TPTP Problem File: SLH0484^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : LP_Duality/0000_Minimum_Maximum/prob_00023_000739__28677128_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1523 ( 460 unt; 241 typ; 0 def)
% Number of atoms : 4190 (1022 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 10572 ( 355 ~; 52 |; 287 &;7960 @)
% ( 0 <=>;1918 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 7 avg)
% Number of types : 10 ( 9 usr)
% Number of type conns : 2227 (2227 >; 0 *; 0 +; 0 <<)
% Number of symbols : 235 ( 232 usr; 13 con; 0-4 aty)
% Number of variables : 3782 ( 651 ^;3060 !; 71 ?;3782 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 13:49:22.517
%------------------------------------------------------------------------------
% Could-be-implicit typings (9)
thf(ty_n_t__Set__Oset_I_062_It__Set__Oset_I_Eo_J_M_Eo_J_J,type,
set_set_o_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_Eo_M_062_I_Eo_M_Eo_J_J_J,type,
set_o_o_o: $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_Mtf__a_J_J,type,
set_o_a: $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_tf__a,type,
a: $tType ).
% Explicit typings (232)
thf(sy_c_BNF__Composition_ODEADID_Opred__DEADID_001t__Set__Oset_I_Eo_J,type,
bNF_pr4134575714447080524_set_o: set_o > $o ).
thf(sy_c_BNF__Composition_ODEADID_Opred__DEADID_001tf__a,type,
bNF_pred_DEADID_a: a > $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__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_Itf__a_J,type,
comple6135023378680113637_set_a: set_set_a > set_a ).
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__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_Itf__a_J,type,
comple2307003609928055243_set_a: set_set_a > set_a ).
thf(sy_c_Complete__Partial__Order_Occpo_Oadmissible_001_Eo,type,
comple8949206149834442853ible_o: ( set_o > $o ) > ( $o > $o > $o ) > ( $o > $o ) > $o ).
thf(sy_c_Complete__Partial__Order_Occpo__class_Ofixp_001_Eo,type,
comple2713996627985145509fixp_o: ( $o > $o ) > $o ).
thf(sy_c_Complete__Partial__Order_Ochain_001_Eo,type,
comple520228465662580424hain_o: ( $o > $o > $o ) > set_o > $o ).
thf(sy_c_Complete__Partial__Order_Ochain_001tf__a,type,
comple1697357536187991598hain_a: ( a > a > $o ) > set_a > $o ).
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_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_001tf__a,type,
condit5901475214736682318elow_a: set_a > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_001tf__a,type,
condit4103000493307248661_bdd_a: ( a > a > $o ) > ( a > a > $o ) > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_Obdd_001tf__a,type,
condit6541519642617408243_bdd_a: ( a > a > $o ) > set_a > $o ).
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__Set__Oset_I_Eo_J,type,
inj_on_o_set_o: ( $o > set_o ) > set_o > $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_001tf__a,type,
inj_on_a_a: ( a > a ) > set_a > $o ).
thf(sy_c_Fun_Omonotone__on_001_062_It__Set__Oset_I_Eo_J_M_Eo_J_001_062_It__Set__Oset_I_Eo_J_M_Eo_J,type,
monoto2155102285175209587et_o_o: set_set_o_o > ( ( set_o > $o ) > ( set_o > $o ) > $o ) > ( ( set_o > $o ) > ( set_o > $o ) > $o ) > ( ( set_o > $o ) > set_o > $o ) > $o ).
thf(sy_c_Fun_Omonotone__on_001_Eo_001_Eo,type,
monotone_on_o_o: set_o > ( $o > $o > $o ) > ( $o > $o > $o ) > ( $o > $o ) > $o ).
thf(sy_c_Fun_Omonotone__on_001_Eo_001tf__a,type,
monotone_on_o_a: set_o > ( $o > $o > $o ) > ( a > a > $o ) > ( $o > a ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
monoto7172710143293369831_set_a: set_set_a > ( set_a > set_a > $o ) > ( set_a > set_a > $o ) > ( set_a > set_a ) > $o ).
thf(sy_c_Fun_Omonotone__on_001tf__a_001tf__a,type,
monotone_on_a_a: set_a > ( a > a > $o ) > ( a > a > $o ) > ( a > a ) > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_Itf__a_M_Eo_J,type,
minus_minus_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_Eo_J,type,
minus_minus_set_o: set_o > set_o > set_o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
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_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_Itf__a_J,type,
uminus_uminus_set_a: set_a > set_a ).
thf(sy_c_HOL_OEx1_001_Eo,type,
ex1_o: ( $o > $o ) > $o ).
thf(sy_c_HOL_OEx1_001tf__a,type,
ex1_a: ( a > $o ) > $o ).
thf(sy_c_HOL_OThe_001_062_I_Eo_M_062_I_Eo_M_Eo_J_J,type,
the_o_o_o: ( ( $o > $o > $o ) > $o ) > $o > $o > $o ).
thf(sy_c_HOL_OThe_001_062_I_Eo_Mtf__a_J,type,
the_o_a: ( ( $o > a ) > $o ) > $o > a ).
thf(sy_c_HOL_OThe_001_062_It__Set__Oset_I_Eo_J_M_Eo_J,type,
the_set_o_o: ( ( set_o > $o ) > $o ) > set_o > $o ).
thf(sy_c_HOL_OThe_001_062_Itf__a_M_Eo_J,type,
the_a_o: ( ( a > $o ) > $o ) > a > $o ).
thf(sy_c_HOL_OThe_001_Eo,type,
the_o: ( $o > $o ) > $o ).
thf(sy_c_HOL_OThe_001t__Set__Oset_Itf__a_J,type,
the_set_a: ( set_a > $o ) > set_a ).
thf(sy_c_HOL_OThe_001tf__a,type,
the_a: ( a > $o ) > a ).
thf(sy_c_HOL_OUniq_001_Eo,type,
uniq_o: ( $o > $o ) > $o ).
thf(sy_c_HOL_OUniq_001tf__a,type,
uniq_a: ( a > $o ) > $o ).
thf(sy_c_HOL_Oinduct__false,type,
induct_false: $o ).
thf(sy_c_If_001_Eo,type,
if_o: $o > $o > $o > $o ).
thf(sy_c_If_001tf__a,type,
if_a: $o > a > a > a ).
thf(sy_c_Inductive_Ocomplete__lattice__class_Ogfp_001_Eo,type,
comple1228283932920895894_gfp_o: ( $o > $o ) > $o ).
thf(sy_c_Inductive_Ocomplete__lattice__class_Ogfp_001t__Set__Oset_Itf__a_J,type,
comple3341859861669737308_set_a: ( set_a > set_a ) > set_a ).
thf(sy_c_Inductive_Ocomplete__lattice__class_Olfp_001_Eo,type,
comple5737750096767067345_lfp_o: ( $o > $o ) > $o ).
thf(sy_c_Inductive_Ocomplete__lattice__class_Olfp_001t__Set__Oset_Itf__a_J,type,
comple1558298011288954135_set_a: ( set_a > set_a ) > set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_Itf__a_M_Eo_J,type,
inf_inf_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_Eo,type,
inf_inf_o: $o > $o > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_Eo_J,type,
inf_inf_set_o: set_o > set_o > set_o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_Osemilattice__neutr__order_001t__Set__Oset_I_Eo_J,type,
semila2554085542299052326_set_o: ( set_o > set_o > set_o ) > set_o > ( set_o > set_o > $o ) > ( set_o > set_o > $o ) > $o ).
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_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_List_OBleast_001_Eo,type,
bleast_o: set_o > ( $o > $o ) > $o ).
thf(sy_c_List_OBleast_001t__Set__Oset_I_Eo_J,type,
bleast_set_o: set_set_o > ( set_o > $o ) > set_o ).
thf(sy_c_List_OBleast_001t__Set__Oset_Itf__a_J,type,
bleast_set_a: set_set_a > ( set_a > $o ) > set_a ).
thf(sy_c_List_OBleast_001tf__a,type,
bleast_a: set_a > ( a > $o ) > a ).
thf(sy_c_List_Oabort__Bleast_001_Eo,type,
abort_Bleast_o: set_o > ( $o > $o ) > $o ).
thf(sy_c_List_Oabort__Bleast_001t__Set__Oset_I_Eo_J,type,
abort_Bleast_set_o: set_set_o > ( set_o > $o ) > set_o ).
thf(sy_c_List_Oabort__Bleast_001t__Set__Oset_Itf__a_J,type,
abort_Bleast_set_a: set_set_a > ( set_a > $o ) > set_a ).
thf(sy_c_List_Oabort__Bleast_001tf__a,type,
abort_Bleast_a: set_a > ( a > $o ) > a ).
thf(sy_c_List_Ofolding__insort__key_001_Eo_001_Eo,type,
foldin4841961211520082085ey_o_o: ( $o > $o > $o ) > ( $o > $o > $o ) > set_o > ( $o > $o ) > $o ).
thf(sy_c_List_Ofolding__insort__key_001tf__a_001tf__a,type,
foldin4382019238405368997ey_a_a: ( a > a > $o ) > ( a > a > $o ) > set_a > ( a > a ) > $o ).
thf(sy_c_Minimum__Maximum_OMaximum_001_Eo,type,
minimum_Maximum_o: set_o > $o ).
thf(sy_c_Minimum__Maximum_OMaximum_001tf__a,type,
minimum_Maximum_a: set_a > a ).
thf(sy_c_Minimum__Maximum_OMinimum_001_Eo,type,
minimum_Minimum_o: set_o > $o ).
thf(sy_c_Minimum__Maximum_OMinimum_001tf__a,type,
minimum_Minimum_a: set_a > a ).
thf(sy_c_Minimum__Maximum_Ohas__Maximum_001_062_I_Eo_M_062_I_Eo_M_Eo_J_J,type,
minimu3168151026475980457_o_o_o: set_o_o_o > $o ).
thf(sy_c_Minimum__Maximum_Ohas__Maximum_001_062_I_Eo_Mtf__a_J,type,
minimu315547183909508560um_o_a: set_o_a > $o ).
thf(sy_c_Minimum__Maximum_Ohas__Maximum_001_062_It__Set__Oset_I_Eo_J_M_Eo_J,type,
minimu1222600986477729290et_o_o: set_set_o_o > $o ).
thf(sy_c_Minimum__Maximum_Ohas__Maximum_001_062_Itf__a_M_Eo_J,type,
minimu1594658540777173508um_a_o: set_a_o > $o ).
thf(sy_c_Minimum__Maximum_Ohas__Maximum_001_Eo,type,
minimu3250747243564136147imum_o: set_o > $o ).
thf(sy_c_Minimum__Maximum_Ohas__Maximum_001t__Set__Oset_I_Eo_J,type,
minimu5048523218729521587_set_o: set_set_o > $o ).
thf(sy_c_Minimum__Maximum_Ohas__Maximum_001t__Set__Oset_Itf__a_J,type,
minimu8775777210878807577_set_a: set_set_a > $o ).
thf(sy_c_Minimum__Maximum_Ohas__Maximum_001tf__a,type,
minimu6197867597544231097imum_a: set_a > $o ).
thf(sy_c_Minimum__Maximum_Ohas__Minimum_001_062_I_Eo_M_062_I_Eo_M_Eo_J_J,type,
minimu4395757616390794811_o_o_o: set_o_o_o > $o ).
thf(sy_c_Minimum__Maximum_Ohas__Minimum_001_062_I_Eo_Mtf__a_J,type,
minimu4657282916794952894um_o_a: set_o_a > $o ).
thf(sy_c_Minimum__Maximum_Ohas__Minimum_001_062_It__Set__Oset_I_Eo_J_M_Eo_J,type,
minimu7721965553910500344et_o_o: set_set_o_o > $o ).
thf(sy_c_Minimum__Maximum_Ohas__Minimum_001_062_Itf__a_M_Eo_J,type,
minimu5936394273662617842um_a_o: set_a_o > $o ).
thf(sy_c_Minimum__Maximum_Ohas__Minimum_001_Eo,type,
minimu7793368684116157285imum_o: set_o > $o ).
thf(sy_c_Minimum__Maximum_Ohas__Minimum_001t__Set__Oset_I_Eo_J,type,
minimu842366695515436613_set_o: set_set_o > $o ).
thf(sy_c_Minimum__Maximum_Ohas__Minimum_001t__Set__Oset_Itf__a_J,type,
minimu6896447672505010603_set_a: set_set_a > $o ).
thf(sy_c_Minimum__Maximum_Ohas__Minimum_001tf__a,type,
minimu7473987258551571531imum_a: set_a > $o ).
thf(sy_c_Nunchaku_OThe__unsafe_001_Eo,type,
the_unsafe_o: ( $o > $o ) > $o ).
thf(sy_c_Nunchaku_OThe__unsafe_001tf__a,type,
the_unsafe_a: ( a > $o ) > a ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_I_Eo_M_062_I_Eo_M_Eo_J_J,type,
bot_bot_o_o_o: $o > $o > $o ).
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_Itf__a_M_Eo_J,type,
bot_bot_a_o: a > $o ).
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_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord_OLeast_001_Eo,type,
least_o: ( $o > $o > $o ) > ( $o > $o ) > $o ).
thf(sy_c_Orderings_Oord_OLeast_001tf__a,type,
least_a: ( a > a > $o ) > ( a > $o ) > a ).
thf(sy_c_Orderings_Oord__class_OLeast_001_062_I_Eo_M_062_I_Eo_M_Eo_J_J,type,
ord_Least_o_o_o: ( ( $o > $o > $o ) > $o ) > $o > $o > $o ).
thf(sy_c_Orderings_Oord__class_OLeast_001_062_I_Eo_Mtf__a_J,type,
ord_Least_o_a: ( ( $o > a ) > $o ) > $o > a ).
thf(sy_c_Orderings_Oord__class_OLeast_001_062_It__Set__Oset_I_Eo_J_M_Eo_J,type,
ord_Least_set_o_o: ( ( set_o > $o ) > $o ) > set_o > $o ).
thf(sy_c_Orderings_Oord__class_OLeast_001_062_Itf__a_M_Eo_J,type,
ord_Least_a_o: ( ( a > $o ) > $o ) > a > $o ).
thf(sy_c_Orderings_Oord__class_OLeast_001_Eo,type,
ord_Least_o: ( $o > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_OLeast_001t__Set__Oset_I_Eo_J,type,
ord_Least_set_o: ( set_o > $o ) > set_o ).
thf(sy_c_Orderings_Oord__class_OLeast_001t__Set__Oset_Itf__a_J,type,
ord_Least_set_a: ( set_a > $o ) > set_a ).
thf(sy_c_Orderings_Oord__class_OLeast_001tf__a,type,
ord_Least_a: ( a > $o ) > a ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_I_Eo_M_062_I_Eo_M_Eo_J_J,type,
ord_less_o_o_o: ( $o > $o > $o ) > ( $o > $o > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_I_Eo_M_Eo_J,type,
ord_less_o_o: ( $o > $o ) > ( $o > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_I_Eo_Mtf__a_J,type,
ord_less_o_a: ( $o > a ) > ( $o > a ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Set__Oset_I_Eo_J_M_Eo_J,type,
ord_less_set_o_o: ( set_o > $o ) > ( set_o > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_Itf__a_M_Eo_J,type,
ord_less_a_o: ( a > $o ) > ( a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_Eo,type,
ord_less_o: $o > $o > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_062_It__Set__Oset_I_Eo_J_M_Eo_J_J,type,
ord_less_set_set_o_o: set_set_o_o > set_set_o_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_Eo_J,type,
ord_less_set_o: set_o > set_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_less_set_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001tf__a,type,
ord_less_a: a > a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_Eo_J_J_J,type,
ord_less_eq_o_o_o_o: ( $o > $o > $o > $o ) > ( $o > $o > $o > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_062_I_Eo_M_Eo_J_J,type,
ord_less_eq_o_o_o: ( $o > $o > $o ) > ( $o > $o > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_062_I_Eo_Mtf__a_J_J,type,
ord_less_eq_o_o_a: ( $o > $o > a ) > ( $o > $o > a ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_062_It__Set__Oset_I_Eo_J_M_Eo_J_J,type,
ord_le1909991985842454446et_o_o: ( $o > set_o > $o ) > ( $o > set_o > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_062_Itf__a_M_Eo_J_J,type,
ord_less_eq_o_a_o: ( $o > a > $o ) > ( $o > a > $o ) > $o ).
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_I_Eo_Mt__Set__Oset_Itf__a_J_J,type,
ord_less_eq_o_set_a: ( $o > set_a ) > ( $o > set_a ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mtf__a_J,type,
ord_less_eq_o_a: ( $o > a ) > ( $o > a ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_I_Eo_J_M_Eo_J,type,
ord_less_eq_set_o_o: ( set_o > $o ) > ( set_o > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_It__Set__Oset_I_Eo_J_J_M_Eo_J,type,
ord_le8367510561069133573et_o_o: ( set_set_o > $o ) > ( set_set_o > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_Itf__a_J_M_Eo_J,type,
ord_less_eq_set_a_o: ( set_a > $o ) > ( set_a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_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__Set__Oset_I_062_It__Set__Oset_I_Eo_J_M_Eo_J_J,type,
ord_le4904625296160870427et_o_o: set_set_o_o > set_set_o_o > $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__Set__Oset_I_Eo_J_J,type,
ord_le4374716579403074808_set_o: set_set_o > set_set_o > $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_Orderings_Oorder__class_OGreatest_001_062_I_Eo_M_062_I_Eo_M_Eo_J_J,type,
order_Greatest_o_o_o: ( ( $o > $o > $o ) > $o ) > $o > $o > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001_062_I_Eo_Mtf__a_J,type,
order_Greatest_o_a: ( ( $o > a ) > $o ) > $o > a ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001_062_It__Set__Oset_I_Eo_J_M_Eo_J,type,
order_5805555182909007980et_o_o: ( ( set_o > $o ) > $o ) > set_o > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001_062_Itf__a_M_Eo_J,type,
order_Greatest_a_o: ( ( a > $o ) > $o ) > a > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001_Eo,type,
order_Greatest_o: ( $o > $o ) > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_Itf__a_J,type,
order_Greatest_set_a: ( set_a > $o ) > set_a ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001tf__a,type,
order_Greatest_a: ( a > $o ) > a ).
thf(sy_c_Orderings_Oordering_001_062_I_Eo_M_062_I_Eo_M_Eo_J_J,type,
ordering_o_o_o: ( ( $o > $o > $o ) > ( $o > $o > $o ) > $o ) > ( ( $o > $o > $o ) > ( $o > $o > $o ) > $o ) > $o ).
thf(sy_c_Orderings_Oordering_001_062_I_Eo_Mtf__a_J,type,
ordering_o_a: ( ( $o > a ) > ( $o > a ) > $o ) > ( ( $o > a ) > ( $o > a ) > $o ) > $o ).
thf(sy_c_Orderings_Oordering_001_062_It__Set__Oset_I_Eo_J_M_Eo_J,type,
ordering_set_o_o: ( ( set_o > $o ) > ( set_o > $o ) > $o ) > ( ( set_o > $o ) > ( set_o > $o ) > $o ) > $o ).
thf(sy_c_Orderings_Oordering_001_062_Itf__a_M_Eo_J,type,
ordering_a_o: ( ( a > $o ) > ( a > $o ) > $o ) > ( ( a > $o ) > ( a > $o ) > $o ) > $o ).
thf(sy_c_Orderings_Oordering_001_Eo,type,
ordering_o: ( $o > $o > $o ) > ( $o > $o > $o ) > $o ).
thf(sy_c_Orderings_Oordering_001t__Set__Oset_I_Eo_J,type,
ordering_set_o: ( set_o > set_o > $o ) > ( set_o > set_o > $o ) > $o ).
thf(sy_c_Orderings_Oordering_001t__Set__Oset_Itf__a_J,type,
ordering_set_a: ( set_a > set_a > $o ) > ( set_a > set_a > $o ) > $o ).
thf(sy_c_Orderings_Oordering_001tf__a,type,
ordering_a: ( a > a > $o ) > ( a > a > $o ) > $o ).
thf(sy_c_Orderings_Oordering__axioms_001tf__a,type,
ordering_axioms_a: ( a > a > $o ) > ( a > a > $o ) > $o ).
thf(sy_c_Orderings_Oordering__top_001_062_I_Eo_M_062_I_Eo_M_Eo_J_J,type,
ordering_top_o_o_o: ( ( $o > $o > $o ) > ( $o > $o > $o ) > $o ) > ( ( $o > $o > $o ) > ( $o > $o > $o ) > $o ) > ( $o > $o > $o ) > $o ).
thf(sy_c_Orderings_Oordering__top_001_062_I_Eo_M_Eo_J,type,
ordering_top_o_o: ( ( $o > $o ) > ( $o > $o ) > $o ) > ( ( $o > $o ) > ( $o > $o ) > $o ) > ( $o > $o ) > $o ).
thf(sy_c_Orderings_Oordering__top_001_062_It__Set__Oset_I_Eo_J_M_Eo_J,type,
ordering_top_set_o_o: ( ( set_o > $o ) > ( set_o > $o ) > $o ) > ( ( set_o > $o ) > ( set_o > $o ) > $o ) > ( set_o > $o ) > $o ).
thf(sy_c_Orderings_Oordering__top_001_062_Itf__a_M_Eo_J,type,
ordering_top_a_o: ( ( a > $o ) > ( a > $o ) > $o ) > ( ( a > $o ) > ( a > $o ) > $o ) > ( a > $o ) > $o ).
thf(sy_c_Orderings_Oordering__top_001_Eo,type,
ordering_top_o: ( $o > $o > $o ) > ( $o > $o > $o ) > $o > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_I_062_It__Set__Oset_I_Eo_J_M_Eo_J_J,type,
orderi1143446957891364042et_o_o: ( set_set_o_o > set_set_o_o > $o ) > ( set_set_o_o > set_set_o_o > $o ) > set_set_o_o > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_I_Eo_J,type,
ordering_top_set_o: ( set_o > set_o > $o ) > ( set_o > set_o > $o ) > set_o > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
orderi5875812994216768367_set_a: ( set_set_a > set_set_a > $o ) > ( set_set_a > set_set_a > $o ) > set_set_a > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_Itf__a_J,type,
ordering_top_set_a: ( set_a > set_a > $o ) > ( set_a > set_a > $o ) > set_a > $o ).
thf(sy_c_Orderings_Oordering__top_001tf__a,type,
ordering_top_a: ( a > a > $o ) > ( a > a > $o ) > a > $o ).
thf(sy_c_Orderings_Opartial__preordering_001_062_I_Eo_M_062_I_Eo_M_Eo_J_J,type,
partia1881799573076113956_o_o_o: ( ( $o > $o > $o ) > ( $o > $o > $o ) > $o ) > $o ).
thf(sy_c_Orderings_Opartial__preordering_001_062_I_Eo_Mtf__a_J,type,
partia5423788306336055317ng_o_a: ( ( $o > a ) > ( $o > a ) > $o ) > $o ).
thf(sy_c_Orderings_Opartial__preordering_001_062_It__Set__Oset_I_Eo_J_M_Eo_J,type,
partia4811167007327223503et_o_o: ( ( set_o > $o ) > ( set_o > $o ) > $o ) > $o ).
thf(sy_c_Orderings_Opartial__preordering_001_062_Itf__a_M_Eo_J,type,
partia6702899663203720265ng_a_o: ( ( a > $o ) > ( a > $o ) > $o ) > $o ).
thf(sy_c_Orderings_Opartial__preordering_001t__Set__Oset_Itf__a_J,type,
partia6602192050731689876_set_a: ( set_a > set_a > $o ) > $o ).
thf(sy_c_Orderings_Opartial__preordering_001tf__a,type,
partia125584492769400372ring_a: ( a > a > $o ) > $o ).
thf(sy_c_Orderings_Opreordering_001tf__a,type,
preordering_a: ( a > a > $o ) > ( a > a > $o ) > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_I_Eo_M_062_I_Eo_M_Eo_J_J,type,
top_top_o_o_o: $o > $o > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_I_Eo_M_Eo_J,type,
top_top_o_o: $o > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Set__Oset_I_Eo_J_M_Eo_J,type,
top_top_set_o_o: set_o > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_Itf__a_M_Eo_J,type,
top_top_a_o: a > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_Eo,type,
top_top_o: $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Set__Oset_I_Eo_J_M_Eo_J_J,type,
top_top_set_set_o_o: set_set_o_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
top_top_set_o: set_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
top_top_set_set_o: set_set_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
top_top_set_set_a: set_set_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__a_J,type,
top_top_set_a: set_a ).
thf(sy_c_Partial__Function_Oflat__lub_001_Eo,type,
partial_flat_lub_o: $o > set_o > $o ).
thf(sy_c_Partial__Function_Oflat__lub_001tf__a,type,
partial_flat_lub_a: a > set_a > a ).
thf(sy_c_Relation_OPowp_001_Eo,type,
powp_o: ( $o > $o ) > set_o > $o ).
thf(sy_c_Relation_OPowp_001t__Set__Oset_I_Eo_J,type,
powp_set_o: ( set_o > $o ) > set_set_o > $o ).
thf(sy_c_Relation_OPowp_001tf__a,type,
powp_a: ( a > $o ) > set_a > $o ).
thf(sy_c_Relation_Oantisymp__on_001_Eo,type,
antisymp_on_o: set_o > ( $o > $o > $o ) > $o ).
thf(sy_c_Relation_Oantisymp__on_001tf__a,type,
antisymp_on_a: set_a > ( a > a > $o ) > $o ).
thf(sy_c_Relation_Osingle__valuedp_001_Eo_001_Eo,type,
single_valuedp_o_o: ( $o > $o > $o ) > $o ).
thf(sy_c_Set_OBall_001_Eo,type,
ball_o: set_o > ( $o > $o ) > $o ).
thf(sy_c_Set_OBall_001t__Set__Oset_I_Eo_J,type,
ball_set_o: set_set_o > ( set_o > $o ) > $o ).
thf(sy_c_Set_OBall_001t__Set__Oset_Itf__a_J,type,
ball_set_a: set_set_a > ( set_a > $o ) > $o ).
thf(sy_c_Set_OBall_001tf__a,type,
ball_a: set_a > ( a > $o ) > $o ).
thf(sy_c_Set_OCollect_001_062_It__Set__Oset_I_Eo_J_M_Eo_J,type,
collect_set_o_o: ( ( set_o > $o ) > $o ) > set_set_o_o ).
thf(sy_c_Set_OCollect_001_Eo,type,
collect_o: ( $o > $o ) > set_o ).
thf(sy_c_Set_OCollect_001t__Set__Oset_I_Eo_J,type,
collect_set_o: ( set_o > $o ) > set_set_o ).
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_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__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_001tf__a,type,
image_o_a: ( $o > a ) > set_o > set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_Eo_J_001t__Set__Oset_I_Eo_J,type,
image_set_o_set_o: ( set_o > set_o ) > set_set_o > set_set_o ).
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_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__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_Itf__a_J,type,
image_a_set_a: ( a > set_a ) > set_a > set_set_a ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set_Oinsert_001_Eo,type,
insert_o: $o > set_o > set_o ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Ois__empty_001_Eo,type,
is_empty_o: set_o > $o ).
thf(sy_c_Set_Ois__singleton_001_Eo,type,
is_singleton_o: set_o > $o ).
thf(sy_c_Set_Ois__singleton_001tf__a,type,
is_singleton_a: set_a > $o ).
thf(sy_c_Set_Oremove_001_Eo,type,
remove_o: $o > set_o > set_o ).
thf(sy_c_Set_Oremove_001tf__a,type,
remove_a: a > set_a > set_a ).
thf(sy_c_Set_Othe__elem_001_Eo,type,
the_elem_o: set_o > $o ).
thf(sy_c_Set_Othe__elem_001tf__a,type,
the_elem_a: set_a > a ).
thf(sy_c_Set_Ovimage_001_Eo_001_Eo,type,
vimage_o_o: ( $o > $o ) > set_o > set_o ).
thf(sy_c_Set_Ovimage_001_Eo_001tf__a,type,
vimage_o_a: ( $o > a ) > set_a > set_o ).
thf(sy_c_Set_Ovimage_001tf__a_001_Eo,type,
vimage_a_o: ( a > $o ) > set_o > set_a ).
thf(sy_c_Set_Ovimage_001tf__a_001tf__a,type,
vimage_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_member_001_062_I_Eo_M_062_I_Eo_M_Eo_J_J,type,
member_o_o_o: ( $o > $o > $o ) > set_o_o_o > $o ).
thf(sy_c_member_001_062_I_Eo_Mtf__a_J,type,
member_o_a: ( $o > a ) > set_o_a > $o ).
thf(sy_c_member_001_062_It__Set__Oset_I_Eo_J_M_Eo_J,type,
member_set_o_o: ( set_o > $o ) > set_set_o_o > $o ).
thf(sy_c_member_001_062_Itf__a_M_Eo_J,type,
member_a_o: ( a > $o ) > set_a_o > $o ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $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_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_S,type,
s: set_a ).
thf(sy_v_x,type,
x: a ).
% Relevant facts (1276)
thf(fact_0_assms_I1_J,axiom,
member_a @ x @ s ).
% assms(1)
thf(fact_1_assms_I2_J,axiom,
! [Y: a] :
( ( member_a @ Y @ s )
=> ( ord_less_eq_a @ Y @ x ) ) ).
% assms(2)
thf(fact_2_Maximum__def,axiom,
( minimum_Maximum_o
= ( ^ [S: set_o] :
( the_o
@ ^ [X: $o] :
( ( member_o @ X @ S )
& ! [Y2: $o] :
( ( member_o @ Y2 @ S )
=> ( ord_less_eq_o @ Y2 @ X ) ) ) ) ) ) ).
% Maximum_def
thf(fact_3_Maximum__def,axiom,
( minimum_Maximum_a
= ( ^ [S: set_a] :
( the_a
@ ^ [X: a] :
( ( member_a @ X @ S )
& ! [Y2: a] :
( ( member_a @ Y2 @ S )
=> ( ord_less_eq_a @ Y2 @ X ) ) ) ) ) ) ).
% Maximum_def
thf(fact_4_Minimum__def,axiom,
( minimum_Minimum_o
= ( ^ [S: set_o] :
( the_o
@ ^ [X: $o] :
( ( member_o @ X @ S )
& ! [Y2: $o] :
( ( member_o @ Y2 @ S )
=> ( ord_less_eq_o @ X @ Y2 ) ) ) ) ) ) ).
% Minimum_def
thf(fact_5_Minimum__def,axiom,
( minimum_Minimum_a
= ( ^ [S: set_a] :
( the_a
@ ^ [X: a] :
( ( member_a @ X @ S )
& ! [Y2: a] :
( ( member_a @ Y2 @ S )
=> ( ord_less_eq_a @ X @ Y2 ) ) ) ) ) ) ).
% Minimum_def
thf(fact_6_the__equality,axiom,
! [P: $o > $o,A: $o] :
( ( P @ A )
=> ( ! [X2: $o] :
( ( P @ X2 )
=> ( X2 = A ) )
=> ( ( the_o @ P )
= A ) ) ) ).
% the_equality
thf(fact_7_the__equality,axiom,
! [P: a > $o,A: a] :
( ( P @ A )
=> ( ! [X2: a] :
( ( P @ X2 )
=> ( X2 = A ) )
=> ( ( the_a @ P )
= A ) ) ) ).
% the_equality
thf(fact_8_the__eq__trivial,axiom,
! [A: $o] :
( ( the_o
@ ^ [X: $o] : ( X = A ) )
= A ) ).
% the_eq_trivial
thf(fact_9_the__eq__trivial,axiom,
! [A: a] :
( ( the_a
@ ^ [X: a] : ( X = A ) )
= A ) ).
% the_eq_trivial
thf(fact_10_the__sym__eq__trivial,axiom,
! [X3: $o] :
( ( the_o
@ ( ^ [Y3: $o,Z: $o] : ( Y3 = Z )
@ X3 ) )
= X3 ) ).
% the_sym_eq_trivial
thf(fact_11_the__sym__eq__trivial,axiom,
! [X3: a] :
( ( the_a
@ ( ^ [Y3: a,Z: a] : ( Y3 = Z )
@ X3 ) )
= X3 ) ).
% the_sym_eq_trivial
thf(fact_12_order__refl,axiom,
! [X3: set_a] : ( ord_less_eq_set_a @ X3 @ X3 ) ).
% order_refl
thf(fact_13_order__refl,axiom,
! [X3: $o > $o > $o] : ( ord_less_eq_o_o_o @ X3 @ X3 ) ).
% order_refl
thf(fact_14_order__refl,axiom,
! [X3: $o > a] : ( ord_less_eq_o_a @ X3 @ X3 ) ).
% order_refl
thf(fact_15_order__refl,axiom,
! [X3: set_o > $o] : ( ord_less_eq_set_o_o @ X3 @ X3 ) ).
% order_refl
thf(fact_16_order__refl,axiom,
! [X3: a > $o] : ( ord_less_eq_a_o @ X3 @ X3 ) ).
% order_refl
thf(fact_17_order__refl,axiom,
! [X3: a] : ( ord_less_eq_a @ X3 @ X3 ) ).
% order_refl
thf(fact_18_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_19_dual__order_Orefl,axiom,
! [A: $o > $o > $o] : ( ord_less_eq_o_o_o @ A @ A ) ).
% dual_order.refl
thf(fact_20_dual__order_Orefl,axiom,
! [A: $o > a] : ( ord_less_eq_o_a @ A @ A ) ).
% dual_order.refl
thf(fact_21_dual__order_Orefl,axiom,
! [A: set_o > $o] : ( ord_less_eq_set_o_o @ A @ A ) ).
% dual_order.refl
thf(fact_22_dual__order_Orefl,axiom,
! [A: a > $o] : ( ord_less_eq_a_o @ A @ A ) ).
% dual_order.refl
thf(fact_23_dual__order_Orefl,axiom,
! [A: a] : ( ord_less_eq_a @ A @ A ) ).
% dual_order.refl
thf(fact_24_has__Maximum__def,axiom,
( minimu3250747243564136147imum_o
= ( ^ [S: set_o] :
? [X: $o] :
( ( member_o @ X @ S )
& ! [Y2: $o] :
( ( member_o @ Y2 @ S )
=> ( ord_less_eq_o @ Y2 @ X ) ) ) ) ) ).
% has_Maximum_def
thf(fact_25_has__Maximum__def,axiom,
( minimu5048523218729521587_set_o
= ( ^ [S: set_set_o] :
? [X: set_o] :
( ( member_set_o @ X @ S )
& ! [Y2: set_o] :
( ( member_set_o @ Y2 @ S )
=> ( ord_less_eq_set_o @ Y2 @ X ) ) ) ) ) ).
% has_Maximum_def
thf(fact_26_has__Maximum__def,axiom,
( minimu8775777210878807577_set_a
= ( ^ [S: set_set_a] :
? [X: set_a] :
( ( member_set_a @ X @ S )
& ! [Y2: set_a] :
( ( member_set_a @ Y2 @ S )
=> ( ord_less_eq_set_a @ Y2 @ X ) ) ) ) ) ).
% has_Maximum_def
thf(fact_27_has__Maximum__def,axiom,
( minimu3168151026475980457_o_o_o
= ( ^ [S: set_o_o_o] :
? [X: $o > $o > $o] :
( ( member_o_o_o @ X @ S )
& ! [Y2: $o > $o > $o] :
( ( member_o_o_o @ Y2 @ S )
=> ( ord_less_eq_o_o_o @ Y2 @ X ) ) ) ) ) ).
% has_Maximum_def
thf(fact_28_has__Maximum__def,axiom,
( minimu315547183909508560um_o_a
= ( ^ [S: set_o_a] :
? [X: $o > a] :
( ( member_o_a @ X @ S )
& ! [Y2: $o > a] :
( ( member_o_a @ Y2 @ S )
=> ( ord_less_eq_o_a @ Y2 @ X ) ) ) ) ) ).
% has_Maximum_def
thf(fact_29_has__Maximum__def,axiom,
( minimu1222600986477729290et_o_o
= ( ^ [S: set_set_o_o] :
? [X: set_o > $o] :
( ( member_set_o_o @ X @ S )
& ! [Y2: set_o > $o] :
( ( member_set_o_o @ Y2 @ S )
=> ( ord_less_eq_set_o_o @ Y2 @ X ) ) ) ) ) ).
% has_Maximum_def
thf(fact_30_has__Maximum__def,axiom,
( minimu1594658540777173508um_a_o
= ( ^ [S: set_a_o] :
? [X: a > $o] :
( ( member_a_o @ X @ S )
& ! [Y2: a > $o] :
( ( member_a_o @ Y2 @ S )
=> ( ord_less_eq_a_o @ Y2 @ X ) ) ) ) ) ).
% has_Maximum_def
thf(fact_31_has__Maximum__def,axiom,
( minimu6197867597544231097imum_a
= ( ^ [S: set_a] :
? [X: a] :
( ( member_a @ X @ S )
& ! [Y2: a] :
( ( member_a @ Y2 @ S )
=> ( ord_less_eq_a @ Y2 @ X ) ) ) ) ) ).
% has_Maximum_def
thf(fact_32_has__Minimum__def,axiom,
( minimu7793368684116157285imum_o
= ( ^ [S: set_o] :
? [X: $o] :
( ( member_o @ X @ S )
& ! [Y2: $o] :
( ( member_o @ Y2 @ S )
=> ( ord_less_eq_o @ X @ Y2 ) ) ) ) ) ).
% has_Minimum_def
thf(fact_33_has__Minimum__def,axiom,
( minimu842366695515436613_set_o
= ( ^ [S: set_set_o] :
? [X: set_o] :
( ( member_set_o @ X @ S )
& ! [Y2: set_o] :
( ( member_set_o @ Y2 @ S )
=> ( ord_less_eq_set_o @ X @ Y2 ) ) ) ) ) ).
% has_Minimum_def
thf(fact_34_has__Minimum__def,axiom,
( minimu6896447672505010603_set_a
= ( ^ [S: set_set_a] :
? [X: set_a] :
( ( member_set_a @ X @ S )
& ! [Y2: set_a] :
( ( member_set_a @ Y2 @ S )
=> ( ord_less_eq_set_a @ X @ Y2 ) ) ) ) ) ).
% has_Minimum_def
thf(fact_35_has__Minimum__def,axiom,
( minimu4395757616390794811_o_o_o
= ( ^ [S: set_o_o_o] :
? [X: $o > $o > $o] :
( ( member_o_o_o @ X @ S )
& ! [Y2: $o > $o > $o] :
( ( member_o_o_o @ Y2 @ S )
=> ( ord_less_eq_o_o_o @ X @ Y2 ) ) ) ) ) ).
% has_Minimum_def
thf(fact_36_has__Minimum__def,axiom,
( minimu4657282916794952894um_o_a
= ( ^ [S: set_o_a] :
? [X: $o > a] :
( ( member_o_a @ X @ S )
& ! [Y2: $o > a] :
( ( member_o_a @ Y2 @ S )
=> ( ord_less_eq_o_a @ X @ Y2 ) ) ) ) ) ).
% has_Minimum_def
thf(fact_37_has__Minimum__def,axiom,
( minimu7721965553910500344et_o_o
= ( ^ [S: set_set_o_o] :
? [X: set_o > $o] :
( ( member_set_o_o @ X @ S )
& ! [Y2: set_o > $o] :
( ( member_set_o_o @ Y2 @ S )
=> ( ord_less_eq_set_o_o @ X @ Y2 ) ) ) ) ) ).
% has_Minimum_def
thf(fact_38_has__Minimum__def,axiom,
( minimu5936394273662617842um_a_o
= ( ^ [S: set_a_o] :
? [X: a > $o] :
( ( member_a_o @ X @ S )
& ! [Y2: a > $o] :
( ( member_a_o @ Y2 @ S )
=> ( ord_less_eq_a_o @ X @ Y2 ) ) ) ) ) ).
% has_Minimum_def
thf(fact_39_has__Minimum__def,axiom,
( minimu7473987258551571531imum_a
= ( ^ [S: set_a] :
? [X: a] :
( ( member_a @ X @ S )
& ! [Y2: a] :
( ( member_a @ Y2 @ S )
=> ( ord_less_eq_a @ X @ Y2 ) ) ) ) ) ).
% has_Minimum_def
thf(fact_40_theI,axiom,
! [P: $o > $o,A: $o] :
( ( P @ A )
=> ( ! [X2: $o] :
( ( P @ X2 )
=> ( X2 = A ) )
=> ( P @ ( the_o @ P ) ) ) ) ).
% theI
thf(fact_41_theI,axiom,
! [P: a > $o,A: a] :
( ( P @ A )
=> ( ! [X2: a] :
( ( P @ X2 )
=> ( X2 = A ) )
=> ( P @ ( the_a @ P ) ) ) ) ).
% theI
thf(fact_42_theI_H,axiom,
! [P: $o > $o] :
( ? [X4: $o] :
( ( P @ X4 )
& ! [Y4: $o] :
( ( P @ Y4 )
=> ( Y4 = X4 ) ) )
=> ( P @ ( the_o @ P ) ) ) ).
% theI'
thf(fact_43_theI_H,axiom,
! [P: a > $o] :
( ? [X4: a] :
( ( P @ X4 )
& ! [Y4: a] :
( ( P @ Y4 )
=> ( Y4 = X4 ) ) )
=> ( P @ ( the_a @ P ) ) ) ).
% theI'
thf(fact_44_theI2,axiom,
! [P: $o > $o,A: $o,Q: $o > $o] :
( ( P @ A )
=> ( ! [X2: $o] :
( ( P @ X2 )
=> ( X2 = A ) )
=> ( ! [X2: $o] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( Q @ ( the_o @ P ) ) ) ) ) ).
% theI2
thf(fact_45_theI2,axiom,
! [P: a > $o,A: a,Q: a > $o] :
( ( P @ A )
=> ( ! [X2: a] :
( ( P @ X2 )
=> ( X2 = A ) )
=> ( ! [X2: a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( Q @ ( the_a @ P ) ) ) ) ) ).
% theI2
thf(fact_46_If__def,axiom,
( if_o
= ( ^ [P2: $o,X: $o,Y2: $o] :
( the_o
@ ^ [Z2: $o] :
( ( P2
=> ( Z2 = X ) )
& ( ~ P2
=> ( Z2 = Y2 ) ) ) ) ) ) ).
% If_def
thf(fact_47_If__def,axiom,
( if_a
= ( ^ [P2: $o,X: a,Y2: a] :
( the_a
@ ^ [Z2: a] :
( ( P2
=> ( Z2 = X ) )
& ( ~ P2
=> ( Z2 = Y2 ) ) ) ) ) ) ).
% If_def
thf(fact_48_order__antisym__conv,axiom,
! [Y: set_a,X3: set_a] :
( ( ord_less_eq_set_a @ Y @ X3 )
=> ( ( ord_less_eq_set_a @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_49_order__antisym__conv,axiom,
! [Y: $o > $o > $o,X3: $o > $o > $o] :
( ( ord_less_eq_o_o_o @ Y @ X3 )
=> ( ( ord_less_eq_o_o_o @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_50_order__antisym__conv,axiom,
! [Y: $o > a,X3: $o > a] :
( ( ord_less_eq_o_a @ Y @ X3 )
=> ( ( ord_less_eq_o_a @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_51_order__antisym__conv,axiom,
! [Y: set_o > $o,X3: set_o > $o] :
( ( ord_less_eq_set_o_o @ Y @ X3 )
=> ( ( ord_less_eq_set_o_o @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_52_order__antisym__conv,axiom,
! [Y: a > $o,X3: a > $o] :
( ( ord_less_eq_a_o @ Y @ X3 )
=> ( ( ord_less_eq_a_o @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_53_order__antisym__conv,axiom,
! [Y: a,X3: a] :
( ( ord_less_eq_a @ Y @ X3 )
=> ( ( ord_less_eq_a @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_54_linorder__le__cases,axiom,
! [X3: a,Y: a] :
( ~ ( ord_less_eq_a @ X3 @ Y )
=> ( ord_less_eq_a @ Y @ X3 ) ) ).
% linorder_le_cases
thf(fact_55_ord__le__eq__subst,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_56_ord__le__eq__subst,axiom,
! [A: a,B: a,F: a > set_a,C: set_a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_57_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > a,C: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_58_ord__le__eq__subst,axiom,
! [A: a,B: a,F: a > $o > a,C: $o > a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_less_eq_o_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_o_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_59_ord__le__eq__subst,axiom,
! [A: a,B: a,F: a > a > $o,C: a > $o] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_less_eq_a_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a_o @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_60_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_61_ord__le__eq__subst,axiom,
! [A: $o > a,B: $o > a,F: ( $o > a ) > a,C: a] :
( ( ord_less_eq_o_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: $o > a,Y4: $o > a] :
( ( ord_less_eq_o_a @ X2 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_62_ord__le__eq__subst,axiom,
! [A: a > $o,B: a > $o,F: ( a > $o ) > a,C: a] :
( ( ord_less_eq_a_o @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: a > $o,Y4: a > $o] :
( ( ord_less_eq_a_o @ X2 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_63_ord__le__eq__subst,axiom,
! [A: a,B: a,F: a > set_o > $o,C: set_o > $o] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_less_eq_set_o_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_o_o @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_64_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > $o > a,C: $o > a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_o_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_o_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_65_ord__eq__le__subst,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_66_ord__eq__le__subst,axiom,
! [A: set_a,F: a > set_a,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_67_ord__eq__le__subst,axiom,
! [A: a,F: set_a > a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_68_ord__eq__le__subst,axiom,
! [A: $o > a,F: a > $o > a,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_less_eq_o_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_o_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_69_ord__eq__le__subst,axiom,
! [A: a > $o,F: a > a > $o,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_less_eq_a_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a_o @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_70_ord__eq__le__subst,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_71_ord__eq__le__subst,axiom,
! [A: a,F: ( $o > a ) > a,B: $o > a,C: $o > a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_o_a @ B @ C )
=> ( ! [X2: $o > a,Y4: $o > a] :
( ( ord_less_eq_o_a @ X2 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_72_ord__eq__le__subst,axiom,
! [A: a,F: ( a > $o ) > a,B: a > $o,C: a > $o] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a_o @ B @ C )
=> ( ! [X2: a > $o,Y4: a > $o] :
( ( ord_less_eq_a_o @ X2 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_73_ord__eq__le__subst,axiom,
! [A: set_o > $o,F: a > set_o > $o,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_less_eq_set_o_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_o_o @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_74_ord__eq__le__subst,axiom,
! [A: $o > a,F: set_a > $o > a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_o_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_o_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_75_linorder__linear,axiom,
! [X3: a,Y: a] :
( ( ord_less_eq_a @ X3 @ Y )
| ( ord_less_eq_a @ Y @ X3 ) ) ).
% linorder_linear
thf(fact_76_order__eq__refl,axiom,
! [X3: set_a,Y: set_a] :
( ( X3 = Y )
=> ( ord_less_eq_set_a @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_77_order__eq__refl,axiom,
! [X3: $o > $o > $o,Y: $o > $o > $o] :
( ( X3 = Y )
=> ( ord_less_eq_o_o_o @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_78_order__eq__refl,axiom,
! [X3: $o > a,Y: $o > a] :
( ( X3 = Y )
=> ( ord_less_eq_o_a @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_79_order__eq__refl,axiom,
! [X3: set_o > $o,Y: set_o > $o] :
( ( X3 = Y )
=> ( ord_less_eq_set_o_o @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_80_order__eq__refl,axiom,
! [X3: a > $o,Y: a > $o] :
( ( X3 = Y )
=> ( ord_less_eq_a_o @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_81_order__eq__refl,axiom,
! [X3: a,Y: a] :
( ( X3 = Y )
=> ( ord_less_eq_a @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_82_order__subst2,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_83_order__subst2,axiom,
! [A: a,B: a,F: a > set_a,C: set_a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_84_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > a,C: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_85_order__subst2,axiom,
! [A: a,B: a,F: a > $o > a,C: $o > a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_o_a @ ( F @ B ) @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_less_eq_o_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_o_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_86_order__subst2,axiom,
! [A: a,B: a,F: a > a > $o,C: a > $o] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a_o @ ( F @ B ) @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_less_eq_a_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a_o @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_87_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_88_order__subst2,axiom,
! [A: $o > a,B: $o > a,F: ( $o > a ) > a,C: a] :
( ( ord_less_eq_o_a @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X2: $o > a,Y4: $o > a] :
( ( ord_less_eq_o_a @ X2 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_89_order__subst2,axiom,
! [A: a > $o,B: a > $o,F: ( a > $o ) > a,C: a] :
( ( ord_less_eq_a_o @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X2: a > $o,Y4: a > $o] :
( ( ord_less_eq_a_o @ X2 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_90_order__subst2,axiom,
! [A: a,B: a,F: a > set_o > $o,C: set_o > $o] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_set_o_o @ ( F @ B ) @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_less_eq_set_o_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_o_o @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_91_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > $o > a,C: $o > a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_o_a @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_o_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_o_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_92_order__subst1,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_93_order__subst1,axiom,
! [A: a,F: set_a > a,B: set_a,C: set_a] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_94_order__subst1,axiom,
! [A: set_a,F: a > set_a,B: a,C: a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_95_order__subst1,axiom,
! [A: a,F: ( $o > a ) > a,B: $o > a,C: $o > a] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_o_a @ B @ C )
=> ( ! [X2: $o > a,Y4: $o > a] :
( ( ord_less_eq_o_a @ X2 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_96_order__subst1,axiom,
! [A: a,F: ( a > $o ) > a,B: a > $o,C: a > $o] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a_o @ B @ C )
=> ( ! [X2: a > $o,Y4: a > $o] :
( ( ord_less_eq_a_o @ X2 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_97_order__subst1,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_98_order__subst1,axiom,
! [A: $o > a,F: a > $o > a,B: a,C: a] :
( ( ord_less_eq_o_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_less_eq_o_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_o_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_99_order__subst1,axiom,
! [A: a > $o,F: a > a > $o,B: a,C: a] :
( ( ord_less_eq_a_o @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_less_eq_a_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a_o @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_100_order__subst1,axiom,
! [A: a,F: ( set_o > $o ) > a,B: set_o > $o,C: set_o > $o] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_o_o @ B @ C )
=> ( ! [X2: set_o > $o,Y4: set_o > $o] :
( ( ord_less_eq_set_o_o @ X2 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_101_order__subst1,axiom,
! [A: set_a,F: ( $o > a ) > set_a,B: $o > a,C: $o > a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_o_a @ B @ C )
=> ( ! [X2: $o > a,Y4: $o > a] :
( ( ord_less_eq_o_a @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_102_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
= ( ^ [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
& ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_103_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: $o > $o > $o,Z: $o > $o > $o] : ( Y3 = Z ) )
= ( ^ [A2: $o > $o > $o,B2: $o > $o > $o] :
( ( ord_less_eq_o_o_o @ A2 @ B2 )
& ( ord_less_eq_o_o_o @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_104_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: $o > a,Z: $o > a] : ( Y3 = Z ) )
= ( ^ [A2: $o > a,B2: $o > a] :
( ( ord_less_eq_o_a @ A2 @ B2 )
& ( ord_less_eq_o_a @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_105_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_o > $o,Z: set_o > $o] : ( Y3 = Z ) )
= ( ^ [A2: set_o > $o,B2: set_o > $o] :
( ( ord_less_eq_set_o_o @ A2 @ B2 )
& ( ord_less_eq_set_o_o @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_106_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: a > $o,Z: a > $o] : ( Y3 = Z ) )
= ( ^ [A2: a > $o,B2: a > $o] :
( ( ord_less_eq_a_o @ A2 @ B2 )
& ( ord_less_eq_a_o @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_107_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: a,Z: a] : ( Y3 = Z ) )
= ( ^ [A2: a,B2: a] :
( ( ord_less_eq_a @ A2 @ B2 )
& ( ord_less_eq_a @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_108_le__fun__def,axiom,
( ord_less_eq_o_o_o
= ( ^ [F2: $o > $o > $o,G: $o > $o > $o] :
! [X: $o] : ( ord_less_eq_o_o @ ( F2 @ X ) @ ( G @ X ) ) ) ) ).
% le_fun_def
thf(fact_109_le__fun__def,axiom,
( ord_less_eq_o_a
= ( ^ [F2: $o > a,G: $o > a] :
! [X: $o] : ( ord_less_eq_a @ ( F2 @ X ) @ ( G @ X ) ) ) ) ).
% le_fun_def
thf(fact_110_le__fun__def,axiom,
( ord_less_eq_set_o_o
= ( ^ [F2: set_o > $o,G: set_o > $o] :
! [X: set_o] : ( ord_less_eq_o @ ( F2 @ X ) @ ( G @ X ) ) ) ) ).
% le_fun_def
thf(fact_111_le__fun__def,axiom,
( ord_less_eq_a_o
= ( ^ [F2: a > $o,G: a > $o] :
! [X: a] : ( ord_less_eq_o @ ( F2 @ X ) @ ( G @ X ) ) ) ) ).
% le_fun_def
thf(fact_112_le__funI,axiom,
! [F: $o > $o > $o,G2: $o > $o > $o] :
( ! [X2: $o] : ( ord_less_eq_o_o @ ( F @ X2 ) @ ( G2 @ X2 ) )
=> ( ord_less_eq_o_o_o @ F @ G2 ) ) ).
% le_funI
thf(fact_113_le__funI,axiom,
! [F: set_o > $o,G2: set_o > $o] :
( ! [X2: set_o] : ( ord_less_eq_o @ ( F @ X2 ) @ ( G2 @ X2 ) )
=> ( ord_less_eq_set_o_o @ F @ G2 ) ) ).
% le_funI
thf(fact_114_le__funI,axiom,
! [F: a > $o,G2: a > $o] :
( ! [X2: a] : ( ord_less_eq_o @ ( F @ X2 ) @ ( G2 @ X2 ) )
=> ( ord_less_eq_a_o @ F @ G2 ) ) ).
% le_funI
thf(fact_115_le__funI,axiom,
! [F: $o > a,G2: $o > a] :
( ! [X2: $o] : ( ord_less_eq_a @ ( F @ X2 ) @ ( G2 @ X2 ) )
=> ( ord_less_eq_o_a @ F @ G2 ) ) ).
% le_funI
thf(fact_116_le__funE,axiom,
! [F: $o > $o > $o,G2: $o > $o > $o,X3: $o] :
( ( ord_less_eq_o_o_o @ F @ G2 )
=> ( ord_less_eq_o_o @ ( F @ X3 ) @ ( G2 @ X3 ) ) ) ).
% le_funE
thf(fact_117_le__funE,axiom,
! [F: $o > a,G2: $o > a,X3: $o] :
( ( ord_less_eq_o_a @ F @ G2 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( G2 @ X3 ) ) ) ).
% le_funE
thf(fact_118_le__funE,axiom,
! [F: set_o > $o,G2: set_o > $o,X3: set_o] :
( ( ord_less_eq_set_o_o @ F @ G2 )
=> ( ord_less_eq_o @ ( F @ X3 ) @ ( G2 @ X3 ) ) ) ).
% le_funE
thf(fact_119_le__funE,axiom,
! [F: a > $o,G2: a > $o,X3: a] :
( ( ord_less_eq_a_o @ F @ G2 )
=> ( ord_less_eq_o @ ( F @ X3 ) @ ( G2 @ X3 ) ) ) ).
% le_funE
thf(fact_120_le__funD,axiom,
! [F: $o > $o > $o,G2: $o > $o > $o,X3: $o] :
( ( ord_less_eq_o_o_o @ F @ G2 )
=> ( ord_less_eq_o_o @ ( F @ X3 ) @ ( G2 @ X3 ) ) ) ).
% le_funD
thf(fact_121_le__funD,axiom,
! [F: $o > a,G2: $o > a,X3: $o] :
( ( ord_less_eq_o_a @ F @ G2 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( G2 @ X3 ) ) ) ).
% le_funD
thf(fact_122_le__funD,axiom,
! [F: set_o > $o,G2: set_o > $o,X3: set_o] :
( ( ord_less_eq_set_o_o @ F @ G2 )
=> ( ord_less_eq_o @ ( F @ X3 ) @ ( G2 @ X3 ) ) ) ).
% le_funD
thf(fact_123_le__funD,axiom,
! [F: a > $o,G2: a > $o,X3: a] :
( ( ord_less_eq_a_o @ F @ G2 )
=> ( ord_less_eq_o @ ( F @ X3 ) @ ( G2 @ X3 ) ) ) ).
% le_funD
thf(fact_124_antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_125_antisym,axiom,
! [A: $o > $o > $o,B: $o > $o > $o] :
( ( ord_less_eq_o_o_o @ A @ B )
=> ( ( ord_less_eq_o_o_o @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_126_antisym,axiom,
! [A: $o > a,B: $o > a] :
( ( ord_less_eq_o_a @ A @ B )
=> ( ( ord_less_eq_o_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_127_antisym,axiom,
! [A: set_o > $o,B: set_o > $o] :
( ( ord_less_eq_set_o_o @ A @ B )
=> ( ( ord_less_eq_set_o_o @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_128_antisym,axiom,
! [A: a > $o,B: a > $o] :
( ( ord_less_eq_a_o @ A @ B )
=> ( ( ord_less_eq_a_o @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_129_antisym,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_130_dual__order_Otrans,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_131_dual__order_Otrans,axiom,
! [B: $o > $o > $o,A: $o > $o > $o,C: $o > $o > $o] :
( ( ord_less_eq_o_o_o @ B @ A )
=> ( ( ord_less_eq_o_o_o @ C @ B )
=> ( ord_less_eq_o_o_o @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_132_dual__order_Otrans,axiom,
! [B: $o > a,A: $o > a,C: $o > a] :
( ( ord_less_eq_o_a @ B @ A )
=> ( ( ord_less_eq_o_a @ C @ B )
=> ( ord_less_eq_o_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_133_dual__order_Otrans,axiom,
! [B: set_o > $o,A: set_o > $o,C: set_o > $o] :
( ( ord_less_eq_set_o_o @ B @ A )
=> ( ( ord_less_eq_set_o_o @ C @ B )
=> ( ord_less_eq_set_o_o @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_134_dual__order_Otrans,axiom,
! [B: a > $o,A: a > $o,C: a > $o] :
( ( ord_less_eq_a_o @ B @ A )
=> ( ( ord_less_eq_a_o @ C @ B )
=> ( ord_less_eq_a_o @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_135_dual__order_Otrans,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( ord_less_eq_a @ C @ B )
=> ( ord_less_eq_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_136_dual__order_Oantisym,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_137_dual__order_Oantisym,axiom,
! [B: $o > $o > $o,A: $o > $o > $o] :
( ( ord_less_eq_o_o_o @ B @ A )
=> ( ( ord_less_eq_o_o_o @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_138_dual__order_Oantisym,axiom,
! [B: $o > a,A: $o > a] :
( ( ord_less_eq_o_a @ B @ A )
=> ( ( ord_less_eq_o_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_139_dual__order_Oantisym,axiom,
! [B: set_o > $o,A: set_o > $o] :
( ( ord_less_eq_set_o_o @ B @ A )
=> ( ( ord_less_eq_set_o_o @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_140_dual__order_Oantisym,axiom,
! [B: a > $o,A: a > $o] :
( ( ord_less_eq_a_o @ B @ A )
=> ( ( ord_less_eq_a_o @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_141_dual__order_Oantisym,axiom,
! [B: a,A: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( ord_less_eq_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_142_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
= ( ^ [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
& ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_143_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: $o > $o > $o,Z: $o > $o > $o] : ( Y3 = Z ) )
= ( ^ [A2: $o > $o > $o,B2: $o > $o > $o] :
( ( ord_less_eq_o_o_o @ B2 @ A2 )
& ( ord_less_eq_o_o_o @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_144_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: $o > a,Z: $o > a] : ( Y3 = Z ) )
= ( ^ [A2: $o > a,B2: $o > a] :
( ( ord_less_eq_o_a @ B2 @ A2 )
& ( ord_less_eq_o_a @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_145_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_o > $o,Z: set_o > $o] : ( Y3 = Z ) )
= ( ^ [A2: set_o > $o,B2: set_o > $o] :
( ( ord_less_eq_set_o_o @ B2 @ A2 )
& ( ord_less_eq_set_o_o @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_146_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: a > $o,Z: a > $o] : ( Y3 = Z ) )
= ( ^ [A2: a > $o,B2: a > $o] :
( ( ord_less_eq_a_o @ B2 @ A2 )
& ( ord_less_eq_a_o @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_147_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: a,Z: a] : ( Y3 = Z ) )
= ( ^ [A2: a,B2: a] :
( ( ord_less_eq_a @ B2 @ A2 )
& ( ord_less_eq_a @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_148_linorder__wlog,axiom,
! [P: a > a > $o,A: a,B: a] :
( ! [A3: a,B3: a] :
( ( ord_less_eq_a @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: a,B3: a] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_149_order__trans,axiom,
! [X3: set_a,Y: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z3 )
=> ( ord_less_eq_set_a @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_150_order__trans,axiom,
! [X3: $o > $o > $o,Y: $o > $o > $o,Z3: $o > $o > $o] :
( ( ord_less_eq_o_o_o @ X3 @ Y )
=> ( ( ord_less_eq_o_o_o @ Y @ Z3 )
=> ( ord_less_eq_o_o_o @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_151_order__trans,axiom,
! [X3: $o > a,Y: $o > a,Z3: $o > a] :
( ( ord_less_eq_o_a @ X3 @ Y )
=> ( ( ord_less_eq_o_a @ Y @ Z3 )
=> ( ord_less_eq_o_a @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_152_order__trans,axiom,
! [X3: set_o > $o,Y: set_o > $o,Z3: set_o > $o] :
( ( ord_less_eq_set_o_o @ X3 @ Y )
=> ( ( ord_less_eq_set_o_o @ Y @ Z3 )
=> ( ord_less_eq_set_o_o @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_153_order__trans,axiom,
! [X3: a > $o,Y: a > $o,Z3: a > $o] :
( ( ord_less_eq_a_o @ X3 @ Y )
=> ( ( ord_less_eq_a_o @ Y @ Z3 )
=> ( ord_less_eq_a_o @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_154_order__trans,axiom,
! [X3: a,Y: a,Z3: a] :
( ( ord_less_eq_a @ X3 @ Y )
=> ( ( ord_less_eq_a @ Y @ Z3 )
=> ( ord_less_eq_a @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_155_order_Otrans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_156_order_Otrans,axiom,
! [A: $o > $o > $o,B: $o > $o > $o,C: $o > $o > $o] :
( ( ord_less_eq_o_o_o @ A @ B )
=> ( ( ord_less_eq_o_o_o @ B @ C )
=> ( ord_less_eq_o_o_o @ A @ C ) ) ) ).
% order.trans
thf(fact_157_order_Otrans,axiom,
! [A: $o > a,B: $o > a,C: $o > a] :
( ( ord_less_eq_o_a @ A @ B )
=> ( ( ord_less_eq_o_a @ B @ C )
=> ( ord_less_eq_o_a @ A @ C ) ) ) ).
% order.trans
thf(fact_158_order_Otrans,axiom,
! [A: set_o > $o,B: set_o > $o,C: set_o > $o] :
( ( ord_less_eq_set_o_o @ A @ B )
=> ( ( ord_less_eq_set_o_o @ B @ C )
=> ( ord_less_eq_set_o_o @ A @ C ) ) ) ).
% order.trans
thf(fact_159_order_Otrans,axiom,
! [A: a > $o,B: a > $o,C: a > $o] :
( ( ord_less_eq_a_o @ A @ B )
=> ( ( ord_less_eq_a_o @ B @ C )
=> ( ord_less_eq_a_o @ A @ C ) ) ) ).
% order.trans
thf(fact_160_order_Otrans,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_eq_a @ A @ C ) ) ) ).
% order.trans
thf(fact_161_order__antisym,axiom,
! [X3: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y )
=> ( ( ord_less_eq_set_a @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_162_order__antisym,axiom,
! [X3: $o > $o > $o,Y: $o > $o > $o] :
( ( ord_less_eq_o_o_o @ X3 @ Y )
=> ( ( ord_less_eq_o_o_o @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_163_order__antisym,axiom,
! [X3: $o > a,Y: $o > a] :
( ( ord_less_eq_o_a @ X3 @ Y )
=> ( ( ord_less_eq_o_a @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_164_order__antisym,axiom,
! [X3: set_o > $o,Y: set_o > $o] :
( ( ord_less_eq_set_o_o @ X3 @ Y )
=> ( ( ord_less_eq_set_o_o @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_165_order__antisym,axiom,
! [X3: a > $o,Y: a > $o] :
( ( ord_less_eq_a_o @ X3 @ Y )
=> ( ( ord_less_eq_a_o @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_166_order__antisym,axiom,
! [X3: a,Y: a] :
( ( ord_less_eq_a @ X3 @ Y )
=> ( ( ord_less_eq_a @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_167_ord__le__eq__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_168_ord__le__eq__trans,axiom,
! [A: $o > $o > $o,B: $o > $o > $o,C: $o > $o > $o] :
( ( ord_less_eq_o_o_o @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_o_o_o @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_169_ord__le__eq__trans,axiom,
! [A: $o > a,B: $o > a,C: $o > a] :
( ( ord_less_eq_o_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_o_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_170_ord__le__eq__trans,axiom,
! [A: set_o > $o,B: set_o > $o,C: set_o > $o] :
( ( ord_less_eq_set_o_o @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_o_o @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_171_ord__le__eq__trans,axiom,
! [A: a > $o,B: a > $o,C: a > $o] :
( ( ord_less_eq_a_o @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_a_o @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_172_ord__le__eq__trans,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_173_ord__eq__le__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( A = B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_174_ord__eq__le__trans,axiom,
! [A: $o > $o > $o,B: $o > $o > $o,C: $o > $o > $o] :
( ( A = B )
=> ( ( ord_less_eq_o_o_o @ B @ C )
=> ( ord_less_eq_o_o_o @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_175_ord__eq__le__trans,axiom,
! [A: $o > a,B: $o > a,C: $o > a] :
( ( A = B )
=> ( ( ord_less_eq_o_a @ B @ C )
=> ( ord_less_eq_o_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_176_ord__eq__le__trans,axiom,
! [A: set_o > $o,B: set_o > $o,C: set_o > $o] :
( ( A = B )
=> ( ( ord_less_eq_set_o_o @ B @ C )
=> ( ord_less_eq_set_o_o @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_177_ord__eq__le__trans,axiom,
! [A: a > $o,B: a > $o,C: a > $o] :
( ( A = B )
=> ( ( ord_less_eq_a_o @ B @ C )
=> ( ord_less_eq_a_o @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_178_ord__eq__le__trans,axiom,
! [A: a,B: a,C: a] :
( ( A = B )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_eq_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_179_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
= ( ^ [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
& ( ord_less_eq_set_a @ Y2 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_180_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: $o > $o > $o,Z: $o > $o > $o] : ( Y3 = Z ) )
= ( ^ [X: $o > $o > $o,Y2: $o > $o > $o] :
( ( ord_less_eq_o_o_o @ X @ Y2 )
& ( ord_less_eq_o_o_o @ Y2 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_181_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: $o > a,Z: $o > a] : ( Y3 = Z ) )
= ( ^ [X: $o > a,Y2: $o > a] :
( ( ord_less_eq_o_a @ X @ Y2 )
& ( ord_less_eq_o_a @ Y2 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_182_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_o > $o,Z: set_o > $o] : ( Y3 = Z ) )
= ( ^ [X: set_o > $o,Y2: set_o > $o] :
( ( ord_less_eq_set_o_o @ X @ Y2 )
& ( ord_less_eq_set_o_o @ Y2 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_183_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: a > $o,Z: a > $o] : ( Y3 = Z ) )
= ( ^ [X: a > $o,Y2: a > $o] :
( ( ord_less_eq_a_o @ X @ Y2 )
& ( ord_less_eq_a_o @ Y2 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_184_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: a,Z: a] : ( Y3 = Z ) )
= ( ^ [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
& ( ord_less_eq_a @ Y2 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_185_le__cases3,axiom,
! [X3: a,Y: a,Z3: a] :
( ( ( ord_less_eq_a @ X3 @ Y )
=> ~ ( ord_less_eq_a @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_a @ Y @ X3 )
=> ~ ( ord_less_eq_a @ X3 @ Z3 ) )
=> ( ( ( ord_less_eq_a @ X3 @ Z3 )
=> ~ ( ord_less_eq_a @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_a @ Z3 @ Y )
=> ~ ( ord_less_eq_a @ Y @ X3 ) )
=> ( ( ( ord_less_eq_a @ Y @ Z3 )
=> ~ ( ord_less_eq_a @ Z3 @ X3 ) )
=> ~ ( ( ord_less_eq_a @ Z3 @ X3 )
=> ~ ( ord_less_eq_a @ X3 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_186_nle__le,axiom,
! [A: a,B: a] :
( ( ~ ( ord_less_eq_a @ A @ B ) )
= ( ( ord_less_eq_a @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_187_the1__equality,axiom,
! [P: $o > $o,A: $o] :
( ? [X4: $o] :
( ( P @ X4 )
& ! [Y4: $o] :
( ( P @ Y4 )
=> ( Y4 = X4 ) ) )
=> ( ( P @ A )
=> ( ( the_o @ P )
= A ) ) ) ).
% the1_equality
thf(fact_188_the1__equality,axiom,
! [P: a > $o,A: a] :
( ? [X4: a] :
( ( P @ X4 )
& ! [Y4: a] :
( ( P @ Y4 )
=> ( Y4 = X4 ) ) )
=> ( ( P @ A )
=> ( ( the_a @ P )
= A ) ) ) ).
% the1_equality
thf(fact_189_the1I2,axiom,
! [P: $o > $o,Q: $o > $o] :
( ? [X4: $o] :
( ( P @ X4 )
& ! [Y4: $o] :
( ( P @ Y4 )
=> ( Y4 = X4 ) ) )
=> ( ! [X2: $o] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( Q @ ( the_o @ P ) ) ) ) ).
% the1I2
thf(fact_190_the1I2,axiom,
! [P: a > $o,Q: a > $o] :
( ? [X4: a] :
( ( P @ X4 )
& ! [Y4: a] :
( ( P @ Y4 )
=> ( Y4 = X4 ) ) )
=> ( ! [X2: a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( Q @ ( the_a @ P ) ) ) ) ).
% the1I2
thf(fact_191_theI__unique,axiom,
! [P: $o > $o,X3: $o] :
( ? [X4: $o] :
( ( P @ X4 )
& ! [Y4: $o] :
( ( P @ Y4 )
=> ( Y4 = X4 ) ) )
=> ( ( P @ X3 )
= ( X3
= ( the_o @ P ) ) ) ) ).
% theI_unique
thf(fact_192_theI__unique,axiom,
! [P: a > $o,X3: a] :
( ? [X4: a] :
( ( P @ X4 )
& ! [Y4: a] :
( ( P @ Y4 )
=> ( Y4 = X4 ) ) )
=> ( ( P @ X3 )
= ( X3
= ( the_a @ P ) ) ) ) ).
% theI_unique
thf(fact_193_Nitpick_OThe__psimp,axiom,
! [P: $o > $o,X3: $o] :
( ( P
= ( ^ [Y3: $o,Z: $o] : ( Y3 = Z )
@ X3 ) )
=> ( ( the_o @ P )
= X3 ) ) ).
% Nitpick.The_psimp
thf(fact_194_Nitpick_OThe__psimp,axiom,
! [P: a > $o,X3: a] :
( ( P
= ( ^ [Y3: a,Z: a] : ( Y3 = Z )
@ X3 ) )
=> ( ( the_a @ P )
= X3 ) ) ).
% Nitpick.The_psimp
thf(fact_195_mem__Collect__eq,axiom,
! [A: set_o,P: set_o > $o] :
( ( member_set_o @ A @ ( collect_set_o @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_196_mem__Collect__eq,axiom,
! [A: set_a,P: set_a > $o] :
( ( member_set_a @ A @ ( collect_set_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_197_mem__Collect__eq,axiom,
! [A: $o,P: $o > $o] :
( ( member_o @ A @ ( collect_o @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_198_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_199_Collect__mem__eq,axiom,
! [A4: set_set_o] :
( ( collect_set_o
@ ^ [X: set_o] : ( member_set_o @ X @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_200_Collect__mem__eq,axiom,
! [A4: set_set_a] :
( ( collect_set_a
@ ^ [X: set_a] : ( member_set_a @ X @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_201_Collect__mem__eq,axiom,
! [A4: set_o] :
( ( collect_o
@ ^ [X: $o] : ( member_o @ X @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_202_Collect__mem__eq,axiom,
! [A4: set_a] :
( ( collect_a
@ ^ [X: a] : ( member_a @ X @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_203_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_204_Collect__cong,axiom,
! [P: $o > $o,Q: $o > $o] :
( ! [X2: $o] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_o @ P )
= ( collect_o @ Q ) ) ) ).
% Collect_cong
thf(fact_205_Greatest__equality,axiom,
! [P: set_a > $o,X3: set_a] :
( ( P @ X3 )
=> ( ! [Y4: set_a] :
( ( P @ Y4 )
=> ( ord_less_eq_set_a @ Y4 @ X3 ) )
=> ( ( order_Greatest_set_a @ P )
= X3 ) ) ) ).
% Greatest_equality
thf(fact_206_Greatest__equality,axiom,
! [P: ( $o > $o > $o ) > $o,X3: $o > $o > $o] :
( ( P @ X3 )
=> ( ! [Y4: $o > $o > $o] :
( ( P @ Y4 )
=> ( ord_less_eq_o_o_o @ Y4 @ X3 ) )
=> ( ( order_Greatest_o_o_o @ P )
= X3 ) ) ) ).
% Greatest_equality
thf(fact_207_Greatest__equality,axiom,
! [P: ( $o > a ) > $o,X3: $o > a] :
( ( P @ X3 )
=> ( ! [Y4: $o > a] :
( ( P @ Y4 )
=> ( ord_less_eq_o_a @ Y4 @ X3 ) )
=> ( ( order_Greatest_o_a @ P )
= X3 ) ) ) ).
% Greatest_equality
thf(fact_208_Greatest__equality,axiom,
! [P: ( set_o > $o ) > $o,X3: set_o > $o] :
( ( P @ X3 )
=> ( ! [Y4: set_o > $o] :
( ( P @ Y4 )
=> ( ord_less_eq_set_o_o @ Y4 @ X3 ) )
=> ( ( order_5805555182909007980et_o_o @ P )
= X3 ) ) ) ).
% Greatest_equality
thf(fact_209_Greatest__equality,axiom,
! [P: ( a > $o ) > $o,X3: a > $o] :
( ( P @ X3 )
=> ( ! [Y4: a > $o] :
( ( P @ Y4 )
=> ( ord_less_eq_a_o @ Y4 @ X3 ) )
=> ( ( order_Greatest_a_o @ P )
= X3 ) ) ) ).
% Greatest_equality
thf(fact_210_Greatest__equality,axiom,
! [P: a > $o,X3: a] :
( ( P @ X3 )
=> ( ! [Y4: a] :
( ( P @ Y4 )
=> ( ord_less_eq_a @ Y4 @ X3 ) )
=> ( ( order_Greatest_a @ P )
= X3 ) ) ) ).
% Greatest_equality
thf(fact_211_GreatestI2__order,axiom,
! [P: set_a > $o,X3: set_a,Q: set_a > $o] :
( ( P @ X3 )
=> ( ! [Y4: set_a] :
( ( P @ Y4 )
=> ( ord_less_eq_set_a @ Y4 @ X3 ) )
=> ( ! [X2: set_a] :
( ( P @ X2 )
=> ( ! [Y5: set_a] :
( ( P @ Y5 )
=> ( ord_less_eq_set_a @ Y5 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_Greatest_set_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_212_GreatestI2__order,axiom,
! [P: ( $o > $o > $o ) > $o,X3: $o > $o > $o,Q: ( $o > $o > $o ) > $o] :
( ( P @ X3 )
=> ( ! [Y4: $o > $o > $o] :
( ( P @ Y4 )
=> ( ord_less_eq_o_o_o @ Y4 @ X3 ) )
=> ( ! [X2: $o > $o > $o] :
( ( P @ X2 )
=> ( ! [Y5: $o > $o > $o] :
( ( P @ Y5 )
=> ( ord_less_eq_o_o_o @ Y5 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_Greatest_o_o_o @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_213_GreatestI2__order,axiom,
! [P: ( $o > a ) > $o,X3: $o > a,Q: ( $o > a ) > $o] :
( ( P @ X3 )
=> ( ! [Y4: $o > a] :
( ( P @ Y4 )
=> ( ord_less_eq_o_a @ Y4 @ X3 ) )
=> ( ! [X2: $o > a] :
( ( P @ X2 )
=> ( ! [Y5: $o > a] :
( ( P @ Y5 )
=> ( ord_less_eq_o_a @ Y5 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_Greatest_o_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_214_GreatestI2__order,axiom,
! [P: ( set_o > $o ) > $o,X3: set_o > $o,Q: ( set_o > $o ) > $o] :
( ( P @ X3 )
=> ( ! [Y4: set_o > $o] :
( ( P @ Y4 )
=> ( ord_less_eq_set_o_o @ Y4 @ X3 ) )
=> ( ! [X2: set_o > $o] :
( ( P @ X2 )
=> ( ! [Y5: set_o > $o] :
( ( P @ Y5 )
=> ( ord_less_eq_set_o_o @ Y5 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_5805555182909007980et_o_o @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_215_GreatestI2__order,axiom,
! [P: ( a > $o ) > $o,X3: a > $o,Q: ( a > $o ) > $o] :
( ( P @ X3 )
=> ( ! [Y4: a > $o] :
( ( P @ Y4 )
=> ( ord_less_eq_a_o @ Y4 @ X3 ) )
=> ( ! [X2: a > $o] :
( ( P @ X2 )
=> ( ! [Y5: a > $o] :
( ( P @ Y5 )
=> ( ord_less_eq_a_o @ Y5 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_Greatest_a_o @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_216_GreatestI2__order,axiom,
! [P: a > $o,X3: a,Q: a > $o] :
( ( P @ X3 )
=> ( ! [Y4: a] :
( ( P @ Y4 )
=> ( ord_less_eq_a @ Y4 @ X3 ) )
=> ( ! [X2: a] :
( ( P @ X2 )
=> ( ! [Y5: a] :
( ( P @ Y5 )
=> ( ord_less_eq_a @ Y5 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_Greatest_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_217_Ball__Collect,axiom,
( ball_o
= ( ^ [A5: set_o,P2: $o > $o] : ( ord_less_eq_set_o @ A5 @ ( collect_o @ P2 ) ) ) ) ).
% Ball_Collect
thf(fact_218_Ball__Collect,axiom,
( ball_a
= ( ^ [A5: set_a,P2: a > $o] : ( ord_less_eq_set_a @ A5 @ ( collect_a @ P2 ) ) ) ) ).
% Ball_Collect
thf(fact_219_ball__reg,axiom,
! [R: set_o,P: $o > $o,Q: $o > $o] :
( ! [X2: $o] :
( ( member_o @ X2 @ R )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ R )
=> ( P @ X2 ) )
=> ! [X4: $o] :
( ( member_o @ X4 @ R )
=> ( Q @ X4 ) ) ) ) ).
% ball_reg
thf(fact_220_ball__reg,axiom,
! [R: set_set_o,P: set_o > $o,Q: set_o > $o] :
( ! [X2: set_o] :
( ( member_set_o @ X2 @ R )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ! [X2: set_o] :
( ( member_set_o @ X2 @ R )
=> ( P @ X2 ) )
=> ! [X4: set_o] :
( ( member_set_o @ X4 @ R )
=> ( Q @ X4 ) ) ) ) ).
% ball_reg
thf(fact_221_ball__reg,axiom,
! [R: set_set_a,P: set_a > $o,Q: set_a > $o] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ R )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ R )
=> ( P @ X2 ) )
=> ! [X4: set_a] :
( ( member_set_a @ X4 @ R )
=> ( Q @ X4 ) ) ) ) ).
% ball_reg
thf(fact_222_ball__reg,axiom,
! [R: set_a,P: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( member_a @ X2 @ R )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ R )
=> ( P @ X2 ) )
=> ! [X4: a] :
( ( member_a @ X4 @ R )
=> ( Q @ X4 ) ) ) ) ).
% ball_reg
thf(fact_223_Ball__def,axiom,
( ball_o
= ( ^ [A5: set_o,P2: $o > $o] :
! [X: $o] :
( ( member_o @ X @ A5 )
=> ( P2 @ X ) ) ) ) ).
% Ball_def
thf(fact_224_Ball__def,axiom,
( ball_set_o
= ( ^ [A5: set_set_o,P2: set_o > $o] :
! [X: set_o] :
( ( member_set_o @ X @ A5 )
=> ( P2 @ X ) ) ) ) ).
% Ball_def
thf(fact_225_Ball__def,axiom,
( ball_set_a
= ( ^ [A5: set_set_a,P2: set_a > $o] :
! [X: set_a] :
( ( member_set_a @ X @ A5 )
=> ( P2 @ X ) ) ) ) ).
% Ball_def
thf(fact_226_Ball__def,axiom,
( ball_a
= ( ^ [A5: set_a,P2: a > $o] :
! [X: a] :
( ( member_a @ X @ A5 )
=> ( P2 @ X ) ) ) ) ).
% Ball_def
thf(fact_227_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_set_a
= ( ^ [X5: $o > set_a,Y6: $o > set_a] :
( ( ord_less_eq_set_a @ ( X5 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_set_a @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_228_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_o_o_o
= ( ^ [X5: $o > $o > $o > $o,Y6: $o > $o > $o > $o] :
( ( ord_less_eq_o_o_o @ ( X5 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_o_o_o @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_229_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_o_a
= ( ^ [X5: $o > $o > a,Y6: $o > $o > a] :
( ( ord_less_eq_o_a @ ( X5 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_o_a @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_230_le__rel__bool__arg__iff,axiom,
( ord_le1909991985842454446et_o_o
= ( ^ [X5: $o > set_o > $o,Y6: $o > set_o > $o] :
( ( ord_less_eq_set_o_o @ ( X5 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_set_o_o @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_231_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_a_o
= ( ^ [X5: $o > a > $o,Y6: $o > a > $o] :
( ( ord_less_eq_a_o @ ( X5 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_a_o @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_232_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_o_o
= ( ^ [X5: $o > $o > $o,Y6: $o > $o > $o] :
( ( ord_less_eq_o_o @ ( X5 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_o_o @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_233_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_a
= ( ^ [X5: $o > a,Y6: $o > a] :
( ( ord_less_eq_a @ ( X5 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_a @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_234_verit__la__disequality,axiom,
! [A: a,B: a] :
( ( A = B )
| ~ ( ord_less_eq_a @ A @ B )
| ~ ( ord_less_eq_a @ B @ A ) ) ).
% verit_la_disequality
thf(fact_235_verit__comp__simplify1_I2_J,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_236_verit__comp__simplify1_I2_J,axiom,
! [A: $o > $o > $o] : ( ord_less_eq_o_o_o @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_237_verit__comp__simplify1_I2_J,axiom,
! [A: $o > a] : ( ord_less_eq_o_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_238_verit__comp__simplify1_I2_J,axiom,
! [A: set_o > $o] : ( ord_less_eq_set_o_o @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_239_verit__comp__simplify1_I2_J,axiom,
! [A: a > $o] : ( ord_less_eq_a_o @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_240_verit__comp__simplify1_I2_J,axiom,
! [A: a] : ( ord_less_eq_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_241_the1__equality_H,axiom,
! [P: $o > $o,A: $o] :
( ( uniq_o @ P )
=> ( ( P @ A )
=> ( ( the_o @ P )
= A ) ) ) ).
% the1_equality'
thf(fact_242_the1__equality_H,axiom,
! [P: a > $o,A: a] :
( ( uniq_a @ P )
=> ( ( P @ A )
=> ( ( the_a @ P )
= A ) ) ) ).
% the1_equality'
thf(fact_243_dual__order_Opartial__preordering__axioms,axiom,
( partia6602192050731689876_set_a
@ ^ [X: set_a,Y2: set_a] : ( ord_less_eq_set_a @ Y2 @ X ) ) ).
% dual_order.partial_preordering_axioms
thf(fact_244_dual__order_Opartial__preordering__axioms,axiom,
( partia1881799573076113956_o_o_o
@ ^ [X: $o > $o > $o,Y2: $o > $o > $o] : ( ord_less_eq_o_o_o @ Y2 @ X ) ) ).
% dual_order.partial_preordering_axioms
thf(fact_245_dual__order_Opartial__preordering__axioms,axiom,
( partia5423788306336055317ng_o_a
@ ^ [X: $o > a,Y2: $o > a] : ( ord_less_eq_o_a @ Y2 @ X ) ) ).
% dual_order.partial_preordering_axioms
thf(fact_246_dual__order_Opartial__preordering__axioms,axiom,
( partia4811167007327223503et_o_o
@ ^ [X: set_o > $o,Y2: set_o > $o] : ( ord_less_eq_set_o_o @ Y2 @ X ) ) ).
% dual_order.partial_preordering_axioms
thf(fact_247_dual__order_Opartial__preordering__axioms,axiom,
( partia6702899663203720265ng_a_o
@ ^ [X: a > $o,Y2: a > $o] : ( ord_less_eq_a_o @ Y2 @ X ) ) ).
% dual_order.partial_preordering_axioms
thf(fact_248_dual__order_Opartial__preordering__axioms,axiom,
( partia125584492769400372ring_a
@ ^ [X: a,Y2: a] : ( ord_less_eq_a @ Y2 @ X ) ) ).
% dual_order.partial_preordering_axioms
thf(fact_249_subset__antisym,axiom,
! [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ A4 )
=> ( A4 = B4 ) ) ) ).
% subset_antisym
thf(fact_250_subsetI,axiom,
! [A4: set_o,B4: set_o] :
( ! [X2: $o] :
( ( member_o @ X2 @ A4 )
=> ( member_o @ X2 @ B4 ) )
=> ( ord_less_eq_set_o @ A4 @ B4 ) ) ).
% subsetI
thf(fact_251_subsetI,axiom,
! [A4: set_set_o,B4: set_set_o] :
( ! [X2: set_o] :
( ( member_set_o @ X2 @ A4 )
=> ( member_set_o @ X2 @ B4 ) )
=> ( ord_le4374716579403074808_set_o @ A4 @ B4 ) ) ).
% subsetI
thf(fact_252_subsetI,axiom,
! [A4: set_set_a,B4: set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A4 )
=> ( member_set_a @ X2 @ B4 ) )
=> ( ord_le3724670747650509150_set_a @ A4 @ B4 ) ) ).
% subsetI
thf(fact_253_subsetI,axiom,
! [A4: set_a,B4: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A4 )
=> ( member_a @ X2 @ B4 ) )
=> ( ord_less_eq_set_a @ A4 @ B4 ) ) ).
% subsetI
thf(fact_254_Uniq__def,axiom,
( uniq_a
= ( ^ [P2: a > $o] :
! [X: a,Y2: a] :
( ( P2 @ X )
=> ( ( P2 @ Y2 )
=> ( Y2 = X ) ) ) ) ) ).
% Uniq_def
thf(fact_255_Uniq__def,axiom,
( uniq_o
= ( ^ [P2: $o > $o] :
! [X: $o,Y2: $o] :
( ( P2 @ X )
=> ( ( P2 @ Y2 )
=> ( Y2 = X ) ) ) ) ) ).
% Uniq_def
thf(fact_256_Uniq__I,axiom,
! [P: a > $o] :
( ! [X2: a,Y4: a] :
( ( P @ X2 )
=> ( ( P @ Y4 )
=> ( Y4 = X2 ) ) )
=> ( uniq_a @ P ) ) ).
% Uniq_I
thf(fact_257_Uniq__I,axiom,
! [P: $o > $o] :
( ! [X2: $o,Y4: $o] :
( ( P @ X2 )
=> ( ( P @ Y4 )
=> ( Y4 = X2 ) ) )
=> ( uniq_o @ P ) ) ).
% Uniq_I
thf(fact_258_Uniq__D,axiom,
! [P: a > $o,A: a,B: a] :
( ( uniq_a @ P )
=> ( ( P @ A )
=> ( ( P @ B )
=> ( A = B ) ) ) ) ).
% Uniq_D
thf(fact_259_Uniq__D,axiom,
! [P: $o > $o,A: $o,B: $o] :
( ( uniq_o @ P )
=> ( ( P @ A )
=> ( ( P @ B )
=> ( A = B ) ) ) ) ).
% Uniq_D
thf(fact_260_Collect__mono__iff,axiom,
! [P: $o > $o,Q: $o > $o] :
( ( ord_less_eq_set_o @ ( collect_o @ P ) @ ( collect_o @ Q ) )
= ( ! [X: $o] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_261_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X: a] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_262_less__eq__set__def,axiom,
( ord_less_eq_set_o
= ( ^ [A5: set_o,B5: set_o] :
( ord_less_eq_o_o
@ ^ [X: $o] : ( member_o @ X @ A5 )
@ ^ [X: $o] : ( member_o @ X @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_263_less__eq__set__def,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
( ord_less_eq_set_a_o
@ ^ [X: set_a] : ( member_set_a @ X @ A5 )
@ ^ [X: set_a] : ( member_set_a @ X @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_264_less__eq__set__def,axiom,
( ord_le4374716579403074808_set_o
= ( ^ [A5: set_set_o,B5: set_set_o] :
( ord_less_eq_set_o_o
@ ^ [X: set_o] : ( member_set_o @ X @ A5 )
@ ^ [X: set_o] : ( member_set_o @ X @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_265_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ord_less_eq_a_o
@ ^ [X: a] : ( member_a @ X @ A5 )
@ ^ [X: a] : ( member_a @ X @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_266_Collect__subset,axiom,
! [A4: set_set_o,P: set_o > $o] :
( ord_le4374716579403074808_set_o
@ ( collect_set_o
@ ^ [X: set_o] :
( ( member_set_o @ X @ A4 )
& ( P @ X ) ) )
@ A4 ) ).
% Collect_subset
thf(fact_267_Collect__subset,axiom,
! [A4: set_set_a,P: set_a > $o] :
( ord_le3724670747650509150_set_a
@ ( collect_set_a
@ ^ [X: set_a] :
( ( member_set_a @ X @ A4 )
& ( P @ X ) ) )
@ A4 ) ).
% Collect_subset
thf(fact_268_Collect__subset,axiom,
! [A4: set_o,P: $o > $o] :
( ord_less_eq_set_o
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A4 )
& ( P @ X ) ) )
@ A4 ) ).
% Collect_subset
thf(fact_269_Collect__subset,axiom,
! [A4: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A4 )
& ( P @ X ) ) )
@ A4 ) ).
% Collect_subset
thf(fact_270_set__eq__subset,axiom,
( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_271_subset__trans,axiom,
! [A4: set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ord_less_eq_set_a @ A4 @ C2 ) ) ) ).
% subset_trans
thf(fact_272_Collect__mono,axiom,
! [P: $o > $o,Q: $o > $o] :
( ! [X2: $o] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_o @ ( collect_o @ P ) @ ( collect_o @ Q ) ) ) ).
% Collect_mono
thf(fact_273_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_274_subset__refl,axiom,
! [A4: set_a] : ( ord_less_eq_set_a @ A4 @ A4 ) ).
% subset_refl
thf(fact_275_subset__iff,axiom,
( ord_less_eq_set_o
= ( ^ [A5: set_o,B5: set_o] :
! [T: $o] :
( ( member_o @ T @ A5 )
=> ( member_o @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_276_subset__iff,axiom,
( ord_le4374716579403074808_set_o
= ( ^ [A5: set_set_o,B5: set_set_o] :
! [T: set_o] :
( ( member_set_o @ T @ A5 )
=> ( member_set_o @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_277_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
! [T: set_a] :
( ( member_set_a @ T @ A5 )
=> ( member_set_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_278_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [T: a] :
( ( member_a @ T @ A5 )
=> ( member_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_279_equalityD2,axiom,
! [A4: set_a,B4: set_a] :
( ( A4 = B4 )
=> ( ord_less_eq_set_a @ B4 @ A4 ) ) ).
% equalityD2
thf(fact_280_equalityD1,axiom,
! [A4: set_a,B4: set_a] :
( ( A4 = B4 )
=> ( ord_less_eq_set_a @ A4 @ B4 ) ) ).
% equalityD1
thf(fact_281_subset__eq,axiom,
( ord_less_eq_set_o
= ( ^ [A5: set_o,B5: set_o] :
! [X: $o] :
( ( member_o @ X @ A5 )
=> ( member_o @ X @ B5 ) ) ) ) ).
% subset_eq
thf(fact_282_subset__eq,axiom,
( ord_le4374716579403074808_set_o
= ( ^ [A5: set_set_o,B5: set_set_o] :
! [X: set_o] :
( ( member_set_o @ X @ A5 )
=> ( member_set_o @ X @ B5 ) ) ) ) ).
% subset_eq
thf(fact_283_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
! [X: set_a] :
( ( member_set_a @ X @ A5 )
=> ( member_set_a @ X @ B5 ) ) ) ) ).
% subset_eq
thf(fact_284_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [X: a] :
( ( member_a @ X @ A5 )
=> ( member_a @ X @ B5 ) ) ) ) ).
% subset_eq
thf(fact_285_equalityE,axiom,
! [A4: set_a,B4: set_a] :
( ( A4 = B4 )
=> ~ ( ( ord_less_eq_set_a @ A4 @ B4 )
=> ~ ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ).
% equalityE
thf(fact_286_subsetD,axiom,
! [A4: set_o,B4: set_o,C: $o] :
( ( ord_less_eq_set_o @ A4 @ B4 )
=> ( ( member_o @ C @ A4 )
=> ( member_o @ C @ B4 ) ) ) ).
% subsetD
thf(fact_287_subsetD,axiom,
! [A4: set_set_o,B4: set_set_o,C: set_o] :
( ( ord_le4374716579403074808_set_o @ A4 @ B4 )
=> ( ( member_set_o @ C @ A4 )
=> ( member_set_o @ C @ B4 ) ) ) ).
% subsetD
thf(fact_288_subsetD,axiom,
! [A4: set_set_a,B4: set_set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ B4 )
=> ( ( member_set_a @ C @ A4 )
=> ( member_set_a @ C @ B4 ) ) ) ).
% subsetD
thf(fact_289_subsetD,axiom,
! [A4: set_a,B4: set_a,C: a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
=> ( ( member_a @ C @ A4 )
=> ( member_a @ C @ B4 ) ) ) ).
% subsetD
thf(fact_290_in__mono,axiom,
! [A4: set_o,B4: set_o,X3: $o] :
( ( ord_less_eq_set_o @ A4 @ B4 )
=> ( ( member_o @ X3 @ A4 )
=> ( member_o @ X3 @ B4 ) ) ) ).
% in_mono
thf(fact_291_in__mono,axiom,
! [A4: set_set_o,B4: set_set_o,X3: set_o] :
( ( ord_le4374716579403074808_set_o @ A4 @ B4 )
=> ( ( member_set_o @ X3 @ A4 )
=> ( member_set_o @ X3 @ B4 ) ) ) ).
% in_mono
thf(fact_292_in__mono,axiom,
! [A4: set_set_a,B4: set_set_a,X3: set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ B4 )
=> ( ( member_set_a @ X3 @ A4 )
=> ( member_set_a @ X3 @ B4 ) ) ) ).
% in_mono
thf(fact_293_in__mono,axiom,
! [A4: set_a,B4: set_a,X3: a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
=> ( ( member_a @ X3 @ A4 )
=> ( member_a @ X3 @ B4 ) ) ) ).
% in_mono
thf(fact_294_partial__preordering_Orefl,axiom,
! [Less_eq: a > a > $o,A: a] :
( ( partia125584492769400372ring_a @ Less_eq )
=> ( Less_eq @ A @ A ) ) ).
% partial_preordering.refl
thf(fact_295_partial__preordering_Ointro,axiom,
! [Less_eq: a > a > $o] :
( ! [A3: a] : ( Less_eq @ A3 @ A3 )
=> ( ! [A3: a,B3: a,C3: a] :
( ( Less_eq @ A3 @ B3 )
=> ( ( Less_eq @ B3 @ C3 )
=> ( Less_eq @ A3 @ C3 ) ) )
=> ( partia125584492769400372ring_a @ Less_eq ) ) ) ).
% partial_preordering.intro
thf(fact_296_partial__preordering_Otrans,axiom,
! [Less_eq: a > a > $o,A: a,B: a,C: a] :
( ( partia125584492769400372ring_a @ Less_eq )
=> ( ( Less_eq @ A @ B )
=> ( ( Less_eq @ B @ C )
=> ( Less_eq @ A @ C ) ) ) ) ).
% partial_preordering.trans
thf(fact_297_partial__preordering__def,axiom,
( partia125584492769400372ring_a
= ( ^ [Less_eq2: a > a > $o] :
( ! [A2: a] : ( Less_eq2 @ A2 @ A2 )
& ! [A2: a,B2: a,C4: a] :
( ( Less_eq2 @ A2 @ B2 )
=> ( ( Less_eq2 @ B2 @ C4 )
=> ( Less_eq2 @ A2 @ C4 ) ) ) ) ) ) ).
% partial_preordering_def
thf(fact_298_ex1__iff__ex__Uniq,axiom,
( ex1_a
= ( ^ [P2: a > $o] :
( ? [X5: a] : ( P2 @ X5 )
& ( uniq_a @ P2 ) ) ) ) ).
% ex1_iff_ex_Uniq
thf(fact_299_ex1__iff__ex__Uniq,axiom,
( ex1_o
= ( ^ [P2: $o > $o] :
( ? [X5: $o] : ( P2 @ X5 )
& ( uniq_o @ P2 ) ) ) ) ).
% ex1_iff_ex_Uniq
thf(fact_300_alt__ex1E_H,axiom,
! [P: a > $o] :
( ? [X4: a] :
( ( P @ X4 )
& ! [Y4: a] :
( ( P @ Y4 )
=> ( Y4 = X4 ) ) )
=> ~ ( ? [X_1: a] : ( P @ X_1 )
=> ~ ( uniq_a @ P ) ) ) ).
% alt_ex1E'
thf(fact_301_alt__ex1E_H,axiom,
! [P: $o > $o] :
( ? [X4: $o] :
( ( P @ X4 )
& ! [Y4: $o] :
( ( P @ Y4 )
=> ( Y4 = X4 ) ) )
=> ~ ( ? [X_1: $o] : ( P @ X_1 )
=> ~ ( uniq_o @ P ) ) ) ).
% alt_ex1E'
thf(fact_302_order_Opartial__preordering__axioms,axiom,
partia6602192050731689876_set_a @ ord_less_eq_set_a ).
% order.partial_preordering_axioms
thf(fact_303_order_Opartial__preordering__axioms,axiom,
partia1881799573076113956_o_o_o @ ord_less_eq_o_o_o ).
% order.partial_preordering_axioms
thf(fact_304_order_Opartial__preordering__axioms,axiom,
partia5423788306336055317ng_o_a @ ord_less_eq_o_a ).
% order.partial_preordering_axioms
thf(fact_305_order_Opartial__preordering__axioms,axiom,
partia4811167007327223503et_o_o @ ord_less_eq_set_o_o ).
% order.partial_preordering_axioms
thf(fact_306_order_Opartial__preordering__axioms,axiom,
partia6702899663203720265ng_a_o @ ord_less_eq_a_o ).
% order.partial_preordering_axioms
thf(fact_307_order_Opartial__preordering__axioms,axiom,
partia125584492769400372ring_a @ ord_less_eq_a ).
% order.partial_preordering_axioms
thf(fact_308_subset__Collect__iff,axiom,
! [B4: set_set_o,A4: set_set_o,P: set_o > $o] :
( ( ord_le4374716579403074808_set_o @ B4 @ A4 )
=> ( ( ord_le4374716579403074808_set_o @ B4
@ ( collect_set_o
@ ^ [X: set_o] :
( ( member_set_o @ X @ A4 )
& ( P @ X ) ) ) )
= ( ! [X: set_o] :
( ( member_set_o @ X @ B4 )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_309_subset__Collect__iff,axiom,
! [B4: set_set_a,A4: set_set_a,P: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ B4 @ A4 )
=> ( ( ord_le3724670747650509150_set_a @ B4
@ ( collect_set_a
@ ^ [X: set_a] :
( ( member_set_a @ X @ A4 )
& ( P @ X ) ) ) )
= ( ! [X: set_a] :
( ( member_set_a @ X @ B4 )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_310_subset__Collect__iff,axiom,
! [B4: set_o,A4: set_o,P: $o > $o] :
( ( ord_less_eq_set_o @ B4 @ A4 )
=> ( ( ord_less_eq_set_o @ B4
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A4 )
& ( P @ X ) ) ) )
= ( ! [X: $o] :
( ( member_o @ X @ B4 )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_311_subset__Collect__iff,axiom,
! [B4: set_a,A4: set_a,P: a > $o] :
( ( ord_less_eq_set_a @ B4 @ A4 )
=> ( ( ord_less_eq_set_a @ B4
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A4 )
& ( P @ X ) ) ) )
= ( ! [X: a] :
( ( member_a @ X @ B4 )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_312_subset__CollectI,axiom,
! [B4: set_set_o,A4: set_set_o,Q: set_o > $o,P: set_o > $o] :
( ( ord_le4374716579403074808_set_o @ B4 @ A4 )
=> ( ! [X2: set_o] :
( ( member_set_o @ X2 @ B4 )
=> ( ( Q @ X2 )
=> ( P @ X2 ) ) )
=> ( ord_le4374716579403074808_set_o
@ ( collect_set_o
@ ^ [X: set_o] :
( ( member_set_o @ X @ B4 )
& ( Q @ X ) ) )
@ ( collect_set_o
@ ^ [X: set_o] :
( ( member_set_o @ X @ A4 )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_313_subset__CollectI,axiom,
! [B4: set_set_a,A4: set_set_a,Q: set_a > $o,P: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ B4 @ A4 )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ B4 )
=> ( ( Q @ X2 )
=> ( P @ X2 ) ) )
=> ( ord_le3724670747650509150_set_a
@ ( collect_set_a
@ ^ [X: set_a] :
( ( member_set_a @ X @ B4 )
& ( Q @ X ) ) )
@ ( collect_set_a
@ ^ [X: set_a] :
( ( member_set_a @ X @ A4 )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_314_subset__CollectI,axiom,
! [B4: set_o,A4: set_o,Q: $o > $o,P: $o > $o] :
( ( ord_less_eq_set_o @ B4 @ A4 )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ B4 )
=> ( ( Q @ X2 )
=> ( P @ X2 ) ) )
=> ( ord_less_eq_set_o
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ B4 )
& ( Q @ X ) ) )
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A4 )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_315_subset__CollectI,axiom,
! [B4: set_a,A4: set_a,Q: a > $o,P: a > $o] :
( ( ord_less_eq_set_a @ B4 @ A4 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B4 )
=> ( ( Q @ X2 )
=> ( P @ X2 ) ) )
=> ( ord_less_eq_set_a
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ B4 )
& ( Q @ X ) ) )
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A4 )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_316_Collect__restrict,axiom,
! [X6: set_set_o,P: set_o > $o] :
( ord_le4374716579403074808_set_o
@ ( collect_set_o
@ ^ [X: set_o] :
( ( member_set_o @ X @ X6 )
& ( P @ X ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_317_Collect__restrict,axiom,
! [X6: set_set_a,P: set_a > $o] :
( ord_le3724670747650509150_set_a
@ ( collect_set_a
@ ^ [X: set_a] :
( ( member_set_a @ X @ X6 )
& ( P @ X ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_318_Collect__restrict,axiom,
! [X6: set_o,P: $o > $o] :
( ord_less_eq_set_o
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ X6 )
& ( P @ X ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_319_Collect__restrict,axiom,
! [X6: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ X6 )
& ( P @ X ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_320_prop__restrict,axiom,
! [X3: set_o,Z4: set_set_o,X6: set_set_o,P: set_o > $o] :
( ( member_set_o @ X3 @ Z4 )
=> ( ( ord_le4374716579403074808_set_o @ Z4
@ ( collect_set_o
@ ^ [X: set_o] :
( ( member_set_o @ X @ X6 )
& ( P @ X ) ) ) )
=> ( P @ X3 ) ) ) ).
% prop_restrict
thf(fact_321_prop__restrict,axiom,
! [X3: set_a,Z4: set_set_a,X6: set_set_a,P: set_a > $o] :
( ( member_set_a @ X3 @ Z4 )
=> ( ( ord_le3724670747650509150_set_a @ Z4
@ ( collect_set_a
@ ^ [X: set_a] :
( ( member_set_a @ X @ X6 )
& ( P @ X ) ) ) )
=> ( P @ X3 ) ) ) ).
% prop_restrict
thf(fact_322_prop__restrict,axiom,
! [X3: $o,Z4: set_o,X6: set_o,P: $o > $o] :
( ( member_o @ X3 @ Z4 )
=> ( ( ord_less_eq_set_o @ Z4
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ X6 )
& ( P @ X ) ) ) )
=> ( P @ X3 ) ) ) ).
% prop_restrict
thf(fact_323_prop__restrict,axiom,
! [X3: a,Z4: set_a,X6: set_a,P: a > $o] :
( ( member_a @ X3 @ Z4 )
=> ( ( ord_less_eq_set_a @ Z4
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ X6 )
& ( P @ X ) ) ) )
=> ( P @ X3 ) ) ) ).
% prop_restrict
thf(fact_324_conj__subset__def,axiom,
! [A4: set_o,P: $o > $o,Q: $o > $o] :
( ( ord_less_eq_set_o @ A4
@ ( collect_o
@ ^ [X: $o] :
( ( P @ X )
& ( Q @ X ) ) ) )
= ( ( ord_less_eq_set_o @ A4 @ ( collect_o @ P ) )
& ( ord_less_eq_set_o @ A4 @ ( collect_o @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_325_conj__subset__def,axiom,
! [A4: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A4
@ ( collect_a
@ ^ [X: a] :
( ( P @ X )
& ( Q @ X ) ) ) )
= ( ( ord_less_eq_set_a @ A4 @ ( collect_a @ P ) )
& ( ord_less_eq_set_a @ A4 @ ( collect_a @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_326_pred__subset__eq,axiom,
! [R: set_o,S2: set_o] :
( ( ord_less_eq_o_o
@ ^ [X: $o] : ( member_o @ X @ R )
@ ^ [X: $o] : ( member_o @ X @ S2 ) )
= ( ord_less_eq_set_o @ R @ S2 ) ) ).
% pred_subset_eq
thf(fact_327_pred__subset__eq,axiom,
! [R: set_set_a,S2: set_set_a] :
( ( ord_less_eq_set_a_o
@ ^ [X: set_a] : ( member_set_a @ X @ R )
@ ^ [X: set_a] : ( member_set_a @ X @ S2 ) )
= ( ord_le3724670747650509150_set_a @ R @ S2 ) ) ).
% pred_subset_eq
thf(fact_328_pred__subset__eq,axiom,
! [R: set_set_o,S2: set_set_o] :
( ( ord_less_eq_set_o_o
@ ^ [X: set_o] : ( member_set_o @ X @ R )
@ ^ [X: set_o] : ( member_set_o @ X @ S2 ) )
= ( ord_le4374716579403074808_set_o @ R @ S2 ) ) ).
% pred_subset_eq
thf(fact_329_pred__subset__eq,axiom,
! [R: set_a,S2: set_a] :
( ( ord_less_eq_a_o
@ ^ [X: a] : ( member_a @ X @ R )
@ ^ [X: a] : ( member_a @ X @ S2 ) )
= ( ord_less_eq_set_a @ R @ S2 ) ) ).
% pred_subset_eq
thf(fact_330_Greatest__def,axiom,
( order_Greatest_o
= ( ^ [P2: $o > $o] :
( the_o
@ ^ [X: $o] :
( ( P2 @ X )
& ! [Y2: $o] :
( ( P2 @ Y2 )
=> ( ord_less_eq_o @ Y2 @ X ) ) ) ) ) ) ).
% Greatest_def
thf(fact_331_Greatest__def,axiom,
( order_Greatest_set_a
= ( ^ [P2: set_a > $o] :
( the_set_a
@ ^ [X: set_a] :
( ( P2 @ X )
& ! [Y2: set_a] :
( ( P2 @ Y2 )
=> ( ord_less_eq_set_a @ Y2 @ X ) ) ) ) ) ) ).
% Greatest_def
thf(fact_332_Greatest__def,axiom,
( order_Greatest_o_o_o
= ( ^ [P2: ( $o > $o > $o ) > $o] :
( the_o_o_o
@ ^ [X: $o > $o > $o] :
( ( P2 @ X )
& ! [Y2: $o > $o > $o] :
( ( P2 @ Y2 )
=> ( ord_less_eq_o_o_o @ Y2 @ X ) ) ) ) ) ) ).
% Greatest_def
thf(fact_333_Greatest__def,axiom,
( order_Greatest_o_a
= ( ^ [P2: ( $o > a ) > $o] :
( the_o_a
@ ^ [X: $o > a] :
( ( P2 @ X )
& ! [Y2: $o > a] :
( ( P2 @ Y2 )
=> ( ord_less_eq_o_a @ Y2 @ X ) ) ) ) ) ) ).
% Greatest_def
thf(fact_334_Greatest__def,axiom,
( order_5805555182909007980et_o_o
= ( ^ [P2: ( set_o > $o ) > $o] :
( the_set_o_o
@ ^ [X: set_o > $o] :
( ( P2 @ X )
& ! [Y2: set_o > $o] :
( ( P2 @ Y2 )
=> ( ord_less_eq_set_o_o @ Y2 @ X ) ) ) ) ) ) ).
% Greatest_def
thf(fact_335_Greatest__def,axiom,
( order_Greatest_a_o
= ( ^ [P2: ( a > $o ) > $o] :
( the_a_o
@ ^ [X: a > $o] :
( ( P2 @ X )
& ! [Y2: a > $o] :
( ( P2 @ Y2 )
=> ( ord_less_eq_a_o @ Y2 @ X ) ) ) ) ) ) ).
% Greatest_def
thf(fact_336_Greatest__def,axiom,
( order_Greatest_a
= ( ^ [P2: a > $o] :
( the_a
@ ^ [X: a] :
( ( P2 @ X )
& ! [Y2: a] :
( ( P2 @ Y2 )
=> ( ord_less_eq_a @ Y2 @ X ) ) ) ) ) ) ).
% Greatest_def
thf(fact_337_The__unsafe__def,axiom,
the_unsafe_o = the_o ).
% The_unsafe_def
thf(fact_338_The__unsafe__def,axiom,
the_unsafe_a = the_a ).
% The_unsafe_def
thf(fact_339_HOL_Oinduct__false__def,axiom,
~ induct_false ).
% HOL.induct_false_def
thf(fact_340_ordering_Oaxioms_I1_J,axiom,
! [Less_eq: a > a > $o,Less: a > a > $o] :
( ( ordering_a @ Less_eq @ Less )
=> ( partia125584492769400372ring_a @ Less_eq ) ) ).
% ordering.axioms(1)
thf(fact_341_predicate1I,axiom,
! [P: set_o > $o,Q: set_o > $o] :
( ! [X2: set_o] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_o_o @ P @ Q ) ) ).
% predicate1I
thf(fact_342_predicate1I,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_a_o @ P @ Q ) ) ).
% predicate1I
thf(fact_343_predicate1D,axiom,
! [P: set_o > $o,Q: set_o > $o,X3: set_o] :
( ( ord_less_eq_set_o_o @ P @ Q )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) ) ).
% predicate1D
thf(fact_344_predicate1D,axiom,
! [P: a > $o,Q: a > $o,X3: a] :
( ( ord_less_eq_a_o @ P @ Q )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) ) ).
% predicate1D
thf(fact_345_rev__predicate1D,axiom,
! [P: set_o > $o,X3: set_o,Q: set_o > $o] :
( ( P @ X3 )
=> ( ( ord_less_eq_set_o_o @ P @ Q )
=> ( Q @ X3 ) ) ) ).
% rev_predicate1D
thf(fact_346_rev__predicate1D,axiom,
! [P: a > $o,X3: a,Q: a > $o] :
( ( P @ X3 )
=> ( ( ord_less_eq_a_o @ P @ Q )
=> ( Q @ X3 ) ) ) ).
% rev_predicate1D
thf(fact_347_ordering_Oeq__iff,axiom,
! [Less_eq: a > a > $o,Less: a > a > $o,A: a,B: a] :
( ( ordering_a @ Less_eq @ Less )
=> ( ( A = B )
= ( ( Less_eq @ A @ B )
& ( Less_eq @ B @ A ) ) ) ) ).
% ordering.eq_iff
thf(fact_348_ordering_Oantisym,axiom,
! [Less_eq: a > a > $o,Less: a > a > $o,A: a,B: a] :
( ( ordering_a @ Less_eq @ Less )
=> ( ( Less_eq @ A @ B )
=> ( ( Less_eq @ B @ A )
=> ( A = B ) ) ) ) ).
% ordering.antisym
thf(fact_349_ordering_Oorder__iff__strict,axiom,
! [Less_eq: a > a > $o,Less: a > a > $o,A: a,B: a] :
( ( ordering_a @ Less_eq @ Less )
=> ( ( Less_eq @ A @ B )
= ( ( Less @ A @ B )
| ( A = B ) ) ) ) ).
% ordering.order_iff_strict
thf(fact_350_ordering_Ostrict__iff__order,axiom,
! [Less_eq: a > a > $o,Less: a > a > $o,A: a,B: a] :
( ( ordering_a @ Less_eq @ Less )
=> ( ( Less @ A @ B )
= ( ( Less_eq @ A @ B )
& ( A != B ) ) ) ) ).
% ordering.strict_iff_order
thf(fact_351_ordering_Ostrict__implies__not__eq,axiom,
! [Less_eq: a > a > $o,Less: a > a > $o,A: a,B: a] :
( ( ordering_a @ Less_eq @ Less )
=> ( ( Less @ A @ B )
=> ( A != B ) ) ) ).
% ordering.strict_implies_not_eq
thf(fact_352_ordering_Onot__eq__order__implies__strict,axiom,
! [Less_eq: a > a > $o,Less: a > a > $o,A: a,B: a] :
( ( ordering_a @ Less_eq @ Less )
=> ( ( A != B )
=> ( ( Less_eq @ A @ B )
=> ( Less @ A @ B ) ) ) ) ).
% ordering.not_eq_order_implies_strict
thf(fact_353_ordering__strictI,axiom,
! [Less_eq: a > a > $o,Less: a > a > $o] :
( ! [A3: a,B3: a] :
( ( Less_eq @ A3 @ B3 )
= ( ( Less @ A3 @ B3 )
| ( A3 = B3 ) ) )
=> ( ! [A3: a,B3: a] :
( ( Less @ A3 @ B3 )
=> ~ ( Less @ B3 @ A3 ) )
=> ( ! [A3: a] :
~ ( Less @ A3 @ A3 )
=> ( ! [A3: a,B3: a,C3: a] :
( ( Less @ A3 @ B3 )
=> ( ( Less @ B3 @ C3 )
=> ( Less @ A3 @ C3 ) ) )
=> ( ordering_a @ Less_eq @ Less ) ) ) ) ) ).
% ordering_strictI
thf(fact_354_ordering__dualI,axiom,
! [Less_eq: a > a > $o,Less: a > a > $o] :
( ( ordering_a
@ ^ [A2: a,B2: a] : ( Less_eq @ B2 @ A2 )
@ ^ [A2: a,B2: a] : ( Less @ B2 @ A2 ) )
=> ( ordering_a @ Less_eq @ Less ) ) ).
% ordering_dualI
thf(fact_355_ball__reg__right,axiom,
! [R: set_o,P: $o > $o,Q: $o > $o] :
( ! [X2: $o] :
( ( member_o @ X2 @ R )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ! [X_1: $o] : ( P @ X_1 )
=> ! [X4: $o] :
( ( member_o @ X4 @ R )
=> ( Q @ X4 ) ) ) ) ).
% ball_reg_right
thf(fact_356_ball__reg__right,axiom,
! [R: set_set_o,P: set_o > $o,Q: set_o > $o] :
( ! [X2: set_o] :
( ( member_set_o @ X2 @ R )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ! [X_1: set_o] : ( P @ X_1 )
=> ! [X4: set_o] :
( ( member_set_o @ X4 @ R )
=> ( Q @ X4 ) ) ) ) ).
% ball_reg_right
thf(fact_357_ball__reg__right,axiom,
! [R: set_set_a,P: set_a > $o,Q: set_a > $o] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ R )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ! [X_1: set_a] : ( P @ X_1 )
=> ! [X4: set_a] :
( ( member_set_a @ X4 @ R )
=> ( Q @ X4 ) ) ) ) ).
% ball_reg_right
thf(fact_358_ball__reg__right,axiom,
! [R: set_a,P: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( member_a @ X2 @ R )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ! [X_1: a] : ( P @ X_1 )
=> ! [X4: a] :
( ( member_a @ X4 @ R )
=> ( Q @ X4 ) ) ) ) ).
% ball_reg_right
thf(fact_359_Ball__def__raw,axiom,
( ball_o
= ( ^ [A5: set_o,P2: $o > $o] :
! [X: $o] :
( ( member_o @ X @ A5 )
=> ( P2 @ X ) ) ) ) ).
% Ball_def_raw
thf(fact_360_Ball__def__raw,axiom,
( ball_set_o
= ( ^ [A5: set_set_o,P2: set_o > $o] :
! [X: set_o] :
( ( member_set_o @ X @ A5 )
=> ( P2 @ X ) ) ) ) ).
% Ball_def_raw
thf(fact_361_Ball__def__raw,axiom,
( ball_set_a
= ( ^ [A5: set_set_a,P2: set_a > $o] :
! [X: set_a] :
( ( member_set_a @ X @ A5 )
=> ( P2 @ X ) ) ) ) ).
% Ball_def_raw
thf(fact_362_Ball__def__raw,axiom,
( ball_a
= ( ^ [A5: set_a,P2: a > $o] :
! [X: a] :
( ( member_a @ X @ A5 )
=> ( P2 @ X ) ) ) ) ).
% Ball_def_raw
thf(fact_363_ord_OLeast__def,axiom,
( least_o
= ( ^ [Less_eq2: $o > $o > $o,P2: $o > $o] :
( the_o
@ ^ [X: $o] :
( ( P2 @ X )
& ! [Y2: $o] :
( ( P2 @ Y2 )
=> ( Less_eq2 @ X @ Y2 ) ) ) ) ) ) ).
% ord.Least_def
thf(fact_364_ord_OLeast__def,axiom,
( least_a
= ( ^ [Less_eq2: a > a > $o,P2: a > $o] :
( the_a
@ ^ [X: a] :
( ( P2 @ X )
& ! [Y2: a] :
( ( P2 @ Y2 )
=> ( Less_eq2 @ X @ Y2 ) ) ) ) ) ) ).
% ord.Least_def
thf(fact_365_ordering_Ointro,axiom,
! [Less_eq: a > a > $o,Less: a > a > $o] :
( ( partia125584492769400372ring_a @ Less_eq )
=> ( ( ordering_axioms_a @ Less_eq @ Less )
=> ( ordering_a @ Less_eq @ Less ) ) ) ).
% ordering.intro
thf(fact_366_ordering__def,axiom,
( ordering_a
= ( ^ [Less_eq2: a > a > $o,Less2: a > a > $o] :
( ( partia125584492769400372ring_a @ Less_eq2 )
& ( ordering_axioms_a @ Less_eq2 @ Less2 ) ) ) ) ).
% ordering_def
thf(fact_367_Least__def,axiom,
( ord_Least_o
= ( ^ [P2: $o > $o] :
( the_o
@ ^ [X: $o] :
( ( P2 @ X )
& ! [Y2: $o] :
( ( P2 @ Y2 )
=> ( ord_less_eq_o @ X @ Y2 ) ) ) ) ) ) ).
% Least_def
thf(fact_368_Least__def,axiom,
( ord_Least_set_a
= ( ^ [P2: set_a > $o] :
( the_set_a
@ ^ [X: set_a] :
( ( P2 @ X )
& ! [Y2: set_a] :
( ( P2 @ Y2 )
=> ( ord_less_eq_set_a @ X @ Y2 ) ) ) ) ) ) ).
% Least_def
thf(fact_369_Least__def,axiom,
( ord_Least_o_o_o
= ( ^ [P2: ( $o > $o > $o ) > $o] :
( the_o_o_o
@ ^ [X: $o > $o > $o] :
( ( P2 @ X )
& ! [Y2: $o > $o > $o] :
( ( P2 @ Y2 )
=> ( ord_less_eq_o_o_o @ X @ Y2 ) ) ) ) ) ) ).
% Least_def
thf(fact_370_Least__def,axiom,
( ord_Least_o_a
= ( ^ [P2: ( $o > a ) > $o] :
( the_o_a
@ ^ [X: $o > a] :
( ( P2 @ X )
& ! [Y2: $o > a] :
( ( P2 @ Y2 )
=> ( ord_less_eq_o_a @ X @ Y2 ) ) ) ) ) ) ).
% Least_def
thf(fact_371_Least__def,axiom,
( ord_Least_set_o_o
= ( ^ [P2: ( set_o > $o ) > $o] :
( the_set_o_o
@ ^ [X: set_o > $o] :
( ( P2 @ X )
& ! [Y2: set_o > $o] :
( ( P2 @ Y2 )
=> ( ord_less_eq_set_o_o @ X @ Y2 ) ) ) ) ) ) ).
% Least_def
thf(fact_372_Least__def,axiom,
( ord_Least_a_o
= ( ^ [P2: ( a > $o ) > $o] :
( the_a_o
@ ^ [X: a > $o] :
( ( P2 @ X )
& ! [Y2: a > $o] :
( ( P2 @ Y2 )
=> ( ord_less_eq_a_o @ X @ Y2 ) ) ) ) ) ) ).
% Least_def
thf(fact_373_Least__def,axiom,
( ord_Least_a
= ( ^ [P2: a > $o] :
( the_a
@ ^ [X: a] :
( ( P2 @ X )
& ! [Y2: a] :
( ( P2 @ Y2 )
=> ( ord_less_eq_a @ X @ Y2 ) ) ) ) ) ) ).
% Least_def
thf(fact_374_Powp__mono,axiom,
! [A4: $o > $o,B4: $o > $o] :
( ( ord_less_eq_o_o @ A4 @ B4 )
=> ( ord_less_eq_set_o_o @ ( powp_o @ A4 ) @ ( powp_o @ B4 ) ) ) ).
% Powp_mono
thf(fact_375_Powp__mono,axiom,
! [A4: set_o > $o,B4: set_o > $o] :
( ( ord_less_eq_set_o_o @ A4 @ B4 )
=> ( ord_le8367510561069133573et_o_o @ ( powp_set_o @ A4 ) @ ( powp_set_o @ B4 ) ) ) ).
% Powp_mono
thf(fact_376_Powp__mono,axiom,
! [A4: a > $o,B4: a > $o] :
( ( ord_less_eq_a_o @ A4 @ B4 )
=> ( ord_less_eq_set_a_o @ ( powp_a @ A4 ) @ ( powp_a @ B4 ) ) ) ).
% Powp_mono
thf(fact_377_ordering_Oaxioms_I2_J,axiom,
! [Less_eq: a > a > $o,Less: a > a > $o] :
( ( ordering_a @ Less_eq @ Less )
=> ( ordering_axioms_a @ Less_eq @ Less ) ) ).
% ordering.axioms(2)
thf(fact_378_single__valuedp__less__eq,axiom,
! [R2: $o > $o > $o,S3: $o > $o > $o] :
( ( ord_less_eq_o_o_o @ R2 @ S3 )
=> ( ( single_valuedp_o_o @ S3 )
=> ( single_valuedp_o_o @ R2 ) ) ) ).
% single_valuedp_less_eq
thf(fact_379_ord_OLeast_Ocong,axiom,
least_a = least_a ).
% ord.Least.cong
thf(fact_380_Least1I,axiom,
! [P: $o > $o] :
( ? [X4: $o] :
( ( P @ X4 )
& ! [Y4: $o] :
( ( P @ Y4 )
=> ( ord_less_eq_o @ X4 @ Y4 ) )
& ! [Y4: $o] :
( ( ( P @ Y4 )
& ! [Ya: $o] :
( ( P @ Ya )
=> ( ord_less_eq_o @ Y4 @ Ya ) ) )
=> ( Y4 = X4 ) ) )
=> ( P @ ( ord_Least_o @ P ) ) ) ).
% Least1I
thf(fact_381_Least1I,axiom,
! [P: set_a > $o] :
( ? [X4: set_a] :
( ( P @ X4 )
& ! [Y4: set_a] :
( ( P @ Y4 )
=> ( ord_less_eq_set_a @ X4 @ Y4 ) )
& ! [Y4: set_a] :
( ( ( P @ Y4 )
& ! [Ya: set_a] :
( ( P @ Ya )
=> ( ord_less_eq_set_a @ Y4 @ Ya ) ) )
=> ( Y4 = X4 ) ) )
=> ( P @ ( ord_Least_set_a @ P ) ) ) ).
% Least1I
thf(fact_382_Least1I,axiom,
! [P: ( $o > $o > $o ) > $o] :
( ? [X4: $o > $o > $o] :
( ( P @ X4 )
& ! [Y4: $o > $o > $o] :
( ( P @ Y4 )
=> ( ord_less_eq_o_o_o @ X4 @ Y4 ) )
& ! [Y4: $o > $o > $o] :
( ( ( P @ Y4 )
& ! [Ya: $o > $o > $o] :
( ( P @ Ya )
=> ( ord_less_eq_o_o_o @ Y4 @ Ya ) ) )
=> ( Y4 = X4 ) ) )
=> ( P @ ( ord_Least_o_o_o @ P ) ) ) ).
% Least1I
thf(fact_383_Least1I,axiom,
! [P: ( $o > a ) > $o] :
( ? [X4: $o > a] :
( ( P @ X4 )
& ! [Y4: $o > a] :
( ( P @ Y4 )
=> ( ord_less_eq_o_a @ X4 @ Y4 ) )
& ! [Y4: $o > a] :
( ( ( P @ Y4 )
& ! [Ya: $o > a] :
( ( P @ Ya )
=> ( ord_less_eq_o_a @ Y4 @ Ya ) ) )
=> ( Y4 = X4 ) ) )
=> ( P @ ( ord_Least_o_a @ P ) ) ) ).
% Least1I
thf(fact_384_Least1I,axiom,
! [P: ( set_o > $o ) > $o] :
( ? [X4: set_o > $o] :
( ( P @ X4 )
& ! [Y4: set_o > $o] :
( ( P @ Y4 )
=> ( ord_less_eq_set_o_o @ X4 @ Y4 ) )
& ! [Y4: set_o > $o] :
( ( ( P @ Y4 )
& ! [Ya: set_o > $o] :
( ( P @ Ya )
=> ( ord_less_eq_set_o_o @ Y4 @ Ya ) ) )
=> ( Y4 = X4 ) ) )
=> ( P @ ( ord_Least_set_o_o @ P ) ) ) ).
% Least1I
thf(fact_385_Least1I,axiom,
! [P: ( a > $o ) > $o] :
( ? [X4: a > $o] :
( ( P @ X4 )
& ! [Y4: a > $o] :
( ( P @ Y4 )
=> ( ord_less_eq_a_o @ X4 @ Y4 ) )
& ! [Y4: a > $o] :
( ( ( P @ Y4 )
& ! [Ya: a > $o] :
( ( P @ Ya )
=> ( ord_less_eq_a_o @ Y4 @ Ya ) ) )
=> ( Y4 = X4 ) ) )
=> ( P @ ( ord_Least_a_o @ P ) ) ) ).
% Least1I
thf(fact_386_Least1I,axiom,
! [P: a > $o] :
( ? [X4: a] :
( ( P @ X4 )
& ! [Y4: a] :
( ( P @ Y4 )
=> ( ord_less_eq_a @ X4 @ Y4 ) )
& ! [Y4: a] :
( ( ( P @ Y4 )
& ! [Ya: a] :
( ( P @ Ya )
=> ( ord_less_eq_a @ Y4 @ Ya ) ) )
=> ( Y4 = X4 ) ) )
=> ( P @ ( ord_Least_a @ P ) ) ) ).
% Least1I
thf(fact_387_Least1__le,axiom,
! [P: $o > $o,Z3: $o] :
( ? [X4: $o] :
( ( P @ X4 )
& ! [Y4: $o] :
( ( P @ Y4 )
=> ( ord_less_eq_o @ X4 @ Y4 ) )
& ! [Y4: $o] :
( ( ( P @ Y4 )
& ! [Ya: $o] :
( ( P @ Ya )
=> ( ord_less_eq_o @ Y4 @ Ya ) ) )
=> ( Y4 = X4 ) ) )
=> ( ( P @ Z3 )
=> ( ord_less_eq_o @ ( ord_Least_o @ P ) @ Z3 ) ) ) ).
% Least1_le
thf(fact_388_Least1__le,axiom,
! [P: set_a > $o,Z3: set_a] :
( ? [X4: set_a] :
( ( P @ X4 )
& ! [Y4: set_a] :
( ( P @ Y4 )
=> ( ord_less_eq_set_a @ X4 @ Y4 ) )
& ! [Y4: set_a] :
( ( ( P @ Y4 )
& ! [Ya: set_a] :
( ( P @ Ya )
=> ( ord_less_eq_set_a @ Y4 @ Ya ) ) )
=> ( Y4 = X4 ) ) )
=> ( ( P @ Z3 )
=> ( ord_less_eq_set_a @ ( ord_Least_set_a @ P ) @ Z3 ) ) ) ).
% Least1_le
thf(fact_389_Least1__le,axiom,
! [P: ( $o > $o > $o ) > $o,Z3: $o > $o > $o] :
( ? [X4: $o > $o > $o] :
( ( P @ X4 )
& ! [Y4: $o > $o > $o] :
( ( P @ Y4 )
=> ( ord_less_eq_o_o_o @ X4 @ Y4 ) )
& ! [Y4: $o > $o > $o] :
( ( ( P @ Y4 )
& ! [Ya: $o > $o > $o] :
( ( P @ Ya )
=> ( ord_less_eq_o_o_o @ Y4 @ Ya ) ) )
=> ( Y4 = X4 ) ) )
=> ( ( P @ Z3 )
=> ( ord_less_eq_o_o_o @ ( ord_Least_o_o_o @ P ) @ Z3 ) ) ) ).
% Least1_le
thf(fact_390_Least1__le,axiom,
! [P: ( $o > a ) > $o,Z3: $o > a] :
( ? [X4: $o > a] :
( ( P @ X4 )
& ! [Y4: $o > a] :
( ( P @ Y4 )
=> ( ord_less_eq_o_a @ X4 @ Y4 ) )
& ! [Y4: $o > a] :
( ( ( P @ Y4 )
& ! [Ya: $o > a] :
( ( P @ Ya )
=> ( ord_less_eq_o_a @ Y4 @ Ya ) ) )
=> ( Y4 = X4 ) ) )
=> ( ( P @ Z3 )
=> ( ord_less_eq_o_a @ ( ord_Least_o_a @ P ) @ Z3 ) ) ) ).
% Least1_le
thf(fact_391_Least1__le,axiom,
! [P: ( set_o > $o ) > $o,Z3: set_o > $o] :
( ? [X4: set_o > $o] :
( ( P @ X4 )
& ! [Y4: set_o > $o] :
( ( P @ Y4 )
=> ( ord_less_eq_set_o_o @ X4 @ Y4 ) )
& ! [Y4: set_o > $o] :
( ( ( P @ Y4 )
& ! [Ya: set_o > $o] :
( ( P @ Ya )
=> ( ord_less_eq_set_o_o @ Y4 @ Ya ) ) )
=> ( Y4 = X4 ) ) )
=> ( ( P @ Z3 )
=> ( ord_less_eq_set_o_o @ ( ord_Least_set_o_o @ P ) @ Z3 ) ) ) ).
% Least1_le
thf(fact_392_Least1__le,axiom,
! [P: ( a > $o ) > $o,Z3: a > $o] :
( ? [X4: a > $o] :
( ( P @ X4 )
& ! [Y4: a > $o] :
( ( P @ Y4 )
=> ( ord_less_eq_a_o @ X4 @ Y4 ) )
& ! [Y4: a > $o] :
( ( ( P @ Y4 )
& ! [Ya: a > $o] :
( ( P @ Ya )
=> ( ord_less_eq_a_o @ Y4 @ Ya ) ) )
=> ( Y4 = X4 ) ) )
=> ( ( P @ Z3 )
=> ( ord_less_eq_a_o @ ( ord_Least_a_o @ P ) @ Z3 ) ) ) ).
% Least1_le
thf(fact_393_Least1__le,axiom,
! [P: a > $o,Z3: a] :
( ? [X4: a] :
( ( P @ X4 )
& ! [Y4: a] :
( ( P @ Y4 )
=> ( ord_less_eq_a @ X4 @ Y4 ) )
& ! [Y4: a] :
( ( ( P @ Y4 )
& ! [Ya: a] :
( ( P @ Ya )
=> ( ord_less_eq_a @ Y4 @ Ya ) ) )
=> ( Y4 = X4 ) ) )
=> ( ( P @ Z3 )
=> ( ord_less_eq_a @ ( ord_Least_a @ P ) @ Z3 ) ) ) ).
% Least1_le
thf(fact_394_LeastI2__order,axiom,
! [P: $o > $o,X3: $o,Q: $o > $o] :
( ( P @ X3 )
=> ( ! [Y4: $o] :
( ( P @ Y4 )
=> ( ord_less_eq_o @ X3 @ Y4 ) )
=> ( ! [X2: $o] :
( ( P @ X2 )
=> ( ! [Y5: $o] :
( ( P @ Y5 )
=> ( ord_less_eq_o @ X2 @ Y5 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( ord_Least_o @ P ) ) ) ) ) ).
% LeastI2_order
thf(fact_395_LeastI2__order,axiom,
! [P: set_a > $o,X3: set_a,Q: set_a > $o] :
( ( P @ X3 )
=> ( ! [Y4: set_a] :
( ( P @ Y4 )
=> ( ord_less_eq_set_a @ X3 @ Y4 ) )
=> ( ! [X2: set_a] :
( ( P @ X2 )
=> ( ! [Y5: set_a] :
( ( P @ Y5 )
=> ( ord_less_eq_set_a @ X2 @ Y5 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( ord_Least_set_a @ P ) ) ) ) ) ).
% LeastI2_order
thf(fact_396_LeastI2__order,axiom,
! [P: ( $o > $o > $o ) > $o,X3: $o > $o > $o,Q: ( $o > $o > $o ) > $o] :
( ( P @ X3 )
=> ( ! [Y4: $o > $o > $o] :
( ( P @ Y4 )
=> ( ord_less_eq_o_o_o @ X3 @ Y4 ) )
=> ( ! [X2: $o > $o > $o] :
( ( P @ X2 )
=> ( ! [Y5: $o > $o > $o] :
( ( P @ Y5 )
=> ( ord_less_eq_o_o_o @ X2 @ Y5 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( ord_Least_o_o_o @ P ) ) ) ) ) ).
% LeastI2_order
thf(fact_397_LeastI2__order,axiom,
! [P: ( $o > a ) > $o,X3: $o > a,Q: ( $o > a ) > $o] :
( ( P @ X3 )
=> ( ! [Y4: $o > a] :
( ( P @ Y4 )
=> ( ord_less_eq_o_a @ X3 @ Y4 ) )
=> ( ! [X2: $o > a] :
( ( P @ X2 )
=> ( ! [Y5: $o > a] :
( ( P @ Y5 )
=> ( ord_less_eq_o_a @ X2 @ Y5 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( ord_Least_o_a @ P ) ) ) ) ) ).
% LeastI2_order
thf(fact_398_LeastI2__order,axiom,
! [P: ( set_o > $o ) > $o,X3: set_o > $o,Q: ( set_o > $o ) > $o] :
( ( P @ X3 )
=> ( ! [Y4: set_o > $o] :
( ( P @ Y4 )
=> ( ord_less_eq_set_o_o @ X3 @ Y4 ) )
=> ( ! [X2: set_o > $o] :
( ( P @ X2 )
=> ( ! [Y5: set_o > $o] :
( ( P @ Y5 )
=> ( ord_less_eq_set_o_o @ X2 @ Y5 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( ord_Least_set_o_o @ P ) ) ) ) ) ).
% LeastI2_order
thf(fact_399_LeastI2__order,axiom,
! [P: ( a > $o ) > $o,X3: a > $o,Q: ( a > $o ) > $o] :
( ( P @ X3 )
=> ( ! [Y4: a > $o] :
( ( P @ Y4 )
=> ( ord_less_eq_a_o @ X3 @ Y4 ) )
=> ( ! [X2: a > $o] :
( ( P @ X2 )
=> ( ! [Y5: a > $o] :
( ( P @ Y5 )
=> ( ord_less_eq_a_o @ X2 @ Y5 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( ord_Least_a_o @ P ) ) ) ) ) ).
% LeastI2_order
thf(fact_400_LeastI2__order,axiom,
! [P: a > $o,X3: a,Q: a > $o] :
( ( P @ X3 )
=> ( ! [Y4: a] :
( ( P @ Y4 )
=> ( ord_less_eq_a @ X3 @ Y4 ) )
=> ( ! [X2: a] :
( ( P @ X2 )
=> ( ! [Y5: a] :
( ( P @ Y5 )
=> ( ord_less_eq_a @ X2 @ Y5 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( ord_Least_a @ P ) ) ) ) ) ).
% LeastI2_order
thf(fact_401_Least__equality,axiom,
! [P: $o > $o,X3: $o] :
( ( P @ X3 )
=> ( ! [Y4: $o] :
( ( P @ Y4 )
=> ( ord_less_eq_o @ X3 @ Y4 ) )
=> ( ( ord_Least_o @ P )
= X3 ) ) ) ).
% Least_equality
thf(fact_402_Least__equality,axiom,
! [P: set_a > $o,X3: set_a] :
( ( P @ X3 )
=> ( ! [Y4: set_a] :
( ( P @ Y4 )
=> ( ord_less_eq_set_a @ X3 @ Y4 ) )
=> ( ( ord_Least_set_a @ P )
= X3 ) ) ) ).
% Least_equality
thf(fact_403_Least__equality,axiom,
! [P: ( $o > $o > $o ) > $o,X3: $o > $o > $o] :
( ( P @ X3 )
=> ( ! [Y4: $o > $o > $o] :
( ( P @ Y4 )
=> ( ord_less_eq_o_o_o @ X3 @ Y4 ) )
=> ( ( ord_Least_o_o_o @ P )
= X3 ) ) ) ).
% Least_equality
thf(fact_404_Least__equality,axiom,
! [P: ( $o > a ) > $o,X3: $o > a] :
( ( P @ X3 )
=> ( ! [Y4: $o > a] :
( ( P @ Y4 )
=> ( ord_less_eq_o_a @ X3 @ Y4 ) )
=> ( ( ord_Least_o_a @ P )
= X3 ) ) ) ).
% Least_equality
thf(fact_405_Least__equality,axiom,
! [P: ( set_o > $o ) > $o,X3: set_o > $o] :
( ( P @ X3 )
=> ( ! [Y4: set_o > $o] :
( ( P @ Y4 )
=> ( ord_less_eq_set_o_o @ X3 @ Y4 ) )
=> ( ( ord_Least_set_o_o @ P )
= X3 ) ) ) ).
% Least_equality
thf(fact_406_Least__equality,axiom,
! [P: ( a > $o ) > $o,X3: a > $o] :
( ( P @ X3 )
=> ( ! [Y4: a > $o] :
( ( P @ Y4 )
=> ( ord_less_eq_a_o @ X3 @ Y4 ) )
=> ( ( ord_Least_a_o @ P )
= X3 ) ) ) ).
% Least_equality
thf(fact_407_Least__equality,axiom,
! [P: a > $o,X3: a] :
( ( P @ X3 )
=> ( ! [Y4: a] :
( ( P @ Y4 )
=> ( ord_less_eq_a @ X3 @ Y4 ) )
=> ( ( ord_Least_a @ P )
= X3 ) ) ) ).
% Least_equality
thf(fact_408_ball__UNIV,axiom,
! [P: a > $o] :
( ( ! [X: a] :
( ( member_a @ X @ top_top_set_a )
=> ( P @ X ) ) )
= ( ! [X5: a] : ( P @ X5 ) ) ) ).
% ball_UNIV
thf(fact_409_ball__UNIV,axiom,
! [P: set_a > $o] :
( ( ! [X: set_a] :
( ( member_set_a @ X @ top_top_set_set_a )
=> ( P @ X ) ) )
= ( ! [X5: set_a] : ( P @ X5 ) ) ) ).
% ball_UNIV
thf(fact_410_ball__UNIV,axiom,
! [P: ( set_o > $o ) > $o] :
( ( ! [X: set_o > $o] :
( ( member_set_o_o @ X @ top_top_set_set_o_o )
=> ( P @ X ) ) )
= ( ! [X5: set_o > $o] : ( P @ X5 ) ) ) ).
% ball_UNIV
thf(fact_411_ball__UNIV,axiom,
! [P: $o > $o] :
( ( ! [X: $o] :
( ( member_o @ X @ top_top_set_o )
=> ( P @ X ) ) )
= ( ! [X5: $o] : ( P @ X5 ) ) ) ).
% ball_UNIV
thf(fact_412_dual__order_Oordering__axioms,axiom,
( ordering_set_o
@ ^ [X: set_o,Y2: set_o] : ( ord_less_eq_set_o @ Y2 @ X )
@ ^ [X: set_o,Y2: set_o] : ( ord_less_set_o @ Y2 @ X ) ) ).
% dual_order.ordering_axioms
thf(fact_413_dual__order_Oordering__axioms,axiom,
( ordering_o
@ ^ [X: $o,Y2: $o] : ( ord_less_eq_o @ Y2 @ X )
@ ^ [X: $o,Y2: $o] : ( ord_less_o @ Y2 @ X ) ) ).
% dual_order.ordering_axioms
thf(fact_414_dual__order_Oordering__axioms,axiom,
( ordering_set_a
@ ^ [X: set_a,Y2: set_a] : ( ord_less_eq_set_a @ Y2 @ X )
@ ^ [X: set_a,Y2: set_a] : ( ord_less_set_a @ Y2 @ X ) ) ).
% dual_order.ordering_axioms
thf(fact_415_dual__order_Oordering__axioms,axiom,
( ordering_o_o_o
@ ^ [X: $o > $o > $o,Y2: $o > $o > $o] : ( ord_less_eq_o_o_o @ Y2 @ X )
@ ^ [X: $o > $o > $o,Y2: $o > $o > $o] : ( ord_less_o_o_o @ Y2 @ X ) ) ).
% dual_order.ordering_axioms
thf(fact_416_dual__order_Oordering__axioms,axiom,
( ordering_o_a
@ ^ [X: $o > a,Y2: $o > a] : ( ord_less_eq_o_a @ Y2 @ X )
@ ^ [X: $o > a,Y2: $o > a] : ( ord_less_o_a @ Y2 @ X ) ) ).
% dual_order.ordering_axioms
thf(fact_417_dual__order_Oordering__axioms,axiom,
( ordering_set_o_o
@ ^ [X: set_o > $o,Y2: set_o > $o] : ( ord_less_eq_set_o_o @ Y2 @ X )
@ ^ [X: set_o > $o,Y2: set_o > $o] : ( ord_less_set_o_o @ Y2 @ X ) ) ).
% dual_order.ordering_axioms
thf(fact_418_dual__order_Oordering__axioms,axiom,
( ordering_a_o
@ ^ [X: a > $o,Y2: a > $o] : ( ord_less_eq_a_o @ Y2 @ X )
@ ^ [X: a > $o,Y2: a > $o] : ( ord_less_a_o @ Y2 @ X ) ) ).
% dual_order.ordering_axioms
thf(fact_419_dual__order_Oordering__axioms,axiom,
( ordering_a
@ ^ [X: a,Y2: a] : ( ord_less_eq_a @ Y2 @ X )
@ ^ [X: a,Y2: a] : ( ord_less_a @ Y2 @ X ) ) ).
% dual_order.ordering_axioms
thf(fact_420_Bleast__def,axiom,
( bleast_set_o
= ( ^ [S: set_set_o,P2: set_o > $o] :
( ord_Least_set_o
@ ^ [X: set_o] :
( ( member_set_o @ X @ S )
& ( P2 @ X ) ) ) ) ) ).
% Bleast_def
thf(fact_421_Bleast__def,axiom,
( bleast_set_a
= ( ^ [S: set_set_a,P2: set_a > $o] :
( ord_Least_set_a
@ ^ [X: set_a] :
( ( member_set_a @ X @ S )
& ( P2 @ X ) ) ) ) ) ).
% Bleast_def
thf(fact_422_Bleast__def,axiom,
( bleast_o
= ( ^ [S: set_o,P2: $o > $o] :
( ord_Least_o
@ ^ [X: $o] :
( ( member_o @ X @ S )
& ( P2 @ X ) ) ) ) ) ).
% Bleast_def
thf(fact_423_Bleast__def,axiom,
( bleast_a
= ( ^ [S: set_a,P2: a > $o] :
( ord_Least_a
@ ^ [X: a] :
( ( member_a @ X @ S )
& ( P2 @ X ) ) ) ) ) ).
% Bleast_def
thf(fact_424_abort__Bleast__def,axiom,
( abort_Bleast_set_o
= ( ^ [S: set_set_o,P2: set_o > $o] :
( ord_Least_set_o
@ ^ [X: set_o] :
( ( member_set_o @ X @ S )
& ( P2 @ X ) ) ) ) ) ).
% abort_Bleast_def
thf(fact_425_abort__Bleast__def,axiom,
( abort_Bleast_set_a
= ( ^ [S: set_set_a,P2: set_a > $o] :
( ord_Least_set_a
@ ^ [X: set_a] :
( ( member_set_a @ X @ S )
& ( P2 @ X ) ) ) ) ) ).
% abort_Bleast_def
thf(fact_426_abort__Bleast__def,axiom,
( abort_Bleast_o
= ( ^ [S: set_o,P2: $o > $o] :
( ord_Least_o
@ ^ [X: $o] :
( ( member_o @ X @ S )
& ( P2 @ X ) ) ) ) ) ).
% abort_Bleast_def
thf(fact_427_abort__Bleast__def,axiom,
( abort_Bleast_a
= ( ^ [S: set_a,P2: a > $o] :
( ord_Least_a
@ ^ [X: a] :
( ( member_a @ X @ S )
& ( P2 @ X ) ) ) ) ) ).
% abort_Bleast_def
thf(fact_428_ordering__top_Oaxioms_I1_J,axiom,
! [Less_eq: a > a > $o,Less: a > a > $o,Top: a] :
( ( ordering_top_a @ Less_eq @ Less @ Top )
=> ( ordering_a @ Less_eq @ Less ) ) ).
% ordering_top.axioms(1)
thf(fact_429_ordering__top_Oaxioms_I1_J,axiom,
! [Less_eq: set_o > set_o > $o,Less: set_o > set_o > $o,Top: set_o] :
( ( ordering_top_set_o @ Less_eq @ Less @ Top )
=> ( ordering_set_o @ Less_eq @ Less ) ) ).
% ordering_top.axioms(1)
thf(fact_430_order_Oordering__axioms,axiom,
ordering_set_o @ ord_less_eq_set_o @ ord_less_set_o ).
% order.ordering_axioms
thf(fact_431_order_Oordering__axioms,axiom,
ordering_o @ ord_less_eq_o @ ord_less_o ).
% order.ordering_axioms
thf(fact_432_order_Oordering__axioms,axiom,
ordering_set_a @ ord_less_eq_set_a @ ord_less_set_a ).
% order.ordering_axioms
thf(fact_433_order_Oordering__axioms,axiom,
ordering_o_o_o @ ord_less_eq_o_o_o @ ord_less_o_o_o ).
% order.ordering_axioms
thf(fact_434_order_Oordering__axioms,axiom,
ordering_o_a @ ord_less_eq_o_a @ ord_less_o_a ).
% order.ordering_axioms
thf(fact_435_order_Oordering__axioms,axiom,
ordering_set_o_o @ ord_less_eq_set_o_o @ ord_less_set_o_o ).
% order.ordering_axioms
thf(fact_436_order_Oordering__axioms,axiom,
ordering_a_o @ ord_less_eq_a_o @ ord_less_a_o ).
% order.ordering_axioms
thf(fact_437_order_Oordering__axioms,axiom,
ordering_a @ ord_less_eq_a @ ord_less_a ).
% order.ordering_axioms
thf(fact_438_DEADID_Opred__mono,axiom,
ord_less_eq_set_o_o @ bNF_pr4134575714447080524_set_o @ bNF_pr4134575714447080524_set_o ).
% DEADID.pred_mono
thf(fact_439_DEADID_Opred__mono,axiom,
ord_less_eq_a_o @ bNF_pred_DEADID_a @ bNF_pred_DEADID_a ).
% DEADID.pred_mono
thf(fact_440_top__apply,axiom,
( top_top_o_o
= ( ^ [X: $o] : top_top_o ) ) ).
% top_apply
thf(fact_441_top__apply,axiom,
( top_top_a_o
= ( ^ [X: a] : top_top_o ) ) ).
% top_apply
thf(fact_442_UNIV__I,axiom,
! [X3: set_o] : ( member_set_o @ X3 @ top_top_set_set_o ) ).
% UNIV_I
thf(fact_443_UNIV__I,axiom,
! [X3: set_a] : ( member_set_a @ X3 @ top_top_set_set_a ) ).
% UNIV_I
thf(fact_444_UNIV__I,axiom,
! [X3: set_o > $o] : ( member_set_o_o @ X3 @ top_top_set_set_o_o ) ).
% UNIV_I
thf(fact_445_UNIV__I,axiom,
! [X3: a] : ( member_a @ X3 @ top_top_set_a ) ).
% UNIV_I
thf(fact_446_UNIV__I,axiom,
! [X3: $o] : ( member_o @ X3 @ top_top_set_o ) ).
% UNIV_I
thf(fact_447_psubsetI,axiom,
! [A4: set_o,B4: set_o] :
( ( ord_less_eq_set_o @ A4 @ B4 )
=> ( ( A4 != B4 )
=> ( ord_less_set_o @ A4 @ B4 ) ) ) ).
% psubsetI
thf(fact_448_psubsetI,axiom,
! [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
=> ( ( A4 != B4 )
=> ( ord_less_set_a @ A4 @ B4 ) ) ) ).
% psubsetI
thf(fact_449_predicate2I,axiom,
! [P: $o > $o > $o,Q: $o > $o > $o] :
( ! [X2: $o,Y4: $o] :
( ( P @ X2 @ Y4 )
=> ( Q @ X2 @ Y4 ) )
=> ( ord_less_eq_o_o_o @ P @ Q ) ) ).
% predicate2I
thf(fact_450_top_Oordering__top__axioms,axiom,
ordering_top_o_o @ ord_less_eq_o_o @ ord_less_o_o @ top_top_o_o ).
% top.ordering_top_axioms
thf(fact_451_top_Oordering__top__axioms,axiom,
orderi5875812994216768367_set_a @ ord_le3724670747650509150_set_a @ ord_less_set_set_a @ top_top_set_set_a ).
% top.ordering_top_axioms
thf(fact_452_top_Oordering__top__axioms,axiom,
orderi1143446957891364042et_o_o @ ord_le4904625296160870427et_o_o @ ord_less_set_set_o_o @ top_top_set_set_o_o ).
% top.ordering_top_axioms
thf(fact_453_top_Oordering__top__axioms,axiom,
ordering_top_o @ ord_less_eq_o @ ord_less_o @ top_top_o ).
% top.ordering_top_axioms
thf(fact_454_top_Oordering__top__axioms,axiom,
ordering_top_set_a @ ord_less_eq_set_a @ ord_less_set_a @ top_top_set_a ).
% top.ordering_top_axioms
thf(fact_455_top_Oordering__top__axioms,axiom,
ordering_top_o_o_o @ ord_less_eq_o_o_o @ ord_less_o_o_o @ top_top_o_o_o ).
% top.ordering_top_axioms
thf(fact_456_top_Oordering__top__axioms,axiom,
ordering_top_set_o_o @ ord_less_eq_set_o_o @ ord_less_set_o_o @ top_top_set_o_o ).
% top.ordering_top_axioms
thf(fact_457_top_Oordering__top__axioms,axiom,
ordering_top_a_o @ ord_less_eq_a_o @ ord_less_a_o @ top_top_a_o ).
% top.ordering_top_axioms
thf(fact_458_top_Oordering__top__axioms,axiom,
ordering_top_set_o @ ord_less_eq_set_o @ ord_less_set_o @ top_top_set_o ).
% top.ordering_top_axioms
thf(fact_459_predicate2D,axiom,
! [P: $o > $o > $o,Q: $o > $o > $o,X3: $o,Y: $o] :
( ( ord_less_eq_o_o_o @ P @ Q )
=> ( ( P @ X3 @ Y )
=> ( Q @ X3 @ Y ) ) ) ).
% predicate2D
thf(fact_460_rev__predicate2D,axiom,
! [P: $o > $o > $o,X3: $o,Y: $o,Q: $o > $o > $o] :
( ( P @ X3 @ Y )
=> ( ( ord_less_eq_o_o_o @ P @ Q )
=> ( Q @ X3 @ Y ) ) ) ).
% rev_predicate2D
thf(fact_461_refl__ge__eq,axiom,
! [R: $o > $o > $o] :
( ! [X2: $o] : ( R @ X2 @ X2 )
=> ( ord_less_eq_o_o_o
@ ^ [Y3: $o,Z: $o] : ( Y3 = Z )
@ R ) ) ).
% refl_ge_eq
thf(fact_462_ge__eq__refl,axiom,
! [R: $o > $o > $o,X3: $o] :
( ( ord_less_eq_o_o_o
@ ^ [Y3: $o,Z: $o] : ( Y3 = Z )
@ R )
=> ( R @ X3 @ X3 ) ) ).
% ge_eq_refl
thf(fact_463_verit__comp__simplify1_I1_J,axiom,
! [A: a] :
~ ( ord_less_a @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_464_verit__comp__simplify1_I1_J,axiom,
! [A: set_o] :
~ ( ord_less_set_o @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_465_verit__comp__simplify1_I1_J,axiom,
! [A: $o] :
~ ( ord_less_o @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_466_verit__comp__simplify1_I1_J,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_467_verit__comp__simplify1_I1_J,axiom,
! [A: a > $o] :
~ ( ord_less_a_o @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_468_UNIV__eq__I,axiom,
! [A4: set_set_o] :
( ! [X2: set_o] : ( member_set_o @ X2 @ A4 )
=> ( top_top_set_set_o = A4 ) ) ).
% UNIV_eq_I
thf(fact_469_UNIV__eq__I,axiom,
! [A4: set_set_a] :
( ! [X2: set_a] : ( member_set_a @ X2 @ A4 )
=> ( top_top_set_set_a = A4 ) ) ).
% UNIV_eq_I
thf(fact_470_UNIV__eq__I,axiom,
! [A4: set_set_o_o] :
( ! [X2: set_o > $o] : ( member_set_o_o @ X2 @ A4 )
=> ( top_top_set_set_o_o = A4 ) ) ).
% UNIV_eq_I
thf(fact_471_UNIV__eq__I,axiom,
! [A4: set_a] :
( ! [X2: a] : ( member_a @ X2 @ A4 )
=> ( top_top_set_a = A4 ) ) ).
% UNIV_eq_I
thf(fact_472_UNIV__eq__I,axiom,
! [A4: set_o] :
( ! [X2: $o] : ( member_o @ X2 @ A4 )
=> ( top_top_set_o = A4 ) ) ).
% UNIV_eq_I
thf(fact_473_UNIV__witness,axiom,
? [X2: set_o] : ( member_set_o @ X2 @ top_top_set_set_o ) ).
% UNIV_witness
thf(fact_474_UNIV__witness,axiom,
? [X2: set_a] : ( member_set_a @ X2 @ top_top_set_set_a ) ).
% UNIV_witness
thf(fact_475_UNIV__witness,axiom,
? [X2: set_o > $o] : ( member_set_o_o @ X2 @ top_top_set_set_o_o ) ).
% UNIV_witness
thf(fact_476_UNIV__witness,axiom,
? [X2: a] : ( member_a @ X2 @ top_top_set_a ) ).
% UNIV_witness
thf(fact_477_UNIV__witness,axiom,
? [X2: $o] : ( member_o @ X2 @ top_top_set_o ) ).
% UNIV_witness
thf(fact_478_ordering__top_Oextremum,axiom,
! [Less_eq: set_o > set_o > $o,Less: set_o > set_o > $o,Top: set_o,A: set_o] :
( ( ordering_top_set_o @ Less_eq @ Less @ Top )
=> ( Less_eq @ A @ Top ) ) ).
% ordering_top.extremum
thf(fact_479_ordering__top_Oextremum__strict,axiom,
! [Less_eq: set_o > set_o > $o,Less: set_o > set_o > $o,Top: set_o,A: set_o] :
( ( ordering_top_set_o @ Less_eq @ Less @ Top )
=> ~ ( Less @ Top @ A ) ) ).
% ordering_top.extremum_strict
thf(fact_480_ordering__top_Oextremum__unique,axiom,
! [Less_eq: set_o > set_o > $o,Less: set_o > set_o > $o,Top: set_o,A: set_o] :
( ( ordering_top_set_o @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A )
= ( A = Top ) ) ) ).
% ordering_top.extremum_unique
thf(fact_481_ordering__top_Onot__eq__extremum,axiom,
! [Less_eq: set_o > set_o > $o,Less: set_o > set_o > $o,Top: set_o,A: set_o] :
( ( ordering_top_set_o @ Less_eq @ Less @ Top )
=> ( ( A != Top )
= ( Less @ A @ Top ) ) ) ).
% ordering_top.not_eq_extremum
thf(fact_482_ordering__top_Oextremum__uniqueI,axiom,
! [Less_eq: set_o > set_o > $o,Less: set_o > set_o > $o,Top: set_o,A: set_o] :
( ( ordering_top_set_o @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A )
=> ( A = Top ) ) ) ).
% ordering_top.extremum_uniqueI
thf(fact_483_less__imp__neq,axiom,
! [X3: a,Y: a] :
( ( ord_less_a @ X3 @ Y )
=> ( X3 != Y ) ) ).
% less_imp_neq
thf(fact_484_less__imp__neq,axiom,
! [X3: set_o,Y: set_o] :
( ( ord_less_set_o @ X3 @ Y )
=> ( X3 != Y ) ) ).
% less_imp_neq
thf(fact_485_less__imp__neq,axiom,
! [X3: $o,Y: $o] :
( ( ord_less_o @ X3 @ Y )
=> ( X3 != Y ) ) ).
% less_imp_neq
thf(fact_486_less__imp__neq,axiom,
! [X3: set_a,Y: set_a] :
( ( ord_less_set_a @ X3 @ Y )
=> ( X3 != Y ) ) ).
% less_imp_neq
thf(fact_487_less__imp__neq,axiom,
! [X3: a > $o,Y: a > $o] :
( ( ord_less_a_o @ X3 @ Y )
=> ( X3 != Y ) ) ).
% less_imp_neq
thf(fact_488_order_Oasym,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ B )
=> ~ ( ord_less_a @ B @ A ) ) ).
% order.asym
thf(fact_489_order_Oasym,axiom,
! [A: set_o,B: set_o] :
( ( ord_less_set_o @ A @ B )
=> ~ ( ord_less_set_o @ B @ A ) ) ).
% order.asym
thf(fact_490_order_Oasym,axiom,
! [A: $o,B: $o] :
( ( ord_less_o @ A @ B )
=> ~ ( ord_less_o @ B @ A ) ) ).
% order.asym
thf(fact_491_order_Oasym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ~ ( ord_less_set_a @ B @ A ) ) ).
% order.asym
thf(fact_492_order_Oasym,axiom,
! [A: a > $o,B: a > $o] :
( ( ord_less_a_o @ A @ B )
=> ~ ( ord_less_a_o @ B @ A ) ) ).
% order.asym
thf(fact_493_ord__eq__less__trans,axiom,
! [A: a,B: a,C: a] :
( ( A = B )
=> ( ( ord_less_a @ B @ C )
=> ( ord_less_a @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_494_ord__eq__less__trans,axiom,
! [A: set_o,B: set_o,C: set_o] :
( ( A = B )
=> ( ( ord_less_set_o @ B @ C )
=> ( ord_less_set_o @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_495_ord__eq__less__trans,axiom,
! [A: $o,B: $o,C: $o] :
( ( A = B )
=> ( ( ord_less_o @ B @ C )
=> ( ord_less_o @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_496_ord__eq__less__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( A = B )
=> ( ( ord_less_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_497_ord__eq__less__trans,axiom,
! [A: a > $o,B: a > $o,C: a > $o] :
( ( A = B )
=> ( ( ord_less_a_o @ B @ C )
=> ( ord_less_a_o @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_498_ord__less__eq__trans,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_a @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_499_ord__less__eq__trans,axiom,
! [A: set_o,B: set_o,C: set_o] :
( ( ord_less_set_o @ A @ B )
=> ( ( B = C )
=> ( ord_less_set_o @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_500_ord__less__eq__trans,axiom,
! [A: $o,B: $o,C: $o] :
( ( ord_less_o @ A @ B )
=> ( ( B = C )
=> ( ord_less_o @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_501_ord__less__eq__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_502_ord__less__eq__trans,axiom,
! [A: a > $o,B: a > $o,C: a > $o] :
( ( ord_less_a_o @ A @ B )
=> ( ( B = C )
=> ( ord_less_a_o @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_503_antisym__conv3,axiom,
! [Y: a,X3: a] :
( ~ ( ord_less_a @ Y @ X3 )
=> ( ( ~ ( ord_less_a @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv3
thf(fact_504_antisym__conv3,axiom,
! [Y: $o,X3: $o] :
( ~ ( ord_less_o @ Y @ X3 )
=> ( ( ~ ( ord_less_o @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv3
thf(fact_505_linorder__cases,axiom,
! [X3: a,Y: a] :
( ~ ( ord_less_a @ X3 @ Y )
=> ( ( X3 != Y )
=> ( ord_less_a @ Y @ X3 ) ) ) ).
% linorder_cases
thf(fact_506_linorder__cases,axiom,
! [X3: $o,Y: $o] :
( ~ ( ord_less_o @ X3 @ Y )
=> ( ( X3 = (~ Y) )
=> ( ord_less_o @ Y @ X3 ) ) ) ).
% linorder_cases
thf(fact_507_dual__order_Oasym,axiom,
! [B: a,A: a] :
( ( ord_less_a @ B @ A )
=> ~ ( ord_less_a @ A @ B ) ) ).
% dual_order.asym
thf(fact_508_dual__order_Oasym,axiom,
! [B: set_o,A: set_o] :
( ( ord_less_set_o @ B @ A )
=> ~ ( ord_less_set_o @ A @ B ) ) ).
% dual_order.asym
thf(fact_509_dual__order_Oasym,axiom,
! [B: $o,A: $o] :
( ( ord_less_o @ B @ A )
=> ~ ( ord_less_o @ A @ B ) ) ).
% dual_order.asym
thf(fact_510_dual__order_Oasym,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_set_a @ B @ A )
=> ~ ( ord_less_set_a @ A @ B ) ) ).
% dual_order.asym
thf(fact_511_dual__order_Oasym,axiom,
! [B: a > $o,A: a > $o] :
( ( ord_less_a_o @ B @ A )
=> ~ ( ord_less_a_o @ A @ B ) ) ).
% dual_order.asym
thf(fact_512_dual__order_Oirrefl,axiom,
! [A: a] :
~ ( ord_less_a @ A @ A ) ).
% dual_order.irrefl
thf(fact_513_dual__order_Oirrefl,axiom,
! [A: set_o] :
~ ( ord_less_set_o @ A @ A ) ).
% dual_order.irrefl
thf(fact_514_dual__order_Oirrefl,axiom,
! [A: $o] :
~ ( ord_less_o @ A @ A ) ).
% dual_order.irrefl
thf(fact_515_dual__order_Oirrefl,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ A @ A ) ).
% dual_order.irrefl
thf(fact_516_dual__order_Oirrefl,axiom,
! [A: a > $o] :
~ ( ord_less_a_o @ A @ A ) ).
% dual_order.irrefl
thf(fact_517_linorder__less__wlog,axiom,
! [P: a > a > $o,A: a,B: a] :
( ! [A3: a,B3: a] :
( ( ord_less_a @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: a] : ( P @ A3 @ A3 )
=> ( ! [A3: a,B3: a] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_518_linorder__less__wlog,axiom,
! [P: $o > $o > $o,A: $o,B: $o] :
( ! [A3: $o,B3: $o] :
( ( ord_less_o @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: $o] : ( P @ A3 @ A3 )
=> ( ! [A3: $o,B3: $o] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_519_order_Ostrict__trans,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_a @ B @ C )
=> ( ord_less_a @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_520_order_Ostrict__trans,axiom,
! [A: set_o,B: set_o,C: set_o] :
( ( ord_less_set_o @ A @ B )
=> ( ( ord_less_set_o @ B @ C )
=> ( ord_less_set_o @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_521_order_Ostrict__trans,axiom,
! [A: $o,B: $o,C: $o] :
( ( ord_less_o @ A @ B )
=> ( ( ord_less_o @ B @ C )
=> ( ord_less_o @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_522_order_Ostrict__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_523_order_Ostrict__trans,axiom,
! [A: a > $o,B: a > $o,C: a > $o] :
( ( ord_less_a_o @ A @ B )
=> ( ( ord_less_a_o @ B @ C )
=> ( ord_less_a_o @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_524_top_Oextremum__strict,axiom,
! [A: $o > $o] :
~ ( ord_less_o_o @ top_top_o_o @ A ) ).
% top.extremum_strict
thf(fact_525_top_Oextremum__strict,axiom,
! [A: set_set_a] :
~ ( ord_less_set_set_a @ top_top_set_set_a @ A ) ).
% top.extremum_strict
thf(fact_526_top_Oextremum__strict,axiom,
! [A: set_set_o_o] :
~ ( ord_less_set_set_o_o @ top_top_set_set_o_o @ A ) ).
% top.extremum_strict
thf(fact_527_top_Oextremum__strict,axiom,
! [A: $o] :
~ ( ord_less_o @ top_top_o @ A ) ).
% top.extremum_strict
thf(fact_528_top_Oextremum__strict,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ top_top_set_a @ A ) ).
% top.extremum_strict
thf(fact_529_top_Oextremum__strict,axiom,
! [A: a > $o] :
~ ( ord_less_a_o @ top_top_a_o @ A ) ).
% top.extremum_strict
thf(fact_530_top_Oextremum__strict,axiom,
! [A: set_o] :
~ ( ord_less_set_o @ top_top_set_o @ A ) ).
% top.extremum_strict
thf(fact_531_top_Onot__eq__extremum,axiom,
! [A: $o > $o] :
( ( A != top_top_o_o )
= ( ord_less_o_o @ A @ top_top_o_o ) ) ).
% top.not_eq_extremum
thf(fact_532_top_Onot__eq__extremum,axiom,
! [A: set_set_a] :
( ( A != top_top_set_set_a )
= ( ord_less_set_set_a @ A @ top_top_set_set_a ) ) ).
% top.not_eq_extremum
thf(fact_533_top_Onot__eq__extremum,axiom,
! [A: set_set_o_o] :
( ( A != top_top_set_set_o_o )
= ( ord_less_set_set_o_o @ A @ top_top_set_set_o_o ) ) ).
% top.not_eq_extremum
thf(fact_534_top_Onot__eq__extremum,axiom,
! [A: $o] :
( ( A != top_top_o )
= ( ord_less_o @ A @ top_top_o ) ) ).
% top.not_eq_extremum
thf(fact_535_top_Onot__eq__extremum,axiom,
! [A: set_a] :
( ( A != top_top_set_a )
= ( ord_less_set_a @ A @ top_top_set_a ) ) ).
% top.not_eq_extremum
thf(fact_536_top_Onot__eq__extremum,axiom,
! [A: a > $o] :
( ( A != top_top_a_o )
= ( ord_less_a_o @ A @ top_top_a_o ) ) ).
% top.not_eq_extremum
thf(fact_537_top_Onot__eq__extremum,axiom,
! [A: set_o] :
( ( A != top_top_set_o )
= ( ord_less_set_o @ A @ top_top_set_o ) ) ).
% top.not_eq_extremum
thf(fact_538_not__less__iff__gr__or__eq,axiom,
! [X3: a,Y: a] :
( ( ~ ( ord_less_a @ X3 @ Y ) )
= ( ( ord_less_a @ Y @ X3 )
| ( X3 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_539_not__less__iff__gr__or__eq,axiom,
! [X3: $o,Y: $o] :
( ( ~ ( ord_less_o @ X3 @ Y ) )
= ( ( ord_less_o @ Y @ X3 )
| ( X3 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_540_dual__order_Ostrict__trans,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_a @ B @ A )
=> ( ( ord_less_a @ C @ B )
=> ( ord_less_a @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_541_dual__order_Ostrict__trans,axiom,
! [B: set_o,A: set_o,C: set_o] :
( ( ord_less_set_o @ B @ A )
=> ( ( ord_less_set_o @ C @ B )
=> ( ord_less_set_o @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_542_dual__order_Ostrict__trans,axiom,
! [B: $o,A: $o,C: $o] :
( ( ord_less_o @ B @ A )
=> ( ( ord_less_o @ C @ B )
=> ( ord_less_o @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_543_dual__order_Ostrict__trans,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ( ord_less_set_a @ C @ B )
=> ( ord_less_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_544_dual__order_Ostrict__trans,axiom,
! [B: a > $o,A: a > $o,C: a > $o] :
( ( ord_less_a_o @ B @ A )
=> ( ( ord_less_a_o @ C @ B )
=> ( ord_less_a_o @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_545_order_Ostrict__implies__not__eq,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_546_order_Ostrict__implies__not__eq,axiom,
! [A: set_o,B: set_o] :
( ( ord_less_set_o @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_547_order_Ostrict__implies__not__eq,axiom,
! [A: $o,B: $o] :
( ( ord_less_o @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_548_order_Ostrict__implies__not__eq,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_549_order_Ostrict__implies__not__eq,axiom,
! [A: a > $o,B: a > $o] :
( ( ord_less_a_o @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_550_dual__order_Ostrict__implies__not__eq,axiom,
! [B: a,A: a] :
( ( ord_less_a @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_551_dual__order_Ostrict__implies__not__eq,axiom,
! [B: set_o,A: set_o] :
( ( ord_less_set_o @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_552_dual__order_Ostrict__implies__not__eq,axiom,
! [B: $o,A: $o] :
( ( ord_less_o @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_553_dual__order_Ostrict__implies__not__eq,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_554_dual__order_Ostrict__implies__not__eq,axiom,
! [B: a > $o,A: a > $o] :
( ( ord_less_a_o @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_555_linorder__neqE,axiom,
! [X3: a,Y: a] :
( ( X3 != Y )
=> ( ~ ( ord_less_a @ X3 @ Y )
=> ( ord_less_a @ Y @ X3 ) ) ) ).
% linorder_neqE
thf(fact_556_linorder__neqE,axiom,
! [X3: $o,Y: $o] :
( ( X3 = (~ Y) )
=> ( ~ ( ord_less_o @ X3 @ Y )
=> ( ord_less_o @ Y @ X3 ) ) ) ).
% linorder_neqE
thf(fact_557_order__less__asym,axiom,
! [X3: a,Y: a] :
( ( ord_less_a @ X3 @ Y )
=> ~ ( ord_less_a @ Y @ X3 ) ) ).
% order_less_asym
thf(fact_558_order__less__asym,axiom,
! [X3: set_o,Y: set_o] :
( ( ord_less_set_o @ X3 @ Y )
=> ~ ( ord_less_set_o @ Y @ X3 ) ) ).
% order_less_asym
thf(fact_559_order__less__asym,axiom,
! [X3: $o,Y: $o] :
( ( ord_less_o @ X3 @ Y )
=> ~ ( ord_less_o @ Y @ X3 ) ) ).
% order_less_asym
thf(fact_560_order__less__asym,axiom,
! [X3: set_a,Y: set_a] :
( ( ord_less_set_a @ X3 @ Y )
=> ~ ( ord_less_set_a @ Y @ X3 ) ) ).
% order_less_asym
thf(fact_561_order__less__asym,axiom,
! [X3: a > $o,Y: a > $o] :
( ( ord_less_a_o @ X3 @ Y )
=> ~ ( ord_less_a_o @ Y @ X3 ) ) ).
% order_less_asym
thf(fact_562_linorder__neq__iff,axiom,
! [X3: a,Y: a] :
( ( X3 != Y )
= ( ( ord_less_a @ X3 @ Y )
| ( ord_less_a @ Y @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_563_linorder__neq__iff,axiom,
! [X3: $o,Y: $o] :
( ( X3 != Y )
= ( ( ord_less_o @ X3 @ Y )
| ( ord_less_o @ Y @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_564_order__less__asym_H,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ B )
=> ~ ( ord_less_a @ B @ A ) ) ).
% order_less_asym'
thf(fact_565_order__less__asym_H,axiom,
! [A: set_o,B: set_o] :
( ( ord_less_set_o @ A @ B )
=> ~ ( ord_less_set_o @ B @ A ) ) ).
% order_less_asym'
thf(fact_566_order__less__asym_H,axiom,
! [A: $o,B: $o] :
( ( ord_less_o @ A @ B )
=> ~ ( ord_less_o @ B @ A ) ) ).
% order_less_asym'
thf(fact_567_order__less__asym_H,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ~ ( ord_less_set_a @ B @ A ) ) ).
% order_less_asym'
thf(fact_568_order__less__asym_H,axiom,
! [A: a > $o,B: a > $o] :
( ( ord_less_a_o @ A @ B )
=> ~ ( ord_less_a_o @ B @ A ) ) ).
% order_less_asym'
thf(fact_569_order__less__trans,axiom,
! [X3: a,Y: a,Z3: a] :
( ( ord_less_a @ X3 @ Y )
=> ( ( ord_less_a @ Y @ Z3 )
=> ( ord_less_a @ X3 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_570_order__less__trans,axiom,
! [X3: set_o,Y: set_o,Z3: set_o] :
( ( ord_less_set_o @ X3 @ Y )
=> ( ( ord_less_set_o @ Y @ Z3 )
=> ( ord_less_set_o @ X3 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_571_order__less__trans,axiom,
! [X3: $o,Y: $o,Z3: $o] :
( ( ord_less_o @ X3 @ Y )
=> ( ( ord_less_o @ Y @ Z3 )
=> ( ord_less_o @ X3 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_572_order__less__trans,axiom,
! [X3: set_a,Y: set_a,Z3: set_a] :
( ( ord_less_set_a @ X3 @ Y )
=> ( ( ord_less_set_a @ Y @ Z3 )
=> ( ord_less_set_a @ X3 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_573_order__less__trans,axiom,
! [X3: a > $o,Y: a > $o,Z3: a > $o] :
( ( ord_less_a_o @ X3 @ Y )
=> ( ( ord_less_a_o @ Y @ Z3 )
=> ( ord_less_a_o @ X3 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_574_ord__eq__less__subst,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_a @ X2 @ Y4 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_575_ord__eq__less__subst,axiom,
! [A: $o,F: a > $o,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_a @ X2 @ Y4 )
=> ( ord_less_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_o @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_576_ord__eq__less__subst,axiom,
! [A: a,F: $o > a,B: $o,C: $o] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_o @ B @ C )
=> ( ! [X2: $o,Y4: $o] :
( ( ord_less_o @ X2 @ Y4 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_577_ord__eq__less__subst,axiom,
! [A: $o,F: $o > $o,B: $o,C: $o] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_o @ B @ C )
=> ( ! [X2: $o,Y4: $o] :
( ( ord_less_o @ X2 @ Y4 )
=> ( ord_less_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_o @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_578_ord__eq__less__subst,axiom,
! [A: set_o,F: a > set_o,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_a @ X2 @ Y4 )
=> ( ord_less_set_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_o @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_579_ord__eq__less__subst,axiom,
! [A: set_a,F: a > set_a,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_a @ X2 @ Y4 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_580_ord__eq__less__subst,axiom,
! [A: a,F: set_o > a,B: set_o,C: set_o] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_set_o @ B @ C )
=> ( ! [X2: set_o,Y4: set_o] :
( ( ord_less_set_o @ X2 @ Y4 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_581_ord__eq__less__subst,axiom,
! [A: $o,F: set_o > $o,B: set_o,C: set_o] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_set_o @ B @ C )
=> ( ! [X2: set_o,Y4: set_o] :
( ( ord_less_set_o @ X2 @ Y4 )
=> ( ord_less_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_o @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_582_ord__eq__less__subst,axiom,
! [A: set_o,F: $o > set_o,B: $o,C: $o] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_o @ B @ C )
=> ( ! [X2: $o,Y4: $o] :
( ( ord_less_o @ X2 @ Y4 )
=> ( ord_less_set_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_o @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_583_ord__eq__less__subst,axiom,
! [A: set_a,F: $o > set_a,B: $o,C: $o] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_o @ B @ C )
=> ( ! [X2: $o,Y4: $o] :
( ( ord_less_o @ X2 @ Y4 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_584_ord__less__eq__subst,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_a @ X2 @ Y4 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_585_ord__less__eq__subst,axiom,
! [A: a,B: a,F: a > $o,C: $o] :
( ( ord_less_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_a @ X2 @ Y4 )
=> ( ord_less_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_o @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_586_ord__less__eq__subst,axiom,
! [A: $o,B: $o,F: $o > a,C: a] :
( ( ord_less_o @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: $o,Y4: $o] :
( ( ord_less_o @ X2 @ Y4 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_587_ord__less__eq__subst,axiom,
! [A: $o,B: $o,F: $o > $o,C: $o] :
( ( ord_less_o @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: $o,Y4: $o] :
( ( ord_less_o @ X2 @ Y4 )
=> ( ord_less_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_o @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_588_ord__less__eq__subst,axiom,
! [A: a,B: a,F: a > set_o,C: set_o] :
( ( ord_less_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_a @ X2 @ Y4 )
=> ( ord_less_set_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_o @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_589_ord__less__eq__subst,axiom,
! [A: a,B: a,F: a > set_a,C: set_a] :
( ( ord_less_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_a @ X2 @ Y4 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_590_ord__less__eq__subst,axiom,
! [A: set_o,B: set_o,F: set_o > a,C: a] :
( ( ord_less_set_o @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_o,Y4: set_o] :
( ( ord_less_set_o @ X2 @ Y4 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_591_ord__less__eq__subst,axiom,
! [A: set_o,B: set_o,F: set_o > $o,C: $o] :
( ( ord_less_set_o @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_o,Y4: set_o] :
( ( ord_less_set_o @ X2 @ Y4 )
=> ( ord_less_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_o @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_592_ord__less__eq__subst,axiom,
! [A: $o,B: $o,F: $o > set_o,C: set_o] :
( ( ord_less_o @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: $o,Y4: $o] :
( ( ord_less_o @ X2 @ Y4 )
=> ( ord_less_set_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_o @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_593_ord__less__eq__subst,axiom,
! [A: $o,B: $o,F: $o > set_a,C: set_a] :
( ( ord_less_o @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: $o,Y4: $o] :
( ( ord_less_o @ X2 @ Y4 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_594_order__less__irrefl,axiom,
! [X3: a] :
~ ( ord_less_a @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_595_order__less__irrefl,axiom,
! [X3: set_o] :
~ ( ord_less_set_o @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_596_order__less__irrefl,axiom,
! [X3: $o] :
~ ( ord_less_o @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_597_order__less__irrefl,axiom,
! [X3: set_a] :
~ ( ord_less_set_a @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_598_order__less__irrefl,axiom,
! [X3: a > $o] :
~ ( ord_less_a_o @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_599_order__less__subst1,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( ord_less_a @ A @ ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_a @ X2 @ Y4 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_600_order__less__subst1,axiom,
! [A: a,F: $o > a,B: $o,C: $o] :
( ( ord_less_a @ A @ ( F @ B ) )
=> ( ( ord_less_o @ B @ C )
=> ( ! [X2: $o,Y4: $o] :
( ( ord_less_o @ X2 @ Y4 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_601_order__less__subst1,axiom,
! [A: $o,F: a > $o,B: a,C: a] :
( ( ord_less_o @ A @ ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_a @ X2 @ Y4 )
=> ( ord_less_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_o @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_602_order__less__subst1,axiom,
! [A: $o,F: $o > $o,B: $o,C: $o] :
( ( ord_less_o @ A @ ( F @ B ) )
=> ( ( ord_less_o @ B @ C )
=> ( ! [X2: $o,Y4: $o] :
( ( ord_less_o @ X2 @ Y4 )
=> ( ord_less_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_o @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_603_order__less__subst1,axiom,
! [A: a,F: set_o > a,B: set_o,C: set_o] :
( ( ord_less_a @ A @ ( F @ B ) )
=> ( ( ord_less_set_o @ B @ C )
=> ( ! [X2: set_o,Y4: set_o] :
( ( ord_less_set_o @ X2 @ Y4 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_604_order__less__subst1,axiom,
! [A: a,F: set_a > a,B: set_a,C: set_a] :
( ( ord_less_a @ A @ ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_set_a @ X2 @ Y4 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_605_order__less__subst1,axiom,
! [A: set_o,F: a > set_o,B: a,C: a] :
( ( ord_less_set_o @ A @ ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_a @ X2 @ Y4 )
=> ( ord_less_set_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_o @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_606_order__less__subst1,axiom,
! [A: set_o,F: $o > set_o,B: $o,C: $o] :
( ( ord_less_set_o @ A @ ( F @ B ) )
=> ( ( ord_less_o @ B @ C )
=> ( ! [X2: $o,Y4: $o] :
( ( ord_less_o @ X2 @ Y4 )
=> ( ord_less_set_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_o @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_607_order__less__subst1,axiom,
! [A: $o,F: set_o > $o,B: set_o,C: set_o] :
( ( ord_less_o @ A @ ( F @ B ) )
=> ( ( ord_less_set_o @ B @ C )
=> ( ! [X2: set_o,Y4: set_o] :
( ( ord_less_set_o @ X2 @ Y4 )
=> ( ord_less_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_o @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_608_order__less__subst1,axiom,
! [A: $o,F: set_a > $o,B: set_a,C: set_a] :
( ( ord_less_o @ A @ ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_set_a @ X2 @ Y4 )
=> ( ord_less_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_o @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_609_order__less__subst2,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_a @ ( F @ B ) @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_a @ X2 @ Y4 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_610_order__less__subst2,axiom,
! [A: a,B: a,F: a > $o,C: $o] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_o @ ( F @ B ) @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_a @ X2 @ Y4 )
=> ( ord_less_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_o @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_611_order__less__subst2,axiom,
! [A: $o,B: $o,F: $o > a,C: a] :
( ( ord_less_o @ A @ B )
=> ( ( ord_less_a @ ( F @ B ) @ C )
=> ( ! [X2: $o,Y4: $o] :
( ( ord_less_o @ X2 @ Y4 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_612_order__less__subst2,axiom,
! [A: $o,B: $o,F: $o > $o,C: $o] :
( ( ord_less_o @ A @ B )
=> ( ( ord_less_o @ ( F @ B ) @ C )
=> ( ! [X2: $o,Y4: $o] :
( ( ord_less_o @ X2 @ Y4 )
=> ( ord_less_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_o @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_613_order__less__subst2,axiom,
! [A: a,B: a,F: a > set_o,C: set_o] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_set_o @ ( F @ B ) @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_a @ X2 @ Y4 )
=> ( ord_less_set_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_o @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_614_order__less__subst2,axiom,
! [A: a,B: a,F: a > set_a,C: set_a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_a @ X2 @ Y4 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_615_order__less__subst2,axiom,
! [A: set_o,B: set_o,F: set_o > a,C: a] :
( ( ord_less_set_o @ A @ B )
=> ( ( ord_less_a @ ( F @ B ) @ C )
=> ( ! [X2: set_o,Y4: set_o] :
( ( ord_less_set_o @ X2 @ Y4 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_616_order__less__subst2,axiom,
! [A: set_o,B: set_o,F: set_o > $o,C: $o] :
( ( ord_less_set_o @ A @ B )
=> ( ( ord_less_o @ ( F @ B ) @ C )
=> ( ! [X2: set_o,Y4: set_o] :
( ( ord_less_set_o @ X2 @ Y4 )
=> ( ord_less_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_o @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_617_order__less__subst2,axiom,
! [A: $o,B: $o,F: $o > set_o,C: set_o] :
( ( ord_less_o @ A @ B )
=> ( ( ord_less_set_o @ ( F @ B ) @ C )
=> ( ! [X2: $o,Y4: $o] :
( ( ord_less_o @ X2 @ Y4 )
=> ( ord_less_set_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_o @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_618_order__less__subst2,axiom,
! [A: $o,B: $o,F: $o > set_a,C: set_a] :
( ( ord_less_o @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X2: $o,Y4: $o] :
( ( ord_less_o @ X2 @ Y4 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_619_order__less__not__sym,axiom,
! [X3: a,Y: a] :
( ( ord_less_a @ X3 @ Y )
=> ~ ( ord_less_a @ Y @ X3 ) ) ).
% order_less_not_sym
thf(fact_620_order__less__not__sym,axiom,
! [X3: set_o,Y: set_o] :
( ( ord_less_set_o @ X3 @ Y )
=> ~ ( ord_less_set_o @ Y @ X3 ) ) ).
% order_less_not_sym
thf(fact_621_order__less__not__sym,axiom,
! [X3: $o,Y: $o] :
( ( ord_less_o @ X3 @ Y )
=> ~ ( ord_less_o @ Y @ X3 ) ) ).
% order_less_not_sym
thf(fact_622_order__less__not__sym,axiom,
! [X3: set_a,Y: set_a] :
( ( ord_less_set_a @ X3 @ Y )
=> ~ ( ord_less_set_a @ Y @ X3 ) ) ).
% order_less_not_sym
thf(fact_623_order__less__not__sym,axiom,
! [X3: a > $o,Y: a > $o] :
( ( ord_less_a_o @ X3 @ Y )
=> ~ ( ord_less_a_o @ Y @ X3 ) ) ).
% order_less_not_sym
thf(fact_624_order__less__imp__triv,axiom,
! [X3: a,Y: a,P: $o] :
( ( ord_less_a @ X3 @ Y )
=> ( ( ord_less_a @ Y @ X3 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_625_order__less__imp__triv,axiom,
! [X3: set_o,Y: set_o,P: $o] :
( ( ord_less_set_o @ X3 @ Y )
=> ( ( ord_less_set_o @ Y @ X3 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_626_order__less__imp__triv,axiom,
! [X3: $o,Y: $o,P: $o] :
( ( ord_less_o @ X3 @ Y )
=> ( ( ord_less_o @ Y @ X3 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_627_order__less__imp__triv,axiom,
! [X3: set_a,Y: set_a,P: $o] :
( ( ord_less_set_a @ X3 @ Y )
=> ( ( ord_less_set_a @ Y @ X3 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_628_order__less__imp__triv,axiom,
! [X3: a > $o,Y: a > $o,P: $o] :
( ( ord_less_a_o @ X3 @ Y )
=> ( ( ord_less_a_o @ Y @ X3 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_629_linorder__less__linear,axiom,
! [X3: a,Y: a] :
( ( ord_less_a @ X3 @ Y )
| ( X3 = Y )
| ( ord_less_a @ Y @ X3 ) ) ).
% linorder_less_linear
thf(fact_630_linorder__less__linear,axiom,
! [X3: $o,Y: $o] :
( ( ord_less_o @ X3 @ Y )
| ( X3 = Y )
| ( ord_less_o @ Y @ X3 ) ) ).
% linorder_less_linear
thf(fact_631_order__less__imp__not__eq,axiom,
! [X3: a,Y: a] :
( ( ord_less_a @ X3 @ Y )
=> ( X3 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_632_order__less__imp__not__eq,axiom,
! [X3: set_o,Y: set_o] :
( ( ord_less_set_o @ X3 @ Y )
=> ( X3 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_633_order__less__imp__not__eq,axiom,
! [X3: $o,Y: $o] :
( ( ord_less_o @ X3 @ Y )
=> ( X3 = (~ Y) ) ) ).
% order_less_imp_not_eq
thf(fact_634_order__less__imp__not__eq,axiom,
! [X3: set_a,Y: set_a] :
( ( ord_less_set_a @ X3 @ Y )
=> ( X3 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_635_order__less__imp__not__eq,axiom,
! [X3: a > $o,Y: a > $o] :
( ( ord_less_a_o @ X3 @ Y )
=> ( X3 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_636_order__less__imp__not__eq2,axiom,
! [X3: a,Y: a] :
( ( ord_less_a @ X3 @ Y )
=> ( Y != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_637_order__less__imp__not__eq2,axiom,
! [X3: set_o,Y: set_o] :
( ( ord_less_set_o @ X3 @ Y )
=> ( Y != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_638_order__less__imp__not__eq2,axiom,
! [X3: $o,Y: $o] :
( ( ord_less_o @ X3 @ Y )
=> ( Y = (~ X3) ) ) ).
% order_less_imp_not_eq2
thf(fact_639_order__less__imp__not__eq2,axiom,
! [X3: set_a,Y: set_a] :
( ( ord_less_set_a @ X3 @ Y )
=> ( Y != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_640_order__less__imp__not__eq2,axiom,
! [X3: a > $o,Y: a > $o] :
( ( ord_less_a_o @ X3 @ Y )
=> ( Y != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_641_order__less__imp__not__less,axiom,
! [X3: a,Y: a] :
( ( ord_less_a @ X3 @ Y )
=> ~ ( ord_less_a @ Y @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_642_order__less__imp__not__less,axiom,
! [X3: set_o,Y: set_o] :
( ( ord_less_set_o @ X3 @ Y )
=> ~ ( ord_less_set_o @ Y @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_643_order__less__imp__not__less,axiom,
! [X3: $o,Y: $o] :
( ( ord_less_o @ X3 @ Y )
=> ~ ( ord_less_o @ Y @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_644_order__less__imp__not__less,axiom,
! [X3: set_a,Y: set_a] :
( ( ord_less_set_a @ X3 @ Y )
=> ~ ( ord_less_set_a @ Y @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_645_order__less__imp__not__less,axiom,
! [X3: a > $o,Y: a > $o] :
( ( ord_less_a_o @ X3 @ Y )
=> ~ ( ord_less_a_o @ Y @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_646_UNIV__def,axiom,
( top_top_set_a
= ( collect_a
@ ^ [X: a] : $true ) ) ).
% UNIV_def
thf(fact_647_UNIV__def,axiom,
( top_top_set_set_a
= ( collect_set_a
@ ^ [X: set_a] : $true ) ) ).
% UNIV_def
thf(fact_648_UNIV__def,axiom,
( top_top_set_set_o_o
= ( collect_set_o_o
@ ^ [X: set_o > $o] : $true ) ) ).
% UNIV_def
thf(fact_649_UNIV__def,axiom,
( top_top_set_o
= ( collect_o
@ ^ [X: $o] : $true ) ) ).
% UNIV_def
thf(fact_650_top_Oextremum__uniqueI,axiom,
! [A: $o > $o] :
( ( ord_less_eq_o_o @ top_top_o_o @ A )
=> ( A = top_top_o_o ) ) ).
% top.extremum_uniqueI
thf(fact_651_top_Oextremum__uniqueI,axiom,
! [A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ top_top_set_set_a @ A )
=> ( A = top_top_set_set_a ) ) ).
% top.extremum_uniqueI
thf(fact_652_top_Oextremum__uniqueI,axiom,
! [A: set_set_o_o] :
( ( ord_le4904625296160870427et_o_o @ top_top_set_set_o_o @ A )
=> ( A = top_top_set_set_o_o ) ) ).
% top.extremum_uniqueI
thf(fact_653_top_Oextremum__uniqueI,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A )
=> ( A = top_top_set_a ) ) ).
% top.extremum_uniqueI
thf(fact_654_top_Oextremum__uniqueI,axiom,
! [A: $o > $o > $o] :
( ( ord_less_eq_o_o_o @ top_top_o_o_o @ A )
=> ( A = top_top_o_o_o ) ) ).
% top.extremum_uniqueI
thf(fact_655_top_Oextremum__uniqueI,axiom,
! [A: set_o > $o] :
( ( ord_less_eq_set_o_o @ top_top_set_o_o @ A )
=> ( A = top_top_set_o_o ) ) ).
% top.extremum_uniqueI
thf(fact_656_top_Oextremum__uniqueI,axiom,
! [A: a > $o] :
( ( ord_less_eq_a_o @ top_top_a_o @ A )
=> ( A = top_top_a_o ) ) ).
% top.extremum_uniqueI
thf(fact_657_top_Oextremum__uniqueI,axiom,
! [A: set_o] :
( ( ord_less_eq_set_o @ top_top_set_o @ A )
=> ( A = top_top_set_o ) ) ).
% top.extremum_uniqueI
thf(fact_658_top_Oextremum__unique,axiom,
! [A: $o > $o] :
( ( ord_less_eq_o_o @ top_top_o_o @ A )
= ( A = top_top_o_o ) ) ).
% top.extremum_unique
thf(fact_659_top_Oextremum__unique,axiom,
! [A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ top_top_set_set_a @ A )
= ( A = top_top_set_set_a ) ) ).
% top.extremum_unique
thf(fact_660_top_Oextremum__unique,axiom,
! [A: set_set_o_o] :
( ( ord_le4904625296160870427et_o_o @ top_top_set_set_o_o @ A )
= ( A = top_top_set_set_o_o ) ) ).
% top.extremum_unique
thf(fact_661_top_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A )
= ( A = top_top_set_a ) ) ).
% top.extremum_unique
thf(fact_662_top_Oextremum__unique,axiom,
! [A: $o > $o > $o] :
( ( ord_less_eq_o_o_o @ top_top_o_o_o @ A )
= ( A = top_top_o_o_o ) ) ).
% top.extremum_unique
thf(fact_663_top_Oextremum__unique,axiom,
! [A: set_o > $o] :
( ( ord_less_eq_set_o_o @ top_top_set_o_o @ A )
= ( A = top_top_set_o_o ) ) ).
% top.extremum_unique
thf(fact_664_top_Oextremum__unique,axiom,
! [A: a > $o] :
( ( ord_less_eq_a_o @ top_top_a_o @ A )
= ( A = top_top_a_o ) ) ).
% top.extremum_unique
thf(fact_665_top_Oextremum__unique,axiom,
! [A: set_o] :
( ( ord_less_eq_set_o @ top_top_set_o @ A )
= ( A = top_top_set_o ) ) ).
% top.extremum_unique
thf(fact_666_top__greatest,axiom,
! [A: $o > $o] : ( ord_less_eq_o_o @ A @ top_top_o_o ) ).
% top_greatest
thf(fact_667_top__greatest,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ top_top_set_set_a ) ).
% top_greatest
thf(fact_668_top__greatest,axiom,
! [A: set_set_o_o] : ( ord_le4904625296160870427et_o_o @ A @ top_top_set_set_o_o ) ).
% top_greatest
thf(fact_669_top__greatest,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ top_top_set_a ) ).
% top_greatest
thf(fact_670_top__greatest,axiom,
! [A: $o > $o > $o] : ( ord_less_eq_o_o_o @ A @ top_top_o_o_o ) ).
% top_greatest
thf(fact_671_top__greatest,axiom,
! [A: set_o > $o] : ( ord_less_eq_set_o_o @ A @ top_top_set_o_o ) ).
% top_greatest
thf(fact_672_top__greatest,axiom,
! [A: a > $o] : ( ord_less_eq_a_o @ A @ top_top_a_o ) ).
% top_greatest
thf(fact_673_top__greatest,axiom,
! [A: set_o] : ( ord_less_eq_set_o @ A @ top_top_set_o ) ).
% top_greatest
thf(fact_674_order__le__imp__less__or__eq,axiom,
! [X3: set_o,Y: set_o] :
( ( ord_less_eq_set_o @ X3 @ Y )
=> ( ( ord_less_set_o @ X3 @ Y )
| ( X3 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_675_order__le__imp__less__or__eq,axiom,
! [X3: $o,Y: $o] :
( ( ord_less_eq_o @ X3 @ Y )
=> ( ( ord_less_o @ X3 @ Y )
| ( X3 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_676_order__le__imp__less__or__eq,axiom,
! [X3: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y )
=> ( ( ord_less_set_a @ X3 @ Y )
| ( X3 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_677_order__le__imp__less__or__eq,axiom,
! [X3: $o > $o > $o,Y: $o > $o > $o] :
( ( ord_less_eq_o_o_o @ X3 @ Y )
=> ( ( ord_less_o_o_o @ X3 @ Y )
| ( X3 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_678_order__le__imp__less__or__eq,axiom,
! [X3: $o > a,Y: $o > a] :
( ( ord_less_eq_o_a @ X3 @ Y )
=> ( ( ord_less_o_a @ X3 @ Y )
| ( X3 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_679_order__le__imp__less__or__eq,axiom,
! [X3: set_o > $o,Y: set_o > $o] :
( ( ord_less_eq_set_o_o @ X3 @ Y )
=> ( ( ord_less_set_o_o @ X3 @ Y )
| ( X3 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_680_order__le__imp__less__or__eq,axiom,
! [X3: a > $o,Y: a > $o] :
( ( ord_less_eq_a_o @ X3 @ Y )
=> ( ( ord_less_a_o @ X3 @ Y )
| ( X3 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_681_order__le__imp__less__or__eq,axiom,
! [X3: a,Y: a] :
( ( ord_less_eq_a @ X3 @ Y )
=> ( ( ord_less_a @ X3 @ Y )
| ( X3 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_682_linorder__le__less__linear,axiom,
! [X3: $o,Y: $o] :
( ( ord_less_eq_o @ X3 @ Y )
| ( ord_less_o @ Y @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_683_linorder__le__less__linear,axiom,
! [X3: a,Y: a] :
( ( ord_less_eq_a @ X3 @ Y )
| ( ord_less_a @ Y @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_684_order__less__le__subst2,axiom,
! [A: a,B: a,F: a > $o,C: $o] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_eq_o @ ( F @ B ) @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_a @ X2 @ Y4 )
=> ( ord_less_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_o @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_685_order__less__le__subst2,axiom,
! [A: $o,B: $o,F: $o > $o,C: $o] :
( ( ord_less_o @ A @ B )
=> ( ( ord_less_eq_o @ ( F @ B ) @ C )
=> ( ! [X2: $o,Y4: $o] :
( ( ord_less_o @ X2 @ Y4 )
=> ( ord_less_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_o @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_686_order__less__le__subst2,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_a @ X2 @ Y4 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_687_order__less__le__subst2,axiom,
! [A: $o,B: $o,F: $o > a,C: a] :
( ( ord_less_o @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X2: $o,Y4: $o] :
( ( ord_less_o @ X2 @ Y4 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_688_order__less__le__subst2,axiom,
! [A: a,B: a,F: a > set_o,C: set_o] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_eq_set_o @ ( F @ B ) @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_a @ X2 @ Y4 )
=> ( ord_less_set_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_o @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_689_order__less__le__subst2,axiom,
! [A: set_o,B: set_o,F: set_o > $o,C: $o] :
( ( ord_less_set_o @ A @ B )
=> ( ( ord_less_eq_o @ ( F @ B ) @ C )
=> ( ! [X2: set_o,Y4: set_o] :
( ( ord_less_set_o @ X2 @ Y4 )
=> ( ord_less_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_o @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_690_order__less__le__subst2,axiom,
! [A: $o,B: $o,F: $o > set_o,C: set_o] :
( ( ord_less_o @ A @ B )
=> ( ( ord_less_eq_set_o @ ( F @ B ) @ C )
=> ( ! [X2: $o,Y4: $o] :
( ( ord_less_o @ X2 @ Y4 )
=> ( ord_less_set_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_o @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_691_order__less__le__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > $o,C: $o] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_eq_o @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_set_a @ X2 @ Y4 )
=> ( ord_less_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_o @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_692_order__less__le__subst2,axiom,
! [A: set_o,B: set_o,F: set_o > a,C: a] :
( ( ord_less_set_o @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X2: set_o,Y4: set_o] :
( ( ord_less_set_o @ X2 @ Y4 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_693_order__less__le__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > a,C: a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_set_a @ X2 @ Y4 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_694_order__less__le__subst1,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( ord_less_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_695_order__less__le__subst1,axiom,
! [A: $o,F: a > $o,B: a,C: a] :
( ( ord_less_o @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_less_eq_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_o @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_696_order__less__le__subst1,axiom,
! [A: set_o,F: a > set_o,B: a,C: a] :
( ( ord_less_set_o @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_less_eq_set_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_o @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_697_order__less__le__subst1,axiom,
! [A: set_a,F: a > set_a,B: a,C: a] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_698_order__less__le__subst1,axiom,
! [A: $o,F: set_a > $o,B: set_a,C: set_a] :
( ( ord_less_o @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_o @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_699_order__less__le__subst1,axiom,
! [A: a,F: set_a > a,B: set_a,C: set_a] :
( ( ord_less_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_700_order__less__le__subst1,axiom,
! [A: $o > a,F: a > $o > a,B: a,C: a] :
( ( ord_less_o_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_less_eq_o_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_o_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_701_order__less__le__subst1,axiom,
! [A: a > $o,F: a > a > $o,B: a,C: a] :
( ( ord_less_a_o @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_less_eq_a_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a_o @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_702_order__less__le__subst1,axiom,
! [A: set_o,F: set_a > set_o,B: set_a,C: set_a] :
( ( ord_less_set_o @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_set_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_o @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_703_order__less__le__subst1,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_704_order__le__less__subst2,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_a @ ( F @ B ) @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_705_order__le__less__subst2,axiom,
! [A: a,B: a,F: a > $o,C: $o] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_o @ ( F @ B ) @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_less_eq_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_o @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_706_order__le__less__subst2,axiom,
! [A: a,B: a,F: a > set_o,C: set_o] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_set_o @ ( F @ B ) @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_less_eq_set_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_o @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_707_order__le__less__subst2,axiom,
! [A: a,B: a,F: a > set_a,C: set_a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_708_order__le__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > $o,C: $o] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_o @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_o @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_709_order__le__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > a,C: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_a @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_710_order__le__less__subst2,axiom,
! [A: a,B: a,F: a > $o > a,C: $o > a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_o_a @ ( F @ B ) @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_less_eq_o_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_o_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_711_order__le__less__subst2,axiom,
! [A: a,B: a,F: a > a > $o,C: a > $o] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_a_o @ ( F @ B ) @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_less_eq_a_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a_o @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_712_order__le__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_o,C: set_o] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_set_o @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_set_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_o @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_713_order__le__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_714_order__le__less__subst1,axiom,
! [A: $o,F: a > $o,B: a,C: a] :
( ( ord_less_eq_o @ A @ ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_a @ X2 @ Y4 )
=> ( ord_less_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_o @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_715_order__le__less__subst1,axiom,
! [A: $o,F: $o > $o,B: $o,C: $o] :
( ( ord_less_eq_o @ A @ ( F @ B ) )
=> ( ( ord_less_o @ B @ C )
=> ( ! [X2: $o,Y4: $o] :
( ( ord_less_o @ X2 @ Y4 )
=> ( ord_less_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_o @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_716_order__le__less__subst1,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_a @ X2 @ Y4 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_717_order__le__less__subst1,axiom,
! [A: a,F: $o > a,B: $o,C: $o] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_o @ B @ C )
=> ( ! [X2: $o,Y4: $o] :
( ( ord_less_o @ X2 @ Y4 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_718_order__le__less__subst1,axiom,
! [A: set_o,F: a > set_o,B: a,C: a] :
( ( ord_less_eq_set_o @ A @ ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_a @ X2 @ Y4 )
=> ( ord_less_set_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_o @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_719_order__le__less__subst1,axiom,
! [A: $o,F: set_o > $o,B: set_o,C: set_o] :
( ( ord_less_eq_o @ A @ ( F @ B ) )
=> ( ( ord_less_set_o @ B @ C )
=> ( ! [X2: set_o,Y4: set_o] :
( ( ord_less_set_o @ X2 @ Y4 )
=> ( ord_less_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_o @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_720_order__le__less__subst1,axiom,
! [A: set_o,F: $o > set_o,B: $o,C: $o] :
( ( ord_less_eq_set_o @ A @ ( F @ B ) )
=> ( ( ord_less_o @ B @ C )
=> ( ! [X2: $o,Y4: $o] :
( ( ord_less_o @ X2 @ Y4 )
=> ( ord_less_set_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_o @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_721_order__le__less__subst1,axiom,
! [A: $o,F: set_a > $o,B: set_a,C: set_a] :
( ( ord_less_eq_o @ A @ ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_set_a @ X2 @ Y4 )
=> ( ord_less_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_o @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_722_order__le__less__subst1,axiom,
! [A: a,F: set_o > a,B: set_o,C: set_o] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_set_o @ B @ C )
=> ( ! [X2: set_o,Y4: set_o] :
( ( ord_less_set_o @ X2 @ Y4 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_723_order__le__less__subst1,axiom,
! [A: a,F: set_a > a,B: set_a,C: set_a] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_set_a @ X2 @ Y4 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_724_order__less__le__trans,axiom,
! [X3: set_o,Y: set_o,Z3: set_o] :
( ( ord_less_set_o @ X3 @ Y )
=> ( ( ord_less_eq_set_o @ Y @ Z3 )
=> ( ord_less_set_o @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_725_order__less__le__trans,axiom,
! [X3: $o,Y: $o,Z3: $o] :
( ( ord_less_o @ X3 @ Y )
=> ( ( ord_less_eq_o @ Y @ Z3 )
=> ( ord_less_o @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_726_order__less__le__trans,axiom,
! [X3: set_a,Y: set_a,Z3: set_a] :
( ( ord_less_set_a @ X3 @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z3 )
=> ( ord_less_set_a @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_727_order__less__le__trans,axiom,
! [X3: $o > $o > $o,Y: $o > $o > $o,Z3: $o > $o > $o] :
( ( ord_less_o_o_o @ X3 @ Y )
=> ( ( ord_less_eq_o_o_o @ Y @ Z3 )
=> ( ord_less_o_o_o @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_728_order__less__le__trans,axiom,
! [X3: $o > a,Y: $o > a,Z3: $o > a] :
( ( ord_less_o_a @ X3 @ Y )
=> ( ( ord_less_eq_o_a @ Y @ Z3 )
=> ( ord_less_o_a @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_729_order__less__le__trans,axiom,
! [X3: set_o > $o,Y: set_o > $o,Z3: set_o > $o] :
( ( ord_less_set_o_o @ X3 @ Y )
=> ( ( ord_less_eq_set_o_o @ Y @ Z3 )
=> ( ord_less_set_o_o @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_730_order__less__le__trans,axiom,
! [X3: a > $o,Y: a > $o,Z3: a > $o] :
( ( ord_less_a_o @ X3 @ Y )
=> ( ( ord_less_eq_a_o @ Y @ Z3 )
=> ( ord_less_a_o @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_731_order__less__le__trans,axiom,
! [X3: a,Y: a,Z3: a] :
( ( ord_less_a @ X3 @ Y )
=> ( ( ord_less_eq_a @ Y @ Z3 )
=> ( ord_less_a @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_732_order__le__less__trans,axiom,
! [X3: set_o,Y: set_o,Z3: set_o] :
( ( ord_less_eq_set_o @ X3 @ Y )
=> ( ( ord_less_set_o @ Y @ Z3 )
=> ( ord_less_set_o @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_733_order__le__less__trans,axiom,
! [X3: $o,Y: $o,Z3: $o] :
( ( ord_less_eq_o @ X3 @ Y )
=> ( ( ord_less_o @ Y @ Z3 )
=> ( ord_less_o @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_734_order__le__less__trans,axiom,
! [X3: set_a,Y: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y )
=> ( ( ord_less_set_a @ Y @ Z3 )
=> ( ord_less_set_a @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_735_order__le__less__trans,axiom,
! [X3: $o > $o > $o,Y: $o > $o > $o,Z3: $o > $o > $o] :
( ( ord_less_eq_o_o_o @ X3 @ Y )
=> ( ( ord_less_o_o_o @ Y @ Z3 )
=> ( ord_less_o_o_o @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_736_order__le__less__trans,axiom,
! [X3: $o > a,Y: $o > a,Z3: $o > a] :
( ( ord_less_eq_o_a @ X3 @ Y )
=> ( ( ord_less_o_a @ Y @ Z3 )
=> ( ord_less_o_a @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_737_order__le__less__trans,axiom,
! [X3: set_o > $o,Y: set_o > $o,Z3: set_o > $o] :
( ( ord_less_eq_set_o_o @ X3 @ Y )
=> ( ( ord_less_set_o_o @ Y @ Z3 )
=> ( ord_less_set_o_o @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_738_order__le__less__trans,axiom,
! [X3: a > $o,Y: a > $o,Z3: a > $o] :
( ( ord_less_eq_a_o @ X3 @ Y )
=> ( ( ord_less_a_o @ Y @ Z3 )
=> ( ord_less_a_o @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_739_order__le__less__trans,axiom,
! [X3: a,Y: a,Z3: a] :
( ( ord_less_eq_a @ X3 @ Y )
=> ( ( ord_less_a @ Y @ Z3 )
=> ( ord_less_a @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_740_order__neq__le__trans,axiom,
! [A: set_o,B: set_o] :
( ( A != B )
=> ( ( ord_less_eq_set_o @ A @ B )
=> ( ord_less_set_o @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_741_order__neq__le__trans,axiom,
! [A: $o,B: $o] :
( ( A != B )
=> ( ( ord_less_eq_o @ A @ B )
=> ( ord_less_o @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_742_order__neq__le__trans,axiom,
! [A: set_a,B: set_a] :
( ( A != B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_set_a @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_743_order__neq__le__trans,axiom,
! [A: $o > $o > $o,B: $o > $o > $o] :
( ( A != B )
=> ( ( ord_less_eq_o_o_o @ A @ B )
=> ( ord_less_o_o_o @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_744_order__neq__le__trans,axiom,
! [A: $o > a,B: $o > a] :
( ( A != B )
=> ( ( ord_less_eq_o_a @ A @ B )
=> ( ord_less_o_a @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_745_order__neq__le__trans,axiom,
! [A: set_o > $o,B: set_o > $o] :
( ( A != B )
=> ( ( ord_less_eq_set_o_o @ A @ B )
=> ( ord_less_set_o_o @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_746_order__neq__le__trans,axiom,
! [A: a > $o,B: a > $o] :
( ( A != B )
=> ( ( ord_less_eq_a_o @ A @ B )
=> ( ord_less_a_o @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_747_order__neq__le__trans,axiom,
! [A: a,B: a] :
( ( A != B )
=> ( ( ord_less_eq_a @ A @ B )
=> ( ord_less_a @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_748_order__le__neq__trans,axiom,
! [A: set_o,B: set_o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_o @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_749_order__le__neq__trans,axiom,
! [A: $o,B: $o] :
( ( ord_less_eq_o @ A @ B )
=> ( ( A != B )
=> ( ord_less_o @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_750_order__le__neq__trans,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_a @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_751_order__le__neq__trans,axiom,
! [A: $o > $o > $o,B: $o > $o > $o] :
( ( ord_less_eq_o_o_o @ A @ B )
=> ( ( A != B )
=> ( ord_less_o_o_o @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_752_order__le__neq__trans,axiom,
! [A: $o > a,B: $o > a] :
( ( ord_less_eq_o_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_o_a @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_753_order__le__neq__trans,axiom,
! [A: set_o > $o,B: set_o > $o] :
( ( ord_less_eq_set_o_o @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_o_o @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_754_order__le__neq__trans,axiom,
! [A: a > $o,B: a > $o] :
( ( ord_less_eq_a_o @ A @ B )
=> ( ( A != B )
=> ( ord_less_a_o @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_755_order__le__neq__trans,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_a @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_756_order__less__imp__le,axiom,
! [X3: set_o,Y: set_o] :
( ( ord_less_set_o @ X3 @ Y )
=> ( ord_less_eq_set_o @ X3 @ Y ) ) ).
% order_less_imp_le
thf(fact_757_order__less__imp__le,axiom,
! [X3: $o,Y: $o] :
( ( ord_less_o @ X3 @ Y )
=> ( ord_less_eq_o @ X3 @ Y ) ) ).
% order_less_imp_le
thf(fact_758_order__less__imp__le,axiom,
! [X3: set_a,Y: set_a] :
( ( ord_less_set_a @ X3 @ Y )
=> ( ord_less_eq_set_a @ X3 @ Y ) ) ).
% order_less_imp_le
thf(fact_759_order__less__imp__le,axiom,
! [X3: $o > $o > $o,Y: $o > $o > $o] :
( ( ord_less_o_o_o @ X3 @ Y )
=> ( ord_less_eq_o_o_o @ X3 @ Y ) ) ).
% order_less_imp_le
thf(fact_760_order__less__imp__le,axiom,
! [X3: $o > a,Y: $o > a] :
( ( ord_less_o_a @ X3 @ Y )
=> ( ord_less_eq_o_a @ X3 @ Y ) ) ).
% order_less_imp_le
thf(fact_761_order__less__imp__le,axiom,
! [X3: set_o > $o,Y: set_o > $o] :
( ( ord_less_set_o_o @ X3 @ Y )
=> ( ord_less_eq_set_o_o @ X3 @ Y ) ) ).
% order_less_imp_le
thf(fact_762_order__less__imp__le,axiom,
! [X3: a > $o,Y: a > $o] :
( ( ord_less_a_o @ X3 @ Y )
=> ( ord_less_eq_a_o @ X3 @ Y ) ) ).
% order_less_imp_le
thf(fact_763_order__less__imp__le,axiom,
! [X3: a,Y: a] :
( ( ord_less_a @ X3 @ Y )
=> ( ord_less_eq_a @ X3 @ Y ) ) ).
% order_less_imp_le
thf(fact_764_linorder__not__less,axiom,
! [X3: $o,Y: $o] :
( ( ~ ( ord_less_o @ X3 @ Y ) )
= ( ord_less_eq_o @ Y @ X3 ) ) ).
% linorder_not_less
thf(fact_765_linorder__not__less,axiom,
! [X3: a,Y: a] :
( ( ~ ( ord_less_a @ X3 @ Y ) )
= ( ord_less_eq_a @ Y @ X3 ) ) ).
% linorder_not_less
thf(fact_766_linorder__not__le,axiom,
! [X3: $o,Y: $o] :
( ( ~ ( ord_less_eq_o @ X3 @ Y ) )
= ( ord_less_o @ Y @ X3 ) ) ).
% linorder_not_le
thf(fact_767_linorder__not__le,axiom,
! [X3: a,Y: a] :
( ( ~ ( ord_less_eq_a @ X3 @ Y ) )
= ( ord_less_a @ Y @ X3 ) ) ).
% linorder_not_le
thf(fact_768_order__less__le,axiom,
( ord_less_set_o
= ( ^ [X: set_o,Y2: set_o] :
( ( ord_less_eq_set_o @ X @ Y2 )
& ( X != Y2 ) ) ) ) ).
% order_less_le
thf(fact_769_order__less__le,axiom,
( ord_less_o
= ( ^ [X: $o,Y2: $o] :
( ( ord_less_eq_o @ X @ Y2 )
& ( X != Y2 ) ) ) ) ).
% order_less_le
thf(fact_770_order__less__le,axiom,
( ord_less_set_a
= ( ^ [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
& ( X != Y2 ) ) ) ) ).
% order_less_le
thf(fact_771_order__less__le,axiom,
( ord_less_o_o_o
= ( ^ [X: $o > $o > $o,Y2: $o > $o > $o] :
( ( ord_less_eq_o_o_o @ X @ Y2 )
& ( X != Y2 ) ) ) ) ).
% order_less_le
thf(fact_772_order__less__le,axiom,
( ord_less_o_a
= ( ^ [X: $o > a,Y2: $o > a] :
( ( ord_less_eq_o_a @ X @ Y2 )
& ( X != Y2 ) ) ) ) ).
% order_less_le
thf(fact_773_order__less__le,axiom,
( ord_less_set_o_o
= ( ^ [X: set_o > $o,Y2: set_o > $o] :
( ( ord_less_eq_set_o_o @ X @ Y2 )
& ( X != Y2 ) ) ) ) ).
% order_less_le
thf(fact_774_order__less__le,axiom,
( ord_less_a_o
= ( ^ [X: a > $o,Y2: a > $o] :
( ( ord_less_eq_a_o @ X @ Y2 )
& ( X != Y2 ) ) ) ) ).
% order_less_le
thf(fact_775_order__less__le,axiom,
( ord_less_a
= ( ^ [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
& ( X != Y2 ) ) ) ) ).
% order_less_le
thf(fact_776_order__le__less,axiom,
( ord_less_eq_set_o
= ( ^ [X: set_o,Y2: set_o] :
( ( ord_less_set_o @ X @ Y2 )
| ( X = Y2 ) ) ) ) ).
% order_le_less
thf(fact_777_order__le__less,axiom,
( ord_less_eq_o
= ( ^ [X: $o,Y2: $o] :
( ( ord_less_o @ X @ Y2 )
| ( X = Y2 ) ) ) ) ).
% order_le_less
thf(fact_778_order__le__less,axiom,
( ord_less_eq_set_a
= ( ^ [X: set_a,Y2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
| ( X = Y2 ) ) ) ) ).
% order_le_less
thf(fact_779_order__le__less,axiom,
( ord_less_eq_o_o_o
= ( ^ [X: $o > $o > $o,Y2: $o > $o > $o] :
( ( ord_less_o_o_o @ X @ Y2 )
| ( X = Y2 ) ) ) ) ).
% order_le_less
thf(fact_780_order__le__less,axiom,
( ord_less_eq_o_a
= ( ^ [X: $o > a,Y2: $o > a] :
( ( ord_less_o_a @ X @ Y2 )
| ( X = Y2 ) ) ) ) ).
% order_le_less
thf(fact_781_order__le__less,axiom,
( ord_less_eq_set_o_o
= ( ^ [X: set_o > $o,Y2: set_o > $o] :
( ( ord_less_set_o_o @ X @ Y2 )
| ( X = Y2 ) ) ) ) ).
% order_le_less
thf(fact_782_order__le__less,axiom,
( ord_less_eq_a_o
= ( ^ [X: a > $o,Y2: a > $o] :
( ( ord_less_a_o @ X @ Y2 )
| ( X = Y2 ) ) ) ) ).
% order_le_less
thf(fact_783_order__le__less,axiom,
( ord_less_eq_a
= ( ^ [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
| ( X = Y2 ) ) ) ) ).
% order_le_less
thf(fact_784_dual__order_Ostrict__implies__order,axiom,
! [B: set_o,A: set_o] :
( ( ord_less_set_o @ B @ A )
=> ( ord_less_eq_set_o @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_785_dual__order_Ostrict__implies__order,axiom,
! [B: $o,A: $o] :
( ( ord_less_o @ B @ A )
=> ( ord_less_eq_o @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_786_dual__order_Ostrict__implies__order,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_787_dual__order_Ostrict__implies__order,axiom,
! [B: $o > $o > $o,A: $o > $o > $o] :
( ( ord_less_o_o_o @ B @ A )
=> ( ord_less_eq_o_o_o @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_788_dual__order_Ostrict__implies__order,axiom,
! [B: $o > a,A: $o > a] :
( ( ord_less_o_a @ B @ A )
=> ( ord_less_eq_o_a @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_789_dual__order_Ostrict__implies__order,axiom,
! [B: set_o > $o,A: set_o > $o] :
( ( ord_less_set_o_o @ B @ A )
=> ( ord_less_eq_set_o_o @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_790_dual__order_Ostrict__implies__order,axiom,
! [B: a > $o,A: a > $o] :
( ( ord_less_a_o @ B @ A )
=> ( ord_less_eq_a_o @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_791_dual__order_Ostrict__implies__order,axiom,
! [B: a,A: a] :
( ( ord_less_a @ B @ A )
=> ( ord_less_eq_a @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_792_order_Ostrict__implies__order,axiom,
! [A: set_o,B: set_o] :
( ( ord_less_set_o @ A @ B )
=> ( ord_less_eq_set_o @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_793_order_Ostrict__implies__order,axiom,
! [A: $o,B: $o] :
( ( ord_less_o @ A @ B )
=> ( ord_less_eq_o @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_794_order_Ostrict__implies__order,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_795_order_Ostrict__implies__order,axiom,
! [A: $o > $o > $o,B: $o > $o > $o] :
( ( ord_less_o_o_o @ A @ B )
=> ( ord_less_eq_o_o_o @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_796_order_Ostrict__implies__order,axiom,
! [A: $o > a,B: $o > a] :
( ( ord_less_o_a @ A @ B )
=> ( ord_less_eq_o_a @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_797_order_Ostrict__implies__order,axiom,
! [A: set_o > $o,B: set_o > $o] :
( ( ord_less_set_o_o @ A @ B )
=> ( ord_less_eq_set_o_o @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_798_order_Ostrict__implies__order,axiom,
! [A: a > $o,B: a > $o] :
( ( ord_less_a_o @ A @ B )
=> ( ord_less_eq_a_o @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_799_order_Ostrict__implies__order,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ B )
=> ( ord_less_eq_a @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_800_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_o
= ( ^ [B2: set_o,A2: set_o] :
( ( ord_less_eq_set_o @ B2 @ A2 )
& ~ ( ord_less_eq_set_o @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_801_dual__order_Ostrict__iff__not,axiom,
( ord_less_o
= ( ^ [B2: $o,A2: $o] :
( ( ord_less_eq_o @ B2 @ A2 )
& ~ ( ord_less_eq_o @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_802_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
& ~ ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_803_dual__order_Ostrict__iff__not,axiom,
( ord_less_o_o_o
= ( ^ [B2: $o > $o > $o,A2: $o > $o > $o] :
( ( ord_less_eq_o_o_o @ B2 @ A2 )
& ~ ( ord_less_eq_o_o_o @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_804_dual__order_Ostrict__iff__not,axiom,
( ord_less_o_a
= ( ^ [B2: $o > a,A2: $o > a] :
( ( ord_less_eq_o_a @ B2 @ A2 )
& ~ ( ord_less_eq_o_a @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_805_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_o_o
= ( ^ [B2: set_o > $o,A2: set_o > $o] :
( ( ord_less_eq_set_o_o @ B2 @ A2 )
& ~ ( ord_less_eq_set_o_o @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_806_dual__order_Ostrict__iff__not,axiom,
( ord_less_a_o
= ( ^ [B2: a > $o,A2: a > $o] :
( ( ord_less_eq_a_o @ B2 @ A2 )
& ~ ( ord_less_eq_a_o @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_807_dual__order_Ostrict__iff__not,axiom,
( ord_less_a
= ( ^ [B2: a,A2: a] :
( ( ord_less_eq_a @ B2 @ A2 )
& ~ ( ord_less_eq_a @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_808_dual__order_Ostrict__trans2,axiom,
! [B: set_o,A: set_o,C: set_o] :
( ( ord_less_set_o @ B @ A )
=> ( ( ord_less_eq_set_o @ C @ B )
=> ( ord_less_set_o @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_809_dual__order_Ostrict__trans2,axiom,
! [B: $o,A: $o,C: $o] :
( ( ord_less_o @ B @ A )
=> ( ( ord_less_eq_o @ C @ B )
=> ( ord_less_o @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_810_dual__order_Ostrict__trans2,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_811_dual__order_Ostrict__trans2,axiom,
! [B: $o > $o > $o,A: $o > $o > $o,C: $o > $o > $o] :
( ( ord_less_o_o_o @ B @ A )
=> ( ( ord_less_eq_o_o_o @ C @ B )
=> ( ord_less_o_o_o @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_812_dual__order_Ostrict__trans2,axiom,
! [B: $o > a,A: $o > a,C: $o > a] :
( ( ord_less_o_a @ B @ A )
=> ( ( ord_less_eq_o_a @ C @ B )
=> ( ord_less_o_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_813_dual__order_Ostrict__trans2,axiom,
! [B: set_o > $o,A: set_o > $o,C: set_o > $o] :
( ( ord_less_set_o_o @ B @ A )
=> ( ( ord_less_eq_set_o_o @ C @ B )
=> ( ord_less_set_o_o @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_814_dual__order_Ostrict__trans2,axiom,
! [B: a > $o,A: a > $o,C: a > $o] :
( ( ord_less_a_o @ B @ A )
=> ( ( ord_less_eq_a_o @ C @ B )
=> ( ord_less_a_o @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_815_dual__order_Ostrict__trans2,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_a @ B @ A )
=> ( ( ord_less_eq_a @ C @ B )
=> ( ord_less_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_816_dual__order_Ostrict__trans1,axiom,
! [B: set_o,A: set_o,C: set_o] :
( ( ord_less_eq_set_o @ B @ A )
=> ( ( ord_less_set_o @ C @ B )
=> ( ord_less_set_o @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_817_dual__order_Ostrict__trans1,axiom,
! [B: $o,A: $o,C: $o] :
( ( ord_less_eq_o @ B @ A )
=> ( ( ord_less_o @ C @ B )
=> ( ord_less_o @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_818_dual__order_Ostrict__trans1,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_set_a @ C @ B )
=> ( ord_less_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_819_dual__order_Ostrict__trans1,axiom,
! [B: $o > $o > $o,A: $o > $o > $o,C: $o > $o > $o] :
( ( ord_less_eq_o_o_o @ B @ A )
=> ( ( ord_less_o_o_o @ C @ B )
=> ( ord_less_o_o_o @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_820_dual__order_Ostrict__trans1,axiom,
! [B: $o > a,A: $o > a,C: $o > a] :
( ( ord_less_eq_o_a @ B @ A )
=> ( ( ord_less_o_a @ C @ B )
=> ( ord_less_o_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_821_dual__order_Ostrict__trans1,axiom,
! [B: set_o > $o,A: set_o > $o,C: set_o > $o] :
( ( ord_less_eq_set_o_o @ B @ A )
=> ( ( ord_less_set_o_o @ C @ B )
=> ( ord_less_set_o_o @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_822_dual__order_Ostrict__trans1,axiom,
! [B: a > $o,A: a > $o,C: a > $o] :
( ( ord_less_eq_a_o @ B @ A )
=> ( ( ord_less_a_o @ C @ B )
=> ( ord_less_a_o @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_823_dual__order_Ostrict__trans1,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( ord_less_a @ C @ B )
=> ( ord_less_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_824_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_o
= ( ^ [B2: set_o,A2: set_o] :
( ( ord_less_eq_set_o @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_825_dual__order_Ostrict__iff__order,axiom,
( ord_less_o
= ( ^ [B2: $o,A2: $o] :
( ( ord_less_eq_o @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_826_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_827_dual__order_Ostrict__iff__order,axiom,
( ord_less_o_o_o
= ( ^ [B2: $o > $o > $o,A2: $o > $o > $o] :
( ( ord_less_eq_o_o_o @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_828_dual__order_Ostrict__iff__order,axiom,
( ord_less_o_a
= ( ^ [B2: $o > a,A2: $o > a] :
( ( ord_less_eq_o_a @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_829_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_o_o
= ( ^ [B2: set_o > $o,A2: set_o > $o] :
( ( ord_less_eq_set_o_o @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_830_dual__order_Ostrict__iff__order,axiom,
( ord_less_a_o
= ( ^ [B2: a > $o,A2: a > $o] :
( ( ord_less_eq_a_o @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_831_dual__order_Ostrict__iff__order,axiom,
( ord_less_a
= ( ^ [B2: a,A2: a] :
( ( ord_less_eq_a @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_832_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_o
= ( ^ [B2: set_o,A2: set_o] :
( ( ord_less_set_o @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_833_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_o
= ( ^ [B2: $o,A2: $o] :
( ( ord_less_o @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_834_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [B2: set_a,A2: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_835_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_o_o_o
= ( ^ [B2: $o > $o > $o,A2: $o > $o > $o] :
( ( ord_less_o_o_o @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_836_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_o_a
= ( ^ [B2: $o > a,A2: $o > a] :
( ( ord_less_o_a @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_837_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_o_o
= ( ^ [B2: set_o > $o,A2: set_o > $o] :
( ( ord_less_set_o_o @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_838_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_a_o
= ( ^ [B2: a > $o,A2: a > $o] :
( ( ord_less_a_o @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_839_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_a
= ( ^ [B2: a,A2: a] :
( ( ord_less_a @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_840_order_Ostrict__iff__not,axiom,
( ord_less_set_o
= ( ^ [A2: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
& ~ ( ord_less_eq_set_o @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_841_order_Ostrict__iff__not,axiom,
( ord_less_o
= ( ^ [A2: $o,B2: $o] :
( ( ord_less_eq_o @ A2 @ B2 )
& ~ ( ord_less_eq_o @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_842_order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
& ~ ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_843_order_Ostrict__iff__not,axiom,
( ord_less_o_o_o
= ( ^ [A2: $o > $o > $o,B2: $o > $o > $o] :
( ( ord_less_eq_o_o_o @ A2 @ B2 )
& ~ ( ord_less_eq_o_o_o @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_844_order_Ostrict__iff__not,axiom,
( ord_less_o_a
= ( ^ [A2: $o > a,B2: $o > a] :
( ( ord_less_eq_o_a @ A2 @ B2 )
& ~ ( ord_less_eq_o_a @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_845_order_Ostrict__iff__not,axiom,
( ord_less_set_o_o
= ( ^ [A2: set_o > $o,B2: set_o > $o] :
( ( ord_less_eq_set_o_o @ A2 @ B2 )
& ~ ( ord_less_eq_set_o_o @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_846_order_Ostrict__iff__not,axiom,
( ord_less_a_o
= ( ^ [A2: a > $o,B2: a > $o] :
( ( ord_less_eq_a_o @ A2 @ B2 )
& ~ ( ord_less_eq_a_o @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_847_order_Ostrict__iff__not,axiom,
( ord_less_a
= ( ^ [A2: a,B2: a] :
( ( ord_less_eq_a @ A2 @ B2 )
& ~ ( ord_less_eq_a @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_848_order_Ostrict__trans2,axiom,
! [A: set_o,B: set_o,C: set_o] :
( ( ord_less_set_o @ A @ B )
=> ( ( ord_less_eq_set_o @ B @ C )
=> ( ord_less_set_o @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_849_order_Ostrict__trans2,axiom,
! [A: $o,B: $o,C: $o] :
( ( ord_less_o @ A @ B )
=> ( ( ord_less_eq_o @ B @ C )
=> ( ord_less_o @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_850_order_Ostrict__trans2,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_851_order_Ostrict__trans2,axiom,
! [A: $o > $o > $o,B: $o > $o > $o,C: $o > $o > $o] :
( ( ord_less_o_o_o @ A @ B )
=> ( ( ord_less_eq_o_o_o @ B @ C )
=> ( ord_less_o_o_o @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_852_order_Ostrict__trans2,axiom,
! [A: $o > a,B: $o > a,C: $o > a] :
( ( ord_less_o_a @ A @ B )
=> ( ( ord_less_eq_o_a @ B @ C )
=> ( ord_less_o_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_853_order_Ostrict__trans2,axiom,
! [A: set_o > $o,B: set_o > $o,C: set_o > $o] :
( ( ord_less_set_o_o @ A @ B )
=> ( ( ord_less_eq_set_o_o @ B @ C )
=> ( ord_less_set_o_o @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_854_order_Ostrict__trans2,axiom,
! [A: a > $o,B: a > $o,C: a > $o] :
( ( ord_less_a_o @ A @ B )
=> ( ( ord_less_eq_a_o @ B @ C )
=> ( ord_less_a_o @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_855_order_Ostrict__trans2,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_856_order_Ostrict__trans1,axiom,
! [A: set_o,B: set_o,C: set_o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( ord_less_set_o @ B @ C )
=> ( ord_less_set_o @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_857_order_Ostrict__trans1,axiom,
! [A: $o,B: $o,C: $o] :
( ( ord_less_eq_o @ A @ B )
=> ( ( ord_less_o @ B @ C )
=> ( ord_less_o @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_858_order_Ostrict__trans1,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_859_order_Ostrict__trans1,axiom,
! [A: $o > $o > $o,B: $o > $o > $o,C: $o > $o > $o] :
( ( ord_less_eq_o_o_o @ A @ B )
=> ( ( ord_less_o_o_o @ B @ C )
=> ( ord_less_o_o_o @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_860_order_Ostrict__trans1,axiom,
! [A: $o > a,B: $o > a,C: $o > a] :
( ( ord_less_eq_o_a @ A @ B )
=> ( ( ord_less_o_a @ B @ C )
=> ( ord_less_o_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_861_order_Ostrict__trans1,axiom,
! [A: set_o > $o,B: set_o > $o,C: set_o > $o] :
( ( ord_less_eq_set_o_o @ A @ B )
=> ( ( ord_less_set_o_o @ B @ C )
=> ( ord_less_set_o_o @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_862_order_Ostrict__trans1,axiom,
! [A: a > $o,B: a > $o,C: a > $o] :
( ( ord_less_eq_a_o @ A @ B )
=> ( ( ord_less_a_o @ B @ C )
=> ( ord_less_a_o @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_863_order_Ostrict__trans1,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_a @ B @ C )
=> ( ord_less_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_864_order_Ostrict__iff__order,axiom,
( ord_less_set_o
= ( ^ [A2: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_865_order_Ostrict__iff__order,axiom,
( ord_less_o
= ( ^ [A2: $o,B2: $o] :
( ( ord_less_eq_o @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_866_order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_867_order_Ostrict__iff__order,axiom,
( ord_less_o_o_o
= ( ^ [A2: $o > $o > $o,B2: $o > $o > $o] :
( ( ord_less_eq_o_o_o @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_868_order_Ostrict__iff__order,axiom,
( ord_less_o_a
= ( ^ [A2: $o > a,B2: $o > a] :
( ( ord_less_eq_o_a @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_869_order_Ostrict__iff__order,axiom,
( ord_less_set_o_o
= ( ^ [A2: set_o > $o,B2: set_o > $o] :
( ( ord_less_eq_set_o_o @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_870_order_Ostrict__iff__order,axiom,
( ord_less_a_o
= ( ^ [A2: a > $o,B2: a > $o] :
( ( ord_less_eq_a_o @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_871_order_Ostrict__iff__order,axiom,
( ord_less_a
= ( ^ [A2: a,B2: a] :
( ( ord_less_eq_a @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_872_order_Oorder__iff__strict,axiom,
( ord_less_eq_a_o
= ( ^ [A2: a > $o,B2: a > $o] :
( ( ord_less_a_o @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_873_order_Oorder__iff__strict,axiom,
( ord_less_eq_a
= ( ^ [A2: a,B2: a] :
( ( ord_less_a @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_874_not__le__imp__less,axiom,
! [Y: a,X3: a] :
( ~ ( ord_less_eq_a @ Y @ X3 )
=> ( ord_less_a @ X3 @ Y ) ) ).
% not_le_imp_less
thf(fact_875_less__le__not__le,axiom,
( ord_less_a
= ( ^ [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
& ~ ( ord_less_eq_a @ Y2 @ X ) ) ) ) ).
% less_le_not_le
thf(fact_876_antisym__conv2,axiom,
! [X3: a,Y: a] :
( ( ord_less_eq_a @ X3 @ Y )
=> ( ( ~ ( ord_less_a @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv2
thf(fact_877_antisym__conv1,axiom,
! [X3: a,Y: a] :
( ~ ( ord_less_a @ X3 @ Y )
=> ( ( ord_less_eq_a @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% antisym_conv1
thf(fact_878_nless__le,axiom,
! [A: a,B: a] :
( ( ~ ( ord_less_a @ A @ B ) )
= ( ~ ( ord_less_eq_a @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_879_leI,axiom,
! [X3: a,Y: a] :
( ~ ( ord_less_a @ X3 @ Y )
=> ( ord_less_eq_a @ Y @ X3 ) ) ).
% leI
thf(fact_880_leD,axiom,
! [Y: a,X3: a] :
( ( ord_less_eq_a @ Y @ X3 )
=> ~ ( ord_less_a @ X3 @ Y ) ) ).
% leD
thf(fact_881_verit__comp__simplify1_I3_J,axiom,
! [B6: a,A6: a] :
( ( ~ ( ord_less_eq_a @ B6 @ A6 ) )
= ( ord_less_a @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_882_subset__UNIV,axiom,
! [A4: set_o] : ( ord_less_eq_set_o @ A4 @ top_top_set_o ) ).
% subset_UNIV
thf(fact_883_minf_I8_J,axiom,
! [T2: a] :
? [Z5: a] :
! [X4: a] :
( ( ord_less_a @ X4 @ Z5 )
=> ~ ( ord_less_eq_a @ T2 @ X4 ) ) ).
% minf(8)
thf(fact_884_minf_I6_J,axiom,
! [T2: a] :
? [Z5: a] :
! [X4: a] :
( ( ord_less_a @ X4 @ Z5 )
=> ( ord_less_eq_a @ X4 @ T2 ) ) ).
% minf(6)
thf(fact_885_pinf_I8_J,axiom,
! [T2: a] :
? [Z5: a] :
! [X4: a] :
( ( ord_less_a @ Z5 @ X4 )
=> ( ord_less_eq_a @ T2 @ X4 ) ) ).
% pinf(8)
thf(fact_886_pinf_I6_J,axiom,
! [T2: a] :
? [Z5: a] :
! [X4: a] :
( ( ord_less_a @ Z5 @ X4 )
=> ~ ( ord_less_eq_a @ X4 @ T2 ) ) ).
% pinf(6)
thf(fact_887_sorted__list__of__set_Ofolding__insort__key__axioms,axiom,
( foldin4841961211520082085ey_o_o @ ord_less_eq_o @ ord_less_o @ top_top_set_o
@ ^ [X: $o] : X ) ).
% sorted_list_of_set.folding_insort_key_axioms
thf(fact_888_sorted__list__of__set_Ofolding__insort__key__axioms,axiom,
( foldin4382019238405368997ey_a_a @ ord_less_eq_a @ ord_less_a @ top_top_set_a
@ ^ [X: a] : X ) ).
% sorted_list_of_set.folding_insort_key_axioms
thf(fact_889_psubsetD,axiom,
! [A4: set_a,B4: set_a,C: a] :
( ( ord_less_set_a @ A4 @ B4 )
=> ( ( member_a @ C @ A4 )
=> ( member_a @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_890_less__set__def,axiom,
( ord_less_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ord_less_a_o
@ ^ [X: a] : ( member_a @ X @ A5 )
@ ^ [X: a] : ( member_a @ X @ B5 ) ) ) ) ).
% less_set_def
thf(fact_891_top__empty__eq,axiom,
( top_top_a_o
= ( ^ [X: a] : ( member_a @ X @ top_top_set_a ) ) ) ).
% top_empty_eq
thf(fact_892_top__empty__eq,axiom,
( top_top_o_o
= ( ^ [X: $o] : ( member_o @ X @ top_top_set_o ) ) ) ).
% top_empty_eq
thf(fact_893_top__set__def,axiom,
( top_top_set_o
= ( collect_o @ top_top_o_o ) ) ).
% top_set_def
thf(fact_894_bdd__below_Opreordering__bdd__axioms,axiom,
( condit4103000493307248661_bdd_a
@ ^ [X: a,Y2: a] : ( ord_less_eq_a @ Y2 @ X )
@ ^ [X: a,Y2: a] : ( ord_less_a @ Y2 @ X ) ) ).
% bdd_below.preordering_bdd_axioms
thf(fact_895_bot_Oordering__top__axioms,axiom,
( ordering_top_set_o
@ ^ [X: set_o,Y2: set_o] : ( ord_less_eq_set_o @ Y2 @ X )
@ ^ [X: set_o,Y2: set_o] : ( ord_less_set_o @ Y2 @ X )
@ bot_bot_set_o ) ).
% bot.ordering_top_axioms
thf(fact_896_dual__order_Opreordering__axioms,axiom,
( preordering_a
@ ^ [X: a,Y2: a] : ( ord_less_eq_a @ Y2 @ X )
@ ^ [X: a,Y2: a] : ( ord_less_a @ Y2 @ X ) ) ).
% dual_order.preordering_axioms
thf(fact_897_bdd__above_Opreordering__bdd__axioms,axiom,
condit4103000493307248661_bdd_a @ ord_less_eq_a @ ord_less_a ).
% bdd_above.preordering_bdd_axioms
thf(fact_898_subset__empty,axiom,
! [A4: set_o] :
( ( ord_less_eq_set_o @ A4 @ bot_bot_set_o )
= ( A4 = bot_bot_set_o ) ) ).
% subset_empty
thf(fact_899_empty__subsetI,axiom,
! [A4: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A4 ) ).
% empty_subsetI
thf(fact_900_ball__empty,axiom,
! [P: $o > $o,X4: $o] :
( ( member_o @ X4 @ bot_bot_set_o )
=> ( P @ X4 ) ) ).
% ball_empty
thf(fact_901_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collect_o
@ ^ [S4: $o] : P )
= top_top_set_o ) )
& ( ~ P
=> ( ( collect_o
@ ^ [S4: $o] : P )
= bot_bot_set_o ) ) ) ).
% Collect_const
thf(fact_902_bot_Oextremum__uniqueI,axiom,
! [A: set_o] :
( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
=> ( A = bot_bot_set_o ) ) ).
% bot.extremum_uniqueI
thf(fact_903_bot_Oextremum__unique,axiom,
! [A: set_o] :
( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
= ( A = bot_bot_set_o ) ) ).
% bot.extremum_unique
thf(fact_904_bot_Oextremum,axiom,
! [A: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A ) ).
% bot.extremum
thf(fact_905_bot_Oextremum__strict,axiom,
! [A: set_o] :
~ ( ord_less_set_o @ A @ bot_bot_set_o ) ).
% bot.extremum_strict
thf(fact_906_bot_Onot__eq__extremum,axiom,
! [A: set_o] :
( ( A != bot_bot_set_o )
= ( ord_less_set_o @ bot_bot_set_o @ A ) ) ).
% bot.not_eq_extremum
thf(fact_907_empty__not__UNIV,axiom,
bot_bot_set_o != top_top_set_o ).
% empty_not_UNIV
thf(fact_908_subset__emptyI,axiom,
! [A4: set_a] :
( ! [X2: a] :
~ ( member_a @ X2 @ A4 )
=> ( ord_less_eq_set_a @ A4 @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_909_subset__emptyI,axiom,
! [A4: set_o] :
( ! [X2: $o] :
~ ( member_o @ X2 @ A4 )
=> ( ord_less_eq_set_o @ A4 @ bot_bot_set_o ) ) ).
% subset_emptyI
thf(fact_910_not__psubset__empty,axiom,
! [A4: set_o] :
~ ( ord_less_set_o @ A4 @ bot_bot_set_o ) ).
% not_psubset_empty
thf(fact_911_order_Opreordering__axioms,axiom,
preordering_a @ ord_less_eq_a @ ord_less_a ).
% order.preordering_axioms
thf(fact_912_antisymp__less__eq,axiom,
! [R2: $o > $o > $o,S3: $o > $o > $o] :
( ( ord_less_eq_o_o_o @ R2 @ S3 )
=> ( ( antisymp_on_o @ top_top_set_o @ S3 )
=> ( antisymp_on_o @ top_top_set_o @ R2 ) ) ) ).
% antisymp_less_eq
thf(fact_913_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_914_empty__iff,axiom,
! [C: $o] :
~ ( member_o @ C @ bot_bot_set_o ) ).
% empty_iff
thf(fact_915_all__not__in__conv,axiom,
! [A4: set_a] :
( ( ! [X: a] :
~ ( member_a @ X @ A4 ) )
= ( A4 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_916_all__not__in__conv,axiom,
! [A4: set_o] :
( ( ! [X: $o] :
~ ( member_o @ X @ A4 ) )
= ( A4 = bot_bot_set_o ) ) ).
% all_not_in_conv
thf(fact_917_Collect__empty__eq,axiom,
! [P: $o > $o] :
( ( ( collect_o @ P )
= bot_bot_set_o )
= ( ! [X: $o] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_918_empty__Collect__eq,axiom,
! [P: $o > $o] :
( ( bot_bot_set_o
= ( collect_o @ P ) )
= ( ! [X: $o] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_919_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_920_emptyE,axiom,
! [A: $o] :
~ ( member_o @ A @ bot_bot_set_o ) ).
% emptyE
thf(fact_921_equals0D,axiom,
! [A4: set_a,A: a] :
( ( A4 = bot_bot_set_a )
=> ~ ( member_a @ A @ A4 ) ) ).
% equals0D
thf(fact_922_equals0D,axiom,
! [A4: set_o,A: $o] :
( ( A4 = bot_bot_set_o )
=> ~ ( member_o @ A @ A4 ) ) ).
% equals0D
thf(fact_923_equals0I,axiom,
! [A4: set_a] :
( ! [Y4: a] :
~ ( member_a @ Y4 @ A4 )
=> ( A4 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_924_equals0I,axiom,
! [A4: set_o] :
( ! [Y4: $o] :
~ ( member_o @ Y4 @ A4 )
=> ( A4 = bot_bot_set_o ) ) ).
% equals0I
thf(fact_925_ex__in__conv,axiom,
! [A4: set_a] :
( ( ? [X: a] : ( member_a @ X @ A4 ) )
= ( A4 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_926_ex__in__conv,axiom,
! [A4: set_o] :
( ( ? [X: $o] : ( member_o @ X @ A4 ) )
= ( A4 != bot_bot_set_o ) ) ).
% ex_in_conv
thf(fact_927_bot__set__def,axiom,
( bot_bot_set_o
= ( collect_o @ bot_bot_o_o ) ) ).
% bot_set_def
thf(fact_928_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X: a] : ( member_a @ X @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_929_bot__empty__eq,axiom,
( bot_bot_o_o
= ( ^ [X: $o] : ( member_o @ X @ bot_bot_set_o ) ) ) ).
% bot_empty_eq
thf(fact_930_empty__def,axiom,
( bot_bot_set_o
= ( collect_o
@ ^ [X: $o] : $false ) ) ).
% empty_def
thf(fact_931_antisymp__onI,axiom,
! [A4: set_a,R: a > a > $o] :
( ! [X2: a,Y4: a] :
( ( member_a @ X2 @ A4 )
=> ( ( member_a @ Y4 @ A4 )
=> ( ( R @ X2 @ Y4 )
=> ( ( R @ Y4 @ X2 )
=> ( X2 = Y4 ) ) ) ) )
=> ( antisymp_on_a @ A4 @ R ) ) ).
% antisymp_onI
thf(fact_932_antisymp__onD,axiom,
! [A4: set_a,R: a > a > $o,X3: a,Y: a] :
( ( antisymp_on_a @ A4 @ R )
=> ( ( member_a @ X3 @ A4 )
=> ( ( member_a @ Y @ A4 )
=> ( ( R @ X3 @ Y )
=> ( ( R @ Y @ X3 )
=> ( X3 = Y ) ) ) ) ) ) ).
% antisymp_onD
thf(fact_933_antisymp__on__le,axiom,
! [A4: set_a] : ( antisymp_on_a @ A4 @ ord_less_eq_a ) ).
% antisymp_on_le
thf(fact_934_antisymp__equality,axiom,
( antisymp_on_o @ top_top_set_o
@ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) ).
% antisymp_equality
thf(fact_935_antisympI,axiom,
! [R: $o > $o > $o] :
( ! [X2: $o,Y4: $o] :
( ( R @ X2 @ Y4 )
=> ( ( R @ Y4 @ X2 )
=> ( X2 = Y4 ) ) )
=> ( antisymp_on_o @ top_top_set_o @ R ) ) ).
% antisympI
thf(fact_936_antisympD,axiom,
! [R: $o > $o > $o,X3: $o,Y: $o] :
( ( antisymp_on_o @ top_top_set_o @ R )
=> ( ( R @ X3 @ Y )
=> ( ( R @ Y @ X3 )
=> ( X3 = Y ) ) ) ) ).
% antisympD
thf(fact_937_antisym__bot,axiom,
antisymp_on_o @ top_top_set_o @ bot_bot_o_o_o ).
% antisym_bot
thf(fact_938_antisymp__on__ge,axiom,
! [A4: set_a] :
( antisymp_on_a @ A4
@ ^ [X: a,Y2: a] : ( ord_less_eq_a @ Y2 @ X ) ) ).
% antisymp_on_ge
thf(fact_939_Set_Ois__empty__def,axiom,
( is_empty_o
= ( ^ [A5: set_o] : ( A5 = bot_bot_set_o ) ) ) ).
% Set.is_empty_def
thf(fact_940_subset__singleton__iff__Uniq,axiom,
! [A4: set_a] :
( ( ? [A2: a] : ( ord_less_eq_set_a @ A4 @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
= ( uniq_a
@ ^ [X: a] : ( member_a @ X @ A4 ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_941_subset__singleton__iff__Uniq,axiom,
! [A4: set_o] :
( ( ? [A2: $o] : ( ord_less_eq_set_o @ A4 @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
= ( uniq_o
@ ^ [X: $o] : ( member_o @ X @ A4 ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_942_bdd__below__primitive__def,axiom,
( condit5901475214736682318elow_a
= ( condit6541519642617408243_bdd_a
@ ^ [X: a,Y2: a] : ( ord_less_eq_a @ Y2 @ X ) ) ) ).
% bdd_below_primitive_def
thf(fact_943_is__singletonI_H,axiom,
! [A4: set_a] :
( ( A4 != bot_bot_set_a )
=> ( ! [X2: a,Y4: a] :
( ( member_a @ X2 @ A4 )
=> ( ( member_a @ Y4 @ A4 )
=> ( X2 = Y4 ) ) )
=> ( is_singleton_a @ A4 ) ) ) ).
% is_singletonI'
thf(fact_944_is__singletonI_H,axiom,
! [A4: set_o] :
( ( A4 != bot_bot_set_o )
=> ( ! [X2: $o,Y4: $o] :
( ( member_o @ X2 @ A4 )
=> ( ( member_o @ Y4 @ A4 )
=> ( X2 = Y4 ) ) )
=> ( is_singleton_o @ A4 ) ) ) ).
% is_singletonI'
thf(fact_945_vimage__const,axiom,
! [C: a,A4: set_a] :
( ( ( member_a @ C @ A4 )
=> ( ( vimage_o_a
@ ^ [X: $o] : C
@ A4 )
= top_top_set_o ) )
& ( ~ ( member_a @ C @ A4 )
=> ( ( vimage_o_a
@ ^ [X: $o] : C
@ A4 )
= bot_bot_set_o ) ) ) ).
% vimage_const
thf(fact_946_insertCI,axiom,
! [A: $o,B4: set_o,B: $o] :
( ( ~ ( member_o @ A @ B4 )
=> ( A = B ) )
=> ( member_o @ A @ ( insert_o @ B @ B4 ) ) ) ).
% insertCI
thf(fact_947_insertCI,axiom,
! [A: a,B4: set_a,B: a] :
( ( ~ ( member_a @ A @ B4 )
=> ( A = B ) )
=> ( member_a @ A @ ( insert_a @ B @ B4 ) ) ) ).
% insertCI
thf(fact_948_insert__iff,axiom,
! [A: $o,B: $o,A4: set_o] :
( ( member_o @ A @ ( insert_o @ B @ A4 ) )
= ( ( A = B )
| ( member_o @ A @ A4 ) ) ) ).
% insert_iff
thf(fact_949_insert__iff,axiom,
! [A: a,B: a,A4: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A4 ) )
= ( ( A = B )
| ( member_a @ A @ A4 ) ) ) ).
% insert_iff
thf(fact_950_insert__absorb2,axiom,
! [X3: $o,A4: set_o] :
( ( insert_o @ X3 @ ( insert_o @ X3 @ A4 ) )
= ( insert_o @ X3 @ A4 ) ) ).
% insert_absorb2
thf(fact_951_vimageI,axiom,
! [F: a > a,A: a,B: a,B4: set_a] :
( ( ( F @ A )
= B )
=> ( ( member_a @ B @ B4 )
=> ( member_a @ A @ ( vimage_a_a @ F @ B4 ) ) ) ) ).
% vimageI
thf(fact_952_vimage__eq,axiom,
! [A: a,F: a > a,B4: set_a] :
( ( member_a @ A @ ( vimage_a_a @ F @ B4 ) )
= ( member_a @ ( F @ A ) @ B4 ) ) ).
% vimage_eq
thf(fact_953_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_954_singletonI,axiom,
! [A: $o] : ( member_o @ A @ ( insert_o @ A @ bot_bot_set_o ) ) ).
% singletonI
thf(fact_955_insert__subset,axiom,
! [X3: $o,A4: set_o,B4: set_o] :
( ( ord_less_eq_set_o @ ( insert_o @ X3 @ A4 ) @ B4 )
= ( ( member_o @ X3 @ B4 )
& ( ord_less_eq_set_o @ A4 @ B4 ) ) ) ).
% insert_subset
thf(fact_956_insert__subset,axiom,
! [X3: a,A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X3 @ A4 ) @ B4 )
= ( ( member_a @ X3 @ B4 )
& ( ord_less_eq_set_a @ A4 @ B4 ) ) ) ).
% insert_subset
thf(fact_957_bdd__below_OI,axiom,
! [A4: set_a,M: a] :
( ! [X2: a] :
( ( member_a @ X2 @ A4 )
=> ( ord_less_eq_a @ M @ X2 ) )
=> ( condit5901475214736682318elow_a @ A4 ) ) ).
% bdd_below.I
thf(fact_958_bdd__belowI,axiom,
! [A4: set_a,M2: a] :
( ! [X2: a] :
( ( member_a @ X2 @ A4 )
=> ( ord_less_eq_a @ M2 @ X2 ) )
=> ( condit5901475214736682318elow_a @ A4 ) ) ).
% bdd_belowI
thf(fact_959_vimage__UNIV,axiom,
! [F: $o > $o] :
( ( vimage_o_o @ F @ top_top_set_o )
= top_top_set_o ) ).
% vimage_UNIV
thf(fact_960_vimage__empty,axiom,
! [F: $o > $o] :
( ( vimage_o_o @ F @ bot_bot_set_o )
= bot_bot_set_o ) ).
% vimage_empty
thf(fact_961_singleton__conv2,axiom,
! [A: $o] :
( ( collect_o
@ ( ^ [Y3: $o,Z: $o] : ( Y3 = Z )
@ A ) )
= ( insert_o @ A @ bot_bot_set_o ) ) ).
% singleton_conv2
thf(fact_962_singleton__conv,axiom,
! [A: $o] :
( ( collect_o
@ ^ [X: $o] : ( X = A ) )
= ( insert_o @ A @ bot_bot_set_o ) ) ).
% singleton_conv
thf(fact_963_singleton__insert__inj__eq_H,axiom,
! [A: $o,A4: set_o,B: $o] :
( ( ( insert_o @ A @ A4 )
= ( insert_o @ B @ bot_bot_set_o ) )
= ( ( A = B )
& ( ord_less_eq_set_o @ A4 @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_964_singleton__insert__inj__eq,axiom,
! [B: $o,A: $o,A4: set_o] :
( ( ( insert_o @ B @ bot_bot_set_o )
= ( insert_o @ A @ A4 ) )
= ( ( A = B )
& ( ord_less_eq_set_o @ A4 @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_965_is__singletonI,axiom,
! [X3: $o] : ( is_singleton_o @ ( insert_o @ X3 @ bot_bot_set_o ) ) ).
% is_singletonI
thf(fact_966_is__singletonE,axiom,
! [A4: set_o] :
( ( is_singleton_o @ A4 )
=> ~ ! [X2: $o] :
( A4
!= ( insert_o @ X2 @ bot_bot_set_o ) ) ) ).
% is_singletonE
thf(fact_967_is__singleton__def,axiom,
( is_singleton_o
= ( ^ [A5: set_o] :
? [X: $o] :
( A5
= ( insert_o @ X @ bot_bot_set_o ) ) ) ) ).
% is_singleton_def
thf(fact_968_vimage__singleton__eq,axiom,
! [A: a,F: a > $o,B: $o] :
( ( member_a @ A @ ( vimage_a_o @ F @ ( insert_o @ B @ bot_bot_set_o ) ) )
= ( ( F @ A )
= B ) ) ).
% vimage_singleton_eq
thf(fact_969_insert__Collect,axiom,
! [A: $o,P: $o > $o] :
( ( insert_o @ A @ ( collect_o @ P ) )
= ( collect_o
@ ^ [U: $o] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_970_insert__compr,axiom,
( insert_o
= ( ^ [A2: $o,B5: set_o] :
( collect_o
@ ^ [X: $o] :
( ( X = A2 )
| ( member_o @ X @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_971_insert__compr,axiom,
( insert_a
= ( ^ [A2: a,B5: set_a] :
( collect_a
@ ^ [X: a] :
( ( X = A2 )
| ( member_a @ X @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_972_insertE,axiom,
! [A: $o,B: $o,A4: set_o] :
( ( member_o @ A @ ( insert_o @ B @ A4 ) )
=> ( ( A = (~ B) )
=> ( member_o @ A @ A4 ) ) ) ).
% insertE
thf(fact_973_insertE,axiom,
! [A: a,B: a,A4: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A4 ) )
=> ( ( A != B )
=> ( member_a @ A @ A4 ) ) ) ).
% insertE
thf(fact_974_vimageD,axiom,
! [A: a,F: a > a,A4: set_a] :
( ( member_a @ A @ ( vimage_a_a @ F @ A4 ) )
=> ( member_a @ ( F @ A ) @ A4 ) ) ).
% vimageD
thf(fact_975_vimageE,axiom,
! [A: a,F: a > a,B4: set_a] :
( ( member_a @ A @ ( vimage_a_a @ F @ B4 ) )
=> ( member_a @ ( F @ A ) @ B4 ) ) ).
% vimageE
thf(fact_976_insertI1,axiom,
! [A: $o,B4: set_o] : ( member_o @ A @ ( insert_o @ A @ B4 ) ) ).
% insertI1
thf(fact_977_insertI1,axiom,
! [A: a,B4: set_a] : ( member_a @ A @ ( insert_a @ A @ B4 ) ) ).
% insertI1
thf(fact_978_insertI2,axiom,
! [A: $o,B4: set_o,B: $o] :
( ( member_o @ A @ B4 )
=> ( member_o @ A @ ( insert_o @ B @ B4 ) ) ) ).
% insertI2
thf(fact_979_insertI2,axiom,
! [A: a,B4: set_a,B: a] :
( ( member_a @ A @ B4 )
=> ( member_a @ A @ ( insert_a @ B @ B4 ) ) ) ).
% insertI2
thf(fact_980_vimageI2,axiom,
! [F: a > a,A: a,A4: set_a] :
( ( member_a @ ( F @ A ) @ A4 )
=> ( member_a @ A @ ( vimage_a_a @ F @ A4 ) ) ) ).
% vimageI2
thf(fact_981_Set_Oset__insert,axiom,
! [X3: $o,A4: set_o] :
( ( member_o @ X3 @ A4 )
=> ~ ! [B7: set_o] :
( ( A4
= ( insert_o @ X3 @ B7 ) )
=> ( member_o @ X3 @ B7 ) ) ) ).
% Set.set_insert
thf(fact_982_Set_Oset__insert,axiom,
! [X3: a,A4: set_a] :
( ( member_a @ X3 @ A4 )
=> ~ ! [B7: set_a] :
( ( A4
= ( insert_a @ X3 @ B7 ) )
=> ( member_a @ X3 @ B7 ) ) ) ).
% Set.set_insert
thf(fact_983_insert__ident,axiom,
! [X3: $o,A4: set_o,B4: set_o] :
( ~ ( member_o @ X3 @ A4 )
=> ( ~ ( member_o @ X3 @ B4 )
=> ( ( ( insert_o @ X3 @ A4 )
= ( insert_o @ X3 @ B4 ) )
= ( A4 = B4 ) ) ) ) ).
% insert_ident
thf(fact_984_insert__ident,axiom,
! [X3: a,A4: set_a,B4: set_a] :
( ~ ( member_a @ X3 @ A4 )
=> ( ~ ( member_a @ X3 @ B4 )
=> ( ( ( insert_a @ X3 @ A4 )
= ( insert_a @ X3 @ B4 ) )
= ( A4 = B4 ) ) ) ) ).
% insert_ident
thf(fact_985_insert__absorb,axiom,
! [A: $o,A4: set_o] :
( ( member_o @ A @ A4 )
=> ( ( insert_o @ A @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_986_insert__absorb,axiom,
! [A: a,A4: set_a] :
( ( member_a @ A @ A4 )
=> ( ( insert_a @ A @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_987_insert__eq__iff,axiom,
! [A: $o,A4: set_o,B: $o,B4: set_o] :
( ~ ( member_o @ A @ A4 )
=> ( ~ ( member_o @ B @ B4 )
=> ( ( ( insert_o @ A @ A4 )
= ( insert_o @ B @ B4 ) )
= ( ( ( A = B )
=> ( A4 = B4 ) )
& ( ( A = (~ B) )
=> ? [C5: set_o] :
( ( A4
= ( insert_o @ B @ C5 ) )
& ~ ( member_o @ B @ C5 )
& ( B4
= ( insert_o @ A @ C5 ) )
& ~ ( member_o @ A @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_988_insert__eq__iff,axiom,
! [A: a,A4: set_a,B: a,B4: set_a] :
( ~ ( member_a @ A @ A4 )
=> ( ~ ( member_a @ B @ B4 )
=> ( ( ( insert_a @ A @ A4 )
= ( insert_a @ B @ B4 ) )
= ( ( ( A = B )
=> ( A4 = B4 ) )
& ( ( A != B )
=> ? [C5: set_a] :
( ( A4
= ( insert_a @ B @ C5 ) )
& ~ ( member_a @ B @ C5 )
& ( B4
= ( insert_a @ A @ C5 ) )
& ~ ( member_a @ A @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_989_insert__commute,axiom,
! [X3: $o,Y: $o,A4: set_o] :
( ( insert_o @ X3 @ ( insert_o @ Y @ A4 ) )
= ( insert_o @ Y @ ( insert_o @ X3 @ A4 ) ) ) ).
% insert_commute
thf(fact_990_mk__disjoint__insert,axiom,
! [A: $o,A4: set_o] :
( ( member_o @ A @ A4 )
=> ? [B7: set_o] :
( ( A4
= ( insert_o @ A @ B7 ) )
& ~ ( member_o @ A @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_991_mk__disjoint__insert,axiom,
! [A: a,A4: set_a] :
( ( member_a @ A @ A4 )
=> ? [B7: set_a] :
( ( A4
= ( insert_a @ A @ B7 ) )
& ~ ( member_a @ A @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_992_bdd__below_Ounfold,axiom,
( condit5901475214736682318elow_a
= ( ^ [A5: set_a] :
? [M3: a] :
! [X: a] :
( ( member_a @ X @ A5 )
=> ( ord_less_eq_a @ M3 @ X ) ) ) ) ).
% bdd_below.unfold
thf(fact_993_bdd__below_OE,axiom,
! [A4: set_a] :
( ( condit5901475214736682318elow_a @ A4 )
=> ~ ! [M4: a] :
~ ! [X4: a] :
( ( member_a @ X4 @ A4 )
=> ( ord_less_eq_a @ M4 @ X4 ) ) ) ).
% bdd_below.E
thf(fact_994_insert__UNIV,axiom,
! [X3: $o] :
( ( insert_o @ X3 @ top_top_set_o )
= top_top_set_o ) ).
% insert_UNIV
thf(fact_995_singleton__inject,axiom,
! [A: $o,B: $o] :
( ( ( insert_o @ A @ bot_bot_set_o )
= ( insert_o @ B @ bot_bot_set_o ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_996_insert__not__empty,axiom,
! [A: $o,A4: set_o] :
( ( insert_o @ A @ A4 )
!= bot_bot_set_o ) ).
% insert_not_empty
thf(fact_997_doubleton__eq__iff,axiom,
! [A: $o,B: $o,C: $o,D: $o] :
( ( ( insert_o @ A @ ( insert_o @ B @ bot_bot_set_o ) )
= ( insert_o @ C @ ( insert_o @ D @ bot_bot_set_o ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_998_singleton__iff,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_999_singleton__iff,axiom,
! [B: $o,A: $o] :
( ( member_o @ B @ ( insert_o @ A @ bot_bot_set_o ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_1000_singletonD,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_1001_singletonD,axiom,
! [B: $o,A: $o] :
( ( member_o @ B @ ( insert_o @ A @ bot_bot_set_o ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_1002_insert__subsetI,axiom,
! [X3: $o,A4: set_o,X6: set_o] :
( ( member_o @ X3 @ A4 )
=> ( ( ord_less_eq_set_o @ X6 @ A4 )
=> ( ord_less_eq_set_o @ ( insert_o @ X3 @ X6 ) @ A4 ) ) ) ).
% insert_subsetI
thf(fact_1003_insert__subsetI,axiom,
! [X3: a,A4: set_a,X6: set_a] :
( ( member_a @ X3 @ A4 )
=> ( ( ord_less_eq_set_a @ X6 @ A4 )
=> ( ord_less_eq_set_a @ ( insert_a @ X3 @ X6 ) @ A4 ) ) ) ).
% insert_subsetI
thf(fact_1004_insert__mono,axiom,
! [C2: set_o,D2: set_o,A: $o] :
( ( ord_less_eq_set_o @ C2 @ D2 )
=> ( ord_less_eq_set_o @ ( insert_o @ A @ C2 ) @ ( insert_o @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_1005_subset__insert,axiom,
! [X3: $o,A4: set_o,B4: set_o] :
( ~ ( member_o @ X3 @ A4 )
=> ( ( ord_less_eq_set_o @ A4 @ ( insert_o @ X3 @ B4 ) )
= ( ord_less_eq_set_o @ A4 @ B4 ) ) ) ).
% subset_insert
thf(fact_1006_subset__insert,axiom,
! [X3: a,A4: set_a,B4: set_a] :
( ~ ( member_a @ X3 @ A4 )
=> ( ( ord_less_eq_set_a @ A4 @ ( insert_a @ X3 @ B4 ) )
= ( ord_less_eq_set_a @ A4 @ B4 ) ) ) ).
% subset_insert
thf(fact_1007_subset__insertI,axiom,
! [B4: set_o,A: $o] : ( ord_less_eq_set_o @ B4 @ ( insert_o @ A @ B4 ) ) ).
% subset_insertI
thf(fact_1008_subset__insertI2,axiom,
! [A4: set_o,B4: set_o,B: $o] :
( ( ord_less_eq_set_o @ A4 @ B4 )
=> ( ord_less_eq_set_o @ A4 @ ( insert_o @ B @ B4 ) ) ) ).
% subset_insertI2
thf(fact_1009_Collect__conv__if2,axiom,
! [P: $o > $o,A: $o] :
( ( ( P @ A )
=> ( ( collect_o
@ ^ [X: $o] :
( ( A = X )
& ( P @ X ) ) )
= ( insert_o @ A @ bot_bot_set_o ) ) )
& ( ~ ( P @ A )
=> ( ( collect_o
@ ^ [X: $o] :
( ( A = X )
& ( P @ X ) ) )
= bot_bot_set_o ) ) ) ).
% Collect_conv_if2
thf(fact_1010_Collect__conv__if,axiom,
! [P: $o > $o,A: $o] :
( ( ( P @ A )
=> ( ( collect_o
@ ^ [X: $o] :
( ( X = A )
& ( P @ X ) ) )
= ( insert_o @ A @ bot_bot_set_o ) ) )
& ( ~ ( P @ A )
=> ( ( collect_o
@ ^ [X: $o] :
( ( X = A )
& ( P @ X ) ) )
= bot_bot_set_o ) ) ) ).
% Collect_conv_if
thf(fact_1011_subset__singleton__iff,axiom,
! [X6: set_o,A: $o] :
( ( ord_less_eq_set_o @ X6 @ ( insert_o @ A @ bot_bot_set_o ) )
= ( ( X6 = bot_bot_set_o )
| ( X6
= ( insert_o @ A @ bot_bot_set_o ) ) ) ) ).
% subset_singleton_iff
thf(fact_1012_subset__singletonD,axiom,
! [A4: set_o,X3: $o] :
( ( ord_less_eq_set_o @ A4 @ ( insert_o @ X3 @ bot_bot_set_o ) )
=> ( ( A4 = bot_bot_set_o )
| ( A4
= ( insert_o @ X3 @ bot_bot_set_o ) ) ) ) ).
% subset_singletonD
thf(fact_1013_the__elem__def,axiom,
( the_elem_a
= ( ^ [X5: set_a] :
( the_a
@ ^ [X: a] :
( X5
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% the_elem_def
thf(fact_1014_the__elem__def,axiom,
( the_elem_o
= ( ^ [X5: set_o] :
( the_o
@ ^ [X: $o] :
( X5
= ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ).
% the_elem_def
thf(fact_1015_is__singleton__the__elem,axiom,
( is_singleton_o
= ( ^ [A5: set_o] :
( A5
= ( insert_o @ ( the_elem_o @ A5 ) @ bot_bot_set_o ) ) ) ) ).
% is_singleton_the_elem
thf(fact_1016_the__elem__eq,axiom,
! [X3: $o] :
( ( the_elem_o @ ( insert_o @ X3 @ bot_bot_set_o ) )
= X3 ) ).
% the_elem_eq
thf(fact_1017_inj__vimage__singleton,axiom,
! [F: a > $o,A: $o] :
( ( inj_on_a_o @ F @ top_top_set_a )
=> ( ord_less_eq_set_a @ ( vimage_a_o @ F @ ( insert_o @ A @ bot_bot_set_o ) )
@ ( insert_a
@ ( the_a
@ ^ [X: a] :
( ( F @ X )
= A ) )
@ bot_bot_set_a ) ) ) ).
% inj_vimage_singleton
thf(fact_1018_inj__vimage__singleton,axiom,
! [F: $o > $o,A: $o] :
( ( inj_on_o_o @ F @ top_top_set_o )
=> ( ord_less_eq_set_o @ ( vimage_o_o @ F @ ( insert_o @ A @ bot_bot_set_o ) )
@ ( insert_o
@ ( the_o
@ ^ [X: $o] :
( ( F @ X )
= A ) )
@ bot_bot_set_o ) ) ) ).
% inj_vimage_singleton
thf(fact_1019_vimage__if,axiom,
! [C: a,A4: set_a,D: a,B4: set_a] :
( ( ( member_a @ C @ A4 )
=> ( ( ( member_a @ D @ A4 )
=> ( ( vimage_a_a
@ ^ [X: a] : ( if_a @ ( member_a @ X @ B4 ) @ C @ D )
@ A4 )
= top_top_set_a ) )
& ( ~ ( member_a @ D @ A4 )
=> ( ( vimage_a_a
@ ^ [X: a] : ( if_a @ ( member_a @ X @ B4 ) @ C @ D )
@ A4 )
= B4 ) ) ) )
& ( ~ ( member_a @ C @ A4 )
=> ( ( ( member_a @ D @ A4 )
=> ( ( vimage_a_a
@ ^ [X: a] : ( if_a @ ( member_a @ X @ B4 ) @ C @ D )
@ A4 )
= ( uminus_uminus_set_a @ B4 ) ) )
& ( ~ ( member_a @ D @ A4 )
=> ( ( vimage_a_a
@ ^ [X: a] : ( if_a @ ( member_a @ X @ B4 ) @ C @ D )
@ A4 )
= bot_bot_set_a ) ) ) ) ) ).
% vimage_if
thf(fact_1020_vimage__if,axiom,
! [C: a,A4: set_a,D: a,B4: set_o] :
( ( ( member_a @ C @ A4 )
=> ( ( ( member_a @ D @ A4 )
=> ( ( vimage_o_a
@ ^ [X: $o] : ( if_a @ ( member_o @ X @ B4 ) @ C @ D )
@ A4 )
= top_top_set_o ) )
& ( ~ ( member_a @ D @ A4 )
=> ( ( vimage_o_a
@ ^ [X: $o] : ( if_a @ ( member_o @ X @ B4 ) @ C @ D )
@ A4 )
= B4 ) ) ) )
& ( ~ ( member_a @ C @ A4 )
=> ( ( ( member_a @ D @ A4 )
=> ( ( vimage_o_a
@ ^ [X: $o] : ( if_a @ ( member_o @ X @ B4 ) @ C @ D )
@ A4 )
= ( uminus_uminus_set_o @ B4 ) ) )
& ( ~ ( member_a @ D @ A4 )
=> ( ( vimage_o_a
@ ^ [X: $o] : ( if_a @ ( member_o @ X @ B4 ) @ C @ D )
@ A4 )
= bot_bot_set_o ) ) ) ) ) ).
% vimage_if
thf(fact_1021_Compl__iff,axiom,
! [C: a,A4: set_a] :
( ( member_a @ C @ ( uminus_uminus_set_a @ A4 ) )
= ( ~ ( member_a @ C @ A4 ) ) ) ).
% Compl_iff
thf(fact_1022_ComplI,axiom,
! [C: a,A4: set_a] :
( ~ ( member_a @ C @ A4 )
=> ( member_a @ C @ ( uminus_uminus_set_a @ A4 ) ) ) ).
% ComplI
thf(fact_1023_subset__Compl__singleton,axiom,
! [A4: set_a,B: a] :
( ( ord_less_eq_set_a @ A4 @ ( uminus_uminus_set_a @ ( insert_a @ B @ bot_bot_set_a ) ) )
= ( ~ ( member_a @ B @ A4 ) ) ) ).
% subset_Compl_singleton
thf(fact_1024_subset__Compl__singleton,axiom,
! [A4: set_o,B: $o] :
( ( ord_less_eq_set_o @ A4 @ ( uminus_uminus_set_o @ ( insert_o @ B @ bot_bot_set_o ) ) )
= ( ~ ( member_o @ B @ A4 ) ) ) ).
% subset_Compl_singleton
thf(fact_1025_Compl__eq,axiom,
( uminus_uminus_set_a
= ( ^ [A5: set_a] :
( collect_a
@ ^ [X: a] :
~ ( member_a @ X @ A5 ) ) ) ) ).
% Compl_eq
thf(fact_1026_ComplD,axiom,
! [C: a,A4: set_a] :
( ( member_a @ C @ ( uminus_uminus_set_a @ A4 ) )
=> ~ ( member_a @ C @ A4 ) ) ).
% ComplD
thf(fact_1027_sorted__list__of__set_Oinj__on,axiom,
( inj_on_o_o
@ ^ [X: $o] : X
@ top_top_set_o ) ).
% sorted_list_of_set.inj_on
thf(fact_1028_inj__singleton,axiom,
! [A4: set_o] :
( inj_on_o_set_o
@ ^ [X: $o] : ( insert_o @ X @ bot_bot_set_o )
@ A4 ) ).
% inj_singleton
thf(fact_1029_Compl__empty__eq,axiom,
( ( uminus_uminus_set_o @ bot_bot_set_o )
= top_top_set_o ) ).
% Compl_empty_eq
thf(fact_1030_Compl__UNIV__eq,axiom,
( ( uminus_uminus_set_o @ top_top_set_o )
= bot_bot_set_o ) ).
% Compl_UNIV_eq
thf(fact_1031_subset__Compl__self__eq,axiom,
! [A4: set_o] :
( ( ord_less_eq_set_o @ A4 @ ( uminus_uminus_set_o @ A4 ) )
= ( A4 = bot_bot_set_o ) ) ).
% subset_Compl_self_eq
thf(fact_1032_uminus__set__def,axiom,
( uminus_uminus_set_a
= ( ^ [A5: set_a] :
( collect_a
@ ( uminus_uminus_a_o
@ ^ [X: a] : ( member_a @ X @ A5 ) ) ) ) ) ).
% uminus_set_def
thf(fact_1033_inj__on__vimage__singleton,axiom,
! [F: a > $o,A4: set_a,A: $o] :
( ( inj_on_a_o @ F @ A4 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ ( vimage_a_o @ F @ ( insert_o @ A @ bot_bot_set_o ) ) @ A4 )
@ ( insert_a
@ ( the_a
@ ^ [X: a] :
( ( member_a @ X @ A4 )
& ( ( F @ X )
= A ) ) )
@ bot_bot_set_a ) ) ) ).
% inj_on_vimage_singleton
thf(fact_1034_inj__on__vimage__singleton,axiom,
! [F: $o > $o,A4: set_o,A: $o] :
( ( inj_on_o_o @ F @ A4 )
=> ( ord_less_eq_set_o @ ( inf_inf_set_o @ ( vimage_o_o @ F @ ( insert_o @ A @ bot_bot_set_o ) ) @ A4 )
@ ( insert_o
@ ( the_o
@ ^ [X: $o] :
( ( member_o @ X @ A4 )
& ( ( F @ X )
= A ) ) )
@ bot_bot_set_o ) ) ) ).
% inj_on_vimage_singleton
thf(fact_1035_psubset__insert__iff,axiom,
! [A4: set_a,X3: a,B4: set_a] :
( ( ord_less_set_a @ A4 @ ( insert_a @ X3 @ B4 ) )
= ( ( ( member_a @ X3 @ B4 )
=> ( ord_less_set_a @ A4 @ B4 ) )
& ( ~ ( member_a @ X3 @ B4 )
=> ( ( ( member_a @ X3 @ A4 )
=> ( ord_less_set_a @ ( minus_minus_set_a @ A4 @ ( insert_a @ X3 @ bot_bot_set_a ) ) @ B4 ) )
& ( ~ ( member_a @ X3 @ A4 )
=> ( ord_less_eq_set_a @ A4 @ B4 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1036_psubset__insert__iff,axiom,
! [A4: set_o,X3: $o,B4: set_o] :
( ( ord_less_set_o @ A4 @ ( insert_o @ X3 @ B4 ) )
= ( ( ( member_o @ X3 @ B4 )
=> ( ord_less_set_o @ A4 @ B4 ) )
& ( ~ ( member_o @ X3 @ B4 )
=> ( ( ( member_o @ X3 @ A4 )
=> ( ord_less_set_o @ ( minus_minus_set_o @ A4 @ ( insert_o @ X3 @ bot_bot_set_o ) ) @ B4 ) )
& ( ~ ( member_o @ X3 @ A4 )
=> ( ord_less_eq_set_o @ A4 @ B4 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1037_image__eqI,axiom,
! [B: a,F: a > a,X3: a,A4: set_a] :
( ( B
= ( F @ X3 ) )
=> ( ( member_a @ X3 @ A4 )
=> ( member_a @ B @ ( image_a_a @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_1038_Int__iff,axiom,
! [C: a,A4: set_a,B4: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A4 @ B4 ) )
= ( ( member_a @ C @ A4 )
& ( member_a @ C @ B4 ) ) ) ).
% Int_iff
thf(fact_1039_IntI,axiom,
! [C: a,A4: set_a,B4: set_a] :
( ( member_a @ C @ A4 )
=> ( ( member_a @ C @ B4 )
=> ( member_a @ C @ ( inf_inf_set_a @ A4 @ B4 ) ) ) ) ).
% IntI
thf(fact_1040_Diff__iff,axiom,
! [C: a,A4: set_a,B4: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A4 @ B4 ) )
= ( ( member_a @ C @ A4 )
& ~ ( member_a @ C @ B4 ) ) ) ).
% Diff_iff
thf(fact_1041_DiffI,axiom,
! [C: a,A4: set_a,B4: set_a] :
( ( member_a @ C @ A4 )
=> ( ~ ( member_a @ C @ B4 )
=> ( member_a @ C @ ( minus_minus_set_a @ A4 @ B4 ) ) ) ) ).
% DiffI
thf(fact_1042_image__is__empty,axiom,
! [F: $o > $o,A4: set_o] :
( ( ( image_o_o @ F @ A4 )
= bot_bot_set_o )
= ( A4 = bot_bot_set_o ) ) ).
% image_is_empty
thf(fact_1043_empty__is__image,axiom,
! [F: $o > $o,A4: set_o] :
( ( bot_bot_set_o
= ( image_o_o @ F @ A4 ) )
= ( A4 = bot_bot_set_o ) ) ).
% empty_is_image
thf(fact_1044_image__empty,axiom,
! [F: $o > $o] :
( ( image_o_o @ F @ bot_bot_set_o )
= bot_bot_set_o ) ).
% image_empty
thf(fact_1045_insert__image,axiom,
! [X3: a,A4: set_a,F: a > $o] :
( ( member_a @ X3 @ A4 )
=> ( ( insert_o @ ( F @ X3 ) @ ( image_a_o @ F @ A4 ) )
= ( image_a_o @ F @ A4 ) ) ) ).
% insert_image
thf(fact_1046_image__insert,axiom,
! [F: $o > $o,A: $o,B4: set_o] :
( ( image_o_o @ F @ ( insert_o @ A @ B4 ) )
= ( insert_o @ ( F @ A ) @ ( image_o_o @ F @ B4 ) ) ) ).
% image_insert
thf(fact_1047_Int__UNIV,axiom,
! [A4: set_o,B4: set_o] :
( ( ( inf_inf_set_o @ A4 @ B4 )
= top_top_set_o )
= ( ( A4 = top_top_set_o )
& ( B4 = top_top_set_o ) ) ) ).
% Int_UNIV
thf(fact_1048_Diff__empty,axiom,
! [A4: set_o] :
( ( minus_minus_set_o @ A4 @ bot_bot_set_o )
= A4 ) ).
% Diff_empty
thf(fact_1049_empty__Diff,axiom,
! [A4: set_o] :
( ( minus_minus_set_o @ bot_bot_set_o @ A4 )
= bot_bot_set_o ) ).
% empty_Diff
thf(fact_1050_Diff__cancel,axiom,
! [A4: set_o] :
( ( minus_minus_set_o @ A4 @ A4 )
= bot_bot_set_o ) ).
% Diff_cancel
thf(fact_1051_Int__insert__right__if1,axiom,
! [A: $o,A4: set_o,B4: set_o] :
( ( member_o @ A @ A4 )
=> ( ( inf_inf_set_o @ A4 @ ( insert_o @ A @ B4 ) )
= ( insert_o @ A @ ( inf_inf_set_o @ A4 @ B4 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1052_Int__insert__right__if1,axiom,
! [A: a,A4: set_a,B4: set_a] :
( ( member_a @ A @ A4 )
=> ( ( inf_inf_set_a @ A4 @ ( insert_a @ A @ B4 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A4 @ B4 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1053_Int__insert__right__if0,axiom,
! [A: $o,A4: set_o,B4: set_o] :
( ~ ( member_o @ A @ A4 )
=> ( ( inf_inf_set_o @ A4 @ ( insert_o @ A @ B4 ) )
= ( inf_inf_set_o @ A4 @ B4 ) ) ) ).
% Int_insert_right_if0
thf(fact_1054_Int__insert__right__if0,axiom,
! [A: a,A4: set_a,B4: set_a] :
( ~ ( member_a @ A @ A4 )
=> ( ( inf_inf_set_a @ A4 @ ( insert_a @ A @ B4 ) )
= ( inf_inf_set_a @ A4 @ B4 ) ) ) ).
% Int_insert_right_if0
thf(fact_1055_insert__inter__insert,axiom,
! [A: $o,A4: set_o,B4: set_o] :
( ( inf_inf_set_o @ ( insert_o @ A @ A4 ) @ ( insert_o @ A @ B4 ) )
= ( insert_o @ A @ ( inf_inf_set_o @ A4 @ B4 ) ) ) ).
% insert_inter_insert
thf(fact_1056_Int__insert__left__if1,axiom,
! [A: $o,C2: set_o,B4: set_o] :
( ( member_o @ A @ C2 )
=> ( ( inf_inf_set_o @ ( insert_o @ A @ B4 ) @ C2 )
= ( insert_o @ A @ ( inf_inf_set_o @ B4 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1057_Int__insert__left__if1,axiom,
! [A: a,C2: set_a,B4: set_a] :
( ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B4 ) @ C2 )
= ( insert_a @ A @ ( inf_inf_set_a @ B4 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1058_Int__insert__left__if0,axiom,
! [A: $o,C2: set_o,B4: set_o] :
( ~ ( member_o @ A @ C2 )
=> ( ( inf_inf_set_o @ ( insert_o @ A @ B4 ) @ C2 )
= ( inf_inf_set_o @ B4 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_1059_Int__insert__left__if0,axiom,
! [A: a,C2: set_a,B4: set_a] :
( ~ ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B4 ) @ C2 )
= ( inf_inf_set_a @ B4 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_1060_insert__Diff1,axiom,
! [X3: $o,B4: set_o,A4: set_o] :
( ( member_o @ X3 @ B4 )
=> ( ( minus_minus_set_o @ ( insert_o @ X3 @ A4 ) @ B4 )
= ( minus_minus_set_o @ A4 @ B4 ) ) ) ).
% insert_Diff1
thf(fact_1061_insert__Diff1,axiom,
! [X3: a,B4: set_a,A4: set_a] :
( ( member_a @ X3 @ B4 )
=> ( ( minus_minus_set_a @ ( insert_a @ X3 @ A4 ) @ B4 )
= ( minus_minus_set_a @ A4 @ B4 ) ) ) ).
% insert_Diff1
thf(fact_1062_Diff__insert0,axiom,
! [X3: $o,A4: set_o,B4: set_o] :
( ~ ( member_o @ X3 @ A4 )
=> ( ( minus_minus_set_o @ A4 @ ( insert_o @ X3 @ B4 ) )
= ( minus_minus_set_o @ A4 @ B4 ) ) ) ).
% Diff_insert0
thf(fact_1063_Diff__insert0,axiom,
! [X3: a,A4: set_a,B4: set_a] :
( ~ ( member_a @ X3 @ A4 )
=> ( ( minus_minus_set_a @ A4 @ ( insert_a @ X3 @ B4 ) )
= ( minus_minus_set_a @ A4 @ B4 ) ) ) ).
% Diff_insert0
thf(fact_1064_Diff__UNIV,axiom,
! [A4: set_o] :
( ( minus_minus_set_o @ A4 @ top_top_set_o )
= bot_bot_set_o ) ).
% Diff_UNIV
thf(fact_1065_insert__disjoint_I1_J,axiom,
! [A: a,A4: set_a,B4: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A @ A4 ) @ B4 )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B4 )
& ( ( inf_inf_set_a @ A4 @ B4 )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_1066_insert__disjoint_I1_J,axiom,
! [A: $o,A4: set_o,B4: set_o] :
( ( ( inf_inf_set_o @ ( insert_o @ A @ A4 ) @ B4 )
= bot_bot_set_o )
= ( ~ ( member_o @ A @ B4 )
& ( ( inf_inf_set_o @ A4 @ B4 )
= bot_bot_set_o ) ) ) ).
% insert_disjoint(1)
thf(fact_1067_insert__disjoint_I2_J,axiom,
! [A: a,A4: set_a,B4: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A @ A4 ) @ B4 ) )
= ( ~ ( member_a @ A @ B4 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A4 @ B4 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1068_insert__disjoint_I2_J,axiom,
! [A: $o,A4: set_o,B4: set_o] :
( ( bot_bot_set_o
= ( inf_inf_set_o @ ( insert_o @ A @ A4 ) @ B4 ) )
= ( ~ ( member_o @ A @ B4 )
& ( bot_bot_set_o
= ( inf_inf_set_o @ A4 @ B4 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1069_disjoint__insert_I1_J,axiom,
! [B4: set_a,A: a,A4: set_a] :
( ( ( inf_inf_set_a @ B4 @ ( insert_a @ A @ A4 ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B4 )
& ( ( inf_inf_set_a @ B4 @ A4 )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_1070_disjoint__insert_I1_J,axiom,
! [B4: set_o,A: $o,A4: set_o] :
( ( ( inf_inf_set_o @ B4 @ ( insert_o @ A @ A4 ) )
= bot_bot_set_o )
= ( ~ ( member_o @ A @ B4 )
& ( ( inf_inf_set_o @ B4 @ A4 )
= bot_bot_set_o ) ) ) ).
% disjoint_insert(1)
thf(fact_1071_disjoint__insert_I2_J,axiom,
! [A4: set_a,B: a,B4: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A4 @ ( insert_a @ B @ B4 ) ) )
= ( ~ ( member_a @ B @ A4 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A4 @ B4 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1072_disjoint__insert_I2_J,axiom,
! [A4: set_o,B: $o,B4: set_o] :
( ( bot_bot_set_o
= ( inf_inf_set_o @ A4 @ ( insert_o @ B @ B4 ) ) )
= ( ~ ( member_o @ B @ A4 )
& ( bot_bot_set_o
= ( inf_inf_set_o @ A4 @ B4 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1073_Diff__eq__empty__iff,axiom,
! [A4: set_o,B4: set_o] :
( ( ( minus_minus_set_o @ A4 @ B4 )
= bot_bot_set_o )
= ( ord_less_eq_set_o @ A4 @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_1074_insert__Diff__single,axiom,
! [A: $o,A4: set_o] :
( ( insert_o @ A @ ( minus_minus_set_o @ A4 @ ( insert_o @ A @ bot_bot_set_o ) ) )
= ( insert_o @ A @ A4 ) ) ).
% insert_Diff_single
thf(fact_1075_Diff__disjoint,axiom,
! [A4: set_o,B4: set_o] :
( ( inf_inf_set_o @ A4 @ ( minus_minus_set_o @ B4 @ A4 ) )
= bot_bot_set_o ) ).
% Diff_disjoint
thf(fact_1076_Compl__disjoint,axiom,
! [A4: set_o] :
( ( inf_inf_set_o @ A4 @ ( uminus_uminus_set_o @ A4 ) )
= bot_bot_set_o ) ).
% Compl_disjoint
thf(fact_1077_Compl__disjoint2,axiom,
! [A4: set_o] :
( ( inf_inf_set_o @ ( uminus_uminus_set_o @ A4 ) @ A4 )
= bot_bot_set_o ) ).
% Compl_disjoint2
thf(fact_1078_range__constant,axiom,
! [X3: $o] :
( ( image_o_o
@ ^ [Uu: $o] : X3
@ top_top_set_o )
= ( insert_o @ X3 @ bot_bot_set_o ) ) ).
% range_constant
thf(fact_1079_Diff__triv,axiom,
! [A4: set_o,B4: set_o] :
( ( ( inf_inf_set_o @ A4 @ B4 )
= bot_bot_set_o )
=> ( ( minus_minus_set_o @ A4 @ B4 )
= A4 ) ) ).
% Diff_triv
thf(fact_1080_Int__Diff__disjoint,axiom,
! [A4: set_o,B4: set_o] :
( ( inf_inf_set_o @ ( inf_inf_set_o @ A4 @ B4 ) @ ( minus_minus_set_o @ A4 @ B4 ) )
= bot_bot_set_o ) ).
% Int_Diff_disjoint
thf(fact_1081_in__image__insert__iff,axiom,
! [B4: set_set_a,X3: a,A4: set_a] :
( ! [C6: set_a] :
( ( member_set_a @ C6 @ B4 )
=> ~ ( member_a @ X3 @ C6 ) )
=> ( ( member_set_a @ A4 @ ( image_set_a_set_a @ ( insert_a @ X3 ) @ B4 ) )
= ( ( member_a @ X3 @ A4 )
& ( member_set_a @ ( minus_minus_set_a @ A4 @ ( insert_a @ X3 @ bot_bot_set_a ) ) @ B4 ) ) ) ) ).
% in_image_insert_iff
thf(fact_1082_in__image__insert__iff,axiom,
! [B4: set_set_o,X3: $o,A4: set_o] :
( ! [C6: set_o] :
( ( member_set_o @ C6 @ B4 )
=> ~ ( member_o @ X3 @ C6 ) )
=> ( ( member_set_o @ A4 @ ( image_set_o_set_o @ ( insert_o @ X3 ) @ B4 ) )
= ( ( member_o @ X3 @ A4 )
& ( member_set_o @ ( minus_minus_set_o @ A4 @ ( insert_o @ X3 @ bot_bot_set_o ) ) @ B4 ) ) ) ) ).
% in_image_insert_iff
thf(fact_1083_set__diff__eq,axiom,
( minus_minus_set_a
= ( ^ [A5: set_a,B5: set_a] :
( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A5 )
& ~ ( member_a @ X @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1084_rev__image__eqI,axiom,
! [X3: a,A4: set_a,B: a,F: a > a] :
( ( member_a @ X3 @ A4 )
=> ( ( B
= ( F @ X3 ) )
=> ( member_a @ B @ ( image_a_a @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_1085_imageI,axiom,
! [X3: a,A4: set_a,F: a > a] :
( ( member_a @ X3 @ A4 )
=> ( member_a @ ( F @ X3 ) @ ( image_a_a @ F @ A4 ) ) ) ).
% imageI
thf(fact_1086_DiffD2,axiom,
! [C: a,A4: set_a,B4: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A4 @ B4 ) )
=> ~ ( member_a @ C @ B4 ) ) ).
% DiffD2
thf(fact_1087_DiffD1,axiom,
! [C: a,A4: set_a,B4: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A4 @ B4 ) )
=> ( member_a @ C @ A4 ) ) ).
% DiffD1
thf(fact_1088_IntD2,axiom,
! [C: a,A4: set_a,B4: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A4 @ B4 ) )
=> ( member_a @ C @ B4 ) ) ).
% IntD2
thf(fact_1089_IntD1,axiom,
! [C: a,A4: set_a,B4: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A4 @ B4 ) )
=> ( member_a @ C @ A4 ) ) ).
% IntD1
thf(fact_1090_DiffE,axiom,
! [C: a,A4: set_a,B4: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A4 @ B4 ) )
=> ~ ( ( member_a @ C @ A4 )
=> ( member_a @ C @ B4 ) ) ) ).
% DiffE
thf(fact_1091_IntE,axiom,
! [C: a,A4: set_a,B4: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A4 @ B4 ) )
=> ~ ( ( member_a @ C @ A4 )
=> ~ ( member_a @ C @ B4 ) ) ) ).
% IntE
thf(fact_1092_Int__def,axiom,
( inf_inf_set_a
= ( ^ [A5: set_a,B5: set_a] :
( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A5 )
& ( member_a @ X @ B5 ) ) ) ) ) ).
% Int_def
thf(fact_1093_Int__Collect,axiom,
! [X3: a,A4: set_a,P: a > $o] :
( ( member_a @ X3 @ ( inf_inf_set_a @ A4 @ ( collect_a @ P ) ) )
= ( ( member_a @ X3 @ A4 )
& ( P @ X3 ) ) ) ).
% Int_Collect
thf(fact_1094_imageE,axiom,
! [B: a,F: a > a,A4: set_a] :
( ( member_a @ B @ ( image_a_a @ F @ A4 ) )
=> ~ ! [X2: a] :
( ( B
= ( F @ X2 ) )
=> ~ ( member_a @ X2 @ A4 ) ) ) ).
% imageE
thf(fact_1095_Compr__image__eq,axiom,
! [F: a > a,A4: set_a,P: a > $o] :
( ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( image_a_a @ F @ A4 ) )
& ( P @ X ) ) )
= ( image_a_a @ F
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A4 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1096_psubset__imp__ex__mem,axiom,
! [A4: set_a,B4: set_a] :
( ( ord_less_set_a @ A4 @ B4 )
=> ? [B3: a] : ( member_a @ B3 @ ( minus_minus_set_a @ B4 @ A4 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1097_insert__Diff__if,axiom,
! [X3: $o,B4: set_o,A4: set_o] :
( ( ( member_o @ X3 @ B4 )
=> ( ( minus_minus_set_o @ ( insert_o @ X3 @ A4 ) @ B4 )
= ( minus_minus_set_o @ A4 @ B4 ) ) )
& ( ~ ( member_o @ X3 @ B4 )
=> ( ( minus_minus_set_o @ ( insert_o @ X3 @ A4 ) @ B4 )
= ( insert_o @ X3 @ ( minus_minus_set_o @ A4 @ B4 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1098_insert__Diff__if,axiom,
! [X3: a,B4: set_a,A4: set_a] :
( ( ( member_a @ X3 @ B4 )
=> ( ( minus_minus_set_a @ ( insert_a @ X3 @ A4 ) @ B4 )
= ( minus_minus_set_a @ A4 @ B4 ) ) )
& ( ~ ( member_a @ X3 @ B4 )
=> ( ( minus_minus_set_a @ ( insert_a @ X3 @ A4 ) @ B4 )
= ( insert_a @ X3 @ ( minus_minus_set_a @ A4 @ B4 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1099_Int__insert__left,axiom,
! [A: $o,C2: set_o,B4: set_o] :
( ( ( member_o @ A @ C2 )
=> ( ( inf_inf_set_o @ ( insert_o @ A @ B4 ) @ C2 )
= ( insert_o @ A @ ( inf_inf_set_o @ B4 @ C2 ) ) ) )
& ( ~ ( member_o @ A @ C2 )
=> ( ( inf_inf_set_o @ ( insert_o @ A @ B4 ) @ C2 )
= ( inf_inf_set_o @ B4 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1100_Int__insert__left,axiom,
! [A: a,C2: set_a,B4: set_a] :
( ( ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B4 ) @ C2 )
= ( insert_a @ A @ ( inf_inf_set_a @ B4 @ C2 ) ) ) )
& ( ~ ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B4 ) @ C2 )
= ( inf_inf_set_a @ B4 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1101_Int__insert__right,axiom,
! [A: $o,A4: set_o,B4: set_o] :
( ( ( member_o @ A @ A4 )
=> ( ( inf_inf_set_o @ A4 @ ( insert_o @ A @ B4 ) )
= ( insert_o @ A @ ( inf_inf_set_o @ A4 @ B4 ) ) ) )
& ( ~ ( member_o @ A @ A4 )
=> ( ( inf_inf_set_o @ A4 @ ( insert_o @ A @ B4 ) )
= ( inf_inf_set_o @ A4 @ B4 ) ) ) ) ).
% Int_insert_right
thf(fact_1102_Int__insert__right,axiom,
! [A: a,A4: set_a,B4: set_a] :
( ( ( member_a @ A @ A4 )
=> ( ( inf_inf_set_a @ A4 @ ( insert_a @ A @ B4 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A4 @ B4 ) ) ) )
& ( ~ ( member_a @ A @ A4 )
=> ( ( inf_inf_set_a @ A4 @ ( insert_a @ A @ B4 ) )
= ( inf_inf_set_a @ A4 @ B4 ) ) ) ) ).
% Int_insert_right
thf(fact_1103_Int__Collect__mono,axiom,
! [A4: set_a,B4: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A4 @ B4 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A4 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A4 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B4 @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1104_disjoint__iff__not__equal,axiom,
! [A4: set_o,B4: set_o] :
( ( ( inf_inf_set_o @ A4 @ B4 )
= bot_bot_set_o )
= ( ! [X: $o] :
( ( member_o @ X @ A4 )
=> ! [Y2: $o] :
( ( member_o @ Y2 @ B4 )
=> ( X = (~ Y2) ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1105_Int__empty__right,axiom,
! [A4: set_o] :
( ( inf_inf_set_o @ A4 @ bot_bot_set_o )
= bot_bot_set_o ) ).
% Int_empty_right
thf(fact_1106_Int__empty__left,axiom,
! [B4: set_o] :
( ( inf_inf_set_o @ bot_bot_set_o @ B4 )
= bot_bot_set_o ) ).
% Int_empty_left
thf(fact_1107_disjoint__iff,axiom,
! [A4: set_a,B4: set_a] :
( ( ( inf_inf_set_a @ A4 @ B4 )
= bot_bot_set_a )
= ( ! [X: a] :
( ( member_a @ X @ A4 )
=> ~ ( member_a @ X @ B4 ) ) ) ) ).
% disjoint_iff
thf(fact_1108_disjoint__iff,axiom,
! [A4: set_o,B4: set_o] :
( ( ( inf_inf_set_o @ A4 @ B4 )
= bot_bot_set_o )
= ( ! [X: $o] :
( ( member_o @ X @ A4 )
=> ~ ( member_o @ X @ B4 ) ) ) ) ).
% disjoint_iff
thf(fact_1109_Int__emptyI,axiom,
! [A4: set_a,B4: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A4 )
=> ~ ( member_a @ X2 @ B4 ) )
=> ( ( inf_inf_set_a @ A4 @ B4 )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_1110_Int__emptyI,axiom,
! [A4: set_o,B4: set_o] :
( ! [X2: $o] :
( ( member_o @ X2 @ A4 )
=> ~ ( member_o @ X2 @ B4 ) )
=> ( ( inf_inf_set_o @ A4 @ B4 )
= bot_bot_set_o ) ) ).
% Int_emptyI
thf(fact_1111_Int__UNIV__left,axiom,
! [B4: set_o] :
( ( inf_inf_set_o @ top_top_set_o @ B4 )
= B4 ) ).
% Int_UNIV_left
thf(fact_1112_Int__UNIV__right,axiom,
! [A4: set_o] :
( ( inf_inf_set_o @ A4 @ top_top_set_o )
= A4 ) ).
% Int_UNIV_right
thf(fact_1113_image__subsetI,axiom,
! [A4: set_a,F: a > a,B4: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A4 )
=> ( member_a @ ( F @ X2 ) @ B4 ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A4 ) @ B4 ) ) ).
% image_subsetI
thf(fact_1114_rangeI,axiom,
! [F: $o > a,X3: $o] : ( member_a @ ( F @ X3 ) @ ( image_o_a @ F @ top_top_set_o ) ) ).
% rangeI
thf(fact_1115_range__eqI,axiom,
! [B: a,F: $o > a,X3: $o] :
( ( B
= ( F @ X3 ) )
=> ( member_a @ B @ ( image_o_a @ F @ top_top_set_o ) ) ) ).
% range_eqI
thf(fact_1116_rangeE,axiom,
! [B: a,F: $o > a] :
( ( member_a @ B @ ( image_o_a @ F @ top_top_set_o ) )
=> ~ ! [X2: $o] :
( B
!= ( F @ X2 ) ) ) ).
% rangeE
thf(fact_1117_range__subsetD,axiom,
! [F: $o > a,B4: set_a,I: $o] :
( ( ord_less_eq_set_a @ ( image_o_a @ F @ top_top_set_o ) @ B4 )
=> ( member_a @ ( F @ I ) @ B4 ) ) ).
% range_subsetD
thf(fact_1118_diff__shunt__var,axiom,
! [X3: set_o,Y: set_o] :
( ( ( minus_minus_set_o @ X3 @ Y )
= bot_bot_set_o )
= ( ord_less_eq_set_o @ X3 @ Y ) ) ).
% diff_shunt_var
thf(fact_1119_inj__on__image__mem__iff,axiom,
! [F: a > a,B4: set_a,A: a,A4: set_a] :
( ( inj_on_a_a @ F @ B4 )
=> ( ( member_a @ A @ B4 )
=> ( ( ord_less_eq_set_a @ A4 @ B4 )
=> ( ( member_a @ ( F @ A ) @ ( image_a_a @ F @ A4 ) )
= ( member_a @ A @ A4 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1120_bdd__belowI2,axiom,
! [A4: set_a,M2: a,F: a > a] :
( ! [X2: a] :
( ( member_a @ X2 @ A4 )
=> ( ord_less_eq_a @ M2 @ ( F @ X2 ) ) )
=> ( condit5901475214736682318elow_a @ ( image_a_a @ F @ A4 ) ) ) ).
% bdd_belowI2
thf(fact_1121_bdd__below_OI2,axiom,
! [A4: set_a,M: a,F: a > a] :
( ! [X2: a] :
( ( member_a @ X2 @ A4 )
=> ( ord_less_eq_a @ M @ ( F @ X2 ) ) )
=> ( condit5901475214736682318elow_a @ ( image_a_a @ F @ A4 ) ) ) ).
% bdd_below.I2
thf(fact_1122_Diff__insert__absorb,axiom,
! [X3: a,A4: set_a] :
( ~ ( member_a @ X3 @ A4 )
=> ( ( minus_minus_set_a @ ( insert_a @ X3 @ A4 ) @ ( insert_a @ X3 @ bot_bot_set_a ) )
= A4 ) ) ).
% Diff_insert_absorb
thf(fact_1123_Diff__insert__absorb,axiom,
! [X3: $o,A4: set_o] :
( ~ ( member_o @ X3 @ A4 )
=> ( ( minus_minus_set_o @ ( insert_o @ X3 @ A4 ) @ ( insert_o @ X3 @ bot_bot_set_o ) )
= A4 ) ) ).
% Diff_insert_absorb
thf(fact_1124_Diff__insert2,axiom,
! [A4: set_o,A: $o,B4: set_o] :
( ( minus_minus_set_o @ A4 @ ( insert_o @ A @ B4 ) )
= ( minus_minus_set_o @ ( minus_minus_set_o @ A4 @ ( insert_o @ A @ bot_bot_set_o ) ) @ B4 ) ) ).
% Diff_insert2
thf(fact_1125_insert__Diff,axiom,
! [A: a,A4: set_a] :
( ( member_a @ A @ A4 )
=> ( ( insert_a @ A @ ( minus_minus_set_a @ A4 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= A4 ) ) ).
% insert_Diff
thf(fact_1126_insert__Diff,axiom,
! [A: $o,A4: set_o] :
( ( member_o @ A @ A4 )
=> ( ( insert_o @ A @ ( minus_minus_set_o @ A4 @ ( insert_o @ A @ bot_bot_set_o ) ) )
= A4 ) ) ).
% insert_Diff
thf(fact_1127_Diff__insert,axiom,
! [A4: set_o,A: $o,B4: set_o] :
( ( minus_minus_set_o @ A4 @ ( insert_o @ A @ B4 ) )
= ( minus_minus_set_o @ ( minus_minus_set_o @ A4 @ B4 ) @ ( insert_o @ A @ bot_bot_set_o ) ) ) ).
% Diff_insert
thf(fact_1128_subset__Diff__insert,axiom,
! [A4: set_o,B4: set_o,X3: $o,C2: set_o] :
( ( ord_less_eq_set_o @ A4 @ ( minus_minus_set_o @ B4 @ ( insert_o @ X3 @ C2 ) ) )
= ( ( ord_less_eq_set_o @ A4 @ ( minus_minus_set_o @ B4 @ C2 ) )
& ~ ( member_o @ X3 @ A4 ) ) ) ).
% subset_Diff_insert
thf(fact_1129_subset__Diff__insert,axiom,
! [A4: set_a,B4: set_a,X3: a,C2: set_a] :
( ( ord_less_eq_set_a @ A4 @ ( minus_minus_set_a @ B4 @ ( insert_a @ X3 @ C2 ) ) )
= ( ( ord_less_eq_set_a @ A4 @ ( minus_minus_set_a @ B4 @ C2 ) )
& ~ ( member_a @ X3 @ A4 ) ) ) ).
% subset_Diff_insert
thf(fact_1130_Compl__eq__Diff__UNIV,axiom,
( uminus_uminus_set_o
= ( minus_minus_set_o @ top_top_set_o ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_1131_image__constant__conv,axiom,
! [A4: set_o,C: $o] :
( ( ( A4 = bot_bot_set_o )
=> ( ( image_o_o
@ ^ [X: $o] : C
@ A4 )
= bot_bot_set_o ) )
& ( ( A4 != bot_bot_set_o )
=> ( ( image_o_o
@ ^ [X: $o] : C
@ A4 )
= ( insert_o @ C @ bot_bot_set_o ) ) ) ) ).
% image_constant_conv
thf(fact_1132_image__constant,axiom,
! [X3: a,A4: set_a,C: $o] :
( ( member_a @ X3 @ A4 )
=> ( ( image_a_o
@ ^ [X: a] : C
@ A4 )
= ( insert_o @ C @ bot_bot_set_o ) ) ) ).
% image_constant
thf(fact_1133_inf__shunt,axiom,
! [X3: set_o,Y: set_o] :
( ( ( inf_inf_set_o @ X3 @ Y )
= bot_bot_set_o )
= ( ord_less_eq_set_o @ X3 @ ( uminus_uminus_set_o @ Y ) ) ) ).
% inf_shunt
thf(fact_1134_range__eq__singletonD,axiom,
! [F: $o > $o,A: $o,X3: $o] :
( ( ( image_o_o @ F @ top_top_set_o )
= ( insert_o @ A @ bot_bot_set_o ) )
=> ( ( F @ X3 )
= A ) ) ).
% range_eq_singletonD
thf(fact_1135_vimage__subsetD,axiom,
! [F: $o > $o,B4: set_o,A4: set_o] :
( ( ( image_o_o @ F @ top_top_set_o )
= top_top_set_o )
=> ( ( ord_less_eq_set_o @ ( vimage_o_o @ F @ B4 ) @ A4 )
=> ( ord_less_eq_set_o @ B4 @ ( image_o_o @ F @ A4 ) ) ) ) ).
% vimage_subsetD
thf(fact_1136_surj__Compl__image__subset,axiom,
! [F: $o > $o,A4: set_o] :
( ( ( image_o_o @ F @ top_top_set_o )
= top_top_set_o )
=> ( ord_less_eq_set_o @ ( uminus_uminus_set_o @ ( image_o_o @ F @ A4 ) ) @ ( image_o_o @ F @ ( uminus_uminus_set_o @ A4 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_1137_Diff__single__insert,axiom,
! [A4: set_o,X3: $o,B4: set_o] :
( ( ord_less_eq_set_o @ ( minus_minus_set_o @ A4 @ ( insert_o @ X3 @ bot_bot_set_o ) ) @ B4 )
=> ( ord_less_eq_set_o @ A4 @ ( insert_o @ X3 @ B4 ) ) ) ).
% Diff_single_insert
thf(fact_1138_subset__insert__iff,axiom,
! [A4: set_a,X3: a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ ( insert_a @ X3 @ B4 ) )
= ( ( ( member_a @ X3 @ A4 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A4 @ ( insert_a @ X3 @ bot_bot_set_a ) ) @ B4 ) )
& ( ~ ( member_a @ X3 @ A4 )
=> ( ord_less_eq_set_a @ A4 @ B4 ) ) ) ) ).
% subset_insert_iff
thf(fact_1139_subset__insert__iff,axiom,
! [A4: set_o,X3: $o,B4: set_o] :
( ( ord_less_eq_set_o @ A4 @ ( insert_o @ X3 @ B4 ) )
= ( ( ( member_o @ X3 @ A4 )
=> ( ord_less_eq_set_o @ ( minus_minus_set_o @ A4 @ ( insert_o @ X3 @ bot_bot_set_o ) ) @ B4 ) )
& ( ~ ( member_o @ X3 @ A4 )
=> ( ord_less_eq_set_o @ A4 @ B4 ) ) ) ) ).
% subset_insert_iff
thf(fact_1140_disjoint__eq__subset__Compl,axiom,
! [A4: set_o,B4: set_o] :
( ( ( inf_inf_set_o @ A4 @ B4 )
= bot_bot_set_o )
= ( ord_less_eq_set_o @ A4 @ ( uminus_uminus_set_o @ B4 ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_1141_Compl__insert,axiom,
! [X3: $o,A4: set_o] :
( ( uminus_uminus_set_o @ ( insert_o @ X3 @ A4 ) )
= ( minus_minus_set_o @ ( uminus_uminus_set_o @ A4 ) @ ( insert_o @ X3 @ bot_bot_set_o ) ) ) ).
% Compl_insert
thf(fact_1142_inf__Int__eq,axiom,
! [R: set_a,S2: set_a] :
( ( inf_inf_a_o
@ ^ [X: a] : ( member_a @ X @ R )
@ ^ [X: a] : ( member_a @ X @ S2 ) )
= ( ^ [X: a] : ( member_a @ X @ ( inf_inf_set_a @ R @ S2 ) ) ) ) ).
% inf_Int_eq
thf(fact_1143_inf__set__def,axiom,
( inf_inf_set_a
= ( ^ [A5: set_a,B5: set_a] :
( collect_a
@ ( inf_inf_a_o
@ ^ [X: a] : ( member_a @ X @ A5 )
@ ^ [X: a] : ( member_a @ X @ B5 ) ) ) ) ) ).
% inf_set_def
thf(fact_1144_minus__set__def,axiom,
( minus_minus_set_a
= ( ^ [A5: set_a,B5: set_a] :
( collect_a
@ ( minus_minus_a_o
@ ^ [X: a] : ( member_a @ X @ A5 )
@ ^ [X: a] : ( member_a @ X @ B5 ) ) ) ) ) ).
% minus_set_def
thf(fact_1145_inf__top_Osemilattice__neutr__order__axioms,axiom,
semila2554085542299052326_set_o @ inf_inf_set_o @ top_top_set_o @ ord_less_eq_set_o @ ord_less_set_o ).
% inf_top.semilattice_neutr_order_axioms
thf(fact_1146_remove__def,axiom,
( remove_o
= ( ^ [X: $o,A5: set_o] : ( minus_minus_set_o @ A5 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ).
% remove_def
thf(fact_1147_flat__lub__def,axiom,
( partial_flat_lub_a
= ( ^ [B2: a,A5: set_a] :
( if_a @ ( ord_less_eq_set_a @ A5 @ ( insert_a @ B2 @ bot_bot_set_a ) ) @ B2
@ ( the_a
@ ^ [X: a] : ( member_a @ X @ ( minus_minus_set_a @ A5 @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ) ) ) ).
% flat_lub_def
thf(fact_1148_flat__lub__def,axiom,
( partial_flat_lub_o
= ( ^ [B2: $o,A5: set_o] :
( ( ( ord_less_eq_set_o @ A5 @ ( insert_o @ B2 @ bot_bot_set_o ) )
=> B2 )
& ( ~ ( ord_less_eq_set_o @ A5 @ ( insert_o @ B2 @ bot_bot_set_o ) )
=> ( the_o
@ ^ [X: $o] : ( member_o @ X @ ( minus_minus_set_o @ A5 @ ( insert_o @ B2 @ bot_bot_set_o ) ) ) ) ) ) ) ) ).
% flat_lub_def
thf(fact_1149_member__remove,axiom,
! [X3: a,Y: a,A4: set_a] :
( ( member_a @ X3 @ ( remove_a @ Y @ A4 ) )
= ( ( member_a @ X3 @ A4 )
& ( X3 != Y ) ) ) ).
% member_remove
thf(fact_1150_INT__simps_I3_J,axiom,
! [C2: set_o,A4: $o > set_o,B4: set_o] :
( ( ( C2 = bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( minus_minus_set_o @ ( A4 @ X ) @ B4 )
@ C2 ) )
= top_top_set_o ) )
& ( ( C2 != bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( minus_minus_set_o @ ( A4 @ X ) @ B4 )
@ C2 ) )
= ( minus_minus_set_o @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ A4 @ C2 ) ) @ B4 ) ) ) ) ).
% INT_simps(3)
thf(fact_1151_INT__simps_I2_J,axiom,
! [C2: set_o,A4: set_o,B4: $o > set_o] :
( ( ( C2 = bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( inf_inf_set_o @ A4 @ ( B4 @ X ) )
@ C2 ) )
= top_top_set_o ) )
& ( ( C2 != bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( inf_inf_set_o @ A4 @ ( B4 @ X ) )
@ C2 ) )
= ( inf_inf_set_o @ A4 @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ B4 @ C2 ) ) ) ) ) ) ).
% INT_simps(2)
thf(fact_1152_INT__simps_I1_J,axiom,
! [C2: set_o,A4: $o > set_o,B4: set_o] :
( ( ( C2 = bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( inf_inf_set_o @ ( A4 @ X ) @ B4 )
@ C2 ) )
= top_top_set_o ) )
& ( ( C2 != bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( inf_inf_set_o @ ( A4 @ X ) @ B4 )
@ C2 ) )
= ( inf_inf_set_o @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ A4 @ C2 ) ) @ B4 ) ) ) ) ).
% INT_simps(1)
thf(fact_1153_INT__I,axiom,
! [A4: set_a,B: a,B4: a > set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A4 )
=> ( member_a @ B @ ( B4 @ X2 ) ) )
=> ( member_a @ B @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ B4 @ A4 ) ) ) ) ).
% INT_I
thf(fact_1154_INT__constant,axiom,
! [A4: set_o,C: set_o] :
( ( ( A4 = bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [Y2: $o] : C
@ A4 ) )
= top_top_set_o ) )
& ( ( A4 != bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [Y2: $o] : C
@ A4 ) )
= C ) ) ) ).
% INT_constant
thf(fact_1155_Inter__eq,axiom,
( comple6135023378680113637_set_a
= ( ^ [A5: set_set_a] :
( collect_a
@ ^ [X: a] :
! [Y2: set_a] :
( ( member_set_a @ Y2 @ A5 )
=> ( member_a @ X @ Y2 ) ) ) ) ) ).
% Inter_eq
thf(fact_1156_INT__E,axiom,
! [B: a,B4: a > set_a,A4: set_a,A: a] :
( ( member_a @ B @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ B4 @ A4 ) ) )
=> ( ~ ( member_a @ B @ ( B4 @ A ) )
=> ~ ( member_a @ A @ A4 ) ) ) ).
% INT_E
thf(fact_1157_INT__D,axiom,
! [B: a,B4: a > set_a,A4: set_a,A: a] :
( ( member_a @ B @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ B4 @ A4 ) ) )
=> ( ( member_a @ A @ A4 )
=> ( member_a @ B @ ( B4 @ A ) ) ) ) ).
% INT_D
thf(fact_1158_Inf__less__eq,axiom,
! [A4: set_o,U2: $o] :
( ! [V: $o] :
( ( member_o @ V @ A4 )
=> ( ord_less_eq_o @ V @ U2 ) )
=> ( ( A4 != bot_bot_set_o )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A4 ) @ U2 ) ) ) ).
% Inf_less_eq
thf(fact_1159_cInf__greatest,axiom,
! [X6: set_o,Z3: $o] :
( ( X6 != bot_bot_set_o )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ X6 )
=> ( ord_less_eq_o @ Z3 @ X2 ) )
=> ( ord_less_eq_o @ Z3 @ ( complete_Inf_Inf_o @ X6 ) ) ) ) ).
% cInf_greatest
thf(fact_1160_cInf__eq__non__empty,axiom,
! [X6: set_o,A: $o] :
( ( X6 != bot_bot_set_o )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ X6 )
=> ( ord_less_eq_o @ A @ X2 ) )
=> ( ! [Y4: $o] :
( ! [X4: $o] :
( ( member_o @ X4 @ X6 )
=> ( ord_less_eq_o @ Y4 @ X4 ) )
=> ( ord_less_eq_o @ Y4 @ A ) )
=> ( ( complete_Inf_Inf_o @ X6 )
= A ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_1161_INT__insert__distrib,axiom,
! [U2: a,A4: set_a,A: $o,B4: a > set_o] :
( ( member_a @ U2 @ A4 )
=> ( ( comple3063163877087187839_set_o
@ ( image_a_set_o
@ ^ [X: a] : ( insert_o @ A @ ( B4 @ X ) )
@ A4 ) )
= ( insert_o @ A @ ( comple3063163877087187839_set_o @ ( image_a_set_o @ B4 @ A4 ) ) ) ) ) ).
% INT_insert_distrib
thf(fact_1162_cInf__mono,axiom,
! [B4: set_o,A4: set_o] :
( ( B4 != bot_bot_set_o )
=> ( ( condit5413489452508810728elow_o @ A4 )
=> ( ! [B3: $o] :
( ( member_o @ B3 @ B4 )
=> ? [X4: $o] :
( ( member_o @ X4 @ A4 )
& ( ord_less_eq_o @ X4 @ B3 ) ) )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A4 ) @ ( complete_Inf_Inf_o @ B4 ) ) ) ) ) ).
% cInf_mono
thf(fact_1163_le__cInf__iff,axiom,
! [S2: set_o,A: $o] :
( ( S2 != bot_bot_set_o )
=> ( ( condit5413489452508810728elow_o @ S2 )
=> ( ( ord_less_eq_o @ A @ ( complete_Inf_Inf_o @ S2 ) )
= ( ! [X: $o] :
( ( member_o @ X @ S2 )
=> ( ord_less_eq_o @ A @ X ) ) ) ) ) ) ).
% le_cInf_iff
thf(fact_1164_INF__constant,axiom,
! [A4: set_o,C: set_o] :
( ( ( A4 = bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [Y2: $o] : C
@ A4 ) )
= top_top_set_o ) )
& ( ( A4 != bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [Y2: $o] : C
@ A4 ) )
= C ) ) ) ).
% INF_constant
thf(fact_1165_INF__empty,axiom,
! [F: $o > set_o] :
( ( comple3063163877087187839_set_o @ ( image_o_set_o @ F @ bot_bot_set_o ) )
= top_top_set_o ) ).
% INF_empty
thf(fact_1166_INT__empty,axiom,
! [B4: $o > set_o] :
( ( comple3063163877087187839_set_o @ ( image_o_set_o @ B4 @ bot_bot_set_o ) )
= top_top_set_o ) ).
% INT_empty
thf(fact_1167_cInf__superset__mono,axiom,
! [A4: set_o,B4: set_o] :
( ( A4 != bot_bot_set_o )
=> ( ( condit5413489452508810728elow_o @ B4 )
=> ( ( ord_less_eq_set_o @ A4 @ B4 )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ B4 ) @ ( complete_Inf_Inf_o @ A4 ) ) ) ) ) ).
% cInf_superset_mono
thf(fact_1168_INT__extend__simps_I3_J,axiom,
! [C2: set_o,A4: $o > set_o,B4: set_o] :
( ( ( C2 = bot_bot_set_o )
=> ( ( minus_minus_set_o @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ A4 @ C2 ) ) @ B4 )
= ( minus_minus_set_o @ top_top_set_o @ B4 ) ) )
& ( ( C2 != bot_bot_set_o )
=> ( ( minus_minus_set_o @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ A4 @ C2 ) ) @ B4 )
= ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( minus_minus_set_o @ ( A4 @ X ) @ B4 )
@ C2 ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_1169_less__eq__cInf__inter,axiom,
! [A4: set_o,B4: set_o] :
( ( condit5413489452508810728elow_o @ A4 )
=> ( ( condit5413489452508810728elow_o @ B4 )
=> ( ( ( inf_inf_set_o @ A4 @ B4 )
!= bot_bot_set_o )
=> ( ord_less_eq_o @ ( inf_inf_o @ ( complete_Inf_Inf_o @ A4 ) @ ( complete_Inf_Inf_o @ B4 ) ) @ ( complete_Inf_Inf_o @ ( inf_inf_set_o @ A4 @ B4 ) ) ) ) ) ) ).
% less_eq_cInf_inter
thf(fact_1170_INT__simps_I4_J,axiom,
! [C2: set_o,A4: set_o,B4: $o > set_o] :
( ( ( C2 = bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( minus_minus_set_o @ A4 @ ( B4 @ X ) )
@ C2 ) )
= top_top_set_o ) )
& ( ( C2 != bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( minus_minus_set_o @ A4 @ ( B4 @ X ) )
@ C2 ) )
= ( minus_minus_set_o @ A4 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B4 @ C2 ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_1171_UN__I,axiom,
! [A: a,A4: set_a,B: a,B4: a > set_a] :
( ( member_a @ A @ A4 )
=> ( ( member_a @ B @ ( B4 @ A ) )
=> ( member_a @ B @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B4 @ A4 ) ) ) ) ) ).
% UN_I
thf(fact_1172_UN__constant,axiom,
! [A4: set_o,C: set_o] :
( ( ( A4 = bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [Y2: $o] : C
@ A4 ) )
= bot_bot_set_o ) )
& ( ( A4 != bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [Y2: $o] : C
@ A4 ) )
= C ) ) ) ).
% UN_constant
thf(fact_1173_UN__simps_I1_J,axiom,
! [C2: set_o,A: $o,B4: $o > set_o] :
( ( ( C2 = bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( insert_o @ A @ ( B4 @ X ) )
@ C2 ) )
= bot_bot_set_o ) )
& ( ( C2 != bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( insert_o @ A @ ( B4 @ X ) )
@ C2 ) )
= ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B4 @ C2 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_1174_UN__singleton,axiom,
! [A4: set_o] :
( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( insert_o @ X @ bot_bot_set_o )
@ A4 ) )
= A4 ) ).
% UN_singleton
thf(fact_1175_Inf__INT__eq,axiom,
( complete_Inf_Inf_a_o
= ( ^ [S: set_a_o,X: a] : ( member_a @ X @ ( comple6135023378680113637_set_a @ ( image_a_o_set_a @ collect_a @ S ) ) ) ) ) ).
% Inf_INT_eq
thf(fact_1176_Inf__set__def,axiom,
( comple6135023378680113637_set_a
= ( ^ [A5: set_set_a] :
( collect_a
@ ^ [X: a] : ( complete_Inf_Inf_o @ ( image_set_a_o @ ( member_a @ X ) @ A5 ) ) ) ) ) ).
% Inf_set_def
thf(fact_1177_UN__E,axiom,
! [B: a,B4: a > set_a,A4: set_a] :
( ( member_a @ B @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B4 @ A4 ) ) )
=> ~ ! [X2: a] :
( ( member_a @ X2 @ A4 )
=> ~ ( member_a @ B @ ( B4 @ X2 ) ) ) ) ).
% UN_E
thf(fact_1178_INF__Int__eq,axiom,
! [S2: set_set_a] :
( ( complete_Inf_Inf_a_o
@ ( image_set_a_a_o
@ ^ [I2: set_a,X: a] : ( member_a @ X @ I2 )
@ S2 ) )
= ( ^ [X: a] : ( member_a @ X @ ( comple6135023378680113637_set_a @ S2 ) ) ) ) ).
% INF_Int_eq
thf(fact_1179_UNIV__bool,axiom,
( top_top_set_o
= ( insert_o @ $false @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).
% UNIV_bool
thf(fact_1180_strict__mono__on__imp__mono__on,axiom,
! [A4: set_a,F: a > a] :
( ( monotone_on_a_a @ A4 @ ord_less_a @ ord_less_a @ F )
=> ( monotone_on_a_a @ A4 @ ord_less_eq_a @ ord_less_eq_a @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_1181_strict__mono__on__leD,axiom,
! [A4: set_a,F: a > a,X3: a,Y: a] :
( ( monotone_on_a_a @ A4 @ ord_less_a @ ord_less_a @ F )
=> ( ( member_a @ X3 @ A4 )
=> ( ( member_a @ Y @ A4 )
=> ( ( ord_less_eq_a @ X3 @ Y )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_1182_mono__on__greaterD,axiom,
! [A4: set_a,G2: a > a,X3: a,Y: a] :
( ( monotone_on_a_a @ A4 @ ord_less_eq_a @ ord_less_eq_a @ G2 )
=> ( ( member_a @ X3 @ A4 )
=> ( ( member_a @ Y @ A4 )
=> ( ( ord_less_a @ ( G2 @ Y ) @ ( G2 @ X3 ) )
=> ( ord_less_a @ Y @ X3 ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_1183_ord_Omono__onD,axiom,
! [A4: set_a,Less_eq: a > a > $o,F: a > a,R2: a,S3: a] :
( ( monotone_on_a_a @ A4 @ Less_eq @ ord_less_eq_a @ F )
=> ( ( member_a @ R2 @ A4 )
=> ( ( member_a @ S3 @ A4 )
=> ( ( Less_eq @ R2 @ S3 )
=> ( ord_less_eq_a @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_1184_ord_Omono__onI,axiom,
! [A4: set_a,Less_eq: a > a > $o,F: a > a] :
( ! [R3: a,S5: a] :
( ( member_a @ R3 @ A4 )
=> ( ( member_a @ S5 @ A4 )
=> ( ( Less_eq @ R3 @ S5 )
=> ( ord_less_eq_a @ ( F @ R3 ) @ ( F @ S5 ) ) ) ) )
=> ( monotone_on_a_a @ A4 @ Less_eq @ ord_less_eq_a @ F ) ) ).
% ord.mono_onI
thf(fact_1185_ord_Omono__on__def,axiom,
! [A4: set_a,Less_eq: a > a > $o,F: a > a] :
( ( monotone_on_a_a @ A4 @ Less_eq @ ord_less_eq_a @ F )
= ( ! [R4: a,S4: a] :
( ( ( member_a @ R4 @ A4 )
& ( member_a @ S4 @ A4 )
& ( Less_eq @ R4 @ S4 ) )
=> ( ord_less_eq_a @ ( F @ R4 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_1186_mono__onD,axiom,
! [A4: set_a,F: a > a,R2: a,S3: a] :
( ( monotone_on_a_a @ A4 @ ord_less_eq_a @ ord_less_eq_a @ F )
=> ( ( member_a @ R2 @ A4 )
=> ( ( member_a @ S3 @ A4 )
=> ( ( ord_less_eq_a @ R2 @ S3 )
=> ( ord_less_eq_a @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) ) ) ).
% mono_onD
thf(fact_1187_mono__onI,axiom,
! [A4: set_a,F: a > a] :
( ! [R3: a,S5: a] :
( ( member_a @ R3 @ A4 )
=> ( ( member_a @ S5 @ A4 )
=> ( ( ord_less_eq_a @ R3 @ S5 )
=> ( ord_less_eq_a @ ( F @ R3 ) @ ( F @ S5 ) ) ) ) )
=> ( monotone_on_a_a @ A4 @ ord_less_eq_a @ ord_less_eq_a @ F ) ) ).
% mono_onI
thf(fact_1188_mono__on__subset,axiom,
! [A4: set_a,F: a > a,B4: set_a] :
( ( monotone_on_a_a @ A4 @ ord_less_eq_a @ ord_less_eq_a @ F )
=> ( ( ord_less_eq_set_a @ B4 @ A4 )
=> ( monotone_on_a_a @ B4 @ ord_less_eq_a @ ord_less_eq_a @ F ) ) ) ).
% mono_on_subset
thf(fact_1189_mono__imp__mono__on,axiom,
! [F: $o > a,A4: set_o] :
( ( monotone_on_o_a @ top_top_set_o @ ord_less_eq_o @ ord_less_eq_a @ F )
=> ( monotone_on_o_a @ A4 @ ord_less_eq_o @ ord_less_eq_a @ F ) ) ).
% mono_imp_mono_on
thf(fact_1190_mono__imp__mono__on,axiom,
! [F: a > a,A4: set_a] :
( ( monotone_on_a_a @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_a @ F )
=> ( monotone_on_a_a @ A4 @ ord_less_eq_a @ ord_less_eq_a @ F ) ) ).
% mono_imp_mono_on
thf(fact_1191_monoI,axiom,
! [F: $o > a] :
( ! [X2: $o,Y4: $o] :
( ( ord_less_eq_o @ X2 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( monotone_on_o_a @ top_top_set_o @ ord_less_eq_o @ ord_less_eq_a @ F ) ) ).
% monoI
thf(fact_1192_monoI,axiom,
! [F: a > a] :
( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( monotone_on_a_a @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_a @ F ) ) ).
% monoI
thf(fact_1193_monoE,axiom,
! [F: $o > a,X3: $o,Y: $o] :
( ( monotone_on_o_a @ top_top_set_o @ ord_less_eq_o @ ord_less_eq_a @ F )
=> ( ( ord_less_eq_o @ X3 @ Y )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_1194_monoE,axiom,
! [F: a > a,X3: a,Y: a] :
( ( monotone_on_a_a @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_a @ F )
=> ( ( ord_less_eq_a @ X3 @ Y )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_1195_monoD,axiom,
! [F: $o > a,X3: $o,Y: $o] :
( ( monotone_on_o_a @ top_top_set_o @ ord_less_eq_o @ ord_less_eq_a @ F )
=> ( ( ord_less_eq_o @ X3 @ Y )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_1196_monoD,axiom,
! [F: a > a,X3: a,Y: a] :
( ( monotone_on_a_a @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_a @ F )
=> ( ( ord_less_eq_a @ X3 @ Y )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_1197_cSup__eq__non__empty,axiom,
! [X6: set_o,A: $o] :
( ( X6 != bot_bot_set_o )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ X6 )
=> ( ord_less_eq_o @ X2 @ A ) )
=> ( ! [Y4: $o] :
( ! [X4: $o] :
( ( member_o @ X4 @ X6 )
=> ( ord_less_eq_o @ X4 @ Y4 ) )
=> ( ord_less_eq_o @ A @ Y4 ) )
=> ( ( complete_Sup_Sup_o @ X6 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_1198_cSup__least,axiom,
! [X6: set_o,Z3: $o] :
( ( X6 != bot_bot_set_o )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ X6 )
=> ( ord_less_eq_o @ X2 @ Z3 ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ X6 ) @ Z3 ) ) ) ).
% cSup_least
thf(fact_1199_less__eq__Sup,axiom,
! [A4: set_o,U2: $o] :
( ! [V: $o] :
( ( member_o @ V @ A4 )
=> ( ord_less_eq_o @ U2 @ V ) )
=> ( ( A4 != bot_bot_set_o )
=> ( ord_less_eq_o @ U2 @ ( complete_Sup_Sup_o @ A4 ) ) ) ) ).
% less_eq_Sup
thf(fact_1200_antimonoI,axiom,
! [F: $o > a] :
( ! [X2: $o,Y4: $o] :
( ( ord_less_eq_o @ X2 @ Y4 )
=> ( ord_less_eq_a @ ( F @ Y4 ) @ ( F @ X2 ) ) )
=> ( monotone_on_o_a @ top_top_set_o @ ord_less_eq_o
@ ^ [X: a,Y2: a] : ( ord_less_eq_a @ Y2 @ X )
@ F ) ) ).
% antimonoI
thf(fact_1201_antimonoI,axiom,
! [F: a > a] :
( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_less_eq_a @ ( F @ Y4 ) @ ( F @ X2 ) ) )
=> ( monotone_on_a_a @ top_top_set_a @ ord_less_eq_a
@ ^ [X: a,Y2: a] : ( ord_less_eq_a @ Y2 @ X )
@ F ) ) ).
% antimonoI
thf(fact_1202_antimonoE,axiom,
! [F: $o > a,X3: $o,Y: $o] :
( ( monotone_on_o_a @ top_top_set_o @ ord_less_eq_o
@ ^ [X: a,Y2: a] : ( ord_less_eq_a @ Y2 @ X )
@ F )
=> ( ( ord_less_eq_o @ X3 @ Y )
=> ( ord_less_eq_a @ ( F @ Y ) @ ( F @ X3 ) ) ) ) ).
% antimonoE
thf(fact_1203_antimonoE,axiom,
! [F: a > a,X3: a,Y: a] :
( ( monotone_on_a_a @ top_top_set_a @ ord_less_eq_a
@ ^ [X: a,Y2: a] : ( ord_less_eq_a @ Y2 @ X )
@ F )
=> ( ( ord_less_eq_a @ X3 @ Y )
=> ( ord_less_eq_a @ ( F @ Y ) @ ( F @ X3 ) ) ) ) ).
% antimonoE
thf(fact_1204_antimonoD,axiom,
! [F: $o > a,X3: $o,Y: $o] :
( ( monotone_on_o_a @ top_top_set_o @ ord_less_eq_o
@ ^ [X: a,Y2: a] : ( ord_less_eq_a @ Y2 @ X )
@ F )
=> ( ( ord_less_eq_o @ X3 @ Y )
=> ( ord_less_eq_a @ ( F @ Y ) @ ( F @ X3 ) ) ) ) ).
% antimonoD
thf(fact_1205_antimonoD,axiom,
! [F: a > a,X3: a,Y: a] :
( ( monotone_on_a_a @ top_top_set_a @ ord_less_eq_a
@ ^ [X: a,Y2: a] : ( ord_less_eq_a @ Y2 @ X )
@ F )
=> ( ( ord_less_eq_a @ X3 @ Y )
=> ( ord_less_eq_a @ ( F @ Y ) @ ( F @ X3 ) ) ) ) ).
% antimonoD
thf(fact_1206_UN__empty,axiom,
! [B4: $o > set_o] :
( ( comple90263536869209701_set_o @ ( image_o_set_o @ B4 @ bot_bot_set_o ) )
= bot_bot_set_o ) ).
% UN_empty
thf(fact_1207_UN__insert__distrib,axiom,
! [U2: a,A4: set_a,A: $o,B4: a > set_o] :
( ( member_a @ U2 @ A4 )
=> ( ( comple90263536869209701_set_o
@ ( image_a_set_o
@ ^ [X: a] : ( insert_o @ A @ ( B4 @ X ) )
@ A4 ) )
= ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_a_set_o @ B4 @ A4 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_1208_strict__mono__less__eq,axiom,
! [F: $o > a,X3: $o,Y: $o] :
( ( monotone_on_o_a @ top_top_set_o @ ord_less_o @ ord_less_a @ F )
=> ( ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y ) )
= ( ord_less_eq_o @ X3 @ Y ) ) ) ).
% strict_mono_less_eq
thf(fact_1209_strict__mono__less__eq,axiom,
! [F: a > a,X3: a,Y: a] :
( ( monotone_on_a_a @ top_top_set_a @ ord_less_a @ ord_less_a @ F )
=> ( ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y ) )
= ( ord_less_eq_a @ X3 @ Y ) ) ) ).
% strict_mono_less_eq
thf(fact_1210_mono__strict__invE,axiom,
! [F: $o > a,X3: $o,Y: $o] :
( ( monotone_on_o_a @ top_top_set_o @ ord_less_eq_o @ ord_less_eq_a @ F )
=> ( ( ord_less_a @ ( F @ X3 ) @ ( F @ Y ) )
=> ( ord_less_o @ X3 @ Y ) ) ) ).
% mono_strict_invE
thf(fact_1211_mono__strict__invE,axiom,
! [F: a > a,X3: a,Y: a] :
( ( monotone_on_a_a @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_a @ F )
=> ( ( ord_less_a @ ( F @ X3 ) @ ( F @ Y ) )
=> ( ord_less_a @ X3 @ Y ) ) ) ).
% mono_strict_invE
thf(fact_1212_strict__mono__mono,axiom,
! [F: $o > a] :
( ( monotone_on_o_a @ top_top_set_o @ ord_less_o @ ord_less_a @ F )
=> ( monotone_on_o_a @ top_top_set_o @ ord_less_eq_o @ ord_less_eq_a @ F ) ) ).
% strict_mono_mono
thf(fact_1213_strict__mono__mono,axiom,
! [F: a > a] :
( ( monotone_on_a_a @ top_top_set_a @ ord_less_a @ ord_less_a @ F )
=> ( monotone_on_a_a @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_a @ F ) ) ).
% strict_mono_mono
thf(fact_1214_mono__invE,axiom,
! [F: $o > a,X3: $o,Y: $o] :
( ( monotone_on_o_a @ top_top_set_o @ ord_less_eq_o @ ord_less_eq_a @ F )
=> ( ( ord_less_a @ ( F @ X3 ) @ ( F @ Y ) )
=> ( ord_less_eq_o @ X3 @ Y ) ) ) ).
% mono_invE
thf(fact_1215_mono__invE,axiom,
! [F: a > a,X3: a,Y: a] :
( ( monotone_on_a_a @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_a @ F )
=> ( ( ord_less_a @ ( F @ X3 ) @ ( F @ Y ) )
=> ( ord_less_eq_a @ X3 @ Y ) ) ) ).
% mono_invE
thf(fact_1216_Inf__le__Sup,axiom,
! [A4: set_o] :
( ( A4 != bot_bot_set_o )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A4 ) @ ( complete_Sup_Sup_o @ A4 ) ) ) ).
% Inf_le_Sup
thf(fact_1217_SUP__empty,axiom,
! [F: $o > set_o] :
( ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ bot_bot_set_o ) )
= bot_bot_set_o ) ).
% SUP_empty
thf(fact_1218_SUP__constant,axiom,
! [A4: set_o,C: set_o] :
( ( ( A4 = bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [Y2: $o] : C
@ A4 ) )
= bot_bot_set_o ) )
& ( ( A4 != bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [Y2: $o] : C
@ A4 ) )
= C ) ) ) ).
% SUP_constant
thf(fact_1219_UN__extend__simps_I1_J,axiom,
! [C2: set_o,A: $o,B4: $o > set_o] :
( ( ( C2 = bot_bot_set_o )
=> ( ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B4 @ C2 ) ) )
= ( insert_o @ A @ bot_bot_set_o ) ) )
& ( ( C2 != bot_bot_set_o )
=> ( ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B4 @ C2 ) ) )
= ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( insert_o @ A @ ( B4 @ X ) )
@ C2 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_1220_bdd__below__image__mono,axiom,
! [F: $o > a,A4: set_o] :
( ( monotone_on_o_a @ top_top_set_o @ ord_less_eq_o @ ord_less_eq_a @ F )
=> ( ( condit5413489452508810728elow_o @ A4 )
=> ( condit5901475214736682318elow_a @ ( image_o_a @ F @ A4 ) ) ) ) ).
% bdd_below_image_mono
thf(fact_1221_bdd__below__image__mono,axiom,
! [F: a > a,A4: set_a] :
( ( monotone_on_a_a @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_a @ F )
=> ( ( condit5901475214736682318elow_a @ A4 )
=> ( condit5901475214736682318elow_a @ ( image_a_a @ F @ A4 ) ) ) ) ).
% bdd_below_image_mono
thf(fact_1222_Least__mono,axiom,
! [F: $o > a,S2: set_o] :
( ( monotone_on_o_a @ top_top_set_o @ ord_less_eq_o @ ord_less_eq_a @ F )
=> ( ? [X4: $o] :
( ( member_o @ X4 @ S2 )
& ! [Xa: $o] :
( ( member_o @ Xa @ S2 )
=> ( ord_less_eq_o @ X4 @ Xa ) ) )
=> ( ( ord_Least_a
@ ^ [Y2: a] : ( member_a @ Y2 @ ( image_o_a @ F @ S2 ) ) )
= ( F
@ ( ord_Least_o
@ ^ [X: $o] : ( member_o @ X @ S2 ) ) ) ) ) ) ).
% Least_mono
thf(fact_1223_Least__mono,axiom,
! [F: a > a,S2: set_a] :
( ( monotone_on_a_a @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_a @ F )
=> ( ? [X4: a] :
( ( member_a @ X4 @ S2 )
& ! [Xa: a] :
( ( member_a @ Xa @ S2 )
=> ( ord_less_eq_a @ X4 @ Xa ) ) )
=> ( ( ord_Least_a
@ ^ [Y2: a] : ( member_a @ Y2 @ ( image_a_a @ F @ S2 ) ) )
= ( F
@ ( ord_Least_a
@ ^ [X: a] : ( member_a @ X @ S2 ) ) ) ) ) ) ).
% Least_mono
thf(fact_1224_cInf__cSup,axiom,
! [S2: set_o] :
( ( S2 != bot_bot_set_o )
=> ( ( condit5413489452508810728elow_o @ S2 )
=> ( ( complete_Inf_Inf_o @ S2 )
= ( complete_Sup_Sup_o
@ ( collect_o
@ ^ [X: $o] :
! [Y2: $o] :
( ( member_o @ Y2 @ S2 )
=> ( ord_less_eq_o @ X @ Y2 ) ) ) ) ) ) ) ).
% cInf_cSup
thf(fact_1225_bdd__above_OI,axiom,
! [A4: set_a,M: a] :
( ! [X2: a] :
( ( member_a @ X2 @ A4 )
=> ( ord_less_eq_a @ X2 @ M ) )
=> ( condit5209368051240477026bove_a @ A4 ) ) ).
% bdd_above.I
thf(fact_1226_Sup__SUP__eq,axiom,
( complete_Sup_Sup_a_o
= ( ^ [S: set_a_o,X: a] : ( member_a @ X @ ( comple2307003609928055243_set_a @ ( image_a_o_set_a @ collect_a @ S ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_1227_Sup__set__def,axiom,
( comple2307003609928055243_set_a
= ( ^ [A5: set_set_a] :
( collect_a
@ ^ [X: a] : ( complete_Sup_Sup_o @ ( image_set_a_o @ ( member_a @ X ) @ A5 ) ) ) ) ) ).
% Sup_set_def
thf(fact_1228_SUP__Sup__eq,axiom,
! [S2: set_set_a] :
( ( complete_Sup_Sup_a_o
@ ( image_set_a_a_o
@ ^ [I2: set_a,X: a] : ( member_a @ X @ I2 )
@ S2 ) )
= ( ^ [X: a] : ( member_a @ X @ ( comple2307003609928055243_set_a @ S2 ) ) ) ) ).
% SUP_Sup_eq
thf(fact_1229_bdd__above_OI2,axiom,
! [A4: set_a,F: a > a,M: a] :
( ! [X2: a] :
( ( member_a @ X2 @ A4 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ M ) )
=> ( condit5209368051240477026bove_a @ ( image_a_a @ F @ A4 ) ) ) ).
% bdd_above.I2
thf(fact_1230_bdd__above_OE,axiom,
! [A4: set_a] :
( ( condit5209368051240477026bove_a @ A4 )
=> ~ ! [M4: a] :
~ ! [X4: a] :
( ( member_a @ X4 @ A4 )
=> ( ord_less_eq_a @ X4 @ M4 ) ) ) ).
% bdd_above.E
thf(fact_1231_bdd__above_Ounfold,axiom,
( condit5209368051240477026bove_a
= ( ^ [A5: set_a] :
? [M3: a] :
! [X: a] :
( ( member_a @ X @ A5 )
=> ( ord_less_eq_a @ X @ M3 ) ) ) ) ).
% bdd_above.unfold
thf(fact_1232_bdd__above__primitive__def,axiom,
( condit5209368051240477026bove_a
= ( condit6541519642617408243_bdd_a @ ord_less_eq_a ) ) ).
% bdd_above_primitive_def
thf(fact_1233_cSup__mono,axiom,
! [B4: set_o,A4: set_o] :
( ( B4 != bot_bot_set_o )
=> ( ( condit5488710616941104124bove_o @ A4 )
=> ( ! [B3: $o] :
( ( member_o @ B3 @ B4 )
=> ? [X4: $o] :
( ( member_o @ X4 @ A4 )
& ( ord_less_eq_o @ B3 @ X4 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ B4 ) @ ( complete_Sup_Sup_o @ A4 ) ) ) ) ) ).
% cSup_mono
thf(fact_1234_cSup__le__iff,axiom,
! [S2: set_o,A: $o] :
( ( S2 != bot_bot_set_o )
=> ( ( condit5488710616941104124bove_o @ S2 )
=> ( ( ord_less_eq_o @ ( complete_Sup_Sup_o @ S2 ) @ A )
= ( ! [X: $o] :
( ( member_o @ X @ S2 )
=> ( ord_less_eq_o @ X @ A ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_1235_cSup__subset__mono,axiom,
! [A4: set_o,B4: set_o] :
( ( A4 != bot_bot_set_o )
=> ( ( condit5488710616941104124bove_o @ B4 )
=> ( ( ord_less_eq_set_o @ A4 @ B4 )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A4 ) @ ( complete_Sup_Sup_o @ B4 ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_1236_bdd__above__image__mono,axiom,
! [F: $o > a,A4: set_o] :
( ( monotone_on_o_a @ top_top_set_o @ ord_less_eq_o @ ord_less_eq_a @ F )
=> ( ( condit5488710616941104124bove_o @ A4 )
=> ( condit5209368051240477026bove_a @ ( image_o_a @ F @ A4 ) ) ) ) ).
% bdd_above_image_mono
thf(fact_1237_bdd__above__image__mono,axiom,
! [F: a > a,A4: set_a] :
( ( monotone_on_a_a @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_a @ F )
=> ( ( condit5209368051240477026bove_a @ A4 )
=> ( condit5209368051240477026bove_a @ ( image_a_a @ F @ A4 ) ) ) ) ).
% bdd_above_image_mono
thf(fact_1238_cInf__le__cSup,axiom,
! [A4: set_o] :
( ( A4 != bot_bot_set_o )
=> ( ( condit5488710616941104124bove_o @ A4 )
=> ( ( condit5413489452508810728elow_o @ A4 )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A4 ) @ ( complete_Sup_Sup_o @ A4 ) ) ) ) ) ).
% cInf_le_cSup
thf(fact_1239_cSup__cInf,axiom,
! [S2: set_o] :
( ( S2 != bot_bot_set_o )
=> ( ( condit5488710616941104124bove_o @ S2 )
=> ( ( complete_Sup_Sup_o @ S2 )
= ( complete_Inf_Inf_o
@ ( collect_o
@ ^ [X: $o] :
! [Y2: $o] :
( ( member_o @ Y2 @ S2 )
=> ( ord_less_eq_o @ Y2 @ X ) ) ) ) ) ) ) ).
% cSup_cInf
thf(fact_1240_bdd__above__image__antimono,axiom,
! [F: $o > a,A4: set_o] :
( ( monotone_on_o_a @ top_top_set_o @ ord_less_eq_o
@ ^ [X: a,Y2: a] : ( ord_less_eq_a @ Y2 @ X )
@ F )
=> ( ( condit5413489452508810728elow_o @ A4 )
=> ( condit5209368051240477026bove_a @ ( image_o_a @ F @ A4 ) ) ) ) ).
% bdd_above_image_antimono
thf(fact_1241_bdd__above__image__antimono,axiom,
! [F: a > a,A4: set_a] :
( ( monotone_on_a_a @ top_top_set_a @ ord_less_eq_a
@ ^ [X: a,Y2: a] : ( ord_less_eq_a @ Y2 @ X )
@ F )
=> ( ( condit5901475214736682318elow_a @ A4 )
=> ( condit5209368051240477026bove_a @ ( image_a_a @ F @ A4 ) ) ) ) ).
% bdd_above_image_antimono
thf(fact_1242_bdd__below__image__antimono,axiom,
! [F: $o > a,A4: set_o] :
( ( monotone_on_o_a @ top_top_set_o @ ord_less_eq_o
@ ^ [X: a,Y2: a] : ( ord_less_eq_a @ Y2 @ X )
@ F )
=> ( ( condit5488710616941104124bove_o @ A4 )
=> ( condit5901475214736682318elow_a @ ( image_o_a @ F @ A4 ) ) ) ) ).
% bdd_below_image_antimono
thf(fact_1243_bdd__below__image__antimono,axiom,
! [F: a > a,A4: set_a] :
( ( monotone_on_a_a @ top_top_set_a @ ord_less_eq_a
@ ^ [X: a,Y2: a] : ( ord_less_eq_a @ Y2 @ X )
@ F )
=> ( ( condit5209368051240477026bove_a @ A4 )
=> ( condit5901475214736682318elow_a @ ( image_a_a @ F @ A4 ) ) ) ) ).
% bdd_below_image_antimono
thf(fact_1244_def__lfp__induct__set,axiom,
! [A4: set_a,F: set_a > set_a,A: a,P: a > $o] :
( ( A4
= ( comple1558298011288954135_set_a @ F ) )
=> ( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ( member_a @ A @ A4 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( F @ ( inf_inf_set_a @ A4 @ ( collect_a @ P ) ) ) )
=> ( P @ X2 ) )
=> ( P @ A ) ) ) ) ) ).
% def_lfp_induct_set
thf(fact_1245_lfp__induct__set,axiom,
! [A: a,F: set_a > set_a,P: a > $o] :
( ( member_a @ A @ ( comple1558298011288954135_set_a @ F ) )
=> ( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( F @ ( inf_inf_set_a @ ( comple1558298011288954135_set_a @ F ) @ ( collect_a @ P ) ) ) )
=> ( P @ X2 ) )
=> ( P @ A ) ) ) ) ).
% lfp_induct_set
thf(fact_1246_def__lfp__unfold,axiom,
! [H: $o,F: $o > $o] :
( ( H
= ( comple5737750096767067345_lfp_o @ F ) )
=> ( ( monotone_on_o_o @ top_top_set_o @ ord_less_eq_o @ ord_less_eq_o @ F )
=> ( H
= ( F @ H ) ) ) ) ).
% def_lfp_unfold
thf(fact_1247_lfp__fixpoint,axiom,
! [F: $o > $o] :
( ( monotone_on_o_o @ top_top_set_o @ ord_less_eq_o @ ord_less_eq_o @ F )
=> ( ( F @ ( comple5737750096767067345_lfp_o @ F ) )
= ( comple5737750096767067345_lfp_o @ F ) ) ) ).
% lfp_fixpoint
thf(fact_1248_lfp__unfold,axiom,
! [F: $o > $o] :
( ( monotone_on_o_o @ top_top_set_o @ ord_less_eq_o @ ord_less_eq_o @ F )
=> ( ( comple5737750096767067345_lfp_o @ F )
= ( F @ ( comple5737750096767067345_lfp_o @ F ) ) ) ) ).
% lfp_unfold
thf(fact_1249_lfp__eqI,axiom,
! [F3: $o > $o,X3: $o] :
( ( monotone_on_o_o @ top_top_set_o @ ord_less_eq_o @ ord_less_eq_o @ F3 )
=> ( ( ( F3 @ X3 )
= X3 )
=> ( ! [Z5: $o] :
( ( ( F3 @ Z5 )
= Z5 )
=> ( ord_less_eq_o @ X3 @ Z5 ) )
=> ( ( comple5737750096767067345_lfp_o @ F3 )
= X3 ) ) ) ) ).
% lfp_eqI
thf(fact_1250_lfp__rolling,axiom,
! [G2: $o > $o,F: $o > $o] :
( ( monotone_on_o_o @ top_top_set_o @ ord_less_eq_o @ ord_less_eq_o @ G2 )
=> ( ( monotone_on_o_o @ top_top_set_o @ ord_less_eq_o @ ord_less_eq_o @ F )
=> ( ( G2
@ ( comple5737750096767067345_lfp_o
@ ^ [X: $o] : ( F @ ( G2 @ X ) ) ) )
= ( comple5737750096767067345_lfp_o
@ ^ [X: $o] : ( G2 @ ( F @ X ) ) ) ) ) ) ).
% lfp_rolling
thf(fact_1251_def__lfp__induct,axiom,
! [A4: $o,F: $o > $o,P: $o] :
( ( A4
= ( comple5737750096767067345_lfp_o @ F ) )
=> ( ( monotone_on_o_o @ top_top_set_o @ ord_less_eq_o @ ord_less_eq_o @ F )
=> ( ( ord_less_eq_o @ ( F @ ( inf_inf_o @ A4 @ P ) ) @ P )
=> ( ord_less_eq_o @ A4 @ P ) ) ) ) ).
% def_lfp_induct
thf(fact_1252_lfp__induct,axiom,
! [F: $o > $o,P: $o] :
( ( monotone_on_o_o @ top_top_set_o @ ord_less_eq_o @ ord_less_eq_o @ F )
=> ( ( ord_less_eq_o @ ( F @ ( inf_inf_o @ ( comple5737750096767067345_lfp_o @ F ) @ P ) ) @ P )
=> ( ord_less_eq_o @ ( comple5737750096767067345_lfp_o @ F ) @ P ) ) ) ).
% lfp_induct
thf(fact_1253_lfp__ordinal__induct,axiom,
! [F: $o > $o,P: $o > $o] :
( ( monotone_on_o_o @ top_top_set_o @ ord_less_eq_o @ ord_less_eq_o @ F )
=> ( ! [S6: $o] :
( ( P @ S6 )
=> ( ( ord_less_eq_o @ S6 @ ( comple5737750096767067345_lfp_o @ F ) )
=> ( P @ ( F @ S6 ) ) ) )
=> ( ! [M4: set_o] :
( ! [X4: $o] :
( ( member_o @ X4 @ M4 )
=> ( P @ X4 ) )
=> ( P @ ( complete_Sup_Sup_o @ M4 ) ) )
=> ( P @ ( comple5737750096767067345_lfp_o @ F ) ) ) ) ) ).
% lfp_ordinal_induct
thf(fact_1254_finite_Omono,axiom,
( monoto2155102285175209587et_o_o @ top_top_set_set_o_o @ ord_less_eq_set_o_o @ ord_less_eq_set_o_o
@ ^ [P3: set_o > $o,X: set_o] :
( ( X = bot_bot_set_o )
| ? [A5: set_o,A2: $o] :
( ( X
= ( insert_o @ A2 @ A5 ) )
& ( P3 @ A5 ) ) ) ) ).
% finite.mono
thf(fact_1255_lfp__le__gfp,axiom,
! [F: $o > $o] :
( ( monotone_on_o_o @ top_top_set_o @ ord_less_eq_o @ ord_less_eq_o @ F )
=> ( ord_less_eq_o @ ( comple5737750096767067345_lfp_o @ F ) @ ( comple1228283932920895894_gfp_o @ F ) ) ) ).
% lfp_le_gfp
thf(fact_1256_weak__coinduct__image,axiom,
! [A: a,X6: set_a,G2: a > a,F: set_a > set_a] :
( ( member_a @ A @ X6 )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ G2 @ X6 ) @ ( F @ ( image_a_a @ G2 @ X6 ) ) )
=> ( member_a @ ( G2 @ A ) @ ( comple3341859861669737308_set_a @ F ) ) ) ) ).
% weak_coinduct_image
thf(fact_1257_chain__singleton,axiom,
! [X3: $o] : ( comple520228465662580424hain_o @ ord_less_eq_o @ ( insert_o @ X3 @ bot_bot_set_o ) ) ).
% chain_singleton
thf(fact_1258_chain__compr,axiom,
! [Ord: a > a > $o,A4: set_a,P: a > $o] :
( ( comple1697357536187991598hain_a @ Ord @ A4 )
=> ( comple1697357536187991598hain_a @ Ord
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A4 )
& ( P @ X ) ) ) ) ) ).
% chain_compr
thf(fact_1259_weak__coinduct,axiom,
! [A: a,X6: set_a,F: set_a > set_a] :
( ( member_a @ A @ X6 )
=> ( ( ord_less_eq_set_a @ X6 @ ( F @ X6 ) )
=> ( member_a @ A @ ( comple3341859861669737308_set_a @ F ) ) ) ) ).
% weak_coinduct
thf(fact_1260_def__gfp__unfold,axiom,
! [A4: $o,F: $o > $o] :
( ( A4
= ( comple1228283932920895894_gfp_o @ F ) )
=> ( ( monotone_on_o_o @ top_top_set_o @ ord_less_eq_o @ ord_less_eq_o @ F )
=> ( A4
= ( F @ A4 ) ) ) ) ).
% def_gfp_unfold
thf(fact_1261_gfp__fixpoint,axiom,
! [F: $o > $o] :
( ( monotone_on_o_o @ top_top_set_o @ ord_less_eq_o @ ord_less_eq_o @ F )
=> ( ( F @ ( comple1228283932920895894_gfp_o @ F ) )
= ( comple1228283932920895894_gfp_o @ F ) ) ) ).
% gfp_fixpoint
thf(fact_1262_gfp__unfold,axiom,
! [F: $o > $o] :
( ( monotone_on_o_o @ top_top_set_o @ ord_less_eq_o @ ord_less_eq_o @ F )
=> ( ( comple1228283932920895894_gfp_o @ F )
= ( F @ ( comple1228283932920895894_gfp_o @ F ) ) ) ) ).
% gfp_unfold
thf(fact_1263_gfp__eqI,axiom,
! [F3: $o > $o,X3: $o] :
( ( monotone_on_o_o @ top_top_set_o @ ord_less_eq_o @ ord_less_eq_o @ F3 )
=> ( ( ( F3 @ X3 )
= X3 )
=> ( ! [Z5: $o] :
( ( ( F3 @ Z5 )
= Z5 )
=> ( ord_less_eq_o @ Z5 @ X3 ) )
=> ( ( comple1228283932920895894_gfp_o @ F3 )
= X3 ) ) ) ) ).
% gfp_eqI
thf(fact_1264_gfp__rolling,axiom,
! [G2: $o > $o,F: $o > $o] :
( ( monotone_on_o_o @ top_top_set_o @ ord_less_eq_o @ ord_less_eq_o @ G2 )
=> ( ( monotone_on_o_o @ top_top_set_o @ ord_less_eq_o @ ord_less_eq_o @ F )
=> ( ( G2
@ ( comple1228283932920895894_gfp_o
@ ^ [X: $o] : ( F @ ( G2 @ X ) ) ) )
= ( comple1228283932920895894_gfp_o
@ ^ [X: $o] : ( G2 @ ( F @ X ) ) ) ) ) ) ).
% gfp_rolling
thf(fact_1265_gfp__ordinal__induct,axiom,
! [F: $o > $o,P: $o > $o] :
( ( monotone_on_o_o @ top_top_set_o @ ord_less_eq_o @ ord_less_eq_o @ F )
=> ( ! [S6: $o] :
( ( P @ S6 )
=> ( ( ord_less_eq_o @ ( comple1228283932920895894_gfp_o @ F ) @ S6 )
=> ( P @ ( F @ S6 ) ) ) )
=> ( ! [M4: set_o] :
( ! [X4: $o] :
( ( member_o @ X4 @ M4 )
=> ( P @ X4 ) )
=> ( P @ ( complete_Inf_Inf_o @ M4 ) ) )
=> ( P @ ( comple1228283932920895894_gfp_o @ F ) ) ) ) ) ).
% gfp_ordinal_induct
thf(fact_1266_lfp__eq__fixp,axiom,
! [F: $o > $o] :
( ( monotone_on_o_o @ top_top_set_o @ ord_less_eq_o @ ord_less_eq_o @ F )
=> ( ( comple5737750096767067345_lfp_o @ F )
= ( comple2713996627985145509fixp_o @ F ) ) ) ).
% lfp_eq_fixp
thf(fact_1267_fixp__lowerbound,axiom,
! [F: $o > $o,Z3: $o] :
( ( monotone_on_o_o @ top_top_set_o @ ord_less_eq_o @ ord_less_eq_o @ F )
=> ( ( ord_less_eq_o @ ( F @ Z3 ) @ Z3 )
=> ( ord_less_eq_o @ ( comple2713996627985145509fixp_o @ F ) @ Z3 ) ) ) ).
% fixp_lowerbound
thf(fact_1268_fixp__unfold,axiom,
! [F: $o > $o] :
( ( monotone_on_o_o @ top_top_set_o @ ord_less_eq_o @ ord_less_eq_o @ F )
=> ( ( comple2713996627985145509fixp_o @ F )
= ( F @ ( comple2713996627985145509fixp_o @ F ) ) ) ) ).
% fixp_unfold
thf(fact_1269_fixp__induct,axiom,
! [P: $o > $o,F: $o > $o] :
( ( comple8949206149834442853ible_o @ complete_Sup_Sup_o @ ord_less_eq_o @ P )
=> ( ( monotone_on_o_o @ top_top_set_o @ ord_less_eq_o @ ord_less_eq_o @ F )
=> ( ( P @ ( complete_Sup_Sup_o @ bot_bot_set_o ) )
=> ( ! [X2: $o] :
( ( P @ X2 )
=> ( P @ ( F @ X2 ) ) )
=> ( P @ ( comple2713996627985145509fixp_o @ F ) ) ) ) ) ) ).
% fixp_induct
thf(fact_1270_coinduct3,axiom,
! [F: set_a > set_a,A: a,X6: set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ( member_a @ A @ X6 )
=> ( ( ord_less_eq_set_a @ X6
@ ( F
@ ( comple1558298011288954135_set_a
@ ^ [X: set_a] : ( sup_sup_set_a @ ( sup_sup_set_a @ ( F @ X ) @ X6 ) @ ( comple3341859861669737308_set_a @ F ) ) ) ) )
=> ( member_a @ A @ ( comple3341859861669737308_set_a @ F ) ) ) ) ) ).
% coinduct3
thf(fact_1271_UnCI,axiom,
! [C: a,B4: set_a,A4: set_a] :
( ( ~ ( member_a @ C @ B4 )
=> ( member_a @ C @ A4 ) )
=> ( member_a @ C @ ( sup_sup_set_a @ A4 @ B4 ) ) ) ).
% UnCI
thf(fact_1272_Un__iff,axiom,
! [C: a,A4: set_a,B4: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A4 @ B4 ) )
= ( ( member_a @ C @ A4 )
| ( member_a @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_1273_Un__empty,axiom,
! [A4: set_o,B4: set_o] :
( ( ( sup_sup_set_o @ A4 @ B4 )
= bot_bot_set_o )
= ( ( A4 = bot_bot_set_o )
& ( B4 = bot_bot_set_o ) ) ) ).
% Un_empty
thf(fact_1274_Un__insert__left,axiom,
! [A: $o,B4: set_o,C2: set_o] :
( ( sup_sup_set_o @ ( insert_o @ A @ B4 ) @ C2 )
= ( insert_o @ A @ ( sup_sup_set_o @ B4 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_1275_Un__insert__right,axiom,
! [A4: set_o,A: $o,B4: set_o] :
( ( sup_sup_set_o @ A4 @ ( insert_o @ A @ B4 ) )
= ( insert_o @ A @ ( sup_sup_set_o @ A4 @ B4 ) ) ) ).
% Un_insert_right
% Helper facts (5)
thf(help_If_2_1_If_001_Eo_T,axiom,
! [X3: $o,Y: $o] :
( ( if_o @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001_Eo_T,axiom,
! [X3: $o,Y: $o] :
( ( if_o @ $true @ X3 @ Y )
= X3 ) ).
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,Y: a] :
( ( if_a @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X3: a,Y: a] :
( ( if_a @ $true @ X3 @ Y )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( minimum_Maximum_a @ s )
= x ) ).
%------------------------------------------------------------------------------