( ( member_a @ I3 @ I )
=> ( ( F2 @ I3 )
= X3 ) )
=> ( ( complete_Inf_Inf_o @ ( image_a_o @ F2 @ I ) )
= X3 ) ) ) ).
% INF_eq_const
thf(fact_1150_INF__eq__const,axiom,
! [I: set_nat,F2: nat > $o,X3: $o] :
( ( I != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I )
=> ( ( F2 @ I3 )
= X3 ) )
=> ( ( complete_Inf_Inf_o @ ( image_nat_o @ F2 @ I ) )
= X3 ) ) ) ).
% INF_eq_const
thf(fact_1151_INF__eq__const,axiom,
! [I: set_o,F2: $o > $o,X3: $o] :
( ( I != bot_bot_set_o )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I )
=> ( ( F2 @ I3 )
= X3 ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ F2 @ I ) )
= X3 ) ) ) ).
% INF_eq_const
thf(fact_1152_Inter__subset,axiom,
! [A: set_set_nat,B: set_nat] :
( ! [X7: set_nat] :
( ( member_set_nat @ X7 @ A )
=> ( ord_less_eq_set_nat @ X7 @ B ) )
=> ( ( A != bot_bot_set_set_nat )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A ) @ B ) ) ) ).
% Inter_subset
thf(fact_1153_cov_Ononfree,axiom,
! [X3: nat,Q: relational_fmla_a_b] :
( ~ ( member_nat @ X3 @ ( relational_fv_a_b @ Q ) )
=> ( relational_cov_a_b @ X3 @ Q @ bot_bo4495933725496725865la_a_b ) ) ).
% cov.nonfree
thf(fact_1154_uminus__Sup,axiom,
! [A: set_o] :
( ( uminus_uminus_o @ ( complete_Sup_Sup_o @ A ) )
= ( complete_Inf_Inf_o @ ( image_o_o @ uminus_uminus_o @ A ) ) ) ).
% uminus_Sup
thf(fact_1155_uminus__Inf,axiom,
! [A: set_o] :
( ( uminus_uminus_o @ ( complete_Inf_Inf_o @ A ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ uminus_uminus_o @ A ) ) ) ).
% uminus_Inf
thf(fact_1156_SUP__eq__iff,axiom,
! [I: set_a,C: set_nat,F2: a > set_nat] :
( ( I != bot_bot_set_a )
=> ( ! [I3: a] :
( ( member_a @ I3 @ I )
=> ( ord_less_eq_set_nat @ C @ ( F2 @ I3 ) ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_a_set_nat @ F2 @ I ) )
= C )
= ( ! [X2: a] :
( ( member_a @ X2 @ I )
=> ( ( F2 @ X2 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1157_SUP__eq__iff,axiom,
! [I: set_nat,C: set_nat,F2: nat > set_nat] :
( ( I != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I )
=> ( ord_less_eq_set_nat @ C @ ( F2 @ I3 ) ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ I ) )
= C )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ I )
=> ( ( F2 @ X2 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1158_SUP__eq__iff,axiom,
! [I: set_o,C: set_nat,F2: $o > set_nat] :
( ( I != bot_bot_set_o )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I )
=> ( ord_less_eq_set_nat @ C @ ( F2 @ I3 ) ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F2 @ I ) )
= C )
= ( ! [X2: $o] :
( ( member_o @ X2 @ I )
=> ( ( F2 @ X2 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1159_SUP__eq__iff,axiom,
! [I: set_a,C: $o,F2: a > $o] :
( ( I != bot_bot_set_a )
=> ( ! [I3: a] :
( ( member_a @ I3 @ I )
=> ( ord_less_eq_o @ C @ ( F2 @ I3 ) ) )
=> ( ( ( complete_Sup_Sup_o @ ( image_a_o @ F2 @ I ) )
= C )
= ( ! [X2: a] :
( ( member_a @ X2 @ I )
=> ( ( F2 @ X2 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1160_SUP__eq__iff,axiom,
! [I: set_nat,C: $o,F2: nat > $o] :
( ( I != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I )
=> ( ord_less_eq_o @ C @ ( F2 @ I3 ) ) )
=> ( ( ( complete_Sup_Sup_o @ ( image_nat_o @ F2 @ I ) )
= C )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ I )
=> ( ( F2 @ X2 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1161_SUP__eq__iff,axiom,
! [I: set_o,C: $o,F2: $o > $o] :
( ( I != bot_bot_set_o )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I )
=> ( ord_less_eq_o @ C @ ( F2 @ I3 ) ) )
=> ( ( ( complete_Sup_Sup_o @ ( image_o_o @ F2 @ I ) )
= C )
= ( ! [X2: $o] :
( ( member_o @ X2 @ I )
=> ( ( F2 @ X2 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1162_cSUP__least,axiom,
! [A: set_a,F2: a > set_nat,M: set_nat] :
( ( A != bot_bot_set_a )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_set_nat @ ( F2 @ X ) @ M ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ F2 @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_1163_cSUP__least,axiom,
! [A: set_nat,F2: nat > set_nat,M: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_set_nat @ ( F2 @ X ) @ M ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_1164_cSUP__least,axiom,
! [A: set_o,F2: $o > set_nat,M: set_nat] :
( ( A != bot_bot_set_o )
=> ( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_set_nat @ ( F2 @ X ) @ M ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F2 @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_1165_cSUP__least,axiom,
! [A: set_a,F2: a > nat,M: nat] :
( ( A != bot_bot_set_a )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ M ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_a_nat @ F2 @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_1166_cSUP__least,axiom,
! [A: set_nat,F2: nat > nat,M: nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ M ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F2 @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_1167_cSUP__least,axiom,
! [A: set_o,F2: $o > nat,M: nat] :
( ( A != bot_bot_set_o )
=> ( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ M ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_o_nat @ F2 @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_1168_cSUP__least,axiom,
! [A: set_a,F2: a > $o,M: $o] :
( ( A != bot_bot_set_a )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_o @ ( F2 @ X ) @ M ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_a_o @ F2 @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_1169_cSUP__least,axiom,
! [A: set_nat,F2: nat > $o,M: $o] :
( ( A != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_o @ ( F2 @ X ) @ M ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_nat_o @ F2 @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_1170_cSUP__least,axiom,
! [A: set_o,F2: $o > $o,M: $o] :
( ( A != bot_bot_set_o )
=> ( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_o @ ( F2 @ X ) @ M ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F2 @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_1171_INF__eq__iff,axiom,
! [I: set_a,F2: a > set_nat,C: set_nat] :
( ( I != bot_bot_set_a )
=> ( ! [I3: a] :
( ( member_a @ I3 @ I )
=> ( ord_less_eq_set_nat @ ( F2 @ I3 ) @ C ) )
=> ( ( ( comple7806235888213564991et_nat @ ( image_a_set_nat @ F2 @ I ) )
= C )
= ( ! [X2: a] :
( ( member_a @ X2 @ I )
=> ( ( F2 @ X2 )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_1172_INF__eq__iff,axiom,
! [I: set_nat,F2: nat > set_nat,C: set_nat] :
( ( I != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I )
=> ( ord_less_eq_set_nat @ ( F2 @ I3 ) @ C ) )
=> ( ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F2 @ I ) )
= C )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ I )
=> ( ( F2 @ X2 )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_1173_INF__eq__iff,axiom,
! [I: set_o,F2: $o > set_nat,C: set_nat] :
( ( I != bot_bot_set_o )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I )
=> ( ord_less_eq_set_nat @ ( F2 @ I3 ) @ C ) )
=> ( ( ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F2 @ I ) )
= C )
= ( ! [X2: $o] :
( ( member_o @ X2 @ I )
=> ( ( F2 @ X2 )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_1174_INF__eq__iff,axiom,
! [I: set_a,F2: a > $o,C: $o] :
( ( I != bot_bot_set_a )
=> ( ! [I3: a] :
( ( member_a @ I3 @ I )
=> ( ord_less_eq_o @ ( F2 @ I3 ) @ C ) )
=> ( ( ( complete_Inf_Inf_o @ ( image_a_o @ F2 @ I ) )
= C )
= ( ! [X2: a] :
( ( member_a @ X2 @ I )
=> ( ( F2 @ X2 )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_1175_INF__eq__iff,axiom,
! [I: set_nat,F2: nat > $o,C: $o] :
( ( I != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I )
=> ( ord_less_eq_o @ ( F2 @ I3 ) @ C ) )
=> ( ( ( complete_Inf_Inf_o @ ( image_nat_o @ F2 @ I ) )
= C )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ I )
=> ( ( F2 @ X2 )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_1176_INF__eq__iff,axiom,
! [I: set_o,F2: $o > $o,C: $o] :
( ( I != bot_bot_set_o )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I )
=> ( ord_less_eq_o @ ( F2 @ I3 ) @ C ) )
=> ( ( ( complete_Inf_Inf_o @ ( image_o_o @ F2 @ I ) )
= C )
= ( ! [X2: $o] :
( ( member_o @ X2 @ I )
=> ( ( F2 @ X2 )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_1177_cINF__greatest,axiom,
! [A: set_a,M2: set_nat,F2: a > set_nat] :
( ( A != bot_bot_set_a )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_set_nat @ M2 @ ( F2 @ X ) ) )
=> ( ord_less_eq_set_nat @ M2 @ ( comple7806235888213564991et_nat @ ( image_a_set_nat @ F2 @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_1178_cINF__greatest,axiom,
! [A: set_nat,M2: set_nat,F2: nat > set_nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_set_nat @ M2 @ ( F2 @ X ) ) )
=> ( ord_less_eq_set_nat @ M2 @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F2 @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_1179_cINF__greatest,axiom,
! [A: set_o,M2: set_nat,F2: $o > set_nat] :
( ( A != bot_bot_set_o )
=> ( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_set_nat @ M2 @ ( F2 @ X ) ) )
=> ( ord_less_eq_set_nat @ M2 @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F2 @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_1180_cINF__greatest,axiom,
! [A: set_a,M2: nat,F2: a > nat] :
( ( A != bot_bot_set_a )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_nat @ M2 @ ( F2 @ X ) ) )
=> ( ord_less_eq_nat @ M2 @ ( complete_Inf_Inf_nat @ ( image_a_nat @ F2 @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_1181_cINF__greatest,axiom,
! [A: set_nat,M2: nat,F2: nat > nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_nat @ M2 @ ( F2 @ X ) ) )
=> ( ord_less_eq_nat @ M2 @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ F2 @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_1182_cINF__greatest,axiom,
! [A: set_o,M2: nat,F2: $o > nat] :
( ( A != bot_bot_set_o )
=> ( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_nat @ M2 @ ( F2 @ X ) ) )
=> ( ord_less_eq_nat @ M2 @ ( complete_Inf_Inf_nat @ ( image_o_nat @ F2 @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_1183_cINF__greatest,axiom,
! [A: set_a,M2: $o,F2: a > $o] :
( ( A != bot_bot_set_a )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_o @ M2 @ ( F2 @ X ) ) )
=> ( ord_less_eq_o @ M2 @ ( complete_Inf_Inf_o @ ( image_a_o @ F2 @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_1184_cINF__greatest,axiom,
! [A: set_nat,M2: $o,F2: nat > $o] :
( ( A != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_o @ M2 @ ( F2 @ X ) ) )
=> ( ord_less_eq_o @ M2 @ ( complete_Inf_Inf_o @ ( image_nat_o @ F2 @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_1185_cINF__greatest,axiom,
! [A: set_o,M2: $o,F2: $o > $o] :
( ( A != bot_bot_set_o )
=> ( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_o @ M2 @ ( F2 @ X ) ) )
=> ( ord_less_eq_o @ M2 @ ( complete_Inf_Inf_o @ ( image_o_o @ F2 @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_1186_Inf__le__Sup,axiom,
! [A: set_set_nat] :
( ( A != bot_bot_set_set_nat )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A ) @ ( comple7399068483239264473et_nat @ A ) ) ) ).
% Inf_le_Sup
thf(fact_1187_Inf__le__Sup,axiom,
! [A: set_o] :
( ( A != bot_bot_set_o )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A ) @ ( complete_Sup_Sup_o @ A ) ) ) ).
% Inf_le_Sup
thf(fact_1188_finite__Sup__in,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ! [X: set_a,Y5: set_a] :
( ( member_set_a @ X @ A )
=> ( ( member_set_a @ Y5 @ A )
=> ( member_set_a @ ( sup_sup_set_a @ X @ Y5 ) @ A ) ) )
=> ( member_set_a @ ( comple2307003609928055243_set_a @ A ) @ A ) ) ) ) ).
% finite_Sup_in
thf(fact_1189_finite__Sup__in,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ( ! [X: set_nat,Y5: set_nat] :
( ( member_set_nat @ X @ A )
=> ( ( member_set_nat @ Y5 @ A )
=> ( member_set_nat @ ( sup_sup_set_nat @ X @ Y5 ) @ A ) ) )
=> ( member_set_nat @ ( comple7399068483239264473et_nat @ A ) @ A ) ) ) ) ).
% finite_Sup_in
thf(fact_1190_finite__Sup__in,axiom,
! [A: set_o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ! [X: $o,Y5: $o] :
( ( member_o @ X @ A )
=> ( ( member_o @ Y5 @ A )
=> ( member_o @ ( sup_sup_o @ X @ Y5 ) @ A ) ) )
=> ( member_o @ ( complete_Sup_Sup_o @ A ) @ A ) ) ) ) ).
% finite_Sup_in
thf(fact_1191_inj__on__iff__surj,axiom,
! [A: set_nat,A7: set_a] :
( ( A != bot_bot_set_nat )
=> ( ( ? [F4: nat > a] :
( ( inj_on_nat_a @ F4 @ A )
& ( ord_less_eq_set_a @ ( image_nat_a @ F4 @ A ) @ A7 ) ) )
= ( ? [G4: a > nat] :
( ( image_a_nat @ G4 @ A7 )
= A ) ) ) ) ).
% inj_on_iff_surj
thf(fact_1192_inj__on__iff__surj,axiom,
! [A: set_a,A7: set_a] :
( ( A != bot_bot_set_a )
=> ( ( ? [F4: a > a] :
( ( inj_on_a_a @ F4 @ A )
& ( ord_less_eq_set_a @ ( image_a_a @ F4 @ A ) @ A7 ) ) )
= ( ? [G4: a > a] :
( ( image_a_a @ G4 @ A7 )
= A ) ) ) ) ).
% inj_on_iff_surj
thf(fact_1193_inj__on__iff__surj,axiom,
! [A: set_nat,A7: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ( ? [F4: nat > nat] :
( ( inj_on_nat_nat @ F4 @ A )
& ( ord_less_eq_set_nat @ ( image_nat_nat @ F4 @ A ) @ A7 ) ) )
= ( ? [G4: nat > nat] :
( ( image_nat_nat @ G4 @ A7 )
= A ) ) ) ) ).
% inj_on_iff_surj
thf(fact_1194_inj__on__iff__surj,axiom,
! [A: set_a,A7: set_nat] :
( ( A != bot_bot_set_a )
=> ( ( ? [F4: a > nat] :
( ( inj_on_a_nat @ F4 @ A )
& ( ord_less_eq_set_nat @ ( image_a_nat @ F4 @ A ) @ A7 ) ) )
= ( ? [G4: nat > a] :
( ( image_nat_a @ G4 @ A7 )
= A ) ) ) ) ).
% inj_on_iff_surj
thf(fact_1195_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X2: a] : ( member_a @ X2 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_1196_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X2: nat] : ( member_nat @ X2 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_1197_bot__empty__eq,axiom,
( bot_bot_o_o
= ( ^ [X2: $o] : ( member_o @ X2 @ bot_bot_set_o ) ) ) ).
% bot_empty_eq
thf(fact_1198_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_1199_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_1200_uminus__set__def,axiom,
( uminus_uminus_set_o
= ( ^ [A3: set_o] :
( collect_o
@ ( uminus_uminus_o_o
@ ^ [X2: $o] : ( member_o @ X2 @ A3 ) ) ) ) ) ).
% uminus_set_def
thf(fact_1201_uminus__set__def,axiom,
( uminus_uminus_set_a
= ( ^ [A3: set_a] :
( collect_a
@ ( uminus_uminus_a_o
@ ^ [X2: a] : ( member_a @ X2 @ A3 ) ) ) ) ) ).
% uminus_set_def
thf(fact_1202_uminus__set__def,axiom,
( uminus5710092332889474511et_nat
= ( ^ [A3: set_nat] :
( collect_nat
@ ( uminus_uminus_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A3 ) ) ) ) ) ).
% uminus_set_def
thf(fact_1203_cSUP__UNION,axiom,
! [A: set_a,B: a > set_nat,F2: nat > nat] :
( ( A != bot_bot_set_a )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ( B @ X )
!= bot_bot_set_nat ) )
=> ( ( condit2214826472909112428ve_nat
@ ( comple7399068483239264473et_nat
@ ( image_a_set_nat
@ ^ [X2: a] : ( image_nat_nat @ F2 @ ( B @ X2 ) )
@ A ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat @ F2 @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ B @ A ) ) ) )
= ( complete_Sup_Sup_nat
@ ( image_a_nat
@ ^ [X2: a] : ( complete_Sup_Sup_nat @ ( image_nat_nat @ F2 @ ( B @ X2 ) ) )
@ A ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_1204_cSUP__UNION,axiom,
! [A: set_a,B: a > set_a,F2: a > nat] :
( ( A != bot_bot_set_a )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ( B @ X )
!= bot_bot_set_a ) )
=> ( ( condit2214826472909112428ve_nat
@ ( comple7399068483239264473et_nat
@ ( image_a_set_nat
@ ^ [X2: a] : ( image_a_nat @ F2 @ ( B @ X2 ) )
@ A ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_a_nat @ F2 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A ) ) ) )
= ( complete_Sup_Sup_nat
@ ( image_a_nat
@ ^ [X2: a] : ( complete_Sup_Sup_nat @ ( image_a_nat @ F2 @ ( B @ X2 ) ) )
@ A ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_1205_cSUP__UNION,axiom,
! [A: set_nat,B: nat > set_nat,F2: nat > nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ( B @ X )
!= bot_bot_set_nat ) )
=> ( ( condit2214826472909112428ve_nat
@ ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( image_nat_nat @ F2 @ ( B @ X2 ) )
@ A ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat @ F2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) ) )
= ( complete_Sup_Sup_nat
@ ( image_nat_nat
@ ^ [X2: nat] : ( complete_Sup_Sup_nat @ ( image_nat_nat @ F2 @ ( B @ X2 ) ) )
@ A ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_1206_cSUP__UNION,axiom,
! [A: set_nat,B: nat > set_a,F2: a > nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ( B @ X )
!= bot_bot_set_a ) )
=> ( ( condit2214826472909112428ve_nat
@ ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( image_a_nat @ F2 @ ( B @ X2 ) )
@ A ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_a_nat @ F2 @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ B @ A ) ) ) )
= ( complete_Sup_Sup_nat
@ ( image_nat_nat
@ ^ [X2: nat] : ( complete_Sup_Sup_nat @ ( image_a_nat @ F2 @ ( B @ X2 ) ) )
@ A ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_1207_cSUP__UNION,axiom,
! [A: set_o,B: $o > set_nat,F2: nat > nat] :
( ( A != bot_bot_set_o )
=> ( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ( B @ X )
!= bot_bot_set_nat ) )
=> ( ( condit2214826472909112428ve_nat
@ ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [X2: $o] : ( image_nat_nat @ F2 @ ( B @ X2 ) )
@ A ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat @ F2 @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B @ A ) ) ) )
= ( complete_Sup_Sup_nat
@ ( image_o_nat
@ ^ [X2: $o] : ( complete_Sup_Sup_nat @ ( image_nat_nat @ F2 @ ( B @ X2 ) ) )
@ A ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_1208_cSUP__UNION,axiom,
! [A: set_o,B: $o > set_a,F2: a > nat] :
( ( A != bot_bot_set_o )
=> ( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ( B @ X )
!= bot_bot_set_a ) )
=> ( ( condit2214826472909112428ve_nat
@ ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [X2: $o] : ( image_a_nat @ F2 @ ( B @ X2 ) )
@ A ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_a_nat @ F2 @ ( comple2307003609928055243_set_a @ ( image_o_set_a @ B @ A ) ) ) )
= ( complete_Sup_Sup_nat
@ ( image_o_nat
@ ^ [X2: $o] : ( complete_Sup_Sup_nat @ ( image_a_nat @ F2 @ ( B @ X2 ) ) )
@ A ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_1209_cINF__UNION,axiom,
! [A: set_a,B: a > set_nat,F2: nat > nat] :
( ( A != bot_bot_set_a )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ( B @ X )
!= bot_bot_set_nat ) )
=> ( ( condit1738341127787009408ow_nat
@ ( comple7399068483239264473et_nat
@ ( image_a_set_nat
@ ^ [X2: a] : ( image_nat_nat @ F2 @ ( B @ X2 ) )
@ A ) ) )
=> ( ( complete_Inf_Inf_nat @ ( image_nat_nat @ F2 @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ B @ A ) ) ) )
= ( complete_Inf_Inf_nat
@ ( image_a_nat
@ ^ [X2: a] : ( complete_Inf_Inf_nat @ ( image_nat_nat @ F2 @ ( B @ X2 ) ) )
@ A ) ) ) ) ) ) ).
% cINF_UNION
thf(fact_1210_cINF__UNION,axiom,
! [A: set_a,B: a > set_a,F2: a > nat] :
( ( A != bot_bot_set_a )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ( B @ X )
!= bot_bot_set_a ) )
=> ( ( condit1738341127787009408ow_nat
@ ( comple7399068483239264473et_nat
@ ( image_a_set_nat
@ ^ [X2: a] : ( image_a_nat @ F2 @ ( B @ X2 ) )
@ A ) ) )
=> ( ( complete_Inf_Inf_nat @ ( image_a_nat @ F2 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A ) ) ) )
= ( complete_Inf_Inf_nat
@ ( image_a_nat
@ ^ [X2: a] : ( complete_Inf_Inf_nat @ ( image_a_nat @ F2 @ ( B @ X2 ) ) )
@ A ) ) ) ) ) ) ).
% cINF_UNION
thf(fact_1211_cINF__UNION,axiom,
! [A: set_nat,B: nat > set_nat,F2: nat > nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ( B @ X )
!= bot_bot_set_nat ) )
=> ( ( condit1738341127787009408ow_nat
@ ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( image_nat_nat @ F2 @ ( B @ X2 ) )
@ A ) ) )
=> ( ( complete_Inf_Inf_nat @ ( image_nat_nat @ F2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) ) )
= ( complete_Inf_Inf_nat
@ ( image_nat_nat
@ ^ [X2: nat] : ( complete_Inf_Inf_nat @ ( image_nat_nat @ F2 @ ( B @ X2 ) ) )
@ A ) ) ) ) ) ) ).
% cINF_UNION
thf(fact_1212_cINF__UNION,axiom,
! [A: set_nat,B: nat > set_a,F2: a > nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ( B @ X )
!= bot_bot_set_a ) )
=> ( ( condit1738341127787009408ow_nat
@ ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( image_a_nat @ F2 @ ( B @ X2 ) )
@ A ) ) )
=> ( ( complete_Inf_Inf_nat @ ( image_a_nat @ F2 @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ B @ A ) ) ) )
= ( complete_Inf_Inf_nat
@ ( image_nat_nat
@ ^ [X2: nat] : ( complete_Inf_Inf_nat @ ( image_a_nat @ F2 @ ( B @ X2 ) ) )
@ A ) ) ) ) ) ) ).
% cINF_UNION
thf(fact_1213_cINF__UNION,axiom,
! [A: set_o,B: $o > set_nat,F2: nat > nat] :
( ( A != bot_bot_set_o )
=> ( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ( B @ X )
!= bot_bot_set_nat ) )
=> ( ( condit1738341127787009408ow_nat
@ ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [X2: $o] : ( image_nat_nat @ F2 @ ( B @ X2 ) )
@ A ) ) )
=> ( ( complete_Inf_Inf_nat @ ( image_nat_nat @ F2 @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B @ A ) ) ) )
= ( complete_Inf_Inf_nat
@ ( image_o_nat
@ ^ [X2: $o] : ( complete_Inf_Inf_nat @ ( image_nat_nat @ F2 @ ( B @ X2 ) ) )
@ A ) ) ) ) ) ) ).
% cINF_UNION
thf(fact_1214_cINF__UNION,axiom,
! [A: set_o,B: $o > set_a,F2: a > nat] :
( ( A != bot_bot_set_o )
=> ( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ( B @ X )
!= bot_bot_set_a ) )
=> ( ( condit1738341127787009408ow_nat
@ ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [X2: $o] : ( image_a_nat @ F2 @ ( B @ X2 ) )
@ A ) ) )
=> ( ( complete_Inf_Inf_nat @ ( image_a_nat @ F2 @ ( comple2307003609928055243_set_a @ ( image_o_set_a @ B @ A ) ) ) )
= ( complete_Inf_Inf_nat
@ ( image_o_nat
@ ^ [X2: $o] : ( complete_Inf_Inf_nat @ ( image_a_nat @ F2 @ ( B @ X2 ) ) )
@ A ) ) ) ) ) ) ).
% cINF_UNION
thf(fact_1215_bdd__belowI,axiom,
! [A: set_a,M2: a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_a @ M2 @ X ) )
=> ( condit5901475214736682318elow_a @ A ) ) ).
% bdd_belowI
thf(fact_1216_bdd__belowI,axiom,
! [A: set_o,M2: $o] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_o @ M2 @ X ) )
=> ( condit5413489452508810728elow_o @ A ) ) ).
% bdd_belowI
thf(fact_1217_bdd__belowI,axiom,
! [A: set_set_nat,M2: set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ( ord_less_eq_set_nat @ M2 @ X ) )
=> ( condit68592940725977398et_nat @ A ) ) ).
% bdd_belowI
thf(fact_1218_bdd__belowI,axiom,
! [A: set_nat,M2: nat] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_nat @ M2 @ X ) )
=> ( condit1738341127787009408ow_nat @ A ) ) ).
% bdd_belowI
thf(fact_1219_bdd__below_OI,axiom,
! [A: set_a,M: a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_a @ M @ X ) )
=> ( condit5901475214736682318elow_a @ A ) ) ).
% bdd_below.I
thf(fact_1220_bdd__below_OI,axiom,
! [A: set_o,M: $o] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_o @ M @ X ) )
=> ( condit5413489452508810728elow_o @ A ) ) ).
% bdd_below.I
thf(fact_1221_bdd__below_OI,axiom,
! [A: set_set_nat,M: set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ( ord_less_eq_set_nat @ M @ X ) )
=> ( condit68592940725977398et_nat @ A ) ) ).
% bdd_below.I
thf(fact_1222_bdd__below_OI,axiom,
! [A: set_nat,M: nat] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_nat @ M @ X ) )
=> ( condit1738341127787009408ow_nat @ A ) ) ).
% bdd_below.I
thf(fact_1223_bdd__above_OI,axiom,
! [A: set_a,M: a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_a @ X @ M ) )
=> ( condit5209368051240477026bove_a @ A ) ) ).
% bdd_above.I
thf(fact_1224_bdd__above_OI,axiom,
! [A: set_o,M: $o] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_o @ X @ M ) )
=> ( condit5488710616941104124bove_o @ A ) ) ).
% bdd_above.I
thf(fact_1225_bdd__above_OI,axiom,
! [A: set_set_nat,M: set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ( ord_less_eq_set_nat @ X @ M ) )
=> ( condit5477540289124974626et_nat @ A ) ) ).
% bdd_above.I
thf(fact_1226_bdd__above_OI,axiom,
! [A: set_nat,M: nat] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_nat @ X @ M ) )
=> ( condit2214826472909112428ve_nat @ A ) ) ).
% bdd_above.I
thf(fact_1227_bdd__below__Un,axiom,
! [A: set_nat,B: set_nat] :
( ( condit1738341127787009408ow_nat @ ( sup_sup_set_nat @ A @ B ) )
= ( ( condit1738341127787009408ow_nat @ A )
& ( condit1738341127787009408ow_nat @ B ) ) ) ).
% bdd_below_Un
thf(fact_1228_bdd__above__Un,axiom,
! [A: set_nat,B: set_nat] :
( ( condit2214826472909112428ve_nat @ ( sup_sup_set_nat @ A @ B ) )
= ( ( condit2214826472909112428ve_nat @ A )
& ( condit2214826472909112428ve_nat @ B ) ) ) ).
% bdd_above_Un
thf(fact_1229_bdd__above__image__sup,axiom,
! [F2: nat > nat,G2: nat > nat,A: set_nat] :
( ( condit2214826472909112428ve_nat
@ ( image_nat_nat
@ ^ [X2: nat] : ( sup_sup_nat @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
@ A ) )
= ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F2 @ A ) )
& ( condit2214826472909112428ve_nat @ ( image_nat_nat @ G2 @ A ) ) ) ) ).
% bdd_above_image_sup
thf(fact_1230_bdd__above__image__sup,axiom,
! [F2: a > nat,G2: a > nat,A: set_a] :
( ( condit2214826472909112428ve_nat
@ ( image_a_nat
@ ^ [X2: a] : ( sup_sup_nat @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
@ A ) )
= ( ( condit2214826472909112428ve_nat @ ( image_a_nat @ F2 @ A ) )
& ( condit2214826472909112428ve_nat @ ( image_a_nat @ G2 @ A ) ) ) ) ).
% bdd_above_image_sup
thf(fact_1231_bdd__above__UN,axiom,
! [I: set_a,A: a > set_nat] :
( ( finite_finite_a @ I )
=> ( ( condit2214826472909112428ve_nat @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ A @ I ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ I )
=> ( condit2214826472909112428ve_nat @ ( A @ X2 ) ) ) ) ) ) ).
% bdd_above_UN
thf(fact_1232_bdd__above__UN,axiom,
! [I: set_nat,A: nat > set_nat] :
( ( finite_finite_nat @ I )
=> ( ( condit2214826472909112428ve_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A @ I ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ I )
=> ( condit2214826472909112428ve_nat @ ( A @ X2 ) ) ) ) ) ) ).
% bdd_above_UN
thf(fact_1233_bdd__above__mono,axiom,
! [B: set_nat,A: set_nat] :
( ( condit2214826472909112428ve_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( condit2214826472909112428ve_nat @ A ) ) ) ).
% bdd_above_mono
thf(fact_1234_bdd__below__mono,axiom,
! [B: set_nat,A: set_nat] :
( ( condit1738341127787009408ow_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( condit1738341127787009408ow_nat @ A ) ) ) ).
% bdd_below_mono
thf(fact_1235_bdd__above_Ounfold,axiom,
( condit5477540289124974626et_nat
= ( ^ [A3: set_set_nat] :
? [M6: set_nat] :
! [X2: set_nat] :
( ( member_set_nat @ X2 @ A3 )
=> ( ord_less_eq_set_nat @ X2 @ M6 ) ) ) ) ).
% bdd_above.unfold
thf(fact_1236_bdd__above_Ounfold,axiom,
( condit2214826472909112428ve_nat
= ( ^ [A3: set_nat] :
? [M6: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A3 )
=> ( ord_less_eq_nat @ X2 @ M6 ) ) ) ) ).
% bdd_above.unfold
thf(fact_1237_bdd__above_OE,axiom,
! [A: set_a] :
( ( condit5209368051240477026bove_a @ A )
=> ~ ! [M5: a] :
~ ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( ord_less_eq_a @ X4 @ M5 ) ) ) ).
% bdd_above.E
thf(fact_1238_bdd__above_OE,axiom,
! [A: set_o] :
( ( condit5488710616941104124bove_o @ A )
=> ~ ! [M5: $o] :
~ ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( ord_less_eq_o @ X4 @ M5 ) ) ) ).
% bdd_above.E
thf(fact_1239_bdd__above_OE,axiom,
! [A: set_set_nat] :
( ( condit5477540289124974626et_nat @ A )
=> ~ ! [M5: set_nat] :
~ ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( ord_less_eq_set_nat @ X4 @ M5 ) ) ) ).
% bdd_above.E
thf(fact_1240_bdd__above_OE,axiom,
! [A: set_nat] :
( ( condit2214826472909112428ve_nat @ A )
=> ~ ! [M5: nat] :
~ ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ord_less_eq_nat @ X4 @ M5 ) ) ) ).
% bdd_above.E
thf(fact_1241_bdd__below_Ounfold,axiom,
( condit68592940725977398et_nat
= ( ^ [A3: set_set_nat] :
? [M6: set_nat] :
! [X2: set_nat] :
( ( member_set_nat @ X2 @ A3 )
=> ( ord_less_eq_set_nat @ M6 @ X2 ) ) ) ) ).
% bdd_below.unfold
thf(fact_1242_bdd__below_Ounfold,axiom,
( condit1738341127787009408ow_nat
= ( ^ [A3: set_nat] :
? [M6: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A3 )
=> ( ord_less_eq_nat @ M6 @ X2 ) ) ) ) ).
% bdd_below.unfold
thf(fact_1243_bdd__below_OE,axiom,
! [A: set_a] :
( ( condit5901475214736682318elow_a @ A )
=> ~ ! [M5: a] :
~ ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( ord_less_eq_a @ M5 @ X4 ) ) ) ).
% bdd_below.E
thf(fact_1244_bdd__below_OE,axiom,
! [A: set_o] :
( ( condit5413489452508810728elow_o @ A )
=> ~ ! [M5: $o] :
~ ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( ord_less_eq_o @ M5 @ X4 ) ) ) ).
% bdd_below.E
thf(fact_1245_bdd__below_OE,axiom,
! [A: set_set_nat] :
( ( condit68592940725977398et_nat @ A )
=> ~ ! [M5: set_nat] :
~ ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( ord_less_eq_set_nat @ M5 @ X4 ) ) ) ).
% bdd_below.E
thf(fact_1246_bdd__below_OE,axiom,
! [A: set_nat] :
( ( condit1738341127787009408ow_nat @ A )
=> ~ ! [M5: nat] :
~ ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ord_less_eq_nat @ M5 @ X4 ) ) ) ).
% bdd_below.E
thf(fact_1247_cSUP__eq__cINF__D,axiom,
! [F2: a > $o,A: set_a,A2: a] :
( ( ( complete_Sup_Sup_o @ ( image_a_o @ F2 @ A ) )
= ( complete_Inf_Inf_o @ ( image_a_o @ F2 @ A ) ) )
=> ( ( condit5488710616941104124bove_o @ ( image_a_o @ F2 @ A ) )
=> ( ( condit5413489452508810728elow_o @ ( image_a_o @ F2 @ A ) )
=> ( ( member_a @ A2 @ A )
=> ( ( F2 @ A2 )
= ( complete_Inf_Inf_o @ ( image_a_o @ F2 @ A ) ) ) ) ) ) ) ).
% cSUP_eq_cINF_D
thf(fact_1248_cSUP__eq__cINF__D,axiom,
! [F2: nat > $o,A: set_nat,A2: nat] :
( ( ( complete_Sup_Sup_o @ ( image_nat_o @ F2 @ A ) )
= ( complete_Inf_Inf_o @ ( image_nat_o @ F2 @ A ) ) )
=> ( ( condit5488710616941104124bove_o @ ( image_nat_o @ F2 @ A ) )
=> ( ( condit5413489452508810728elow_o @ ( image_nat_o @ F2 @ A ) )
=> ( ( member_nat @ A2 @ A )
=> ( ( F2 @ A2 )
= ( complete_Inf_Inf_o @ ( image_nat_o @ F2 @ A ) ) ) ) ) ) ) ).
% cSUP_eq_cINF_D
thf(fact_1249_cSUP__eq__cINF__D,axiom,
! [F2: $o > $o,A: set_o,A2: $o] :
( ( ( complete_Sup_Sup_o @ ( image_o_o @ F2 @ A ) )
= ( complete_Inf_Inf_o @ ( image_o_o @ F2 @ A ) ) )
=> ( ( condit5488710616941104124bove_o @ ( image_o_o @ F2 @ A ) )
=> ( ( condit5413489452508810728elow_o @ ( image_o_o @ F2 @ A ) )
=> ( ( member_o @ A2 @ A )
=> ( ( F2 @ A2 )
= ( complete_Inf_Inf_o @ ( image_o_o @ F2 @ A ) ) ) ) ) ) ) ).
% cSUP_eq_cINF_D
thf(fact_1250_cSUP__eq__cINF__D,axiom,
! [F2: a > nat,A: set_a,A2: a] :
( ( ( complete_Sup_Sup_nat @ ( image_a_nat @ F2 @ A ) )
= ( complete_Inf_Inf_nat @ ( image_a_nat @ F2 @ A ) ) )
=> ( ( condit2214826472909112428ve_nat @ ( image_a_nat @ F2 @ A ) )
=> ( ( condit1738341127787009408ow_nat @ ( image_a_nat @ F2 @ A ) )
=> ( ( member_a @ A2 @ A )
=> ( ( F2 @ A2 )
= ( complete_Inf_Inf_nat @ ( image_a_nat @ F2 @ A ) ) ) ) ) ) ) ).
% cSUP_eq_cINF_D
thf(fact_1251_cSUP__eq__cINF__D,axiom,
! [F2: nat > nat,A: set_nat,A2: nat] :
( ( ( complete_Sup_Sup_nat @ ( image_nat_nat @ F2 @ A ) )
= ( complete_Inf_Inf_nat @ ( image_nat_nat @ F2 @ A ) ) )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F2 @ A ) )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F2 @ A ) )
=> ( ( member_nat @ A2 @ A )
=> ( ( F2 @ A2 )
= ( complete_Inf_Inf_nat @ ( image_nat_nat @ F2 @ A ) ) ) ) ) ) ) ).
% cSUP_eq_cINF_D
thf(fact_1252_cSUP__eq__cINF__D,axiom,
! [F2: $o > nat,A: set_o,A2: $o] :
( ( ( complete_Sup_Sup_nat @ ( image_o_nat @ F2 @ A ) )
= ( complete_Inf_Inf_nat @ ( image_o_nat @ F2 @ A ) ) )
=> ( ( condit2214826472909112428ve_nat @ ( image_o_nat @ F2 @ A ) )
=> ( ( condit1738341127787009408ow_nat @ ( image_o_nat @ F2 @ A ) )
=> ( ( member_o @ A2 @ A )
=> ( ( F2 @ A2 )
= ( complete_Inf_Inf_nat @ ( image_o_nat @ F2 @ A ) ) ) ) ) ) ) ).
% cSUP_eq_cINF_D
thf(fact_1253_bdd__below__finite,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( condit1738341127787009408ow_nat @ A ) ) ).
% bdd_below_finite
thf(fact_1254_bdd__above__finite,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( condit2214826472909112428ve_nat @ A ) ) ).
% bdd_above_finite
thf(fact_1255_bdd__above__nat,axiom,
condit2214826472909112428ve_nat = finite_finite_nat ).
% bdd_above_nat
thf(fact_1256_cInf__le__cSup,axiom,
! [A: set_set_nat] :
( ( A != bot_bot_set_set_nat )
=> ( ( condit5477540289124974626et_nat @ A )
=> ( ( condit68592940725977398et_nat @ A )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A ) @ ( comple7399068483239264473et_nat @ A ) ) ) ) ) ).
% cInf_le_cSup
thf(fact_1257_cInf__le__cSup,axiom,
! [A: set_o] :
( ( A != bot_bot_set_o )
=> ( ( condit5488710616941104124bove_o @ A )
=> ( ( condit5413489452508810728elow_o @ A )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A ) @ ( complete_Sup_Sup_o @ A ) ) ) ) ) ).
% cInf_le_cSup
thf(fact_1258_cInf__le__cSup,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ A )
=> ( ( condit1738341127787009408ow_nat @ A )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ A ) @ ( complete_Sup_Sup_nat @ A ) ) ) ) ) ).
% cInf_le_cSup
thf(fact_1259_bdd__below_OI2,axiom,
! [A: set_a,M: a,F2: a > a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_a @ M @ ( F2 @ X ) ) )
=> ( condit5901475214736682318elow_a @ ( image_a_a @ F2 @ A ) ) ) ).
% bdd_below.I2
thf(fact_1260_bdd__below_OI2,axiom,
! [A: set_nat,M: a,F2: nat > a] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_a @ M @ ( F2 @ X ) ) )
=> ( condit5901475214736682318elow_a @ ( image_nat_a @ F2 @ A ) ) ) ).
% bdd_below.I2
thf(fact_1261_bdd__below_OI2,axiom,
! [A: set_a,M: set_nat,F2: a > set_nat] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_set_nat @ M @ ( F2 @ X ) ) )
=> ( condit68592940725977398et_nat @ ( image_a_set_nat @ F2 @ A ) ) ) ).
% bdd_below.I2
thf(fact_1262_bdd__below_OI2,axiom,
! [A: set_nat,M: set_nat,F2: nat > set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_set_nat @ M @ ( F2 @ X ) ) )
=> ( condit68592940725977398et_nat @ ( image_nat_set_nat @ F2 @ A ) ) ) ).
% bdd_below.I2
thf(fact_1263_bdd__below_OI2,axiom,
! [A: set_o,M: set_nat,F2: $o > set_nat] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_set_nat @ M @ ( F2 @ X ) ) )
=> ( condit68592940725977398et_nat @ ( image_o_set_nat @ F2 @ A ) ) ) ).
% bdd_below.I2
thf(fact_1264_bdd__below_OI2,axiom,
! [A: set_a,M: nat,F2: a > nat] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_nat @ M @ ( F2 @ X ) ) )
=> ( condit1738341127787009408ow_nat @ ( image_a_nat @ F2 @ A ) ) ) ).
% bdd_below.I2
thf(fact_1265_bdd__below_OI2,axiom,
! [A: set_nat,M: nat,F2: nat > nat] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_nat @ M @ ( F2 @ X ) ) )
=> ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F2 @ A ) ) ) ).
% bdd_below.I2
thf(fact_1266_bdd__below_OI2,axiom,
! [A: set_o,M: nat,F2: $o > nat] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_nat @ M @ ( F2 @ X ) ) )
=> ( condit1738341127787009408ow_nat @ ( image_o_nat @ F2 @ A ) ) ) ).
% bdd_below.I2
thf(fact_1267_bdd__belowI2,axiom,
! [A: set_a,M2: set_nat,F2: a > set_nat] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_set_nat @ M2 @ ( F2 @ X ) ) )
=> ( condit68592940725977398et_nat @ ( image_a_set_nat @ F2 @ A ) ) ) ).
% bdd_belowI2
thf(fact_1268_bdd__belowI2,axiom,
! [A: set_nat,M2: set_nat,F2: nat > set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_set_nat @ M2 @ ( F2 @ X ) ) )
=> ( condit68592940725977398et_nat @ ( image_nat_set_nat @ F2 @ A ) ) ) ).
% bdd_belowI2
thf(fact_1269_bdd__belowI2,axiom,
! [A: set_o,M2: set_nat,F2: $o > set_nat] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_set_nat @ M2 @ ( F2 @ X ) ) )
=> ( condit68592940725977398et_nat @ ( image_o_set_nat @ F2 @ A ) ) ) ).
% bdd_belowI2
thf(fact_1270_bdd__belowI2,axiom,
! [A: set_a,M2: nat,F2: a > nat] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_nat @ M2 @ ( F2 @ X ) ) )
=> ( condit1738341127787009408ow_nat @ ( image_a_nat @ F2 @ A ) ) ) ).
% bdd_belowI2
thf(fact_1271_bdd__belowI2,axiom,
! [A: set_nat,M2: nat,F2: nat > nat] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_nat @ M2 @ ( F2 @ X ) ) )
=> ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F2 @ A ) ) ) ).
% bdd_belowI2
thf(fact_1272_bdd__belowI2,axiom,
! [A: set_o,M2: nat,F2: $o > nat] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_nat @ M2 @ ( F2 @ X ) ) )
=> ( condit1738341127787009408ow_nat @ ( image_o_nat @ F2 @ A ) ) ) ).
% bdd_belowI2
thf(fact_1273_diff__diff__cancel,axiom,
! [I5: nat,N2: nat] :
( ( ord_less_eq_nat @ I5 @ N2 )
=> ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I5 ) )
= I5 ) ) ).
% diff_diff_cancel
% Helper facts (6)
thf(help_If_3_1_If_001tf__a_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X3: a,Y2: a] :
( ( if_a @ $false @ X3 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X3: a,Y2: a] :
( ( if_a @ $true @ X3 @ Y2 )
= X3 ) ).
thf(help_fChoice_1_1_fChoice_001_Eo_T,axiom,
! [P: $o > $o] :
( ( P @ ( fChoice_o @ P ) )
= ( ? [X6: $o] : ( P @ X6 ) ) ) ).
thf(help_fChoice_1_1_fChoice_001tf__a_T,axiom,
! [P: a > $o] :
( ( P @ ( fChoice_a @ P ) )
= ( ? [X6: a] : ( P @ X6 ) ) ) ).
thf(help_fChoice_1_1_fChoice_001t__Nat__Onat_T,axiom,
! [P: nat > $o] :
( ( P @ ( fChoice_nat @ P ) )
= ( ? [X6: nat] : ( P @ X6 ) ) ) ).
% Conjectures (1)
thf(conj_0,conjecture,
~ ( finite_finite_a
@ ( collect_a
@ ^ [D3: a] :
~ ( member_a @ D3 @ ( sup_sup_set_a @ ( sup_sup_set_a @ ( relational_adom_b_a @ i ) @ ( relational_csts_a_b @ q ) ) @ ( image_nat_a @ sigma @ ( relational_fv_a_b @ q ) ) ) ) ) ) ).
%------------------------------------------------------------------------------