TPTP Problem File: SLH0825^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Undirected_Graph_Theory/0015_Graph_Triangles/prob_00184_006827__13157842_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1509 ( 624 unt; 233 typ; 0 def)
% Number of atoms : 3708 (1310 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 11591 ( 221 ~; 3 |; 302 &;9516 @)
% ( 0 <=>;1549 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 6 avg)
% Number of types : 26 ( 25 usr)
% Number of type conns : 1471 (1471 >; 0 *; 0 +; 0 <<)
% Number of symbols : 211 ( 208 usr; 23 con; 0-4 aty)
% Number of variables : 4218 ( 497 ^;3540 !; 181 ?;4218 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:33:08.236
%------------------------------------------------------------------------------
% Could-be-implicit typings (25)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_J,type,
set_se9027383378080648592od_a_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_M_Eo_J_J,type,
set_Pr5684138772510291555_a_a_o: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
set_Pr5530083903271594800od_a_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
produc4044097585999906000od_a_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
set_se5735800977113168103od_a_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Product____Type__Oprod_Itf__a_Mtf__a_J_M_Eo_J_J,type,
set_Pr974637816708178892_a_a_o: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
set_Product_prod_a_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
set_set_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_I_Eo_J_J_J,type,
set_set_set_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Set__Oset_Itf__a_J_M_Eo_J_J,type,
set_set_a_o: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_Eo_J_J,type,
set_nat_o: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
product_prod_a_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
set_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
set_set_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_M_Eo_J_J,type,
set_a_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_Eo_M_Eo_J_J,type,
set_o_o: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_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_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (208)
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_I_Eo_M_Eo_J,type,
complete_Sup_Sup_o_o: set_o_o > $o > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Nat__Onat_M_Eo_J,type,
comple8317665133742190828_nat_o: set_nat_o > nat > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_M_Eo_J,type,
comple2673673910019652224_a_a_o: set_Pr5684138772510291555_a_a_o > produc4044097585999906000od_a_a > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Product____Type__Oprod_Itf__a_Mtf__a_J_M_Eo_J,type,
comple9027675562681848937_a_a_o: set_Pr974637816708178892_a_a_o > product_prod_a_a > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Set__Oset_Itf__a_J_M_Eo_J,type,
comple7256090232125724530et_a_o: set_set_a_o > set_a > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_Itf__a_M_Eo_J,type,
complete_Sup_Sup_a_o: set_a_o > a > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_Eo,type,
complete_Sup_Sup_o: set_o > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
complete_Sup_Sup_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_I_Eo_J,type,
comple90263536869209701_set_o: set_set_o > set_o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
comple7399068483239264473et_nat: set_set_nat > set_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
comple2978350343072902813od_a_a: set_se9027383378080648592od_a_a > set_Pr5530083903271594800od_a_a ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
comple8421679170691845492od_a_a: set_se5735800977113168103od_a_a > set_Product_prod_a_a ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
comple4436988014476444997_set_o: set_set_set_o > set_set_o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
comple3958522678809307947_set_a: set_set_set_a > set_set_a ).
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_Finite__Set_Ocard_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
finite6893194910719049976od_a_a: set_Pr5530083903271594800od_a_a > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
finite4795055649997197647od_a_a: set_Product_prod_a_a > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_Itf__a_J,type,
finite_card_set_a: set_set_a > nat ).
thf(sy_c_Finite__Set_Ocard_001tf__a,type,
finite_card_a: set_a > nat ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
finite6544458595007987280od_a_a: set_Product_prod_a_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
finite_finite_set_a: set_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Set__Oset_Itf__a_J,type,
inj_on4851796814176604264_set_a: ( product_prod_a_a > set_a ) > set_Product_prod_a_a > $o ).
thf(sy_c_Graph__Triangles_Osgraph_Otriangle__in__graph_001tf__a,type,
graph_4582152751571636272raph_a: set_set_a > a > a > a > $o ).
thf(sy_c_Graph__Triangles_Osgraph_Otriangle__set_001tf__a,type,
graph_triangle_set_a: set_set_a > set_set_a ).
thf(sy_c_Graph__Triangles_Osgraph_Otriangle__triples_001tf__a,type,
graph_4774508486909600516ples_a: set_set_a > set_a > set_a > set_a > set_Pr5530083903271594800od_a_a ).
thf(sy_c_Graph__Triangles_Osgraph_Ounique__triangles_001tf__a,type,
graph_6144490306505338871gles_a: set_set_a > $o ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001tf__a_001t__Nat__Onat,type,
groups6334556678337121940_a_nat: ( a > nat ) > set_a > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
inf_inf_set_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
sup_su3048258781599657691od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
sup_sup_set_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_Multiset__Permutations_Opermutations__of__set_001tf__a,type,
multis2428024204330136193_set_a: set_a > set_list_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_Eo_J,type,
bot_bot_set_o: set_o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
bot_bo4436838304982128028od_a_a: set_Pr5530083903271594800od_a_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
bot_bo3357376287454694259od_a_a: set_Product_prod_a_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
bot_bot_set_set_o: set_set_o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bot_bot_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_J,type,
bot_bo5623065384236484604od_a_a: set_se9027383378080648592od_a_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
bot_bo777872063958040403od_a_a: set_se5735800977113168103od_a_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
bot_bo3380559777022489994_set_a: set_set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
bot_bot_set_set_a: set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_Eo_J,type,
ord_less_eq_o_o: ( $o > $o ) > ( $o > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_M_Eo_J,type,
ord_le4133739015287644173_a_a_o: ( produc4044097585999906000od_a_a > $o ) > ( produc4044097585999906000od_a_a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_Itf__a_Mtf__a_J_M_Eo_J,type,
ord_le1591150415168442102_a_a_o: ( product_prod_a_a > $o ) > ( product_prod_a_a > $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_062_It__Product____Type__Oprod_Itf__a_Mtf__a_J_M_Eo_J_J,type,
ord_le5507034481861240613_a_a_o: ( a > product_prod_a_a > $o ) > ( a > product_prod_a_a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_M_062_Itf__a_M_Eo_J_J,type,
ord_less_eq_a_a_o: ( a > a > $o ) > ( a > 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__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
ord_less_eq_set_o: set_o > set_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
ord_le114883831454073552od_a_a: set_Pr5530083903271594800od_a_a > set_Pr5530083903271594800od_a_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
ord_le746702958409616551od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > $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_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_J,type,
ord_le5596166698269566256od_a_a: set_se9027383378080648592od_a_a > set_se9027383378080648592od_a_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
ord_le1995061765932249223od_a_a: set_se5735800977113168103od_a_a > set_se5735800977113168103od_a_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
ord_le5722252365846178494_set_a: set_set_set_a > set_set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_le3724670747650509150_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Product__Type_OPair_001tf__a_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
produc431845341423274048od_a_a: a > product_prod_a_a > produc4044097585999906000od_a_a ).
thf(sy_c_Product__Type_OPair_001tf__a_001tf__a,type,
product_Pair_a_a: a > a > product_prod_a_a ).
thf(sy_c_Product__Type_OSigma_001tf__a_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
produc6342321021181284593od_a_a: set_a > ( a > set_Product_prod_a_a ) > set_Pr5530083903271594800od_a_a ).
thf(sy_c_Product__Type_OSigma_001tf__a_001tf__a,type,
product_Sigma_a_a: set_a > ( a > set_a ) > set_Product_prod_a_a ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001tf__a_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001_Eo,type,
produc5856822985862792195_a_a_o: ( a > product_prod_a_a > $o ) > produc4044097585999906000od_a_a > $o ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001tf__a_001tf__a_001_Eo,type,
produc6436628058953941356_a_a_o: ( a > a > $o ) > product_prod_a_a > $o ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001tf__a_001tf__a_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
produc408267641121961211od_a_a: ( a > a > product_prod_a_a ) > product_prod_a_a > product_prod_a_a ).
thf(sy_c_Set_OCollect_001_Eo,type,
collect_o: ( $o > $o ) > set_o ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
collec5045780995415420475od_a_a: ( produc4044097585999906000od_a_a > $o ) > set_Pr5530083903271594800od_a_a ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
collec3336397797384452498od_a_a: ( product_prod_a_a > $o ) > set_Product_prod_a_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_OPow_001tf__a,type,
pow_a: set_a > set_set_a ).
thf(sy_c_Set_Oimage_001_062_I_Eo_M_Eo_J_001t__Set__Oset_I_Eo_J,type,
image_o_o_set_o: ( ( $o > $o ) > set_o ) > set_o_o > set_set_o ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_M_Eo_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_o_set_nat: ( ( nat > $o ) > set_nat ) > set_nat_o > set_set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_M_Eo_J_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
image_3003369154657308540od_a_a: ( ( produc4044097585999906000od_a_a > $o ) > set_Pr5530083903271594800od_a_a ) > set_Pr5684138772510291555_a_a_o > set_se9027383378080648592od_a_a ).
thf(sy_c_Set_Oimage_001_062_It__Product____Type__Oprod_Itf__a_Mtf__a_J_M_Eo_J_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
image_7999903765548794282od_a_a: ( ( product_prod_a_a > $o ) > set_Product_prod_a_a ) > set_Pr974637816708178892_a_a_o > set_se5735800977113168103od_a_a ).
thf(sy_c_Set_Oimage_001_062_It__Set__Oset_Itf__a_J_M_Eo_J_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
image_5433501368419010456_set_a: ( ( set_a > $o ) > set_set_a ) > set_set_a_o > set_set_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__Nat__Onat,type,
image_o_nat: ( $o > nat ) > set_o > set_nat ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_I_Eo_J,type,
image_o_set_o: ( $o > set_o ) > set_o > set_set_o ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
image_4840794370999913844od_a_a: ( $o > set_Product_prod_a_a ) > set_o > set_se5735800977113168103od_a_a ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
image_o_set_set_a: ( $o > set_set_a ) > set_o > set_set_set_a ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_Itf__a_J,type,
image_o_set_a: ( $o > set_a ) > set_o > set_set_a ).
thf(sy_c_Set_Oimage_001_Eo_001tf__a,type,
image_o_a: ( $o > a ) > set_o > set_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
image_1195025184546981201od_a_a: ( list_a > produc4044097585999906000od_a_a ) > set_list_a > set_Pr5530083903271594800od_a_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_Eo,type,
image_nat_o: ( nat > $o ) > set_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_I_Eo_J,type,
image_nat_set_o: ( nat > set_o ) > set_nat > set_set_o ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
image_2330413017338827248od_a_a: ( nat > set_Product_prod_a_a ) > set_nat > set_se5735800977113168103od_a_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
image_nat_set_set_a: ( nat > set_set_a ) > set_nat > set_set_set_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_Itf__a_J,type,
image_nat_set_a: ( nat > set_a ) > set_nat > set_set_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001tf__a,type,
image_nat_a: ( nat > a ) > set_nat > set_a ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_001t__Set__Oset_Itf__a_J,type,
image_3577654474136104851_set_a: ( produc4044097585999906000od_a_a > set_a ) > set_Pr5530083903271594800od_a_a > set_set_a ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
image_4636654165204879301od_a_a: ( product_prod_a_a > product_prod_a_a ) > set_Product_prod_a_a > set_Product_prod_a_a ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Set__Oset_I_Eo_J,type,
image_7872567118757839382_set_o: ( product_prod_a_a > set_o ) > set_Product_prod_a_a > set_set_o ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Set__Oset_Itf__a_J,type,
image_9052089385058188540_set_a: ( product_prod_a_a > set_a ) > set_Product_prod_a_a > set_set_a ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001tf__a,type,
image_3437945252899457948_a_a_a: ( product_prod_a_a > a ) > set_Product_prod_a_a > set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_Eo_J_001_062_I_Eo_M_Eo_J,type,
image_set_o_o_o: ( set_o > $o > $o ) > set_set_o > set_o_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_Eo_J_001_Eo,type,
image_set_o_o: ( set_o > $o ) > set_set_o > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_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_I_Eo_J_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
image_1650781834251077268od_a_a: ( set_o > set_Product_prod_a_a ) > set_set_o > set_se5735800977113168103od_a_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_Eo_J_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
image_7612453856319013963_set_a: ( set_o > set_set_a ) > set_set_o > set_set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_Eo_J_001t__Set__Oset_Itf__a_J,type,
image_set_o_set_a: ( set_o > set_a ) > set_set_o > set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001_Eo,type,
image_set_nat_o: ( set_nat > $o ) > set_set_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_001_062_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_M_Eo_J,type,
image_2231517904016305432_a_a_o: ( set_Pr5530083903271594800od_a_a > produc4044097585999906000od_a_a > $o ) > set_se9027383378080648592od_a_a > set_Pr5684138772510291555_a_a_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_001_Eo,type,
image_329807791404844781_a_a_o: ( set_Pr5530083903271594800od_a_a > $o ) > set_se9027383378080648592od_a_a > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
image_5986082258655243781od_a_a: ( set_Pr5530083903271594800od_a_a > set_Pr5530083903271594800od_a_a ) > set_se9027383378080648592od_a_a > set_se9027383378080648592od_a_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_001_062_It__Product____Type__Oprod_Itf__a_Mtf__a_J_M_Eo_J,type,
image_1002440501707978072_a_a_o: ( set_Product_prod_a_a > product_prod_a_a > $o ) > set_se5735800977113168103od_a_a > set_Pr974637816708178892_a_a_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_001_Eo,type,
image_3158127254886349654_a_a_o: ( set_Product_prod_a_a > $o ) > set_se5735800977113168103od_a_a > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
image_4506799131697958853od_a_a: ( set_Product_prod_a_a > set_Product_prod_a_a ) > set_se5735800977113168103od_a_a > set_se5735800977113168103od_a_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_Itf__a_J_J_001_062_It__Set__Oset_Itf__a_J_M_Eo_J,type,
image_7036300546039719832et_a_o: ( set_set_a > set_a > $o ) > set_set_set_a > set_set_a_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_Itf__a_J_J_001_Eo,type,
image_set_set_a_o: ( set_set_a > $o ) > set_set_set_a > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_Itf__a_J_J_001t__Set__Oset_I_Eo_J,type,
image_4406776271737083839_set_o: ( set_set_a > set_o ) > set_set_set_a > set_set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_Itf__a_J_J_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
image_1042221919965026181_set_a: ( set_set_a > set_set_a ) > set_set_set_a > set_set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_Itf__a_J_J_001t__Set__Oset_Itf__a_J,type,
image_6061375613820669477_set_a: ( set_set_a > set_a ) > set_set_set_a > set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001_062_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_M_Eo_J,type,
image_2136885688481450310_a_a_o: ( set_a > produc4044097585999906000od_a_a > $o ) > set_set_a > set_Pr5684138772510291555_a_a_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001_062_It__Product____Type__Oprod_Itf__a_Mtf__a_J_M_Eo_J,type,
image_1140608277309626671_a_a_o: ( set_a > product_prod_a_a > $o ) > set_set_a > set_Pr974637816708178892_a_a_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__Nat__Onat,type,
image_set_a_nat: ( set_a > nat ) > set_set_a > set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_I_Eo_J,type,
image_set_a_set_o: ( set_a > set_o ) > set_set_a > set_set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_set_a_set_nat: ( set_a > set_nat ) > set_set_a > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
image_7562202058474640471od_a_a: ( set_a > set_Pr5530083903271594800od_a_a ) > set_set_a > set_se9027383378080648592od_a_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
image_6165024369500519726od_a_a: ( set_a > set_Product_prod_a_a ) > set_set_a > set_se5735800977113168103od_a_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
image_8955363097146037247_set_o: ( set_a > set_set_o ) > set_set_a > set_set_set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
image_4955109552351689957_set_a: ( set_a > set_set_a ) > set_set_a > set_set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
image_set_a_set_a: ( set_a > set_a ) > set_set_a > set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001tf__a,type,
image_set_a_a: ( set_a > a ) > set_set_a > set_a ).
thf(sy_c_Set_Oimage_001tf__a_001_062_I_Eo_M_Eo_J,type,
image_a_o_o: ( a > $o > $o ) > set_a > set_o_o ).
thf(sy_c_Set_Oimage_001tf__a_001_062_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_M_Eo_J,type,
image_7327113576949061606_a_a_o: ( a > produc4044097585999906000od_a_a > $o ) > set_a > set_Pr5684138772510291555_a_a_o ).
thf(sy_c_Set_Oimage_001tf__a_001_062_It__Product____Type__Oprod_Itf__a_Mtf__a_J_M_Eo_J,type,
image_5603332478075979727_a_a_o: ( a > product_prod_a_a > $o ) > set_a > set_Pr974637816708178892_a_a_o ).
thf(sy_c_Set_Oimage_001tf__a_001_062_It__Set__Oset_Itf__a_J_M_Eo_J,type,
image_a_set_a_o: ( a > set_a > $o ) > set_a > set_set_a_o ).
thf(sy_c_Set_Oimage_001tf__a_001_062_Itf__a_M_Eo_J,type,
image_a_a_o: ( a > a > $o ) > set_a > set_a_o ).
thf(sy_c_Set_Oimage_001tf__a_001_Eo,type,
image_a_o: ( a > $o ) > set_a > set_o ).
thf(sy_c_Set_Oimage_001tf__a_001t__Nat__Onat,type,
image_a_nat: ( a > nat ) > set_a > set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
image_7400625782589995694od_a_a: ( a > product_prod_a_a ) > set_a > set_Product_prod_a_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_I_Eo_J,type,
image_a_set_o: ( a > set_o ) > set_a > set_set_o ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_It__Nat__Onat_J,type,
image_a_set_nat: ( a > set_nat ) > set_a > set_set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
image_5653227685612666295od_a_a: ( a > set_Pr5530083903271594800od_a_a ) > set_a > set_se9027383378080648592od_a_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
image_4421510592991446670od_a_a: ( a > set_Product_prod_a_a ) > set_a > set_se5735800977113168103od_a_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
image_a_set_set_o: ( a > set_set_o ) > set_a > set_set_set_o ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
image_a_set_set_a: ( a > set_set_a ) > set_a > set_set_set_a ).
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_001t__Set__Oset_Itf__a_J,type,
insert_set_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Undirected__Graph__Basics_Oall__edges_001tf__a,type,
undire2918257014606996450dges_a: set_a > set_set_a ).
thf(sy_c_Undirected__Graph__Basics_Ofin__graph__system_001tf__a,type,
undire945497512398942277stem_a: set_a > set_set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Ofin__sgraph_001tf__a,type,
undire5670813279940215357raph_a: set_a > set_set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Ofin__ulgraph_001tf__a,type,
undire7599193295422474955raph_a: set_a > set_set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_001tf__a,type,
undire2554140024507503526stem_a: set_a > set_set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oedge__adj_001tf__a,type,
undire4022703626023482010_adj_a: set_set_a > set_a > set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oincident_001tf__a,type,
undire1521409233611534436dent_a: a > set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oincident__edges_001tf__a,type,
undire3231912044278729248dges_a: set_set_a > a > set_set_a ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oinduced__edges_001tf__a,type,
undire7777452895879145676dges_a: set_set_a > set_a > set_set_a ).
thf(sy_c_Undirected__Graph__Basics_Omk__edge_001tf__a,type,
undire6670514144573423676edge_a: product_prod_a_a > set_a ).
thf(sy_c_Undirected__Graph__Basics_Omk__triangle__set_001tf__a,type,
undire8536760333753235943_set_a: produc4044097585999906000od_a_a > set_a ).
thf(sy_c_Undirected__Graph__Basics_Osgraph_001tf__a,type,
undire3507641187627840796raph_a: set_a > set_set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Osgraph_Odegree__set_001tf__a,type,
undire5448620520650870808_set_a: set_set_a > set_a > nat ).
thf(sy_c_Undirected__Graph__Basics_Osubgraph_001tf__a,type,
undire7103218114511261257raph_a: set_a > set_set_a > set_a > set_set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oall__edges__between_001tf__a,type,
undire8383842906760478443ween_a: set_set_a > set_a > set_a > set_Product_prod_a_a ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Odegree_001tf__a,type,
undire8867928226783802224gree_a: set_set_a > a > nat ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oedge__density_001tf__a,type,
undire297304480579013331sity_a: set_set_a > set_a > set_a > real ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ohas__loop_001tf__a,type,
undire3617971648856834880loop_a: set_set_a > a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oincident__loops_001tf__a,type,
undire4753905205749729249oops_a: set_set_a > a > set_set_a ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oincident__sedges_001tf__a,type,
undire1270416042309875431dges_a: set_set_a > a > set_set_a ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__edge__between_001tf__a,type,
undire8544646567961481629ween_a: set_a > set_a > set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__isolated__vertex_001tf__a,type,
undire8931668460104145173rtex_a: set_a > set_set_a > a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__loop_001tf__a,type,
undire2905028936066782638loop_a: set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__sedge_001tf__a,type,
undire4917966558017083288edge_a: set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oneighborhood_001tf__a,type,
undire8504279938402040014hood_a: set_a > set_set_a > a > set_a ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oneighbors__ss_001tf__a,type,
undire401937927514038589s_ss_a: set_set_a > a > set_a > set_a ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Overt__adj_001tf__a,type,
undire397441198561214472_adj_a: set_set_a > a > a > $o ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
member3071122053849602553od_a_a: produc4044097585999906000od_a_a > set_Pr5530083903271594800od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_member_001t__Set__Oset_I_Eo_J,type,
member_set_o: set_o > set_set_o > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
member4210947715425868889od_a_a: set_Pr5530083903271594800od_a_a > set_se9027383378080648592od_a_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
member1816616512716248880od_a_a: set_Product_prod_a_a > set_se5735800977113168103od_a_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
member_set_set_a: set_set_a > set_set_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_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_X,type,
x: set_a ).
thf(sy_v_Y,type,
y: set_a ).
thf(sy_v_Z,type,
z: set_a ).
thf(sy_v_edges,type,
edges: set_set_a ).
thf(sy_v_tofl____,type,
tofl: list_a > produc4044097585999906000od_a_a ).
thf(sy_v_vertices,type,
vertices: set_a ).
% Relevant facts (1275)
thf(fact_0_triangle__commu1,axiom,
! [X: a,Y: a,Z: a] :
( ( graph_4582152751571636272raph_a @ edges @ X @ Y @ Z )
=> ( graph_4582152751571636272raph_a @ edges @ Y @ X @ Z ) ) ).
% triangle_commu1
thf(fact_1_triangle__vertices__distinct1,axiom,
! [X: a,Y: a,Z: a] :
( ( graph_4582152751571636272raph_a @ edges @ X @ Y @ Z )
=> ( X != Y ) ) ).
% triangle_vertices_distinct1
thf(fact_2_triangle__vertices__distinct2,axiom,
! [X: a,Y: a,Z: a] :
( ( graph_4582152751571636272raph_a @ edges @ X @ Y @ Z )
=> ( Y != Z ) ) ).
% triangle_vertices_distinct2
thf(fact_3_triangle__vertices__distinct3,axiom,
! [X: a,Y: a,Z: a] :
( ( graph_4582152751571636272raph_a @ edges @ X @ Y @ Z )
=> ( Z != X ) ) ).
% triangle_vertices_distinct3
thf(fact_4_edge__adj__inE,axiom,
! [E1: set_a,E2: set_a] :
( ( undire4022703626023482010_adj_a @ edges @ E1 @ E2 )
=> ( ( member_set_a @ E1 @ edges )
& ( member_set_a @ E2 @ edges ) ) ) ).
% edge_adj_inE
thf(fact_5_sgraph_Otriangle__set_Ocong,axiom,
graph_triangle_set_a = graph_triangle_set_a ).
% sgraph.triangle_set.cong
thf(fact_6_sgraph_Otriangle__in__graph_Ocong,axiom,
graph_4582152751571636272raph_a = graph_4582152751571636272raph_a ).
% sgraph.triangle_in_graph.cong
thf(fact_7_UN__I,axiom,
! [A: $o,A2: set_o,B: $o,B2: $o > set_o] :
( ( member_o @ A @ A2 )
=> ( ( member_o @ B @ ( B2 @ A ) )
=> ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_8_UN__I,axiom,
! [A: a,A2: set_a,B: $o,B2: a > set_o] :
( ( member_a @ A @ A2 )
=> ( ( member_o @ B @ ( B2 @ A ) )
=> ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_a_set_o @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_9_UN__I,axiom,
! [A: $o,A2: set_o,B: a,B2: $o > set_a] :
( ( member_o @ A @ A2 )
=> ( ( member_a @ B @ ( B2 @ A ) )
=> ( member_a @ B @ ( comple2307003609928055243_set_a @ ( image_o_set_a @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_10_UN__I,axiom,
! [A: a,A2: set_a,B: a,B2: a > set_a] :
( ( member_a @ A @ A2 )
=> ( ( member_a @ B @ ( B2 @ A ) )
=> ( member_a @ B @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_11_UN__I,axiom,
! [A: set_a,A2: set_set_a,B: $o,B2: set_a > set_o] :
( ( member_set_a @ A @ A2 )
=> ( ( member_o @ B @ ( B2 @ A ) )
=> ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_set_a_set_o @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_12_UN__I,axiom,
! [A: $o,A2: set_o,B: set_a,B2: $o > set_set_a] :
( ( member_o @ A @ A2 )
=> ( ( member_set_a @ B @ ( B2 @ A ) )
=> ( member_set_a @ B @ ( comple3958522678809307947_set_a @ ( image_o_set_set_a @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_13_UN__I,axiom,
! [A: a,A2: set_a,B: set_a,B2: a > set_set_a] :
( ( member_a @ A @ A2 )
=> ( ( member_set_a @ B @ ( B2 @ A ) )
=> ( member_set_a @ B @ ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_14_UN__I,axiom,
! [A: set_a,A2: set_set_a,B: a,B2: set_a > set_a] :
( ( member_set_a @ A @ A2 )
=> ( ( member_a @ B @ ( B2 @ A ) )
=> ( member_a @ B @ ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_15_UN__I,axiom,
! [A: product_prod_a_a,A2: set_Product_prod_a_a,B: $o,B2: product_prod_a_a > set_o] :
( ( member1426531477525435216od_a_a @ A @ A2 )
=> ( ( member_o @ B @ ( B2 @ A ) )
=> ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_7872567118757839382_set_o @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_16_UN__I,axiom,
! [A: $o,A2: set_o,B: product_prod_a_a,B2: $o > set_Product_prod_a_a] :
( ( member_o @ A @ A2 )
=> ( ( member1426531477525435216od_a_a @ B @ ( B2 @ A ) )
=> ( member1426531477525435216od_a_a @ B @ ( comple8421679170691845492od_a_a @ ( image_4840794370999913844od_a_a @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_17_UN__iff,axiom,
! [B: produc4044097585999906000od_a_a,B2: a > set_Pr5530083903271594800od_a_a,A2: set_a] :
( ( member3071122053849602553od_a_a @ B @ ( comple2978350343072902813od_a_a @ ( image_5653227685612666295od_a_a @ B2 @ A2 ) ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ( member3071122053849602553od_a_a @ B @ ( B2 @ X2 ) ) ) ) ) ).
% UN_iff
thf(fact_18_UN__iff,axiom,
! [B: product_prod_a_a,B2: a > set_Product_prod_a_a,A2: set_a] :
( ( member1426531477525435216od_a_a @ B @ ( comple8421679170691845492od_a_a @ ( image_4421510592991446670od_a_a @ B2 @ A2 ) ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ( member1426531477525435216od_a_a @ B @ ( B2 @ X2 ) ) ) ) ) ).
% UN_iff
thf(fact_19_UN__iff,axiom,
! [B: a,B2: set_a > set_a,A2: set_set_a] :
( ( member_a @ B @ ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ B2 @ A2 ) ) )
= ( ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ( member_a @ B @ ( B2 @ X2 ) ) ) ) ) ).
% UN_iff
thf(fact_20_UN__iff,axiom,
! [B: $o,B2: a > set_o,A2: set_a] :
( ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_a_set_o @ B2 @ A2 ) ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ( member_o @ B @ ( B2 @ X2 ) ) ) ) ) ).
% UN_iff
thf(fact_21_UN__iff,axiom,
! [B: produc4044097585999906000od_a_a,B2: set_a > set_Pr5530083903271594800od_a_a,A2: set_set_a] :
( ( member3071122053849602553od_a_a @ B @ ( comple2978350343072902813od_a_a @ ( image_7562202058474640471od_a_a @ B2 @ A2 ) ) )
= ( ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ( member3071122053849602553od_a_a @ B @ ( B2 @ X2 ) ) ) ) ) ).
% UN_iff
thf(fact_22_UN__iff,axiom,
! [B: product_prod_a_a,B2: set_a > set_Product_prod_a_a,A2: set_set_a] :
( ( member1426531477525435216od_a_a @ B @ ( comple8421679170691845492od_a_a @ ( image_6165024369500519726od_a_a @ B2 @ A2 ) ) )
= ( ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ( member1426531477525435216od_a_a @ B @ ( B2 @ X2 ) ) ) ) ) ).
% UN_iff
thf(fact_23_UN__iff,axiom,
! [B: set_a,B2: a > set_set_a,A2: set_a] :
( ( member_set_a @ B @ ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ B2 @ A2 ) ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ( member_set_a @ B @ ( B2 @ X2 ) ) ) ) ) ).
% UN_iff
thf(fact_24_UN__iff,axiom,
! [B: a,B2: a > set_a,A2: set_a] :
( ( member_a @ B @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B2 @ A2 ) ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ( member_a @ B @ ( B2 @ X2 ) ) ) ) ) ).
% UN_iff
thf(fact_25_SUP__identity__eq,axiom,
! [A2: set_set_o] :
( ( comple90263536869209701_set_o
@ ( image_set_o_set_o
@ ^ [X2: set_o] : X2
@ A2 ) )
= ( comple90263536869209701_set_o @ A2 ) ) ).
% SUP_identity_eq
thf(fact_26_SUP__identity__eq,axiom,
! [A2: set_se9027383378080648592od_a_a] :
( ( comple2978350343072902813od_a_a
@ ( image_5986082258655243781od_a_a
@ ^ [X2: set_Pr5530083903271594800od_a_a] : X2
@ A2 ) )
= ( comple2978350343072902813od_a_a @ A2 ) ) ).
% SUP_identity_eq
thf(fact_27_SUP__identity__eq,axiom,
! [A2: set_se5735800977113168103od_a_a] :
( ( comple8421679170691845492od_a_a
@ ( image_4506799131697958853od_a_a
@ ^ [X2: set_Product_prod_a_a] : X2
@ A2 ) )
= ( comple8421679170691845492od_a_a @ A2 ) ) ).
% SUP_identity_eq
thf(fact_28_SUP__identity__eq,axiom,
! [A2: set_set_set_a] :
( ( comple3958522678809307947_set_a
@ ( image_1042221919965026181_set_a
@ ^ [X2: set_set_a] : X2
@ A2 ) )
= ( comple3958522678809307947_set_a @ A2 ) ) ).
% SUP_identity_eq
thf(fact_29_SUP__identity__eq,axiom,
! [A2: set_set_a] :
( ( comple2307003609928055243_set_a
@ ( image_set_a_set_a
@ ^ [X2: set_a] : X2
@ A2 ) )
= ( comple2307003609928055243_set_a @ A2 ) ) ).
% SUP_identity_eq
thf(fact_30_SUP__identity__eq,axiom,
! [A2: set_nat] :
( ( complete_Sup_Sup_nat
@ ( image_nat_nat
@ ^ [X2: nat] : X2
@ A2 ) )
= ( complete_Sup_Sup_nat @ A2 ) ) ).
% SUP_identity_eq
thf(fact_31_UN__ball__bex__simps_I4_J,axiom,
! [B2: set_a > set_Pr5530083903271594800od_a_a,A2: set_set_a,P: produc4044097585999906000od_a_a > $o] :
( ( ? [X2: produc4044097585999906000od_a_a] :
( ( member3071122053849602553od_a_a @ X2 @ ( comple2978350343072902813od_a_a @ ( image_7562202058474640471od_a_a @ B2 @ A2 ) ) )
& ( P @ X2 ) ) )
= ( ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ? [Y2: produc4044097585999906000od_a_a] :
( ( member3071122053849602553od_a_a @ Y2 @ ( B2 @ X2 ) )
& ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_32_UN__ball__bex__simps_I4_J,axiom,
! [B2: a > set_Pr5530083903271594800od_a_a,A2: set_a,P: produc4044097585999906000od_a_a > $o] :
( ( ? [X2: produc4044097585999906000od_a_a] :
( ( member3071122053849602553od_a_a @ X2 @ ( comple2978350343072902813od_a_a @ ( image_5653227685612666295od_a_a @ B2 @ A2 ) ) )
& ( P @ X2 ) ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ? [Y2: produc4044097585999906000od_a_a] :
( ( member3071122053849602553od_a_a @ Y2 @ ( B2 @ X2 ) )
& ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_33_UN__ball__bex__simps_I4_J,axiom,
! [B2: set_a > set_Product_prod_a_a,A2: set_set_a,P: product_prod_a_a > $o] :
( ( ? [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ ( comple8421679170691845492od_a_a @ ( image_6165024369500519726od_a_a @ B2 @ A2 ) ) )
& ( P @ X2 ) ) )
= ( ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ? [Y2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y2 @ ( B2 @ X2 ) )
& ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_34_UN__ball__bex__simps_I4_J,axiom,
! [B2: a > set_Product_prod_a_a,A2: set_a,P: product_prod_a_a > $o] :
( ( ? [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ ( comple8421679170691845492od_a_a @ ( image_4421510592991446670od_a_a @ B2 @ A2 ) ) )
& ( P @ X2 ) ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ? [Y2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y2 @ ( B2 @ X2 ) )
& ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_35_UN__ball__bex__simps_I4_J,axiom,
! [B2: a > set_set_a,A2: set_a,P: set_a > $o] :
( ( ? [X2: set_a] :
( ( member_set_a @ X2 @ ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ B2 @ A2 ) ) )
& ( P @ X2 ) ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ? [Y2: set_a] :
( ( member_set_a @ Y2 @ ( B2 @ X2 ) )
& ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_36_UN__ball__bex__simps_I4_J,axiom,
! [B2: a > set_a,A2: set_a,P: a > $o] :
( ( ? [X2: a] :
( ( member_a @ X2 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B2 @ A2 ) ) )
& ( P @ X2 ) ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ? [Y2: a] :
( ( member_a @ Y2 @ ( B2 @ X2 ) )
& ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_37_UN__ball__bex__simps_I4_J,axiom,
! [B2: set_a > set_a,A2: set_set_a,P: a > $o] :
( ( ? [X2: a] :
( ( member_a @ X2 @ ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ B2 @ A2 ) ) )
& ( P @ X2 ) ) )
= ( ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ? [Y2: a] :
( ( member_a @ Y2 @ ( B2 @ X2 ) )
& ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_38_UN__ball__bex__simps_I4_J,axiom,
! [B2: a > set_o,A2: set_a,P: $o > $o] :
( ( ? [X2: $o] :
( ( member_o @ X2 @ ( comple90263536869209701_set_o @ ( image_a_set_o @ B2 @ A2 ) ) )
& ( P @ X2 ) ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ? [Y2: $o] :
( ( member_o @ Y2 @ ( B2 @ X2 ) )
& ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_39_UN__ball__bex__simps_I2_J,axiom,
! [B2: set_a > set_Pr5530083903271594800od_a_a,A2: set_set_a,P: produc4044097585999906000od_a_a > $o] :
( ( ! [X2: produc4044097585999906000od_a_a] :
( ( member3071122053849602553od_a_a @ X2 @ ( comple2978350343072902813od_a_a @ ( image_7562202058474640471od_a_a @ B2 @ A2 ) ) )
=> ( P @ X2 ) ) )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ! [Y2: produc4044097585999906000od_a_a] :
( ( member3071122053849602553od_a_a @ Y2 @ ( B2 @ X2 ) )
=> ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_40_UN__ball__bex__simps_I2_J,axiom,
! [B2: a > set_Pr5530083903271594800od_a_a,A2: set_a,P: produc4044097585999906000od_a_a > $o] :
( ( ! [X2: produc4044097585999906000od_a_a] :
( ( member3071122053849602553od_a_a @ X2 @ ( comple2978350343072902813od_a_a @ ( image_5653227685612666295od_a_a @ B2 @ A2 ) ) )
=> ( P @ X2 ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ! [Y2: produc4044097585999906000od_a_a] :
( ( member3071122053849602553od_a_a @ Y2 @ ( B2 @ X2 ) )
=> ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_41_UN__ball__bex__simps_I2_J,axiom,
! [B2: set_a > set_Product_prod_a_a,A2: set_set_a,P: product_prod_a_a > $o] :
( ( ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ ( comple8421679170691845492od_a_a @ ( image_6165024369500519726od_a_a @ B2 @ A2 ) ) )
=> ( P @ X2 ) ) )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ! [Y2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y2 @ ( B2 @ X2 ) )
=> ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_42_UN__ball__bex__simps_I2_J,axiom,
! [B2: a > set_Product_prod_a_a,A2: set_a,P: product_prod_a_a > $o] :
( ( ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ ( comple8421679170691845492od_a_a @ ( image_4421510592991446670od_a_a @ B2 @ A2 ) ) )
=> ( P @ X2 ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ! [Y2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y2 @ ( B2 @ X2 ) )
=> ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_43_UN__ball__bex__simps_I2_J,axiom,
! [B2: a > set_set_a,A2: set_a,P: set_a > $o] :
( ( ! [X2: set_a] :
( ( member_set_a @ X2 @ ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ B2 @ A2 ) ) )
=> ( P @ X2 ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ! [Y2: set_a] :
( ( member_set_a @ Y2 @ ( B2 @ X2 ) )
=> ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_44_UN__ball__bex__simps_I2_J,axiom,
! [B2: a > set_a,A2: set_a,P: a > $o] :
( ( ! [X2: a] :
( ( member_a @ X2 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B2 @ A2 ) ) )
=> ( P @ X2 ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ! [Y2: a] :
( ( member_a @ Y2 @ ( B2 @ X2 ) )
=> ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_45_UN__ball__bex__simps_I2_J,axiom,
! [B2: set_a > set_a,A2: set_set_a,P: a > $o] :
( ( ! [X2: a] :
( ( member_a @ X2 @ ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ B2 @ A2 ) ) )
=> ( P @ X2 ) ) )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ! [Y2: a] :
( ( member_a @ Y2 @ ( B2 @ X2 ) )
=> ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_46_UN__ball__bex__simps_I2_J,axiom,
! [B2: a > set_o,A2: set_a,P: $o > $o] :
( ( ! [X2: $o] :
( ( member_o @ X2 @ ( comple90263536869209701_set_o @ ( image_a_set_o @ B2 @ A2 ) ) )
=> ( P @ X2 ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ! [Y2: $o] :
( ( member_o @ Y2 @ ( B2 @ X2 ) )
=> ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_47_bex__UN,axiom,
! [B2: set_a > set_Pr5530083903271594800od_a_a,A2: set_set_a,P: produc4044097585999906000od_a_a > $o] :
( ( ? [X2: produc4044097585999906000od_a_a] :
( ( member3071122053849602553od_a_a @ X2 @ ( comple2978350343072902813od_a_a @ ( image_7562202058474640471od_a_a @ B2 @ A2 ) ) )
& ( P @ X2 ) ) )
= ( ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ? [Y2: produc4044097585999906000od_a_a] :
( ( member3071122053849602553od_a_a @ Y2 @ ( B2 @ X2 ) )
& ( P @ Y2 ) ) ) ) ) ).
% bex_UN
thf(fact_48_bex__UN,axiom,
! [B2: a > set_Pr5530083903271594800od_a_a,A2: set_a,P: produc4044097585999906000od_a_a > $o] :
( ( ? [X2: produc4044097585999906000od_a_a] :
( ( member3071122053849602553od_a_a @ X2 @ ( comple2978350343072902813od_a_a @ ( image_5653227685612666295od_a_a @ B2 @ A2 ) ) )
& ( P @ X2 ) ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ? [Y2: produc4044097585999906000od_a_a] :
( ( member3071122053849602553od_a_a @ Y2 @ ( B2 @ X2 ) )
& ( P @ Y2 ) ) ) ) ) ).
% bex_UN
thf(fact_49_bex__UN,axiom,
! [B2: set_a > set_Product_prod_a_a,A2: set_set_a,P: product_prod_a_a > $o] :
( ( ? [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ ( comple8421679170691845492od_a_a @ ( image_6165024369500519726od_a_a @ B2 @ A2 ) ) )
& ( P @ X2 ) ) )
= ( ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ? [Y2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y2 @ ( B2 @ X2 ) )
& ( P @ Y2 ) ) ) ) ) ).
% bex_UN
thf(fact_50_bex__UN,axiom,
! [B2: a > set_Product_prod_a_a,A2: set_a,P: product_prod_a_a > $o] :
( ( ? [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ ( comple8421679170691845492od_a_a @ ( image_4421510592991446670od_a_a @ B2 @ A2 ) ) )
& ( P @ X2 ) ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ? [Y2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y2 @ ( B2 @ X2 ) )
& ( P @ Y2 ) ) ) ) ) ).
% bex_UN
thf(fact_51_bex__UN,axiom,
! [B2: a > set_set_a,A2: set_a,P: set_a > $o] :
( ( ? [X2: set_a] :
( ( member_set_a @ X2 @ ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ B2 @ A2 ) ) )
& ( P @ X2 ) ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ? [Y2: set_a] :
( ( member_set_a @ Y2 @ ( B2 @ X2 ) )
& ( P @ Y2 ) ) ) ) ) ).
% bex_UN
thf(fact_52_bex__UN,axiom,
! [B2: a > set_a,A2: set_a,P: a > $o] :
( ( ? [X2: a] :
( ( member_a @ X2 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B2 @ A2 ) ) )
& ( P @ X2 ) ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ? [Y2: a] :
( ( member_a @ Y2 @ ( B2 @ X2 ) )
& ( P @ Y2 ) ) ) ) ) ).
% bex_UN
thf(fact_53_bex__UN,axiom,
! [B2: set_a > set_a,A2: set_set_a,P: a > $o] :
( ( ? [X2: a] :
( ( member_a @ X2 @ ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ B2 @ A2 ) ) )
& ( P @ X2 ) ) )
= ( ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ? [Y2: a] :
( ( member_a @ Y2 @ ( B2 @ X2 ) )
& ( P @ Y2 ) ) ) ) ) ).
% bex_UN
thf(fact_54_bex__UN,axiom,
! [B2: a > set_o,A2: set_a,P: $o > $o] :
( ( ? [X2: $o] :
( ( member_o @ X2 @ ( comple90263536869209701_set_o @ ( image_a_set_o @ B2 @ A2 ) ) )
& ( P @ X2 ) ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ? [Y2: $o] :
( ( member_o @ Y2 @ ( B2 @ X2 ) )
& ( P @ Y2 ) ) ) ) ) ).
% bex_UN
thf(fact_55_ball__UN,axiom,
! [B2: set_a > set_Pr5530083903271594800od_a_a,A2: set_set_a,P: produc4044097585999906000od_a_a > $o] :
( ( ! [X2: produc4044097585999906000od_a_a] :
( ( member3071122053849602553od_a_a @ X2 @ ( comple2978350343072902813od_a_a @ ( image_7562202058474640471od_a_a @ B2 @ A2 ) ) )
=> ( P @ X2 ) ) )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ! [Y2: produc4044097585999906000od_a_a] :
( ( member3071122053849602553od_a_a @ Y2 @ ( B2 @ X2 ) )
=> ( P @ Y2 ) ) ) ) ) ).
% ball_UN
thf(fact_56_ball__UN,axiom,
! [B2: a > set_Pr5530083903271594800od_a_a,A2: set_a,P: produc4044097585999906000od_a_a > $o] :
( ( ! [X2: produc4044097585999906000od_a_a] :
( ( member3071122053849602553od_a_a @ X2 @ ( comple2978350343072902813od_a_a @ ( image_5653227685612666295od_a_a @ B2 @ A2 ) ) )
=> ( P @ X2 ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ! [Y2: produc4044097585999906000od_a_a] :
( ( member3071122053849602553od_a_a @ Y2 @ ( B2 @ X2 ) )
=> ( P @ Y2 ) ) ) ) ) ).
% ball_UN
thf(fact_57_ball__UN,axiom,
! [B2: set_a > set_Product_prod_a_a,A2: set_set_a,P: product_prod_a_a > $o] :
( ( ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ ( comple8421679170691845492od_a_a @ ( image_6165024369500519726od_a_a @ B2 @ A2 ) ) )
=> ( P @ X2 ) ) )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ! [Y2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y2 @ ( B2 @ X2 ) )
=> ( P @ Y2 ) ) ) ) ) ).
% ball_UN
thf(fact_58_ball__UN,axiom,
! [B2: a > set_Product_prod_a_a,A2: set_a,P: product_prod_a_a > $o] :
( ( ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ ( comple8421679170691845492od_a_a @ ( image_4421510592991446670od_a_a @ B2 @ A2 ) ) )
=> ( P @ X2 ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ! [Y2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y2 @ ( B2 @ X2 ) )
=> ( P @ Y2 ) ) ) ) ) ).
% ball_UN
thf(fact_59_ball__UN,axiom,
! [B2: a > set_set_a,A2: set_a,P: set_a > $o] :
( ( ! [X2: set_a] :
( ( member_set_a @ X2 @ ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ B2 @ A2 ) ) )
=> ( P @ X2 ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ! [Y2: set_a] :
( ( member_set_a @ Y2 @ ( B2 @ X2 ) )
=> ( P @ Y2 ) ) ) ) ) ).
% ball_UN
thf(fact_60_ball__UN,axiom,
! [B2: a > set_a,A2: set_a,P: a > $o] :
( ( ! [X2: a] :
( ( member_a @ X2 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B2 @ A2 ) ) )
=> ( P @ X2 ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ! [Y2: a] :
( ( member_a @ Y2 @ ( B2 @ X2 ) )
=> ( P @ Y2 ) ) ) ) ) ).
% ball_UN
thf(fact_61_ball__UN,axiom,
! [B2: set_a > set_a,A2: set_set_a,P: a > $o] :
( ( ! [X2: a] :
( ( member_a @ X2 @ ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ B2 @ A2 ) ) )
=> ( P @ X2 ) ) )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ! [Y2: a] :
( ( member_a @ Y2 @ ( B2 @ X2 ) )
=> ( P @ Y2 ) ) ) ) ) ).
% ball_UN
thf(fact_62_ball__UN,axiom,
! [B2: a > set_o,A2: set_a,P: $o > $o] :
( ( ! [X2: $o] :
( ( member_o @ X2 @ ( comple90263536869209701_set_o @ ( image_a_set_o @ B2 @ A2 ) ) )
=> ( P @ X2 ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ! [Y2: $o] :
( ( member_o @ Y2 @ ( B2 @ X2 ) )
=> ( P @ Y2 ) ) ) ) ) ).
% ball_UN
thf(fact_63_SUP__UNION,axiom,
! [F: a > set_a,G: a > set_a,A2: set_a] :
( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ G @ A2 ) ) ) )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [Y2: a] : ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ ( G @ Y2 ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_64_SUP__UNION,axiom,
! [F: $o > set_a,G: a > set_o,A2: set_a] :
( ( comple2307003609928055243_set_a @ ( image_o_set_a @ F @ ( comple90263536869209701_set_o @ ( image_a_set_o @ G @ A2 ) ) ) )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [Y2: a] : ( comple2307003609928055243_set_a @ ( image_o_set_a @ F @ ( G @ Y2 ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_65_SUP__UNION,axiom,
! [F: a > set_o,G: a > set_a,A2: set_a] :
( ( comple90263536869209701_set_o @ ( image_a_set_o @ F @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ G @ A2 ) ) ) )
= ( comple90263536869209701_set_o
@ ( image_a_set_o
@ ^ [Y2: a] : ( comple90263536869209701_set_o @ ( image_a_set_o @ F @ ( G @ Y2 ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_66_SUP__UNION,axiom,
! [F: $o > set_o,G: a > set_o,A2: set_a] :
( ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ ( comple90263536869209701_set_o @ ( image_a_set_o @ G @ A2 ) ) ) )
= ( comple90263536869209701_set_o
@ ( image_a_set_o
@ ^ [Y2: a] : ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ ( G @ Y2 ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_67_SUP__UNION,axiom,
! [F: a > set_set_a,G: a > set_a,A2: set_a] :
( ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ F @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ G @ A2 ) ) ) )
= ( comple3958522678809307947_set_a
@ ( image_a_set_set_a
@ ^ [Y2: a] : ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ F @ ( G @ Y2 ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_68_SUP__UNION,axiom,
! [F: $o > set_set_a,G: a > set_o,A2: set_a] :
( ( comple3958522678809307947_set_a @ ( image_o_set_set_a @ F @ ( comple90263536869209701_set_o @ ( image_a_set_o @ G @ A2 ) ) ) )
= ( comple3958522678809307947_set_a
@ ( image_a_set_set_a
@ ^ [Y2: a] : ( comple3958522678809307947_set_a @ ( image_o_set_set_a @ F @ ( G @ Y2 ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_69_SUP__UNION,axiom,
! [F: set_a > set_a,G: a > set_set_a,A2: set_a] :
( ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ F @ ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ G @ A2 ) ) ) )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [Y2: a] : ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ F @ ( G @ Y2 ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_70_SUP__UNION,axiom,
! [F: a > set_a,G: set_a > set_a,A2: set_set_a] :
( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ G @ A2 ) ) ) )
= ( comple2307003609928055243_set_a
@ ( image_set_a_set_a
@ ^ [Y2: set_a] : ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ ( G @ Y2 ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_71_SUP__UNION,axiom,
! [F: $o > set_a,G: set_a > set_o,A2: set_set_a] :
( ( comple2307003609928055243_set_a @ ( image_o_set_a @ F @ ( comple90263536869209701_set_o @ ( image_set_a_set_o @ G @ A2 ) ) ) )
= ( comple2307003609928055243_set_a
@ ( image_set_a_set_a
@ ^ [Y2: set_a] : ( comple2307003609928055243_set_a @ ( image_o_set_a @ F @ ( G @ Y2 ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_72_SUP__UNION,axiom,
! [F: set_a > set_o,G: a > set_set_a,A2: set_a] :
( ( comple90263536869209701_set_o @ ( image_set_a_set_o @ F @ ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ G @ A2 ) ) ) )
= ( comple90263536869209701_set_o
@ ( image_a_set_o
@ ^ [Y2: a] : ( comple90263536869209701_set_o @ ( image_set_a_set_o @ F @ ( G @ Y2 ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_73_SUP__subset__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > set_a,G: $o > set_a] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_o_set_a @ F @ A2 ) ) @ ( comple2307003609928055243_set_a @ ( image_o_set_a @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_74_SUP__subset__mono,axiom,
! [A2: set_a,B2: set_a,F: a > set_a,G: a > set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A2 ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_75_SUP__subset__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > set_o,G: $o > set_o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_less_eq_set_o @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ A2 ) ) @ ( comple90263536869209701_set_o @ ( image_o_set_o @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_76_SUP__subset__mono,axiom,
! [A2: set_a,B2: set_a,F: a > set_o,G: a > set_o] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ord_less_eq_set_o @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_a_set_o @ F @ A2 ) ) @ ( comple90263536869209701_set_o @ ( image_a_set_o @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_77_SUP__subset__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > set_set_a,G: $o > set_set_a] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ ( image_o_set_set_a @ F @ A2 ) ) @ ( comple3958522678809307947_set_a @ ( image_o_set_set_a @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_78_SUP__subset__mono,axiom,
! [A2: set_a,B2: set_a,F: a > set_set_a,G: a > set_set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ F @ A2 ) ) @ ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_79_SUP__subset__mono,axiom,
! [A2: set_set_a,B2: set_set_a,F: set_a > set_a,G: set_a > set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ F @ A2 ) ) @ ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_80_SUP__subset__mono,axiom,
! [A2: set_set_a,B2: set_set_a,F: set_a > set_o,G: set_a > set_o] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( ord_less_eq_set_o @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_set_a_set_o @ F @ A2 ) ) @ ( comple90263536869209701_set_o @ ( image_set_a_set_o @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_81_SUP__subset__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > set_Product_prod_a_a,G: $o > set_Product_prod_a_a] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_le746702958409616551od_a_a @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_le746702958409616551od_a_a @ ( comple8421679170691845492od_a_a @ ( image_4840794370999913844od_a_a @ F @ A2 ) ) @ ( comple8421679170691845492od_a_a @ ( image_4840794370999913844od_a_a @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_82_SUP__subset__mono,axiom,
! [A2: set_a,B2: set_a,F: a > set_Product_prod_a_a,G: a > set_Product_prod_a_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ord_le746702958409616551od_a_a @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_le746702958409616551od_a_a @ ( comple8421679170691845492od_a_a @ ( image_4421510592991446670od_a_a @ F @ A2 ) ) @ ( comple8421679170691845492od_a_a @ ( image_4421510592991446670od_a_a @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_83_split__part,axiom,
! [P: $o,Q: a > product_prod_a_a > $o] :
( ( produc5856822985862792195_a_a_o
@ ^ [A3: a,B3: product_prod_a_a] :
( P
& ( Q @ A3 @ B3 ) ) )
= ( ^ [Ab: produc4044097585999906000od_a_a] :
( P
& ( produc5856822985862792195_a_a_o @ Q @ Ab ) ) ) ) ).
% split_part
thf(fact_84_split__part,axiom,
! [P: $o,Q: a > a > $o] :
( ( produc6436628058953941356_a_a_o
@ ^ [A3: a,B3: a] :
( P
& ( Q @ A3 @ B3 ) ) )
= ( ^ [Ab: product_prod_a_a] :
( P
& ( produc6436628058953941356_a_a_o @ Q @ Ab ) ) ) ) ).
% split_part
thf(fact_85_UN__extend__simps_I8_J,axiom,
! [B2: a > set_a,A2: set_set_a] :
( ( comple2307003609928055243_set_a
@ ( image_set_a_set_a
@ ^ [Y2: set_a] : ( comple2307003609928055243_set_a @ ( image_a_set_a @ B2 @ Y2 ) )
@ A2 ) )
= ( comple2307003609928055243_set_a @ ( image_a_set_a @ B2 @ ( comple2307003609928055243_set_a @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_86_UN__extend__simps_I8_J,axiom,
! [B2: $o > set_a,A2: set_set_o] :
( ( comple2307003609928055243_set_a
@ ( image_set_o_set_a
@ ^ [Y2: set_o] : ( comple2307003609928055243_set_a @ ( image_o_set_a @ B2 @ Y2 ) )
@ A2 ) )
= ( comple2307003609928055243_set_a @ ( image_o_set_a @ B2 @ ( comple90263536869209701_set_o @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_87_UN__extend__simps_I8_J,axiom,
! [B2: a > set_o,A2: set_set_a] :
( ( comple90263536869209701_set_o
@ ( image_set_a_set_o
@ ^ [Y2: set_a] : ( comple90263536869209701_set_o @ ( image_a_set_o @ B2 @ Y2 ) )
@ A2 ) )
= ( comple90263536869209701_set_o @ ( image_a_set_o @ B2 @ ( comple2307003609928055243_set_a @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_88_UN__extend__simps_I8_J,axiom,
! [B2: $o > set_o,A2: set_set_o] :
( ( comple90263536869209701_set_o
@ ( image_set_o_set_o
@ ^ [Y2: set_o] : ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ Y2 ) )
@ A2 ) )
= ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ ( comple90263536869209701_set_o @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_89_UN__extend__simps_I8_J,axiom,
! [B2: a > set_set_a,A2: set_set_a] :
( ( comple3958522678809307947_set_a
@ ( image_4955109552351689957_set_a
@ ^ [Y2: set_a] : ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ B2 @ Y2 ) )
@ A2 ) )
= ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ B2 @ ( comple2307003609928055243_set_a @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_90_UN__extend__simps_I8_J,axiom,
! [B2: $o > set_set_a,A2: set_set_o] :
( ( comple3958522678809307947_set_a
@ ( image_7612453856319013963_set_a
@ ^ [Y2: set_o] : ( comple3958522678809307947_set_a @ ( image_o_set_set_a @ B2 @ Y2 ) )
@ A2 ) )
= ( comple3958522678809307947_set_a @ ( image_o_set_set_a @ B2 @ ( comple90263536869209701_set_o @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_91_UN__extend__simps_I8_J,axiom,
! [B2: set_a > set_a,A2: set_set_set_a] :
( ( comple2307003609928055243_set_a
@ ( image_6061375613820669477_set_a
@ ^ [Y2: set_set_a] : ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ B2 @ Y2 ) )
@ A2 ) )
= ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ B2 @ ( comple3958522678809307947_set_a @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_92_UN__extend__simps_I8_J,axiom,
! [B2: set_a > set_o,A2: set_set_set_a] :
( ( comple90263536869209701_set_o
@ ( image_4406776271737083839_set_o
@ ^ [Y2: set_set_a] : ( comple90263536869209701_set_o @ ( image_set_a_set_o @ B2 @ Y2 ) )
@ A2 ) )
= ( comple90263536869209701_set_o @ ( image_set_a_set_o @ B2 @ ( comple3958522678809307947_set_a @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_93_UN__extend__simps_I8_J,axiom,
! [B2: a > set_Product_prod_a_a,A2: set_set_a] :
( ( comple8421679170691845492od_a_a
@ ( image_6165024369500519726od_a_a
@ ^ [Y2: set_a] : ( comple8421679170691845492od_a_a @ ( image_4421510592991446670od_a_a @ B2 @ Y2 ) )
@ A2 ) )
= ( comple8421679170691845492od_a_a @ ( image_4421510592991446670od_a_a @ B2 @ ( comple2307003609928055243_set_a @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_94_UN__extend__simps_I8_J,axiom,
! [B2: $o > set_Product_prod_a_a,A2: set_set_o] :
( ( comple8421679170691845492od_a_a
@ ( image_1650781834251077268od_a_a
@ ^ [Y2: set_o] : ( comple8421679170691845492od_a_a @ ( image_4840794370999913844od_a_a @ B2 @ Y2 ) )
@ A2 ) )
= ( comple8421679170691845492od_a_a @ ( image_4840794370999913844od_a_a @ B2 @ ( comple90263536869209701_set_o @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_95_triangle__in__graph__ss,axiom,
! [E: set_set_a,X: a,Y: a,Z: a] :
( ( ord_le3724670747650509150_set_a @ E @ edges )
=> ( ( graph_4582152751571636272raph_a @ E @ X @ Y @ Z )
=> ( graph_4582152751571636272raph_a @ edges @ X @ Y @ Z ) ) ) ).
% triangle_in_graph_ss
thf(fact_96_triangle__set__graph__edge__ss,axiom,
! [E: set_set_a] :
( ( ord_le3724670747650509150_set_a @ E @ edges )
=> ( ord_le3724670747650509150_set_a @ ( graph_triangle_set_a @ E ) @ ( graph_triangle_set_a @ edges ) ) ) ).
% triangle_set_graph_edge_ss
thf(fact_97_Union__iff,axiom,
! [A2: produc4044097585999906000od_a_a,C: set_se9027383378080648592od_a_a] :
( ( member3071122053849602553od_a_a @ A2 @ ( comple2978350343072902813od_a_a @ C ) )
= ( ? [X2: set_Pr5530083903271594800od_a_a] :
( ( member4210947715425868889od_a_a @ X2 @ C )
& ( member3071122053849602553od_a_a @ A2 @ X2 ) ) ) ) ).
% Union_iff
thf(fact_98_Union__iff,axiom,
! [A2: product_prod_a_a,C: set_se5735800977113168103od_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ ( comple8421679170691845492od_a_a @ C ) )
= ( ? [X2: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X2 @ C )
& ( member1426531477525435216od_a_a @ A2 @ X2 ) ) ) ) ).
% Union_iff
thf(fact_99_Union__iff,axiom,
! [A2: set_a,C: set_set_set_a] :
( ( member_set_a @ A2 @ ( comple3958522678809307947_set_a @ C ) )
= ( ? [X2: set_set_a] :
( ( member_set_set_a @ X2 @ C )
& ( member_set_a @ A2 @ X2 ) ) ) ) ).
% Union_iff
thf(fact_100_Union__iff,axiom,
! [A2: a,C: set_set_a] :
( ( member_a @ A2 @ ( comple2307003609928055243_set_a @ C ) )
= ( ? [X2: set_a] :
( ( member_set_a @ X2 @ C )
& ( member_a @ A2 @ X2 ) ) ) ) ).
% Union_iff
thf(fact_101_Union__iff,axiom,
! [A2: $o,C: set_set_o] :
( ( member_o @ A2 @ ( comple90263536869209701_set_o @ C ) )
= ( ? [X2: set_o] :
( ( member_set_o @ X2 @ C )
& ( member_o @ A2 @ X2 ) ) ) ) ).
% Union_iff
thf(fact_102_UnionI,axiom,
! [X4: set_Pr5530083903271594800od_a_a,C: set_se9027383378080648592od_a_a,A2: produc4044097585999906000od_a_a] :
( ( member4210947715425868889od_a_a @ X4 @ C )
=> ( ( member3071122053849602553od_a_a @ A2 @ X4 )
=> ( member3071122053849602553od_a_a @ A2 @ ( comple2978350343072902813od_a_a @ C ) ) ) ) ).
% UnionI
thf(fact_103_UnionI,axiom,
! [X4: set_Product_prod_a_a,C: set_se5735800977113168103od_a_a,A2: product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X4 @ C )
=> ( ( member1426531477525435216od_a_a @ A2 @ X4 )
=> ( member1426531477525435216od_a_a @ A2 @ ( comple8421679170691845492od_a_a @ C ) ) ) ) ).
% UnionI
thf(fact_104_UnionI,axiom,
! [X4: set_set_a,C: set_set_set_a,A2: set_a] :
( ( member_set_set_a @ X4 @ C )
=> ( ( member_set_a @ A2 @ X4 )
=> ( member_set_a @ A2 @ ( comple3958522678809307947_set_a @ C ) ) ) ) ).
% UnionI
thf(fact_105_UnionI,axiom,
! [X4: set_a,C: set_set_a,A2: a] :
( ( member_set_a @ X4 @ C )
=> ( ( member_a @ A2 @ X4 )
=> ( member_a @ A2 @ ( comple2307003609928055243_set_a @ C ) ) ) ) ).
% UnionI
thf(fact_106_UnionI,axiom,
! [X4: set_o,C: set_set_o,A2: $o] :
( ( member_set_o @ X4 @ C )
=> ( ( member_o @ A2 @ X4 )
=> ( member_o @ A2 @ ( comple90263536869209701_set_o @ C ) ) ) ) ).
% UnionI
thf(fact_107_UN__ball__bex__simps_I1_J,axiom,
! [A2: set_se9027383378080648592od_a_a,P: produc4044097585999906000od_a_a > $o] :
( ( ! [X2: produc4044097585999906000od_a_a] :
( ( member3071122053849602553od_a_a @ X2 @ ( comple2978350343072902813od_a_a @ A2 ) )
=> ( P @ X2 ) ) )
= ( ! [X2: set_Pr5530083903271594800od_a_a] :
( ( member4210947715425868889od_a_a @ X2 @ A2 )
=> ! [Y2: produc4044097585999906000od_a_a] :
( ( member3071122053849602553od_a_a @ Y2 @ X2 )
=> ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_108_UN__ball__bex__simps_I1_J,axiom,
! [A2: set_se5735800977113168103od_a_a,P: product_prod_a_a > $o] :
( ( ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ ( comple8421679170691845492od_a_a @ A2 ) )
=> ( P @ X2 ) ) )
= ( ! [X2: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X2 @ A2 )
=> ! [Y2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y2 @ X2 )
=> ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_109_UN__ball__bex__simps_I1_J,axiom,
! [A2: set_set_set_a,P: set_a > $o] :
( ( ! [X2: set_a] :
( ( member_set_a @ X2 @ ( comple3958522678809307947_set_a @ A2 ) )
=> ( P @ X2 ) ) )
= ( ! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ A2 )
=> ! [Y2: set_a] :
( ( member_set_a @ Y2 @ X2 )
=> ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_110_UN__ball__bex__simps_I1_J,axiom,
! [A2: set_set_a,P: a > $o] :
( ( ! [X2: a] :
( ( member_a @ X2 @ ( comple2307003609928055243_set_a @ A2 ) )
=> ( P @ X2 ) ) )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ! [Y2: a] :
( ( member_a @ Y2 @ X2 )
=> ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_111_UN__ball__bex__simps_I1_J,axiom,
! [A2: set_set_o,P: $o > $o] :
( ( ! [X2: $o] :
( ( member_o @ X2 @ ( comple90263536869209701_set_o @ A2 ) )
=> ( P @ X2 ) ) )
= ( ! [X2: set_o] :
( ( member_set_o @ X2 @ A2 )
=> ! [Y2: $o] :
( ( member_o @ Y2 @ X2 )
=> ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_112_UN__ball__bex__simps_I3_J,axiom,
! [A2: set_se9027383378080648592od_a_a,P: produc4044097585999906000od_a_a > $o] :
( ( ? [X2: produc4044097585999906000od_a_a] :
( ( member3071122053849602553od_a_a @ X2 @ ( comple2978350343072902813od_a_a @ A2 ) )
& ( P @ X2 ) ) )
= ( ? [X2: set_Pr5530083903271594800od_a_a] :
( ( member4210947715425868889od_a_a @ X2 @ A2 )
& ? [Y2: produc4044097585999906000od_a_a] :
( ( member3071122053849602553od_a_a @ Y2 @ X2 )
& ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_113_UN__ball__bex__simps_I3_J,axiom,
! [A2: set_se5735800977113168103od_a_a,P: product_prod_a_a > $o] :
( ( ? [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ ( comple8421679170691845492od_a_a @ A2 ) )
& ( P @ X2 ) ) )
= ( ? [X2: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X2 @ A2 )
& ? [Y2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y2 @ X2 )
& ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_114_UN__ball__bex__simps_I3_J,axiom,
! [A2: set_set_set_a,P: set_a > $o] :
( ( ? [X2: set_a] :
( ( member_set_a @ X2 @ ( comple3958522678809307947_set_a @ A2 ) )
& ( P @ X2 ) ) )
= ( ? [X2: set_set_a] :
( ( member_set_set_a @ X2 @ A2 )
& ? [Y2: set_a] :
( ( member_set_a @ Y2 @ X2 )
& ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_115_UN__ball__bex__simps_I3_J,axiom,
! [A2: set_set_a,P: a > $o] :
( ( ? [X2: a] :
( ( member_a @ X2 @ ( comple2307003609928055243_set_a @ A2 ) )
& ( P @ X2 ) ) )
= ( ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ? [Y2: a] :
( ( member_a @ Y2 @ X2 )
& ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_116_UN__ball__bex__simps_I3_J,axiom,
! [A2: set_set_o,P: $o > $o] :
( ( ? [X2: $o] :
( ( member_o @ X2 @ ( comple90263536869209701_set_o @ A2 ) )
& ( P @ X2 ) ) )
= ( ? [X2: set_o] :
( ( member_set_o @ X2 @ A2 )
& ? [Y2: $o] :
( ( member_o @ Y2 @ X2 )
& ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_117_Sup__set__def,axiom,
( comple7399068483239264473et_nat
= ( ^ [A4: set_set_nat] :
( collect_nat
@ ^ [X2: nat] : ( complete_Sup_Sup_o @ ( image_set_nat_o @ ( member_nat @ X2 ) @ A4 ) ) ) ) ) ).
% Sup_set_def
thf(fact_118_Sup__set__def,axiom,
( comple2978350343072902813od_a_a
= ( ^ [A4: set_se9027383378080648592od_a_a] :
( collec5045780995415420475od_a_a
@ ^ [X2: produc4044097585999906000od_a_a] : ( complete_Sup_Sup_o @ ( image_329807791404844781_a_a_o @ ( member3071122053849602553od_a_a @ X2 ) @ A4 ) ) ) ) ) ).
% Sup_set_def
thf(fact_119_Sup__set__def,axiom,
( comple8421679170691845492od_a_a
= ( ^ [A4: set_se5735800977113168103od_a_a] :
( collec3336397797384452498od_a_a
@ ^ [X2: product_prod_a_a] : ( complete_Sup_Sup_o @ ( image_3158127254886349654_a_a_o @ ( member1426531477525435216od_a_a @ X2 ) @ A4 ) ) ) ) ) ).
% Sup_set_def
thf(fact_120_Sup__set__def,axiom,
( comple3958522678809307947_set_a
= ( ^ [A4: set_set_set_a] :
( collect_set_a
@ ^ [X2: set_a] : ( complete_Sup_Sup_o @ ( image_set_set_a_o @ ( member_set_a @ X2 ) @ A4 ) ) ) ) ) ).
% Sup_set_def
thf(fact_121_Sup__set__def,axiom,
( comple2307003609928055243_set_a
= ( ^ [A4: set_set_a] :
( collect_a
@ ^ [X2: a] : ( complete_Sup_Sup_o @ ( image_set_a_o @ ( member_a @ X2 ) @ A4 ) ) ) ) ) ).
% Sup_set_def
thf(fact_122_Sup__set__def,axiom,
( comple90263536869209701_set_o
= ( ^ [A4: set_set_o] :
( collect_o
@ ^ [X2: $o] : ( complete_Sup_Sup_o @ ( image_set_o_o @ ( member_o @ X2 ) @ A4 ) ) ) ) ) ).
% Sup_set_def
thf(fact_123_Collect__case__prod__mono,axiom,
! [A2: a > product_prod_a_a > $o,B2: a > product_prod_a_a > $o] :
( ( ord_le5507034481861240613_a_a_o @ A2 @ B2 )
=> ( ord_le114883831454073552od_a_a @ ( collec5045780995415420475od_a_a @ ( produc5856822985862792195_a_a_o @ A2 ) ) @ ( collec5045780995415420475od_a_a @ ( produc5856822985862792195_a_a_o @ B2 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_124_Collect__case__prod__mono,axiom,
! [A2: a > a > $o,B2: a > a > $o] :
( ( ord_less_eq_a_a_o @ A2 @ B2 )
=> ( ord_le746702958409616551od_a_a @ ( collec3336397797384452498od_a_a @ ( produc6436628058953941356_a_a_o @ A2 ) ) @ ( collec3336397797384452498od_a_a @ ( produc6436628058953941356_a_a_o @ B2 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_125_Sup_OSUP__cong,axiom,
! [A2: set_list_a,B2: set_list_a,C: list_a > produc4044097585999906000od_a_a,D: list_a > produc4044097585999906000od_a_a,Sup: set_Pr5530083903271594800od_a_a > produc4044097585999906000od_a_a] :
( ( A2 = B2 )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_1195025184546981201od_a_a @ C @ A2 ) )
= ( Sup @ ( image_1195025184546981201od_a_a @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_126_Sup_OSUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C: nat > nat,D: nat > nat,Sup: set_nat > nat] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_nat_nat @ C @ A2 ) )
= ( Sup @ ( image_nat_nat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_127_Sup_OSUP__cong,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_a > set_Pr5530083903271594800od_a_a,D: set_a > set_Pr5530083903271594800od_a_a,Sup: set_se9027383378080648592od_a_a > set_Pr5530083903271594800od_a_a] :
( ( A2 = B2 )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_7562202058474640471od_a_a @ C @ A2 ) )
= ( Sup @ ( image_7562202058474640471od_a_a @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_128_Sup_OSUP__cong,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_a > set_Product_prod_a_a,D: set_a > set_Product_prod_a_a,Sup: set_se5735800977113168103od_a_a > set_Product_prod_a_a] :
( ( A2 = B2 )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_6165024369500519726od_a_a @ C @ A2 ) )
= ( Sup @ ( image_6165024369500519726od_a_a @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_129_Sup_OSUP__cong,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_a > set_a,D: set_a > set_a,Sup: set_set_a > set_a] :
( ( A2 = B2 )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_set_a_set_a @ C @ A2 ) )
= ( Sup @ ( image_set_a_set_a @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_130_Sup_OSUP__cong,axiom,
! [A2: set_a,B2: set_a,C: a > set_set_a,D: a > set_set_a,Sup: set_set_set_a > set_set_a] :
( ( A2 = B2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_a_set_set_a @ C @ A2 ) )
= ( Sup @ ( image_a_set_set_a @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_131_Sup_OSUP__cong,axiom,
! [A2: set_a,B2: set_a,C: a > set_a,D: a > set_a,Sup: set_set_a > set_a] :
( ( A2 = B2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_a_set_a @ C @ A2 ) )
= ( Sup @ ( image_a_set_a @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_132_Sup_OSUP__cong,axiom,
! [A2: set_a,B2: set_a,C: a > set_Pr5530083903271594800od_a_a,D: a > set_Pr5530083903271594800od_a_a,Sup: set_se9027383378080648592od_a_a > set_Pr5530083903271594800od_a_a] :
( ( A2 = B2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_5653227685612666295od_a_a @ C @ A2 ) )
= ( Sup @ ( image_5653227685612666295od_a_a @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_133_Sup_OSUP__cong,axiom,
! [A2: set_a,B2: set_a,C: a > set_Product_prod_a_a,D: a > set_Product_prod_a_a,Sup: set_se5735800977113168103od_a_a > set_Product_prod_a_a] :
( ( A2 = B2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_4421510592991446670od_a_a @ C @ A2 ) )
= ( Sup @ ( image_4421510592991446670od_a_a @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_134_Sup_OSUP__cong,axiom,
! [A2: set_a,B2: set_a,C: a > set_o,D: a > set_o,Sup: set_set_o > set_o] :
( ( A2 = B2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_a_set_o @ C @ A2 ) )
= ( Sup @ ( image_a_set_o @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_135_Inf_OINF__cong,axiom,
! [A2: set_list_a,B2: set_list_a,C: list_a > produc4044097585999906000od_a_a,D: list_a > produc4044097585999906000od_a_a,Inf: set_Pr5530083903271594800od_a_a > produc4044097585999906000od_a_a] :
( ( A2 = B2 )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_1195025184546981201od_a_a @ C @ A2 ) )
= ( Inf @ ( image_1195025184546981201od_a_a @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_136_Inf_OINF__cong,axiom,
! [A2: set_nat,B2: set_nat,C: nat > nat,D: nat > nat,Inf: set_nat > nat] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_nat_nat @ C @ A2 ) )
= ( Inf @ ( image_nat_nat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_137_Inf_OINF__cong,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_a > set_Pr5530083903271594800od_a_a,D: set_a > set_Pr5530083903271594800od_a_a,Inf: set_se9027383378080648592od_a_a > set_Pr5530083903271594800od_a_a] :
( ( A2 = B2 )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_7562202058474640471od_a_a @ C @ A2 ) )
= ( Inf @ ( image_7562202058474640471od_a_a @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_138_Inf_OINF__cong,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_a > set_Product_prod_a_a,D: set_a > set_Product_prod_a_a,Inf: set_se5735800977113168103od_a_a > set_Product_prod_a_a] :
( ( A2 = B2 )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_6165024369500519726od_a_a @ C @ A2 ) )
= ( Inf @ ( image_6165024369500519726od_a_a @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_139_Inf_OINF__cong,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_a > set_a,D: set_a > set_a,Inf: set_set_a > set_a] :
( ( A2 = B2 )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_set_a_set_a @ C @ A2 ) )
= ( Inf @ ( image_set_a_set_a @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_140_Inf_OINF__cong,axiom,
! [A2: set_a,B2: set_a,C: a > set_set_a,D: a > set_set_a,Inf: set_set_set_a > set_set_a] :
( ( A2 = B2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_a_set_set_a @ C @ A2 ) )
= ( Inf @ ( image_a_set_set_a @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_141_Inf_OINF__cong,axiom,
! [A2: set_a,B2: set_a,C: a > set_a,D: a > set_a,Inf: set_set_a > set_a] :
( ( A2 = B2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_a_set_a @ C @ A2 ) )
= ( Inf @ ( image_a_set_a @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_142_Inf_OINF__cong,axiom,
! [A2: set_a,B2: set_a,C: a > set_Pr5530083903271594800od_a_a,D: a > set_Pr5530083903271594800od_a_a,Inf: set_se9027383378080648592od_a_a > set_Pr5530083903271594800od_a_a] :
( ( A2 = B2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_5653227685612666295od_a_a @ C @ A2 ) )
= ( Inf @ ( image_5653227685612666295od_a_a @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_143_Inf_OINF__cong,axiom,
! [A2: set_a,B2: set_a,C: a > set_Product_prod_a_a,D: a > set_Product_prod_a_a,Inf: set_se5735800977113168103od_a_a > set_Product_prod_a_a] :
( ( A2 = B2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_4421510592991446670od_a_a @ C @ A2 ) )
= ( Inf @ ( image_4421510592991446670od_a_a @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_144_Inf_OINF__cong,axiom,
! [A2: set_a,B2: set_a,C: a > set_o,D: a > set_o,Inf: set_set_o > set_o] :
( ( A2 = B2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_a_set_o @ C @ A2 ) )
= ( Inf @ ( image_a_set_o @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_145_UnionE,axiom,
! [A2: produc4044097585999906000od_a_a,C: set_se9027383378080648592od_a_a] :
( ( member3071122053849602553od_a_a @ A2 @ ( comple2978350343072902813od_a_a @ C ) )
=> ~ ! [X5: set_Pr5530083903271594800od_a_a] :
( ( member3071122053849602553od_a_a @ A2 @ X5 )
=> ~ ( member4210947715425868889od_a_a @ X5 @ C ) ) ) ).
% UnionE
thf(fact_146_UnionE,axiom,
! [A2: product_prod_a_a,C: set_se5735800977113168103od_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ ( comple8421679170691845492od_a_a @ C ) )
=> ~ ! [X5: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ X5 )
=> ~ ( member1816616512716248880od_a_a @ X5 @ C ) ) ) ).
% UnionE
thf(fact_147_UnionE,axiom,
! [A2: set_a,C: set_set_set_a] :
( ( member_set_a @ A2 @ ( comple3958522678809307947_set_a @ C ) )
=> ~ ! [X5: set_set_a] :
( ( member_set_a @ A2 @ X5 )
=> ~ ( member_set_set_a @ X5 @ C ) ) ) ).
% UnionE
thf(fact_148_UnionE,axiom,
! [A2: a,C: set_set_a] :
( ( member_a @ A2 @ ( comple2307003609928055243_set_a @ C ) )
=> ~ ! [X5: set_a] :
( ( member_a @ A2 @ X5 )
=> ~ ( member_set_a @ X5 @ C ) ) ) ).
% UnionE
thf(fact_149_UnionE,axiom,
! [A2: $o,C: set_set_o] :
( ( member_o @ A2 @ ( comple90263536869209701_set_o @ C ) )
=> ~ ! [X5: set_o] :
( ( member_o @ A2 @ X5 )
=> ~ ( member_set_o @ X5 @ C ) ) ) ).
% UnionE
thf(fact_150_Sup_OSUP__identity__eq,axiom,
! [Sup: set_set_a > set_a,A2: set_set_a] :
( ( Sup
@ ( image_set_a_set_a
@ ^ [X2: set_a] : X2
@ A2 ) )
= ( Sup @ A2 ) ) ).
% Sup.SUP_identity_eq
thf(fact_151_Sup_OSUP__identity__eq,axiom,
! [Sup: set_nat > nat,A2: set_nat] :
( ( Sup
@ ( image_nat_nat
@ ^ [X2: nat] : X2
@ A2 ) )
= ( Sup @ A2 ) ) ).
% Sup.SUP_identity_eq
thf(fact_152_Inf_OINF__identity__eq,axiom,
! [Inf: set_set_a > set_a,A2: set_set_a] :
( ( Inf
@ ( image_set_a_set_a
@ ^ [X2: set_a] : X2
@ A2 ) )
= ( Inf @ A2 ) ) ).
% Inf.INF_identity_eq
thf(fact_153_Inf_OINF__identity__eq,axiom,
! [Inf: set_nat > nat,A2: set_nat] :
( ( Inf
@ ( image_nat_nat
@ ^ [X2: nat] : X2
@ A2 ) )
= ( Inf @ A2 ) ) ).
% Inf.INF_identity_eq
thf(fact_154_prod_Ocase__distrib,axiom,
! [H: $o > $o,F: a > product_prod_a_a > $o,Prod: produc4044097585999906000od_a_a] :
( ( H @ ( produc5856822985862792195_a_a_o @ F @ Prod ) )
= ( produc5856822985862792195_a_a_o
@ ^ [X1: a,X22: product_prod_a_a] : ( H @ ( F @ X1 @ X22 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_155_prod_Ocase__distrib,axiom,
! [H: $o > $o,F: a > a > $o,Prod: product_prod_a_a] :
( ( H @ ( produc6436628058953941356_a_a_o @ F @ Prod ) )
= ( produc6436628058953941356_a_a_o
@ ^ [X1: a,X22: a] : ( H @ ( F @ X1 @ X22 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_156_prod_Ocase__distrib,axiom,
! [H: $o > product_prod_a_a,F: a > a > $o,Prod: product_prod_a_a] :
( ( H @ ( produc6436628058953941356_a_a_o @ F @ Prod ) )
= ( produc408267641121961211od_a_a
@ ^ [X1: a,X22: a] : ( H @ ( F @ X1 @ X22 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_157_prod_Ocase__distrib,axiom,
! [H: product_prod_a_a > $o,F: a > a > product_prod_a_a,Prod: product_prod_a_a] :
( ( H @ ( produc408267641121961211od_a_a @ F @ Prod ) )
= ( produc6436628058953941356_a_a_o
@ ^ [X1: a,X22: a] : ( H @ ( F @ X1 @ X22 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_158_prod_Ocase__distrib,axiom,
! [H: product_prod_a_a > product_prod_a_a,F: a > a > product_prod_a_a,Prod: product_prod_a_a] :
( ( H @ ( produc408267641121961211od_a_a @ F @ Prod ) )
= ( produc408267641121961211od_a_a
@ ^ [X1: a,X22: a] : ( H @ ( F @ X1 @ X22 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_159_prod_Odisc__eq__case,axiom,
! [Prod: produc4044097585999906000od_a_a] :
( produc5856822985862792195_a_a_o
@ ^ [Uu: a,Uv: product_prod_a_a] : $true
@ Prod ) ).
% prod.disc_eq_case
thf(fact_160_prod_Odisc__eq__case,axiom,
! [Prod: product_prod_a_a] :
( produc6436628058953941356_a_a_o
@ ^ [Uu: a,Uv: a] : $true
@ Prod ) ).
% prod.disc_eq_case
thf(fact_161_Sup__upper2,axiom,
! [U: $o,A2: set_o,V: $o] :
( ( member_o @ U @ A2 )
=> ( ( ord_less_eq_o @ V @ U )
=> ( ord_less_eq_o @ V @ ( complete_Sup_Sup_o @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_162_Sup__upper2,axiom,
! [U: set_Pr5530083903271594800od_a_a,A2: set_se9027383378080648592od_a_a,V: set_Pr5530083903271594800od_a_a] :
( ( member4210947715425868889od_a_a @ U @ A2 )
=> ( ( ord_le114883831454073552od_a_a @ V @ U )
=> ( ord_le114883831454073552od_a_a @ V @ ( comple2978350343072902813od_a_a @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_163_Sup__upper2,axiom,
! [U: set_Product_prod_a_a,A2: set_se5735800977113168103od_a_a,V: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ U @ A2 )
=> ( ( ord_le746702958409616551od_a_a @ V @ U )
=> ( ord_le746702958409616551od_a_a @ V @ ( comple8421679170691845492od_a_a @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_164_Sup__upper2,axiom,
! [U: set_set_a,A2: set_set_set_a,V: set_set_a] :
( ( member_set_set_a @ U @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ V @ U )
=> ( ord_le3724670747650509150_set_a @ V @ ( comple3958522678809307947_set_a @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_165_Sup__upper2,axiom,
! [U: set_a,A2: set_set_a,V: set_a] :
( ( member_set_a @ U @ A2 )
=> ( ( ord_less_eq_set_a @ V @ U )
=> ( ord_less_eq_set_a @ V @ ( comple2307003609928055243_set_a @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_166_Sup__upper2,axiom,
! [U: set_o,A2: set_set_o,V: set_o] :
( ( member_set_o @ U @ A2 )
=> ( ( ord_less_eq_set_o @ V @ U )
=> ( ord_less_eq_set_o @ V @ ( comple90263536869209701_set_o @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_167_Sup__le__iff,axiom,
! [A2: set_se9027383378080648592od_a_a,B: set_Pr5530083903271594800od_a_a] :
( ( ord_le114883831454073552od_a_a @ ( comple2978350343072902813od_a_a @ A2 ) @ B )
= ( ! [X2: set_Pr5530083903271594800od_a_a] :
( ( member4210947715425868889od_a_a @ X2 @ A2 )
=> ( ord_le114883831454073552od_a_a @ X2 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_168_Sup__le__iff,axiom,
! [A2: set_se5735800977113168103od_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( comple8421679170691845492od_a_a @ A2 ) @ B )
= ( ! [X2: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X2 @ A2 )
=> ( ord_le746702958409616551od_a_a @ X2 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_169_Sup__le__iff,axiom,
! [A2: set_set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ A2 ) @ B )
= ( ! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ A2 )
=> ( ord_le3724670747650509150_set_a @ X2 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_170_Sup__le__iff,axiom,
! [A2: set_set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ B )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ X2 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_171_Sup__le__iff,axiom,
! [A2: set_set_o,B: set_o] :
( ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ A2 ) @ B )
= ( ! [X2: set_o] :
( ( member_set_o @ X2 @ A2 )
=> ( ord_less_eq_set_o @ X2 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_172_Sup__upper,axiom,
! [X: $o,A2: set_o] :
( ( member_o @ X @ A2 )
=> ( ord_less_eq_o @ X @ ( complete_Sup_Sup_o @ A2 ) ) ) ).
% Sup_upper
thf(fact_173_Sup__upper,axiom,
! [X: set_Pr5530083903271594800od_a_a,A2: set_se9027383378080648592od_a_a] :
( ( member4210947715425868889od_a_a @ X @ A2 )
=> ( ord_le114883831454073552od_a_a @ X @ ( comple2978350343072902813od_a_a @ A2 ) ) ) ).
% Sup_upper
thf(fact_174_Sup__upper,axiom,
! [X: set_Product_prod_a_a,A2: set_se5735800977113168103od_a_a] :
( ( member1816616512716248880od_a_a @ X @ A2 )
=> ( ord_le746702958409616551od_a_a @ X @ ( comple8421679170691845492od_a_a @ A2 ) ) ) ).
% Sup_upper
thf(fact_175_Sup__upper,axiom,
! [X: set_set_a,A2: set_set_set_a] :
( ( member_set_set_a @ X @ A2 )
=> ( ord_le3724670747650509150_set_a @ X @ ( comple3958522678809307947_set_a @ A2 ) ) ) ).
% Sup_upper
thf(fact_176_Sup__upper,axiom,
! [X: set_a,A2: set_set_a] :
( ( member_set_a @ X @ A2 )
=> ( ord_less_eq_set_a @ X @ ( comple2307003609928055243_set_a @ A2 ) ) ) ).
% Sup_upper
thf(fact_177_Sup__upper,axiom,
! [X: set_o,A2: set_set_o] :
( ( member_set_o @ X @ A2 )
=> ( ord_less_eq_set_o @ X @ ( comple90263536869209701_set_o @ A2 ) ) ) ).
% Sup_upper
thf(fact_178_Sup__least,axiom,
! [A2: set_o,Z: $o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_less_eq_o @ X3 @ Z ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A2 ) @ Z ) ) ).
% Sup_least
thf(fact_179_Sup__least,axiom,
! [A2: set_se9027383378080648592od_a_a,Z: set_Pr5530083903271594800od_a_a] :
( ! [X3: set_Pr5530083903271594800od_a_a] :
( ( member4210947715425868889od_a_a @ X3 @ A2 )
=> ( ord_le114883831454073552od_a_a @ X3 @ Z ) )
=> ( ord_le114883831454073552od_a_a @ ( comple2978350343072902813od_a_a @ A2 ) @ Z ) ) ).
% Sup_least
thf(fact_180_Sup__least,axiom,
! [A2: set_se5735800977113168103od_a_a,Z: set_Product_prod_a_a] :
( ! [X3: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X3 @ A2 )
=> ( ord_le746702958409616551od_a_a @ X3 @ Z ) )
=> ( ord_le746702958409616551od_a_a @ ( comple8421679170691845492od_a_a @ A2 ) @ Z ) ) ).
% Sup_least
thf(fact_181_Sup__least,axiom,
! [A2: set_set_set_a,Z: set_set_a] :
( ! [X3: set_set_a] :
( ( member_set_set_a @ X3 @ A2 )
=> ( ord_le3724670747650509150_set_a @ X3 @ Z ) )
=> ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ A2 ) @ Z ) ) ).
% Sup_least
thf(fact_182_Sup__least,axiom,
! [A2: set_set_a,Z: set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( ord_less_eq_set_a @ X3 @ Z ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ Z ) ) ).
% Sup_least
thf(fact_183_Sup__least,axiom,
! [A2: set_set_o,Z: set_o] :
( ! [X3: set_o] :
( ( member_set_o @ X3 @ A2 )
=> ( ord_less_eq_set_o @ X3 @ Z ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ A2 ) @ Z ) ) ).
% Sup_least
thf(fact_184_Sup__mono,axiom,
! [A2: set_o,B2: set_o] :
( ! [A5: $o] :
( ( member_o @ A5 @ A2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ B2 )
& ( ord_less_eq_o @ A5 @ X6 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A2 ) @ ( complete_Sup_Sup_o @ B2 ) ) ) ).
% Sup_mono
thf(fact_185_Sup__mono,axiom,
! [A2: set_se9027383378080648592od_a_a,B2: set_se9027383378080648592od_a_a] :
( ! [A5: set_Pr5530083903271594800od_a_a] :
( ( member4210947715425868889od_a_a @ A5 @ A2 )
=> ? [X6: set_Pr5530083903271594800od_a_a] :
( ( member4210947715425868889od_a_a @ X6 @ B2 )
& ( ord_le114883831454073552od_a_a @ A5 @ X6 ) ) )
=> ( ord_le114883831454073552od_a_a @ ( comple2978350343072902813od_a_a @ A2 ) @ ( comple2978350343072902813od_a_a @ B2 ) ) ) ).
% Sup_mono
thf(fact_186_Sup__mono,axiom,
! [A2: set_se5735800977113168103od_a_a,B2: set_se5735800977113168103od_a_a] :
( ! [A5: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ A5 @ A2 )
=> ? [X6: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X6 @ B2 )
& ( ord_le746702958409616551od_a_a @ A5 @ X6 ) ) )
=> ( ord_le746702958409616551od_a_a @ ( comple8421679170691845492od_a_a @ A2 ) @ ( comple8421679170691845492od_a_a @ B2 ) ) ) ).
% Sup_mono
thf(fact_187_Sup__mono,axiom,
! [A2: set_set_set_a,B2: set_set_set_a] :
( ! [A5: set_set_a] :
( ( member_set_set_a @ A5 @ A2 )
=> ? [X6: set_set_a] :
( ( member_set_set_a @ X6 @ B2 )
& ( ord_le3724670747650509150_set_a @ A5 @ X6 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ A2 ) @ ( comple3958522678809307947_set_a @ B2 ) ) ) ).
% Sup_mono
thf(fact_188_Sup__mono,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ! [A5: set_a] :
( ( member_set_a @ A5 @ A2 )
=> ? [X6: set_a] :
( ( member_set_a @ X6 @ B2 )
& ( ord_less_eq_set_a @ A5 @ X6 ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ ( comple2307003609928055243_set_a @ B2 ) ) ) ).
% Sup_mono
thf(fact_189_Sup__mono,axiom,
! [A2: set_set_o,B2: set_set_o] :
( ! [A5: set_o] :
( ( member_set_o @ A5 @ A2 )
=> ? [X6: set_o] :
( ( member_set_o @ X6 @ B2 )
& ( ord_less_eq_set_o @ A5 @ X6 ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ A2 ) @ ( comple90263536869209701_set_o @ B2 ) ) ) ).
% Sup_mono
thf(fact_190_Sup__eqI,axiom,
! [A2: set_o,X: $o] :
( ! [Y3: $o] :
( ( member_o @ Y3 @ A2 )
=> ( ord_less_eq_o @ Y3 @ X ) )
=> ( ! [Y3: $o] :
( ! [Z2: $o] :
( ( member_o @ Z2 @ A2 )
=> ( ord_less_eq_o @ Z2 @ Y3 ) )
=> ( ord_less_eq_o @ X @ Y3 ) )
=> ( ( complete_Sup_Sup_o @ A2 )
= X ) ) ) ).
% Sup_eqI
thf(fact_191_Sup__eqI,axiom,
! [A2: set_se9027383378080648592od_a_a,X: set_Pr5530083903271594800od_a_a] :
( ! [Y3: set_Pr5530083903271594800od_a_a] :
( ( member4210947715425868889od_a_a @ Y3 @ A2 )
=> ( ord_le114883831454073552od_a_a @ Y3 @ X ) )
=> ( ! [Y3: set_Pr5530083903271594800od_a_a] :
( ! [Z2: set_Pr5530083903271594800od_a_a] :
( ( member4210947715425868889od_a_a @ Z2 @ A2 )
=> ( ord_le114883831454073552od_a_a @ Z2 @ Y3 ) )
=> ( ord_le114883831454073552od_a_a @ X @ Y3 ) )
=> ( ( comple2978350343072902813od_a_a @ A2 )
= X ) ) ) ).
% Sup_eqI
thf(fact_192_Sup__eqI,axiom,
! [A2: set_se5735800977113168103od_a_a,X: set_Product_prod_a_a] :
( ! [Y3: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ Y3 @ A2 )
=> ( ord_le746702958409616551od_a_a @ Y3 @ X ) )
=> ( ! [Y3: set_Product_prod_a_a] :
( ! [Z2: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ Z2 @ A2 )
=> ( ord_le746702958409616551od_a_a @ Z2 @ Y3 ) )
=> ( ord_le746702958409616551od_a_a @ X @ Y3 ) )
=> ( ( comple8421679170691845492od_a_a @ A2 )
= X ) ) ) ).
% Sup_eqI
thf(fact_193_Sup__eqI,axiom,
! [A2: set_set_set_a,X: set_set_a] :
( ! [Y3: set_set_a] :
( ( member_set_set_a @ Y3 @ A2 )
=> ( ord_le3724670747650509150_set_a @ Y3 @ X ) )
=> ( ! [Y3: set_set_a] :
( ! [Z2: set_set_a] :
( ( member_set_set_a @ Z2 @ A2 )
=> ( ord_le3724670747650509150_set_a @ Z2 @ Y3 ) )
=> ( ord_le3724670747650509150_set_a @ X @ Y3 ) )
=> ( ( comple3958522678809307947_set_a @ A2 )
= X ) ) ) ).
% Sup_eqI
thf(fact_194_Sup__eqI,axiom,
! [A2: set_set_a,X: set_a] :
( ! [Y3: set_a] :
( ( member_set_a @ Y3 @ A2 )
=> ( ord_less_eq_set_a @ Y3 @ X ) )
=> ( ! [Y3: set_a] :
( ! [Z2: set_a] :
( ( member_set_a @ Z2 @ A2 )
=> ( ord_less_eq_set_a @ Z2 @ Y3 ) )
=> ( ord_less_eq_set_a @ X @ Y3 ) )
=> ( ( comple2307003609928055243_set_a @ A2 )
= X ) ) ) ).
% Sup_eqI
thf(fact_195_Sup__eqI,axiom,
! [A2: set_set_o,X: set_o] :
( ! [Y3: set_o] :
( ( member_set_o @ Y3 @ A2 )
=> ( ord_less_eq_set_o @ Y3 @ X ) )
=> ( ! [Y3: set_o] :
( ! [Z2: set_o] :
( ( member_set_o @ Z2 @ A2 )
=> ( ord_less_eq_set_o @ Z2 @ Y3 ) )
=> ( ord_less_eq_set_o @ X @ Y3 ) )
=> ( ( comple90263536869209701_set_o @ A2 )
= X ) ) ) ).
% Sup_eqI
thf(fact_196_SUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C: nat > nat,D: nat > nat] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat @ C @ A2 ) )
= ( complete_Sup_Sup_nat @ ( image_nat_nat @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_197_SUP__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > nat,D: $o > nat] :
( ( A2 = B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_o_nat @ C @ A2 ) )
= ( complete_Sup_Sup_nat @ ( image_o_nat @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_198_SUP__cong,axiom,
! [A2: set_a,B2: set_a,C: a > nat,D: a > nat] :
( ( A2 = B2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_a_nat @ C @ A2 ) )
= ( complete_Sup_Sup_nat @ ( image_a_nat @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_199_SUP__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > set_a,D: $o > set_a] :
( ( A2 = B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( comple2307003609928055243_set_a @ ( image_o_set_a @ C @ A2 ) )
= ( comple2307003609928055243_set_a @ ( image_o_set_a @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_200_SUP__cong,axiom,
! [A2: set_a,B2: set_a,C: a > set_a,D: a > set_a] :
( ( A2 = B2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( comple2307003609928055243_set_a @ ( image_a_set_a @ C @ A2 ) )
= ( comple2307003609928055243_set_a @ ( image_a_set_a @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_201_SUP__cong,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_a > nat,D: set_a > nat] :
( ( A2 = B2 )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_set_a_nat @ C @ A2 ) )
= ( complete_Sup_Sup_nat @ ( image_set_a_nat @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_202_SUP__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > set_o,D: $o > set_o] :
( ( A2 = B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( comple90263536869209701_set_o @ ( image_o_set_o @ C @ A2 ) )
= ( comple90263536869209701_set_o @ ( image_o_set_o @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_203_SUP__cong,axiom,
! [A2: set_a,B2: set_a,C: a > set_o,D: a > set_o] :
( ( A2 = B2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( comple90263536869209701_set_o @ ( image_a_set_o @ C @ A2 ) )
= ( comple90263536869209701_set_o @ ( image_a_set_o @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_204_SUP__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > set_set_a,D: $o > set_set_a] :
( ( A2 = B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( comple3958522678809307947_set_a @ ( image_o_set_set_a @ C @ A2 ) )
= ( comple3958522678809307947_set_a @ ( image_o_set_set_a @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_205_SUP__cong,axiom,
! [A2: set_a,B2: set_a,C: a > set_set_a,D: a > set_set_a] :
( ( A2 = B2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ C @ A2 ) )
= ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_206_Union__subsetI,axiom,
! [A2: set_se9027383378080648592od_a_a,B2: set_se9027383378080648592od_a_a] :
( ! [X3: set_Pr5530083903271594800od_a_a] :
( ( member4210947715425868889od_a_a @ X3 @ A2 )
=> ? [Y4: set_Pr5530083903271594800od_a_a] :
( ( member4210947715425868889od_a_a @ Y4 @ B2 )
& ( ord_le114883831454073552od_a_a @ X3 @ Y4 ) ) )
=> ( ord_le114883831454073552od_a_a @ ( comple2978350343072902813od_a_a @ A2 ) @ ( comple2978350343072902813od_a_a @ B2 ) ) ) ).
% Union_subsetI
thf(fact_207_Union__subsetI,axiom,
! [A2: set_se5735800977113168103od_a_a,B2: set_se5735800977113168103od_a_a] :
( ! [X3: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X3 @ A2 )
=> ? [Y4: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ Y4 @ B2 )
& ( ord_le746702958409616551od_a_a @ X3 @ Y4 ) ) )
=> ( ord_le746702958409616551od_a_a @ ( comple8421679170691845492od_a_a @ A2 ) @ ( comple8421679170691845492od_a_a @ B2 ) ) ) ).
% Union_subsetI
thf(fact_208_Union__subsetI,axiom,
! [A2: set_set_set_a,B2: set_set_set_a] :
( ! [X3: set_set_a] :
( ( member_set_set_a @ X3 @ A2 )
=> ? [Y4: set_set_a] :
( ( member_set_set_a @ Y4 @ B2 )
& ( ord_le3724670747650509150_set_a @ X3 @ Y4 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ A2 ) @ ( comple3958522678809307947_set_a @ B2 ) ) ) ).
% Union_subsetI
thf(fact_209_Union__subsetI,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ? [Y4: set_a] :
( ( member_set_a @ Y4 @ B2 )
& ( ord_less_eq_set_a @ X3 @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ ( comple2307003609928055243_set_a @ B2 ) ) ) ).
% Union_subsetI
thf(fact_210_Union__subsetI,axiom,
! [A2: set_set_o,B2: set_set_o] :
( ! [X3: set_o] :
( ( member_set_o @ X3 @ A2 )
=> ? [Y4: set_o] :
( ( member_set_o @ Y4 @ B2 )
& ( ord_less_eq_set_o @ X3 @ Y4 ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ A2 ) @ ( comple90263536869209701_set_o @ B2 ) ) ) ).
% Union_subsetI
thf(fact_211_Union__upper,axiom,
! [B2: set_Pr5530083903271594800od_a_a,A2: set_se9027383378080648592od_a_a] :
( ( member4210947715425868889od_a_a @ B2 @ A2 )
=> ( ord_le114883831454073552od_a_a @ B2 @ ( comple2978350343072902813od_a_a @ A2 ) ) ) ).
% Union_upper
thf(fact_212_Union__upper,axiom,
! [B2: set_Product_prod_a_a,A2: set_se5735800977113168103od_a_a] :
( ( member1816616512716248880od_a_a @ B2 @ A2 )
=> ( ord_le746702958409616551od_a_a @ B2 @ ( comple8421679170691845492od_a_a @ A2 ) ) ) ).
% Union_upper
thf(fact_213_Union__upper,axiom,
! [B2: set_set_a,A2: set_set_set_a] :
( ( member_set_set_a @ B2 @ A2 )
=> ( ord_le3724670747650509150_set_a @ B2 @ ( comple3958522678809307947_set_a @ A2 ) ) ) ).
% Union_upper
thf(fact_214_Union__upper,axiom,
! [B2: set_a,A2: set_set_a] :
( ( member_set_a @ B2 @ A2 )
=> ( ord_less_eq_set_a @ B2 @ ( comple2307003609928055243_set_a @ A2 ) ) ) ).
% Union_upper
thf(fact_215_Union__upper,axiom,
! [B2: set_o,A2: set_set_o] :
( ( member_set_o @ B2 @ A2 )
=> ( ord_less_eq_set_o @ B2 @ ( comple90263536869209701_set_o @ A2 ) ) ) ).
% Union_upper
thf(fact_216_mem__Collect__eq,axiom,
! [A: $o,P: $o > $o] :
( ( member_o @ A @ ( collect_o @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_217_mem__Collect__eq,axiom,
! [A: produc4044097585999906000od_a_a,P: produc4044097585999906000od_a_a > $o] :
( ( member3071122053849602553od_a_a @ A @ ( collec5045780995415420475od_a_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_218_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_219_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_220_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_221_mem__Collect__eq,axiom,
! [A: product_prod_a_a,P: product_prod_a_a > $o] :
( ( member1426531477525435216od_a_a @ A @ ( collec3336397797384452498od_a_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_222_Collect__mem__eq,axiom,
! [A2: set_o] :
( ( collect_o
@ ^ [X2: $o] : ( member_o @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_223_Collect__mem__eq,axiom,
! [A2: set_Pr5530083903271594800od_a_a] :
( ( collec5045780995415420475od_a_a
@ ^ [X2: produc4044097585999906000od_a_a] : ( member3071122053849602553od_a_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_224_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_225_Collect__mem__eq,axiom,
! [A2: set_set_a] :
( ( collect_set_a
@ ^ [X2: set_a] : ( member_set_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_226_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_227_Collect__mem__eq,axiom,
! [A2: set_Product_prod_a_a] :
( ( collec3336397797384452498od_a_a
@ ^ [X2: product_prod_a_a] : ( member1426531477525435216od_a_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_228_Collect__cong,axiom,
! [P: produc4044097585999906000od_a_a > $o,Q: produc4044097585999906000od_a_a > $o] :
( ! [X3: produc4044097585999906000od_a_a] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collec5045780995415420475od_a_a @ P )
= ( collec5045780995415420475od_a_a @ Q ) ) ) ).
% Collect_cong
thf(fact_229_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_230_Collect__cong,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X3: set_a] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_set_a @ P )
= ( collect_set_a @ Q ) ) ) ).
% Collect_cong
thf(fact_231_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_232_Collect__cong,axiom,
! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ! [X3: product_prod_a_a] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collec3336397797384452498od_a_a @ P )
= ( collec3336397797384452498od_a_a @ Q ) ) ) ).
% Collect_cong
thf(fact_233_Union__least,axiom,
! [A2: set_se9027383378080648592od_a_a,C: set_Pr5530083903271594800od_a_a] :
( ! [X5: set_Pr5530083903271594800od_a_a] :
( ( member4210947715425868889od_a_a @ X5 @ A2 )
=> ( ord_le114883831454073552od_a_a @ X5 @ C ) )
=> ( ord_le114883831454073552od_a_a @ ( comple2978350343072902813od_a_a @ A2 ) @ C ) ) ).
% Union_least
thf(fact_234_Union__least,axiom,
! [A2: set_se5735800977113168103od_a_a,C: set_Product_prod_a_a] :
( ! [X5: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X5 @ A2 )
=> ( ord_le746702958409616551od_a_a @ X5 @ C ) )
=> ( ord_le746702958409616551od_a_a @ ( comple8421679170691845492od_a_a @ A2 ) @ C ) ) ).
% Union_least
thf(fact_235_Union__least,axiom,
! [A2: set_set_set_a,C: set_set_a] :
( ! [X5: set_set_a] :
( ( member_set_set_a @ X5 @ A2 )
=> ( ord_le3724670747650509150_set_a @ X5 @ C ) )
=> ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ A2 ) @ C ) ) ).
% Union_least
thf(fact_236_Union__least,axiom,
! [A2: set_set_a,C: set_a] :
( ! [X5: set_a] :
( ( member_set_a @ X5 @ A2 )
=> ( ord_less_eq_set_a @ X5 @ C ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ C ) ) ).
% Union_least
thf(fact_237_Union__least,axiom,
! [A2: set_set_o,C: set_o] :
( ! [X5: set_o] :
( ( member_set_o @ X5 @ A2 )
=> ( ord_less_eq_set_o @ X5 @ C ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ A2 ) @ C ) ) ).
% Union_least
thf(fact_238_Union__mono,axiom,
! [A2: set_se9027383378080648592od_a_a,B2: set_se9027383378080648592od_a_a] :
( ( ord_le5596166698269566256od_a_a @ A2 @ B2 )
=> ( ord_le114883831454073552od_a_a @ ( comple2978350343072902813od_a_a @ A2 ) @ ( comple2978350343072902813od_a_a @ B2 ) ) ) ).
% Union_mono
thf(fact_239_Union__mono,axiom,
! [A2: set_se5735800977113168103od_a_a,B2: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ A2 @ B2 )
=> ( ord_le746702958409616551od_a_a @ ( comple8421679170691845492od_a_a @ A2 ) @ ( comple8421679170691845492od_a_a @ B2 ) ) ) ).
% Union_mono
thf(fact_240_Union__mono,axiom,
! [A2: set_set_set_a,B2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A2 @ B2 )
=> ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ A2 ) @ ( comple3958522678809307947_set_a @ B2 ) ) ) ).
% Union_mono
thf(fact_241_Union__mono,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ ( comple2307003609928055243_set_a @ B2 ) ) ) ).
% Union_mono
thf(fact_242_Union__mono,axiom,
! [A2: set_set_o,B2: set_set_o] :
( ( ord_le4374716579403074808_set_o @ A2 @ B2 )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ A2 ) @ ( comple90263536869209701_set_o @ B2 ) ) ) ).
% Union_mono
thf(fact_243_SUP__commute,axiom,
! [F: a > a > set_a,B2: set_a,A2: set_a] :
( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [I: a] : ( comple2307003609928055243_set_a @ ( image_a_set_a @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [J: a] :
( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [I: a] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_244_SUP__commute,axiom,
! [F: a > a > set_o,B2: set_a,A2: set_a] :
( ( comple90263536869209701_set_o
@ ( image_a_set_o
@ ^ [I: a] : ( comple90263536869209701_set_o @ ( image_a_set_o @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple90263536869209701_set_o
@ ( image_a_set_o
@ ^ [J: a] :
( comple90263536869209701_set_o
@ ( image_a_set_o
@ ^ [I: a] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_245_SUP__commute,axiom,
! [F: a > a > set_set_a,B2: set_a,A2: set_a] :
( ( comple3958522678809307947_set_a
@ ( image_a_set_set_a
@ ^ [I: a] : ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple3958522678809307947_set_a
@ ( image_a_set_set_a
@ ^ [J: a] :
( comple3958522678809307947_set_a
@ ( image_a_set_set_a
@ ^ [I: a] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_246_SUP__commute,axiom,
! [F: a > set_a > set_a,B2: set_set_a,A2: set_a] :
( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [I: a] : ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple2307003609928055243_set_a
@ ( image_set_a_set_a
@ ^ [J: set_a] :
( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [I: a] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_247_SUP__commute,axiom,
! [F: set_a > a > set_a,B2: set_a,A2: set_set_a] :
( ( comple2307003609928055243_set_a
@ ( image_set_a_set_a
@ ^ [I: set_a] : ( comple2307003609928055243_set_a @ ( image_a_set_a @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [J: a] :
( comple2307003609928055243_set_a
@ ( image_set_a_set_a
@ ^ [I: set_a] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_248_SUP__commute,axiom,
! [F: a > a > set_Product_prod_a_a,B2: set_a,A2: set_a] :
( ( comple8421679170691845492od_a_a
@ ( image_4421510592991446670od_a_a
@ ^ [I: a] : ( comple8421679170691845492od_a_a @ ( image_4421510592991446670od_a_a @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple8421679170691845492od_a_a
@ ( image_4421510592991446670od_a_a
@ ^ [J: a] :
( comple8421679170691845492od_a_a
@ ( image_4421510592991446670od_a_a
@ ^ [I: a] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_249_SUP__commute,axiom,
! [F: set_a > set_a > set_a,B2: set_set_a,A2: set_set_a] :
( ( comple2307003609928055243_set_a
@ ( image_set_a_set_a
@ ^ [I: set_a] : ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple2307003609928055243_set_a
@ ( image_set_a_set_a
@ ^ [J: set_a] :
( comple2307003609928055243_set_a
@ ( image_set_a_set_a
@ ^ [I: set_a] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_250_SUP__commute,axiom,
! [F: set_a > a > set_Product_prod_a_a,B2: set_a,A2: set_set_a] :
( ( comple8421679170691845492od_a_a
@ ( image_6165024369500519726od_a_a
@ ^ [I: set_a] : ( comple8421679170691845492od_a_a @ ( image_4421510592991446670od_a_a @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple8421679170691845492od_a_a
@ ( image_4421510592991446670od_a_a
@ ^ [J: a] :
( comple8421679170691845492od_a_a
@ ( image_6165024369500519726od_a_a
@ ^ [I: set_a] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_251_SUP__commute,axiom,
! [F: a > set_a > set_Product_prod_a_a,B2: set_set_a,A2: set_a] :
( ( comple8421679170691845492od_a_a
@ ( image_4421510592991446670od_a_a
@ ^ [I: a] : ( comple8421679170691845492od_a_a @ ( image_6165024369500519726od_a_a @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple8421679170691845492od_a_a
@ ( image_6165024369500519726od_a_a
@ ^ [J: set_a] :
( comple8421679170691845492od_a_a
@ ( image_4421510592991446670od_a_a
@ ^ [I: a] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_252_SUP__commute,axiom,
! [F: a > a > set_Pr5530083903271594800od_a_a,B2: set_a,A2: set_a] :
( ( comple2978350343072902813od_a_a
@ ( image_5653227685612666295od_a_a
@ ^ [I: a] : ( comple2978350343072902813od_a_a @ ( image_5653227685612666295od_a_a @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple2978350343072902813od_a_a
@ ( image_5653227685612666295od_a_a
@ ^ [J: a] :
( comple2978350343072902813od_a_a
@ ( image_5653227685612666295od_a_a
@ ^ [I: a] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_253_UN__UN__flatten,axiom,
! [C: a > set_a,B2: a > set_a,A2: set_a] :
( ( comple2307003609928055243_set_a @ ( image_a_set_a @ C @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B2 @ A2 ) ) ) )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [Y2: a] : ( comple2307003609928055243_set_a @ ( image_a_set_a @ C @ ( B2 @ Y2 ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_254_UN__UN__flatten,axiom,
! [C: $o > set_a,B2: a > set_o,A2: set_a] :
( ( comple2307003609928055243_set_a @ ( image_o_set_a @ C @ ( comple90263536869209701_set_o @ ( image_a_set_o @ B2 @ A2 ) ) ) )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [Y2: a] : ( comple2307003609928055243_set_a @ ( image_o_set_a @ C @ ( B2 @ Y2 ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_255_UN__UN__flatten,axiom,
! [C: a > set_o,B2: a > set_a,A2: set_a] :
( ( comple90263536869209701_set_o @ ( image_a_set_o @ C @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B2 @ A2 ) ) ) )
= ( comple90263536869209701_set_o
@ ( image_a_set_o
@ ^ [Y2: a] : ( comple90263536869209701_set_o @ ( image_a_set_o @ C @ ( B2 @ Y2 ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_256_UN__UN__flatten,axiom,
! [C: $o > set_o,B2: a > set_o,A2: set_a] :
( ( comple90263536869209701_set_o @ ( image_o_set_o @ C @ ( comple90263536869209701_set_o @ ( image_a_set_o @ B2 @ A2 ) ) ) )
= ( comple90263536869209701_set_o
@ ( image_a_set_o
@ ^ [Y2: a] : ( comple90263536869209701_set_o @ ( image_o_set_o @ C @ ( B2 @ Y2 ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_257_UN__UN__flatten,axiom,
! [C: a > set_set_a,B2: a > set_a,A2: set_a] :
( ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ C @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B2 @ A2 ) ) ) )
= ( comple3958522678809307947_set_a
@ ( image_a_set_set_a
@ ^ [Y2: a] : ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ C @ ( B2 @ Y2 ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_258_UN__UN__flatten,axiom,
! [C: $o > set_set_a,B2: a > set_o,A2: set_a] :
( ( comple3958522678809307947_set_a @ ( image_o_set_set_a @ C @ ( comple90263536869209701_set_o @ ( image_a_set_o @ B2 @ A2 ) ) ) )
= ( comple3958522678809307947_set_a
@ ( image_a_set_set_a
@ ^ [Y2: a] : ( comple3958522678809307947_set_a @ ( image_o_set_set_a @ C @ ( B2 @ Y2 ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_259_UN__UN__flatten,axiom,
! [C: set_a > set_a,B2: a > set_set_a,A2: set_a] :
( ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ C @ ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ B2 @ A2 ) ) ) )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [Y2: a] : ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ C @ ( B2 @ Y2 ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_260_UN__UN__flatten,axiom,
! [C: a > set_a,B2: set_a > set_a,A2: set_set_a] :
( ( comple2307003609928055243_set_a @ ( image_a_set_a @ C @ ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ B2 @ A2 ) ) ) )
= ( comple2307003609928055243_set_a
@ ( image_set_a_set_a
@ ^ [Y2: set_a] : ( comple2307003609928055243_set_a @ ( image_a_set_a @ C @ ( B2 @ Y2 ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_261_UN__UN__flatten,axiom,
! [C: $o > set_a,B2: set_a > set_o,A2: set_set_a] :
( ( comple2307003609928055243_set_a @ ( image_o_set_a @ C @ ( comple90263536869209701_set_o @ ( image_set_a_set_o @ B2 @ A2 ) ) ) )
= ( comple2307003609928055243_set_a
@ ( image_set_a_set_a
@ ^ [Y2: set_a] : ( comple2307003609928055243_set_a @ ( image_o_set_a @ C @ ( B2 @ Y2 ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_262_UN__UN__flatten,axiom,
! [C: set_a > set_o,B2: a > set_set_a,A2: set_a] :
( ( comple90263536869209701_set_o @ ( image_set_a_set_o @ C @ ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ B2 @ A2 ) ) ) )
= ( comple90263536869209701_set_o
@ ( image_a_set_o
@ ^ [Y2: a] : ( comple90263536869209701_set_o @ ( image_set_a_set_o @ C @ ( B2 @ Y2 ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_263_UN__E,axiom,
! [B: a,B2: $o > set_a,A2: set_o] :
( ( member_a @ B @ ( comple2307003609928055243_set_a @ ( image_o_set_a @ B2 @ A2 ) ) )
=> ~ ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ~ ( member_a @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_264_UN__E,axiom,
! [B: a,B2: a > set_a,A2: set_a] :
( ( member_a @ B @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B2 @ A2 ) ) )
=> ~ ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ~ ( member_a @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_265_UN__E,axiom,
! [B: $o,B2: $o > set_o,A2: set_o] :
( ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
=> ~ ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ~ ( member_o @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_266_UN__E,axiom,
! [B: $o,B2: a > set_o,A2: set_a] :
( ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_a_set_o @ B2 @ A2 ) ) )
=> ~ ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ~ ( member_o @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_267_UN__E,axiom,
! [B: set_a,B2: $o > set_set_a,A2: set_o] :
( ( member_set_a @ B @ ( comple3958522678809307947_set_a @ ( image_o_set_set_a @ B2 @ A2 ) ) )
=> ~ ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ~ ( member_set_a @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_268_UN__E,axiom,
! [B: set_a,B2: a > set_set_a,A2: set_a] :
( ( member_set_a @ B @ ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ B2 @ A2 ) ) )
=> ~ ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ~ ( member_set_a @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_269_UN__E,axiom,
! [B: a,B2: set_a > set_a,A2: set_set_a] :
( ( member_a @ B @ ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ B2 @ A2 ) ) )
=> ~ ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ~ ( member_a @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_270_UN__E,axiom,
! [B: $o,B2: set_a > set_o,A2: set_set_a] :
( ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_set_a_set_o @ B2 @ A2 ) ) )
=> ~ ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ~ ( member_o @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_271_UN__E,axiom,
! [B: product_prod_a_a,B2: $o > set_Product_prod_a_a,A2: set_o] :
( ( member1426531477525435216od_a_a @ B @ ( comple8421679170691845492od_a_a @ ( image_4840794370999913844od_a_a @ B2 @ A2 ) ) )
=> ~ ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ~ ( member1426531477525435216od_a_a @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_272_UN__E,axiom,
! [B: product_prod_a_a,B2: a > set_Product_prod_a_a,A2: set_a] :
( ( member1426531477525435216od_a_a @ B @ ( comple8421679170691845492od_a_a @ ( image_4421510592991446670od_a_a @ B2 @ A2 ) ) )
=> ~ ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ~ ( member1426531477525435216od_a_a @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_273_UN__extend__simps_I9_J,axiom,
! [C: a > set_a,B2: a > set_a,A2: set_a] :
( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X2: a] : ( comple2307003609928055243_set_a @ ( image_a_set_a @ C @ ( B2 @ X2 ) ) )
@ A2 ) )
= ( comple2307003609928055243_set_a @ ( image_a_set_a @ C @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_274_UN__extend__simps_I9_J,axiom,
! [C: $o > set_a,B2: a > set_o,A2: set_a] :
( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X2: a] : ( comple2307003609928055243_set_a @ ( image_o_set_a @ C @ ( B2 @ X2 ) ) )
@ A2 ) )
= ( comple2307003609928055243_set_a @ ( image_o_set_a @ C @ ( comple90263536869209701_set_o @ ( image_a_set_o @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_275_UN__extend__simps_I9_J,axiom,
! [C: a > set_o,B2: a > set_a,A2: set_a] :
( ( comple90263536869209701_set_o
@ ( image_a_set_o
@ ^ [X2: a] : ( comple90263536869209701_set_o @ ( image_a_set_o @ C @ ( B2 @ X2 ) ) )
@ A2 ) )
= ( comple90263536869209701_set_o @ ( image_a_set_o @ C @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_276_UN__extend__simps_I9_J,axiom,
! [C: $o > set_o,B2: a > set_o,A2: set_a] :
( ( comple90263536869209701_set_o
@ ( image_a_set_o
@ ^ [X2: a] : ( comple90263536869209701_set_o @ ( image_o_set_o @ C @ ( B2 @ X2 ) ) )
@ A2 ) )
= ( comple90263536869209701_set_o @ ( image_o_set_o @ C @ ( comple90263536869209701_set_o @ ( image_a_set_o @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_277_UN__extend__simps_I9_J,axiom,
! [C: a > set_set_a,B2: a > set_a,A2: set_a] :
( ( comple3958522678809307947_set_a
@ ( image_a_set_set_a
@ ^ [X2: a] : ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ C @ ( B2 @ X2 ) ) )
@ A2 ) )
= ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ C @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_278_UN__extend__simps_I9_J,axiom,
! [C: $o > set_set_a,B2: a > set_o,A2: set_a] :
( ( comple3958522678809307947_set_a
@ ( image_a_set_set_a
@ ^ [X2: a] : ( comple3958522678809307947_set_a @ ( image_o_set_set_a @ C @ ( B2 @ X2 ) ) )
@ A2 ) )
= ( comple3958522678809307947_set_a @ ( image_o_set_set_a @ C @ ( comple90263536869209701_set_o @ ( image_a_set_o @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_279_UN__extend__simps_I9_J,axiom,
! [C: set_a > set_a,B2: a > set_set_a,A2: set_a] :
( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X2: a] : ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ C @ ( B2 @ X2 ) ) )
@ A2 ) )
= ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ C @ ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_280_UN__extend__simps_I9_J,axiom,
! [C: a > set_a,B2: set_a > set_a,A2: set_set_a] :
( ( comple2307003609928055243_set_a
@ ( image_set_a_set_a
@ ^ [X2: set_a] : ( comple2307003609928055243_set_a @ ( image_a_set_a @ C @ ( B2 @ X2 ) ) )
@ A2 ) )
= ( comple2307003609928055243_set_a @ ( image_a_set_a @ C @ ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_281_UN__extend__simps_I9_J,axiom,
! [C: $o > set_a,B2: set_a > set_o,A2: set_set_a] :
( ( comple2307003609928055243_set_a
@ ( image_set_a_set_a
@ ^ [X2: set_a] : ( comple2307003609928055243_set_a @ ( image_o_set_a @ C @ ( B2 @ X2 ) ) )
@ A2 ) )
= ( comple2307003609928055243_set_a @ ( image_o_set_a @ C @ ( comple90263536869209701_set_o @ ( image_set_a_set_o @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_282_UN__extend__simps_I9_J,axiom,
! [C: set_a > set_o,B2: a > set_set_a,A2: set_a] :
( ( comple90263536869209701_set_o
@ ( image_a_set_o
@ ^ [X2: a] : ( comple90263536869209701_set_o @ ( image_set_a_set_o @ C @ ( B2 @ X2 ) ) )
@ A2 ) )
= ( comple90263536869209701_set_o @ ( image_set_a_set_o @ C @ ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_283_SUP__eq,axiom,
! [A2: set_o,B2: set_o,F: $o > set_a,G: $o > set_a] :
( ! [I2: $o] :
( ( member_o @ I2 @ A2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ B2 )
& ( ord_less_eq_set_a @ ( F @ I2 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A2 )
& ( ord_less_eq_set_a @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
=> ( ( comple2307003609928055243_set_a @ ( image_o_set_a @ F @ A2 ) )
= ( comple2307003609928055243_set_a @ ( image_o_set_a @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_284_SUP__eq,axiom,
! [A2: set_o,B2: set_a,F: $o > set_a,G: a > set_a] :
( ! [I2: $o] :
( ( member_o @ I2 @ A2 )
=> ? [X6: a] :
( ( member_a @ X6 @ B2 )
& ( ord_less_eq_set_a @ ( F @ I2 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: a] :
( ( member_a @ J2 @ B2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A2 )
& ( ord_less_eq_set_a @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
=> ( ( comple2307003609928055243_set_a @ ( image_o_set_a @ F @ A2 ) )
= ( comple2307003609928055243_set_a @ ( image_a_set_a @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_285_SUP__eq,axiom,
! [A2: set_a,B2: set_o,F: a > set_a,G: $o > set_a] :
( ! [I2: a] :
( ( member_a @ I2 @ A2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ B2 )
& ( ord_less_eq_set_a @ ( F @ I2 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B2 )
=> ? [X6: a] :
( ( member_a @ X6 @ A2 )
& ( ord_less_eq_set_a @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
=> ( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A2 ) )
= ( comple2307003609928055243_set_a @ ( image_o_set_a @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_286_SUP__eq,axiom,
! [A2: set_a,B2: set_a,F: a > set_a,G: a > set_a] :
( ! [I2: a] :
( ( member_a @ I2 @ A2 )
=> ? [X6: a] :
( ( member_a @ X6 @ B2 )
& ( ord_less_eq_set_a @ ( F @ I2 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: a] :
( ( member_a @ J2 @ B2 )
=> ? [X6: a] :
( ( member_a @ X6 @ A2 )
& ( ord_less_eq_set_a @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
=> ( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A2 ) )
= ( comple2307003609928055243_set_a @ ( image_a_set_a @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_287_SUP__eq,axiom,
! [A2: set_o,B2: set_o,F: $o > set_o,G: $o > set_o] :
( ! [I2: $o] :
( ( member_o @ I2 @ A2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ B2 )
& ( ord_less_eq_set_o @ ( F @ I2 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A2 )
& ( ord_less_eq_set_o @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
=> ( ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ A2 ) )
= ( comple90263536869209701_set_o @ ( image_o_set_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_288_SUP__eq,axiom,
! [A2: set_o,B2: set_a,F: $o > set_o,G: a > set_o] :
( ! [I2: $o] :
( ( member_o @ I2 @ A2 )
=> ? [X6: a] :
( ( member_a @ X6 @ B2 )
& ( ord_less_eq_set_o @ ( F @ I2 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: a] :
( ( member_a @ J2 @ B2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A2 )
& ( ord_less_eq_set_o @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
=> ( ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ A2 ) )
= ( comple90263536869209701_set_o @ ( image_a_set_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_289_SUP__eq,axiom,
! [A2: set_a,B2: set_o,F: a > set_o,G: $o > set_o] :
( ! [I2: a] :
( ( member_a @ I2 @ A2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ B2 )
& ( ord_less_eq_set_o @ ( F @ I2 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B2 )
=> ? [X6: a] :
( ( member_a @ X6 @ A2 )
& ( ord_less_eq_set_o @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
=> ( ( comple90263536869209701_set_o @ ( image_a_set_o @ F @ A2 ) )
= ( comple90263536869209701_set_o @ ( image_o_set_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_290_SUP__eq,axiom,
! [A2: set_a,B2: set_a,F: a > set_o,G: a > set_o] :
( ! [I2: a] :
( ( member_a @ I2 @ A2 )
=> ? [X6: a] :
( ( member_a @ X6 @ B2 )
& ( ord_less_eq_set_o @ ( F @ I2 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: a] :
( ( member_a @ J2 @ B2 )
=> ? [X6: a] :
( ( member_a @ X6 @ A2 )
& ( ord_less_eq_set_o @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
=> ( ( comple90263536869209701_set_o @ ( image_a_set_o @ F @ A2 ) )
= ( comple90263536869209701_set_o @ ( image_a_set_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_291_SUP__eq,axiom,
! [A2: set_o,B2: set_o,F: $o > set_set_a,G: $o > set_set_a] :
( ! [I2: $o] :
( ( member_o @ I2 @ A2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ B2 )
& ( ord_le3724670747650509150_set_a @ ( F @ I2 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A2 )
& ( ord_le3724670747650509150_set_a @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
=> ( ( comple3958522678809307947_set_a @ ( image_o_set_set_a @ F @ A2 ) )
= ( comple3958522678809307947_set_a @ ( image_o_set_set_a @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_292_SUP__eq,axiom,
! [A2: set_o,B2: set_a,F: $o > set_set_a,G: a > set_set_a] :
( ! [I2: $o] :
( ( member_o @ I2 @ A2 )
=> ? [X6: a] :
( ( member_a @ X6 @ B2 )
& ( ord_le3724670747650509150_set_a @ ( F @ I2 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: a] :
( ( member_a @ J2 @ B2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A2 )
& ( ord_le3724670747650509150_set_a @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
=> ( ( comple3958522678809307947_set_a @ ( image_o_set_set_a @ F @ A2 ) )
= ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_293_Sup__subset__mono,axiom,
! [A2: set_se9027383378080648592od_a_a,B2: set_se9027383378080648592od_a_a] :
( ( ord_le5596166698269566256od_a_a @ A2 @ B2 )
=> ( ord_le114883831454073552od_a_a @ ( comple2978350343072902813od_a_a @ A2 ) @ ( comple2978350343072902813od_a_a @ B2 ) ) ) ).
% Sup_subset_mono
thf(fact_294_Sup__subset__mono,axiom,
! [A2: set_se5735800977113168103od_a_a,B2: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ A2 @ B2 )
=> ( ord_le746702958409616551od_a_a @ ( comple8421679170691845492od_a_a @ A2 ) @ ( comple8421679170691845492od_a_a @ B2 ) ) ) ).
% Sup_subset_mono
thf(fact_295_Sup__subset__mono,axiom,
! [A2: set_set_set_a,B2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A2 @ B2 )
=> ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ A2 ) @ ( comple3958522678809307947_set_a @ B2 ) ) ) ).
% Sup_subset_mono
thf(fact_296_Sup__subset__mono,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ ( comple2307003609928055243_set_a @ B2 ) ) ) ).
% Sup_subset_mono
thf(fact_297_Sup__subset__mono,axiom,
! [A2: set_set_o,B2: set_set_o] :
( ( ord_le4374716579403074808_set_o @ A2 @ B2 )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ A2 ) @ ( comple90263536869209701_set_o @ B2 ) ) ) ).
% Sup_subset_mono
thf(fact_298_SUP__upper2,axiom,
! [I3: $o,A2: set_o,U: set_a,F: $o > set_a] :
( ( member_o @ I3 @ A2 )
=> ( ( ord_less_eq_set_a @ U @ ( F @ I3 ) )
=> ( ord_less_eq_set_a @ U @ ( comple2307003609928055243_set_a @ ( image_o_set_a @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_299_SUP__upper2,axiom,
! [I3: a,A2: set_a,U: set_a,F: a > set_a] :
( ( member_a @ I3 @ A2 )
=> ( ( ord_less_eq_set_a @ U @ ( F @ I3 ) )
=> ( ord_less_eq_set_a @ U @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_300_SUP__upper2,axiom,
! [I3: $o,A2: set_o,U: set_o,F: $o > set_o] :
( ( member_o @ I3 @ A2 )
=> ( ( ord_less_eq_set_o @ U @ ( F @ I3 ) )
=> ( ord_less_eq_set_o @ U @ ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_301_SUP__upper2,axiom,
! [I3: a,A2: set_a,U: set_o,F: a > set_o] :
( ( member_a @ I3 @ A2 )
=> ( ( ord_less_eq_set_o @ U @ ( F @ I3 ) )
=> ( ord_less_eq_set_o @ U @ ( comple90263536869209701_set_o @ ( image_a_set_o @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_302_SUP__upper2,axiom,
! [I3: $o,A2: set_o,U: set_set_a,F: $o > set_set_a] :
( ( member_o @ I3 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ U @ ( F @ I3 ) )
=> ( ord_le3724670747650509150_set_a @ U @ ( comple3958522678809307947_set_a @ ( image_o_set_set_a @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_303_SUP__upper2,axiom,
! [I3: a,A2: set_a,U: set_set_a,F: a > set_set_a] :
( ( member_a @ I3 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ U @ ( F @ I3 ) )
=> ( ord_le3724670747650509150_set_a @ U @ ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_304_SUP__upper2,axiom,
! [I3: set_a,A2: set_set_a,U: set_a,F: set_a > set_a] :
( ( member_set_a @ I3 @ A2 )
=> ( ( ord_less_eq_set_a @ U @ ( F @ I3 ) )
=> ( ord_less_eq_set_a @ U @ ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_305_SUP__upper2,axiom,
! [I3: set_a,A2: set_set_a,U: set_o,F: set_a > set_o] :
( ( member_set_a @ I3 @ A2 )
=> ( ( ord_less_eq_set_o @ U @ ( F @ I3 ) )
=> ( ord_less_eq_set_o @ U @ ( comple90263536869209701_set_o @ ( image_set_a_set_o @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_306_SUP__upper2,axiom,
! [I3: $o,A2: set_o,U: set_Product_prod_a_a,F: $o > set_Product_prod_a_a] :
( ( member_o @ I3 @ A2 )
=> ( ( ord_le746702958409616551od_a_a @ U @ ( F @ I3 ) )
=> ( ord_le746702958409616551od_a_a @ U @ ( comple8421679170691845492od_a_a @ ( image_4840794370999913844od_a_a @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_307_SUP__upper2,axiom,
! [I3: a,A2: set_a,U: set_Product_prod_a_a,F: a > set_Product_prod_a_a] :
( ( member_a @ I3 @ A2 )
=> ( ( ord_le746702958409616551od_a_a @ U @ ( F @ I3 ) )
=> ( ord_le746702958409616551od_a_a @ U @ ( comple8421679170691845492od_a_a @ ( image_4421510592991446670od_a_a @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_308_SUP__le__iff,axiom,
! [F: set_a > set_Pr5530083903271594800od_a_a,A2: set_set_a,U: set_Pr5530083903271594800od_a_a] :
( ( ord_le114883831454073552od_a_a @ ( comple2978350343072902813od_a_a @ ( image_7562202058474640471od_a_a @ F @ A2 ) ) @ U )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( ord_le114883831454073552od_a_a @ ( F @ X2 ) @ U ) ) ) ) ).
% SUP_le_iff
thf(fact_309_SUP__le__iff,axiom,
! [F: a > set_Pr5530083903271594800od_a_a,A2: set_a,U: set_Pr5530083903271594800od_a_a] :
( ( ord_le114883831454073552od_a_a @ ( comple2978350343072902813od_a_a @ ( image_5653227685612666295od_a_a @ F @ A2 ) ) @ U )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ord_le114883831454073552od_a_a @ ( F @ X2 ) @ U ) ) ) ) ).
% SUP_le_iff
thf(fact_310_SUP__le__iff,axiom,
! [F: set_a > set_Product_prod_a_a,A2: set_set_a,U: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( comple8421679170691845492od_a_a @ ( image_6165024369500519726od_a_a @ F @ A2 ) ) @ U )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( ord_le746702958409616551od_a_a @ ( F @ X2 ) @ U ) ) ) ) ).
% SUP_le_iff
thf(fact_311_SUP__le__iff,axiom,
! [F: a > set_Product_prod_a_a,A2: set_a,U: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( comple8421679170691845492od_a_a @ ( image_4421510592991446670od_a_a @ F @ A2 ) ) @ U )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ord_le746702958409616551od_a_a @ ( F @ X2 ) @ U ) ) ) ) ).
% SUP_le_iff
thf(fact_312_SUP__le__iff,axiom,
! [F: a > set_set_a,A2: set_a,U: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ F @ A2 ) ) @ U )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ U ) ) ) ) ).
% SUP_le_iff
thf(fact_313_SUP__le__iff,axiom,
! [F: a > set_a,A2: set_a,U: set_a] :
( ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A2 ) ) @ U )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ U ) ) ) ) ).
% SUP_le_iff
thf(fact_314_SUP__le__iff,axiom,
! [F: set_a > set_a,A2: set_set_a,U: set_a] :
( ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ F @ A2 ) ) @ U )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ U ) ) ) ) ).
% SUP_le_iff
thf(fact_315_SUP__le__iff,axiom,
! [F: a > set_o,A2: set_a,U: set_o] :
( ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_a_set_o @ F @ A2 ) ) @ U )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ord_less_eq_set_o @ ( F @ X2 ) @ U ) ) ) ) ).
% SUP_le_iff
thf(fact_316_SUP__upper,axiom,
! [I3: $o,A2: set_o,F: $o > set_a] :
( ( member_o @ I3 @ A2 )
=> ( ord_less_eq_set_a @ ( F @ I3 ) @ ( comple2307003609928055243_set_a @ ( image_o_set_a @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_317_SUP__upper,axiom,
! [I3: a,A2: set_a,F: a > set_a] :
( ( member_a @ I3 @ A2 )
=> ( ord_less_eq_set_a @ ( F @ I3 ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_318_SUP__upper,axiom,
! [I3: $o,A2: set_o,F: $o > set_o] :
( ( member_o @ I3 @ A2 )
=> ( ord_less_eq_set_o @ ( F @ I3 ) @ ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_319_SUP__upper,axiom,
! [I3: a,A2: set_a,F: a > set_o] :
( ( member_a @ I3 @ A2 )
=> ( ord_less_eq_set_o @ ( F @ I3 ) @ ( comple90263536869209701_set_o @ ( image_a_set_o @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_320_SUP__upper,axiom,
! [I3: $o,A2: set_o,F: $o > set_set_a] :
( ( member_o @ I3 @ A2 )
=> ( ord_le3724670747650509150_set_a @ ( F @ I3 ) @ ( comple3958522678809307947_set_a @ ( image_o_set_set_a @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_321_SUP__upper,axiom,
! [I3: a,A2: set_a,F: a > set_set_a] :
( ( member_a @ I3 @ A2 )
=> ( ord_le3724670747650509150_set_a @ ( F @ I3 ) @ ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_322_SUP__upper,axiom,
! [I3: set_a,A2: set_set_a,F: set_a > set_a] :
( ( member_set_a @ I3 @ A2 )
=> ( ord_less_eq_set_a @ ( F @ I3 ) @ ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_323_SUP__upper,axiom,
! [I3: set_a,A2: set_set_a,F: set_a > set_o] :
( ( member_set_a @ I3 @ A2 )
=> ( ord_less_eq_set_o @ ( F @ I3 ) @ ( comple90263536869209701_set_o @ ( image_set_a_set_o @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_324_SUP__upper,axiom,
! [I3: $o,A2: set_o,F: $o > set_Product_prod_a_a] :
( ( member_o @ I3 @ A2 )
=> ( ord_le746702958409616551od_a_a @ ( F @ I3 ) @ ( comple8421679170691845492od_a_a @ ( image_4840794370999913844od_a_a @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_325_SUP__upper,axiom,
! [I3: a,A2: set_a,F: a > set_Product_prod_a_a] :
( ( member_a @ I3 @ A2 )
=> ( ord_le746702958409616551od_a_a @ ( F @ I3 ) @ ( comple8421679170691845492od_a_a @ ( image_4421510592991446670od_a_a @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_326_SUP__mono_H,axiom,
! [F: set_a > set_Pr5530083903271594800od_a_a,G: set_a > set_Pr5530083903271594800od_a_a,A2: set_set_a] :
( ! [X3: set_a] : ( ord_le114883831454073552od_a_a @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_le114883831454073552od_a_a @ ( comple2978350343072902813od_a_a @ ( image_7562202058474640471od_a_a @ F @ A2 ) ) @ ( comple2978350343072902813od_a_a @ ( image_7562202058474640471od_a_a @ G @ A2 ) ) ) ) ).
% SUP_mono'
thf(fact_327_SUP__mono_H,axiom,
! [F: a > set_Pr5530083903271594800od_a_a,G: a > set_Pr5530083903271594800od_a_a,A2: set_a] :
( ! [X3: a] : ( ord_le114883831454073552od_a_a @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_le114883831454073552od_a_a @ ( comple2978350343072902813od_a_a @ ( image_5653227685612666295od_a_a @ F @ A2 ) ) @ ( comple2978350343072902813od_a_a @ ( image_5653227685612666295od_a_a @ G @ A2 ) ) ) ) ).
% SUP_mono'
thf(fact_328_SUP__mono_H,axiom,
! [F: set_a > set_Product_prod_a_a,G: set_a > set_Product_prod_a_a,A2: set_set_a] :
( ! [X3: set_a] : ( ord_le746702958409616551od_a_a @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_le746702958409616551od_a_a @ ( comple8421679170691845492od_a_a @ ( image_6165024369500519726od_a_a @ F @ A2 ) ) @ ( comple8421679170691845492od_a_a @ ( image_6165024369500519726od_a_a @ G @ A2 ) ) ) ) ).
% SUP_mono'
thf(fact_329_SUP__mono_H,axiom,
! [F: a > set_Product_prod_a_a,G: a > set_Product_prod_a_a,A2: set_a] :
( ! [X3: a] : ( ord_le746702958409616551od_a_a @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_le746702958409616551od_a_a @ ( comple8421679170691845492od_a_a @ ( image_4421510592991446670od_a_a @ F @ A2 ) ) @ ( comple8421679170691845492od_a_a @ ( image_4421510592991446670od_a_a @ G @ A2 ) ) ) ) ).
% SUP_mono'
thf(fact_330_SUP__mono_H,axiom,
! [F: a > set_set_a,G: a > set_set_a,A2: set_a] :
( ! [X3: a] : ( ord_le3724670747650509150_set_a @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ F @ A2 ) ) @ ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ G @ A2 ) ) ) ) ).
% SUP_mono'
thf(fact_331_SUP__mono_H,axiom,
! [F: a > set_a,G: a > set_a,A2: set_a] :
( ! [X3: a] : ( ord_less_eq_set_a @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A2 ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ G @ A2 ) ) ) ) ).
% SUP_mono'
thf(fact_332_SUP__mono_H,axiom,
! [F: set_a > set_a,G: set_a > set_a,A2: set_set_a] :
( ! [X3: set_a] : ( ord_less_eq_set_a @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ F @ A2 ) ) @ ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ G @ A2 ) ) ) ) ).
% SUP_mono'
thf(fact_333_SUP__mono_H,axiom,
! [F: a > set_o,G: a > set_o,A2: set_a] :
( ! [X3: a] : ( ord_less_eq_set_o @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_a_set_o @ F @ A2 ) ) @ ( comple90263536869209701_set_o @ ( image_a_set_o @ G @ A2 ) ) ) ) ).
% SUP_mono'
thf(fact_334_SUP__least,axiom,
! [A2: set_o,F: $o > set_a,U: set_a] :
( ! [I2: $o] :
( ( member_o @ I2 @ A2 )
=> ( ord_less_eq_set_a @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_o_set_a @ F @ A2 ) ) @ U ) ) ).
% SUP_least
thf(fact_335_SUP__least,axiom,
! [A2: set_a,F: a > set_a,U: set_a] :
( ! [I2: a] :
( ( member_a @ I2 @ A2 )
=> ( ord_less_eq_set_a @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A2 ) ) @ U ) ) ).
% SUP_least
thf(fact_336_SUP__least,axiom,
! [A2: set_o,F: $o > set_o,U: set_o] :
( ! [I2: $o] :
( ( member_o @ I2 @ A2 )
=> ( ord_less_eq_set_o @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ A2 ) ) @ U ) ) ).
% SUP_least
thf(fact_337_SUP__least,axiom,
! [A2: set_a,F: a > set_o,U: set_o] :
( ! [I2: a] :
( ( member_a @ I2 @ A2 )
=> ( ord_less_eq_set_o @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_a_set_o @ F @ A2 ) ) @ U ) ) ).
% SUP_least
thf(fact_338_SUP__least,axiom,
! [A2: set_o,F: $o > set_set_a,U: set_set_a] :
( ! [I2: $o] :
( ( member_o @ I2 @ A2 )
=> ( ord_le3724670747650509150_set_a @ ( F @ I2 ) @ U ) )
=> ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ ( image_o_set_set_a @ F @ A2 ) ) @ U ) ) ).
% SUP_least
thf(fact_339_SUP__least,axiom,
! [A2: set_a,F: a > set_set_a,U: set_set_a] :
( ! [I2: a] :
( ( member_a @ I2 @ A2 )
=> ( ord_le3724670747650509150_set_a @ ( F @ I2 ) @ U ) )
=> ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ F @ A2 ) ) @ U ) ) ).
% SUP_least
thf(fact_340_SUP__least,axiom,
! [A2: set_set_a,F: set_a > set_a,U: set_a] :
( ! [I2: set_a] :
( ( member_set_a @ I2 @ A2 )
=> ( ord_less_eq_set_a @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ F @ A2 ) ) @ U ) ) ).
% SUP_least
thf(fact_341_SUP__least,axiom,
! [A2: set_set_a,F: set_a > set_o,U: set_o] :
( ! [I2: set_a] :
( ( member_set_a @ I2 @ A2 )
=> ( ord_less_eq_set_o @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_set_a_set_o @ F @ A2 ) ) @ U ) ) ).
% SUP_least
thf(fact_342_SUP__least,axiom,
! [A2: set_o,F: $o > set_Product_prod_a_a,U: set_Product_prod_a_a] :
( ! [I2: $o] :
( ( member_o @ I2 @ A2 )
=> ( ord_le746702958409616551od_a_a @ ( F @ I2 ) @ U ) )
=> ( ord_le746702958409616551od_a_a @ ( comple8421679170691845492od_a_a @ ( image_4840794370999913844od_a_a @ F @ A2 ) ) @ U ) ) ).
% SUP_least
thf(fact_343_SUP__least,axiom,
! [A2: set_a,F: a > set_Product_prod_a_a,U: set_Product_prod_a_a] :
( ! [I2: a] :
( ( member_a @ I2 @ A2 )
=> ( ord_le746702958409616551od_a_a @ ( F @ I2 ) @ U ) )
=> ( ord_le746702958409616551od_a_a @ ( comple8421679170691845492od_a_a @ ( image_4421510592991446670od_a_a @ F @ A2 ) ) @ U ) ) ).
% SUP_least
thf(fact_344_SUP__mono,axiom,
! [A2: set_o,B2: set_a,F: $o > set_a,G: a > set_a] :
( ! [N: $o] :
( ( member_o @ N @ A2 )
=> ? [X6: a] :
( ( member_a @ X6 @ B2 )
& ( ord_less_eq_set_a @ ( F @ N ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_o_set_a @ F @ A2 ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ G @ B2 ) ) ) ) ).
% SUP_mono
thf(fact_345_SUP__mono,axiom,
! [A2: set_a,B2: set_a,F: a > set_a,G: a > set_a] :
( ! [N: a] :
( ( member_a @ N @ A2 )
=> ? [X6: a] :
( ( member_a @ X6 @ B2 )
& ( ord_less_eq_set_a @ ( F @ N ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A2 ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ G @ B2 ) ) ) ) ).
% SUP_mono
thf(fact_346_SUP__mono,axiom,
! [A2: set_o,B2: set_a,F: $o > set_o,G: a > set_o] :
( ! [N: $o] :
( ( member_o @ N @ A2 )
=> ? [X6: a] :
( ( member_a @ X6 @ B2 )
& ( ord_less_eq_set_o @ ( F @ N ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ A2 ) ) @ ( comple90263536869209701_set_o @ ( image_a_set_o @ G @ B2 ) ) ) ) ).
% SUP_mono
thf(fact_347_SUP__mono,axiom,
! [A2: set_a,B2: set_a,F: a > set_o,G: a > set_o] :
( ! [N: a] :
( ( member_a @ N @ A2 )
=> ? [X6: a] :
( ( member_a @ X6 @ B2 )
& ( ord_less_eq_set_o @ ( F @ N ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_a_set_o @ F @ A2 ) ) @ ( comple90263536869209701_set_o @ ( image_a_set_o @ G @ B2 ) ) ) ) ).
% SUP_mono
thf(fact_348_SUP__mono,axiom,
! [A2: set_o,B2: set_a,F: $o > set_set_a,G: a > set_set_a] :
( ! [N: $o] :
( ( member_o @ N @ A2 )
=> ? [X6: a] :
( ( member_a @ X6 @ B2 )
& ( ord_le3724670747650509150_set_a @ ( F @ N ) @ ( G @ X6 ) ) ) )
=> ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ ( image_o_set_set_a @ F @ A2 ) ) @ ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ G @ B2 ) ) ) ) ).
% SUP_mono
thf(fact_349_SUP__mono,axiom,
! [A2: set_a,B2: set_a,F: a > set_set_a,G: a > set_set_a] :
( ! [N: a] :
( ( member_a @ N @ A2 )
=> ? [X6: a] :
( ( member_a @ X6 @ B2 )
& ( ord_le3724670747650509150_set_a @ ( F @ N ) @ ( G @ X6 ) ) ) )
=> ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ F @ A2 ) ) @ ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ G @ B2 ) ) ) ) ).
% SUP_mono
thf(fact_350_SUP__mono,axiom,
! [A2: set_set_a,B2: set_a,F: set_a > set_a,G: a > set_a] :
( ! [N: set_a] :
( ( member_set_a @ N @ A2 )
=> ? [X6: a] :
( ( member_a @ X6 @ B2 )
& ( ord_less_eq_set_a @ ( F @ N ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ F @ A2 ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ G @ B2 ) ) ) ) ).
% SUP_mono
thf(fact_351_SUP__mono,axiom,
! [A2: set_o,B2: set_set_a,F: $o > set_a,G: set_a > set_a] :
( ! [N: $o] :
( ( member_o @ N @ A2 )
=> ? [X6: set_a] :
( ( member_set_a @ X6 @ B2 )
& ( ord_less_eq_set_a @ ( F @ N ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_o_set_a @ F @ A2 ) ) @ ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ G @ B2 ) ) ) ) ).
% SUP_mono
thf(fact_352_SUP__mono,axiom,
! [A2: set_a,B2: set_set_a,F: a > set_a,G: set_a > set_a] :
( ! [N: a] :
( ( member_a @ N @ A2 )
=> ? [X6: set_a] :
( ( member_set_a @ X6 @ B2 )
& ( ord_less_eq_set_a @ ( F @ N ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A2 ) ) @ ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ G @ B2 ) ) ) ) ).
% SUP_mono
thf(fact_353_SUP__mono,axiom,
! [A2: set_set_a,B2: set_a,F: set_a > set_o,G: a > set_o] :
( ! [N: set_a] :
( ( member_set_a @ N @ A2 )
=> ? [X6: a] :
( ( member_a @ X6 @ B2 )
& ( ord_less_eq_set_o @ ( F @ N ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_set_a_set_o @ F @ A2 ) ) @ ( comple90263536869209701_set_o @ ( image_a_set_o @ G @ B2 ) ) ) ) ).
% SUP_mono
thf(fact_354_SUP__eqI,axiom,
! [A2: set_o,F: $o > set_a,X: set_a] :
( ! [I2: $o] :
( ( member_o @ I2 @ A2 )
=> ( ord_less_eq_set_a @ ( F @ I2 ) @ X ) )
=> ( ! [Y3: set_a] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ( ord_less_eq_set_a @ ( F @ I4 ) @ Y3 ) )
=> ( ord_less_eq_set_a @ X @ Y3 ) )
=> ( ( comple2307003609928055243_set_a @ ( image_o_set_a @ F @ A2 ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_355_SUP__eqI,axiom,
! [A2: set_a,F: a > set_a,X: set_a] :
( ! [I2: a] :
( ( member_a @ I2 @ A2 )
=> ( ord_less_eq_set_a @ ( F @ I2 ) @ X ) )
=> ( ! [Y3: set_a] :
( ! [I4: a] :
( ( member_a @ I4 @ A2 )
=> ( ord_less_eq_set_a @ ( F @ I4 ) @ Y3 ) )
=> ( ord_less_eq_set_a @ X @ Y3 ) )
=> ( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A2 ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_356_SUP__eqI,axiom,
! [A2: set_o,F: $o > set_o,X: set_o] :
( ! [I2: $o] :
( ( member_o @ I2 @ A2 )
=> ( ord_less_eq_set_o @ ( F @ I2 ) @ X ) )
=> ( ! [Y3: set_o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ( ord_less_eq_set_o @ ( F @ I4 ) @ Y3 ) )
=> ( ord_less_eq_set_o @ X @ Y3 ) )
=> ( ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ A2 ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_357_SUP__eqI,axiom,
! [A2: set_a,F: a > set_o,X: set_o] :
( ! [I2: a] :
( ( member_a @ I2 @ A2 )
=> ( ord_less_eq_set_o @ ( F @ I2 ) @ X ) )
=> ( ! [Y3: set_o] :
( ! [I4: a] :
( ( member_a @ I4 @ A2 )
=> ( ord_less_eq_set_o @ ( F @ I4 ) @ Y3 ) )
=> ( ord_less_eq_set_o @ X @ Y3 ) )
=> ( ( comple90263536869209701_set_o @ ( image_a_set_o @ F @ A2 ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_358_SUP__eqI,axiom,
! [A2: set_o,F: $o > set_set_a,X: set_set_a] :
( ! [I2: $o] :
( ( member_o @ I2 @ A2 )
=> ( ord_le3724670747650509150_set_a @ ( F @ I2 ) @ X ) )
=> ( ! [Y3: set_set_a] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ( ord_le3724670747650509150_set_a @ ( F @ I4 ) @ Y3 ) )
=> ( ord_le3724670747650509150_set_a @ X @ Y3 ) )
=> ( ( comple3958522678809307947_set_a @ ( image_o_set_set_a @ F @ A2 ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_359_SUP__eqI,axiom,
! [A2: set_a,F: a > set_set_a,X: set_set_a] :
( ! [I2: a] :
( ( member_a @ I2 @ A2 )
=> ( ord_le3724670747650509150_set_a @ ( F @ I2 ) @ X ) )
=> ( ! [Y3: set_set_a] :
( ! [I4: a] :
( ( member_a @ I4 @ A2 )
=> ( ord_le3724670747650509150_set_a @ ( F @ I4 ) @ Y3 ) )
=> ( ord_le3724670747650509150_set_a @ X @ Y3 ) )
=> ( ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ F @ A2 ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_360_SUP__eqI,axiom,
! [A2: set_set_a,F: set_a > set_a,X: set_a] :
( ! [I2: set_a] :
( ( member_set_a @ I2 @ A2 )
=> ( ord_less_eq_set_a @ ( F @ I2 ) @ X ) )
=> ( ! [Y3: set_a] :
( ! [I4: set_a] :
( ( member_set_a @ I4 @ A2 )
=> ( ord_less_eq_set_a @ ( F @ I4 ) @ Y3 ) )
=> ( ord_less_eq_set_a @ X @ Y3 ) )
=> ( ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ F @ A2 ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_361_SUP__eqI,axiom,
! [A2: set_set_a,F: set_a > set_o,X: set_o] :
( ! [I2: set_a] :
( ( member_set_a @ I2 @ A2 )
=> ( ord_less_eq_set_o @ ( F @ I2 ) @ X ) )
=> ( ! [Y3: set_o] :
( ! [I4: set_a] :
( ( member_set_a @ I4 @ A2 )
=> ( ord_less_eq_set_o @ ( F @ I4 ) @ Y3 ) )
=> ( ord_less_eq_set_o @ X @ Y3 ) )
=> ( ( comple90263536869209701_set_o @ ( image_set_a_set_o @ F @ A2 ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_362_SUP__eqI,axiom,
! [A2: set_o,F: $o > set_Product_prod_a_a,X: set_Product_prod_a_a] :
( ! [I2: $o] :
( ( member_o @ I2 @ A2 )
=> ( ord_le746702958409616551od_a_a @ ( F @ I2 ) @ X ) )
=> ( ! [Y3: set_Product_prod_a_a] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ( ord_le746702958409616551od_a_a @ ( F @ I4 ) @ Y3 ) )
=> ( ord_le746702958409616551od_a_a @ X @ Y3 ) )
=> ( ( comple8421679170691845492od_a_a @ ( image_4840794370999913844od_a_a @ F @ A2 ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_363_SUP__eqI,axiom,
! [A2: set_a,F: a > set_Product_prod_a_a,X: set_Product_prod_a_a] :
( ! [I2: a] :
( ( member_a @ I2 @ A2 )
=> ( ord_le746702958409616551od_a_a @ ( F @ I2 ) @ X ) )
=> ( ! [Y3: set_Product_prod_a_a] :
( ! [I4: a] :
( ( member_a @ I4 @ A2 )
=> ( ord_le746702958409616551od_a_a @ ( F @ I4 ) @ Y3 ) )
=> ( ord_le746702958409616551od_a_a @ X @ Y3 ) )
=> ( ( comple8421679170691845492od_a_a @ ( image_4421510592991446670od_a_a @ F @ A2 ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_364_image__UN,axiom,
! [F: a > a,B2: a > set_a,A2: set_a] :
( ( image_a_a @ F @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B2 @ A2 ) ) )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X2: a] : ( image_a_a @ F @ ( B2 @ X2 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_365_image__UN,axiom,
! [F: a > $o,B2: a > set_a,A2: set_a] :
( ( image_a_o @ F @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B2 @ A2 ) ) )
= ( comple90263536869209701_set_o
@ ( image_a_set_o
@ ^ [X2: a] : ( image_a_o @ F @ ( B2 @ X2 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_366_image__UN,axiom,
! [F: $o > a,B2: a > set_o,A2: set_a] :
( ( image_o_a @ F @ ( comple90263536869209701_set_o @ ( image_a_set_o @ B2 @ A2 ) ) )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X2: a] : ( image_o_a @ F @ ( B2 @ X2 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_367_image__UN,axiom,
! [F: $o > $o,B2: a > set_o,A2: set_a] :
( ( image_o_o @ F @ ( comple90263536869209701_set_o @ ( image_a_set_o @ B2 @ A2 ) ) )
= ( comple90263536869209701_set_o
@ ( image_a_set_o
@ ^ [X2: a] : ( image_o_o @ F @ ( B2 @ X2 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_368_image__UN,axiom,
! [F: set_a > a,B2: a > set_set_a,A2: set_a] :
( ( image_set_a_a @ F @ ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ B2 @ A2 ) ) )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X2: a] : ( image_set_a_a @ F @ ( B2 @ X2 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_369_image__UN,axiom,
! [F: set_a > $o,B2: a > set_set_a,A2: set_a] :
( ( image_set_a_o @ F @ ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ B2 @ A2 ) ) )
= ( comple90263536869209701_set_o
@ ( image_a_set_o
@ ^ [X2: a] : ( image_set_a_o @ F @ ( B2 @ X2 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_370_image__UN,axiom,
! [F: a > set_o,B2: a > set_a,A2: set_a] :
( ( image_a_set_o @ F @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B2 @ A2 ) ) )
= ( comple4436988014476444997_set_o
@ ( image_a_set_set_o
@ ^ [X2: a] : ( image_a_set_o @ F @ ( B2 @ X2 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_371_image__UN,axiom,
! [F: a > set_a,B2: a > set_a,A2: set_a] :
( ( image_a_set_a @ F @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B2 @ A2 ) ) )
= ( comple3958522678809307947_set_a
@ ( image_a_set_set_a
@ ^ [X2: a] : ( image_a_set_a @ F @ ( B2 @ X2 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_372_image__UN,axiom,
! [F: a > a,B2: set_a > set_a,A2: set_set_a] :
( ( image_a_a @ F @ ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ B2 @ A2 ) ) )
= ( comple2307003609928055243_set_a
@ ( image_set_a_set_a
@ ^ [X2: set_a] : ( image_a_a @ F @ ( B2 @ X2 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_373_image__UN,axiom,
! [F: a > $o,B2: set_a > set_a,A2: set_set_a] :
( ( image_a_o @ F @ ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ B2 @ A2 ) ) )
= ( comple90263536869209701_set_o
@ ( image_set_a_set_o
@ ^ [X2: set_a] : ( image_a_o @ F @ ( B2 @ X2 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_374_UN__extend__simps_I10_J,axiom,
! [B2: nat > set_a,F: nat > nat,A2: set_nat] :
( ( comple2307003609928055243_set_a
@ ( image_nat_set_a
@ ^ [A3: nat] : ( B2 @ ( F @ A3 ) )
@ A2 ) )
= ( comple2307003609928055243_set_a @ ( image_nat_set_a @ B2 @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_375_UN__extend__simps_I10_J,axiom,
! [B2: a > set_a,F: a > a,A2: set_a] :
( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [A3: a] : ( B2 @ ( F @ A3 ) )
@ A2 ) )
= ( comple2307003609928055243_set_a @ ( image_a_set_a @ B2 @ ( image_a_a @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_376_UN__extend__simps_I10_J,axiom,
! [B2: nat > set_o,F: nat > nat,A2: set_nat] :
( ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [A3: nat] : ( B2 @ ( F @ A3 ) )
@ A2 ) )
= ( comple90263536869209701_set_o @ ( image_nat_set_o @ B2 @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_377_UN__extend__simps_I10_J,axiom,
! [B2: a > set_o,F: a > a,A2: set_a] :
( ( comple90263536869209701_set_o
@ ( image_a_set_o
@ ^ [A3: a] : ( B2 @ ( F @ A3 ) )
@ A2 ) )
= ( comple90263536869209701_set_o @ ( image_a_set_o @ B2 @ ( image_a_a @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_378_UN__extend__simps_I10_J,axiom,
! [B2: nat > set_set_a,F: nat > nat,A2: set_nat] :
( ( comple3958522678809307947_set_a
@ ( image_nat_set_set_a
@ ^ [A3: nat] : ( B2 @ ( F @ A3 ) )
@ A2 ) )
= ( comple3958522678809307947_set_a @ ( image_nat_set_set_a @ B2 @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_379_UN__extend__simps_I10_J,axiom,
! [B2: a > set_set_a,F: a > a,A2: set_a] :
( ( comple3958522678809307947_set_a
@ ( image_a_set_set_a
@ ^ [A3: a] : ( B2 @ ( F @ A3 ) )
@ A2 ) )
= ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ B2 @ ( image_a_a @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_380_UN__extend__simps_I10_J,axiom,
! [B2: set_o > set_a,F: a > set_o,A2: set_a] :
( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [A3: a] : ( B2 @ ( F @ A3 ) )
@ A2 ) )
= ( comple2307003609928055243_set_a @ ( image_set_o_set_a @ B2 @ ( image_a_set_o @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_381_UN__extend__simps_I10_J,axiom,
! [B2: set_a > set_a,F: a > set_a,A2: set_a] :
( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [A3: a] : ( B2 @ ( F @ A3 ) )
@ A2 ) )
= ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ B2 @ ( image_a_set_a @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_382_UN__extend__simps_I10_J,axiom,
! [B2: a > set_a,F: set_a > a,A2: set_set_a] :
( ( comple2307003609928055243_set_a
@ ( image_set_a_set_a
@ ^ [A3: set_a] : ( B2 @ ( F @ A3 ) )
@ A2 ) )
= ( comple2307003609928055243_set_a @ ( image_a_set_a @ B2 @ ( image_set_a_a @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_383_UN__extend__simps_I10_J,axiom,
! [B2: set_a > set_o,F: a > set_a,A2: set_a] :
( ( comple90263536869209701_set_o
@ ( image_a_set_o
@ ^ [A3: a] : ( B2 @ ( F @ A3 ) )
@ A2 ) )
= ( comple90263536869209701_set_o @ ( image_set_a_set_o @ B2 @ ( image_a_set_a @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_384_UN__subset__iff,axiom,
! [A2: set_a > set_Pr5530083903271594800od_a_a,I5: set_set_a,B2: set_Pr5530083903271594800od_a_a] :
( ( ord_le114883831454073552od_a_a @ ( comple2978350343072902813od_a_a @ ( image_7562202058474640471od_a_a @ A2 @ I5 ) ) @ B2 )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ I5 )
=> ( ord_le114883831454073552od_a_a @ ( A2 @ X2 ) @ B2 ) ) ) ) ).
% UN_subset_iff
thf(fact_385_UN__subset__iff,axiom,
! [A2: a > set_Pr5530083903271594800od_a_a,I5: set_a,B2: set_Pr5530083903271594800od_a_a] :
( ( ord_le114883831454073552od_a_a @ ( comple2978350343072902813od_a_a @ ( image_5653227685612666295od_a_a @ A2 @ I5 ) ) @ B2 )
= ( ! [X2: a] :
( ( member_a @ X2 @ I5 )
=> ( ord_le114883831454073552od_a_a @ ( A2 @ X2 ) @ B2 ) ) ) ) ).
% UN_subset_iff
thf(fact_386_UN__subset__iff,axiom,
! [A2: set_a > set_Product_prod_a_a,I5: set_set_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( comple8421679170691845492od_a_a @ ( image_6165024369500519726od_a_a @ A2 @ I5 ) ) @ B2 )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ I5 )
=> ( ord_le746702958409616551od_a_a @ ( A2 @ X2 ) @ B2 ) ) ) ) ).
% UN_subset_iff
thf(fact_387_UN__subset__iff,axiom,
! [A2: a > set_Product_prod_a_a,I5: set_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( comple8421679170691845492od_a_a @ ( image_4421510592991446670od_a_a @ A2 @ I5 ) ) @ B2 )
= ( ! [X2: a] :
( ( member_a @ X2 @ I5 )
=> ( ord_le746702958409616551od_a_a @ ( A2 @ X2 ) @ B2 ) ) ) ) ).
% UN_subset_iff
thf(fact_388_UN__subset__iff,axiom,
! [A2: a > set_set_a,I5: set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ A2 @ I5 ) ) @ B2 )
= ( ! [X2: a] :
( ( member_a @ X2 @ I5 )
=> ( ord_le3724670747650509150_set_a @ ( A2 @ X2 ) @ B2 ) ) ) ) ).
% UN_subset_iff
thf(fact_389_UN__subset__iff,axiom,
! [A2: a > set_a,I5: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ A2 @ I5 ) ) @ B2 )
= ( ! [X2: a] :
( ( member_a @ X2 @ I5 )
=> ( ord_less_eq_set_a @ ( A2 @ X2 ) @ B2 ) ) ) ) ).
% UN_subset_iff
thf(fact_390_UN__subset__iff,axiom,
! [A2: set_a > set_a,I5: set_set_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ A2 @ I5 ) ) @ B2 )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ I5 )
=> ( ord_less_eq_set_a @ ( A2 @ X2 ) @ B2 ) ) ) ) ).
% UN_subset_iff
thf(fact_391_UN__subset__iff,axiom,
! [A2: a > set_o,I5: set_a,B2: set_o] :
( ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_a_set_o @ A2 @ I5 ) ) @ B2 )
= ( ! [X2: a] :
( ( member_a @ X2 @ I5 )
=> ( ord_less_eq_set_o @ ( A2 @ X2 ) @ B2 ) ) ) ) ).
% UN_subset_iff
thf(fact_392_UN__upper,axiom,
! [A: $o,A2: set_o,B2: $o > set_a] :
( ( member_o @ A @ A2 )
=> ( ord_less_eq_set_a @ ( B2 @ A ) @ ( comple2307003609928055243_set_a @ ( image_o_set_a @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_393_UN__upper,axiom,
! [A: a,A2: set_a,B2: a > set_a] :
( ( member_a @ A @ A2 )
=> ( ord_less_eq_set_a @ ( B2 @ A ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_394_UN__upper,axiom,
! [A: $o,A2: set_o,B2: $o > set_o] :
( ( member_o @ A @ A2 )
=> ( ord_less_eq_set_o @ ( B2 @ A ) @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_395_UN__upper,axiom,
! [A: a,A2: set_a,B2: a > set_o] :
( ( member_a @ A @ A2 )
=> ( ord_less_eq_set_o @ ( B2 @ A ) @ ( comple90263536869209701_set_o @ ( image_a_set_o @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_396_UN__upper,axiom,
! [A: $o,A2: set_o,B2: $o > set_set_a] :
( ( member_o @ A @ A2 )
=> ( ord_le3724670747650509150_set_a @ ( B2 @ A ) @ ( comple3958522678809307947_set_a @ ( image_o_set_set_a @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_397_UN__upper,axiom,
! [A: a,A2: set_a,B2: a > set_set_a] :
( ( member_a @ A @ A2 )
=> ( ord_le3724670747650509150_set_a @ ( B2 @ A ) @ ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_398_UN__upper,axiom,
! [A: set_a,A2: set_set_a,B2: set_a > set_a] :
( ( member_set_a @ A @ A2 )
=> ( ord_less_eq_set_a @ ( B2 @ A ) @ ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_399_UN__upper,axiom,
! [A: set_a,A2: set_set_a,B2: set_a > set_o] :
( ( member_set_a @ A @ A2 )
=> ( ord_less_eq_set_o @ ( B2 @ A ) @ ( comple90263536869209701_set_o @ ( image_set_a_set_o @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_400_UN__upper,axiom,
! [A: $o,A2: set_o,B2: $o > set_Product_prod_a_a] :
( ( member_o @ A @ A2 )
=> ( ord_le746702958409616551od_a_a @ ( B2 @ A ) @ ( comple8421679170691845492od_a_a @ ( image_4840794370999913844od_a_a @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_401_UN__upper,axiom,
! [A: a,A2: set_a,B2: a > set_Product_prod_a_a] :
( ( member_a @ A @ A2 )
=> ( ord_le746702958409616551od_a_a @ ( B2 @ A ) @ ( comple8421679170691845492od_a_a @ ( image_4421510592991446670od_a_a @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_402_UN__least,axiom,
! [A2: set_o,B2: $o > set_a,C: set_a] :
( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_less_eq_set_a @ ( B2 @ X3 ) @ C ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_o_set_a @ B2 @ A2 ) ) @ C ) ) ).
% UN_least
thf(fact_403_UN__least,axiom,
! [A2: set_a,B2: a > set_a,C: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ord_less_eq_set_a @ ( B2 @ X3 ) @ C ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B2 @ A2 ) ) @ C ) ) ).
% UN_least
thf(fact_404_UN__least,axiom,
! [A2: set_o,B2: $o > set_o,C: set_o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_less_eq_set_o @ ( B2 @ X3 ) @ C ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) @ C ) ) ).
% UN_least
thf(fact_405_UN__least,axiom,
! [A2: set_a,B2: a > set_o,C: set_o] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ord_less_eq_set_o @ ( B2 @ X3 ) @ C ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_a_set_o @ B2 @ A2 ) ) @ C ) ) ).
% UN_least
thf(fact_406_UN__least,axiom,
! [A2: set_o,B2: $o > set_set_a,C: set_set_a] :
( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_le3724670747650509150_set_a @ ( B2 @ X3 ) @ C ) )
=> ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ ( image_o_set_set_a @ B2 @ A2 ) ) @ C ) ) ).
% UN_least
thf(fact_407_UN__least,axiom,
! [A2: set_a,B2: a > set_set_a,C: set_set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ord_le3724670747650509150_set_a @ ( B2 @ X3 ) @ C ) )
=> ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ B2 @ A2 ) ) @ C ) ) ).
% UN_least
thf(fact_408_UN__least,axiom,
! [A2: set_set_a,B2: set_a > set_a,C: set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( ord_less_eq_set_a @ ( B2 @ X3 ) @ C ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ B2 @ A2 ) ) @ C ) ) ).
% UN_least
thf(fact_409_UN__least,axiom,
! [A2: set_set_a,B2: set_a > set_o,C: set_o] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( ord_less_eq_set_o @ ( B2 @ X3 ) @ C ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_set_a_set_o @ B2 @ A2 ) ) @ C ) ) ).
% UN_least
thf(fact_410_UN__least,axiom,
! [A2: set_o,B2: $o > set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_le746702958409616551od_a_a @ ( B2 @ X3 ) @ C ) )
=> ( ord_le746702958409616551od_a_a @ ( comple8421679170691845492od_a_a @ ( image_4840794370999913844od_a_a @ B2 @ A2 ) ) @ C ) ) ).
% UN_least
thf(fact_411_UN__least,axiom,
! [A2: set_a,B2: a > set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ord_le746702958409616551od_a_a @ ( B2 @ X3 ) @ C ) )
=> ( ord_le746702958409616551od_a_a @ ( comple8421679170691845492od_a_a @ ( image_4421510592991446670od_a_a @ B2 @ A2 ) ) @ C ) ) ).
% UN_least
thf(fact_412_UN__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > set_a,G: $o > set_a] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_o_set_a @ F @ A2 ) ) @ ( comple2307003609928055243_set_a @ ( image_o_set_a @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_413_UN__mono,axiom,
! [A2: set_a,B2: set_a,F: a > set_a,G: a > set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A2 ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_414_UN__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > set_o,G: $o > set_o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_less_eq_set_o @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ A2 ) ) @ ( comple90263536869209701_set_o @ ( image_o_set_o @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_415_UN__mono,axiom,
! [A2: set_a,B2: set_a,F: a > set_o,G: a > set_o] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ord_less_eq_set_o @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_a_set_o @ F @ A2 ) ) @ ( comple90263536869209701_set_o @ ( image_a_set_o @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_416_UN__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > set_set_a,G: $o > set_set_a] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ ( image_o_set_set_a @ F @ A2 ) ) @ ( comple3958522678809307947_set_a @ ( image_o_set_set_a @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_417_UN__mono,axiom,
! [A2: set_a,B2: set_a,F: a > set_set_a,G: a > set_set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ F @ A2 ) ) @ ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_418_UN__mono,axiom,
! [A2: set_set_a,B2: set_set_a,F: set_a > set_a,G: set_a > set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ F @ A2 ) ) @ ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_419_UN__mono,axiom,
! [A2: set_set_a,B2: set_set_a,F: set_a > set_o,G: set_a > set_o] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( ord_less_eq_set_o @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_set_a_set_o @ F @ A2 ) ) @ ( comple90263536869209701_set_o @ ( image_set_a_set_o @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_420_UN__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > set_Product_prod_a_a,G: $o > set_Product_prod_a_a] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_le746702958409616551od_a_a @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_le746702958409616551od_a_a @ ( comple8421679170691845492od_a_a @ ( image_4840794370999913844od_a_a @ F @ A2 ) ) @ ( comple8421679170691845492od_a_a @ ( image_4840794370999913844od_a_a @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_421_UN__mono,axiom,
! [A2: set_a,B2: set_a,F: a > set_Product_prod_a_a,G: a > set_Product_prod_a_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ord_le746702958409616551od_a_a @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_le746702958409616551od_a_a @ ( comple8421679170691845492od_a_a @ ( image_4421510592991446670od_a_a @ F @ A2 ) ) @ ( comple8421679170691845492od_a_a @ ( image_4421510592991446670od_a_a @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_422_image__Union,axiom,
! [F: nat > nat,S: set_set_nat] :
( ( image_nat_nat @ F @ ( comple7399068483239264473et_nat @ S ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ S ) ) ) ).
% image_Union
thf(fact_423_image__Union,axiom,
! [F: a > a,S: set_set_a] :
( ( image_a_a @ F @ ( comple2307003609928055243_set_a @ S ) )
= ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ ( image_a_a @ F ) @ S ) ) ) ).
% image_Union
thf(fact_424_image__Union,axiom,
! [F: a > $o,S: set_set_a] :
( ( image_a_o @ F @ ( comple2307003609928055243_set_a @ S ) )
= ( comple90263536869209701_set_o @ ( image_set_a_set_o @ ( image_a_o @ F ) @ S ) ) ) ).
% image_Union
thf(fact_425_image__Union,axiom,
! [F: $o > a,S: set_set_o] :
( ( image_o_a @ F @ ( comple90263536869209701_set_o @ S ) )
= ( comple2307003609928055243_set_a @ ( image_set_o_set_a @ ( image_o_a @ F ) @ S ) ) ) ).
% image_Union
thf(fact_426_image__Union,axiom,
! [F: $o > $o,S: set_set_o] :
( ( image_o_o @ F @ ( comple90263536869209701_set_o @ S ) )
= ( comple90263536869209701_set_o @ ( image_set_o_set_o @ ( image_o_o @ F ) @ S ) ) ) ).
% image_Union
thf(fact_427_image__Union,axiom,
! [F: set_a > a,S: set_set_set_a] :
( ( image_set_a_a @ F @ ( comple3958522678809307947_set_a @ S ) )
= ( comple2307003609928055243_set_a @ ( image_6061375613820669477_set_a @ ( image_set_a_a @ F ) @ S ) ) ) ).
% image_Union
thf(fact_428_image__Union,axiom,
! [F: set_a > $o,S: set_set_set_a] :
( ( image_set_a_o @ F @ ( comple3958522678809307947_set_a @ S ) )
= ( comple90263536869209701_set_o @ ( image_4406776271737083839_set_o @ ( image_set_a_o @ F ) @ S ) ) ) ).
% image_Union
thf(fact_429_image__Union,axiom,
! [F: a > set_o,S: set_set_a] :
( ( image_a_set_o @ F @ ( comple2307003609928055243_set_a @ S ) )
= ( comple4436988014476444997_set_o @ ( image_8955363097146037247_set_o @ ( image_a_set_o @ F ) @ S ) ) ) ).
% image_Union
thf(fact_430_image__Union,axiom,
! [F: a > set_a,S: set_set_a] :
( ( image_a_set_a @ F @ ( comple2307003609928055243_set_a @ S ) )
= ( comple3958522678809307947_set_a @ ( image_4955109552351689957_set_a @ ( image_a_set_a @ F ) @ S ) ) ) ).
% image_Union
thf(fact_431_image__Union,axiom,
! [F: $o > set_a,S: set_set_o] :
( ( image_o_set_a @ F @ ( comple90263536869209701_set_o @ S ) )
= ( comple3958522678809307947_set_a @ ( image_7612453856319013963_set_a @ ( image_o_set_a @ F ) @ S ) ) ) ).
% image_Union
thf(fact_432_image__ident,axiom,
! [Y5: set_set_a] :
( ( image_set_a_set_a
@ ^ [X2: set_a] : X2
@ Y5 )
= Y5 ) ).
% image_ident
thf(fact_433_image__ident,axiom,
! [Y5: set_nat] :
( ( image_nat_nat
@ ^ [X2: nat] : X2
@ Y5 )
= Y5 ) ).
% image_ident
thf(fact_434_triangle__in__graph__edge__empty,axiom,
! [X: a,Y: a,Z: a] :
( ( edges = bot_bot_set_set_a )
=> ~ ( graph_4582152751571636272raph_a @ edges @ X @ Y @ Z ) ) ).
% triangle_in_graph_edge_empty
thf(fact_435_finite__triangle__set,axiom,
finite_finite_set_a @ ( graph_triangle_set_a @ edges ) ).
% finite_triangle_set
thf(fact_436_subset__antisym,axiom,
! [A2: set_Pr5530083903271594800od_a_a,B2: set_Pr5530083903271594800od_a_a] :
( ( ord_le114883831454073552od_a_a @ A2 @ B2 )
=> ( ( ord_le114883831454073552od_a_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_437_subset__antisym,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B2 )
=> ( ( ord_le746702958409616551od_a_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_438_subset__antisym,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_439_subset__antisym,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_440_subsetI,axiom,
! [A2: set_o,B2: set_o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( member_o @ X3 @ B2 ) )
=> ( ord_less_eq_set_o @ A2 @ B2 ) ) ).
% subsetI
thf(fact_441_subsetI,axiom,
! [A2: set_Pr5530083903271594800od_a_a,B2: set_Pr5530083903271594800od_a_a] :
( ! [X3: produc4044097585999906000od_a_a] :
( ( member3071122053849602553od_a_a @ X3 @ A2 )
=> ( member3071122053849602553od_a_a @ X3 @ B2 ) )
=> ( ord_le114883831454073552od_a_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_442_subsetI,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ! [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A2 )
=> ( member1426531477525435216od_a_a @ X3 @ B2 ) )
=> ( ord_le746702958409616551od_a_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_443_subsetI,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( member_set_a @ X3 @ B2 ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_444_subsetI,axiom,
! [A2: set_a,B2: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_a @ X3 @ B2 ) )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_445_image__eqI,axiom,
! [B: nat,F: nat > nat,X: nat,A2: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_446_image__eqI,axiom,
! [B: $o,F: $o > $o,X: $o,A2: set_o] :
( ( B
= ( F @ X ) )
=> ( ( member_o @ X @ A2 )
=> ( member_o @ B @ ( image_o_o @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_447_image__eqI,axiom,
! [B: a,F: $o > a,X: $o,A2: set_o] :
( ( B
= ( F @ X ) )
=> ( ( member_o @ X @ A2 )
=> ( member_a @ B @ ( image_o_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_448_image__eqI,axiom,
! [B: $o,F: a > $o,X: a,A2: set_a] :
( ( B
= ( F @ X ) )
=> ( ( member_a @ X @ A2 )
=> ( member_o @ B @ ( image_a_o @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_449_image__eqI,axiom,
! [B: a,F: a > a,X: a,A2: set_a] :
( ( B
= ( F @ X ) )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ B @ ( image_a_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_450_image__eqI,axiom,
! [B: $o,F: set_a > $o,X: set_a,A2: set_set_a] :
( ( B
= ( F @ X ) )
=> ( ( member_set_a @ X @ A2 )
=> ( member_o @ B @ ( image_set_a_o @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_451_image__eqI,axiom,
! [B: a,F: set_a > a,X: set_a,A2: set_set_a] :
( ( B
= ( F @ X ) )
=> ( ( member_set_a @ X @ A2 )
=> ( member_a @ B @ ( image_set_a_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_452_image__eqI,axiom,
! [B: set_a,F: $o > set_a,X: $o,A2: set_o] :
( ( B
= ( F @ X ) )
=> ( ( member_o @ X @ A2 )
=> ( member_set_a @ B @ ( image_o_set_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_453_image__eqI,axiom,
! [B: set_o,F: a > set_o,X: a,A2: set_a] :
( ( B
= ( F @ X ) )
=> ( ( member_a @ X @ A2 )
=> ( member_set_o @ B @ ( image_a_set_o @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_454_image__eqI,axiom,
! [B: set_a,F: a > set_a,X: a,A2: set_a] :
( ( B
= ( F @ X ) )
=> ( ( member_a @ X @ A2 )
=> ( member_set_a @ B @ ( image_a_set_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_455_SUP__UN__eq,axiom,
! [R: set_a > set_Pr5530083903271594800od_a_a,S: set_set_a] :
( ( comple2673673910019652224_a_a_o
@ ( image_2136885688481450310_a_a_o
@ ^ [I: set_a,X2: produc4044097585999906000od_a_a] : ( member3071122053849602553od_a_a @ X2 @ ( R @ I ) )
@ S ) )
= ( ^ [X2: produc4044097585999906000od_a_a] : ( member3071122053849602553od_a_a @ X2 @ ( comple2978350343072902813od_a_a @ ( image_7562202058474640471od_a_a @ R @ S ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_456_SUP__UN__eq,axiom,
! [R: a > set_Pr5530083903271594800od_a_a,S: set_a] :
( ( comple2673673910019652224_a_a_o
@ ( image_7327113576949061606_a_a_o
@ ^ [I: a,X2: produc4044097585999906000od_a_a] : ( member3071122053849602553od_a_a @ X2 @ ( R @ I ) )
@ S ) )
= ( ^ [X2: produc4044097585999906000od_a_a] : ( member3071122053849602553od_a_a @ X2 @ ( comple2978350343072902813od_a_a @ ( image_5653227685612666295od_a_a @ R @ S ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_457_SUP__UN__eq,axiom,
! [R: set_a > set_Product_prod_a_a,S: set_set_a] :
( ( comple9027675562681848937_a_a_o
@ ( image_1140608277309626671_a_a_o
@ ^ [I: set_a,X2: product_prod_a_a] : ( member1426531477525435216od_a_a @ X2 @ ( R @ I ) )
@ S ) )
= ( ^ [X2: product_prod_a_a] : ( member1426531477525435216od_a_a @ X2 @ ( comple8421679170691845492od_a_a @ ( image_6165024369500519726od_a_a @ R @ S ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_458_SUP__UN__eq,axiom,
! [R: a > set_Product_prod_a_a,S: set_a] :
( ( comple9027675562681848937_a_a_o
@ ( image_5603332478075979727_a_a_o
@ ^ [I: a,X2: product_prod_a_a] : ( member1426531477525435216od_a_a @ X2 @ ( R @ I ) )
@ S ) )
= ( ^ [X2: product_prod_a_a] : ( member1426531477525435216od_a_a @ X2 @ ( comple8421679170691845492od_a_a @ ( image_4421510592991446670od_a_a @ R @ S ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_459_SUP__UN__eq,axiom,
! [R: a > set_set_a,S: set_a] :
( ( comple7256090232125724530et_a_o
@ ( image_a_set_a_o
@ ^ [I: a,X2: set_a] : ( member_set_a @ X2 @ ( R @ I ) )
@ S ) )
= ( ^ [X2: set_a] : ( member_set_a @ X2 @ ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ R @ S ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_460_SUP__UN__eq,axiom,
! [R: a > set_a,S: set_a] :
( ( complete_Sup_Sup_a_o
@ ( image_a_a_o
@ ^ [I: a,X2: a] : ( member_a @ X2 @ ( R @ I ) )
@ S ) )
= ( ^ [X2: a] : ( member_a @ X2 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ R @ S ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_461_SUP__UN__eq,axiom,
! [R: set_a > set_a,S: set_set_a] :
( ( complete_Sup_Sup_a_o
@ ( image_set_a_a_o
@ ^ [I: set_a,X2: a] : ( member_a @ X2 @ ( R @ I ) )
@ S ) )
= ( ^ [X2: a] : ( member_a @ X2 @ ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ R @ S ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_462_SUP__UN__eq,axiom,
! [R: a > set_o,S: set_a] :
( ( complete_Sup_Sup_o_o
@ ( image_a_o_o
@ ^ [I: a,X2: $o] : ( member_o @ X2 @ ( R @ I ) )
@ S ) )
= ( ^ [X2: $o] : ( member_o @ X2 @ ( comple90263536869209701_set_o @ ( image_a_set_o @ R @ S ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_463_UN__constant__eq,axiom,
! [A: $o,A2: set_o,F: $o > set_a,C2: set_a] :
( ( member_o @ A @ A2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ( F @ X3 )
= C2 ) )
=> ( ( comple2307003609928055243_set_a @ ( image_o_set_a @ F @ A2 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_464_UN__constant__eq,axiom,
! [A: a,A2: set_a,F: a > set_a,C2: set_a] :
( ( member_a @ A @ A2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ( F @ X3 )
= C2 ) )
=> ( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A2 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_465_UN__constant__eq,axiom,
! [A: $o,A2: set_o,F: $o > set_o,C2: set_o] :
( ( member_o @ A @ A2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ( F @ X3 )
= C2 ) )
=> ( ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ A2 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_466_UN__constant__eq,axiom,
! [A: a,A2: set_a,F: a > set_o,C2: set_o] :
( ( member_a @ A @ A2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ( F @ X3 )
= C2 ) )
=> ( ( comple90263536869209701_set_o @ ( image_a_set_o @ F @ A2 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_467_UN__constant__eq,axiom,
! [A: $o,A2: set_o,F: $o > set_set_a,C2: set_set_a] :
( ( member_o @ A @ A2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ( F @ X3 )
= C2 ) )
=> ( ( comple3958522678809307947_set_a @ ( image_o_set_set_a @ F @ A2 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_468_UN__constant__eq,axiom,
! [A: a,A2: set_a,F: a > set_set_a,C2: set_set_a] :
( ( member_a @ A @ A2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ( F @ X3 )
= C2 ) )
=> ( ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ F @ A2 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_469_UN__constant__eq,axiom,
! [A: set_a,A2: set_set_a,F: set_a > set_a,C2: set_a] :
( ( member_set_a @ A @ A2 )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( ( F @ X3 )
= C2 ) )
=> ( ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ F @ A2 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_470_UN__constant__eq,axiom,
! [A: set_a,A2: set_set_a,F: set_a > set_o,C2: set_o] :
( ( member_set_a @ A @ A2 )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( ( F @ X3 )
= C2 ) )
=> ( ( comple90263536869209701_set_o @ ( image_set_a_set_o @ F @ A2 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_471_UN__constant__eq,axiom,
! [A: $o,A2: set_o,F: $o > set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( member_o @ A @ A2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ( F @ X3 )
= C2 ) )
=> ( ( comple8421679170691845492od_a_a @ ( image_4840794370999913844od_a_a @ F @ A2 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_472_UN__constant__eq,axiom,
! [A: a,A2: set_a,F: a > set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( member_a @ A @ A2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ( F @ X3 )
= C2 ) )
=> ( ( comple8421679170691845492od_a_a @ ( image_4421510592991446670od_a_a @ F @ A2 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_473_dual__order_Orefl,axiom,
! [A: set_Pr5530083903271594800od_a_a] : ( ord_le114883831454073552od_a_a @ A @ A ) ).
% dual_order.refl
thf(fact_474_dual__order_Orefl,axiom,
! [A: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ A @ A ) ).
% dual_order.refl
thf(fact_475_dual__order_Orefl,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_476_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_477_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_478_order__refl,axiom,
! [X: set_Pr5530083903271594800od_a_a] : ( ord_le114883831454073552od_a_a @ X @ X ) ).
% order_refl
thf(fact_479_order__refl,axiom,
! [X: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ X @ X ) ).
% order_refl
thf(fact_480_order__refl,axiom,
! [X: set_set_a] : ( ord_le3724670747650509150_set_a @ X @ X ) ).
% order_refl
thf(fact_481_order__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% order_refl
thf(fact_482_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_483_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > nat,B2: set_nat] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( member_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( collect_nat @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_484_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > $o,B2: set_o] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( member_o @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_o @ ( image_nat_o @ F @ ( collect_nat @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_485_image__Collect__subsetI,axiom,
! [P: a > $o,F: a > $o,B2: set_o] :
( ! [X3: a] :
( ( P @ X3 )
=> ( member_o @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_o @ ( image_a_o @ F @ ( collect_a @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_486_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > a,B2: set_a] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( member_a @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_a @ ( image_nat_a @ F @ ( collect_nat @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_487_image__Collect__subsetI,axiom,
! [P: a > $o,F: a > a,B2: set_a] :
( ! [X3: a] :
( ( P @ X3 )
=> ( member_a @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ ( collect_a @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_488_image__Collect__subsetI,axiom,
! [P: set_a > $o,F: set_a > $o,B2: set_o] :
( ! [X3: set_a] :
( ( P @ X3 )
=> ( member_o @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_o @ ( image_set_a_o @ F @ ( collect_set_a @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_489_image__Collect__subsetI,axiom,
! [P: a > $o,F: a > set_o,B2: set_set_o] :
( ! [X3: a] :
( ( P @ X3 )
=> ( member_set_o @ ( F @ X3 ) @ B2 ) )
=> ( ord_le4374716579403074808_set_o @ ( image_a_set_o @ F @ ( collect_a @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_490_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > set_a,B2: set_set_a] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( member_set_a @ ( F @ X3 ) @ B2 ) )
=> ( ord_le3724670747650509150_set_a @ ( image_nat_set_a @ F @ ( collect_nat @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_491_image__Collect__subsetI,axiom,
! [P: a > $o,F: a > set_a,B2: set_set_a] :
( ! [X3: a] :
( ( P @ X3 )
=> ( member_set_a @ ( F @ X3 ) @ B2 ) )
=> ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ ( collect_a @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_492_image__Collect__subsetI,axiom,
! [P: set_a > $o,F: set_a > a,B2: set_a] :
( ! [X3: set_a] :
( ( P @ X3 )
=> ( member_a @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_a @ ( image_set_a_a @ F @ ( collect_set_a @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_493_fin__edges,axiom,
finite_finite_set_a @ edges ).
% fin_edges
thf(fact_494_empty__Collect__eq,axiom,
! [P: produc4044097585999906000od_a_a > $o] :
( ( bot_bo4436838304982128028od_a_a
= ( collec5045780995415420475od_a_a @ P ) )
= ( ! [X2: produc4044097585999906000od_a_a] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_495_empty__Collect__eq,axiom,
! [P: set_a > $o] :
( ( bot_bot_set_set_a
= ( collect_set_a @ P ) )
= ( ! [X2: set_a] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_496_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_497_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_498_empty__Collect__eq,axiom,
! [P: product_prod_a_a > $o] :
( ( bot_bo3357376287454694259od_a_a
= ( collec3336397797384452498od_a_a @ P ) )
= ( ! [X2: product_prod_a_a] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_499_Collect__empty__eq,axiom,
! [P: produc4044097585999906000od_a_a > $o] :
( ( ( collec5045780995415420475od_a_a @ P )
= bot_bo4436838304982128028od_a_a )
= ( ! [X2: produc4044097585999906000od_a_a] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_500_Collect__empty__eq,axiom,
! [P: set_a > $o] :
( ( ( collect_set_a @ P )
= bot_bot_set_set_a )
= ( ! [X2: set_a] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_501_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_502_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_503_Collect__empty__eq,axiom,
! [P: product_prod_a_a > $o] :
( ( ( collec3336397797384452498od_a_a @ P )
= bot_bo3357376287454694259od_a_a )
= ( ! [X2: product_prod_a_a] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_504_all__not__in__conv,axiom,
! [A2: set_o] :
( ( ! [X2: $o] :
~ ( member_o @ X2 @ A2 ) )
= ( A2 = bot_bot_set_o ) ) ).
% all_not_in_conv
thf(fact_505_all__not__in__conv,axiom,
! [A2: set_Pr5530083903271594800od_a_a] :
( ( ! [X2: produc4044097585999906000od_a_a] :
~ ( member3071122053849602553od_a_a @ X2 @ A2 ) )
= ( A2 = bot_bo4436838304982128028od_a_a ) ) ).
% all_not_in_conv
thf(fact_506_all__not__in__conv,axiom,
! [A2: set_set_a] :
( ( ! [X2: set_a] :
~ ( member_set_a @ X2 @ A2 ) )
= ( A2 = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_507_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X2: a] :
~ ( member_a @ X2 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_508_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X2: nat] :
~ ( member_nat @ X2 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_509_all__not__in__conv,axiom,
! [A2: set_Product_prod_a_a] :
( ( ! [X2: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ X2 @ A2 ) )
= ( A2 = bot_bo3357376287454694259od_a_a ) ) ).
% all_not_in_conv
thf(fact_510_empty__iff,axiom,
! [C2: $o] :
~ ( member_o @ C2 @ bot_bot_set_o ) ).
% empty_iff
thf(fact_511_empty__iff,axiom,
! [C2: produc4044097585999906000od_a_a] :
~ ( member3071122053849602553od_a_a @ C2 @ bot_bo4436838304982128028od_a_a ) ).
% empty_iff
thf(fact_512_empty__iff,axiom,
! [C2: set_a] :
~ ( member_set_a @ C2 @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_513_empty__iff,axiom,
! [C2: a] :
~ ( member_a @ C2 @ bot_bot_set_a ) ).
% empty_iff
thf(fact_514_empty__iff,axiom,
! [C2: nat] :
~ ( member_nat @ C2 @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_515_empty__iff,axiom,
! [C2: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ C2 @ bot_bo3357376287454694259od_a_a ) ).
% empty_iff
thf(fact_516_image__empty,axiom,
! [F: a > a] :
( ( image_a_a @ F @ bot_bot_set_a )
= bot_bot_set_a ) ).
% image_empty
thf(fact_517_image__empty,axiom,
! [F: a > nat] :
( ( image_a_nat @ F @ bot_bot_set_a )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_518_image__empty,axiom,
! [F: nat > a] :
( ( image_nat_a @ F @ bot_bot_set_nat )
= bot_bot_set_a ) ).
% image_empty
thf(fact_519_image__empty,axiom,
! [F: nat > nat] :
( ( image_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_520_image__empty,axiom,
! [F: set_a > a] :
( ( image_set_a_a @ F @ bot_bot_set_set_a )
= bot_bot_set_a ) ).
% image_empty
thf(fact_521_image__empty,axiom,
! [F: set_a > nat] :
( ( image_set_a_nat @ F @ bot_bot_set_set_a )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_522_image__empty,axiom,
! [F: a > set_o] :
( ( image_a_set_o @ F @ bot_bot_set_a )
= bot_bot_set_set_o ) ).
% image_empty
thf(fact_523_image__empty,axiom,
! [F: a > set_a] :
( ( image_a_set_a @ F @ bot_bot_set_a )
= bot_bot_set_set_a ) ).
% image_empty
thf(fact_524_image__empty,axiom,
! [F: nat > set_a] :
( ( image_nat_set_a @ F @ bot_bot_set_nat )
= bot_bot_set_set_a ) ).
% image_empty
thf(fact_525_image__empty,axiom,
! [F: set_a > set_a] :
( ( image_set_a_set_a @ F @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% image_empty
thf(fact_526_empty__is__image,axiom,
! [F: a > a,A2: set_a] :
( ( bot_bot_set_a
= ( image_a_a @ F @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_527_empty__is__image,axiom,
! [F: nat > a,A2: set_nat] :
( ( bot_bot_set_a
= ( image_nat_a @ F @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_528_empty__is__image,axiom,
! [F: a > nat,A2: set_a] :
( ( bot_bot_set_nat
= ( image_a_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_529_empty__is__image,axiom,
! [F: nat > nat,A2: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_530_empty__is__image,axiom,
! [F: a > set_o,A2: set_a] :
( ( bot_bot_set_set_o
= ( image_a_set_o @ F @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_531_empty__is__image,axiom,
! [F: a > set_a,A2: set_a] :
( ( bot_bot_set_set_a
= ( image_a_set_a @ F @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_532_empty__is__image,axiom,
! [F: nat > set_a,A2: set_nat] :
( ( bot_bot_set_set_a
= ( image_nat_set_a @ F @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_533_empty__is__image,axiom,
! [F: set_a > a,A2: set_set_a] :
( ( bot_bot_set_a
= ( image_set_a_a @ F @ A2 ) )
= ( A2 = bot_bot_set_set_a ) ) ).
% empty_is_image
thf(fact_534_empty__is__image,axiom,
! [F: set_a > nat,A2: set_set_a] :
( ( bot_bot_set_nat
= ( image_set_a_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_set_a ) ) ).
% empty_is_image
thf(fact_535_empty__is__image,axiom,
! [F: a > set_set_a,A2: set_a] :
( ( bot_bo3380559777022489994_set_a
= ( image_a_set_set_a @ F @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_536_image__is__empty,axiom,
! [F: a > a,A2: set_a] :
( ( ( image_a_a @ F @ A2 )
= bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_537_image__is__empty,axiom,
! [F: nat > a,A2: set_nat] :
( ( ( image_nat_a @ F @ A2 )
= bot_bot_set_a )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_538_image__is__empty,axiom,
! [F: a > nat,A2: set_a] :
( ( ( image_a_nat @ F @ A2 )
= bot_bot_set_nat )
= ( A2 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_539_image__is__empty,axiom,
! [F: nat > nat,A2: set_nat] :
( ( ( image_nat_nat @ F @ A2 )
= bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_540_image__is__empty,axiom,
! [F: a > set_o,A2: set_a] :
( ( ( image_a_set_o @ F @ A2 )
= bot_bot_set_set_o )
= ( A2 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_541_image__is__empty,axiom,
! [F: a > set_a,A2: set_a] :
( ( ( image_a_set_a @ F @ A2 )
= bot_bot_set_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_542_image__is__empty,axiom,
! [F: nat > set_a,A2: set_nat] :
( ( ( image_nat_set_a @ F @ A2 )
= bot_bot_set_set_a )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_543_image__is__empty,axiom,
! [F: set_a > a,A2: set_set_a] :
( ( ( image_set_a_a @ F @ A2 )
= bot_bot_set_a )
= ( A2 = bot_bot_set_set_a ) ) ).
% image_is_empty
thf(fact_544_image__is__empty,axiom,
! [F: set_a > nat,A2: set_set_a] :
( ( ( image_set_a_nat @ F @ A2 )
= bot_bot_set_nat )
= ( A2 = bot_bot_set_set_a ) ) ).
% image_is_empty
thf(fact_545_image__is__empty,axiom,
! [F: a > set_set_a,A2: set_a] :
( ( ( image_a_set_set_a @ F @ A2 )
= bot_bo3380559777022489994_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_546_subset__empty,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_547_subset__empty,axiom,
! [A2: set_Pr5530083903271594800od_a_a] :
( ( ord_le114883831454073552od_a_a @ A2 @ bot_bo4436838304982128028od_a_a )
= ( A2 = bot_bo4436838304982128028od_a_a ) ) ).
% subset_empty
thf(fact_548_subset__empty,axiom,
! [A2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ bot_bo3357376287454694259od_a_a )
= ( A2 = bot_bo3357376287454694259od_a_a ) ) ).
% subset_empty
thf(fact_549_subset__empty,axiom,
! [A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ bot_bot_set_set_a )
= ( A2 = bot_bot_set_set_a ) ) ).
% subset_empty
thf(fact_550_subset__empty,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_551_empty__subsetI,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_552_empty__subsetI,axiom,
! [A2: set_Pr5530083903271594800od_a_a] : ( ord_le114883831454073552od_a_a @ bot_bo4436838304982128028od_a_a @ A2 ) ).
% empty_subsetI
thf(fact_553_empty__subsetI,axiom,
! [A2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ bot_bo3357376287454694259od_a_a @ A2 ) ).
% empty_subsetI
thf(fact_554_empty__subsetI,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A2 ) ).
% empty_subsetI
thf(fact_555_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_556_Sup__bot__conv_I2_J,axiom,
! [A2: set_set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ A2 ) )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( X2 = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_557_Sup__bot__conv_I2_J,axiom,
! [A2: set_se9027383378080648592od_a_a] :
( ( bot_bo4436838304982128028od_a_a
= ( comple2978350343072902813od_a_a @ A2 ) )
= ( ! [X2: set_Pr5530083903271594800od_a_a] :
( ( member4210947715425868889od_a_a @ X2 @ A2 )
=> ( X2 = bot_bo4436838304982128028od_a_a ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_558_Sup__bot__conv_I2_J,axiom,
! [A2: set_se5735800977113168103od_a_a] :
( ( bot_bo3357376287454694259od_a_a
= ( comple8421679170691845492od_a_a @ A2 ) )
= ( ! [X2: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X2 @ A2 )
=> ( X2 = bot_bo3357376287454694259od_a_a ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_559_Sup__bot__conv_I2_J,axiom,
! [A2: set_set_set_a] :
( ( bot_bot_set_set_a
= ( comple3958522678809307947_set_a @ A2 ) )
= ( ! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ A2 )
=> ( X2 = bot_bot_set_set_a ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_560_Sup__bot__conv_I2_J,axiom,
! [A2: set_set_a] :
( ( bot_bot_set_a
= ( comple2307003609928055243_set_a @ A2 ) )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( X2 = bot_bot_set_a ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_561_Sup__bot__conv_I2_J,axiom,
! [A2: set_set_o] :
( ( bot_bot_set_o
= ( comple90263536869209701_set_o @ A2 ) )
= ( ! [X2: set_o] :
( ( member_set_o @ X2 @ A2 )
=> ( X2 = bot_bot_set_o ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_562_Sup__bot__conv_I1_J,axiom,
! [A2: set_set_nat] :
( ( ( comple7399068483239264473et_nat @ A2 )
= bot_bot_set_nat )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( X2 = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_563_Sup__bot__conv_I1_J,axiom,
! [A2: set_se9027383378080648592od_a_a] :
( ( ( comple2978350343072902813od_a_a @ A2 )
= bot_bo4436838304982128028od_a_a )
= ( ! [X2: set_Pr5530083903271594800od_a_a] :
( ( member4210947715425868889od_a_a @ X2 @ A2 )
=> ( X2 = bot_bo4436838304982128028od_a_a ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_564_Sup__bot__conv_I1_J,axiom,
! [A2: set_se5735800977113168103od_a_a] :
( ( ( comple8421679170691845492od_a_a @ A2 )
= bot_bo3357376287454694259od_a_a )
= ( ! [X2: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X2 @ A2 )
=> ( X2 = bot_bo3357376287454694259od_a_a ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_565_Sup__bot__conv_I1_J,axiom,
! [A2: set_set_set_a] :
( ( ( comple3958522678809307947_set_a @ A2 )
= bot_bot_set_set_a )
= ( ! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ A2 )
=> ( X2 = bot_bot_set_set_a ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_566_Sup__bot__conv_I1_J,axiom,
! [A2: set_set_a] :
( ( ( comple2307003609928055243_set_a @ A2 )
= bot_bot_set_a )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( X2 = bot_bot_set_a ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_567_Sup__bot__conv_I1_J,axiom,
! [A2: set_set_o] :
( ( ( comple90263536869209701_set_o @ A2 )
= bot_bot_set_o )
= ( ! [X2: set_o] :
( ( member_set_o @ X2 @ A2 )
=> ( X2 = bot_bot_set_o ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_568_Sup__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% Sup_empty
thf(fact_569_Sup__empty,axiom,
( ( comple2978350343072902813od_a_a @ bot_bo5623065384236484604od_a_a )
= bot_bo4436838304982128028od_a_a ) ).
% Sup_empty
thf(fact_570_Sup__empty,axiom,
( ( comple8421679170691845492od_a_a @ bot_bo777872063958040403od_a_a )
= bot_bo3357376287454694259od_a_a ) ).
% Sup_empty
thf(fact_571_Sup__empty,axiom,
( ( comple3958522678809307947_set_a @ bot_bo3380559777022489994_set_a )
= bot_bot_set_set_a ) ).
% Sup_empty
thf(fact_572_Sup__empty,axiom,
( ( comple2307003609928055243_set_a @ bot_bot_set_set_a )
= bot_bot_set_a ) ).
% Sup_empty
thf(fact_573_Sup__empty,axiom,
( ( comple90263536869209701_set_o @ bot_bot_set_set_o )
= bot_bot_set_o ) ).
% Sup_empty
thf(fact_574_SUP__bot__conv_I2_J,axiom,
! [B2: set_a > set_Pr5530083903271594800od_a_a,A2: set_set_a] :
( ( bot_bo4436838304982128028od_a_a
= ( comple2978350343072902813od_a_a @ ( image_7562202058474640471od_a_a @ B2 @ A2 ) ) )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( ( B2 @ X2 )
= bot_bo4436838304982128028od_a_a ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_575_SUP__bot__conv_I2_J,axiom,
! [B2: a > set_Pr5530083903271594800od_a_a,A2: set_a] :
( ( bot_bo4436838304982128028od_a_a
= ( comple2978350343072902813od_a_a @ ( image_5653227685612666295od_a_a @ B2 @ A2 ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( B2 @ X2 )
= bot_bo4436838304982128028od_a_a ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_576_SUP__bot__conv_I2_J,axiom,
! [B2: set_a > set_Product_prod_a_a,A2: set_set_a] :
( ( bot_bo3357376287454694259od_a_a
= ( comple8421679170691845492od_a_a @ ( image_6165024369500519726od_a_a @ B2 @ A2 ) ) )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( ( B2 @ X2 )
= bot_bo3357376287454694259od_a_a ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_577_SUP__bot__conv_I2_J,axiom,
! [B2: a > set_Product_prod_a_a,A2: set_a] :
( ( bot_bo3357376287454694259od_a_a
= ( comple8421679170691845492od_a_a @ ( image_4421510592991446670od_a_a @ B2 @ A2 ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( B2 @ X2 )
= bot_bo3357376287454694259od_a_a ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_578_SUP__bot__conv_I2_J,axiom,
! [B2: a > set_set_a,A2: set_a] :
( ( bot_bot_set_set_a
= ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ B2 @ A2 ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( B2 @ X2 )
= bot_bot_set_set_a ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_579_SUP__bot__conv_I2_J,axiom,
! [B2: a > set_a,A2: set_a] :
( ( bot_bot_set_a
= ( comple2307003609928055243_set_a @ ( image_a_set_a @ B2 @ A2 ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( B2 @ X2 )
= bot_bot_set_a ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_580_SUP__bot__conv_I2_J,axiom,
! [B2: set_a > set_a,A2: set_set_a] :
( ( bot_bot_set_a
= ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ B2 @ A2 ) ) )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( ( B2 @ X2 )
= bot_bot_set_a ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_581_SUP__bot__conv_I2_J,axiom,
! [B2: a > set_o,A2: set_a] :
( ( bot_bot_set_o
= ( comple90263536869209701_set_o @ ( image_a_set_o @ B2 @ A2 ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( B2 @ X2 )
= bot_bot_set_o ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_582_SUP__bot__conv_I1_J,axiom,
! [B2: set_a > set_Pr5530083903271594800od_a_a,A2: set_set_a] :
( ( ( comple2978350343072902813od_a_a @ ( image_7562202058474640471od_a_a @ B2 @ A2 ) )
= bot_bo4436838304982128028od_a_a )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( ( B2 @ X2 )
= bot_bo4436838304982128028od_a_a ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_583_SUP__bot__conv_I1_J,axiom,
! [B2: a > set_Pr5530083903271594800od_a_a,A2: set_a] :
( ( ( comple2978350343072902813od_a_a @ ( image_5653227685612666295od_a_a @ B2 @ A2 ) )
= bot_bo4436838304982128028od_a_a )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( B2 @ X2 )
= bot_bo4436838304982128028od_a_a ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_584_SUP__bot__conv_I1_J,axiom,
! [B2: set_a > set_Product_prod_a_a,A2: set_set_a] :
( ( ( comple8421679170691845492od_a_a @ ( image_6165024369500519726od_a_a @ B2 @ A2 ) )
= bot_bo3357376287454694259od_a_a )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( ( B2 @ X2 )
= bot_bo3357376287454694259od_a_a ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_585_SUP__bot__conv_I1_J,axiom,
! [B2: a > set_Product_prod_a_a,A2: set_a] :
( ( ( comple8421679170691845492od_a_a @ ( image_4421510592991446670od_a_a @ B2 @ A2 ) )
= bot_bo3357376287454694259od_a_a )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( B2 @ X2 )
= bot_bo3357376287454694259od_a_a ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_586_SUP__bot__conv_I1_J,axiom,
! [B2: a > set_set_a,A2: set_a] :
( ( ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ B2 @ A2 ) )
= bot_bot_set_set_a )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( B2 @ X2 )
= bot_bot_set_set_a ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_587_SUP__bot__conv_I1_J,axiom,
! [B2: a > set_a,A2: set_a] :
( ( ( comple2307003609928055243_set_a @ ( image_a_set_a @ B2 @ A2 ) )
= bot_bot_set_a )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( B2 @ X2 )
= bot_bot_set_a ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_588_SUP__bot__conv_I1_J,axiom,
! [B2: set_a > set_a,A2: set_set_a] :
( ( ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ B2 @ A2 ) )
= bot_bot_set_a )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( ( B2 @ X2 )
= bot_bot_set_a ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_589_SUP__bot__conv_I1_J,axiom,
! [B2: a > set_o,A2: set_a] :
( ( ( comple90263536869209701_set_o @ ( image_a_set_o @ B2 @ A2 ) )
= bot_bot_set_o )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( B2 @ X2 )
= bot_bot_set_o ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_590_SUP__bot,axiom,
! [A2: set_set_a] :
( ( comple2978350343072902813od_a_a
@ ( image_7562202058474640471od_a_a
@ ^ [X2: set_a] : bot_bo4436838304982128028od_a_a
@ A2 ) )
= bot_bo4436838304982128028od_a_a ) ).
% SUP_bot
thf(fact_591_SUP__bot,axiom,
! [A2: set_a] :
( ( comple2978350343072902813od_a_a
@ ( image_5653227685612666295od_a_a
@ ^ [X2: a] : bot_bo4436838304982128028od_a_a
@ A2 ) )
= bot_bo4436838304982128028od_a_a ) ).
% SUP_bot
thf(fact_592_SUP__bot,axiom,
! [A2: set_set_a] :
( ( comple8421679170691845492od_a_a
@ ( image_6165024369500519726od_a_a
@ ^ [X2: set_a] : bot_bo3357376287454694259od_a_a
@ A2 ) )
= bot_bo3357376287454694259od_a_a ) ).
% SUP_bot
thf(fact_593_SUP__bot,axiom,
! [A2: set_a] :
( ( comple8421679170691845492od_a_a
@ ( image_4421510592991446670od_a_a
@ ^ [X2: a] : bot_bo3357376287454694259od_a_a
@ A2 ) )
= bot_bo3357376287454694259od_a_a ) ).
% SUP_bot
thf(fact_594_SUP__bot,axiom,
! [A2: set_a] :
( ( comple3958522678809307947_set_a
@ ( image_a_set_set_a
@ ^ [X2: a] : bot_bot_set_set_a
@ A2 ) )
= bot_bot_set_set_a ) ).
% SUP_bot
thf(fact_595_SUP__bot,axiom,
! [A2: set_a] :
( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X2: a] : bot_bot_set_a
@ A2 ) )
= bot_bot_set_a ) ).
% SUP_bot
thf(fact_596_SUP__bot,axiom,
! [A2: set_set_a] :
( ( comple2307003609928055243_set_a
@ ( image_set_a_set_a
@ ^ [X2: set_a] : bot_bot_set_a
@ A2 ) )
= bot_bot_set_a ) ).
% SUP_bot
thf(fact_597_SUP__bot,axiom,
! [A2: set_a] :
( ( comple90263536869209701_set_o
@ ( image_a_set_o
@ ^ [X2: a] : bot_bot_set_o
@ A2 ) )
= bot_bot_set_o ) ).
% SUP_bot
thf(fact_598_SUP__const,axiom,
! [A2: set_a,F: set_a] :
( ( A2 != bot_bot_set_a )
=> ( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [I: a] : F
@ A2 ) )
= F ) ) ).
% SUP_const
thf(fact_599_SUP__const,axiom,
! [A2: set_nat,F: set_a] :
( ( A2 != bot_bot_set_nat )
=> ( ( comple2307003609928055243_set_a
@ ( image_nat_set_a
@ ^ [I: nat] : F
@ A2 ) )
= F ) ) ).
% SUP_const
thf(fact_600_SUP__const,axiom,
! [A2: set_a,F: set_o] :
( ( A2 != bot_bot_set_a )
=> ( ( comple90263536869209701_set_o
@ ( image_a_set_o
@ ^ [I: a] : F
@ A2 ) )
= F ) ) ).
% SUP_const
thf(fact_601_SUP__const,axiom,
! [A2: set_nat,F: set_o] :
( ( A2 != bot_bot_set_nat )
=> ( ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [I: nat] : F
@ A2 ) )
= F ) ) ).
% SUP_const
thf(fact_602_SUP__const,axiom,
! [A2: set_a,F: set_set_a] :
( ( A2 != bot_bot_set_a )
=> ( ( comple3958522678809307947_set_a
@ ( image_a_set_set_a
@ ^ [I: a] : F
@ A2 ) )
= F ) ) ).
% SUP_const
thf(fact_603_SUP__const,axiom,
! [A2: set_nat,F: set_set_a] :
( ( A2 != bot_bot_set_nat )
=> ( ( comple3958522678809307947_set_a
@ ( image_nat_set_set_a
@ ^ [I: nat] : F
@ A2 ) )
= F ) ) ).
% SUP_const
thf(fact_604_SUP__const,axiom,
! [A2: set_set_a,F: set_a] :
( ( A2 != bot_bot_set_set_a )
=> ( ( comple2307003609928055243_set_a
@ ( image_set_a_set_a
@ ^ [I: set_a] : F
@ A2 ) )
= F ) ) ).
% SUP_const
thf(fact_605_SUP__const,axiom,
! [A2: set_set_a,F: set_o] :
( ( A2 != bot_bot_set_set_a )
=> ( ( comple90263536869209701_set_o
@ ( image_set_a_set_o
@ ^ [I: set_a] : F
@ A2 ) )
= F ) ) ).
% SUP_const
thf(fact_606_SUP__const,axiom,
! [A2: set_a,F: set_Product_prod_a_a] :
( ( A2 != bot_bot_set_a )
=> ( ( comple8421679170691845492od_a_a
@ ( image_4421510592991446670od_a_a
@ ^ [I: a] : F
@ A2 ) )
= F ) ) ).
% SUP_const
thf(fact_607_SUP__const,axiom,
! [A2: set_nat,F: set_Product_prod_a_a] :
( ( A2 != bot_bot_set_nat )
=> ( ( comple8421679170691845492od_a_a
@ ( image_2330413017338827248od_a_a
@ ^ [I: nat] : F
@ A2 ) )
= F ) ) ).
% SUP_const
thf(fact_608_UN__constant,axiom,
! [A2: set_a,C2: set_nat] :
( ( ( A2 = bot_bot_set_a )
=> ( ( comple7399068483239264473et_nat
@ ( image_a_set_nat
@ ^ [Y2: a] : C2
@ A2 ) )
= bot_bot_set_nat ) )
& ( ( A2 != bot_bot_set_a )
=> ( ( comple7399068483239264473et_nat
@ ( image_a_set_nat
@ ^ [Y2: a] : C2
@ A2 ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_609_UN__constant,axiom,
! [A2: set_nat,C2: set_nat] :
( ( ( A2 = bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y2: nat] : C2
@ A2 ) )
= bot_bot_set_nat ) )
& ( ( A2 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y2: nat] : C2
@ A2 ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_610_UN__constant,axiom,
! [A2: set_a,C2: set_a] :
( ( ( A2 = bot_bot_set_a )
=> ( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [Y2: a] : C2
@ A2 ) )
= bot_bot_set_a ) )
& ( ( A2 != bot_bot_set_a )
=> ( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [Y2: a] : C2
@ A2 ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_611_UN__constant,axiom,
! [A2: set_nat,C2: set_a] :
( ( ( A2 = bot_bot_set_nat )
=> ( ( comple2307003609928055243_set_a
@ ( image_nat_set_a
@ ^ [Y2: nat] : C2
@ A2 ) )
= bot_bot_set_a ) )
& ( ( A2 != bot_bot_set_nat )
=> ( ( comple2307003609928055243_set_a
@ ( image_nat_set_a
@ ^ [Y2: nat] : C2
@ A2 ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_612_UN__constant,axiom,
! [A2: set_a,C2: set_o] :
( ( ( A2 = bot_bot_set_a )
=> ( ( comple90263536869209701_set_o
@ ( image_a_set_o
@ ^ [Y2: a] : C2
@ A2 ) )
= bot_bot_set_o ) )
& ( ( A2 != bot_bot_set_a )
=> ( ( comple90263536869209701_set_o
@ ( image_a_set_o
@ ^ [Y2: a] : C2
@ A2 ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_613_UN__constant,axiom,
! [A2: set_nat,C2: set_o] :
( ( ( A2 = bot_bot_set_nat )
=> ( ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [Y2: nat] : C2
@ A2 ) )
= bot_bot_set_o ) )
& ( ( A2 != bot_bot_set_nat )
=> ( ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [Y2: nat] : C2
@ A2 ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_614_UN__constant,axiom,
! [A2: set_set_a,C2: set_nat] :
( ( ( A2 = bot_bot_set_set_a )
=> ( ( comple7399068483239264473et_nat
@ ( image_set_a_set_nat
@ ^ [Y2: set_a] : C2
@ A2 ) )
= bot_bot_set_nat ) )
& ( ( A2 != bot_bot_set_set_a )
=> ( ( comple7399068483239264473et_nat
@ ( image_set_a_set_nat
@ ^ [Y2: set_a] : C2
@ A2 ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_615_UN__constant,axiom,
! [A2: set_a,C2: set_set_a] :
( ( ( A2 = bot_bot_set_a )
=> ( ( comple3958522678809307947_set_a
@ ( image_a_set_set_a
@ ^ [Y2: a] : C2
@ A2 ) )
= bot_bot_set_set_a ) )
& ( ( A2 != bot_bot_set_a )
=> ( ( comple3958522678809307947_set_a
@ ( image_a_set_set_a
@ ^ [Y2: a] : C2
@ A2 ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_616_UN__constant,axiom,
! [A2: set_nat,C2: set_set_a] :
( ( ( A2 = bot_bot_set_nat )
=> ( ( comple3958522678809307947_set_a
@ ( image_nat_set_set_a
@ ^ [Y2: nat] : C2
@ A2 ) )
= bot_bot_set_set_a ) )
& ( ( A2 != bot_bot_set_nat )
=> ( ( comple3958522678809307947_set_a
@ ( image_nat_set_set_a
@ ^ [Y2: nat] : C2
@ A2 ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_617_UN__constant,axiom,
! [A2: set_set_a,C2: set_a] :
( ( ( A2 = bot_bot_set_set_a )
=> ( ( comple2307003609928055243_set_a
@ ( image_set_a_set_a
@ ^ [Y2: set_a] : C2
@ A2 ) )
= bot_bot_set_a ) )
& ( ( A2 != bot_bot_set_set_a )
=> ( ( comple2307003609928055243_set_a
@ ( image_set_a_set_a
@ ^ [Y2: set_a] : C2
@ A2 ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_618_Sup__bool__def,axiom,
( complete_Sup_Sup_o
= ( member_o @ $true ) ) ).
% Sup_bool_def
thf(fact_619_ex__in__conv,axiom,
! [A2: set_o] :
( ( ? [X2: $o] : ( member_o @ X2 @ A2 ) )
= ( A2 != bot_bot_set_o ) ) ).
% ex_in_conv
thf(fact_620_ex__in__conv,axiom,
! [A2: set_Pr5530083903271594800od_a_a] :
( ( ? [X2: produc4044097585999906000od_a_a] : ( member3071122053849602553od_a_a @ X2 @ A2 ) )
= ( A2 != bot_bo4436838304982128028od_a_a ) ) ).
% ex_in_conv
thf(fact_621_ex__in__conv,axiom,
! [A2: set_set_a] :
( ( ? [X2: set_a] : ( member_set_a @ X2 @ A2 ) )
= ( A2 != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_622_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X2: a] : ( member_a @ X2 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_623_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X2: nat] : ( member_nat @ X2 @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_624_ex__in__conv,axiom,
! [A2: set_Product_prod_a_a] :
( ( ? [X2: product_prod_a_a] : ( member1426531477525435216od_a_a @ X2 @ A2 ) )
= ( A2 != bot_bo3357376287454694259od_a_a ) ) ).
% ex_in_conv
thf(fact_625_empty__def,axiom,
( bot_bo4436838304982128028od_a_a
= ( collec5045780995415420475od_a_a
@ ^ [X2: produc4044097585999906000od_a_a] : $false ) ) ).
% empty_def
thf(fact_626_empty__def,axiom,
( bot_bot_set_set_a
= ( collect_set_a
@ ^ [X2: set_a] : $false ) ) ).
% empty_def
thf(fact_627_empty__def,axiom,
( bot_bot_set_a
= ( collect_a
@ ^ [X2: a] : $false ) ) ).
% empty_def
thf(fact_628_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X2: nat] : $false ) ) ).
% empty_def
thf(fact_629_empty__def,axiom,
( bot_bo3357376287454694259od_a_a
= ( collec3336397797384452498od_a_a
@ ^ [X2: product_prod_a_a] : $false ) ) ).
% empty_def
thf(fact_630_equals0I,axiom,
! [A2: set_o] :
( ! [Y3: $o] :
~ ( member_o @ Y3 @ A2 )
=> ( A2 = bot_bot_set_o ) ) ).
% equals0I
thf(fact_631_equals0I,axiom,
! [A2: set_Pr5530083903271594800od_a_a] :
( ! [Y3: produc4044097585999906000od_a_a] :
~ ( member3071122053849602553od_a_a @ Y3 @ A2 )
=> ( A2 = bot_bo4436838304982128028od_a_a ) ) ).
% equals0I
thf(fact_632_equals0I,axiom,
! [A2: set_set_a] :
( ! [Y3: set_a] :
~ ( member_set_a @ Y3 @ A2 )
=> ( A2 = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_633_equals0I,axiom,
! [A2: set_a] :
( ! [Y3: a] :
~ ( member_a @ Y3 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_634_equals0I,axiom,
! [A2: set_nat] :
( ! [Y3: nat] :
~ ( member_nat @ Y3 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_635_equals0I,axiom,
! [A2: set_Product_prod_a_a] :
( ! [Y3: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ Y3 @ A2 )
=> ( A2 = bot_bo3357376287454694259od_a_a ) ) ).
% equals0I
thf(fact_636_equals0D,axiom,
! [A2: set_o,A: $o] :
( ( A2 = bot_bot_set_o )
=> ~ ( member_o @ A @ A2 ) ) ).
% equals0D
thf(fact_637_equals0D,axiom,
! [A2: set_Pr5530083903271594800od_a_a,A: produc4044097585999906000od_a_a] :
( ( A2 = bot_bo4436838304982128028od_a_a )
=> ~ ( member3071122053849602553od_a_a @ A @ A2 ) ) ).
% equals0D
thf(fact_638_equals0D,axiom,
! [A2: set_set_a,A: set_a] :
( ( A2 = bot_bot_set_set_a )
=> ~ ( member_set_a @ A @ A2 ) ) ).
% equals0D
thf(fact_639_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_640_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_641_equals0D,axiom,
! [A2: set_Product_prod_a_a,A: product_prod_a_a] :
( ( A2 = bot_bo3357376287454694259od_a_a )
=> ~ ( member1426531477525435216od_a_a @ A @ A2 ) ) ).
% equals0D
thf(fact_642_emptyE,axiom,
! [A: $o] :
~ ( member_o @ A @ bot_bot_set_o ) ).
% emptyE
thf(fact_643_emptyE,axiom,
! [A: produc4044097585999906000od_a_a] :
~ ( member3071122053849602553od_a_a @ A @ bot_bo4436838304982128028od_a_a ) ).
% emptyE
thf(fact_644_emptyE,axiom,
! [A: set_a] :
~ ( member_set_a @ A @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_645_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_646_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_647_emptyE,axiom,
! [A: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ A @ bot_bo3357376287454694259od_a_a ) ).
% emptyE
thf(fact_648_bot_Oextremum,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% bot.extremum
thf(fact_649_bot_Oextremum,axiom,
! [A: set_Pr5530083903271594800od_a_a] : ( ord_le114883831454073552od_a_a @ bot_bo4436838304982128028od_a_a @ A ) ).
% bot.extremum
thf(fact_650_bot_Oextremum,axiom,
! [A: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ bot_bo3357376287454694259od_a_a @ A ) ).
% bot.extremum
thf(fact_651_bot_Oextremum,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A ) ).
% bot.extremum
thf(fact_652_bot_Oextremum,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% bot.extremum
thf(fact_653_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_654_bot_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_655_bot_Oextremum__unique,axiom,
! [A: set_Pr5530083903271594800od_a_a] :
( ( ord_le114883831454073552od_a_a @ A @ bot_bo4436838304982128028od_a_a )
= ( A = bot_bo4436838304982128028od_a_a ) ) ).
% bot.extremum_unique
thf(fact_656_bot_Oextremum__unique,axiom,
! [A: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ bot_bo3357376287454694259od_a_a )
= ( A = bot_bo3357376287454694259od_a_a ) ) ).
% bot.extremum_unique
thf(fact_657_bot_Oextremum__unique,axiom,
! [A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ bot_bot_set_set_a )
= ( A = bot_bot_set_set_a ) ) ).
% bot.extremum_unique
thf(fact_658_bot_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_659_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_660_bot_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
=> ( A = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_661_bot_Oextremum__uniqueI,axiom,
! [A: set_Pr5530083903271594800od_a_a] :
( ( ord_le114883831454073552od_a_a @ A @ bot_bo4436838304982128028od_a_a )
=> ( A = bot_bo4436838304982128028od_a_a ) ) ).
% bot.extremum_uniqueI
thf(fact_662_bot_Oextremum__uniqueI,axiom,
! [A: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ bot_bo3357376287454694259od_a_a )
=> ( A = bot_bo3357376287454694259od_a_a ) ) ).
% bot.extremum_uniqueI
thf(fact_663_bot_Oextremum__uniqueI,axiom,
! [A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ bot_bot_set_set_a )
=> ( A = bot_bot_set_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_664_bot_Oextremum__uniqueI,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
=> ( A = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_665_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_666_Union__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% Union_empty
thf(fact_667_Union__empty,axiom,
( ( comple2978350343072902813od_a_a @ bot_bo5623065384236484604od_a_a )
= bot_bo4436838304982128028od_a_a ) ).
% Union_empty
thf(fact_668_Union__empty,axiom,
( ( comple8421679170691845492od_a_a @ bot_bo777872063958040403od_a_a )
= bot_bo3357376287454694259od_a_a ) ).
% Union_empty
thf(fact_669_Union__empty,axiom,
( ( comple3958522678809307947_set_a @ bot_bo3380559777022489994_set_a )
= bot_bot_set_set_a ) ).
% Union_empty
thf(fact_670_Union__empty,axiom,
( ( comple2307003609928055243_set_a @ bot_bot_set_set_a )
= bot_bot_set_a ) ).
% Union_empty
thf(fact_671_Union__empty,axiom,
( ( comple90263536869209701_set_o @ bot_bot_set_set_o )
= bot_bot_set_o ) ).
% Union_empty
thf(fact_672_subset__emptyI,axiom,
! [A2: set_o] :
( ! [X3: $o] :
~ ( member_o @ X3 @ A2 )
=> ( ord_less_eq_set_o @ A2 @ bot_bot_set_o ) ) ).
% subset_emptyI
thf(fact_673_subset__emptyI,axiom,
! [A2: set_nat] :
( ! [X3: nat] :
~ ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_674_subset__emptyI,axiom,
! [A2: set_Pr5530083903271594800od_a_a] :
( ! [X3: produc4044097585999906000od_a_a] :
~ ( member3071122053849602553od_a_a @ X3 @ A2 )
=> ( ord_le114883831454073552od_a_a @ A2 @ bot_bo4436838304982128028od_a_a ) ) ).
% subset_emptyI
thf(fact_675_subset__emptyI,axiom,
! [A2: set_Product_prod_a_a] :
( ! [X3: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ X3 @ A2 )
=> ( ord_le746702958409616551od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) ) ).
% subset_emptyI
thf(fact_676_subset__emptyI,axiom,
! [A2: set_set_a] :
( ! [X3: set_a] :
~ ( member_set_a @ X3 @ A2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ bot_bot_set_set_a ) ) ).
% subset_emptyI
thf(fact_677_subset__emptyI,axiom,
! [A2: set_a] :
( ! [X3: a] :
~ ( member_a @ X3 @ A2 )
=> ( ord_less_eq_set_a @ A2 @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_678_SUP__empty,axiom,
! [F: a > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_a_set_nat @ F @ bot_bot_set_a ) )
= bot_bot_set_nat ) ).
% SUP_empty
thf(fact_679_SUP__empty,axiom,
! [F: nat > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ bot_bot_set_nat ) )
= bot_bot_set_nat ) ).
% SUP_empty
thf(fact_680_SUP__empty,axiom,
! [F: a > set_a] :
( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ bot_bot_set_a ) )
= bot_bot_set_a ) ).
% SUP_empty
thf(fact_681_SUP__empty,axiom,
! [F: nat > set_a] :
( ( comple2307003609928055243_set_a @ ( image_nat_set_a @ F @ bot_bot_set_nat ) )
= bot_bot_set_a ) ).
% SUP_empty
thf(fact_682_SUP__empty,axiom,
! [F: a > set_o] :
( ( comple90263536869209701_set_o @ ( image_a_set_o @ F @ bot_bot_set_a ) )
= bot_bot_set_o ) ).
% SUP_empty
thf(fact_683_SUP__empty,axiom,
! [F: nat > set_o] :
( ( comple90263536869209701_set_o @ ( image_nat_set_o @ F @ bot_bot_set_nat ) )
= bot_bot_set_o ) ).
% SUP_empty
thf(fact_684_SUP__empty,axiom,
! [F: set_a > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_set_a_set_nat @ F @ bot_bot_set_set_a ) )
= bot_bot_set_nat ) ).
% SUP_empty
thf(fact_685_SUP__empty,axiom,
! [F: a > set_set_a] :
( ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ F @ bot_bot_set_a ) )
= bot_bot_set_set_a ) ).
% SUP_empty
thf(fact_686_SUP__empty,axiom,
! [F: nat > set_set_a] :
( ( comple3958522678809307947_set_a @ ( image_nat_set_set_a @ F @ bot_bot_set_nat ) )
= bot_bot_set_set_a ) ).
% SUP_empty
thf(fact_687_SUP__empty,axiom,
! [F: set_a > set_a] :
( ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ F @ bot_bot_set_set_a ) )
= bot_bot_set_a ) ).
% SUP_empty
thf(fact_688_SUP__constant,axiom,
! [A2: set_a,C2: set_nat] :
( ( ( A2 = bot_bot_set_a )
=> ( ( comple7399068483239264473et_nat
@ ( image_a_set_nat
@ ^ [Y2: a] : C2
@ A2 ) )
= bot_bot_set_nat ) )
& ( ( A2 != bot_bot_set_a )
=> ( ( comple7399068483239264473et_nat
@ ( image_a_set_nat
@ ^ [Y2: a] : C2
@ A2 ) )
= C2 ) ) ) ).
% SUP_constant
thf(fact_689_SUP__constant,axiom,
! [A2: set_nat,C2: set_nat] :
( ( ( A2 = bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y2: nat] : C2
@ A2 ) )
= bot_bot_set_nat ) )
& ( ( A2 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y2: nat] : C2
@ A2 ) )
= C2 ) ) ) ).
% SUP_constant
thf(fact_690_SUP__constant,axiom,
! [A2: set_a,C2: set_a] :
( ( ( A2 = bot_bot_set_a )
=> ( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [Y2: a] : C2
@ A2 ) )
= bot_bot_set_a ) )
& ( ( A2 != bot_bot_set_a )
=> ( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [Y2: a] : C2
@ A2 ) )
= C2 ) ) ) ).
% SUP_constant
thf(fact_691_SUP__constant,axiom,
! [A2: set_nat,C2: set_a] :
( ( ( A2 = bot_bot_set_nat )
=> ( ( comple2307003609928055243_set_a
@ ( image_nat_set_a
@ ^ [Y2: nat] : C2
@ A2 ) )
= bot_bot_set_a ) )
& ( ( A2 != bot_bot_set_nat )
=> ( ( comple2307003609928055243_set_a
@ ( image_nat_set_a
@ ^ [Y2: nat] : C2
@ A2 ) )
= C2 ) ) ) ).
% SUP_constant
thf(fact_692_SUP__constant,axiom,
! [A2: set_a,C2: set_o] :
( ( ( A2 = bot_bot_set_a )
=> ( ( comple90263536869209701_set_o
@ ( image_a_set_o
@ ^ [Y2: a] : C2
@ A2 ) )
= bot_bot_set_o ) )
& ( ( A2 != bot_bot_set_a )
=> ( ( comple90263536869209701_set_o
@ ( image_a_set_o
@ ^ [Y2: a] : C2
@ A2 ) )
= C2 ) ) ) ).
% SUP_constant
thf(fact_693_SUP__constant,axiom,
! [A2: set_nat,C2: set_o] :
( ( ( A2 = bot_bot_set_nat )
=> ( ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [Y2: nat] : C2
@ A2 ) )
= bot_bot_set_o ) )
& ( ( A2 != bot_bot_set_nat )
=> ( ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [Y2: nat] : C2
@ A2 ) )
= C2 ) ) ) ).
% SUP_constant
thf(fact_694_SUP__constant,axiom,
! [A2: set_set_a,C2: set_nat] :
( ( ( A2 = bot_bot_set_set_a )
=> ( ( comple7399068483239264473et_nat
@ ( image_set_a_set_nat
@ ^ [Y2: set_a] : C2
@ A2 ) )
= bot_bot_set_nat ) )
& ( ( A2 != bot_bot_set_set_a )
=> ( ( comple7399068483239264473et_nat
@ ( image_set_a_set_nat
@ ^ [Y2: set_a] : C2
@ A2 ) )
= C2 ) ) ) ).
% SUP_constant
thf(fact_695_SUP__constant,axiom,
! [A2: set_a,C2: set_set_a] :
( ( ( A2 = bot_bot_set_a )
=> ( ( comple3958522678809307947_set_a
@ ( image_a_set_set_a
@ ^ [Y2: a] : C2
@ A2 ) )
= bot_bot_set_set_a ) )
& ( ( A2 != bot_bot_set_a )
=> ( ( comple3958522678809307947_set_a
@ ( image_a_set_set_a
@ ^ [Y2: a] : C2
@ A2 ) )
= C2 ) ) ) ).
% SUP_constant
thf(fact_696_SUP__constant,axiom,
! [A2: set_nat,C2: set_set_a] :
( ( ( A2 = bot_bot_set_nat )
=> ( ( comple3958522678809307947_set_a
@ ( image_nat_set_set_a
@ ^ [Y2: nat] : C2
@ A2 ) )
= bot_bot_set_set_a ) )
& ( ( A2 != bot_bot_set_nat )
=> ( ( comple3958522678809307947_set_a
@ ( image_nat_set_set_a
@ ^ [Y2: nat] : C2
@ A2 ) )
= C2 ) ) ) ).
% SUP_constant
thf(fact_697_SUP__constant,axiom,
! [A2: set_set_a,C2: set_a] :
( ( ( A2 = bot_bot_set_set_a )
=> ( ( comple2307003609928055243_set_a
@ ( image_set_a_set_a
@ ^ [Y2: set_a] : C2
@ A2 ) )
= bot_bot_set_a ) )
& ( ( A2 != bot_bot_set_set_a )
=> ( ( comple2307003609928055243_set_a
@ ( image_set_a_set_a
@ ^ [Y2: set_a] : C2
@ A2 ) )
= C2 ) ) ) ).
% SUP_constant
thf(fact_698_Union__empty__conv,axiom,
! [A2: set_set_nat] :
( ( ( comple7399068483239264473et_nat @ A2 )
= bot_bot_set_nat )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( X2 = bot_bot_set_nat ) ) ) ) ).
% Union_empty_conv
thf(fact_699_Union__empty__conv,axiom,
! [A2: set_se9027383378080648592od_a_a] :
( ( ( comple2978350343072902813od_a_a @ A2 )
= bot_bo4436838304982128028od_a_a )
= ( ! [X2: set_Pr5530083903271594800od_a_a] :
( ( member4210947715425868889od_a_a @ X2 @ A2 )
=> ( X2 = bot_bo4436838304982128028od_a_a ) ) ) ) ).
% Union_empty_conv
thf(fact_700_Union__empty__conv,axiom,
! [A2: set_se5735800977113168103od_a_a] :
( ( ( comple8421679170691845492od_a_a @ A2 )
= bot_bo3357376287454694259od_a_a )
= ( ! [X2: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X2 @ A2 )
=> ( X2 = bot_bo3357376287454694259od_a_a ) ) ) ) ).
% Union_empty_conv
thf(fact_701_Union__empty__conv,axiom,
! [A2: set_set_set_a] :
( ( ( comple3958522678809307947_set_a @ A2 )
= bot_bot_set_set_a )
= ( ! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ A2 )
=> ( X2 = bot_bot_set_set_a ) ) ) ) ).
% Union_empty_conv
thf(fact_702_Union__empty__conv,axiom,
! [A2: set_set_a] :
( ( ( comple2307003609928055243_set_a @ A2 )
= bot_bot_set_a )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( X2 = bot_bot_set_a ) ) ) ) ).
% Union_empty_conv
thf(fact_703_Union__empty__conv,axiom,
! [A2: set_set_o] :
( ( ( comple90263536869209701_set_o @ A2 )
= bot_bot_set_o )
= ( ! [X2: set_o] :
( ( member_set_o @ X2 @ A2 )
=> ( X2 = bot_bot_set_o ) ) ) ) ).
% Union_empty_conv
thf(fact_704_empty__Union__conv,axiom,
! [A2: set_set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ A2 ) )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( X2 = bot_bot_set_nat ) ) ) ) ).
% empty_Union_conv
thf(fact_705_empty__Union__conv,axiom,
! [A2: set_se9027383378080648592od_a_a] :
( ( bot_bo4436838304982128028od_a_a
= ( comple2978350343072902813od_a_a @ A2 ) )
= ( ! [X2: set_Pr5530083903271594800od_a_a] :
( ( member4210947715425868889od_a_a @ X2 @ A2 )
=> ( X2 = bot_bo4436838304982128028od_a_a ) ) ) ) ).
% empty_Union_conv
thf(fact_706_empty__Union__conv,axiom,
! [A2: set_se5735800977113168103od_a_a] :
( ( bot_bo3357376287454694259od_a_a
= ( comple8421679170691845492od_a_a @ A2 ) )
= ( ! [X2: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X2 @ A2 )
=> ( X2 = bot_bo3357376287454694259od_a_a ) ) ) ) ).
% empty_Union_conv
thf(fact_707_empty__Union__conv,axiom,
! [A2: set_set_set_a] :
( ( bot_bot_set_set_a
= ( comple3958522678809307947_set_a @ A2 ) )
= ( ! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ A2 )
=> ( X2 = bot_bot_set_set_a ) ) ) ) ).
% empty_Union_conv
thf(fact_708_empty__Union__conv,axiom,
! [A2: set_set_a] :
( ( bot_bot_set_a
= ( comple2307003609928055243_set_a @ A2 ) )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( X2 = bot_bot_set_a ) ) ) ) ).
% empty_Union_conv
thf(fact_709_empty__Union__conv,axiom,
! [A2: set_set_o] :
( ( bot_bot_set_o
= ( comple90263536869209701_set_o @ A2 ) )
= ( ! [X2: set_o] :
( ( member_set_o @ X2 @ A2 )
=> ( X2 = bot_bot_set_o ) ) ) ) ).
% empty_Union_conv
thf(fact_710_less__eq__Sup,axiom,
! [A2: set_o,U: $o] :
( ! [V2: $o] :
( ( member_o @ V2 @ A2 )
=> ( ord_less_eq_o @ U @ V2 ) )
=> ( ( A2 != bot_bot_set_o )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ A2 ) ) ) ) ).
% less_eq_Sup
thf(fact_711_less__eq__Sup,axiom,
! [A2: set_se9027383378080648592od_a_a,U: set_Pr5530083903271594800od_a_a] :
( ! [V2: set_Pr5530083903271594800od_a_a] :
( ( member4210947715425868889od_a_a @ V2 @ A2 )
=> ( ord_le114883831454073552od_a_a @ U @ V2 ) )
=> ( ( A2 != bot_bo5623065384236484604od_a_a )
=> ( ord_le114883831454073552od_a_a @ U @ ( comple2978350343072902813od_a_a @ A2 ) ) ) ) ).
% less_eq_Sup
thf(fact_712_less__eq__Sup,axiom,
! [A2: set_se5735800977113168103od_a_a,U: set_Product_prod_a_a] :
( ! [V2: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ V2 @ A2 )
=> ( ord_le746702958409616551od_a_a @ U @ V2 ) )
=> ( ( A2 != bot_bo777872063958040403od_a_a )
=> ( ord_le746702958409616551od_a_a @ U @ ( comple8421679170691845492od_a_a @ A2 ) ) ) ) ).
% less_eq_Sup
thf(fact_713_less__eq__Sup,axiom,
! [A2: set_set_set_a,U: set_set_a] :
( ! [V2: set_set_a] :
( ( member_set_set_a @ V2 @ A2 )
=> ( ord_le3724670747650509150_set_a @ U @ V2 ) )
=> ( ( A2 != bot_bo3380559777022489994_set_a )
=> ( ord_le3724670747650509150_set_a @ U @ ( comple3958522678809307947_set_a @ A2 ) ) ) ) ).
% less_eq_Sup
thf(fact_714_less__eq__Sup,axiom,
! [A2: set_set_a,U: set_a] :
( ! [V2: set_a] :
( ( member_set_a @ V2 @ A2 )
=> ( ord_less_eq_set_a @ U @ V2 ) )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ U @ ( comple2307003609928055243_set_a @ A2 ) ) ) ) ).
% less_eq_Sup
thf(fact_715_less__eq__Sup,axiom,
! [A2: set_set_o,U: set_o] :
( ! [V2: set_o] :
( ( member_set_o @ V2 @ A2 )
=> ( ord_less_eq_set_o @ U @ V2 ) )
=> ( ( A2 != bot_bot_set_set_o )
=> ( ord_less_eq_set_o @ U @ ( comple90263536869209701_set_o @ A2 ) ) ) ) ).
% less_eq_Sup
thf(fact_716_SUP__eq__const,axiom,
! [I5: set_o,F: $o > set_a,X: set_a] :
( ( I5 != bot_bot_set_o )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ I5 )
=> ( ( F @ I2 )
= X ) )
=> ( ( comple2307003609928055243_set_a @ ( image_o_set_a @ F @ I5 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_717_SUP__eq__const,axiom,
! [I5: set_a,F: a > set_a,X: set_a] :
( ( I5 != bot_bot_set_a )
=> ( ! [I2: a] :
( ( member_a @ I2 @ I5 )
=> ( ( F @ I2 )
= X ) )
=> ( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ I5 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_718_SUP__eq__const,axiom,
! [I5: set_nat,F: nat > set_a,X: set_a] :
( ( I5 != bot_bot_set_nat )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I5 )
=> ( ( F @ I2 )
= X ) )
=> ( ( comple2307003609928055243_set_a @ ( image_nat_set_a @ F @ I5 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_719_SUP__eq__const,axiom,
! [I5: set_o,F: $o > set_o,X: set_o] :
( ( I5 != bot_bot_set_o )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ I5 )
=> ( ( F @ I2 )
= X ) )
=> ( ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ I5 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_720_SUP__eq__const,axiom,
! [I5: set_a,F: a > set_o,X: set_o] :
( ( I5 != bot_bot_set_a )
=> ( ! [I2: a] :
( ( member_a @ I2 @ I5 )
=> ( ( F @ I2 )
= X ) )
=> ( ( comple90263536869209701_set_o @ ( image_a_set_o @ F @ I5 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_721_SUP__eq__const,axiom,
! [I5: set_nat,F: nat > set_o,X: set_o] :
( ( I5 != bot_bot_set_nat )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I5 )
=> ( ( F @ I2 )
= X ) )
=> ( ( comple90263536869209701_set_o @ ( image_nat_set_o @ F @ I5 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_722_SUP__eq__const,axiom,
! [I5: set_o,F: $o > set_set_a,X: set_set_a] :
( ( I5 != bot_bot_set_o )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ I5 )
=> ( ( F @ I2 )
= X ) )
=> ( ( comple3958522678809307947_set_a @ ( image_o_set_set_a @ F @ I5 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_723_SUP__eq__const,axiom,
! [I5: set_a,F: a > set_set_a,X: set_set_a] :
( ( I5 != bot_bot_set_a )
=> ( ! [I2: a] :
( ( member_a @ I2 @ I5 )
=> ( ( F @ I2 )
= X ) )
=> ( ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ F @ I5 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_724_SUP__eq__const,axiom,
! [I5: set_nat,F: nat > set_set_a,X: set_set_a] :
( ( I5 != bot_bot_set_nat )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I5 )
=> ( ( F @ I2 )
= X ) )
=> ( ( comple3958522678809307947_set_a @ ( image_nat_set_set_a @ F @ I5 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_725_SUP__eq__const,axiom,
! [I5: set_set_a,F: set_a > set_a,X: set_a] :
( ( I5 != bot_bot_set_set_a )
=> ( ! [I2: set_a] :
( ( member_set_a @ I2 @ I5 )
=> ( ( F @ I2 )
= X ) )
=> ( ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ F @ I5 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_726_UNION__empty__conv_I2_J,axiom,
! [B2: set_a > set_Pr5530083903271594800od_a_a,A2: set_set_a] :
( ( ( comple2978350343072902813od_a_a @ ( image_7562202058474640471od_a_a @ B2 @ A2 ) )
= bot_bo4436838304982128028od_a_a )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( ( B2 @ X2 )
= bot_bo4436838304982128028od_a_a ) ) ) ) ).
% UNION_empty_conv(2)
thf(fact_727_UNION__empty__conv_I2_J,axiom,
! [B2: a > set_Pr5530083903271594800od_a_a,A2: set_a] :
( ( ( comple2978350343072902813od_a_a @ ( image_5653227685612666295od_a_a @ B2 @ A2 ) )
= bot_bo4436838304982128028od_a_a )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( B2 @ X2 )
= bot_bo4436838304982128028od_a_a ) ) ) ) ).
% UNION_empty_conv(2)
thf(fact_728_UNION__empty__conv_I2_J,axiom,
! [B2: set_a > set_Product_prod_a_a,A2: set_set_a] :
( ( ( comple8421679170691845492od_a_a @ ( image_6165024369500519726od_a_a @ B2 @ A2 ) )
= bot_bo3357376287454694259od_a_a )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( ( B2 @ X2 )
= bot_bo3357376287454694259od_a_a ) ) ) ) ).
% UNION_empty_conv(2)
thf(fact_729_UNION__empty__conv_I2_J,axiom,
! [B2: a > set_Product_prod_a_a,A2: set_a] :
( ( ( comple8421679170691845492od_a_a @ ( image_4421510592991446670od_a_a @ B2 @ A2 ) )
= bot_bo3357376287454694259od_a_a )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( B2 @ X2 )
= bot_bo3357376287454694259od_a_a ) ) ) ) ).
% UNION_empty_conv(2)
thf(fact_730_UNION__empty__conv_I2_J,axiom,
! [B2: a > set_set_a,A2: set_a] :
( ( ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ B2 @ A2 ) )
= bot_bot_set_set_a )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( B2 @ X2 )
= bot_bot_set_set_a ) ) ) ) ).
% UNION_empty_conv(2)
thf(fact_731_UNION__empty__conv_I2_J,axiom,
! [B2: a > set_a,A2: set_a] :
( ( ( comple2307003609928055243_set_a @ ( image_a_set_a @ B2 @ A2 ) )
= bot_bot_set_a )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( B2 @ X2 )
= bot_bot_set_a ) ) ) ) ).
% UNION_empty_conv(2)
thf(fact_732_UNION__empty__conv_I2_J,axiom,
! [B2: set_a > set_a,A2: set_set_a] :
( ( ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ B2 @ A2 ) )
= bot_bot_set_a )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( ( B2 @ X2 )
= bot_bot_set_a ) ) ) ) ).
% UNION_empty_conv(2)
thf(fact_733_UNION__empty__conv_I2_J,axiom,
! [B2: a > set_o,A2: set_a] :
( ( ( comple90263536869209701_set_o @ ( image_a_set_o @ B2 @ A2 ) )
= bot_bot_set_o )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( B2 @ X2 )
= bot_bot_set_o ) ) ) ) ).
% UNION_empty_conv(2)
thf(fact_734_UNION__empty__conv_I1_J,axiom,
! [B2: set_a > set_Pr5530083903271594800od_a_a,A2: set_set_a] :
( ( bot_bo4436838304982128028od_a_a
= ( comple2978350343072902813od_a_a @ ( image_7562202058474640471od_a_a @ B2 @ A2 ) ) )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( ( B2 @ X2 )
= bot_bo4436838304982128028od_a_a ) ) ) ) ).
% UNION_empty_conv(1)
thf(fact_735_UNION__empty__conv_I1_J,axiom,
! [B2: a > set_Pr5530083903271594800od_a_a,A2: set_a] :
( ( bot_bo4436838304982128028od_a_a
= ( comple2978350343072902813od_a_a @ ( image_5653227685612666295od_a_a @ B2 @ A2 ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( B2 @ X2 )
= bot_bo4436838304982128028od_a_a ) ) ) ) ).
% UNION_empty_conv(1)
thf(fact_736_UNION__empty__conv_I1_J,axiom,
! [B2: set_a > set_Product_prod_a_a,A2: set_set_a] :
( ( bot_bo3357376287454694259od_a_a
= ( comple8421679170691845492od_a_a @ ( image_6165024369500519726od_a_a @ B2 @ A2 ) ) )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( ( B2 @ X2 )
= bot_bo3357376287454694259od_a_a ) ) ) ) ).
% UNION_empty_conv(1)
thf(fact_737_UNION__empty__conv_I1_J,axiom,
! [B2: a > set_Product_prod_a_a,A2: set_a] :
( ( bot_bo3357376287454694259od_a_a
= ( comple8421679170691845492od_a_a @ ( image_4421510592991446670od_a_a @ B2 @ A2 ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( B2 @ X2 )
= bot_bo3357376287454694259od_a_a ) ) ) ) ).
% UNION_empty_conv(1)
thf(fact_738_UNION__empty__conv_I1_J,axiom,
! [B2: a > set_set_a,A2: set_a] :
( ( bot_bot_set_set_a
= ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ B2 @ A2 ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( B2 @ X2 )
= bot_bot_set_set_a ) ) ) ) ).
% UNION_empty_conv(1)
thf(fact_739_UNION__empty__conv_I1_J,axiom,
! [B2: a > set_a,A2: set_a] :
( ( bot_bot_set_a
= ( comple2307003609928055243_set_a @ ( image_a_set_a @ B2 @ A2 ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( B2 @ X2 )
= bot_bot_set_a ) ) ) ) ).
% UNION_empty_conv(1)
thf(fact_740_UNION__empty__conv_I1_J,axiom,
! [B2: set_a > set_a,A2: set_set_a] :
( ( bot_bot_set_a
= ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ B2 @ A2 ) ) )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( ( B2 @ X2 )
= bot_bot_set_a ) ) ) ) ).
% UNION_empty_conv(1)
thf(fact_741_UNION__empty__conv_I1_J,axiom,
! [B2: a > set_o,A2: set_a] :
( ( bot_bot_set_o
= ( comple90263536869209701_set_o @ ( image_a_set_o @ B2 @ A2 ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( B2 @ X2 )
= bot_bot_set_o ) ) ) ) ).
% UNION_empty_conv(1)
thf(fact_742_UN__empty,axiom,
! [B2: a > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_a_set_nat @ B2 @ bot_bot_set_a ) )
= bot_bot_set_nat ) ).
% UN_empty
thf(fact_743_UN__empty,axiom,
! [B2: nat > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ bot_bot_set_nat ) )
= bot_bot_set_nat ) ).
% UN_empty
thf(fact_744_UN__empty,axiom,
! [B2: a > set_a] :
( ( comple2307003609928055243_set_a @ ( image_a_set_a @ B2 @ bot_bot_set_a ) )
= bot_bot_set_a ) ).
% UN_empty
thf(fact_745_UN__empty,axiom,
! [B2: nat > set_a] :
( ( comple2307003609928055243_set_a @ ( image_nat_set_a @ B2 @ bot_bot_set_nat ) )
= bot_bot_set_a ) ).
% UN_empty
thf(fact_746_UN__empty,axiom,
! [B2: a > set_o] :
( ( comple90263536869209701_set_o @ ( image_a_set_o @ B2 @ bot_bot_set_a ) )
= bot_bot_set_o ) ).
% UN_empty
thf(fact_747_UN__empty,axiom,
! [B2: nat > set_o] :
( ( comple90263536869209701_set_o @ ( image_nat_set_o @ B2 @ bot_bot_set_nat ) )
= bot_bot_set_o ) ).
% UN_empty
thf(fact_748_UN__empty,axiom,
! [B2: set_a > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_set_a_set_nat @ B2 @ bot_bot_set_set_a ) )
= bot_bot_set_nat ) ).
% UN_empty
thf(fact_749_UN__empty,axiom,
! [B2: a > set_set_a] :
( ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ B2 @ bot_bot_set_a ) )
= bot_bot_set_set_a ) ).
% UN_empty
thf(fact_750_UN__empty,axiom,
! [B2: nat > set_set_a] :
( ( comple3958522678809307947_set_a @ ( image_nat_set_set_a @ B2 @ bot_bot_set_nat ) )
= bot_bot_set_set_a ) ).
% UN_empty
thf(fact_751_UN__empty,axiom,
! [B2: set_a > set_a] :
( ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ B2 @ bot_bot_set_set_a ) )
= bot_bot_set_a ) ).
% UN_empty
thf(fact_752_UN__empty2,axiom,
! [A2: set_set_a] :
( ( comple2978350343072902813od_a_a
@ ( image_7562202058474640471od_a_a
@ ^ [X2: set_a] : bot_bo4436838304982128028od_a_a
@ A2 ) )
= bot_bo4436838304982128028od_a_a ) ).
% UN_empty2
thf(fact_753_UN__empty2,axiom,
! [A2: set_a] :
( ( comple2978350343072902813od_a_a
@ ( image_5653227685612666295od_a_a
@ ^ [X2: a] : bot_bo4436838304982128028od_a_a
@ A2 ) )
= bot_bo4436838304982128028od_a_a ) ).
% UN_empty2
thf(fact_754_UN__empty2,axiom,
! [A2: set_set_a] :
( ( comple8421679170691845492od_a_a
@ ( image_6165024369500519726od_a_a
@ ^ [X2: set_a] : bot_bo3357376287454694259od_a_a
@ A2 ) )
= bot_bo3357376287454694259od_a_a ) ).
% UN_empty2
thf(fact_755_UN__empty2,axiom,
! [A2: set_a] :
( ( comple8421679170691845492od_a_a
@ ( image_4421510592991446670od_a_a
@ ^ [X2: a] : bot_bo3357376287454694259od_a_a
@ A2 ) )
= bot_bo3357376287454694259od_a_a ) ).
% UN_empty2
thf(fact_756_UN__empty2,axiom,
! [A2: set_a] :
( ( comple3958522678809307947_set_a
@ ( image_a_set_set_a
@ ^ [X2: a] : bot_bot_set_set_a
@ A2 ) )
= bot_bot_set_set_a ) ).
% UN_empty2
thf(fact_757_UN__empty2,axiom,
! [A2: set_a] :
( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X2: a] : bot_bot_set_a
@ A2 ) )
= bot_bot_set_a ) ).
% UN_empty2
thf(fact_758_UN__empty2,axiom,
! [A2: set_set_a] :
( ( comple2307003609928055243_set_a
@ ( image_set_a_set_a
@ ^ [X2: set_a] : bot_bot_set_a
@ A2 ) )
= bot_bot_set_a ) ).
% UN_empty2
thf(fact_759_UN__empty2,axiom,
! [A2: set_a] :
( ( comple90263536869209701_set_o
@ ( image_a_set_o
@ ^ [X2: a] : bot_bot_set_o
@ A2 ) )
= bot_bot_set_o ) ).
% UN_empty2
thf(fact_760_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_761_le__cases3,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_762_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_Pr5530083903271594800od_a_a,Z3: set_Pr5530083903271594800od_a_a] : ( Y6 = Z3 ) )
= ( ^ [X2: set_Pr5530083903271594800od_a_a,Y2: set_Pr5530083903271594800od_a_a] :
( ( ord_le114883831454073552od_a_a @ X2 @ Y2 )
& ( ord_le114883831454073552od_a_a @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_763_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_Product_prod_a_a,Z3: set_Product_prod_a_a] : ( Y6 = Z3 ) )
= ( ^ [X2: set_Product_prod_a_a,Y2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X2 @ Y2 )
& ( ord_le746702958409616551od_a_a @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_764_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_set_a,Z3: set_set_a] : ( Y6 = Z3 ) )
= ( ^ [X2: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y2 )
& ( ord_le3724670747650509150_set_a @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_765_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_a,Z3: set_a] : ( Y6 = Z3 ) )
= ( ^ [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
& ( ord_less_eq_set_a @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_766_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
& ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_767_ord__eq__le__trans,axiom,
! [A: set_Pr5530083903271594800od_a_a,B: set_Pr5530083903271594800od_a_a,C2: set_Pr5530083903271594800od_a_a] :
( ( A = B )
=> ( ( ord_le114883831454073552od_a_a @ B @ C2 )
=> ( ord_le114883831454073552od_a_a @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_768_ord__eq__le__trans,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( A = B )
=> ( ( ord_le746702958409616551od_a_a @ B @ C2 )
=> ( ord_le746702958409616551od_a_a @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_769_ord__eq__le__trans,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( A = B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C2 )
=> ( ord_le3724670747650509150_set_a @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_770_ord__eq__le__trans,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( A = B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_771_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_772_ord__le__eq__trans,axiom,
! [A: set_Pr5530083903271594800od_a_a,B: set_Pr5530083903271594800od_a_a,C2: set_Pr5530083903271594800od_a_a] :
( ( ord_le114883831454073552od_a_a @ A @ B )
=> ( ( B = C2 )
=> ( ord_le114883831454073552od_a_a @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_773_ord__le__eq__trans,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( ( B = C2 )
=> ( ord_le746702958409616551od_a_a @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_774_ord__le__eq__trans,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( B = C2 )
=> ( ord_le3724670747650509150_set_a @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_775_ord__le__eq__trans,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_776_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_777_order__antisym,axiom,
! [X: set_Pr5530083903271594800od_a_a,Y: set_Pr5530083903271594800od_a_a] :
( ( ord_le114883831454073552od_a_a @ X @ Y )
=> ( ( ord_le114883831454073552od_a_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_778_order__antisym,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X @ Y )
=> ( ( ord_le746702958409616551od_a_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_779_order__antisym,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y )
=> ( ( ord_le3724670747650509150_set_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_780_order__antisym,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_781_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_782_order_Otrans,axiom,
! [A: set_Pr5530083903271594800od_a_a,B: set_Pr5530083903271594800od_a_a,C2: set_Pr5530083903271594800od_a_a] :
( ( ord_le114883831454073552od_a_a @ A @ B )
=> ( ( ord_le114883831454073552od_a_a @ B @ C2 )
=> ( ord_le114883831454073552od_a_a @ A @ C2 ) ) ) ).
% order.trans
thf(fact_783_order_Otrans,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( ( ord_le746702958409616551od_a_a @ B @ C2 )
=> ( ord_le746702958409616551od_a_a @ A @ C2 ) ) ) ).
% order.trans
thf(fact_784_order_Otrans,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C2 )
=> ( ord_le3724670747650509150_set_a @ A @ C2 ) ) ) ).
% order.trans
thf(fact_785_order_Otrans,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% order.trans
thf(fact_786_order_Otrans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% order.trans
thf(fact_787_order__trans,axiom,
! [X: set_Pr5530083903271594800od_a_a,Y: set_Pr5530083903271594800od_a_a,Z: set_Pr5530083903271594800od_a_a] :
( ( ord_le114883831454073552od_a_a @ X @ Y )
=> ( ( ord_le114883831454073552od_a_a @ Y @ Z )
=> ( ord_le114883831454073552od_a_a @ X @ Z ) ) ) ).
% order_trans
thf(fact_788_order__trans,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X @ Y )
=> ( ( ord_le746702958409616551od_a_a @ Y @ Z )
=> ( ord_le746702958409616551od_a_a @ X @ Z ) ) ) ).
% order_trans
thf(fact_789_order__trans,axiom,
! [X: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y )
=> ( ( ord_le3724670747650509150_set_a @ Y @ Z )
=> ( ord_le3724670747650509150_set_a @ X @ Z ) ) ) ).
% order_trans
thf(fact_790_order__trans,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z )
=> ( ord_less_eq_set_a @ X @ Z ) ) ) ).
% order_trans
thf(fact_791_order__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_792_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
=> ( P @ A5 @ B4 ) )
=> ( ! [A5: nat,B4: nat] :
( ( P @ B4 @ A5 )
=> ( P @ A5 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_793_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: set_Pr5530083903271594800od_a_a,Z3: set_Pr5530083903271594800od_a_a] : ( Y6 = Z3 ) )
= ( ^ [A3: set_Pr5530083903271594800od_a_a,B3: set_Pr5530083903271594800od_a_a] :
( ( ord_le114883831454073552od_a_a @ B3 @ A3 )
& ( ord_le114883831454073552od_a_a @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_794_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: set_Product_prod_a_a,Z3: set_Product_prod_a_a] : ( Y6 = Z3 ) )
= ( ^ [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B3 @ A3 )
& ( ord_le746702958409616551od_a_a @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_795_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: set_set_a,Z3: set_set_a] : ( Y6 = Z3 ) )
= ( ^ [A3: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ A3 )
& ( ord_le3724670747650509150_set_a @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_796_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: set_a,Z3: set_a] : ( Y6 = Z3 ) )
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
& ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_797_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_798_dual__order_Oantisym,axiom,
! [B: set_Pr5530083903271594800od_a_a,A: set_Pr5530083903271594800od_a_a] :
( ( ord_le114883831454073552od_a_a @ B @ A )
=> ( ( ord_le114883831454073552od_a_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_799_dual__order_Oantisym,axiom,
! [B: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B @ A )
=> ( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_800_dual__order_Oantisym,axiom,
! [B: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_801_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_802_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_803_dual__order_Otrans,axiom,
! [B: set_Pr5530083903271594800od_a_a,A: set_Pr5530083903271594800od_a_a,C2: set_Pr5530083903271594800od_a_a] :
( ( ord_le114883831454073552od_a_a @ B @ A )
=> ( ( ord_le114883831454073552od_a_a @ C2 @ B )
=> ( ord_le114883831454073552od_a_a @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_804_dual__order_Otrans,axiom,
! [B: set_Product_prod_a_a,A: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B @ A )
=> ( ( ord_le746702958409616551od_a_a @ C2 @ B )
=> ( ord_le746702958409616551od_a_a @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_805_dual__order_Otrans,axiom,
! [B: set_set_a,A: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ( ord_le3724670747650509150_set_a @ C2 @ B )
=> ( ord_le3724670747650509150_set_a @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_806_dual__order_Otrans,axiom,
! [B: set_a,A: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C2 @ B )
=> ( ord_less_eq_set_a @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_807_dual__order_Otrans,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_808_antisym,axiom,
! [A: set_Pr5530083903271594800od_a_a,B: set_Pr5530083903271594800od_a_a] :
( ( ord_le114883831454073552od_a_a @ A @ B )
=> ( ( ord_le114883831454073552od_a_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_809_antisym,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( ( ord_le746702958409616551od_a_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_810_antisym,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_811_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_812_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_813_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_Pr5530083903271594800od_a_a,Z3: set_Pr5530083903271594800od_a_a] : ( Y6 = Z3 ) )
= ( ^ [A3: set_Pr5530083903271594800od_a_a,B3: set_Pr5530083903271594800od_a_a] :
( ( ord_le114883831454073552od_a_a @ A3 @ B3 )
& ( ord_le114883831454073552od_a_a @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_814_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_Product_prod_a_a,Z3: set_Product_prod_a_a] : ( Y6 = Z3 ) )
= ( ^ [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A3 @ B3 )
& ( ord_le746702958409616551od_a_a @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_815_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_set_a,Z3: set_set_a] : ( Y6 = Z3 ) )
= ( ^ [A3: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B3 )
& ( ord_le3724670747650509150_set_a @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_816_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_a,Z3: set_a] : ( Y6 = Z3 ) )
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_817_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_818_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_819_order__subst1,axiom,
! [A: set_a,F: nat > set_a,B: nat,C2: nat] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_820_order__subst1,axiom,
! [A: nat,F: set_a > nat,B: set_a,C2: set_a] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_821_order__subst1,axiom,
! [A: set_set_a,F: nat > set_set_a,B: nat,C2: nat] :
( ( ord_le3724670747650509150_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le3724670747650509150_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_822_order__subst1,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_823_order__subst1,axiom,
! [A: nat,F: set_set_a > nat,B: set_set_a,C2: set_set_a] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le3724670747650509150_set_a @ B @ C2 )
=> ( ! [X3: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_824_order__subst1,axiom,
! [A: set_Product_prod_a_a,F: nat > set_Product_prod_a_a,B: nat,C2: nat] :
( ( ord_le746702958409616551od_a_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le746702958409616551od_a_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le746702958409616551od_a_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_825_order__subst1,axiom,
! [A: set_set_a,F: set_a > set_set_a,B: set_a,C2: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le3724670747650509150_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_826_order__subst1,axiom,
! [A: set_a,F: set_set_a > set_a,B: set_set_a,C2: set_set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_le3724670747650509150_set_a @ B @ C2 )
=> ( ! [X3: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_827_order__subst1,axiom,
! [A: nat,F: set_Product_prod_a_a > nat,B: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le746702958409616551od_a_a @ B @ C2 )
=> ( ! [X3: set_Product_prod_a_a,Y3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_828_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_829_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C2: nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_830_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_a,C2: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_831_order__subst2,axiom,
! [A: set_set_a,B: set_set_a,F: set_set_a > nat,C2: nat] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X3: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_832_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_833_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_set_a,C2: set_set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ ( F @ B ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_834_order__subst2,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,F: set_Product_prod_a_a > nat,C2: nat] :
( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X3: set_Product_prod_a_a,Y3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_835_order__subst2,axiom,
! [A: set_set_a,B: set_set_a,F: set_set_a > set_a,C2: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C2 )
=> ( ! [X3: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_836_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_set_a,C2: set_set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ ( F @ B ) @ C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_837_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le746702958409616551od_a_a @ ( F @ B ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le746702958409616551od_a_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le746702958409616551od_a_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_838_order__eq__refl,axiom,
! [X: set_Pr5530083903271594800od_a_a,Y: set_Pr5530083903271594800od_a_a] :
( ( X = Y )
=> ( ord_le114883831454073552od_a_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_839_order__eq__refl,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( X = Y )
=> ( ord_le746702958409616551od_a_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_840_order__eq__refl,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( X = Y )
=> ( ord_le3724670747650509150_set_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_841_order__eq__refl,axiom,
! [X: set_a,Y: set_a] :
( ( X = Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_842_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_843_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_844_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_845_ord__eq__le__subst,axiom,
! [A: nat,F: set_a > nat,B: set_a,C2: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_846_ord__eq__le__subst,axiom,
! [A: set_a,F: nat > set_a,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_847_ord__eq__le__subst,axiom,
! [A: nat,F: set_set_a > nat,B: set_set_a,C2: set_set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le3724670747650509150_set_a @ B @ C2 )
=> ( ! [X3: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_848_ord__eq__le__subst,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C2: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_849_ord__eq__le__subst,axiom,
! [A: set_set_a,F: nat > set_set_a,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le3724670747650509150_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_850_ord__eq__le__subst,axiom,
! [A: nat,F: set_Product_prod_a_a > nat,B: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le746702958409616551od_a_a @ B @ C2 )
=> ( ! [X3: set_Product_prod_a_a,Y3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_851_ord__eq__le__subst,axiom,
! [A: set_a,F: set_set_a > set_a,B: set_set_a,C2: set_set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le3724670747650509150_set_a @ B @ C2 )
=> ( ! [X3: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_852_ord__eq__le__subst,axiom,
! [A: set_set_a,F: set_a > set_set_a,B: set_a,C2: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le3724670747650509150_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_853_ord__eq__le__subst,axiom,
! [A: set_Product_prod_a_a,F: nat > set_Product_prod_a_a,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le746702958409616551od_a_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le746702958409616551od_a_a @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_854_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_855_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C2: nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_856_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_a,C2: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_857_ord__le__eq__subst,axiom,
! [A: set_set_a,B: set_set_a,F: set_set_a > nat,C2: nat] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_858_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_859_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_set_a,C2: set_set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_860_ord__le__eq__subst,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,F: set_Product_prod_a_a > nat,C2: nat] :
( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: set_Product_prod_a_a,Y3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_861_ord__le__eq__subst,axiom,
! [A: set_set_a,B: set_set_a,F: set_set_a > set_a,C2: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_862_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > set_set_a,C2: set_set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_863_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le746702958409616551od_a_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le746702958409616551od_a_a @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_864_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_865_order__antisym__conv,axiom,
! [Y: set_Pr5530083903271594800od_a_a,X: set_Pr5530083903271594800od_a_a] :
( ( ord_le114883831454073552od_a_a @ Y @ X )
=> ( ( ord_le114883831454073552od_a_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_866_order__antisym__conv,axiom,
! [Y: set_Product_prod_a_a,X: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ Y @ X )
=> ( ( ord_le746702958409616551od_a_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_867_order__antisym__conv,axiom,
! [Y: set_set_a,X: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y @ X )
=> ( ( ord_le3724670747650509150_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_868_order__antisym__conv,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_869_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_870_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_871_imageI,axiom,
! [X: $o,A2: set_o,F: $o > $o] :
( ( member_o @ X @ A2 )
=> ( member_o @ ( F @ X ) @ ( image_o_o @ F @ A2 ) ) ) ).
% imageI
thf(fact_872_imageI,axiom,
! [X: $o,A2: set_o,F: $o > a] :
( ( member_o @ X @ A2 )
=> ( member_a @ ( F @ X ) @ ( image_o_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_873_imageI,axiom,
! [X: a,A2: set_a,F: a > $o] :
( ( member_a @ X @ A2 )
=> ( member_o @ ( F @ X ) @ ( image_a_o @ F @ A2 ) ) ) ).
% imageI
thf(fact_874_imageI,axiom,
! [X: a,A2: set_a,F: a > a] :
( ( member_a @ X @ A2 )
=> ( member_a @ ( F @ X ) @ ( image_a_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_875_imageI,axiom,
! [X: set_a,A2: set_set_a,F: set_a > $o] :
( ( member_set_a @ X @ A2 )
=> ( member_o @ ( F @ X ) @ ( image_set_a_o @ F @ A2 ) ) ) ).
% imageI
thf(fact_876_imageI,axiom,
! [X: set_a,A2: set_set_a,F: set_a > a] :
( ( member_set_a @ X @ A2 )
=> ( member_a @ ( F @ X ) @ ( image_set_a_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_877_imageI,axiom,
! [X: $o,A2: set_o,F: $o > set_a] :
( ( member_o @ X @ A2 )
=> ( member_set_a @ ( F @ X ) @ ( image_o_set_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_878_imageI,axiom,
! [X: a,A2: set_a,F: a > set_o] :
( ( member_a @ X @ A2 )
=> ( member_set_o @ ( F @ X ) @ ( image_a_set_o @ F @ A2 ) ) ) ).
% imageI
thf(fact_879_imageI,axiom,
! [X: a,A2: set_a,F: a > set_a] :
( ( member_a @ X @ A2 )
=> ( member_set_a @ ( F @ X ) @ ( image_a_set_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_880_image__iff,axiom,
! [Z: set_Pr5530083903271594800od_a_a,F: set_a > set_Pr5530083903271594800od_a_a,A2: set_set_a] :
( ( member4210947715425868889od_a_a @ Z @ ( image_7562202058474640471od_a_a @ F @ A2 ) )
= ( ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_881_image__iff,axiom,
! [Z: set_Product_prod_a_a,F: set_a > set_Product_prod_a_a,A2: set_set_a] :
( ( member1816616512716248880od_a_a @ Z @ ( image_6165024369500519726od_a_a @ F @ A2 ) )
= ( ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_882_image__iff,axiom,
! [Z: set_set_a,F: a > set_set_a,A2: set_a] :
( ( member_set_set_a @ Z @ ( image_a_set_set_a @ F @ A2 ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_883_image__iff,axiom,
! [Z: nat,F: nat > nat,A2: set_nat] :
( ( member_nat @ Z @ ( image_nat_nat @ F @ A2 ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_884_image__iff,axiom,
! [Z: set_Pr5530083903271594800od_a_a,F: a > set_Pr5530083903271594800od_a_a,A2: set_a] :
( ( member4210947715425868889od_a_a @ Z @ ( image_5653227685612666295od_a_a @ F @ A2 ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_885_image__iff,axiom,
! [Z: set_Product_prod_a_a,F: a > set_Product_prod_a_a,A2: set_a] :
( ( member1816616512716248880od_a_a @ Z @ ( image_4421510592991446670od_a_a @ F @ A2 ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_886_image__iff,axiom,
! [Z: set_o,F: a > set_o,A2: set_a] :
( ( member_set_o @ Z @ ( image_a_set_o @ F @ A2 ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_887_image__iff,axiom,
! [Z: set_a,F: a > set_a,A2: set_a] :
( ( member_set_a @ Z @ ( image_a_set_a @ F @ A2 ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_888_image__iff,axiom,
! [Z: set_a,F: set_a > set_a,A2: set_set_a] :
( ( member_set_a @ Z @ ( image_set_a_set_a @ F @ A2 ) )
= ( ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_889_image__iff,axiom,
! [Z: produc4044097585999906000od_a_a,F: list_a > produc4044097585999906000od_a_a,A2: set_list_a] :
( ( member3071122053849602553od_a_a @ Z @ ( image_1195025184546981201od_a_a @ F @ A2 ) )
= ( ? [X2: list_a] :
( ( member_list_a @ X2 @ A2 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_890_bex__imageD,axiom,
! [F: list_a > produc4044097585999906000od_a_a,A2: set_list_a,P: produc4044097585999906000od_a_a > $o] :
( ? [X6: produc4044097585999906000od_a_a] :
( ( member3071122053849602553od_a_a @ X6 @ ( image_1195025184546981201od_a_a @ F @ A2 ) )
& ( P @ X6 ) )
=> ? [X3: list_a] :
( ( member_list_a @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_891_bex__imageD,axiom,
! [F: set_a > set_Pr5530083903271594800od_a_a,A2: set_set_a,P: set_Pr5530083903271594800od_a_a > $o] :
( ? [X6: set_Pr5530083903271594800od_a_a] :
( ( member4210947715425868889od_a_a @ X6 @ ( image_7562202058474640471od_a_a @ F @ A2 ) )
& ( P @ X6 ) )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_892_bex__imageD,axiom,
! [F: set_a > set_Product_prod_a_a,A2: set_set_a,P: set_Product_prod_a_a > $o] :
( ? [X6: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X6 @ ( image_6165024369500519726od_a_a @ F @ A2 ) )
& ( P @ X6 ) )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_893_bex__imageD,axiom,
! [F: a > set_set_a,A2: set_a,P: set_set_a > $o] :
( ? [X6: set_set_a] :
( ( member_set_set_a @ X6 @ ( image_a_set_set_a @ F @ A2 ) )
& ( P @ X6 ) )
=> ? [X3: a] :
( ( member_a @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_894_bex__imageD,axiom,
! [F: a > set_a,A2: set_a,P: set_a > $o] :
( ? [X6: set_a] :
( ( member_set_a @ X6 @ ( image_a_set_a @ F @ A2 ) )
& ( P @ X6 ) )
=> ? [X3: a] :
( ( member_a @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_895_bex__imageD,axiom,
! [F: set_a > set_a,A2: set_set_a,P: set_a > $o] :
( ? [X6: set_a] :
( ( member_set_a @ X6 @ ( image_set_a_set_a @ F @ A2 ) )
& ( P @ X6 ) )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_896_bex__imageD,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ? [X6: nat] :
( ( member_nat @ X6 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ X6 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_897_bex__imageD,axiom,
! [F: a > set_Pr5530083903271594800od_a_a,A2: set_a,P: set_Pr5530083903271594800od_a_a > $o] :
( ? [X6: set_Pr5530083903271594800od_a_a] :
( ( member4210947715425868889od_a_a @ X6 @ ( image_5653227685612666295od_a_a @ F @ A2 ) )
& ( P @ X6 ) )
=> ? [X3: a] :
( ( member_a @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_898_bex__imageD,axiom,
! [F: a > set_Product_prod_a_a,A2: set_a,P: set_Product_prod_a_a > $o] :
( ? [X6: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X6 @ ( image_4421510592991446670od_a_a @ F @ A2 ) )
& ( P @ X6 ) )
=> ? [X3: a] :
( ( member_a @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_899_bex__imageD,axiom,
! [F: a > set_o,A2: set_a,P: set_o > $o] :
( ? [X6: set_o] :
( ( member_set_o @ X6 @ ( image_a_set_o @ F @ A2 ) )
& ( P @ X6 ) )
=> ? [X3: a] :
( ( member_a @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_900_image__cong,axiom,
! [M: set_list_a,N2: set_list_a,F: list_a > produc4044097585999906000od_a_a,G: list_a > produc4044097585999906000od_a_a] :
( ( M = N2 )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ N2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_1195025184546981201od_a_a @ F @ M )
= ( image_1195025184546981201od_a_a @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_901_image__cong,axiom,
! [M: set_nat,N2: set_nat,F: nat > nat,G: nat > nat] :
( ( M = N2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_nat_nat @ F @ M )
= ( image_nat_nat @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_902_image__cong,axiom,
! [M: set_set_a,N2: set_set_a,F: set_a > set_Pr5530083903271594800od_a_a,G: set_a > set_Pr5530083903271594800od_a_a] :
( ( M = N2 )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ N2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_7562202058474640471od_a_a @ F @ M )
= ( image_7562202058474640471od_a_a @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_903_image__cong,axiom,
! [M: set_set_a,N2: set_set_a,F: set_a > set_Product_prod_a_a,G: set_a > set_Product_prod_a_a] :
( ( M = N2 )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ N2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_6165024369500519726od_a_a @ F @ M )
= ( image_6165024369500519726od_a_a @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_904_image__cong,axiom,
! [M: set_set_a,N2: set_set_a,F: set_a > set_a,G: set_a > set_a] :
( ( M = N2 )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ N2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_set_a_set_a @ F @ M )
= ( image_set_a_set_a @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_905_image__cong,axiom,
! [M: set_a,N2: set_a,F: a > set_set_a,G: a > set_set_a] :
( ( M = N2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ N2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_a_set_set_a @ F @ M )
= ( image_a_set_set_a @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_906_image__cong,axiom,
! [M: set_a,N2: set_a,F: a > set_a,G: a > set_a] :
( ( M = N2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ N2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_a_set_a @ F @ M )
= ( image_a_set_a @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_907_image__cong,axiom,
! [M: set_a,N2: set_a,F: a > set_Pr5530083903271594800od_a_a,G: a > set_Pr5530083903271594800od_a_a] :
( ( M = N2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ N2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_5653227685612666295od_a_a @ F @ M )
= ( image_5653227685612666295od_a_a @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_908_image__cong,axiom,
! [M: set_a,N2: set_a,F: a > set_Product_prod_a_a,G: a > set_Product_prod_a_a] :
( ( M = N2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ N2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_4421510592991446670od_a_a @ F @ M )
= ( image_4421510592991446670od_a_a @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_909_image__cong,axiom,
! [M: set_a,N2: set_a,F: a > set_o,G: a > set_o] :
( ( M = N2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ N2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_a_set_o @ F @ M )
= ( image_a_set_o @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_910_ball__imageD,axiom,
! [F: list_a > produc4044097585999906000od_a_a,A2: set_list_a,P: produc4044097585999906000od_a_a > $o] :
( ! [X3: produc4044097585999906000od_a_a] :
( ( member3071122053849602553od_a_a @ X3 @ ( image_1195025184546981201od_a_a @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X6: list_a] :
( ( member_list_a @ X6 @ A2 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_911_ball__imageD,axiom,
! [F: set_a > set_Pr5530083903271594800od_a_a,A2: set_set_a,P: set_Pr5530083903271594800od_a_a > $o] :
( ! [X3: set_Pr5530083903271594800od_a_a] :
( ( member4210947715425868889od_a_a @ X3 @ ( image_7562202058474640471od_a_a @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X6: set_a] :
( ( member_set_a @ X6 @ A2 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_912_ball__imageD,axiom,
! [F: set_a > set_Product_prod_a_a,A2: set_set_a,P: set_Product_prod_a_a > $o] :
( ! [X3: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X3 @ ( image_6165024369500519726od_a_a @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X6: set_a] :
( ( member_set_a @ X6 @ A2 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_913_ball__imageD,axiom,
! [F: a > set_set_a,A2: set_a,P: set_set_a > $o] :
( ! [X3: set_set_a] :
( ( member_set_set_a @ X3 @ ( image_a_set_set_a @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X6: a] :
( ( member_a @ X6 @ A2 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_914_ball__imageD,axiom,
! [F: a > set_a,A2: set_a,P: set_a > $o] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ ( image_a_set_a @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X6: a] :
( ( member_a @ X6 @ A2 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_915_ball__imageD,axiom,
! [F: set_a > set_a,A2: set_set_a,P: set_a > $o] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ ( image_set_a_set_a @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X6: set_a] :
( ( member_set_a @ X6 @ A2 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_916_ball__imageD,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( image_nat_nat @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X6: nat] :
( ( member_nat @ X6 @ A2 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_917_ball__imageD,axiom,
! [F: a > set_Pr5530083903271594800od_a_a,A2: set_a,P: set_Pr5530083903271594800od_a_a > $o] :
( ! [X3: set_Pr5530083903271594800od_a_a] :
( ( member4210947715425868889od_a_a @ X3 @ ( image_5653227685612666295od_a_a @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X6: a] :
( ( member_a @ X6 @ A2 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_918_ball__imageD,axiom,
! [F: a > set_Product_prod_a_a,A2: set_a,P: set_Product_prod_a_a > $o] :
( ! [X3: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X3 @ ( image_4421510592991446670od_a_a @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X6: a] :
( ( member_a @ X6 @ A2 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_919_ball__imageD,axiom,
! [F: a > set_o,A2: set_a,P: set_o > $o] :
( ! [X3: set_o] :
( ( member_set_o @ X3 @ ( image_a_set_o @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X6: a] :
( ( member_a @ X6 @ A2 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_920_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: nat,F: nat > nat] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_921_rev__image__eqI,axiom,
! [X: $o,A2: set_o,B: $o,F: $o > $o] :
( ( member_o @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_o @ B @ ( image_o_o @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_922_rev__image__eqI,axiom,
! [X: $o,A2: set_o,B: a,F: $o > a] :
( ( member_o @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_a @ B @ ( image_o_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_923_rev__image__eqI,axiom,
! [X: a,A2: set_a,B: $o,F: a > $o] :
( ( member_a @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_o @ B @ ( image_a_o @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_924_rev__image__eqI,axiom,
! [X: a,A2: set_a,B: a,F: a > a] :
( ( member_a @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_a @ B @ ( image_a_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_925_rev__image__eqI,axiom,
! [X: set_a,A2: set_set_a,B: $o,F: set_a > $o] :
( ( member_set_a @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_o @ B @ ( image_set_a_o @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_926_rev__image__eqI,axiom,
! [X: set_a,A2: set_set_a,B: a,F: set_a > a] :
( ( member_set_a @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_a @ B @ ( image_set_a_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_927_rev__image__eqI,axiom,
! [X: $o,A2: set_o,B: set_a,F: $o > set_a] :
( ( member_o @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_set_a @ B @ ( image_o_set_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_928_rev__image__eqI,axiom,
! [X: a,A2: set_a,B: set_o,F: a > set_o] :
( ( member_a @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_set_o @ B @ ( image_a_set_o @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_929_rev__image__eqI,axiom,
! [X: a,A2: set_a,B: set_a,F: a > set_a] :
( ( member_a @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_set_a @ B @ ( image_a_set_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_930_SUP__eq__iff,axiom,
! [I5: set_o,C2: set_a,F: $o > set_a] :
( ( I5 != bot_bot_set_o )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ I5 )
=> ( ord_less_eq_set_a @ C2 @ ( F @ I2 ) ) )
=> ( ( ( comple2307003609928055243_set_a @ ( image_o_set_a @ F @ I5 ) )
= C2 )
= ( ! [X2: $o] :
( ( member_o @ X2 @ I5 )
=> ( ( F @ X2 )
= C2 ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_931_SUP__eq__iff,axiom,
! [I5: set_a,C2: set_a,F: a > set_a] :
( ( I5 != bot_bot_set_a )
=> ( ! [I2: a] :
( ( member_a @ I2 @ I5 )
=> ( ord_less_eq_set_a @ C2 @ ( F @ I2 ) ) )
=> ( ( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ I5 ) )
= C2 )
= ( ! [X2: a] :
( ( member_a @ X2 @ I5 )
=> ( ( F @ X2 )
= C2 ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_932_SUP__eq__iff,axiom,
! [I5: set_nat,C2: set_a,F: nat > set_a] :
( ( I5 != bot_bot_set_nat )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I5 )
=> ( ord_less_eq_set_a @ C2 @ ( F @ I2 ) ) )
=> ( ( ( comple2307003609928055243_set_a @ ( image_nat_set_a @ F @ I5 ) )
= C2 )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ I5 )
=> ( ( F @ X2 )
= C2 ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_933_SUP__eq__iff,axiom,
! [I5: set_o,C2: set_o,F: $o > set_o] :
( ( I5 != bot_bot_set_o )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ I5 )
=> ( ord_less_eq_set_o @ C2 @ ( F @ I2 ) ) )
=> ( ( ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ I5 ) )
= C2 )
= ( ! [X2: $o] :
( ( member_o @ X2 @ I5 )
=> ( ( F @ X2 )
= C2 ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_934_SUP__eq__iff,axiom,
! [I5: set_a,C2: set_o,F: a > set_o] :
( ( I5 != bot_bot_set_a )
=> ( ! [I2: a] :
( ( member_a @ I2 @ I5 )
=> ( ord_less_eq_set_o @ C2 @ ( F @ I2 ) ) )
=> ( ( ( comple90263536869209701_set_o @ ( image_a_set_o @ F @ I5 ) )
= C2 )
= ( ! [X2: a] :
( ( member_a @ X2 @ I5 )
=> ( ( F @ X2 )
= C2 ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_935_SUP__eq__iff,axiom,
! [I5: set_nat,C2: set_o,F: nat > set_o] :
( ( I5 != bot_bot_set_nat )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I5 )
=> ( ord_less_eq_set_o @ C2 @ ( F @ I2 ) ) )
=> ( ( ( comple90263536869209701_set_o @ ( image_nat_set_o @ F @ I5 ) )
= C2 )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ I5 )
=> ( ( F @ X2 )
= C2 ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_936_SUP__eq__iff,axiom,
! [I5: set_o,C2: set_set_a,F: $o > set_set_a] :
( ( I5 != bot_bot_set_o )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ I5 )
=> ( ord_le3724670747650509150_set_a @ C2 @ ( F @ I2 ) ) )
=> ( ( ( comple3958522678809307947_set_a @ ( image_o_set_set_a @ F @ I5 ) )
= C2 )
= ( ! [X2: $o] :
( ( member_o @ X2 @ I5 )
=> ( ( F @ X2 )
= C2 ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_937_SUP__eq__iff,axiom,
! [I5: set_a,C2: set_set_a,F: a > set_set_a] :
( ( I5 != bot_bot_set_a )
=> ( ! [I2: a] :
( ( member_a @ I2 @ I5 )
=> ( ord_le3724670747650509150_set_a @ C2 @ ( F @ I2 ) ) )
=> ( ( ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ F @ I5 ) )
= C2 )
= ( ! [X2: a] :
( ( member_a @ X2 @ I5 )
=> ( ( F @ X2 )
= C2 ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_938_SUP__eq__iff,axiom,
! [I5: set_nat,C2: set_set_a,F: nat > set_set_a] :
( ( I5 != bot_bot_set_nat )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I5 )
=> ( ord_le3724670747650509150_set_a @ C2 @ ( F @ I2 ) ) )
=> ( ( ( comple3958522678809307947_set_a @ ( image_nat_set_set_a @ F @ I5 ) )
= C2 )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ I5 )
=> ( ( F @ X2 )
= C2 ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_939_SUP__eq__iff,axiom,
! [I5: set_set_a,C2: set_a,F: set_a > set_a] :
( ( I5 != bot_bot_set_set_a )
=> ( ! [I2: set_a] :
( ( member_set_a @ I2 @ I5 )
=> ( ord_less_eq_set_a @ C2 @ ( F @ I2 ) ) )
=> ( ( ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ F @ I5 ) )
= C2 )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ I5 )
=> ( ( F @ X2 )
= C2 ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_940_in__mono,axiom,
! [A2: set_o,B2: set_o,X: $o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ( member_o @ X @ A2 )
=> ( member_o @ X @ B2 ) ) ) ).
% in_mono
thf(fact_941_in__mono,axiom,
! [A2: set_Pr5530083903271594800od_a_a,B2: set_Pr5530083903271594800od_a_a,X: produc4044097585999906000od_a_a] :
( ( ord_le114883831454073552od_a_a @ A2 @ B2 )
=> ( ( member3071122053849602553od_a_a @ X @ A2 )
=> ( member3071122053849602553od_a_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_942_in__mono,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a,X: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B2 )
=> ( ( member1426531477525435216od_a_a @ X @ A2 )
=> ( member1426531477525435216od_a_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_943_in__mono,axiom,
! [A2: set_set_a,B2: set_set_a,X: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( member_set_a @ X @ A2 )
=> ( member_set_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_944_in__mono,axiom,
! [A2: set_a,B2: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_945_subsetD,axiom,
! [A2: set_o,B2: set_o,C2: $o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ( member_o @ C2 @ A2 )
=> ( member_o @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_946_subsetD,axiom,
! [A2: set_Pr5530083903271594800od_a_a,B2: set_Pr5530083903271594800od_a_a,C2: produc4044097585999906000od_a_a] :
( ( ord_le114883831454073552od_a_a @ A2 @ B2 )
=> ( ( member3071122053849602553od_a_a @ C2 @ A2 )
=> ( member3071122053849602553od_a_a @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_947_subsetD,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a,C2: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B2 )
=> ( ( member1426531477525435216od_a_a @ C2 @ A2 )
=> ( member1426531477525435216od_a_a @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_948_subsetD,axiom,
! [A2: set_set_a,B2: set_set_a,C2: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( member_set_a @ C2 @ A2 )
=> ( member_set_a @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_949_subsetD,axiom,
! [A2: set_a,B2: set_a,C2: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_950_equalityE,axiom,
! [A2: set_Pr5530083903271594800od_a_a,B2: set_Pr5530083903271594800od_a_a] :
( ( A2 = B2 )
=> ~ ( ( ord_le114883831454073552od_a_a @ A2 @ B2 )
=> ~ ( ord_le114883831454073552od_a_a @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_951_equalityE,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( A2 = B2 )
=> ~ ( ( ord_le746702958409616551od_a_a @ A2 @ B2 )
=> ~ ( ord_le746702958409616551od_a_a @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_952_equalityE,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( A2 = B2 )
=> ~ ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ~ ( ord_le3724670747650509150_set_a @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_953_equalityE,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ~ ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_954_subset__eq,axiom,
( ord_less_eq_set_o
= ( ^ [A4: set_o,B5: set_o] :
! [X2: $o] :
( ( member_o @ X2 @ A4 )
=> ( member_o @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_955_subset__eq,axiom,
( ord_le114883831454073552od_a_a
= ( ^ [A4: set_Pr5530083903271594800od_a_a,B5: set_Pr5530083903271594800od_a_a] :
! [X2: produc4044097585999906000od_a_a] :
( ( member3071122053849602553od_a_a @ X2 @ A4 )
=> ( member3071122053849602553od_a_a @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_956_subset__eq,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A4: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A4 )
=> ( member1426531477525435216od_a_a @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_957_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B5: set_set_a] :
! [X2: set_a] :
( ( member_set_a @ X2 @ A4 )
=> ( member_set_a @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_958_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B5: set_a] :
! [X2: a] :
( ( member_a @ X2 @ A4 )
=> ( member_a @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_959_equalityD1,axiom,
! [A2: set_Pr5530083903271594800od_a_a,B2: set_Pr5530083903271594800od_a_a] :
( ( A2 = B2 )
=> ( ord_le114883831454073552od_a_a @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_960_equalityD1,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( A2 = B2 )
=> ( ord_le746702958409616551od_a_a @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_961_equalityD1,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( A2 = B2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_962_equalityD1,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_963_equalityD2,axiom,
! [A2: set_Pr5530083903271594800od_a_a,B2: set_Pr5530083903271594800od_a_a] :
( ( A2 = B2 )
=> ( ord_le114883831454073552od_a_a @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_964_equalityD2,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( A2 = B2 )
=> ( ord_le746702958409616551od_a_a @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_965_equalityD2,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( A2 = B2 )
=> ( ord_le3724670747650509150_set_a @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_966_equalityD2,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_967_subset__iff,axiom,
( ord_less_eq_set_o
= ( ^ [A4: set_o,B5: set_o] :
! [T: $o] :
( ( member_o @ T @ A4 )
=> ( member_o @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_968_subset__iff,axiom,
( ord_le114883831454073552od_a_a
= ( ^ [A4: set_Pr5530083903271594800od_a_a,B5: set_Pr5530083903271594800od_a_a] :
! [T: produc4044097585999906000od_a_a] :
( ( member3071122053849602553od_a_a @ T @ A4 )
=> ( member3071122053849602553od_a_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_969_subset__iff,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A4: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
! [T: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ T @ A4 )
=> ( member1426531477525435216od_a_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_970_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B5: set_set_a] :
! [T: set_a] :
( ( member_set_a @ T @ A4 )
=> ( member_set_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_971_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B5: set_a] :
! [T: a] :
( ( member_a @ T @ A4 )
=> ( member_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_972_subset__refl,axiom,
! [A2: set_Pr5530083903271594800od_a_a] : ( ord_le114883831454073552od_a_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_973_subset__refl,axiom,
! [A2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_974_subset__refl,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_975_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_976_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_977_Collect__mono,axiom,
! [P: produc4044097585999906000od_a_a > $o,Q: produc4044097585999906000od_a_a > $o] :
( ! [X3: produc4044097585999906000od_a_a] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le114883831454073552od_a_a @ ( collec5045780995415420475od_a_a @ P ) @ ( collec5045780995415420475od_a_a @ Q ) ) ) ).
% Collect_mono
thf(fact_978_Collect__mono,axiom,
! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ! [X3: product_prod_a_a] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le746702958409616551od_a_a @ ( collec3336397797384452498od_a_a @ P ) @ ( collec3336397797384452498od_a_a @ Q ) ) ) ).
% Collect_mono
thf(fact_979_Collect__mono,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X3: set_a] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).
% Collect_mono
thf(fact_980_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_981_subset__trans,axiom,
! [A2: set_Pr5530083903271594800od_a_a,B2: set_Pr5530083903271594800od_a_a,C: set_Pr5530083903271594800od_a_a] :
( ( ord_le114883831454073552od_a_a @ A2 @ B2 )
=> ( ( ord_le114883831454073552od_a_a @ B2 @ C )
=> ( ord_le114883831454073552od_a_a @ A2 @ C ) ) ) ).
% subset_trans
thf(fact_982_subset__trans,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B2 )
=> ( ( ord_le746702958409616551od_a_a @ B2 @ C )
=> ( ord_le746702958409616551od_a_a @ A2 @ C ) ) ) ).
% subset_trans
thf(fact_983_subset__trans,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ord_le3724670747650509150_set_a @ A2 @ C ) ) ) ).
% subset_trans
thf(fact_984_subset__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% subset_trans
thf(fact_985_set__eq__subset,axiom,
( ( ^ [Y6: set_Pr5530083903271594800od_a_a,Z3: set_Pr5530083903271594800od_a_a] : ( Y6 = Z3 ) )
= ( ^ [A4: set_Pr5530083903271594800od_a_a,B5: set_Pr5530083903271594800od_a_a] :
( ( ord_le114883831454073552od_a_a @ A4 @ B5 )
& ( ord_le114883831454073552od_a_a @ B5 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_986_set__eq__subset,axiom,
( ( ^ [Y6: set_Product_prod_a_a,Z3: set_Product_prod_a_a] : ( Y6 = Z3 ) )
= ( ^ [A4: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A4 @ B5 )
& ( ord_le746702958409616551od_a_a @ B5 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_987_set__eq__subset,axiom,
( ( ^ [Y6: set_set_a,Z3: set_set_a] : ( Y6 = Z3 ) )
= ( ^ [A4: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ B5 )
& ( ord_le3724670747650509150_set_a @ B5 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_988_set__eq__subset,axiom,
( ( ^ [Y6: set_a,Z3: set_a] : ( Y6 = Z3 ) )
= ( ^ [A4: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A4 @ B5 )
& ( ord_less_eq_set_a @ B5 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_989_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X2: nat] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_990_Collect__mono__iff,axiom,
! [P: produc4044097585999906000od_a_a > $o,Q: produc4044097585999906000od_a_a > $o] :
( ( ord_le114883831454073552od_a_a @ ( collec5045780995415420475od_a_a @ P ) @ ( collec5045780995415420475od_a_a @ Q ) )
= ( ! [X2: produc4044097585999906000od_a_a] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_991_Collect__mono__iff,axiom,
! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ( ord_le746702958409616551od_a_a @ ( collec3336397797384452498od_a_a @ P ) @ ( collec3336397797384452498od_a_a @ Q ) )
= ( ! [X2: product_prod_a_a] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_992_Collect__mono__iff,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) )
= ( ! [X2: set_a] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_993_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X2: a] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_994_Sup__SUP__eq,axiom,
( comple8317665133742190828_nat_o
= ( ^ [S2: set_nat_o,X2: nat] : ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ ( image_nat_o_set_nat @ collect_nat @ S2 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_995_Sup__SUP__eq,axiom,
( comple2673673910019652224_a_a_o
= ( ^ [S2: set_Pr5684138772510291555_a_a_o,X2: produc4044097585999906000od_a_a] : ( member3071122053849602553od_a_a @ X2 @ ( comple2978350343072902813od_a_a @ ( image_3003369154657308540od_a_a @ collec5045780995415420475od_a_a @ S2 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_996_Sup__SUP__eq,axiom,
( comple9027675562681848937_a_a_o
= ( ^ [S2: set_Pr974637816708178892_a_a_o,X2: product_prod_a_a] : ( member1426531477525435216od_a_a @ X2 @ ( comple8421679170691845492od_a_a @ ( image_7999903765548794282od_a_a @ collec3336397797384452498od_a_a @ S2 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_997_Sup__SUP__eq,axiom,
( comple7256090232125724530et_a_o
= ( ^ [S2: set_set_a_o,X2: set_a] : ( member_set_a @ X2 @ ( comple3958522678809307947_set_a @ ( image_5433501368419010456_set_a @ collect_set_a @ S2 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_998_Sup__SUP__eq,axiom,
( complete_Sup_Sup_a_o
= ( ^ [S2: set_a_o,X2: a] : ( member_a @ X2 @ ( comple2307003609928055243_set_a @ ( image_a_o_set_a @ collect_a @ S2 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_999_Sup__SUP__eq,axiom,
( complete_Sup_Sup_o_o
= ( ^ [S2: set_o_o,X2: $o] : ( member_o @ X2 @ ( comple90263536869209701_set_o @ ( image_o_o_set_o @ collect_o @ S2 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_1000_imageE,axiom,
! [B: nat,F: nat > nat,A2: set_nat] :
( ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [X3: nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_1001_imageE,axiom,
! [B: $o,F: $o > $o,A2: set_o] :
( ( member_o @ B @ ( image_o_o @ F @ A2 ) )
=> ~ ! [X3: $o] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_o @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_1002_imageE,axiom,
! [B: $o,F: a > $o,A2: set_a] :
( ( member_o @ B @ ( image_a_o @ F @ A2 ) )
=> ~ ! [X3: a] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_a @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_1003_imageE,axiom,
! [B: a,F: $o > a,A2: set_o] :
( ( member_a @ B @ ( image_o_a @ F @ A2 ) )
=> ~ ! [X3: $o] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_o @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_1004_imageE,axiom,
! [B: a,F: a > a,A2: set_a] :
( ( member_a @ B @ ( image_a_a @ F @ A2 ) )
=> ~ ! [X3: a] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_a @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_1005_imageE,axiom,
! [B: set_o,F: a > set_o,A2: set_a] :
( ( member_set_o @ B @ ( image_a_set_o @ F @ A2 ) )
=> ~ ! [X3: a] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_a @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_1006_imageE,axiom,
! [B: set_a,F: $o > set_a,A2: set_o] :
( ( member_set_a @ B @ ( image_o_set_a @ F @ A2 ) )
=> ~ ! [X3: $o] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_o @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_1007_imageE,axiom,
! [B: set_a,F: a > set_a,A2: set_a] :
( ( member_set_a @ B @ ( image_a_set_a @ F @ A2 ) )
=> ~ ! [X3: a] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_a @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_1008_imageE,axiom,
! [B: $o,F: set_a > $o,A2: set_set_a] :
( ( member_o @ B @ ( image_set_a_o @ F @ A2 ) )
=> ~ ! [X3: set_a] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_set_a @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_1009_imageE,axiom,
! [B: a,F: set_a > a,A2: set_set_a] :
( ( member_a @ B @ ( image_set_a_a @ F @ A2 ) )
=> ~ ! [X3: set_a] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_set_a @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_1010_image__image,axiom,
! [F: nat > nat,G: nat > nat,A2: set_nat] :
( ( image_nat_nat @ F @ ( image_nat_nat @ G @ A2 ) )
= ( image_nat_nat
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_1011_image__image,axiom,
! [F: a > set_a,G: a > a,A2: set_a] :
( ( image_a_set_a @ F @ ( image_a_a @ G @ A2 ) )
= ( image_a_set_a
@ ^ [X2: a] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_1012_image__image,axiom,
! [F: a > set_o,G: a > a,A2: set_a] :
( ( image_a_set_o @ F @ ( image_a_a @ G @ A2 ) )
= ( image_a_set_o
@ ^ [X2: a] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_1013_image__image,axiom,
! [F: set_a > set_o,G: a > set_a,A2: set_a] :
( ( image_set_a_set_o @ F @ ( image_a_set_a @ G @ A2 ) )
= ( image_a_set_o
@ ^ [X2: a] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_1014_image__image,axiom,
! [F: set_o > set_a,G: a > set_o,A2: set_a] :
( ( image_set_o_set_a @ F @ ( image_a_set_o @ G @ A2 ) )
= ( image_a_set_a
@ ^ [X2: a] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_1015_image__image,axiom,
! [F: set_o > set_o,G: a > set_o,A2: set_a] :
( ( image_set_o_set_o @ F @ ( image_a_set_o @ G @ A2 ) )
= ( image_a_set_o
@ ^ [X2: a] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_1016_image__image,axiom,
! [F: a > set_set_a,G: a > a,A2: set_a] :
( ( image_a_set_set_a @ F @ ( image_a_a @ G @ A2 ) )
= ( image_a_set_set_a
@ ^ [X2: a] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_1017_image__image,axiom,
! [F: a > set_a,G: set_a > a,A2: set_set_a] :
( ( image_a_set_a @ F @ ( image_set_a_a @ G @ A2 ) )
= ( image_set_a_set_a
@ ^ [X2: set_a] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_1018_image__image,axiom,
! [F: set_a > set_a,G: a > set_a,A2: set_a] :
( ( image_set_a_set_a @ F @ ( image_a_set_a @ G @ A2 ) )
= ( image_a_set_a
@ ^ [X2: a] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_1019_image__image,axiom,
! [F: set_set_a > set_a,G: a > set_set_a,A2: set_a] :
( ( image_6061375613820669477_set_a @ F @ ( image_a_set_set_a @ G @ A2 ) )
= ( image_a_set_a
@ ^ [X2: a] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_1020_Compr__image__eq,axiom,
! [F: $o > $o,A2: set_o,P: $o > $o] :
( ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ ( image_o_o @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_o_o @ F
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1021_Compr__image__eq,axiom,
! [F: nat > $o,A2: set_nat,P: $o > $o] :
( ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ ( image_nat_o @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_nat_o @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1022_Compr__image__eq,axiom,
! [F: a > $o,A2: set_a,P: $o > $o] :
( ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ ( image_a_o @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_a_o @ F
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1023_Compr__image__eq,axiom,
! [F: $o > nat,A2: set_o,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_o_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_o_nat @ F
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1024_Compr__image__eq,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1025_Compr__image__eq,axiom,
! [F: a > nat,A2: set_a,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_a_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_a_nat @ F
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1026_Compr__image__eq,axiom,
! [F: $o > a,A2: set_o,P: a > $o] :
( ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ ( image_o_a @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_o_a @ F
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1027_Compr__image__eq,axiom,
! [F: nat > a,A2: set_nat,P: a > $o] :
( ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ ( image_nat_a @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_nat_a @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1028_Compr__image__eq,axiom,
! [F: a > a,A2: set_a,P: a > $o] :
( ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ ( image_a_a @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_a_a @ F
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1029_Compr__image__eq,axiom,
! [F: set_a > $o,A2: set_set_a,P: $o > $o] :
( ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ ( image_set_a_o @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_set_a_o @ F
@ ( collect_set_a
@ ^ [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1030_Collect__subset,axiom,
! [A2: set_o,P: $o > $o] :
( ord_less_eq_set_o
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_1031_Collect__subset,axiom,
! [A2: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_1032_Collect__subset,axiom,
! [A2: set_Pr5530083903271594800od_a_a,P: produc4044097585999906000od_a_a > $o] :
( ord_le114883831454073552od_a_a
@ ( collec5045780995415420475od_a_a
@ ^ [X2: produc4044097585999906000od_a_a] :
( ( member3071122053849602553od_a_a @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_1033_Collect__subset,axiom,
! [A2: set_Product_prod_a_a,P: product_prod_a_a > $o] :
( ord_le746702958409616551od_a_a
@ ( collec3336397797384452498od_a_a
@ ^ [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_1034_Collect__subset,axiom,
! [A2: set_set_a,P: set_a > $o] :
( ord_le3724670747650509150_set_a
@ ( collect_set_a
@ ^ [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_1035_Collect__subset,axiom,
! [A2: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_1036_conj__subset__def,axiom,
! [A2: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A2
@ ( collect_nat
@ ^ [X2: nat] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) )
= ( ( ord_less_eq_set_nat @ A2 @ ( collect_nat @ P ) )
& ( ord_less_eq_set_nat @ A2 @ ( collect_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_1037_conj__subset__def,axiom,
! [A2: set_Pr5530083903271594800od_a_a,P: produc4044097585999906000od_a_a > $o,Q: produc4044097585999906000od_a_a > $o] :
( ( ord_le114883831454073552od_a_a @ A2
@ ( collec5045780995415420475od_a_a
@ ^ [X2: produc4044097585999906000od_a_a] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) )
= ( ( ord_le114883831454073552od_a_a @ A2 @ ( collec5045780995415420475od_a_a @ P ) )
& ( ord_le114883831454073552od_a_a @ A2 @ ( collec5045780995415420475od_a_a @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_1038_conj__subset__def,axiom,
! [A2: set_Product_prod_a_a,P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ( ord_le746702958409616551od_a_a @ A2
@ ( collec3336397797384452498od_a_a
@ ^ [X2: product_prod_a_a] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) )
= ( ( ord_le746702958409616551od_a_a @ A2 @ ( collec3336397797384452498od_a_a @ P ) )
& ( ord_le746702958409616551od_a_a @ A2 @ ( collec3336397797384452498od_a_a @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_1039_conj__subset__def,axiom,
! [A2: set_set_a,P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ A2
@ ( collect_set_a
@ ^ [X2: set_a] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) )
= ( ( ord_le3724670747650509150_set_a @ A2 @ ( collect_set_a @ P ) )
& ( ord_le3724670747650509150_set_a @ A2 @ ( collect_set_a @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_1040_conj__subset__def,axiom,
! [A2: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A2
@ ( collect_a
@ ^ [X2: a] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) )
= ( ( ord_less_eq_set_a @ A2 @ ( collect_a @ P ) )
& ( ord_less_eq_set_a @ A2 @ ( collect_a @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_1041_prop__restrict,axiom,
! [X: $o,Z4: set_o,X4: set_o,P: $o > $o] :
( ( member_o @ X @ Z4 )
=> ( ( ord_less_eq_set_o @ Z4
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ X4 )
& ( P @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_1042_prop__restrict,axiom,
! [X: nat,Z4: set_nat,X4: set_nat,P: nat > $o] :
( ( member_nat @ X @ Z4 )
=> ( ( ord_less_eq_set_nat @ Z4
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ X4 )
& ( P @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_1043_prop__restrict,axiom,
! [X: produc4044097585999906000od_a_a,Z4: set_Pr5530083903271594800od_a_a,X4: set_Pr5530083903271594800od_a_a,P: produc4044097585999906000od_a_a > $o] :
( ( member3071122053849602553od_a_a @ X @ Z4 )
=> ( ( ord_le114883831454073552od_a_a @ Z4
@ ( collec5045780995415420475od_a_a
@ ^ [X2: produc4044097585999906000od_a_a] :
( ( member3071122053849602553od_a_a @ X2 @ X4 )
& ( P @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_1044_prop__restrict,axiom,
! [X: product_prod_a_a,Z4: set_Product_prod_a_a,X4: set_Product_prod_a_a,P: product_prod_a_a > $o] :
( ( member1426531477525435216od_a_a @ X @ Z4 )
=> ( ( ord_le746702958409616551od_a_a @ Z4
@ ( collec3336397797384452498od_a_a
@ ^ [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ X4 )
& ( P @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_1045_prop__restrict,axiom,
! [X: set_a,Z4: set_set_a,X4: set_set_a,P: set_a > $o] :
( ( member_set_a @ X @ Z4 )
=> ( ( ord_le3724670747650509150_set_a @ Z4
@ ( collect_set_a
@ ^ [X2: set_a] :
( ( member_set_a @ X2 @ X4 )
& ( P @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_1046_prop__restrict,axiom,
! [X: a,Z4: set_a,X4: set_a,P: a > $o] :
( ( member_a @ X @ Z4 )
=> ( ( ord_less_eq_set_a @ Z4
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ X4 )
& ( P @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_1047_Collect__restrict,axiom,
! [X4: set_o,P: $o > $o] :
( ord_less_eq_set_o
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ X4 )
& ( P @ X2 ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_1048_Collect__restrict,axiom,
! [X4: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ X4 )
& ( P @ X2 ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_1049_Collect__restrict,axiom,
! [X4: set_Pr5530083903271594800od_a_a,P: produc4044097585999906000od_a_a > $o] :
( ord_le114883831454073552od_a_a
@ ( collec5045780995415420475od_a_a
@ ^ [X2: produc4044097585999906000od_a_a] :
( ( member3071122053849602553od_a_a @ X2 @ X4 )
& ( P @ X2 ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_1050_Collect__restrict,axiom,
! [X4: set_Product_prod_a_a,P: product_prod_a_a > $o] :
( ord_le746702958409616551od_a_a
@ ( collec3336397797384452498od_a_a
@ ^ [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ X4 )
& ( P @ X2 ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_1051_Collect__restrict,axiom,
! [X4: set_set_a,P: set_a > $o] :
( ord_le3724670747650509150_set_a
@ ( collect_set_a
@ ^ [X2: set_a] :
( ( member_set_a @ X2 @ X4 )
& ( P @ X2 ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_1052_Collect__restrict,axiom,
! [X4: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ X4 )
& ( P @ X2 ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_1053_less__eq__set__def,axiom,
( ord_less_eq_set_o
= ( ^ [A4: set_o,B5: set_o] :
( ord_less_eq_o_o
@ ^ [X2: $o] : ( member_o @ X2 @ A4 )
@ ^ [X2: $o] : ( member_o @ X2 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_1054_less__eq__set__def,axiom,
( ord_le114883831454073552od_a_a
= ( ^ [A4: set_Pr5530083903271594800od_a_a,B5: set_Pr5530083903271594800od_a_a] :
( ord_le4133739015287644173_a_a_o
@ ^ [X2: produc4044097585999906000od_a_a] : ( member3071122053849602553od_a_a @ X2 @ A4 )
@ ^ [X2: produc4044097585999906000od_a_a] : ( member3071122053849602553od_a_a @ X2 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_1055_less__eq__set__def,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A4: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ord_le1591150415168442102_a_a_o
@ ^ [X2: product_prod_a_a] : ( member1426531477525435216od_a_a @ X2 @ A4 )
@ ^ [X2: product_prod_a_a] : ( member1426531477525435216od_a_a @ X2 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_1056_less__eq__set__def,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B5: set_set_a] :
( ord_less_eq_set_a_o
@ ^ [X2: set_a] : ( member_set_a @ X2 @ A4 )
@ ^ [X2: set_a] : ( member_set_a @ X2 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_1057_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B5: set_a] :
( ord_less_eq_a_o
@ ^ [X2: a] : ( member_a @ X2 @ A4 )
@ ^ [X2: a] : ( member_a @ X2 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_1058_pred__subset__eq,axiom,
! [R2: set_o,S: set_o] :
( ( ord_less_eq_o_o
@ ^ [X2: $o] : ( member_o @ X2 @ R2 )
@ ^ [X2: $o] : ( member_o @ X2 @ S ) )
= ( ord_less_eq_set_o @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_1059_pred__subset__eq,axiom,
! [R2: set_Pr5530083903271594800od_a_a,S: set_Pr5530083903271594800od_a_a] :
( ( ord_le4133739015287644173_a_a_o
@ ^ [X2: produc4044097585999906000od_a_a] : ( member3071122053849602553od_a_a @ X2 @ R2 )
@ ^ [X2: produc4044097585999906000od_a_a] : ( member3071122053849602553od_a_a @ X2 @ S ) )
= ( ord_le114883831454073552od_a_a @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_1060_pred__subset__eq,axiom,
! [R2: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( ord_le1591150415168442102_a_a_o
@ ^ [X2: product_prod_a_a] : ( member1426531477525435216od_a_a @ X2 @ R2 )
@ ^ [X2: product_prod_a_a] : ( member1426531477525435216od_a_a @ X2 @ S ) )
= ( ord_le746702958409616551od_a_a @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_1061_pred__subset__eq,axiom,
! [R2: set_set_a,S: set_set_a] :
( ( ord_less_eq_set_a_o
@ ^ [X2: set_a] : ( member_set_a @ X2 @ R2 )
@ ^ [X2: set_a] : ( member_set_a @ X2 @ S ) )
= ( ord_le3724670747650509150_set_a @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_1062_pred__subset__eq,axiom,
! [R2: set_a,S: set_a] :
( ( ord_less_eq_a_o
@ ^ [X2: a] : ( member_a @ X2 @ R2 )
@ ^ [X2: a] : ( member_a @ X2 @ S ) )
= ( ord_less_eq_set_a @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_1063_SUP__Sup__eq,axiom,
! [S: set_se9027383378080648592od_a_a] :
( ( comple2673673910019652224_a_a_o
@ ( image_2231517904016305432_a_a_o
@ ^ [I: set_Pr5530083903271594800od_a_a,X2: produc4044097585999906000od_a_a] : ( member3071122053849602553od_a_a @ X2 @ I )
@ S ) )
= ( ^ [X2: produc4044097585999906000od_a_a] : ( member3071122053849602553od_a_a @ X2 @ ( comple2978350343072902813od_a_a @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_1064_SUP__Sup__eq,axiom,
! [S: set_se5735800977113168103od_a_a] :
( ( comple9027675562681848937_a_a_o
@ ( image_1002440501707978072_a_a_o
@ ^ [I: set_Product_prod_a_a,X2: product_prod_a_a] : ( member1426531477525435216od_a_a @ X2 @ I )
@ S ) )
= ( ^ [X2: product_prod_a_a] : ( member1426531477525435216od_a_a @ X2 @ ( comple8421679170691845492od_a_a @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_1065_SUP__Sup__eq,axiom,
! [S: set_set_set_a] :
( ( comple7256090232125724530et_a_o
@ ( image_7036300546039719832et_a_o
@ ^ [I: set_set_a,X2: set_a] : ( member_set_a @ X2 @ I )
@ S ) )
= ( ^ [X2: set_a] : ( member_set_a @ X2 @ ( comple3958522678809307947_set_a @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_1066_SUP__Sup__eq,axiom,
! [S: set_set_a] :
( ( complete_Sup_Sup_a_o
@ ( image_set_a_a_o
@ ^ [I: set_a,X2: a] : ( member_a @ X2 @ I )
@ S ) )
= ( ^ [X2: a] : ( member_a @ X2 @ ( comple2307003609928055243_set_a @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_1067_SUP__Sup__eq,axiom,
! [S: set_set_o] :
( ( complete_Sup_Sup_o_o
@ ( image_set_o_o_o
@ ^ [I: set_o,X2: $o] : ( member_o @ X2 @ I )
@ S ) )
= ( ^ [X2: $o] : ( member_o @ X2 @ ( comple90263536869209701_set_o @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_1068_image__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_1069_image__mono,axiom,
! [A2: set_a,B2: set_a,F: a > a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A2 ) @ ( image_a_a @ F @ B2 ) ) ) ).
% image_mono
thf(fact_1070_image__mono,axiom,
! [A2: set_set_a,B2: set_set_a,F: set_a > a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ ( image_set_a_a @ F @ A2 ) @ ( image_set_a_a @ F @ B2 ) ) ) ).
% image_mono
thf(fact_1071_image__mono,axiom,
! [A2: set_a,B2: set_a,F: a > set_o] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_le4374716579403074808_set_o @ ( image_a_set_o @ F @ A2 ) @ ( image_a_set_o @ F @ B2 ) ) ) ).
% image_mono
thf(fact_1072_image__mono,axiom,
! [A2: set_a,B2: set_a,F: a > set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A2 ) @ ( image_a_set_a @ F @ B2 ) ) ) ).
% image_mono
thf(fact_1073_image__mono,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a,F: product_prod_a_a > a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ ( image_3437945252899457948_a_a_a @ F @ A2 ) @ ( image_3437945252899457948_a_a_a @ F @ B2 ) ) ) ).
% image_mono
thf(fact_1074_image__mono,axiom,
! [A2: set_set_a,B2: set_set_a,F: set_a > set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ F @ A2 ) @ ( image_set_a_set_a @ F @ B2 ) ) ) ).
% image_mono
thf(fact_1075_image__mono,axiom,
! [A2: set_a,B2: set_a,F: a > set_set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_le5722252365846178494_set_a @ ( image_a_set_set_a @ F @ A2 ) @ ( image_a_set_set_a @ F @ B2 ) ) ) ).
% image_mono
thf(fact_1076_image__mono,axiom,
! [A2: set_a,B2: set_a,F: a > product_prod_a_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_le746702958409616551od_a_a @ ( image_7400625782589995694od_a_a @ F @ A2 ) @ ( image_7400625782589995694od_a_a @ F @ B2 ) ) ) ).
% image_mono
thf(fact_1077_image__mono,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a,F: product_prod_a_a > set_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B2 )
=> ( ord_le3724670747650509150_set_a @ ( image_9052089385058188540_set_a @ F @ A2 ) @ ( image_9052089385058188540_set_a @ F @ B2 ) ) ) ).
% image_mono
thf(fact_1078_image__subsetI,axiom,
! [A2: set_nat,F: nat > nat,B2: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1079_image__subsetI,axiom,
! [A2: set_o,F: $o > $o,B2: set_o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( member_o @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_o @ ( image_o_o @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1080_image__subsetI,axiom,
! [A2: set_a,F: a > $o,B2: set_o] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_o @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_o @ ( image_a_o @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1081_image__subsetI,axiom,
! [A2: set_o,F: $o > a,B2: set_a] :
( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( member_a @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_a @ ( image_o_a @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1082_image__subsetI,axiom,
! [A2: set_a,F: a > a,B2: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_a @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1083_image__subsetI,axiom,
! [A2: set_set_a,F: set_a > $o,B2: set_o] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( member_o @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_o @ ( image_set_a_o @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1084_image__subsetI,axiom,
! [A2: set_a,F: a > set_o,B2: set_set_o] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_set_o @ ( F @ X3 ) @ B2 ) )
=> ( ord_le4374716579403074808_set_o @ ( image_a_set_o @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1085_image__subsetI,axiom,
! [A2: set_o,F: $o > set_a,B2: set_set_a] :
( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( member_set_a @ ( F @ X3 ) @ B2 ) )
=> ( ord_le3724670747650509150_set_a @ ( image_o_set_a @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1086_image__subsetI,axiom,
! [A2: set_a,F: a > set_a,B2: set_set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_set_a @ ( F @ X3 ) @ B2 ) )
=> ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1087_image__subsetI,axiom,
! [A2: set_set_a,F: set_a > a,B2: set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( member_a @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_a @ ( image_set_a_a @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1088_subset__imageE,axiom,
! [B2: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A2 )
=> ( B2
!= ( image_nat_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_1089_subset__imageE,axiom,
! [B2: set_a,F: a > a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F @ A2 ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A2 )
=> ( B2
!= ( image_a_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_1090_subset__imageE,axiom,
! [B2: set_set_o,F: a > set_o,A2: set_a] :
( ( ord_le4374716579403074808_set_o @ B2 @ ( image_a_set_o @ F @ A2 ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A2 )
=> ( B2
!= ( image_a_set_o @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_1091_subset__imageE,axiom,
! [B2: set_set_a,F: a > set_a,A2: set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ ( image_a_set_a @ F @ A2 ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A2 )
=> ( B2
!= ( image_a_set_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_1092_subset__imageE,axiom,
! [B2: set_a,F: set_a > a,A2: set_set_a] :
( ( ord_less_eq_set_a @ B2 @ ( image_set_a_a @ F @ A2 ) )
=> ~ ! [C3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C3 @ A2 )
=> ( B2
!= ( image_set_a_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_1093_subset__imageE,axiom,
! [B2: set_set_set_a,F: a > set_set_a,A2: set_a] :
( ( ord_le5722252365846178494_set_a @ B2 @ ( image_a_set_set_a @ F @ A2 ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A2 )
=> ( B2
!= ( image_a_set_set_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_1094_subset__imageE,axiom,
! [B2: set_Product_prod_a_a,F: a > product_prod_a_a,A2: set_a] :
( ( ord_le746702958409616551od_a_a @ B2 @ ( image_7400625782589995694od_a_a @ F @ A2 ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A2 )
=> ( B2
!= ( image_7400625782589995694od_a_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_1095_subset__imageE,axiom,
! [B2: set_set_a,F: set_a > set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ ( image_set_a_set_a @ F @ A2 ) )
=> ~ ! [C3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C3 @ A2 )
=> ( B2
!= ( image_set_a_set_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_1096_subset__imageE,axiom,
! [B2: set_a,F: product_prod_a_a > a,A2: set_Product_prod_a_a] :
( ( ord_less_eq_set_a @ B2 @ ( image_3437945252899457948_a_a_a @ F @ A2 ) )
=> ~ ! [C3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ C3 @ A2 )
=> ( B2
!= ( image_3437945252899457948_a_a_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_1097_subset__imageE,axiom,
! [B2: set_se5735800977113168103od_a_a,F: a > set_Product_prod_a_a,A2: set_a] :
( ( ord_le1995061765932249223od_a_a @ B2 @ ( image_4421510592991446670od_a_a @ F @ A2 ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A2 )
=> ( B2
!= ( image_4421510592991446670od_a_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_1098_image__subset__iff,axiom,
! [F: set_a > set_Pr5530083903271594800od_a_a,A2: set_set_a,B2: set_se9027383378080648592od_a_a] :
( ( ord_le5596166698269566256od_a_a @ ( image_7562202058474640471od_a_a @ F @ A2 ) @ B2 )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( member4210947715425868889od_a_a @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_1099_image__subset__iff,axiom,
! [F: set_a > set_Product_prod_a_a,A2: set_set_a,B2: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ ( image_6165024369500519726od_a_a @ F @ A2 ) @ B2 )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( member1816616512716248880od_a_a @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_1100_image__subset__iff,axiom,
! [F: a > set_set_a,A2: set_a,B2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ ( image_a_set_set_a @ F @ A2 ) @ B2 )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_set_set_a @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_1101_image__subset__iff,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B2 )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_1102_image__subset__iff,axiom,
! [F: a > set_Pr5530083903271594800od_a_a,A2: set_a,B2: set_se9027383378080648592od_a_a] :
( ( ord_le5596166698269566256od_a_a @ ( image_5653227685612666295od_a_a @ F @ A2 ) @ B2 )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member4210947715425868889od_a_a @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_1103_image__subset__iff,axiom,
! [F: a > set_Product_prod_a_a,A2: set_a,B2: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ ( image_4421510592991446670od_a_a @ F @ A2 ) @ B2 )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member1816616512716248880od_a_a @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_1104_image__subset__iff,axiom,
! [F: a > set_o,A2: set_a,B2: set_set_o] :
( ( ord_le4374716579403074808_set_o @ ( image_a_set_o @ F @ A2 ) @ B2 )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_set_o @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_1105_image__subset__iff,axiom,
! [F: list_a > produc4044097585999906000od_a_a,A2: set_list_a,B2: set_Pr5530083903271594800od_a_a] :
( ( ord_le114883831454073552od_a_a @ ( image_1195025184546981201od_a_a @ F @ A2 ) @ B2 )
= ( ! [X2: list_a] :
( ( member_list_a @ X2 @ A2 )
=> ( member3071122053849602553od_a_a @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_1106_image__subset__iff,axiom,
! [F: a > set_a,A2: set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A2 ) @ B2 )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_set_a @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_1107_image__subset__iff,axiom,
! [F: set_a > set_a,A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ F @ A2 ) @ B2 )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( member_set_a @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_1108_subset__image__iff,axiom,
! [B2: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B2
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1109_subset__image__iff,axiom,
! [B2: set_a,F: a > a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F @ A2 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A2 )
& ( B2
= ( image_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1110_subset__image__iff,axiom,
! [B2: set_set_o,F: a > set_o,A2: set_a] :
( ( ord_le4374716579403074808_set_o @ B2 @ ( image_a_set_o @ F @ A2 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A2 )
& ( B2
= ( image_a_set_o @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1111_subset__image__iff,axiom,
! [B2: set_set_a,F: a > set_a,A2: set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ ( image_a_set_a @ F @ A2 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A2 )
& ( B2
= ( image_a_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1112_subset__image__iff,axiom,
! [B2: set_a,F: set_a > a,A2: set_set_a] :
( ( ord_less_eq_set_a @ B2 @ ( image_set_a_a @ F @ A2 ) )
= ( ? [AA: set_set_a] :
( ( ord_le3724670747650509150_set_a @ AA @ A2 )
& ( B2
= ( image_set_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1113_subset__image__iff,axiom,
! [B2: set_set_set_a,F: a > set_set_a,A2: set_a] :
( ( ord_le5722252365846178494_set_a @ B2 @ ( image_a_set_set_a @ F @ A2 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A2 )
& ( B2
= ( image_a_set_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1114_subset__image__iff,axiom,
! [B2: set_Product_prod_a_a,F: a > product_prod_a_a,A2: set_a] :
( ( ord_le746702958409616551od_a_a @ B2 @ ( image_7400625782589995694od_a_a @ F @ A2 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A2 )
& ( B2
= ( image_7400625782589995694od_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1115_subset__image__iff,axiom,
! [B2: set_set_a,F: set_a > set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ ( image_set_a_set_a @ F @ A2 ) )
= ( ? [AA: set_set_a] :
( ( ord_le3724670747650509150_set_a @ AA @ A2 )
& ( B2
= ( image_set_a_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1116_subset__image__iff,axiom,
! [B2: set_a,F: product_prod_a_a > a,A2: set_Product_prod_a_a] :
( ( ord_less_eq_set_a @ B2 @ ( image_3437945252899457948_a_a_a @ F @ A2 ) )
= ( ? [AA: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ AA @ A2 )
& ( B2
= ( image_3437945252899457948_a_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1117_subset__image__iff,axiom,
! [B2: set_se5735800977113168103od_a_a,F: a > set_Product_prod_a_a,A2: set_a] :
( ( ord_le1995061765932249223od_a_a @ B2 @ ( image_4421510592991446670od_a_a @ F @ A2 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A2 )
& ( B2
= ( image_4421510592991446670od_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1118_ccSUP__empty,axiom,
! [F: a > set_a] :
( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ bot_bot_set_a ) )
= bot_bot_set_a ) ).
% ccSUP_empty
thf(fact_1119_ccSUP__empty,axiom,
! [F: nat > set_a] :
( ( comple2307003609928055243_set_a @ ( image_nat_set_a @ F @ bot_bot_set_nat ) )
= bot_bot_set_a ) ).
% ccSUP_empty
thf(fact_1120_ccSUP__empty,axiom,
! [F: product_prod_a_a > set_a] :
( ( comple2307003609928055243_set_a @ ( image_9052089385058188540_set_a @ F @ bot_bo3357376287454694259od_a_a ) )
= bot_bot_set_a ) ).
% ccSUP_empty
thf(fact_1121_ccSUP__empty,axiom,
! [F: set_a > set_o] :
( ( comple90263536869209701_set_o @ ( image_set_a_set_o @ F @ bot_bot_set_set_a ) )
= bot_bot_set_o ) ).
% ccSUP_empty
thf(fact_1122_ccSUP__empty,axiom,
! [F: a > set_o] :
( ( comple90263536869209701_set_o @ ( image_a_set_o @ F @ bot_bot_set_a ) )
= bot_bot_set_o ) ).
% ccSUP_empty
thf(fact_1123_ccSUP__empty,axiom,
! [F: nat > set_o] :
( ( comple90263536869209701_set_o @ ( image_nat_set_o @ F @ bot_bot_set_nat ) )
= bot_bot_set_o ) ).
% ccSUP_empty
thf(fact_1124_ccSUP__empty,axiom,
! [F: product_prod_a_a > set_o] :
( ( comple90263536869209701_set_o @ ( image_7872567118757839382_set_o @ F @ bot_bo3357376287454694259od_a_a ) )
= bot_bot_set_o ) ).
% ccSUP_empty
thf(fact_1125_finite__inc__sedges,axiom,
! [V: a] :
( ( finite_finite_set_a @ edges )
=> ( finite_finite_set_a @ ( undire1270416042309875431dges_a @ edges @ V ) ) ) ).
% finite_inc_sedges
thf(fact_1126_empty__not__edge,axiom,
~ ( member_set_a @ bot_bot_set_a @ edges ) ).
% empty_not_edge
thf(fact_1127_finite__incident__loops,axiom,
! [V: a] : ( finite_finite_set_a @ ( undire4753905205749729249oops_a @ edges @ V ) ) ).
% finite_incident_loops
thf(fact_1128_finite__incident__edges,axiom,
! [V: a] :
( ( finite_finite_set_a @ edges )
=> ( finite_finite_set_a @ ( undire3231912044278729248dges_a @ edges @ V ) ) ) ).
% finite_incident_edges
thf(fact_1129_triangle__set__graph__edge__ss__bound,axiom,
! [E: set_set_a] :
( ( ord_le3724670747650509150_set_a @ E @ edges )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ ( graph_triangle_set_a @ E ) ) @ ( finite_card_set_a @ ( graph_triangle_set_a @ edges ) ) ) ) ).
% triangle_set_graph_edge_ss_bound
thf(fact_1130_edge__adj__def,axiom,
! [E1: set_a,E2: set_a] :
( ( undire4022703626023482010_adj_a @ edges @ E1 @ E2 )
= ( ( ( inf_inf_set_a @ E1 @ E2 )
!= bot_bot_set_a )
& ( member_set_a @ E1 @ edges )
& ( member_set_a @ E2 @ edges ) ) ) ).
% edge_adj_def
thf(fact_1131_edge__density__commute,axiom,
! [X4: set_a,Y5: set_a] :
( ( undire297304480579013331sity_a @ edges @ X4 @ Y5 )
= ( undire297304480579013331sity_a @ edges @ Y5 @ X4 ) ) ).
% edge_density_commute
thf(fact_1132_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N3: nat] : ( ord_less_eq_nat @ N3 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_1133_degree__set__def,axiom,
! [Vs: set_a] :
( ( undire5448620520650870808_set_a @ edges @ Vs )
= ( finite_card_set_a
@ ( collect_set_a
@ ^ [E3: set_a] :
( ( member_set_a @ E3 @ edges )
& ( ord_less_eq_set_a @ Vs @ E3 ) ) ) ) ) ).
% degree_set_def
thf(fact_1134_incident__loops__simp_I2_J,axiom,
! [V: a] :
( ~ ( undire3617971648856834880loop_a @ edges @ V )
=> ( ( undire4753905205749729249oops_a @ edges @ V )
= bot_bot_set_set_a ) ) ).
% incident_loops_simp(2)
thf(fact_1135_incident__edges__sedges,axiom,
! [V: a] :
( ~ ( undire3617971648856834880loop_a @ edges @ V )
=> ( ( undire3231912044278729248dges_a @ edges @ V )
= ( undire1270416042309875431dges_a @ edges @ V ) ) ) ).
% incident_edges_sedges
thf(fact_1136_card__triangle__triples__rotate,axiom,
! [X4: set_a,Y5: set_a,Z4: set_a] :
( ( finite6893194910719049976od_a_a @ ( graph_4774508486909600516ples_a @ edges @ X4 @ Y5 @ Z4 ) )
= ( finite6893194910719049976od_a_a @ ( graph_4774508486909600516ples_a @ edges @ Y5 @ Z4 @ X4 ) ) ) ).
% card_triangle_triples_rotate
thf(fact_1137_incident__loops__card,axiom,
! [V: a] : ( ord_less_eq_nat @ ( finite_card_set_a @ ( undire4753905205749729249oops_a @ edges @ V ) ) @ one_one_nat ) ).
% incident_loops_card
thf(fact_1138_incident__edges__union,axiom,
! [V: a] :
( ( undire3231912044278729248dges_a @ edges @ V )
= ( sup_sup_set_set_a @ ( undire1270416042309875431dges_a @ edges @ V ) @ ( undire4753905205749729249oops_a @ edges @ V ) ) ) ).
% incident_edges_union
thf(fact_1139_incident__edges__def,axiom,
! [V: a] :
( ( undire3231912044278729248dges_a @ edges @ V )
= ( collect_set_a
@ ^ [E3: set_a] :
( ( member_set_a @ E3 @ edges )
& ( undire1521409233611534436dent_a @ V @ E3 ) ) ) ) ).
% incident_edges_def
thf(fact_1140_edge__density__zero,axiom,
! [Y5: set_a,X4: set_a] :
( ( Y5 = bot_bot_set_a )
=> ( ( undire297304480579013331sity_a @ edges @ X4 @ Y5 )
= zero_zero_real ) ) ).
% edge_density_zero
thf(fact_1141_incident__loops__def,axiom,
! [V: a] :
( ( undire4753905205749729249oops_a @ edges @ V )
= ( collect_set_a
@ ^ [E3: set_a] :
( ( member_set_a @ E3 @ edges )
& ( E3
= ( insert_a @ V @ bot_bot_set_a ) ) ) ) ) ).
% incident_loops_def
thf(fact_1142_induced__edges__def,axiom,
! [V3: set_a] :
( ( undire7777452895879145676dges_a @ edges @ V3 )
= ( collect_set_a
@ ^ [E3: set_a] :
( ( member_set_a @ E3 @ edges )
& ( ord_less_eq_set_a @ E3 @ V3 ) ) ) ) ).
% induced_edges_def
thf(fact_1143_incident__def,axiom,
undire1521409233611534436dent_a = member_a ).
% incident_def
thf(fact_1144_edge__vertices__not__equal,axiom,
! [X: a,Y: a] :
( ( member_set_a @ ( insert_a @ X @ ( insert_a @ Y @ bot_bot_set_a ) ) @ edges )
=> ( X != Y ) ) ).
% edge_vertices_not_equal
thf(fact_1145_singleton__not__edge,axiom,
! [X: a] :
~ ( member_set_a @ ( insert_a @ X @ bot_bot_set_a ) @ edges ) ).
% singleton_not_edge
thf(fact_1146_edge__density__ge0,axiom,
! [X4: set_a,Y5: set_a] : ( ord_less_eq_real @ zero_zero_real @ ( undire297304480579013331sity_a @ edges @ X4 @ Y5 ) ) ).
% edge_density_ge0
thf(fact_1147_card1__incident__imp__vert,axiom,
! [V: a,E4: set_a] :
( ( ( undire1521409233611534436dent_a @ V @ E4 )
& ( ( finite_card_a @ E4 )
= one_one_nat ) )
=> ( E4
= ( insert_a @ V @ bot_bot_set_a ) ) ) ).
% card1_incident_imp_vert
thf(fact_1148_triangle__in__graph__def,axiom,
! [X: a,Y: a,Z: a] :
( ( graph_4582152751571636272raph_a @ edges @ X @ Y @ Z )
= ( ( member_set_a @ ( insert_a @ X @ ( insert_a @ Y @ bot_bot_set_a ) ) @ edges )
& ( member_set_a @ ( insert_a @ Y @ ( insert_a @ Z @ bot_bot_set_a ) ) @ edges )
& ( member_set_a @ ( insert_a @ X @ ( insert_a @ Z @ bot_bot_set_a ) ) @ edges ) ) ) ).
% triangle_in_graph_def
thf(fact_1149_has__loop__def,axiom,
! [V: a] :
( ( undire3617971648856834880loop_a @ edges @ V )
= ( member_set_a @ ( insert_a @ V @ bot_bot_set_a ) @ edges ) ) ).
% has_loop_def
thf(fact_1150_is__edge__between__def,axiom,
( undire8544646567961481629ween_a
= ( ^ [X7: set_a,Y7: set_a,E3: set_a] :
? [X2: a,Y2: a] :
( ( E3
= ( insert_a @ X2 @ ( insert_a @ Y2 @ bot_bot_set_a ) ) )
& ( member_a @ X2 @ X7 )
& ( member_a @ Y2 @ Y7 ) ) ) ) ).
% is_edge_between_def
thf(fact_1151_incident__loops__alt,axiom,
! [V: a] :
( ( undire4753905205749729249oops_a @ edges @ V )
= ( collect_set_a
@ ^ [E3: set_a] :
( ( member_set_a @ E3 @ edges )
& ( undire1521409233611534436dent_a @ V @ E3 )
& ( ( finite_card_a @ E3 )
= one_one_nat ) ) ) ) ).
% incident_loops_alt
thf(fact_1152_unique__triangles__def,axiom,
( ( graph_6144490306505338871gles_a @ edges )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ edges )
=> ? [Y2: set_a] :
( ? [Z5: a,Aa: a,Ab: a] :
( ( Y2
= ( insert_a @ Z5 @ ( insert_a @ Aa @ ( insert_a @ Ab @ bot_bot_set_a ) ) ) )
& ( graph_4582152751571636272raph_a @ edges @ Z5 @ Aa @ Ab )
& ( ord_less_eq_set_a @ X2 @ Y2 ) )
& ! [Z5: set_a] :
( ? [Aa: a,Ab: a,Ac: a] :
( ( Z5
= ( insert_a @ Aa @ ( insert_a @ Ab @ ( insert_a @ Ac @ bot_bot_set_a ) ) ) )
& ( graph_4582152751571636272raph_a @ edges @ Aa @ Ab @ Ac )
& ( ord_less_eq_set_a @ X2 @ Z5 ) )
=> ( Z5 = Y2 ) ) ) ) ) ) ).
% unique_triangles_def
thf(fact_1153_incident__loops__simp_I1_J,axiom,
! [V: a] :
( ( undire3617971648856834880loop_a @ edges @ V )
=> ( ( undire4753905205749729249oops_a @ edges @ V )
= ( insert_set_a @ ( insert_a @ V @ bot_bot_set_a ) @ bot_bot_set_set_a ) ) ) ).
% incident_loops_simp(1)
thf(fact_1154_vert__adj__inc__edge__iff,axiom,
! [V1: a,V22: a] :
( ( undire397441198561214472_adj_a @ edges @ V1 @ V22 )
= ( ( undire1521409233611534436dent_a @ V1 @ ( insert_a @ V1 @ ( insert_a @ V22 @ bot_bot_set_a ) ) )
& ( undire1521409233611534436dent_a @ V22 @ ( insert_a @ V1 @ ( insert_a @ V22 @ bot_bot_set_a ) ) )
& ( member_set_a @ ( insert_a @ V1 @ ( insert_a @ V22 @ bot_bot_set_a ) ) @ edges ) ) ) ).
% vert_adj_inc_edge_iff
thf(fact_1155_triangle__in__graph__edge__point,axiom,
! [X: a,Y: a,Z: a] :
( ( graph_4582152751571636272raph_a @ edges @ X @ Y @ Z )
= ( ( member_set_a @ ( insert_a @ Y @ ( insert_a @ Z @ bot_bot_set_a ) ) @ edges )
& ( undire397441198561214472_adj_a @ edges @ X @ Y )
& ( undire397441198561214472_adj_a @ edges @ X @ Z ) ) ) ).
% triangle_in_graph_edge_point
thf(fact_1156_vert__adj__sym,axiom,
! [V1: a,V22: a] :
( ( undire397441198561214472_adj_a @ edges @ V1 @ V22 )
= ( undire397441198561214472_adj_a @ edges @ V22 @ V1 ) ) ).
% vert_adj_sym
thf(fact_1157_vert__adj__edge__iff2,axiom,
! [V1: a,V22: a] :
( ( V1 != V22 )
=> ( ( undire397441198561214472_adj_a @ edges @ V1 @ V22 )
= ( ? [X2: set_a] :
( ( member_set_a @ X2 @ edges )
& ( undire1521409233611534436dent_a @ V1 @ X2 )
& ( undire1521409233611534436dent_a @ V22 @ X2 ) ) ) ) ) ).
% vert_adj_edge_iff2
thf(fact_1158_vert__adj__def,axiom,
! [V1: a,V22: a] :
( ( undire397441198561214472_adj_a @ edges @ V1 @ V22 )
= ( member_set_a @ ( insert_a @ V1 @ ( insert_a @ V22 @ bot_bot_set_a ) ) @ edges ) ) ).
% vert_adj_def
thf(fact_1159_not__vert__adj,axiom,
! [V: a,U: a] :
( ~ ( undire397441198561214472_adj_a @ edges @ V @ U )
=> ~ ( member_set_a @ ( insert_a @ V @ ( insert_a @ U @ bot_bot_set_a ) ) @ edges ) ) ).
% not_vert_adj
thf(fact_1160_neighbors__ss__def,axiom,
! [X: a,Y5: set_a] :
( ( undire401937927514038589s_ss_a @ edges @ X @ Y5 )
= ( collect_a
@ ^ [Y2: a] :
( ( member_a @ Y2 @ Y5 )
& ( undire397441198561214472_adj_a @ edges @ X @ Y2 ) ) ) ) ).
% neighbors_ss_def
thf(fact_1161_is__loop__def,axiom,
( undire2905028936066782638loop_a
= ( ^ [E3: set_a] :
( ( finite_card_a @ E3 )
= one_one_nat ) ) ) ).
% is_loop_def
thf(fact_1162_induced__edges__union,axiom,
! [VH1: set_a,S: set_a,VH2: set_a,T2: set_a,EH1: set_set_a,EH2: set_set_a] :
( ( ord_less_eq_set_a @ VH1 @ S )
=> ( ( ord_less_eq_set_a @ VH2 @ T2 )
=> ( ( undire2554140024507503526stem_a @ VH1 @ EH1 )
=> ( ( undire2554140024507503526stem_a @ VH2 @ EH2 )
=> ( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ EH1 @ EH2 ) @ ( undire7777452895879145676dges_a @ edges @ ( sup_sup_set_a @ S @ T2 ) ) )
=> ( ord_le3724670747650509150_set_a @ EH1 @ ( undire7777452895879145676dges_a @ edges @ S ) ) ) ) ) ) ) ).
% induced_edges_union
thf(fact_1163_induced__is__graph__sys,axiom,
! [V3: set_a] : ( undire2554140024507503526stem_a @ V3 @ ( undire7777452895879145676dges_a @ edges @ V3 ) ) ).
% induced_is_graph_sys
thf(fact_1164_edges__split__loop,axiom,
( edges
= ( sup_sup_set_set_a
@ ( collect_set_a
@ ^ [E3: set_a] :
( ( member_set_a @ E3 @ edges )
& ( undire2905028936066782638loop_a @ E3 ) ) )
@ ( collect_set_a
@ ^ [E3: set_a] :
( ( member_set_a @ E3 @ edges )
& ( undire4917966558017083288edge_a @ E3 ) ) ) ) ) ).
% edges_split_loop
thf(fact_1165_edges__split__loop__inter__empty,axiom,
( bot_bot_set_set_a
= ( inf_inf_set_set_a
@ ( collect_set_a
@ ^ [E3: set_a] :
( ( member_set_a @ E3 @ edges )
& ( undire2905028936066782638loop_a @ E3 ) ) )
@ ( collect_set_a
@ ^ [E3: set_a] :
( ( member_set_a @ E3 @ edges )
& ( undire4917966558017083288edge_a @ E3 ) ) ) ) ) ).
% edges_split_loop_inter_empty
thf(fact_1166_is__edge__or__loop,axiom,
! [E4: set_a] :
( ( member_set_a @ E4 @ edges )
=> ( ( undire2905028936066782638loop_a @ E4 )
| ( undire4917966558017083288edge_a @ E4 ) ) ) ).
% is_edge_or_loop
thf(fact_1167_Sup__nat__empty,axiom,
( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Sup_nat_empty
thf(fact_1168_degree0__inc__edges__empt__iff,axiom,
! [V: a] :
( ( finite_finite_set_a @ edges )
=> ( ( ( undire8867928226783802224gree_a @ edges @ V )
= zero_zero_nat )
= ( ( undire3231912044278729248dges_a @ edges @ V )
= bot_bot_set_set_a ) ) ) ).
% degree0_inc_edges_empt_iff
thf(fact_1169_induced__union__subgraph,axiom,
! [VH1: set_a,S: set_a,VH2: set_a,T2: set_a,EH1: set_set_a,EH2: set_set_a] :
( ( ord_less_eq_set_a @ VH1 @ S )
=> ( ( ord_less_eq_set_a @ VH2 @ T2 )
=> ( ( undire2554140024507503526stem_a @ VH1 @ EH1 )
=> ( ( undire2554140024507503526stem_a @ VH2 @ EH2 )
=> ( ( ( undire7103218114511261257raph_a @ VH1 @ EH1 @ S @ ( undire7777452895879145676dges_a @ edges @ S ) )
& ( undire7103218114511261257raph_a @ VH2 @ EH2 @ T2 @ ( undire7777452895879145676dges_a @ edges @ T2 ) ) )
= ( undire7103218114511261257raph_a @ ( sup_sup_set_a @ VH1 @ VH2 ) @ ( sup_sup_set_set_a @ EH1 @ EH2 ) @ ( sup_sup_set_a @ S @ T2 ) @ ( undire7777452895879145676dges_a @ edges @ ( sup_sup_set_a @ S @ T2 ) ) ) ) ) ) ) ) ).
% induced_union_subgraph
thf(fact_1170_induced__edges__union__subgraph__single,axiom,
! [VH1: set_a,S: set_a,VH2: set_a,T2: set_a,EH1: set_set_a,EH2: set_set_a] :
( ( ord_less_eq_set_a @ VH1 @ S )
=> ( ( ord_less_eq_set_a @ VH2 @ T2 )
=> ( ( undire2554140024507503526stem_a @ VH1 @ EH1 )
=> ( ( undire2554140024507503526stem_a @ VH2 @ EH2 )
=> ( ( undire7103218114511261257raph_a @ ( sup_sup_set_a @ VH1 @ VH2 ) @ ( sup_sup_set_set_a @ EH1 @ EH2 ) @ ( sup_sup_set_a @ S @ T2 ) @ ( undire7777452895879145676dges_a @ edges @ ( sup_sup_set_a @ S @ T2 ) ) )
=> ( undire7103218114511261257raph_a @ VH1 @ EH1 @ S @ ( undire7777452895879145676dges_a @ edges @ S ) ) ) ) ) ) ) ).
% induced_edges_union_subgraph_single
thf(fact_1171_alt__degree__def,axiom,
! [V: a] :
( ( undire8867928226783802224gree_a @ edges @ V )
= ( finite_card_set_a @ ( undire3231912044278729248dges_a @ edges @ V ) ) ) ).
% alt_degree_def
thf(fact_1172_degree__no__loops,axiom,
! [V: a] :
( ~ ( undire3617971648856834880loop_a @ edges @ V )
=> ( ( undire8867928226783802224gree_a @ edges @ V )
= ( finite_card_set_a @ ( undire3231912044278729248dges_a @ edges @ V ) ) ) ) ).
% degree_no_loops
thf(fact_1173_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1174_le0,axiom,
! [N4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N4 ) ).
% le0
thf(fact_1175_is__loop__set__alt,axiom,
( ( collect_set_a
@ ^ [Uu: set_a] :
? [V4: a] :
( ( Uu
= ( insert_a @ V4 @ bot_bot_set_a ) )
& ( undire3617971648856834880loop_a @ edges @ V4 ) ) )
= ( collect_set_a
@ ^ [E3: set_a] :
( ( member_set_a @ E3 @ edges )
& ( undire2905028936066782638loop_a @ E3 ) ) ) ) ).
% is_loop_set_alt
thf(fact_1176_triangle__set__def,axiom,
( ( graph_triangle_set_a @ edges )
= ( collect_set_a
@ ^ [Uu: set_a] :
? [X2: a,Y2: a,Z5: a] :
( ( Uu
= ( insert_a @ X2 @ ( insert_a @ Y2 @ ( insert_a @ Z5 @ bot_bot_set_a ) ) ) )
& ( graph_4582152751571636272raph_a @ edges @ X2 @ Y2 @ Z5 ) ) ) ) ).
% triangle_set_def
thf(fact_1177_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1178_nat__le__linear,axiom,
! [M2: nat,N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
| ( ord_less_eq_nat @ N4 @ M2 ) ) ).
% nat_le_linear
thf(fact_1179_le__antisym,axiom,
! [M2: nat,N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
=> ( ( ord_less_eq_nat @ N4 @ M2 )
=> ( M2 = N4 ) ) ) ).
% le_antisym
thf(fact_1180_eq__imp__le,axiom,
! [M2: nat,N4: nat] :
( ( M2 = N4 )
=> ( ord_less_eq_nat @ M2 @ N4 ) ) ).
% eq_imp_le
thf(fact_1181_le__trans,axiom,
! [I3: nat,J3: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ( ord_less_eq_nat @ J3 @ K )
=> ( ord_less_eq_nat @ I3 @ K ) ) ) ).
% le_trans
thf(fact_1182_le__refl,axiom,
! [N4: nat] : ( ord_less_eq_nat @ N4 @ N4 ) ).
% le_refl
thf(fact_1183_le__0__eq,axiom,
! [N4: nat] :
( ( ord_less_eq_nat @ N4 @ zero_zero_nat )
= ( N4 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1184_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1185_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1186_less__eq__nat_Osimps_I1_J,axiom,
! [N4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N4 ) ).
% less_eq_nat.simps(1)
thf(fact_1187_edge__density__le1,axiom,
! [X4: set_a,Y5: set_a] : ( ord_less_eq_real @ ( undire297304480579013331sity_a @ edges @ X4 @ Y5 ) @ one_one_real ) ).
% edge_density_le1
thf(fact_1188_all__edges__between__def,axiom,
! [X4: set_a,Y5: set_a] :
( ( undire8383842906760478443ween_a @ edges @ X4 @ Y5 )
= ( collec3336397797384452498od_a_a
@ ( produc6436628058953941356_a_a_o
@ ^ [X2: a,Y2: a] :
( ( member_a @ X2 @ X4 )
& ( member_a @ Y2 @ Y5 )
& ( member_set_a @ ( insert_a @ X2 @ ( insert_a @ Y2 @ bot_bot_set_a ) ) @ edges ) ) ) ) ) ).
% all_edges_between_def
thf(fact_1189_finite__all__edges__between_H,axiom,
! [X4: set_a,Y5: set_a] : ( finite6544458595007987280od_a_a @ ( undire8383842906760478443ween_a @ edges @ X4 @ Y5 ) ) ).
% finite_all_edges_between'
thf(fact_1190_finite__all__edges__between,axiom,
! [X4: set_a,Y5: set_a] :
( ( finite_finite_a @ X4 )
=> ( ( finite_finite_a @ Y5 )
=> ( finite6544458595007987280od_a_a @ ( undire8383842906760478443ween_a @ edges @ X4 @ Y5 ) ) ) ) ).
% finite_all_edges_between
thf(fact_1191_card__all__edges__between__commute,axiom,
! [X4: set_a,Y5: set_a] :
( ( finite4795055649997197647od_a_a @ ( undire8383842906760478443ween_a @ edges @ X4 @ Y5 ) )
= ( finite4795055649997197647od_a_a @ ( undire8383842906760478443ween_a @ edges @ Y5 @ X4 ) ) ) ).
% card_all_edges_between_commute
thf(fact_1192_all__edges__between__Union1,axiom,
! [X8: set_set_a,Y5: set_a] :
( ( undire8383842906760478443ween_a @ edges @ ( comple2307003609928055243_set_a @ X8 ) @ Y5 )
= ( comple8421679170691845492od_a_a
@ ( image_6165024369500519726od_a_a
@ ^ [X7: set_a] : ( undire8383842906760478443ween_a @ edges @ X7 @ Y5 )
@ X8 ) ) ) ).
% all_edges_between_Union1
thf(fact_1193_all__edges__between__Union2,axiom,
! [X4: set_a,Y8: set_set_a] :
( ( undire8383842906760478443ween_a @ edges @ X4 @ ( comple2307003609928055243_set_a @ Y8 ) )
= ( comple8421679170691845492od_a_a @ ( image_6165024369500519726od_a_a @ ( undire8383842906760478443ween_a @ edges @ X4 ) @ Y8 ) ) ) ).
% all_edges_between_Union2
thf(fact_1194_all__edges__between__mono1,axiom,
! [Y5: set_a,Z4: set_a,X4: set_a] :
( ( ord_less_eq_set_a @ Y5 @ Z4 )
=> ( ord_le746702958409616551od_a_a @ ( undire8383842906760478443ween_a @ edges @ Y5 @ X4 ) @ ( undire8383842906760478443ween_a @ edges @ Z4 @ X4 ) ) ) ).
% all_edges_between_mono1
thf(fact_1195_all__edges__between__mono2,axiom,
! [Y5: set_a,Z4: set_a,X4: set_a] :
( ( ord_less_eq_set_a @ Y5 @ Z4 )
=> ( ord_le746702958409616551od_a_a @ ( undire8383842906760478443ween_a @ edges @ X4 @ Y5 ) @ ( undire8383842906760478443ween_a @ edges @ X4 @ Z4 ) ) ) ).
% all_edges_between_mono2
thf(fact_1196_all__edges__between__Un1,axiom,
! [X4: set_a,Y5: set_a,Z4: set_a] :
( ( undire8383842906760478443ween_a @ edges @ ( sup_sup_set_a @ X4 @ Y5 ) @ Z4 )
= ( sup_su3048258781599657691od_a_a @ ( undire8383842906760478443ween_a @ edges @ X4 @ Z4 ) @ ( undire8383842906760478443ween_a @ edges @ Y5 @ Z4 ) ) ) ).
% all_edges_between_Un1
thf(fact_1197_all__edges__between__Un2,axiom,
! [X4: set_a,Y5: set_a,Z4: set_a] :
( ( undire8383842906760478443ween_a @ edges @ X4 @ ( sup_sup_set_a @ Y5 @ Z4 ) )
= ( sup_su3048258781599657691od_a_a @ ( undire8383842906760478443ween_a @ edges @ X4 @ Y5 ) @ ( undire8383842906760478443ween_a @ edges @ X4 @ Z4 ) ) ) ).
% all_edges_between_Un2
thf(fact_1198_edge__density__eq0,axiom,
! [A2: set_a,B2: set_a,X4: set_a,Y5: set_a] :
( ( ( undire8383842906760478443ween_a @ edges @ A2 @ B2 )
= bot_bo3357376287454694259od_a_a )
=> ( ( ord_less_eq_set_a @ X4 @ A2 )
=> ( ( ord_less_eq_set_a @ Y5 @ B2 )
=> ( ( undire297304480579013331sity_a @ edges @ X4 @ Y5 )
= zero_zero_real ) ) ) ) ).
% edge_density_eq0
thf(fact_1199_all__edges__between__empty_I2_J,axiom,
! [Z4: set_a] :
( ( undire8383842906760478443ween_a @ edges @ Z4 @ bot_bot_set_a )
= bot_bo3357376287454694259od_a_a ) ).
% all_edges_between_empty(2)
thf(fact_1200_all__edges__between__empty_I1_J,axiom,
! [Z4: set_a] :
( ( undire8383842906760478443ween_a @ edges @ bot_bot_set_a @ Z4 )
= bot_bo3357376287454694259od_a_a ) ).
% all_edges_between_empty(1)
thf(fact_1201_all__edges__between__set,axiom,
! [X4: set_a,Y5: set_a] :
( ( image_9052089385058188540_set_a @ undire6670514144573423676edge_a @ ( undire8383842906760478443ween_a @ edges @ X4 @ Y5 ) )
= ( collect_set_a
@ ^ [Uu: set_a] :
? [X2: a,Y2: a] :
( ( Uu
= ( insert_a @ X2 @ ( insert_a @ Y2 @ bot_bot_set_a ) ) )
& ( member_a @ X2 @ X4 )
& ( member_a @ Y2 @ Y5 )
& ( member_set_a @ ( insert_a @ X2 @ ( insert_a @ Y2 @ bot_bot_set_a ) ) @ edges ) ) ) ) ).
% all_edges_between_set
thf(fact_1202_all__edges__between__E__ss,axiom,
! [X4: set_a,Y5: set_a] : ( ord_le3724670747650509150_set_a @ ( image_9052089385058188540_set_a @ undire6670514144573423676edge_a @ ( undire8383842906760478443ween_a @ edges @ X4 @ Y5 ) ) @ edges ) ).
% all_edges_between_E_ss
thf(fact_1203_max__all__edges__between,axiom,
! [X4: set_a,Y5: set_a] :
( ( finite_finite_a @ X4 )
=> ( ( finite_finite_a @ Y5 )
=> ( ord_less_eq_nat @ ( finite4795055649997197647od_a_a @ ( undire8383842906760478443ween_a @ edges @ X4 @ Y5 ) ) @ ( times_times_nat @ ( finite_card_a @ X4 ) @ ( finite_card_a @ Y5 ) ) ) ) ) ).
% max_all_edges_between
thf(fact_1204_mult__le__mono2,axiom,
! [I3: nat,J3: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I3 ) @ ( times_times_nat @ K @ J3 ) ) ) ).
% mult_le_mono2
thf(fact_1205_mult__le__mono1,axiom,
! [I3: nat,J3: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I3 @ K ) @ ( times_times_nat @ J3 @ K ) ) ) ).
% mult_le_mono1
thf(fact_1206_mult__le__mono,axiom,
! [I3: nat,J3: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I3 @ K ) @ ( times_times_nat @ J3 @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1207_le__square,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ).
% le_square
thf(fact_1208_le__cube,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ) ).
% le_cube
thf(fact_1209_card__all__edges__betw__neighbor,axiom,
! [X4: set_a,Y5: set_a] :
( ( finite_finite_a @ X4 )
=> ( ( finite_finite_a @ Y5 )
=> ( ( finite4795055649997197647od_a_a @ ( undire8383842906760478443ween_a @ edges @ X4 @ Y5 ) )
= ( groups6334556678337121940_a_nat
@ ^ [X2: a] : ( finite_card_a @ ( undire401937927514038589s_ss_a @ edges @ X2 @ Y5 ) )
@ X4 ) ) ) ) ).
% card_all_edges_betw_neighbor
thf(fact_1210_card__all__edges__between,axiom,
! [Y5: set_a,X4: set_a] :
( ( finite_finite_a @ Y5 )
=> ( ( finite4795055649997197647od_a_a @ ( undire8383842906760478443ween_a @ edges @ X4 @ Y5 ) )
= ( groups6334556678337121940_a_nat
@ ^ [Y2: a] : ( finite4795055649997197647od_a_a @ ( undire8383842906760478443ween_a @ edges @ X4 @ ( insert_a @ Y2 @ bot_bot_set_a ) ) )
@ Y5 ) ) ) ).
% card_all_edges_between
thf(fact_1211_all__edges__between__subset__times,axiom,
! [X4: set_a,Y5: set_a] :
( ord_le746702958409616551od_a_a @ ( undire8383842906760478443ween_a @ edges @ X4 @ Y5 )
@ ( product_Sigma_a_a @ ( inf_inf_set_a @ X4 @ ( comple2307003609928055243_set_a @ edges ) )
@ ^ [Uu: a] : ( inf_inf_set_a @ Y5 @ ( comple2307003609928055243_set_a @ edges ) ) ) ) ).
% all_edges_between_subset_times
thf(fact_1212_all__edges__between__subset,axiom,
! [X4: set_a,Y5: set_a] :
( ord_le746702958409616551od_a_a @ ( undire8383842906760478443ween_a @ edges @ X4 @ Y5 )
@ ( product_Sigma_a_a @ X4
@ ^ [Uu: a] : Y5 ) ) ).
% all_edges_between_subset
thf(fact_1213_all__edges__betw__sigma__neighbor,axiom,
! [X4: set_a,Y5: set_a] :
( ( undire8383842906760478443ween_a @ edges @ X4 @ Y5 )
= ( product_Sigma_a_a @ X4
@ ^ [X2: a] : ( undire401937927514038589s_ss_a @ edges @ X2 @ Y5 ) ) ) ).
% all_edges_betw_sigma_neighbor
thf(fact_1214_all__edges__betw__prod__def__neighbors,axiom,
! [X4: set_a,Y5: set_a] :
( ( undire8383842906760478443ween_a @ edges @ X4 @ Y5 )
= ( collec3336397797384452498od_a_a
@ ( produc6436628058953941356_a_a_o
@ ^ [X2: a,Y2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 )
@ ( product_Sigma_a_a @ X4
@ ^ [Uu: a] : Y5 ) )
& ( undire397441198561214472_adj_a @ edges @ X2 @ Y2 ) ) ) ) ) ).
% all_edges_betw_prod_def_neighbors
thf(fact_1215_local_Oinj__on__mk__edge,axiom,
! [X4: set_a,Y5: set_a] :
( ( ( inf_inf_set_a @ X4 @ Y5 )
= bot_bot_set_a )
=> ( inj_on4851796814176604264_set_a @ undire6670514144573423676edge_a @ ( undire8383842906760478443ween_a @ edges @ X4 @ Y5 ) ) ) ).
% local.inj_on_mk_edge
thf(fact_1216_edge__btw__vertices__not__equal,axiom,
! [X: a,Y: a,X4: set_a,Y5: set_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( undire8383842906760478443ween_a @ edges @ X4 @ Y5 ) )
=> ( X != Y ) ) ).
% edge_btw_vertices_not_equal
thf(fact_1217_all__edges__between__swap,axiom,
! [X4: set_a,Y5: set_a] :
( ( undire8383842906760478443ween_a @ edges @ X4 @ Y5 )
= ( image_4636654165204879301od_a_a
@ ( produc408267641121961211od_a_a
@ ^ [X2: a,Y2: a] : ( product_Pair_a_a @ Y2 @ X2 ) )
@ ( undire8383842906760478443ween_a @ edges @ Y5 @ X4 ) ) ) ).
% all_edges_between_swap
thf(fact_1218_mk__triangle__from__ss__edges,axiom,
! [X: a,Y: a,X4: set_a,Y5: set_a,Z: a,Z4: set_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( undire8383842906760478443ween_a @ edges @ X4 @ Y5 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Z ) @ ( undire8383842906760478443ween_a @ edges @ X4 @ Z4 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ ( undire8383842906760478443ween_a @ edges @ Y5 @ Z4 ) )
=> ( graph_4582152751571636272raph_a @ edges @ X @ Y @ Z ) ) ) ) ).
% mk_triangle_from_ss_edges
thf(fact_1219_all__edges__betw__I,axiom,
! [X: a,X4: set_a,Y: a,Y5: set_a] :
( ( member_a @ X @ X4 )
=> ( ( member_a @ Y @ Y5 )
=> ( ( member_set_a @ ( insert_a @ X @ ( insert_a @ Y @ bot_bot_set_a ) ) @ edges )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( undire8383842906760478443ween_a @ edges @ X4 @ Y5 ) ) ) ) ) ).
% all_edges_betw_I
thf(fact_1220_all__edges__betw__D3,axiom,
! [X: a,Y: a,X4: set_a,Y5: set_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( undire8383842906760478443ween_a @ edges @ X4 @ Y5 ) )
=> ( member_set_a @ ( insert_a @ X @ ( insert_a @ Y @ bot_bot_set_a ) ) @ edges ) ) ).
% all_edges_betw_D3
thf(fact_1221_in__tofl,axiom,
! [X: a,Y: a,Z: a] :
( ( X != Y )
=> ( ( Y != Z )
=> ( ( X != Z )
=> ( member3071122053849602553od_a_a @ ( produc431845341423274048od_a_a @ X @ ( product_Pair_a_a @ Y @ Z ) ) @ ( image_1195025184546981201od_a_a @ tofl @ ( multis2428024204330136193_set_a @ ( insert_a @ X @ ( insert_a @ Y @ ( insert_a @ Z @ bot_bot_set_a ) ) ) ) ) ) ) ) ) ).
% in_tofl
thf(fact_1222_triangle__triples__def,axiom,
! [X4: set_a,Y5: set_a,Z4: set_a] :
( ( graph_4774508486909600516ples_a @ edges @ X4 @ Y5 @ Z4 )
= ( collec5045780995415420475od_a_a
@ ( produc5856822985862792195_a_a_o
@ ^ [X2: a] :
( produc6436628058953941356_a_a_o
@ ^ [Y2: a,Z5: a] :
( ( member3071122053849602553od_a_a @ ( produc431845341423274048od_a_a @ X2 @ ( product_Pair_a_a @ Y2 @ Z5 ) )
@ ( produc6342321021181284593od_a_a @ X4
@ ^ [Uu: a] :
( product_Sigma_a_a @ Y5
@ ^ [Uv: a] : Z4 ) ) )
& ( graph_4582152751571636272raph_a @ edges @ X2 @ Y2 @ Z5 ) ) ) ) ) ) ).
% triangle_triples_def
thf(fact_1223_calculation,axiom,
( ord_le114883831454073552od_a_a
@ ( collec5045780995415420475od_a_a
@ ( produc5856822985862792195_a_a_o
@ ^ [X2: a] :
( produc6436628058953941356_a_a_o
@ ^ [Y2: a,Z5: a] :
( ( member3071122053849602553od_a_a @ ( produc431845341423274048od_a_a @ X2 @ ( product_Pair_a_a @ Y2 @ Z5 ) )
@ ( produc6342321021181284593od_a_a @ x
@ ^ [Uu: a] :
( product_Sigma_a_a @ y
@ ^ [Uv: a] : z ) ) )
& ( graph_4582152751571636272raph_a @ edges @ X2 @ Y2 @ Z5 ) ) ) ) )
@ ( collec5045780995415420475od_a_a
@ ( produc5856822985862792195_a_a_o
@ ^ [X2: a] : ( produc6436628058953941356_a_a_o @ ( graph_4582152751571636272raph_a @ edges @ X2 ) ) ) ) ) ).
% calculation
thf(fact_1224_assms_I3_J,axiom,
ord_less_eq_set_a @ z @ vertices ).
% assms(3)
thf(fact_1225_assms_I2_J,axiom,
ord_less_eq_set_a @ y @ vertices ).
% assms(2)
thf(fact_1226_finV,axiom,
finite_finite_a @ vertices ).
% finV
thf(fact_1227_assms_I1_J,axiom,
ord_less_eq_set_a @ x @ vertices ).
% assms(1)
thf(fact_1228_fin__ulgraph__axioms,axiom,
undire7599193295422474955raph_a @ vertices @ edges ).
% fin_ulgraph_axioms
thf(fact_1229_fin__sgraph__axioms,axiom,
undire5670813279940215357raph_a @ vertices @ edges ).
% fin_sgraph_axioms
thf(fact_1230_wellformed,axiom,
! [E4: set_a] :
( ( member_set_a @ E4 @ edges )
=> ( ord_less_eq_set_a @ E4 @ vertices ) ) ).
% wellformed
thf(fact_1231_triangle__in__graph__verts_I3_J,axiom,
! [X: a,Y: a,Z: a] :
( ( graph_4582152751571636272raph_a @ edges @ X @ Y @ Z )
=> ( member_a @ Z @ vertices ) ) ).
% triangle_in_graph_verts(3)
thf(fact_1232_triangle__in__graph__verts_I2_J,axiom,
! [X: a,Y: a,Z: a] :
( ( graph_4582152751571636272raph_a @ edges @ X @ Y @ Z )
=> ( member_a @ Y @ vertices ) ) ).
% triangle_in_graph_verts(2)
thf(fact_1233_triangle__in__graph__verts_I1_J,axiom,
! [X: a,Y: a,Z: a] :
( ( graph_4582152751571636272raph_a @ edges @ X @ Y @ Z )
=> ( member_a @ X @ vertices ) ) ).
% triangle_in_graph_verts(1)
thf(fact_1234_induced__edges__self,axiom,
( ( undire7777452895879145676dges_a @ edges @ vertices )
= edges ) ).
% induced_edges_self
thf(fact_1235_vert__adj__imp__inV,axiom,
! [V1: a,V22: a] :
( ( undire397441198561214472_adj_a @ edges @ V1 @ V22 )
=> ( ( member_a @ V1 @ vertices )
& ( member_a @ V22 @ vertices ) ) ) ).
% vert_adj_imp_inV
thf(fact_1236_incident__edge__in__wf,axiom,
! [E4: set_a,V: a] :
( ( member_set_a @ E4 @ edges )
=> ( ( undire1521409233611534436dent_a @ V @ E4 )
=> ( member_a @ V @ vertices ) ) ) ).
% incident_edge_in_wf
thf(fact_1237_has__loop__in__verts,axiom,
! [V: a] :
( ( undire3617971648856834880loop_a @ edges @ V )
=> ( member_a @ V @ vertices ) ) ).
% has_loop_in_verts
thf(fact_1238_no__loops,axiom,
! [V: a] :
( ( member_a @ V @ vertices )
=> ~ ( undire3617971648856834880loop_a @ edges @ V ) ) ).
% no_loops
thf(fact_1239_subgraph__refl,axiom,
undire7103218114511261257raph_a @ vertices @ edges @ vertices @ edges ).
% subgraph_refl
thf(fact_1240_graph__system__axioms,axiom,
undire2554140024507503526stem_a @ vertices @ edges ).
% graph_system_axioms
thf(fact_1241_edge__adjacent__alt__def,axiom,
! [E1: set_a,E2: set_a] :
( ( member_set_a @ E1 @ edges )
=> ( ( member_set_a @ E2 @ edges )
=> ( ? [X6: a] :
( ( member_a @ X6 @ vertices )
& ( member_a @ X6 @ E1 )
& ( member_a @ X6 @ E2 ) )
=> ( undire4022703626023482010_adj_a @ edges @ E1 @ E2 ) ) ) ) ).
% edge_adjacent_alt_def
thf(fact_1242_wellformed__alt__snd,axiom,
! [X: a,Y: a] :
( ( member_set_a @ ( insert_a @ X @ ( insert_a @ Y @ bot_bot_set_a ) ) @ edges )
=> ( member_a @ Y @ vertices ) ) ).
% wellformed_alt_snd
thf(fact_1243_wellformed__alt__fst,axiom,
! [X: a,Y: a] :
( ( member_set_a @ ( insert_a @ X @ ( insert_a @ Y @ bot_bot_set_a ) ) @ edges )
=> ( member_a @ X @ vertices ) ) ).
% wellformed_alt_fst
thf(fact_1244_all__edges__between__rem__wf,axiom,
! [X4: set_a,Y5: set_a] :
( ( undire8383842906760478443ween_a @ edges @ X4 @ Y5 )
= ( undire8383842906760478443ween_a @ edges @ ( inf_inf_set_a @ X4 @ vertices ) @ ( inf_inf_set_a @ Y5 @ vertices ) ) ) ).
% all_edges_between_rem_wf
thf(fact_1245_incident__edges__empty,axiom,
! [V: a] :
( ~ ( member_a @ V @ vertices )
=> ( ( undire3231912044278729248dges_a @ edges @ V )
= bot_bot_set_set_a ) ) ).
% incident_edges_empty
thf(fact_1246_incident__loops__union,axiom,
( ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ ( undire4753905205749729249oops_a @ edges ) @ vertices ) )
= ( collect_set_a
@ ^ [E3: set_a] :
( ( member_set_a @ E3 @ edges )
& ( undire2905028936066782638loop_a @ E3 ) ) ) ) ).
% incident_loops_union
thf(fact_1247_incident__sedges__union,axiom,
( ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ ( undire1270416042309875431dges_a @ edges ) @ vertices ) )
= ( collect_set_a
@ ^ [E3: set_a] :
( ( member_set_a @ E3 @ edges )
& ( undire4917966558017083288edge_a @ E3 ) ) ) ) ).
% incident_sedges_union
thf(fact_1248_induced__edges__ss,axiom,
! [V3: set_a] :
( ( ord_less_eq_set_a @ V3 @ vertices )
=> ( ord_le3724670747650509150_set_a @ ( undire7777452895879145676dges_a @ edges @ V3 ) @ edges ) ) ).
% induced_edges_ss
thf(fact_1249_induced__is__subgraph,axiom,
! [V3: set_a] :
( ( ord_less_eq_set_a @ V3 @ vertices )
=> ( undire7103218114511261257raph_a @ V3 @ ( undire7777452895879145676dges_a @ edges @ V3 ) @ vertices @ edges ) ) ).
% induced_is_subgraph
thf(fact_1250_card__is__has__loop__eq,axiom,
( ( finite_card_set_a
@ ( collect_set_a
@ ^ [E3: set_a] :
( ( member_set_a @ E3 @ edges )
& ( undire2905028936066782638loop_a @ E3 ) ) ) )
= ( finite_card_a
@ ( collect_a
@ ^ [V4: a] :
( ( member_a @ V4 @ vertices )
& ( undire3617971648856834880loop_a @ edges @ V4 ) ) ) ) ) ).
% card_is_has_loop_eq
thf(fact_1251_degree__none,axiom,
! [V: a] :
( ~ ( member_a @ V @ vertices )
=> ( ( undire8867928226783802224gree_a @ edges @ V )
= zero_zero_nat ) ) ).
% degree_none
thf(fact_1252_incident__sedges__empty,axiom,
! [V: a] :
( ~ ( member_a @ V @ vertices )
=> ( ( undire1270416042309875431dges_a @ edges @ V )
= bot_bot_set_set_a ) ) ).
% incident_sedges_empty
thf(fact_1253_convert__triangle__rep__ss,axiom,
! [X4: set_a,Y5: set_a,Z4: set_a] :
( ( ord_less_eq_set_a @ X4 @ vertices )
=> ( ( ord_less_eq_set_a @ Y5 @ vertices )
=> ( ( ord_less_eq_set_a @ Z4 @ vertices )
=> ( ord_le3724670747650509150_set_a
@ ( image_3577654474136104851_set_a @ undire8536760333753235943_set_a
@ ( collec5045780995415420475od_a_a
@ ( produc5856822985862792195_a_a_o
@ ^ [X2: a] :
( produc6436628058953941356_a_a_o
@ ^ [Y2: a,Z5: a] :
( ( member3071122053849602553od_a_a @ ( produc431845341423274048od_a_a @ X2 @ ( product_Pair_a_a @ Y2 @ Z5 ) )
@ ( produc6342321021181284593od_a_a @ X4
@ ^ [Uu: a] :
( product_Sigma_a_a @ Y5
@ ^ [Uv: a] : Z4 ) ) )
& ( graph_4582152751571636272raph_a @ edges @ X2 @ Y2 @ Z5 ) ) ) ) ) )
@ ( graph_triangle_set_a @ edges ) ) ) ) ) ).
% convert_triangle_rep_ss
thf(fact_1254_is__isolated__vertex__degree0,axiom,
! [V: a] :
( ( undire8931668460104145173rtex_a @ vertices @ edges @ V )
=> ( ( undire8867928226783802224gree_a @ edges @ V )
= zero_zero_nat ) ) ).
% is_isolated_vertex_degree0
thf(fact_1255_fin__graph__system__axioms,axiom,
undire945497512398942277stem_a @ vertices @ edges ).
% fin_graph_system_axioms
thf(fact_1256_is__isolated__vertex__def,axiom,
! [V: a] :
( ( undire8931668460104145173rtex_a @ vertices @ edges @ V )
= ( ( member_a @ V @ vertices )
& ! [X2: a] :
( ( member_a @ X2 @ vertices )
=> ~ ( undire397441198561214472_adj_a @ edges @ X2 @ V ) ) ) ) ).
% is_isolated_vertex_def
thf(fact_1257_is__isolated__vertex__edge,axiom,
! [V: a,E4: set_a] :
( ( undire8931668460104145173rtex_a @ vertices @ edges @ V )
=> ( ( member_set_a @ E4 @ edges )
=> ~ ( undire1521409233611534436dent_a @ V @ E4 ) ) ) ).
% is_isolated_vertex_edge
thf(fact_1258_is__isolated__vertex__no__loop,axiom,
! [V: a] :
( ( undire8931668460104145173rtex_a @ vertices @ edges @ V )
=> ~ ( undire3617971648856834880loop_a @ edges @ V ) ) ).
% is_isolated_vertex_no_loop
thf(fact_1259_degree0__neighborhood__empt__iff,axiom,
! [V: a] :
( ( finite_finite_set_a @ edges )
=> ( ( ( undire8867928226783802224gree_a @ edges @ V )
= zero_zero_nat )
= ( ( undire8504279938402040014hood_a @ vertices @ edges @ V )
= bot_bot_set_a ) ) ) ).
% degree0_neighborhood_empt_iff
thf(fact_1260_incident__edges__neighbors__img,axiom,
! [V: a] :
( ( undire3231912044278729248dges_a @ edges @ V )
= ( image_a_set_a
@ ^ [U2: a] : ( insert_a @ V @ ( insert_a @ U2 @ bot_bot_set_a ) )
@ ( undire8504279938402040014hood_a @ vertices @ edges @ V ) ) ) ).
% incident_edges_neighbors_img
thf(fact_1261_fin__neighbourhood,axiom,
! [X: a] : ( finite_finite_a @ ( undire8504279938402040014hood_a @ vertices @ edges @ X ) ) ).
% fin_neighbourhood
thf(fact_1262_neighborhood__def,axiom,
! [X: a] :
( ( undire8504279938402040014hood_a @ vertices @ edges @ X )
= ( collect_a
@ ^ [V4: a] :
( ( member_a @ V4 @ vertices )
& ( undire397441198561214472_adj_a @ edges @ X @ V4 ) ) ) ) ).
% neighborhood_def
thf(fact_1263_alt__deg__neighborhood,axiom,
! [V: a] :
( ( undire8867928226783802224gree_a @ edges @ V )
= ( finite_card_a @ ( undire8504279938402040014hood_a @ vertices @ edges @ V ) ) ) ).
% alt_deg_neighborhood
thf(fact_1264_iso__vertex__empty__neighborhood,axiom,
! [V: a] :
( ( undire8931668460104145173rtex_a @ vertices @ edges @ V )
=> ( ( undire8504279938402040014hood_a @ vertices @ edges @ V )
= bot_bot_set_a ) ) ).
% iso_vertex_empty_neighborhood
thf(fact_1265_neighborhood__incident,axiom,
! [U: a,V: a] :
( ( member_a @ U @ ( undire8504279938402040014hood_a @ vertices @ edges @ V ) )
= ( member_set_a @ ( insert_a @ U @ ( insert_a @ V @ bot_bot_set_a ) ) @ ( undire3231912044278729248dges_a @ edges @ V ) ) ) ).
% neighborhood_incident
thf(fact_1266_card__incident__sedges__neighborhood,axiom,
! [V: a] :
( ( finite_card_set_a @ ( undire3231912044278729248dges_a @ edges @ V ) )
= ( finite_card_a @ ( undire8504279938402040014hood_a @ vertices @ edges @ V ) ) ) ).
% card_incident_sedges_neighborhood
thf(fact_1267_triangle__set__power__set__ss,axiom,
ord_le3724670747650509150_set_a @ ( graph_triangle_set_a @ edges ) @ ( pow_a @ vertices ) ).
% triangle_set_power_set_ss
thf(fact_1268_sgraph__axioms,axiom,
undire3507641187627840796raph_a @ vertices @ edges ).
% sgraph_axioms
thf(fact_1269_fin__all__edges,axiom,
finite_finite_set_a @ ( undire2918257014606996450dges_a @ vertices ) ).
% fin_all_edges
thf(fact_1270_e__in__all__edges,axiom,
! [E4: set_a] :
( ( member_set_a @ E4 @ edges )
=> ( member_set_a @ E4 @ ( undire2918257014606996450dges_a @ vertices ) ) ) ).
% e_in_all_edges
thf(fact_1271_e__in__all__edges__ss,axiom,
! [E4: set_a,V3: set_a] :
( ( member_set_a @ E4 @ edges )
=> ( ( ord_less_eq_set_a @ E4 @ V3 )
=> ( ( ord_less_eq_set_a @ V3 @ vertices )
=> ( member_set_a @ E4 @ ( undire2918257014606996450dges_a @ V3 ) ) ) ) ) ).
% e_in_all_edges_ss
thf(fact_1272_wellformed__all__edges,axiom,
ord_le3724670747650509150_set_a @ edges @ ( undire2918257014606996450dges_a @ vertices ) ).
% wellformed_all_edges
thf(fact_1273_subgraph__complete,axiom,
undire7103218114511261257raph_a @ vertices @ edges @ vertices @ ( undire2918257014606996450dges_a @ vertices ) ).
% subgraph_complete
thf(fact_1274_induced__edges__alt,axiom,
! [V3: set_a] :
( ( undire7777452895879145676dges_a @ edges @ V3 )
= ( inf_inf_set_set_a @ edges @ ( undire2918257014606996450dges_a @ V3 ) ) ) ).
% induced_edges_alt
% Conjectures (1)
thf(conj_0,conjecture,
( ord_le114883831454073552od_a_a
@ ( collec5045780995415420475od_a_a
@ ( produc5856822985862792195_a_a_o
@ ^ [X2: a] : ( produc6436628058953941356_a_a_o @ ( graph_4582152751571636272raph_a @ edges @ X2 ) ) ) )
@ ( comple2978350343072902813od_a_a
@ ( image_7562202058474640471od_a_a
@ ^ [T: set_a] : ( image_1195025184546981201od_a_a @ tofl @ ( multis2428024204330136193_set_a @ T ) )
@ ( graph_triangle_set_a @ edges ) ) ) ) ).
%------------------------------------------------------------------------------