TPTP Problem File: SLH0352^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Equivalence_Relation_Enumeration/0007_Equivalence_Relation_Enumeration/prob_00328_012545__11994220_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1660 ( 590 unt; 380 typ; 0 def)
% Number of atoms : 3908 (1838 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 13160 ( 400 ~; 53 |; 271 &;10823 @)
% ( 0 <=>;1613 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 7 avg)
% Number of types : 38 ( 37 usr)
% Number of type conns : 1858 (1858 >; 0 *; 0 +; 0 <<)
% Number of symbols : 346 ( 343 usr; 25 con; 0-4 aty)
% Number of variables : 4261 ( 661 ^;3535 !; 65 ?;4261 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:14:21.550
%------------------------------------------------------------------------------
% Could-be-implicit typings (37)
thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
produc4471711990508489141at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_Pr8693737435421807431at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc859450856879609959at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
set_Pr2539167527615954998at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_Pr7717912310451564380at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J,type,
set_Pr6558897299634722526_nat_o: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_Eo_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_Pr2390076351701138800at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
set_Pr5488025237498180813et_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
produc8373899037510109440at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc7248412053542808358at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
produc652263959640188542_nat_o: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc5146536252030154256at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
set_Pr400265656397884439et_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
produc7819656566062154093et_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J,type,
set_Pr5582243495563764594_nat_o: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_Eo_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
set_Pr5768764684362454787et_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Pr1261947904930325089at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J,type,
produc2400336064389900727et_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_J,type,
set_nat_nat_o: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_M_Eo_J_J,type,
set_Pr3149072824959771635_nat_o: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J_J,type,
set_Pr2101469702781467981_o_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__Set__Oset_It__Nat__Onat_J_J,type,
produc573827679308723021et_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_set_set_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
product_prod_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
set_Product_prod_o_o: $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__Product____Type__Oprod_It__Nat__Onat_M_Eo_J,type,
product_prod_nat_o: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
product_prod_o_nat: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
list_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_Eo_J_J,type,
set_nat_o: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
product_prod_o_o: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
set_set_o: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_I_Eo_J,type,
set_o: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (343)
thf(sy_c_BNF__Def_OGr_001t__Nat__Onat_001t__Nat__Onat,type,
bNF_Gr_nat_nat: set_nat > ( nat > nat ) > set_Pr1261947904930325089at_nat ).
thf(sy_c_BNF__Def_OGr_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
bNF_Gr_nat_set_nat: set_nat > ( nat > set_nat ) > set_Pr400265656397884439et_nat ).
thf(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001t__Nat__Onat_001t__Nat__Onat,type,
bNF_re552426215088010623at_nat: ( $o > $o > $o ) > ( nat > nat > $o ) > ( $o > nat ) > ( $o > nat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
bNF_re8818157018887600274at_nat: ( $o > $o > $o ) > ( nat > product_prod_nat_nat > $o ) > ( $o > nat ) > ( $o > product_prod_nat_nat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
bNF_re5458109625844885172at_nat: ( $o > $o > $o ) > ( product_prod_nat_nat > nat > $o ) > ( $o > product_prod_nat_nat ) > ( $o > nat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
bNF_re45780121106632989at_nat: ( $o > $o > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( $o > product_prod_nat_nat ) > ( $o > product_prod_nat_nat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Nat__Onat_M_Eo_J,type,
bNF_re578469030762574527_nat_o: ( nat > nat > $o ) > ( ( nat > $o ) > ( nat > $o ) > $o ) > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_Eo_001_Eo,type,
bNF_re4705727531993890431at_o_o: ( nat > nat > $o ) > ( $o > $o > $o ) > ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
bNF_re5653821019739307937at_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > ( nat > nat ) > ( nat > nat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
bNF_re969444349294296048at_nat: ( nat > nat > $o ) > ( nat > product_prod_nat_nat > $o ) > ( nat > nat ) > ( nat > product_prod_nat_nat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
bNF_re8140478389947574103et_nat: ( nat > nat > $o ) > ( nat > set_nat > $o ) > ( nat > nat ) > ( nat > set_nat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
bNF_re6832768993106356754at_nat: ( nat > nat > $o ) > ( product_prod_nat_nat > nat > $o ) > ( nat > product_prod_nat_nat ) > ( nat > nat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
bNF_re901410630616637503at_nat: ( nat > nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( nat > product_prod_nat_nat ) > ( nat > product_prod_nat_nat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
bNF_re4574843485292204887at_nat: ( nat > nat > $o ) > ( set_nat > nat > $o ) > ( nat > set_nat ) > ( nat > nat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
bNF_re7960066483578023098at_nat: ( nat > nat > $o ) > ( set_nat > product_prod_nat_nat > $o ) > ( nat > set_nat ) > ( nat > product_prod_nat_nat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
bNF_re4683002380766094093et_nat: ( nat > nat > $o ) > ( set_nat > set_nat > $o ) > ( nat > set_nat ) > ( nat > set_nat ) > $o ).
thf(sy_c_BNF__Greatest__Fixpoint_OrelImage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
bNF_Gr553682781689516860et_nat: set_Pr1261947904930325089at_nat > ( nat > set_nat ) > set_Pr5488025237498180813et_nat ).
thf(sy_c_BNF__Greatest__Fixpoint_OrelInvImage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
bNF_Gr7866296307673166617et_nat: set_nat > set_Pr5488025237498180813et_nat > ( nat > set_nat ) > set_Pr1261947904930325089at_nat ).
thf(sy_c_BNF__Greatest__Fixpoint_Ouniv_001t__Nat__Onat_001t__Nat__Onat,type,
bNF_Gr5483022400647640010at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_BNF__Greatest__Fixpoint_Ouniv_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
bNF_Gr881239092033360007at_nat: ( nat > product_prod_nat_nat ) > set_nat > product_prod_nat_nat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Nat__Onat,type,
complete_Inf_Inf_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
comple3592611370556534995_nat_o: set_nat_nat_o > nat > nat > $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_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
comple1901448594430626575_nat_o: set_Pr5582243495563764594_nat_o > product_prod_nat_nat > $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_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
comple5685304695842803022at_nat: set_se7855581050983116737at_nat > set_Pr1261947904930325089at_nat ).
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_It__Nat__Onat_J_J,type,
comple548664676211718543et_nat: set_set_set_nat > set_set_nat ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001_Eo,type,
condit5488710616941104124bove_o: set_o > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Nat__Onat,type,
condit2214826472909112428ve_nat: set_nat > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Set__Oset_It__Nat__Onat_J,type,
condit5477540289124974626et_nat: set_set_nat > $o ).
thf(sy_c_Equiv__Relations_Ocongruent2_001_Eo_001_Eo_001t__Set__Oset_It__Nat__Onat_J,type,
equiv_2613626581038891459et_nat: set_Product_prod_o_o > set_Product_prod_o_o > ( $o > $o > set_nat ) > $o ).
thf(sy_c_Equiv__Relations_Ocongruent2_001_Eo_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
equiv_8341594750169188083et_nat: set_Product_prod_o_o > set_Pr1261947904930325089at_nat > ( $o > nat > set_nat ) > $o ).
thf(sy_c_Equiv__Relations_Ocongruent2_001_Eo_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
equiv_4406527554178595628et_nat: set_Product_prod_o_o > set_Pr8693737435421807431at_nat > ( $o > product_prod_nat_nat > set_nat ) > $o ).
thf(sy_c_Equiv__Relations_Ocongruent2_001t__Nat__Onat_001_Eo_001t__Set__Oset_It__Nat__Onat_J,type,
equiv_9005645190137227217et_nat: set_Pr1261947904930325089at_nat > set_Product_prod_o_o > ( nat > $o > set_nat ) > $o ).
thf(sy_c_Equiv__Relations_Ocongruent2_001t__Nat__Onat_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
equiv_4377714518549004389et_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > ( nat > nat > set_nat ) > $o ).
thf(sy_c_Equiv__Relations_Ocongruent2_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
equiv_2960386318186113850et_nat: set_Pr1261947904930325089at_nat > set_Pr8693737435421807431at_nat > ( nat > product_prod_nat_nat > set_nat ) > $o ).
thf(sy_c_Equiv__Relations_Ocongruent2_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_Eo_001t__Set__Oset_It__Nat__Onat_J,type,
equiv_1136281463066641178et_nat: set_Pr8693737435421807431at_nat > set_Product_prod_o_o > ( product_prod_nat_nat > $o > set_nat ) > $o ).
thf(sy_c_Equiv__Relations_Ocongruent2_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
equiv_6930685768344799580et_nat: set_Pr8693737435421807431at_nat > set_Pr1261947904930325089at_nat > ( product_prod_nat_nat > nat > set_nat ) > $o ).
thf(sy_c_Equiv__Relations_Ocongruent2_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
equiv_4637926981076332931et_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > ( product_prod_nat_nat > product_prod_nat_nat > set_nat ) > $o ).
thf(sy_c_Equiv__Relations_Ocongruent_001_Eo_001t__Set__Oset_It__Nat__Onat_J,type,
equiv_7035509024949518600et_nat: set_Product_prod_o_o > ( $o > set_nat ) > $o ).
thf(sy_c_Equiv__Relations_Ocongruent_001t__Nat__Onat_001t__Nat__Onat,type,
equiv_5105297490605233976at_nat: set_Pr1261947904930325089at_nat > ( nat > nat ) > $o ).
thf(sy_c_Equiv__Relations_Ocongruent_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
equiv_3178832884793977753at_nat: set_Pr1261947904930325089at_nat > ( nat > product_prod_nat_nat ) > $o ).
thf(sy_c_Equiv__Relations_Ocongruent_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
equiv_4704879099140956910et_nat: set_Pr1261947904930325089at_nat > ( nat > set_nat ) > $o ).
thf(sy_c_Equiv__Relations_Ocongruent_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
equiv_7243916823904213361et_nat: set_Pr8693737435421807431at_nat > ( product_prod_nat_nat > set_nat ) > $o ).
thf(sy_c_Equiv__Relations_Oequiv_001_Eo,type,
equiv_equiv_o: set_o > set_Product_prod_o_o > $o ).
thf(sy_c_Equiv__Relations_Oequiv_001t__Nat__Onat,type,
equiv_equiv_nat: set_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_Equiv__Relations_Oequiv_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
equiv_5407799809235660215at_nat: set_Pr1261947904930325089at_nat > set_Pr8693737435421807431at_nat > $o ).
thf(sy_c_Equiv__Relations_Oproj_001_Eo_001_Eo,type,
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thf(sy_c_Set_Oimage_001_Eo_001_Eo,type,
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thf(sy_c_Set_Oimage_001_Eo_001t__Nat__Onat,type,
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thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
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thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
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image_4566292076127975857et_nat: ( product_prod_nat_nat > produc7819656566062154093et_nat ) > set_Pr1261947904930325089at_nat > set_Pr5488025237498180813et_nat ).
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thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001_Eo,type,
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image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
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thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_6725021117256019401et_nat: ( set_nat > set_set_nat ) > set_set_nat > set_set_set_nat ).
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image_5253903537071781336_nat_o: ( set_Pr1261947904930325089at_nat > product_prod_nat_nat > $o ) > set_se7855581050983116737at_nat > set_Pr5582243495563764594_nat_o ).
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image_3684629450409544005at_nat: ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ) > set_se7855581050983116737at_nat > set_se7855581050983116737at_nat ).
thf(sy_c_Set_Oinsert_001_Eo,type,
insert_o: $o > set_o > set_o ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
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thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
insert7948214739558299847_o_nat: product_prod_o_nat > set_Pr2101469702781467981_o_nat > set_Pr2101469702781467981_o_nat ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Nat__Onat_M_Eo_J,type,
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thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
insert8211810215607154385at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_I_Eo_J,type,
insert_set_o: set_o > set_set_o > set_set_o ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
insert_set_nat: set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Set_Ois__singleton_001_Eo,type,
is_singleton_o: set_o > $o ).
thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
is_singleton_nat: set_nat > $o ).
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thf(sy_c_Set_Othe__elem_001_Eo,type,
the_elem_o: set_o > $o ).
thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
thf(sy_c_Set_Othe__elem_001t__Set__Oset_It__Nat__Onat_J,type,
the_elem_set_nat: set_set_nat > set_nat ).
thf(sy_c_Set_Ovimage_001_Eo_001_Eo,type,
vimage_o_o: ( $o > $o ) > set_o > set_o ).
thf(sy_c_Set_Ovimage_001_Eo_001t__Nat__Onat,type,
vimage_o_nat: ( $o > nat ) > set_nat > set_o ).
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thf(sy_c_Set_Ovimage_001t__Nat__Onat_001_Eo,type,
vimage_nat_o: ( nat > $o ) > set_o > set_nat ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Set__Interval_Oord__class_OlessThan_001_Eo,type,
set_ord_lessThan_o: $o > set_o ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_Wellfounded_Olex__prod_001t__Nat__Onat_001t__Nat__Onat,type,
lex_prod_nat_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr8693737435421807431at_nat ).
thf(sy_c_Wfrec_Osame__fst_001t__Nat__Onat_001t__Nat__Onat,type,
same_fst_nat_nat: ( nat > $o ) > ( nat > set_Pr1261947904930325089at_nat ) > set_Pr8693737435421807431at_nat ).
thf(sy_c_member_001_Eo,type,
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member8206827879206165904at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > $o ).
thf(sy_c_member_001t__Set__Oset_I_Eo_J,type,
member_set_o: set_o > set_set_o > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_v_n,type,
n: nat ).
thf(sy_v_p,type,
p: set_Pr1261947904930325089at_nat ).
thf(sy_v_r____,type,
r: list_set_nat ).
thf(sy_v_u____,type,
u: nat ).
thf(sy_v_v____,type,
v: nat ).
% Relevant facts (1274)
thf(fact_0_prod_Oinject,axiom,
! [X1: nat,X2: nat,Y1: nat,Y2: nat] :
( ( ( product_Pair_nat_nat @ X1 @ X2 )
= ( product_Pair_nat_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_1_old_Oprod_Oinject,axiom,
! [A: nat,B: nat,A2: nat,B2: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_2_nat__gcd_Ocases,axiom,
! [X: product_prod_nat_nat] :
~ ! [X3: nat,Y: nat] :
( X
!= ( product_Pair_nat_nat @ X3 @ Y ) ) ).
% nat_gcd.cases
thf(fact_3_old_Oprod_Oexhaust,axiom,
! [Y3: product_prod_nat_nat] :
~ ! [A3: nat,B3: nat] :
( Y3
!= ( product_Pair_nat_nat @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_4_surj__pair,axiom,
! [P: product_prod_nat_nat] :
? [X3: nat,Y: nat] :
( P
= ( product_Pair_nat_nat @ X3 @ Y ) ) ).
% surj_pair
thf(fact_5_prod__cases,axiom,
! [P2: product_prod_nat_nat > $o,P: product_prod_nat_nat] :
( ! [A3: nat,B3: nat] : ( P2 @ ( product_Pair_nat_nat @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_6_Pair__inject,axiom,
! [A: nat,B: nat,A2: nat,B2: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_7_fold__atLeastAtMost__nat_Ocases,axiom,
! [X: produc4471711990508489141at_nat] :
~ ! [F: nat > nat > nat,A3: nat,B3: nat,Acc: nat] :
( X
!= ( produc3209952032786966637at_nat @ F @ ( produc487386426758144856at_nat @ A3 @ ( product_Pair_nat_nat @ B3 @ Acc ) ) ) ) ).
% fold_atLeastAtMost_nat.cases
thf(fact_8_ssubst__Pair__rhs,axiom,
! [R: nat,S: nat,R2: set_Pr1261947904930325089at_nat,S2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R @ S ) @ R2 )
=> ( ( S2 = S )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R @ S2 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_9_curryI,axiom,
! [F2: product_prod_nat_nat > $o,A: nat,B: nat] :
( ( F2 @ ( product_Pair_nat_nat @ A @ B ) )
=> ( produc1310100445399344235_nat_o @ F2 @ A @ B ) ) ).
% curryI
thf(fact_10_swap__simp,axiom,
! [X: nat,Y3: nat] :
( ( product_swap_nat_nat @ ( product_Pair_nat_nat @ X @ Y3 ) )
= ( product_Pair_nat_nat @ Y3 @ X ) ) ).
% swap_simp
thf(fact_11_apsnd__conv,axiom,
! [F2: nat > nat,X: nat,Y3: nat] :
( ( produc3094765310956027504at_nat @ F2 @ ( product_Pair_nat_nat @ X @ Y3 ) )
= ( product_Pair_nat_nat @ X @ ( F2 @ Y3 ) ) ) ).
% apsnd_conv
thf(fact_12_apfst__conv,axiom,
! [F2: nat > nat,X: nat,Y3: nat] :
( ( produc986720760941809198at_nat @ F2 @ ( product_Pair_nat_nat @ X @ Y3 ) )
= ( product_Pair_nat_nat @ ( F2 @ X ) @ Y3 ) ) ).
% apfst_conv
thf(fact_13_curryD,axiom,
! [F2: product_prod_nat_nat > $o,A: nat,B: nat] :
( ( produc1310100445399344235_nat_o @ F2 @ A @ B )
=> ( F2 @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% curryD
thf(fact_14_curryE,axiom,
! [F2: product_prod_nat_nat > $o,A: nat,B: nat] :
( ( produc1310100445399344235_nat_o @ F2 @ A @ B )
=> ( F2 @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% curryE
thf(fact_15_pair__in__swap__image,axiom,
! [Y3: nat,X: nat,A4: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X ) @ ( image_5168914502847457605at_nat @ product_swap_nat_nat @ A4 ) )
= ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ A4 ) ) ).
% pair_in_swap_image
thf(fact_16_same__fstI,axiom,
! [P2: nat > $o,X: nat,Y4: nat,Y3: nat,R2: nat > set_Pr1261947904930325089at_nat] :
( ( P2 @ X )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ Y3 ) @ ( R2 @ X ) )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X @ Y4 ) @ ( product_Pair_nat_nat @ X @ Y3 ) ) @ ( same_fst_nat_nat @ P2 @ R2 ) ) ) ) ).
% same_fstI
thf(fact_17__092_060open_062_I_Iu_M_Av_J_A_092_060in_062_Akernel__of_Ar_J_A_061_A_I_Iu_M_Av_J_A_092_060in_062_A_123_O_O_060n_125_A_092_060times_062_A_123_O_O_060n_125_A_092_060and_062_Ap_A_096_096_A_123u_125_A_061_Ap_A_096_096_A_123v_125_J_092_060close_062,axiom,
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ u @ v ) @ ( equiva50036288466687814et_nat @ r ) )
= ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ u @ v )
@ ( produc457027306803732586at_nat @ ( set_ord_lessThan_nat @ n )
@ ^ [Uu: nat] : ( set_ord_lessThan_nat @ n ) ) )
& ( ( image_nat_nat @ p @ ( insert_nat @ u @ bot_bot_set_nat ) )
= ( image_nat_nat @ p @ ( insert_nat @ v @ bot_bot_set_nat ) ) ) ) ) ).
% \<open>((u, v) \<in> kernel_of r) = ((u, v) \<in> {..<n} \<times> {..<n} \<and> p `` {u} = p `` {v})\<close>
thf(fact_18_mem__Collect__eq,axiom,
! [A: product_prod_nat_nat,P2: product_prod_nat_nat > $o] :
( ( member8440522571783428010at_nat @ A @ ( collec3392354462482085612at_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_19_mem__Collect__eq,axiom,
! [A: nat,P2: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_20_Collect__mem__eq,axiom,
! [A4: set_Pr1261947904930325089at_nat] :
( ( collec3392354462482085612at_nat
@ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_21_Collect__mem__eq,axiom,
! [A4: set_nat] :
( ( collect_nat
@ ^ [X4: nat] : ( member_nat @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_22_lessThan__eq__iff,axiom,
! [X: nat,Y3: nat] :
( ( ( set_ord_lessThan_nat @ X )
= ( set_ord_lessThan_nat @ Y3 ) )
= ( X = Y3 ) ) ).
% lessThan_eq_iff
thf(fact_23_case__prodI,axiom,
! [F2: nat > nat > $o,A: nat,B: nat] :
( ( F2 @ A @ B )
=> ( produc6081775807080527818_nat_o @ F2 @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% case_prodI
thf(fact_24_case__prodI2,axiom,
! [P: product_prod_nat_nat,C: nat > nat > $o] :
( ! [A3: nat,B3: nat] :
( ( P
= ( product_Pair_nat_nat @ A3 @ B3 ) )
=> ( C @ A3 @ B3 ) )
=> ( produc6081775807080527818_nat_o @ C @ P ) ) ).
% case_prodI2
thf(fact_25_mem__case__prodI,axiom,
! [Z: product_prod_nat_nat,C: nat > nat > set_Pr1261947904930325089at_nat,A: nat,B: nat] :
( ( member8440522571783428010at_nat @ Z @ ( C @ A @ B ) )
=> ( member8440522571783428010at_nat @ Z @ ( produc8197505143624133779at_nat @ C @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_26_mem__case__prodI,axiom,
! [Z: nat,C: nat > nat > set_nat,A: nat,B: nat] :
( ( member_nat @ Z @ ( C @ A @ B ) )
=> ( member_nat @ Z @ ( produc6189476227299908564et_nat @ C @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_27_mem__case__prodI2,axiom,
! [P: product_prod_nat_nat,Z: product_prod_nat_nat,C: nat > nat > set_Pr1261947904930325089at_nat] :
( ! [A3: nat,B3: nat] :
( ( P
= ( product_Pair_nat_nat @ A3 @ B3 ) )
=> ( member8440522571783428010at_nat @ Z @ ( C @ A3 @ B3 ) ) )
=> ( member8440522571783428010at_nat @ Z @ ( produc8197505143624133779at_nat @ C @ P ) ) ) ).
% mem_case_prodI2
thf(fact_28_mem__case__prodI2,axiom,
! [P: product_prod_nat_nat,Z: nat,C: nat > nat > set_nat] :
( ! [A3: nat,B3: nat] :
( ( P
= ( product_Pair_nat_nat @ A3 @ B3 ) )
=> ( member_nat @ Z @ ( C @ A3 @ B3 ) ) )
=> ( member_nat @ Z @ ( produc6189476227299908564et_nat @ C @ P ) ) ) ).
% mem_case_prodI2
thf(fact_29_Collect__case__prod,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [A5: nat,B4: nat] :
( ( P2 @ A5 )
& ( Q @ B4 ) ) ) )
= ( produc457027306803732586at_nat @ ( collect_nat @ P2 )
@ ^ [Uu: nat] : ( collect_nat @ Q ) ) ) ).
% Collect_case_prod
thf(fact_30_mem__Sigma__iff,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( produc2761391749766926216at_nat @ A4 @ B5 ) )
= ( ( member8440522571783428010at_nat @ A @ A4 )
& ( member8440522571783428010at_nat @ B @ ( B5 @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_31_mem__Sigma__iff,axiom,
! [A: product_prod_nat_nat,B: nat,A4: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_nat] :
( ( member3348759134392003351at_nat @ ( produc6350711070570205562at_nat @ A @ B ) @ ( produc7672662199629908489at_nat @ A4 @ B5 ) )
= ( ( member8440522571783428010at_nat @ A @ A4 )
& ( member_nat @ B @ ( B5 @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_32_mem__Sigma__iff,axiom,
! [A: nat,B: product_prod_nat_nat,A4: set_nat,B5: nat > set_Pr1261947904930325089at_nat] :
( ( member2223272150424702269at_nat @ ( produc487386426758144856at_nat @ A @ B ) @ ( produc1809337555817847783at_nat @ A4 @ B5 ) )
= ( ( member_nat @ A @ A4 )
& ( member8440522571783428010at_nat @ B @ ( B5 @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_33_mem__Sigma__iff,axiom,
! [A: nat,B: nat,A4: set_nat,B5: nat > set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( produc457027306803732586at_nat @ A4 @ B5 ) )
= ( ( member_nat @ A @ A4 )
& ( member_nat @ B @ ( B5 @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_34_SigmaI,axiom,
! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,B5: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ A @ A4 )
=> ( ( member8440522571783428010at_nat @ B @ ( B5 @ A ) )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( produc2761391749766926216at_nat @ A4 @ B5 ) ) ) ) ).
% SigmaI
thf(fact_35_SigmaI,axiom,
! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B: nat,B5: product_prod_nat_nat > set_nat] :
( ( member8440522571783428010at_nat @ A @ A4 )
=> ( ( member_nat @ B @ ( B5 @ A ) )
=> ( member3348759134392003351at_nat @ ( produc6350711070570205562at_nat @ A @ B ) @ ( produc7672662199629908489at_nat @ A4 @ B5 ) ) ) ) ).
% SigmaI
thf(fact_36_SigmaI,axiom,
! [A: nat,A4: set_nat,B: product_prod_nat_nat,B5: nat > set_Pr1261947904930325089at_nat] :
( ( member_nat @ A @ A4 )
=> ( ( member8440522571783428010at_nat @ B @ ( B5 @ A ) )
=> ( member2223272150424702269at_nat @ ( produc487386426758144856at_nat @ A @ B ) @ ( produc1809337555817847783at_nat @ A4 @ B5 ) ) ) ) ).
% SigmaI
thf(fact_37_SigmaI,axiom,
! [A: nat,A4: set_nat,B: nat,B5: nat > set_nat] :
( ( member_nat @ A @ A4 )
=> ( ( member_nat @ B @ ( B5 @ A ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( produc457027306803732586at_nat @ A4 @ B5 ) ) ) ) ).
% SigmaI
thf(fact_38_Sigma__empty1,axiom,
! [B5: nat > set_nat] :
( ( produc457027306803732586at_nat @ bot_bot_set_nat @ B5 )
= bot_bo2099793752762293965at_nat ) ).
% Sigma_empty1
thf(fact_39_Sigma__empty2,axiom,
! [A4: set_nat] :
( ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : bot_bot_set_nat )
= bot_bo2099793752762293965at_nat ) ).
% Sigma_empty2
thf(fact_40_Times__empty,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : B5 )
= bot_bo2099793752762293965at_nat )
= ( ( A4 = bot_bot_set_nat )
| ( B5 = bot_bot_set_nat ) ) ) ).
% Times_empty
thf(fact_41_Times__empty,axiom,
! [A4: set_nat,B5: set_o] :
( ( ( product_Sigma_nat_o @ A4
@ ^ [Uu: nat] : B5 )
= bot_bo1114129665703605383_nat_o )
= ( ( A4 = bot_bot_set_nat )
| ( B5 = bot_bot_set_o ) ) ) ).
% Times_empty
thf(fact_42_Times__empty,axiom,
! [A4: set_o,B5: set_nat] :
( ( ( product_Sigma_o_nat @ A4
@ ^ [Uu: $o] : B5 )
= bot_bo66526543525301729_o_nat )
= ( ( A4 = bot_bot_set_o )
| ( B5 = bot_bot_set_nat ) ) ) ).
% Times_empty
thf(fact_43_Times__empty,axiom,
! [A4: set_o,B5: set_o] :
( ( ( product_Sigma_o_o @ A4
@ ^ [Uu: $o] : B5 )
= bot_bo7073875226086086771od_o_o )
= ( ( A4 = bot_bot_set_o )
| ( B5 = bot_bot_set_o ) ) ) ).
% Times_empty
thf(fact_44_prod_Ocollapse,axiom,
! [Prod: product_prod_nat_nat] :
( ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_45_pair__imageI,axiom,
! [A: nat,B: nat,A4: set_Pr1261947904930325089at_nat,F2: nat > nat > product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ A4 )
=> ( member8440522571783428010at_nat @ ( F2 @ A @ B ) @ ( image_5168914502847457605at_nat @ ( produc2626176000494625587at_nat @ F2 ) @ A4 ) ) ) ).
% pair_imageI
thf(fact_46_pair__imageI,axiom,
! [A: nat,B: nat,A4: set_Pr1261947904930325089at_nat,F2: nat > nat > nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ A4 )
=> ( member_nat @ ( F2 @ A @ B ) @ ( image_2486076414777270412at_nat @ ( produc6842872674320459806at_nat @ F2 ) @ A4 ) ) ) ).
% pair_imageI
thf(fact_47__092_060open_062_I_Iu_M_Av_J_A_092_060in_062_A_123_O_O_060n_125_A_092_060times_062_A_123_O_O_060n_125_A_092_060and_062_Ap_A_096_096_A_123u_125_A_061_Ap_A_096_096_A_123v_125_J_A_061_A_I_Iu_M_Av_J_A_092_060in_062_Ap_J_092_060close_062,axiom,
( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ u @ v )
@ ( produc457027306803732586at_nat @ ( set_ord_lessThan_nat @ n )
@ ^ [Uu: nat] : ( set_ord_lessThan_nat @ n ) ) )
& ( ( image_nat_nat @ p @ ( insert_nat @ u @ bot_bot_set_nat ) )
= ( image_nat_nat @ p @ ( insert_nat @ v @ bot_bot_set_nat ) ) ) )
= ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ u @ v ) @ p ) ) ).
% \<open>((u, v) \<in> {..<n} \<times> {..<n} \<and> p `` {u} = p `` {v}) = ((u, v) \<in> p)\<close>
thf(fact_48_fst__image__times,axiom,
! [B5: set_nat,A4: set_nat] :
( ( ( B5 = bot_bot_set_nat )
=> ( ( image_2486076414777270412at_nat @ product_fst_nat_nat
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : B5 ) )
= bot_bot_set_nat ) )
& ( ( B5 != bot_bot_set_nat )
=> ( ( image_2486076414777270412at_nat @ product_fst_nat_nat
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : B5 ) )
= A4 ) ) ) ).
% fst_image_times
thf(fact_49_fst__image__times,axiom,
! [B5: set_nat,A4: set_o] :
( ( ( B5 = bot_bot_set_nat )
=> ( ( image_8105710389281276538_nat_o @ product_fst_o_nat
@ ( product_Sigma_o_nat @ A4
@ ^ [Uu: $o] : B5 ) )
= bot_bot_set_o ) )
& ( ( B5 != bot_bot_set_nat )
=> ( ( image_8105710389281276538_nat_o @ product_fst_o_nat
@ ( product_Sigma_o_nat @ A4
@ ^ [Uu: $o] : B5 ) )
= A4 ) ) ) ).
% fst_image_times
thf(fact_50_fst__image__times,axiom,
! [B5: set_o,A4: set_nat] :
( ( ( B5 = bot_bot_set_o )
=> ( ( image_8629803608462650132_o_nat @ product_fst_nat_o
@ ( product_Sigma_nat_o @ A4
@ ^ [Uu: nat] : B5 ) )
= bot_bot_set_nat ) )
& ( ( B5 != bot_bot_set_o )
=> ( ( image_8629803608462650132_o_nat @ product_fst_nat_o
@ ( product_Sigma_nat_o @ A4
@ ^ [Uu: nat] : B5 ) )
= A4 ) ) ) ).
% fst_image_times
thf(fact_51_fst__image__times,axiom,
! [B5: set_o,A4: set_o] :
( ( ( B5 = bot_bot_set_o )
=> ( ( image_7896445794123959606_o_o_o @ product_fst_o_o
@ ( product_Sigma_o_o @ A4
@ ^ [Uu: $o] : B5 ) )
= bot_bot_set_o ) )
& ( ( B5 != bot_bot_set_o )
=> ( ( image_7896445794123959606_o_o_o @ product_fst_o_o
@ ( product_Sigma_o_o @ A4
@ ^ [Uu: $o] : B5 ) )
= A4 ) ) ) ).
% fst_image_times
thf(fact_52_snd__image__times,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ( A4 = bot_bot_set_nat )
=> ( ( image_2486076414777270412at_nat @ product_snd_nat_nat
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : B5 ) )
= bot_bot_set_nat ) )
& ( ( A4 != bot_bot_set_nat )
=> ( ( image_2486076414777270412at_nat @ product_snd_nat_nat
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : B5 ) )
= B5 ) ) ) ).
% snd_image_times
thf(fact_53_snd__image__times,axiom,
! [A4: set_nat,B5: set_o] :
( ( ( A4 = bot_bot_set_nat )
=> ( ( image_7478392465989118804at_o_o @ product_snd_nat_o
@ ( product_Sigma_nat_o @ A4
@ ^ [Uu: nat] : B5 ) )
= bot_bot_set_o ) )
& ( ( A4 != bot_bot_set_nat )
=> ( ( image_7478392465989118804at_o_o @ product_snd_nat_o
@ ( product_Sigma_nat_o @ A4
@ ^ [Uu: nat] : B5 ) )
= B5 ) ) ) ).
% snd_image_times
thf(fact_54_snd__image__times,axiom,
! [A4: set_o,B5: set_nat] :
( ( ( A4 = bot_bot_set_o )
=> ( ( image_4888532181312399726at_nat @ product_snd_o_nat
@ ( product_Sigma_o_nat @ A4
@ ^ [Uu: $o] : B5 ) )
= bot_bot_set_nat ) )
& ( ( A4 != bot_bot_set_o )
=> ( ( image_4888532181312399726at_nat @ product_snd_o_nat
@ ( product_Sigma_o_nat @ A4
@ ^ [Uu: $o] : B5 ) )
= B5 ) ) ) ).
% snd_image_times
thf(fact_55_snd__image__times,axiom,
! [A4: set_o,B5: set_o] :
( ( ( A4 = bot_bot_set_o )
=> ( ( image_7896445794123959606_o_o_o @ product_snd_o_o
@ ( product_Sigma_o_o @ A4
@ ^ [Uu: $o] : B5 ) )
= bot_bot_set_o ) )
& ( ( A4 != bot_bot_set_o )
=> ( ( image_7896445794123959606_o_o_o @ product_snd_o_o
@ ( product_Sigma_o_o @ A4
@ ^ [Uu: $o] : B5 ) )
= B5 ) ) ) ).
% snd_image_times
thf(fact_56_assms,axiom,
equiv_equiv_nat @ ( set_ord_lessThan_nat @ n ) @ p ).
% assms
thf(fact_57_case__prodD,axiom,
! [F2: nat > nat > $o,A: nat,B: nat] :
( ( produc6081775807080527818_nat_o @ F2 @ ( product_Pair_nat_nat @ A @ B ) )
=> ( F2 @ A @ B ) ) ).
% case_prodD
thf(fact_58_case__prodE,axiom,
! [C: nat > nat > $o,P: product_prod_nat_nat] :
( ( produc6081775807080527818_nat_o @ C @ P )
=> ~ ! [X3: nat,Y: nat] :
( ( P
= ( product_Pair_nat_nat @ X3 @ Y ) )
=> ~ ( C @ X3 @ Y ) ) ) ).
% case_prodE
thf(fact_59_product__swap,axiom,
! [A4: set_nat,B5: set_nat] :
( ( image_5168914502847457605at_nat @ product_swap_nat_nat
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : B5 ) )
= ( produc457027306803732586at_nat @ B5
@ ^ [Uu: nat] : A4 ) ) ).
% product_swap
thf(fact_60_swap__product,axiom,
! [A4: set_nat,B5: set_nat] :
( ( image_5168914502847457605at_nat
@ ( produc2626176000494625587at_nat
@ ^ [I: nat,J: nat] : ( product_Pair_nat_nat @ J @ I ) )
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : B5 ) )
= ( produc457027306803732586at_nat @ B5
@ ^ [Uu: nat] : A4 ) ) ).
% swap_product
thf(fact_61_image__paired__Times,axiom,
! [F2: nat > set_nat,G: nat > set_nat,A4: set_nat,B5: set_nat] :
( ( image_4566292076127975857et_nat
@ ( produc8159404288152011679et_nat
@ ^ [X4: nat,Y5: nat] : ( produc4532415448927165861et_nat @ ( F2 @ X4 ) @ ( G @ Y5 ) ) )
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : B5 ) )
= ( produc105401717395143638et_nat @ ( image_nat_set_nat @ F2 @ A4 )
@ ^ [Uu: set_nat] : ( image_nat_set_nat @ G @ B5 ) ) ) ).
% image_paired_Times
thf(fact_62_image__paired__Times,axiom,
! [F2: nat > nat,G: nat > nat,A4: set_nat,B5: set_nat] :
( ( image_5168914502847457605at_nat
@ ( produc2626176000494625587at_nat
@ ^ [X4: nat,Y5: nat] : ( product_Pair_nat_nat @ ( F2 @ X4 ) @ ( G @ Y5 ) ) )
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : B5 ) )
= ( produc457027306803732586at_nat @ ( image_nat_nat2 @ F2 @ A4 )
@ ^ [Uu: nat] : ( image_nat_nat2 @ G @ B5 ) ) ) ).
% image_paired_Times
thf(fact_63_Iio__eq__empty__iff,axiom,
! [N: $o] :
( ( ( set_ord_lessThan_o @ N )
= bot_bot_set_o )
= ( N = bot_bot_o ) ) ).
% Iio_eq_empty_iff
thf(fact_64_Iio__eq__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan_nat @ N )
= bot_bot_set_nat )
= ( N = bot_bot_nat ) ) ).
% Iio_eq_empty_iff
thf(fact_65_Collect__case__prod__Sigma,axiom,
! [P2: nat > $o,Q: nat > nat > $o] :
( ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [X4: nat,Y5: nat] :
( ( P2 @ X4 )
& ( Q @ X4 @ Y5 ) ) ) )
= ( produc457027306803732586at_nat @ ( collect_nat @ P2 )
@ ^ [X4: nat] : ( collect_nat @ ( Q @ X4 ) ) ) ) ).
% Collect_case_prod_Sigma
thf(fact_66_Product__Type_OCollect__case__prodD,axiom,
! [X: product_prod_nat_nat,A4: nat > nat > $o] :
( ( member8440522571783428010at_nat @ X @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ A4 ) ) )
=> ( A4 @ ( product_fst_nat_nat @ X ) @ ( product_snd_nat_nat @ X ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_67_Times__eq__cancel2,axiom,
! [X: nat,C2: set_nat,A4: set_nat,B5: set_nat] :
( ( member_nat @ X @ C2 )
=> ( ( ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : C2 )
= ( produc457027306803732586at_nat @ B5
@ ^ [Uu: nat] : C2 ) )
= ( A4 = B5 ) ) ) ).
% Times_eq_cancel2
thf(fact_68_fst__image__Sigma,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_nat] :
( ( image_8369444115148354106at_nat @ produc4622195368937614990at_nat @ ( produc7672662199629908489at_nat @ A4 @ B5 ) )
= ( collec3392354462482085612at_nat
@ ^ [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ A4 )
& ( ( B5 @ X4 )
!= bot_bot_set_nat ) ) ) ) ).
% fst_image_Sigma
thf(fact_69_fst__image__Sigma,axiom,
! [A4: set_nat,B5: nat > set_nat] :
( ( image_2486076414777270412at_nat @ product_fst_nat_nat @ ( produc457027306803732586at_nat @ A4 @ B5 ) )
= ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ( ( B5 @ X4 )
!= bot_bot_set_nat ) ) ) ) ).
% fst_image_Sigma
thf(fact_70_fst__image__Sigma,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_o] :
( ( image_2120087853876073608at_nat @ produc6435950281062462810_nat_o @ ( produc203973861852599583_nat_o @ A4 @ B5 ) )
= ( collec3392354462482085612at_nat
@ ^ [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ A4 )
& ( ( B5 @ X4 )
!= bot_bot_set_o ) ) ) ) ).
% fst_image_Sigma
thf(fact_71_fst__image__Sigma,axiom,
! [A4: set_nat,B5: nat > set_o] :
( ( image_8629803608462650132_o_nat @ product_fst_nat_o @ ( product_Sigma_nat_o @ A4 @ B5 ) )
= ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ( ( B5 @ X4 )
!= bot_bot_set_o ) ) ) ) ).
% fst_image_Sigma
thf(fact_72_Sigma__empty__iff,axiom,
! [I2: set_nat,X5: nat > set_nat] :
( ( ( produc457027306803732586at_nat @ I2 @ X5 )
= bot_bo2099793752762293965at_nat )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ I2 )
=> ( ( X5 @ X4 )
= bot_bot_set_nat ) ) ) ) ).
% Sigma_empty_iff
thf(fact_73_mem__Times__iff,axiom,
! [X: produc859450856879609959at_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( member8206827879206165904at_nat @ X
@ ( produc2761391749766926216at_nat @ A4
@ ^ [Uu: product_prod_nat_nat] : B5 ) )
= ( ( member8440522571783428010at_nat @ ( produc3213797794245857475at_nat @ X ) @ A4 )
& ( member8440522571783428010at_nat @ ( produc6408287024330202629at_nat @ X ) @ B5 ) ) ) ).
% mem_Times_iff
thf(fact_74_mem__Times__iff,axiom,
! [X: produc8373899037510109440at_nat,A4: set_Pr1261947904930325089at_nat,B5: set_nat] :
( ( member3348759134392003351at_nat @ X
@ ( produc7672662199629908489at_nat @ A4
@ ^ [Uu: product_prod_nat_nat] : B5 ) )
= ( ( member8440522571783428010at_nat @ ( produc4622195368937614990at_nat @ X ) @ A4 )
& ( member_nat @ ( produc3066998791170923468at_nat @ X ) @ B5 ) ) ) ).
% mem_Times_iff
thf(fact_75_mem__Times__iff,axiom,
! [X: produc7248412053542808358at_nat,A4: set_nat,B5: set_Pr1261947904930325089at_nat] :
( ( member2223272150424702269at_nat @ X
@ ( produc1809337555817847783at_nat @ A4
@ ^ [Uu: nat] : B5 ) )
= ( ( member_nat @ ( produc7982242761980330092at_nat @ X ) @ A4 )
& ( member8440522571783428010at_nat @ ( produc6427046184213638570at_nat @ X ) @ B5 ) ) ) ).
% mem_Times_iff
thf(fact_76_mem__Times__iff,axiom,
! [X: product_prod_nat_nat,A4: set_nat,B5: set_nat] :
( ( member8440522571783428010at_nat @ X
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : B5 ) )
= ( ( member_nat @ ( product_fst_nat_nat @ X ) @ A4 )
& ( member_nat @ ( product_snd_nat_nat @ X ) @ B5 ) ) ) ).
% mem_Times_iff
thf(fact_77_times__eq__iff,axiom,
! [A4: set_nat,B5: set_nat,C2: set_nat,D: set_nat] :
( ( ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : B5 )
= ( produc457027306803732586at_nat @ C2
@ ^ [Uu: nat] : D ) )
= ( ( ( A4 = C2 )
& ( B5 = D ) )
| ( ( ( A4 = bot_bot_set_nat )
| ( B5 = bot_bot_set_nat ) )
& ( ( C2 = bot_bot_set_nat )
| ( D = bot_bot_set_nat ) ) ) ) ) ).
% times_eq_iff
thf(fact_78_times__eq__iff,axiom,
! [A4: set_nat,B5: set_o,C2: set_nat,D: set_o] :
( ( ( product_Sigma_nat_o @ A4
@ ^ [Uu: nat] : B5 )
= ( product_Sigma_nat_o @ C2
@ ^ [Uu: nat] : D ) )
= ( ( ( A4 = C2 )
& ( B5 = D ) )
| ( ( ( A4 = bot_bot_set_nat )
| ( B5 = bot_bot_set_o ) )
& ( ( C2 = bot_bot_set_nat )
| ( D = bot_bot_set_o ) ) ) ) ) ).
% times_eq_iff
thf(fact_79_times__eq__iff,axiom,
! [A4: set_o,B5: set_nat,C2: set_o,D: set_nat] :
( ( ( product_Sigma_o_nat @ A4
@ ^ [Uu: $o] : B5 )
= ( product_Sigma_o_nat @ C2
@ ^ [Uu: $o] : D ) )
= ( ( ( A4 = C2 )
& ( B5 = D ) )
| ( ( ( A4 = bot_bot_set_o )
| ( B5 = bot_bot_set_nat ) )
& ( ( C2 = bot_bot_set_o )
| ( D = bot_bot_set_nat ) ) ) ) ) ).
% times_eq_iff
thf(fact_80_times__eq__iff,axiom,
! [A4: set_o,B5: set_o,C2: set_o,D: set_o] :
( ( ( product_Sigma_o_o @ A4
@ ^ [Uu: $o] : B5 )
= ( product_Sigma_o_o @ C2
@ ^ [Uu: $o] : D ) )
= ( ( ( A4 = C2 )
& ( B5 = D ) )
| ( ( ( A4 = bot_bot_set_o )
| ( B5 = bot_bot_set_o ) )
& ( ( C2 = bot_bot_set_o )
| ( D = bot_bot_set_o ) ) ) ) ) ).
% times_eq_iff
thf(fact_81_Sigma__cong,axiom,
! [A4: set_nat,B5: set_nat,C2: nat > set_nat,D: nat > set_nat] :
( ( A4 = B5 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B5 )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( produc457027306803732586at_nat @ A4 @ C2 )
= ( produc457027306803732586at_nat @ B5 @ D ) ) ) ) ).
% Sigma_cong
thf(fact_82_prod_Oexhaust__sel,axiom,
! [Prod: product_prod_nat_nat] :
( Prod
= ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_83_surjective__pairing,axiom,
! [T: product_prod_nat_nat] :
( T
= ( product_Pair_nat_nat @ ( product_fst_nat_nat @ T ) @ ( product_snd_nat_nat @ T ) ) ) ).
% surjective_pairing
thf(fact_84_SigmaE2,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( produc2761391749766926216at_nat @ A4 @ B5 ) )
=> ~ ( ( member8440522571783428010at_nat @ A @ A4 )
=> ~ ( member8440522571783428010at_nat @ B @ ( B5 @ A ) ) ) ) ).
% SigmaE2
thf(fact_85_SigmaE2,axiom,
! [A: product_prod_nat_nat,B: nat,A4: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_nat] :
( ( member3348759134392003351at_nat @ ( produc6350711070570205562at_nat @ A @ B ) @ ( produc7672662199629908489at_nat @ A4 @ B5 ) )
=> ~ ( ( member8440522571783428010at_nat @ A @ A4 )
=> ~ ( member_nat @ B @ ( B5 @ A ) ) ) ) ).
% SigmaE2
thf(fact_86_SigmaE2,axiom,
! [A: nat,B: product_prod_nat_nat,A4: set_nat,B5: nat > set_Pr1261947904930325089at_nat] :
( ( member2223272150424702269at_nat @ ( produc487386426758144856at_nat @ A @ B ) @ ( produc1809337555817847783at_nat @ A4 @ B5 ) )
=> ~ ( ( member_nat @ A @ A4 )
=> ~ ( member8440522571783428010at_nat @ B @ ( B5 @ A ) ) ) ) ).
% SigmaE2
thf(fact_87_SigmaE2,axiom,
! [A: nat,B: nat,A4: set_nat,B5: nat > set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( produc457027306803732586at_nat @ A4 @ B5 ) )
=> ~ ( ( member_nat @ A @ A4 )
=> ~ ( member_nat @ B @ ( B5 @ A ) ) ) ) ).
% SigmaE2
thf(fact_88_SigmaD2,axiom,
! [A: nat,B: nat,A4: set_nat,B5: nat > set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( produc457027306803732586at_nat @ A4 @ B5 ) )
=> ( member_nat @ B @ ( B5 @ A ) ) ) ).
% SigmaD2
thf(fact_89_SigmaD1,axiom,
! [A: nat,B: nat,A4: set_nat,B5: nat > set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( produc457027306803732586at_nat @ A4 @ B5 ) )
=> ( member_nat @ A @ A4 ) ) ).
% SigmaD1
thf(fact_90_SigmaE,axiom,
! [C: produc859450856879609959at_nat,A4: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
( ( member8206827879206165904at_nat @ C @ ( produc2761391749766926216at_nat @ A4 @ B5 ) )
=> ~ ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A4 )
=> ! [Y: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Y @ ( B5 @ X3 ) )
=> ( C
!= ( produc6161850002892822231at_nat @ X3 @ Y ) ) ) ) ) ).
% SigmaE
thf(fact_91_SigmaE,axiom,
! [C: produc8373899037510109440at_nat,A4: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_nat] :
( ( member3348759134392003351at_nat @ C @ ( produc7672662199629908489at_nat @ A4 @ B5 ) )
=> ~ ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A4 )
=> ! [Y: nat] :
( ( member_nat @ Y @ ( B5 @ X3 ) )
=> ( C
!= ( produc6350711070570205562at_nat @ X3 @ Y ) ) ) ) ) ).
% SigmaE
thf(fact_92_SigmaE,axiom,
! [C: produc7248412053542808358at_nat,A4: set_nat,B5: nat > set_Pr1261947904930325089at_nat] :
( ( member2223272150424702269at_nat @ C @ ( produc1809337555817847783at_nat @ A4 @ B5 ) )
=> ~ ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ! [Y: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Y @ ( B5 @ X3 ) )
=> ( C
!= ( produc487386426758144856at_nat @ X3 @ Y ) ) ) ) ) ).
% SigmaE
thf(fact_93_SigmaE,axiom,
! [C: product_prod_nat_nat,A4: set_nat,B5: nat > set_nat] :
( ( member8440522571783428010at_nat @ C @ ( produc457027306803732586at_nat @ A4 @ B5 ) )
=> ~ ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ! [Y: nat] :
( ( member_nat @ Y @ ( B5 @ X3 ) )
=> ( C
!= ( product_Pair_nat_nat @ X3 @ Y ) ) ) ) ) ).
% SigmaE
thf(fact_94_prod_Oswap__def,axiom,
( product_swap_nat_nat
= ( ^ [P3: product_prod_nat_nat] : ( product_Pair_nat_nat @ ( product_snd_nat_nat @ P3 ) @ ( product_fst_nat_nat @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_95_snd__conv,axiom,
! [X1: nat,X2: nat] :
( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X1 @ X2 ) )
= X2 ) ).
% snd_conv
thf(fact_96_snd__eqD,axiom,
! [X: nat,Y3: nat,A: nat] :
( ( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X @ Y3 ) )
= A )
=> ( Y3 = A ) ) ).
% snd_eqD
thf(fact_97_fst__conv,axiom,
! [X1: nat,X2: nat] :
( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X1 @ X2 ) )
= X1 ) ).
% fst_conv
thf(fact_98_fst__eqD,axiom,
! [X: nat,Y3: nat,A: nat] :
( ( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X @ Y3 ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_99_mem__case__prodE,axiom,
! [Z: product_prod_nat_nat,C: nat > nat > set_Pr1261947904930325089at_nat,P: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Z @ ( produc8197505143624133779at_nat @ C @ P ) )
=> ~ ! [X3: nat,Y: nat] :
( ( P
= ( product_Pair_nat_nat @ X3 @ Y ) )
=> ~ ( member8440522571783428010at_nat @ Z @ ( C @ X3 @ Y ) ) ) ) ).
% mem_case_prodE
thf(fact_100_mem__case__prodE,axiom,
! [Z: nat,C: nat > nat > set_nat,P: product_prod_nat_nat] :
( ( member_nat @ Z @ ( produc6189476227299908564et_nat @ C @ P ) )
=> ~ ! [X3: nat,Y: nat] :
( ( P
= ( product_Pair_nat_nat @ X3 @ Y ) )
=> ~ ( member_nat @ Z @ ( C @ X3 @ Y ) ) ) ) ).
% mem_case_prodE
thf(fact_101_Image__singleton__iff,axiom,
! [B: product_prod_nat_nat,R: set_Pr7717912310451564380at_nat,A: nat] :
( ( member8440522571783428010at_nat @ B @ ( image_8130187405840929308at_nat @ R @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( member2223272150424702269at_nat @ ( produc487386426758144856at_nat @ A @ B ) @ R ) ) ).
% Image_singleton_iff
thf(fact_102_Image__singleton__iff,axiom,
! [B: nat,R: set_Pr1261947904930325089at_nat,A: nat] :
( ( member_nat @ B @ ( image_nat_nat @ R @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R ) ) ).
% Image_singleton_iff
thf(fact_103_Image__singleton__iff,axiom,
! [B: product_prod_nat_nat,R: set_Pr2390076351701138800at_nat,A: $o] :
( ( member8440522571783428010at_nat @ B @ ( image_5697180853172723644at_nat @ R @ ( insert_o @ A @ bot_bot_set_o ) ) )
= ( member6694634564379471929at_nat @ ( produc1017754141990422400at_nat @ A @ B ) @ R ) ) ).
% Image_singleton_iff
thf(fact_104_Image__singleton__iff,axiom,
! [B: nat,R: set_Pr2101469702781467981_o_nat,A: $o] :
( ( member_nat @ B @ ( image_o_nat @ R @ ( insert_o @ A @ bot_bot_set_o ) ) )
= ( member2802428098988154798_o_nat @ ( product_Pair_o_nat @ A @ B ) @ R ) ) ).
% Image_singleton_iff
thf(fact_105_singleton__conv,axiom,
! [A: nat] :
( ( collect_nat
@ ^ [X4: nat] : ( X4 = A ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv
thf(fact_106_singleton__conv,axiom,
! [A: $o] :
( ( collect_o
@ ^ [X4: $o] : ( X4 = A ) )
= ( insert_o @ A @ bot_bot_set_o ) ) ).
% singleton_conv
thf(fact_107_singleton__conv2,axiom,
! [A: nat] :
( ( collect_nat
@ ( ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 )
@ A ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv2
thf(fact_108_singleton__conv2,axiom,
! [A: $o] :
( ( collect_o
@ ( ^ [Y6: $o,Z2: $o] : ( Y6 = Z2 )
@ A ) )
= ( insert_o @ A @ bot_bot_set_o ) ) ).
% singleton_conv2
thf(fact_109_Image__empty1,axiom,
! [X5: set_nat] :
( ( image_nat_nat @ bot_bo2099793752762293965at_nat @ X5 )
= bot_bot_set_nat ) ).
% Image_empty1
thf(fact_110_Image__empty2,axiom,
! [R2: set_Pr1261947904930325089at_nat] :
( ( image_nat_nat @ R2 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% Image_empty2
thf(fact_111_Image__empty2,axiom,
! [R2: set_Pr3149072824959771635_nat_o] :
( ( image_nat_o @ R2 @ bot_bot_set_nat )
= bot_bot_set_o ) ).
% Image_empty2
thf(fact_112_Image__empty2,axiom,
! [R2: set_Pr2101469702781467981_o_nat] :
( ( image_o_nat @ R2 @ bot_bot_set_o )
= bot_bot_set_nat ) ).
% Image_empty2
thf(fact_113_Image__empty2,axiom,
! [R2: set_Product_prod_o_o] :
( ( image_o_o @ R2 @ bot_bot_set_o )
= bot_bot_set_o ) ).
% Image_empty2
thf(fact_114_ImageI,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat,A4: set_Pr1261947904930325089at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R )
=> ( ( member8440522571783428010at_nat @ A @ A4 )
=> ( member8440522571783428010at_nat @ B @ ( image_2484338115069272851at_nat @ R @ A4 ) ) ) ) ).
% ImageI
thf(fact_115_ImageI,axiom,
! [A: product_prod_nat_nat,B: nat,R: set_Pr2539167527615954998at_nat,A4: set_Pr1261947904930325089at_nat] :
( ( member3348759134392003351at_nat @ ( produc6350711070570205562at_nat @ A @ B ) @ R )
=> ( ( member8440522571783428010at_nat @ A @ A4 )
=> ( member_nat @ B @ ( image_4770140012798214206at_nat @ R @ A4 ) ) ) ) ).
% ImageI
thf(fact_116_ImageI,axiom,
! [A: nat,B: product_prod_nat_nat,R: set_Pr7717912310451564380at_nat,A4: set_nat] :
( ( member2223272150424702269at_nat @ ( produc487386426758144856at_nat @ A @ B ) @ R )
=> ( ( member_nat @ A @ A4 )
=> ( member8440522571783428010at_nat @ B @ ( image_8130187405840929308at_nat @ R @ A4 ) ) ) ) ).
% ImageI
thf(fact_117_ImageI,axiom,
! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,A4: set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
=> ( ( member_nat @ A @ A4 )
=> ( member_nat @ B @ ( image_nat_nat @ R @ A4 ) ) ) ) ).
% ImageI
thf(fact_118_singletonI,axiom,
! [A: product_prod_nat_nat] : ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).
% singletonI
thf(fact_119_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_120_singletonI,axiom,
! [A: $o] : ( member_o @ A @ ( insert_o @ A @ bot_bot_set_o ) ) ).
% singletonI
thf(fact_121_image__insert,axiom,
! [F2: nat > set_nat,A: nat,B5: set_nat] :
( ( image_nat_set_nat @ F2 @ ( insert_nat @ A @ B5 ) )
= ( insert_set_nat @ ( F2 @ A ) @ ( image_nat_set_nat @ F2 @ B5 ) ) ) ).
% image_insert
thf(fact_122_image__insert,axiom,
! [F2: nat > nat,A: nat,B5: set_nat] :
( ( image_nat_nat2 @ F2 @ ( insert_nat @ A @ B5 ) )
= ( insert_nat @ ( F2 @ A ) @ ( image_nat_nat2 @ F2 @ B5 ) ) ) ).
% image_insert
thf(fact_123_image__insert,axiom,
! [F2: nat > $o,A: nat,B5: set_nat] :
( ( image_nat_o2 @ F2 @ ( insert_nat @ A @ B5 ) )
= ( insert_o @ ( F2 @ A ) @ ( image_nat_o2 @ F2 @ B5 ) ) ) ).
% image_insert
thf(fact_124_image__insert,axiom,
! [F2: $o > nat,A: $o,B5: set_o] :
( ( image_o_nat2 @ F2 @ ( insert_o @ A @ B5 ) )
= ( insert_nat @ ( F2 @ A ) @ ( image_o_nat2 @ F2 @ B5 ) ) ) ).
% image_insert
thf(fact_125_image__insert,axiom,
! [F2: $o > $o,A: $o,B5: set_o] :
( ( image_o_o2 @ F2 @ ( insert_o @ A @ B5 ) )
= ( insert_o @ ( F2 @ A ) @ ( image_o_o2 @ F2 @ B5 ) ) ) ).
% image_insert
thf(fact_126_insert__image,axiom,
! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > nat] :
( ( member8440522571783428010at_nat @ X @ A4 )
=> ( ( insert_nat @ ( F2 @ X ) @ ( image_2486076414777270412at_nat @ F2 @ A4 ) )
= ( image_2486076414777270412at_nat @ F2 @ A4 ) ) ) ).
% insert_image
thf(fact_127_insert__image,axiom,
! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > $o] :
( ( member8440522571783428010at_nat @ X @ A4 )
=> ( ( insert_o @ ( F2 @ X ) @ ( image_3693632289388996572_nat_o @ F2 @ A4 ) )
= ( image_3693632289388996572_nat_o @ F2 @ A4 ) ) ) ).
% insert_image
thf(fact_128_insert__image,axiom,
! [X: nat,A4: set_nat,F2: nat > set_nat] :
( ( member_nat @ X @ A4 )
=> ( ( insert_set_nat @ ( F2 @ X ) @ ( image_nat_set_nat @ F2 @ A4 ) )
= ( image_nat_set_nat @ F2 @ A4 ) ) ) ).
% insert_image
thf(fact_129_insert__image,axiom,
! [X: nat,A4: set_nat,F2: nat > nat] :
( ( member_nat @ X @ A4 )
=> ( ( insert_nat @ ( F2 @ X ) @ ( image_nat_nat2 @ F2 @ A4 ) )
= ( image_nat_nat2 @ F2 @ A4 ) ) ) ).
% insert_image
thf(fact_130_insert__image,axiom,
! [X: nat,A4: set_nat,F2: nat > $o] :
( ( member_nat @ X @ A4 )
=> ( ( insert_o @ ( F2 @ X ) @ ( image_nat_o2 @ F2 @ A4 ) )
= ( image_nat_o2 @ F2 @ A4 ) ) ) ).
% insert_image
thf(fact_131_image__empty,axiom,
! [F2: nat > set_nat] :
( ( image_nat_set_nat @ F2 @ bot_bot_set_nat )
= bot_bot_set_set_nat ) ).
% image_empty
thf(fact_132_image__empty,axiom,
! [F2: nat > nat] :
( ( image_nat_nat2 @ F2 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_133_image__empty,axiom,
! [F2: nat > $o] :
( ( image_nat_o2 @ F2 @ bot_bot_set_nat )
= bot_bot_set_o ) ).
% image_empty
thf(fact_134_image__empty,axiom,
! [F2: $o > nat] :
( ( image_o_nat2 @ F2 @ bot_bot_set_o )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_135_image__empty,axiom,
! [F2: $o > $o] :
( ( image_o_o2 @ F2 @ bot_bot_set_o )
= bot_bot_set_o ) ).
% image_empty
thf(fact_136_empty__is__image,axiom,
! [F2: nat > set_nat,A4: set_nat] :
( ( bot_bot_set_set_nat
= ( image_nat_set_nat @ F2 @ A4 ) )
= ( A4 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_137_empty__is__image,axiom,
! [F2: nat > nat,A4: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat2 @ F2 @ A4 ) )
= ( A4 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_138_empty__is__image,axiom,
! [F2: $o > nat,A4: set_o] :
( ( bot_bot_set_nat
= ( image_o_nat2 @ F2 @ A4 ) )
= ( A4 = bot_bot_set_o ) ) ).
% empty_is_image
thf(fact_139_empty__is__image,axiom,
! [F2: nat > $o,A4: set_nat] :
( ( bot_bot_set_o
= ( image_nat_o2 @ F2 @ A4 ) )
= ( A4 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_140_empty__is__image,axiom,
! [F2: $o > $o,A4: set_o] :
( ( bot_bot_set_o
= ( image_o_o2 @ F2 @ A4 ) )
= ( A4 = bot_bot_set_o ) ) ).
% empty_is_image
thf(fact_141_image__eqI,axiom,
! [B: product_prod_nat_nat,F2: product_prod_nat_nat > product_prod_nat_nat,X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ( B
= ( F2 @ X ) )
=> ( ( member8440522571783428010at_nat @ X @ A4 )
=> ( member8440522571783428010at_nat @ B @ ( image_5168914502847457605at_nat @ F2 @ A4 ) ) ) ) ).
% image_eqI
thf(fact_142_image__eqI,axiom,
! [B: nat,F2: product_prod_nat_nat > nat,X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ( B
= ( F2 @ X ) )
=> ( ( member8440522571783428010at_nat @ X @ A4 )
=> ( member_nat @ B @ ( image_2486076414777270412at_nat @ F2 @ A4 ) ) ) ) ).
% image_eqI
thf(fact_143_image__eqI,axiom,
! [B: set_nat,F2: nat > set_nat,X: nat,A4: set_nat] :
( ( B
= ( F2 @ X ) )
=> ( ( member_nat @ X @ A4 )
=> ( member_set_nat @ B @ ( image_nat_set_nat @ F2 @ A4 ) ) ) ) ).
% image_eqI
thf(fact_144_image__eqI,axiom,
! [B: product_prod_nat_nat,F2: nat > product_prod_nat_nat,X: nat,A4: set_nat] :
( ( B
= ( F2 @ X ) )
=> ( ( member_nat @ X @ A4 )
=> ( member8440522571783428010at_nat @ B @ ( image_5846123807819985514at_nat @ F2 @ A4 ) ) ) ) ).
% image_eqI
thf(fact_145_image__eqI,axiom,
! [B: nat,F2: nat > nat,X: nat,A4: set_nat] :
( ( B
= ( F2 @ X ) )
=> ( ( member_nat @ X @ A4 )
=> ( member_nat @ B @ ( image_nat_nat2 @ F2 @ A4 ) ) ) ) ).
% image_eqI
thf(fact_146_empty__Collect__eq,axiom,
! [P2: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P2 ) )
= ( ! [X4: nat] :
~ ( P2 @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_147_empty__Collect__eq,axiom,
! [P2: $o > $o] :
( ( bot_bot_set_o
= ( collect_o @ P2 ) )
= ( ! [X4: $o] :
~ ( P2 @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_148_Collect__empty__eq,axiom,
! [P2: nat > $o] :
( ( ( collect_nat @ P2 )
= bot_bot_set_nat )
= ( ! [X4: nat] :
~ ( P2 @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_149_Collect__empty__eq,axiom,
! [P2: $o > $o] :
( ( ( collect_o @ P2 )
= bot_bot_set_o )
= ( ! [X4: $o] :
~ ( P2 @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_150_all__not__in__conv,axiom,
! [A4: set_Pr1261947904930325089at_nat] :
( ( ! [X4: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ X4 @ A4 ) )
= ( A4 = bot_bo2099793752762293965at_nat ) ) ).
% all_not_in_conv
thf(fact_151_all__not__in__conv,axiom,
! [A4: set_nat] :
( ( ! [X4: nat] :
~ ( member_nat @ X4 @ A4 ) )
= ( A4 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_152_all__not__in__conv,axiom,
! [A4: set_o] :
( ( ! [X4: $o] :
~ ( member_o @ X4 @ A4 ) )
= ( A4 = bot_bot_set_o ) ) ).
% all_not_in_conv
thf(fact_153_empty__iff,axiom,
! [C: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ C @ bot_bo2099793752762293965at_nat ) ).
% empty_iff
thf(fact_154_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_155_empty__iff,axiom,
! [C: $o] :
~ ( member_o @ C @ bot_bot_set_o ) ).
% empty_iff
thf(fact_156_insert__absorb2,axiom,
! [X: nat,A4: set_nat] :
( ( insert_nat @ X @ ( insert_nat @ X @ A4 ) )
= ( insert_nat @ X @ A4 ) ) ).
% insert_absorb2
thf(fact_157_insert__absorb2,axiom,
! [X: $o,A4: set_o] :
( ( insert_o @ X @ ( insert_o @ X @ A4 ) )
= ( insert_o @ X @ A4 ) ) ).
% insert_absorb2
thf(fact_158_insert__iff,axiom,
! [A: $o,B: $o,A4: set_o] :
( ( member_o @ A @ ( insert_o @ B @ A4 ) )
= ( ( A = B )
| ( member_o @ A @ A4 ) ) ) ).
% insert_iff
thf(fact_159_insert__iff,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ A4 ) )
= ( ( A = B )
| ( member8440522571783428010at_nat @ A @ A4 ) ) ) ).
% insert_iff
thf(fact_160_insert__iff,axiom,
! [A: nat,B: nat,A4: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A4 ) )
= ( ( A = B )
| ( member_nat @ A @ A4 ) ) ) ).
% insert_iff
thf(fact_161_insertCI,axiom,
! [A: $o,B5: set_o,B: $o] :
( ( ~ ( member_o @ A @ B5 )
=> ( A = B ) )
=> ( member_o @ A @ ( insert_o @ B @ B5 ) ) ) ).
% insertCI
thf(fact_162_insertCI,axiom,
! [A: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
( ( ~ ( member8440522571783428010at_nat @ A @ B5 )
=> ( A = B ) )
=> ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ B5 ) ) ) ).
% insertCI
thf(fact_163_insertCI,axiom,
! [A: nat,B5: set_nat,B: nat] :
( ( ~ ( member_nat @ A @ B5 )
=> ( A = B ) )
=> ( member_nat @ A @ ( insert_nat @ B @ B5 ) ) ) ).
% insertCI
thf(fact_164_image__is__empty,axiom,
! [F2: nat > set_nat,A4: set_nat] :
( ( ( image_nat_set_nat @ F2 @ A4 )
= bot_bot_set_set_nat )
= ( A4 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_165_image__is__empty,axiom,
! [F2: nat > nat,A4: set_nat] :
( ( ( image_nat_nat2 @ F2 @ A4 )
= bot_bot_set_nat )
= ( A4 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_166_image__is__empty,axiom,
! [F2: $o > nat,A4: set_o] :
( ( ( image_o_nat2 @ F2 @ A4 )
= bot_bot_set_nat )
= ( A4 = bot_bot_set_o ) ) ).
% image_is_empty
thf(fact_167_image__is__empty,axiom,
! [F2: nat > $o,A4: set_nat] :
( ( ( image_nat_o2 @ F2 @ A4 )
= bot_bot_set_o )
= ( A4 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_168_image__is__empty,axiom,
! [F2: $o > $o,A4: set_o] :
( ( ( image_o_o2 @ F2 @ A4 )
= bot_bot_set_o )
= ( A4 = bot_bot_set_o ) ) ).
% image_is_empty
thf(fact_169_bot__empty__eq2,axiom,
( bot_bot_nat_nat_o
= ( ^ [X4: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ bot_bo2099793752762293965at_nat ) ) ) ).
% bot_empty_eq2
thf(fact_170_equiv__on__unique,axiom,
! [A4: set_nat,P: set_Pr1261947904930325089at_nat,B5: set_nat] :
( ( equiv_equiv_nat @ A4 @ P )
=> ( ( equiv_equiv_nat @ B5 @ P )
=> ( A4 = B5 ) ) ) ).
% equiv_on_unique
thf(fact_171_bot__empty__eq,axiom,
( bot_bo482883023278783056_nat_o
= ( ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ bot_bo2099793752762293965at_nat ) ) ) ).
% bot_empty_eq
thf(fact_172_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X4: nat] : ( member_nat @ X4 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_173_bot__empty__eq,axiom,
( bot_bot_o_o
= ( ^ [X4: $o] : ( member_o @ X4 @ bot_bot_set_o ) ) ) ).
% bot_empty_eq
thf(fact_174_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_175_bot__set__def,axiom,
( bot_bot_set_o
= ( collect_o @ bot_bot_o_o ) ) ).
% bot_set_def
thf(fact_176_rev__image__eqI,axiom,
! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,F2: product_prod_nat_nat > product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X @ A4 )
=> ( ( B
= ( F2 @ X ) )
=> ( member8440522571783428010at_nat @ B @ ( image_5168914502847457605at_nat @ F2 @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_177_rev__image__eqI,axiom,
! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B: nat,F2: product_prod_nat_nat > nat] :
( ( member8440522571783428010at_nat @ X @ A4 )
=> ( ( B
= ( F2 @ X ) )
=> ( member_nat @ B @ ( image_2486076414777270412at_nat @ F2 @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_178_rev__image__eqI,axiom,
! [X: nat,A4: set_nat,B: set_nat,F2: nat > set_nat] :
( ( member_nat @ X @ A4 )
=> ( ( B
= ( F2 @ X ) )
=> ( member_set_nat @ B @ ( image_nat_set_nat @ F2 @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_179_rev__image__eqI,axiom,
! [X: nat,A4: set_nat,B: product_prod_nat_nat,F2: nat > product_prod_nat_nat] :
( ( member_nat @ X @ A4 )
=> ( ( B
= ( F2 @ X ) )
=> ( member8440522571783428010at_nat @ B @ ( image_5846123807819985514at_nat @ F2 @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_180_rev__image__eqI,axiom,
! [X: nat,A4: set_nat,B: nat,F2: nat > nat] :
( ( member_nat @ X @ A4 )
=> ( ( B
= ( F2 @ X ) )
=> ( member_nat @ B @ ( image_nat_nat2 @ F2 @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_181_ball__imageD,axiom,
! [F2: nat > set_nat,A4: set_nat,P2: set_nat > $o] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ ( image_nat_set_nat @ F2 @ A4 ) )
=> ( P2 @ X3 ) )
=> ! [X6: nat] :
( ( member_nat @ X6 @ A4 )
=> ( P2 @ ( F2 @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_182_image__cong,axiom,
! [M: set_nat,N2: set_nat,F2: nat > set_nat,G: nat > set_nat] :
( ( M = N2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N2 )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_nat_set_nat @ F2 @ M )
= ( image_nat_set_nat @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_183_bex__imageD,axiom,
! [F2: nat > set_nat,A4: set_nat,P2: set_nat > $o] :
( ? [X6: set_nat] :
( ( member_set_nat @ X6 @ ( image_nat_set_nat @ F2 @ A4 ) )
& ( P2 @ X6 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ( P2 @ ( F2 @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_184_image__iff,axiom,
! [Z: set_nat,F2: nat > set_nat,A4: set_nat] :
( ( member_set_nat @ Z @ ( image_nat_set_nat @ F2 @ A4 ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ( Z
= ( F2 @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_185_imageI,axiom,
! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X @ A4 )
=> ( member8440522571783428010at_nat @ ( F2 @ X ) @ ( image_5168914502847457605at_nat @ F2 @ A4 ) ) ) ).
% imageI
thf(fact_186_imageI,axiom,
! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > nat] :
( ( member8440522571783428010at_nat @ X @ A4 )
=> ( member_nat @ ( F2 @ X ) @ ( image_2486076414777270412at_nat @ F2 @ A4 ) ) ) ).
% imageI
thf(fact_187_imageI,axiom,
! [X: nat,A4: set_nat,F2: nat > set_nat] :
( ( member_nat @ X @ A4 )
=> ( member_set_nat @ ( F2 @ X ) @ ( image_nat_set_nat @ F2 @ A4 ) ) ) ).
% imageI
thf(fact_188_imageI,axiom,
! [X: nat,A4: set_nat,F2: nat > product_prod_nat_nat] :
( ( member_nat @ X @ A4 )
=> ( member8440522571783428010at_nat @ ( F2 @ X ) @ ( image_5846123807819985514at_nat @ F2 @ A4 ) ) ) ).
% imageI
thf(fact_189_imageI,axiom,
! [X: nat,A4: set_nat,F2: nat > nat] :
( ( member_nat @ X @ A4 )
=> ( member_nat @ ( F2 @ X ) @ ( image_nat_nat2 @ F2 @ A4 ) ) ) ).
% imageI
thf(fact_190_ex__in__conv,axiom,
! [A4: set_Pr1261947904930325089at_nat] :
( ( ? [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ A4 ) )
= ( A4 != bot_bo2099793752762293965at_nat ) ) ).
% ex_in_conv
thf(fact_191_ex__in__conv,axiom,
! [A4: set_nat] :
( ( ? [X4: nat] : ( member_nat @ X4 @ A4 ) )
= ( A4 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_192_ex__in__conv,axiom,
! [A4: set_o] :
( ( ? [X4: $o] : ( member_o @ X4 @ A4 ) )
= ( A4 != bot_bot_set_o ) ) ).
% ex_in_conv
thf(fact_193_equals0I,axiom,
! [A4: set_Pr1261947904930325089at_nat] :
( ! [Y: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ Y @ A4 )
=> ( A4 = bot_bo2099793752762293965at_nat ) ) ).
% equals0I
thf(fact_194_equals0I,axiom,
! [A4: set_nat] :
( ! [Y: nat] :
~ ( member_nat @ Y @ A4 )
=> ( A4 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_195_equals0I,axiom,
! [A4: set_o] :
( ! [Y: $o] :
~ ( member_o @ Y @ A4 )
=> ( A4 = bot_bot_set_o ) ) ).
% equals0I
thf(fact_196_equals0D,axiom,
! [A4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
( ( A4 = bot_bo2099793752762293965at_nat )
=> ~ ( member8440522571783428010at_nat @ A @ A4 ) ) ).
% equals0D
thf(fact_197_equals0D,axiom,
! [A4: set_nat,A: nat] :
( ( A4 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A4 ) ) ).
% equals0D
thf(fact_198_equals0D,axiom,
! [A4: set_o,A: $o] :
( ( A4 = bot_bot_set_o )
=> ~ ( member_o @ A @ A4 ) ) ).
% equals0D
thf(fact_199_emptyE,axiom,
! [A: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ A @ bot_bo2099793752762293965at_nat ) ).
% emptyE
thf(fact_200_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_201_emptyE,axiom,
! [A: $o] :
~ ( member_o @ A @ bot_bot_set_o ) ).
% emptyE
thf(fact_202_mk__disjoint__insert,axiom,
! [A: $o,A4: set_o] :
( ( member_o @ A @ A4 )
=> ? [B6: set_o] :
( ( A4
= ( insert_o @ A @ B6 ) )
& ~ ( member_o @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_203_mk__disjoint__insert,axiom,
! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ A @ A4 )
=> ? [B6: set_Pr1261947904930325089at_nat] :
( ( A4
= ( insert8211810215607154385at_nat @ A @ B6 ) )
& ~ ( member8440522571783428010at_nat @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_204_mk__disjoint__insert,axiom,
! [A: nat,A4: set_nat] :
( ( member_nat @ A @ A4 )
=> ? [B6: set_nat] :
( ( A4
= ( insert_nat @ A @ B6 ) )
& ~ ( member_nat @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_205_insert__commute,axiom,
! [X: nat,Y3: nat,A4: set_nat] :
( ( insert_nat @ X @ ( insert_nat @ Y3 @ A4 ) )
= ( insert_nat @ Y3 @ ( insert_nat @ X @ A4 ) ) ) ).
% insert_commute
thf(fact_206_insert__commute,axiom,
! [X: $o,Y3: $o,A4: set_o] :
( ( insert_o @ X @ ( insert_o @ Y3 @ A4 ) )
= ( insert_o @ Y3 @ ( insert_o @ X @ A4 ) ) ) ).
% insert_commute
thf(fact_207_insert__eq__iff,axiom,
! [A: $o,A4: set_o,B: $o,B5: set_o] :
( ~ ( member_o @ A @ A4 )
=> ( ~ ( member_o @ B @ B5 )
=> ( ( ( insert_o @ A @ A4 )
= ( insert_o @ B @ B5 ) )
= ( ( ( A = B )
=> ( A4 = B5 ) )
& ( ( A = (~ B) )
=> ? [C3: set_o] :
( ( A4
= ( insert_o @ B @ C3 ) )
& ~ ( member_o @ B @ C3 )
& ( B5
= ( insert_o @ A @ C3 ) )
& ~ ( member_o @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_208_insert__eq__iff,axiom,
! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
( ~ ( member8440522571783428010at_nat @ A @ A4 )
=> ( ~ ( member8440522571783428010at_nat @ B @ B5 )
=> ( ( ( insert8211810215607154385at_nat @ A @ A4 )
= ( insert8211810215607154385at_nat @ B @ B5 ) )
= ( ( ( A = B )
=> ( A4 = B5 ) )
& ( ( A != B )
=> ? [C3: set_Pr1261947904930325089at_nat] :
( ( A4
= ( insert8211810215607154385at_nat @ B @ C3 ) )
& ~ ( member8440522571783428010at_nat @ B @ C3 )
& ( B5
= ( insert8211810215607154385at_nat @ A @ C3 ) )
& ~ ( member8440522571783428010at_nat @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_209_insert__eq__iff,axiom,
! [A: nat,A4: set_nat,B: nat,B5: set_nat] :
( ~ ( member_nat @ A @ A4 )
=> ( ~ ( member_nat @ B @ B5 )
=> ( ( ( insert_nat @ A @ A4 )
= ( insert_nat @ B @ B5 ) )
= ( ( ( A = B )
=> ( A4 = B5 ) )
& ( ( A != B )
=> ? [C3: set_nat] :
( ( A4
= ( insert_nat @ B @ C3 ) )
& ~ ( member_nat @ B @ C3 )
& ( B5
= ( insert_nat @ A @ C3 ) )
& ~ ( member_nat @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_210_insert__absorb,axiom,
! [A: $o,A4: set_o] :
( ( member_o @ A @ A4 )
=> ( ( insert_o @ A @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_211_insert__absorb,axiom,
! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ A @ A4 )
=> ( ( insert8211810215607154385at_nat @ A @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_212_insert__absorb,axiom,
! [A: nat,A4: set_nat] :
( ( member_nat @ A @ A4 )
=> ( ( insert_nat @ A @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_213_insert__ident,axiom,
! [X: $o,A4: set_o,B5: set_o] :
( ~ ( member_o @ X @ A4 )
=> ( ~ ( member_o @ X @ B5 )
=> ( ( ( insert_o @ X @ A4 )
= ( insert_o @ X @ B5 ) )
= ( A4 = B5 ) ) ) ) ).
% insert_ident
thf(fact_214_insert__ident,axiom,
! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ~ ( member8440522571783428010at_nat @ X @ A4 )
=> ( ~ ( member8440522571783428010at_nat @ X @ B5 )
=> ( ( ( insert8211810215607154385at_nat @ X @ A4 )
= ( insert8211810215607154385at_nat @ X @ B5 ) )
= ( A4 = B5 ) ) ) ) ).
% insert_ident
thf(fact_215_insert__ident,axiom,
! [X: nat,A4: set_nat,B5: set_nat] :
( ~ ( member_nat @ X @ A4 )
=> ( ~ ( member_nat @ X @ B5 )
=> ( ( ( insert_nat @ X @ A4 )
= ( insert_nat @ X @ B5 ) )
= ( A4 = B5 ) ) ) ) ).
% insert_ident
thf(fact_216_Set_Oset__insert,axiom,
! [X: $o,A4: set_o] :
( ( member_o @ X @ A4 )
=> ~ ! [B6: set_o] :
( ( A4
= ( insert_o @ X @ B6 ) )
=> ( member_o @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_217_Set_Oset__insert,axiom,
! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ X @ A4 )
=> ~ ! [B6: set_Pr1261947904930325089at_nat] :
( ( A4
= ( insert8211810215607154385at_nat @ X @ B6 ) )
=> ( member8440522571783428010at_nat @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_218_Set_Oset__insert,axiom,
! [X: nat,A4: set_nat] :
( ( member_nat @ X @ A4 )
=> ~ ! [B6: set_nat] :
( ( A4
= ( insert_nat @ X @ B6 ) )
=> ( member_nat @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_219_insertI2,axiom,
! [A: $o,B5: set_o,B: $o] :
( ( member_o @ A @ B5 )
=> ( member_o @ A @ ( insert_o @ B @ B5 ) ) ) ).
% insertI2
thf(fact_220_insertI2,axiom,
! [A: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ A @ B5 )
=> ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ B5 ) ) ) ).
% insertI2
thf(fact_221_insertI2,axiom,
! [A: nat,B5: set_nat,B: nat] :
( ( member_nat @ A @ B5 )
=> ( member_nat @ A @ ( insert_nat @ B @ B5 ) ) ) ).
% insertI2
thf(fact_222_insertI1,axiom,
! [A: $o,B5: set_o] : ( member_o @ A @ ( insert_o @ A @ B5 ) ) ).
% insertI1
thf(fact_223_insertI1,axiom,
! [A: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] : ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ A @ B5 ) ) ).
% insertI1
thf(fact_224_insertI1,axiom,
! [A: nat,B5: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B5 ) ) ).
% insertI1
thf(fact_225_insertE,axiom,
! [A: $o,B: $o,A4: set_o] :
( ( member_o @ A @ ( insert_o @ B @ A4 ) )
=> ( ( A = (~ B) )
=> ( member_o @ A @ A4 ) ) ) ).
% insertE
thf(fact_226_insertE,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ A4 ) )
=> ( ( A != B )
=> ( member8440522571783428010at_nat @ A @ A4 ) ) ) ).
% insertE
thf(fact_227_insertE,axiom,
! [A: nat,B: nat,A4: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A4 ) )
=> ( ( A != B )
=> ( member_nat @ A @ A4 ) ) ) ).
% insertE
thf(fact_228_pred__equals__eq2,axiom,
! [R2: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
( ( ( ^ [X4: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ R2 ) )
= ( ^ [X4: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ S3 ) ) )
= ( R2 = S3 ) ) ).
% pred_equals_eq2
thf(fact_229_Compr__image__eq,axiom,
! [F2: nat > set_nat,A4: set_nat,P2: set_nat > $o] :
( ( collect_set_nat
@ ^ [X4: set_nat] :
( ( member_set_nat @ X4 @ ( image_nat_set_nat @ F2 @ A4 ) )
& ( P2 @ X4 ) ) )
= ( image_nat_set_nat @ F2
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ( P2 @ ( F2 @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_230_Compr__image__eq,axiom,
! [F2: product_prod_nat_nat > product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,P2: product_prod_nat_nat > $o] :
( ( collec3392354462482085612at_nat
@ ^ [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ ( image_5168914502847457605at_nat @ F2 @ A4 ) )
& ( P2 @ X4 ) ) )
= ( image_5168914502847457605at_nat @ F2
@ ( collec3392354462482085612at_nat
@ ^ [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ A4 )
& ( P2 @ ( F2 @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_231_Compr__image__eq,axiom,
! [F2: nat > product_prod_nat_nat,A4: set_nat,P2: product_prod_nat_nat > $o] :
( ( collec3392354462482085612at_nat
@ ^ [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ ( image_5846123807819985514at_nat @ F2 @ A4 ) )
& ( P2 @ X4 ) ) )
= ( image_5846123807819985514at_nat @ F2
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ( P2 @ ( F2 @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_232_Compr__image__eq,axiom,
! [F2: product_prod_nat_nat > nat,A4: set_Pr1261947904930325089at_nat,P2: nat > $o] :
( ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ ( image_2486076414777270412at_nat @ F2 @ A4 ) )
& ( P2 @ X4 ) ) )
= ( image_2486076414777270412at_nat @ F2
@ ( collec3392354462482085612at_nat
@ ^ [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ A4 )
& ( P2 @ ( F2 @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_233_Compr__image__eq,axiom,
! [F2: nat > nat,A4: set_nat,P2: nat > $o] :
( ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ ( image_nat_nat2 @ F2 @ A4 ) )
& ( P2 @ X4 ) ) )
= ( image_nat_nat2 @ F2
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ( P2 @ ( F2 @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_234_image__image,axiom,
! [F2: set_nat > set_nat,G: nat > set_nat,A4: set_nat] :
( ( image_7916887816326733075et_nat @ F2 @ ( image_nat_set_nat @ G @ A4 ) )
= ( image_nat_set_nat
@ ^ [X4: nat] : ( F2 @ ( G @ X4 ) )
@ A4 ) ) ).
% image_image
thf(fact_235_image__image,axiom,
! [F2: nat > set_nat,G: nat > nat,A4: set_nat] :
( ( image_nat_set_nat @ F2 @ ( image_nat_nat2 @ G @ A4 ) )
= ( image_nat_set_nat
@ ^ [X4: nat] : ( F2 @ ( G @ X4 ) )
@ A4 ) ) ).
% image_image
thf(fact_236_imageE,axiom,
! [B: set_nat,F2: nat > set_nat,A4: set_nat] :
( ( member_set_nat @ B @ ( image_nat_set_nat @ F2 @ A4 ) )
=> ~ ! [X3: nat] :
( ( B
= ( F2 @ X3 ) )
=> ~ ( member_nat @ X3 @ A4 ) ) ) ).
% imageE
thf(fact_237_imageE,axiom,
! [B: product_prod_nat_nat,F2: product_prod_nat_nat > product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ B @ ( image_5168914502847457605at_nat @ F2 @ A4 ) )
=> ~ ! [X3: product_prod_nat_nat] :
( ( B
= ( F2 @ X3 ) )
=> ~ ( member8440522571783428010at_nat @ X3 @ A4 ) ) ) ).
% imageE
thf(fact_238_imageE,axiom,
! [B: product_prod_nat_nat,F2: nat > product_prod_nat_nat,A4: set_nat] :
( ( member8440522571783428010at_nat @ B @ ( image_5846123807819985514at_nat @ F2 @ A4 ) )
=> ~ ! [X3: nat] :
( ( B
= ( F2 @ X3 ) )
=> ~ ( member_nat @ X3 @ A4 ) ) ) ).
% imageE
thf(fact_239_imageE,axiom,
! [B: nat,F2: product_prod_nat_nat > nat,A4: set_Pr1261947904930325089at_nat] :
( ( member_nat @ B @ ( image_2486076414777270412at_nat @ F2 @ A4 ) )
=> ~ ! [X3: product_prod_nat_nat] :
( ( B
= ( F2 @ X3 ) )
=> ~ ( member8440522571783428010at_nat @ X3 @ A4 ) ) ) ).
% imageE
thf(fact_240_imageE,axiom,
! [B: nat,F2: nat > nat,A4: set_nat] :
( ( member_nat @ B @ ( image_nat_nat2 @ F2 @ A4 ) )
=> ~ ! [X3: nat] :
( ( B
= ( F2 @ X3 ) )
=> ~ ( member_nat @ X3 @ A4 ) ) ) ).
% imageE
thf(fact_241_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X4: nat] : $false ) ) ).
% empty_def
thf(fact_242_empty__def,axiom,
( bot_bot_set_o
= ( collect_o
@ ^ [X4: $o] : $false ) ) ).
% empty_def
thf(fact_243_insert__Collect,axiom,
! [A: nat,P2: nat > $o] :
( ( insert_nat @ A @ ( collect_nat @ P2 ) )
= ( collect_nat
@ ^ [U: nat] :
( ( U != A )
=> ( P2 @ U ) ) ) ) ).
% insert_Collect
thf(fact_244_insert__Collect,axiom,
! [A: $o,P2: $o > $o] :
( ( insert_o @ A @ ( collect_o @ P2 ) )
= ( collect_o
@ ^ [U: $o] :
( ( U != A )
=> ( P2 @ U ) ) ) ) ).
% insert_Collect
thf(fact_245_insert__compr,axiom,
( insert_o
= ( ^ [A5: $o,B7: set_o] :
( collect_o
@ ^ [X4: $o] :
( ( X4 = A5 )
| ( member_o @ X4 @ B7 ) ) ) ) ) ).
% insert_compr
thf(fact_246_insert__compr,axiom,
( insert8211810215607154385at_nat
= ( ^ [A5: product_prod_nat_nat,B7: set_Pr1261947904930325089at_nat] :
( collec3392354462482085612at_nat
@ ^ [X4: product_prod_nat_nat] :
( ( X4 = A5 )
| ( member8440522571783428010at_nat @ X4 @ B7 ) ) ) ) ) ).
% insert_compr
thf(fact_247_insert__compr,axiom,
( insert_nat
= ( ^ [A5: nat,B7: set_nat] :
( collect_nat
@ ^ [X4: nat] :
( ( X4 = A5 )
| ( member_nat @ X4 @ B7 ) ) ) ) ) ).
% insert_compr
thf(fact_248_singleton__inject,axiom,
! [A: nat,B: nat] :
( ( ( insert_nat @ A @ bot_bot_set_nat )
= ( insert_nat @ B @ bot_bot_set_nat ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_249_singleton__inject,axiom,
! [A: $o,B: $o] :
( ( ( insert_o @ A @ bot_bot_set_o )
= ( insert_o @ B @ bot_bot_set_o ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_250_insert__not__empty,axiom,
! [A: nat,A4: set_nat] :
( ( insert_nat @ A @ A4 )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_251_insert__not__empty,axiom,
! [A: $o,A4: set_o] :
( ( insert_o @ A @ A4 )
!= bot_bot_set_o ) ).
% insert_not_empty
thf(fact_252_doubleton__eq__iff,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) )
= ( insert_nat @ C @ ( insert_nat @ D2 @ bot_bot_set_nat ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_253_doubleton__eq__iff,axiom,
! [A: $o,B: $o,C: $o,D2: $o] :
( ( ( insert_o @ A @ ( insert_o @ B @ bot_bot_set_o ) )
= ( insert_o @ C @ ( insert_o @ D2 @ bot_bot_set_o ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_254_singleton__iff,axiom,
! [B: product_prod_nat_nat,A: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ B @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_255_singleton__iff,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_256_singleton__iff,axiom,
! [B: $o,A: $o] :
( ( member_o @ B @ ( insert_o @ A @ bot_bot_set_o ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_257_singletonD,axiom,
! [B: product_prod_nat_nat,A: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ B @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_258_singletonD,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_259_singletonD,axiom,
! [B: $o,A: $o] :
( ( member_o @ B @ ( insert_o @ A @ bot_bot_set_o ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_260_ImageE,axiom,
! [B: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat,A4: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ B @ ( image_2484338115069272851at_nat @ R @ A4 ) )
=> ~ ! [X3: product_prod_nat_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ B ) @ R )
=> ~ ( member8440522571783428010at_nat @ X3 @ A4 ) ) ) ).
% ImageE
thf(fact_261_ImageE,axiom,
! [B: product_prod_nat_nat,R: set_Pr7717912310451564380at_nat,A4: set_nat] :
( ( member8440522571783428010at_nat @ B @ ( image_8130187405840929308at_nat @ R @ A4 ) )
=> ~ ! [X3: nat] :
( ( member2223272150424702269at_nat @ ( produc487386426758144856at_nat @ X3 @ B ) @ R )
=> ~ ( member_nat @ X3 @ A4 ) ) ) ).
% ImageE
thf(fact_262_ImageE,axiom,
! [B: nat,R: set_Pr2539167527615954998at_nat,A4: set_Pr1261947904930325089at_nat] :
( ( member_nat @ B @ ( image_4770140012798214206at_nat @ R @ A4 ) )
=> ~ ! [X3: product_prod_nat_nat] :
( ( member3348759134392003351at_nat @ ( produc6350711070570205562at_nat @ X3 @ B ) @ R )
=> ~ ( member8440522571783428010at_nat @ X3 @ A4 ) ) ) ).
% ImageE
thf(fact_263_ImageE,axiom,
! [B: nat,R: set_Pr1261947904930325089at_nat,A4: set_nat] :
( ( member_nat @ B @ ( image_nat_nat @ R @ A4 ) )
=> ~ ! [X3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ B ) @ R )
=> ~ ( member_nat @ X3 @ A4 ) ) ) ).
% ImageE
thf(fact_264_Image__iff,axiom,
! [B: nat,R: set_Pr1261947904930325089at_nat,A4: set_nat] :
( ( member_nat @ B @ ( image_nat_nat @ R @ A4 ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ B ) @ R ) ) ) ) ).
% Image_iff
thf(fact_265_rev__ImageI,axiom,
! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat] :
( ( member8440522571783428010at_nat @ A @ A4 )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R )
=> ( member8440522571783428010at_nat @ B @ ( image_2484338115069272851at_nat @ R @ A4 ) ) ) ) ).
% rev_ImageI
thf(fact_266_rev__ImageI,axiom,
! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B: nat,R: set_Pr2539167527615954998at_nat] :
( ( member8440522571783428010at_nat @ A @ A4 )
=> ( ( member3348759134392003351at_nat @ ( produc6350711070570205562at_nat @ A @ B ) @ R )
=> ( member_nat @ B @ ( image_4770140012798214206at_nat @ R @ A4 ) ) ) ) ).
% rev_ImageI
thf(fact_267_rev__ImageI,axiom,
! [A: nat,A4: set_nat,B: product_prod_nat_nat,R: set_Pr7717912310451564380at_nat] :
( ( member_nat @ A @ A4 )
=> ( ( member2223272150424702269at_nat @ ( produc487386426758144856at_nat @ A @ B ) @ R )
=> ( member8440522571783428010at_nat @ B @ ( image_8130187405840929308at_nat @ R @ A4 ) ) ) ) ).
% rev_ImageI
thf(fact_268_rev__ImageI,axiom,
! [A: nat,A4: set_nat,B: nat,R: set_Pr1261947904930325089at_nat] :
( ( member_nat @ A @ A4 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
=> ( member_nat @ B @ ( image_nat_nat @ R @ A4 ) ) ) ) ).
% rev_ImageI
thf(fact_269_Collect__conv__if2,axiom,
! [P2: nat > $o,A: nat] :
( ( ( P2 @ A )
=> ( ( collect_nat
@ ^ [X4: nat] :
( ( A = X4 )
& ( P2 @ X4 ) ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) )
& ( ~ ( P2 @ A )
=> ( ( collect_nat
@ ^ [X4: nat] :
( ( A = X4 )
& ( P2 @ X4 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if2
thf(fact_270_Collect__conv__if2,axiom,
! [P2: $o > $o,A: $o] :
( ( ( P2 @ A )
=> ( ( collect_o
@ ^ [X4: $o] :
( ( A = X4 )
& ( P2 @ X4 ) ) )
= ( insert_o @ A @ bot_bot_set_o ) ) )
& ( ~ ( P2 @ A )
=> ( ( collect_o
@ ^ [X4: $o] :
( ( A = X4 )
& ( P2 @ X4 ) ) )
= bot_bot_set_o ) ) ) ).
% Collect_conv_if2
thf(fact_271_Collect__conv__if,axiom,
! [P2: nat > $o,A: nat] :
( ( ( P2 @ A )
=> ( ( collect_nat
@ ^ [X4: nat] :
( ( X4 = A )
& ( P2 @ X4 ) ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) )
& ( ~ ( P2 @ A )
=> ( ( collect_nat
@ ^ [X4: nat] :
( ( X4 = A )
& ( P2 @ X4 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if
thf(fact_272_Collect__conv__if,axiom,
! [P2: $o > $o,A: $o] :
( ( ( P2 @ A )
=> ( ( collect_o
@ ^ [X4: $o] :
( ( X4 = A )
& ( P2 @ X4 ) ) )
= ( insert_o @ A @ bot_bot_set_o ) ) )
& ( ~ ( P2 @ A )
=> ( ( collect_o
@ ^ [X4: $o] :
( ( X4 = A )
& ( P2 @ X4 ) ) )
= bot_bot_set_o ) ) ) ).
% Collect_conv_if
thf(fact_273_image__constant__conv,axiom,
! [A4: set_nat,C: set_nat] :
( ( ( A4 = bot_bot_set_nat )
=> ( ( image_nat_set_nat
@ ^ [X4: nat] : C
@ A4 )
= bot_bot_set_set_nat ) )
& ( ( A4 != bot_bot_set_nat )
=> ( ( image_nat_set_nat
@ ^ [X4: nat] : C
@ A4 )
= ( insert_set_nat @ C @ bot_bot_set_set_nat ) ) ) ) ).
% image_constant_conv
thf(fact_274_image__constant__conv,axiom,
! [A4: set_nat,C: nat] :
( ( ( A4 = bot_bot_set_nat )
=> ( ( image_nat_nat2
@ ^ [X4: nat] : C
@ A4 )
= bot_bot_set_nat ) )
& ( ( A4 != bot_bot_set_nat )
=> ( ( image_nat_nat2
@ ^ [X4: nat] : C
@ A4 )
= ( insert_nat @ C @ bot_bot_set_nat ) ) ) ) ).
% image_constant_conv
thf(fact_275_image__constant__conv,axiom,
! [A4: set_nat,C: $o] :
( ( ( A4 = bot_bot_set_nat )
=> ( ( image_nat_o2
@ ^ [X4: nat] : C
@ A4 )
= bot_bot_set_o ) )
& ( ( A4 != bot_bot_set_nat )
=> ( ( image_nat_o2
@ ^ [X4: nat] : C
@ A4 )
= ( insert_o @ C @ bot_bot_set_o ) ) ) ) ).
% image_constant_conv
thf(fact_276_image__constant__conv,axiom,
! [A4: set_o,C: nat] :
( ( ( A4 = bot_bot_set_o )
=> ( ( image_o_nat2
@ ^ [X4: $o] : C
@ A4 )
= bot_bot_set_nat ) )
& ( ( A4 != bot_bot_set_o )
=> ( ( image_o_nat2
@ ^ [X4: $o] : C
@ A4 )
= ( insert_nat @ C @ bot_bot_set_nat ) ) ) ) ).
% image_constant_conv
thf(fact_277_image__constant__conv,axiom,
! [A4: set_o,C: $o] :
( ( ( A4 = bot_bot_set_o )
=> ( ( image_o_o2
@ ^ [X4: $o] : C
@ A4 )
= bot_bot_set_o ) )
& ( ( A4 != bot_bot_set_o )
=> ( ( image_o_o2
@ ^ [X4: $o] : C
@ A4 )
= ( insert_o @ C @ bot_bot_set_o ) ) ) ) ).
% image_constant_conv
thf(fact_278_image__constant,axiom,
! [X: nat,A4: set_nat,C: set_nat] :
( ( member_nat @ X @ A4 )
=> ( ( image_nat_set_nat
@ ^ [X4: nat] : C
@ A4 )
= ( insert_set_nat @ C @ bot_bot_set_set_nat ) ) ) ).
% image_constant
thf(fact_279_image__constant,axiom,
! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,C: nat] :
( ( member8440522571783428010at_nat @ X @ A4 )
=> ( ( image_2486076414777270412at_nat
@ ^ [X4: product_prod_nat_nat] : C
@ A4 )
= ( insert_nat @ C @ bot_bot_set_nat ) ) ) ).
% image_constant
thf(fact_280_image__constant,axiom,
! [X: nat,A4: set_nat,C: nat] :
( ( member_nat @ X @ A4 )
=> ( ( image_nat_nat2
@ ^ [X4: nat] : C
@ A4 )
= ( insert_nat @ C @ bot_bot_set_nat ) ) ) ).
% image_constant
thf(fact_281_image__constant,axiom,
! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,C: $o] :
( ( member8440522571783428010at_nat @ X @ A4 )
=> ( ( image_3693632289388996572_nat_o
@ ^ [X4: product_prod_nat_nat] : C
@ A4 )
= ( insert_o @ C @ bot_bot_set_o ) ) ) ).
% image_constant
thf(fact_282_image__constant,axiom,
! [X: nat,A4: set_nat,C: $o] :
( ( member_nat @ X @ A4 )
=> ( ( image_nat_o2
@ ^ [X4: nat] : C
@ A4 )
= ( insert_o @ C @ bot_bot_set_o ) ) ) ).
% image_constant
thf(fact_283_Image__singleton,axiom,
! [R: set_Pr1261947904930325089at_nat,A: nat] :
( ( image_nat_nat @ R @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( collect_nat
@ ^ [B4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B4 ) @ R ) ) ) ).
% Image_singleton
thf(fact_284_equiv__class__eq__iff,axiom,
! [A4: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat,X: product_prod_nat_nat,Y3: product_prod_nat_nat] :
( ( equiv_5407799809235660215at_nat @ A4 @ R )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ R )
= ( ( ( image_2484338115069272851at_nat @ R @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
= ( image_2484338115069272851at_nat @ R @ ( insert8211810215607154385at_nat @ Y3 @ bot_bo2099793752762293965at_nat ) ) )
& ( member8440522571783428010at_nat @ X @ A4 )
& ( member8440522571783428010at_nat @ Y3 @ A4 ) ) ) ) ).
% equiv_class_eq_iff
thf(fact_285_equiv__class__eq__iff,axiom,
! [A4: set_nat,R: set_Pr1261947904930325089at_nat,X: nat,Y3: nat] :
( ( equiv_equiv_nat @ A4 @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R )
= ( ( ( image_nat_nat @ R @ ( insert_nat @ X @ bot_bot_set_nat ) )
= ( image_nat_nat @ R @ ( insert_nat @ Y3 @ bot_bot_set_nat ) ) )
& ( member_nat @ X @ A4 )
& ( member_nat @ Y3 @ A4 ) ) ) ) ).
% equiv_class_eq_iff
thf(fact_286_equiv__class__eq__iff,axiom,
! [A4: set_o,R: set_Product_prod_o_o,X: $o,Y3: $o] :
( ( equiv_equiv_o @ A4 @ R )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y3 ) @ R )
= ( ( ( image_o_o @ R @ ( insert_o @ X @ bot_bot_set_o ) )
= ( image_o_o @ R @ ( insert_o @ Y3 @ bot_bot_set_o ) ) )
& ( member_o @ X @ A4 )
& ( member_o @ Y3 @ A4 ) ) ) ) ).
% equiv_class_eq_iff
thf(fact_287_eq__equiv__class__iff,axiom,
! [A4: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat,X: product_prod_nat_nat,Y3: product_prod_nat_nat] :
( ( equiv_5407799809235660215at_nat @ A4 @ R )
=> ( ( member8440522571783428010at_nat @ X @ A4 )
=> ( ( member8440522571783428010at_nat @ Y3 @ A4 )
=> ( ( ( image_2484338115069272851at_nat @ R @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
= ( image_2484338115069272851at_nat @ R @ ( insert8211810215607154385at_nat @ Y3 @ bot_bo2099793752762293965at_nat ) ) )
= ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ R ) ) ) ) ) ).
% eq_equiv_class_iff
thf(fact_288_eq__equiv__class__iff,axiom,
! [A4: set_nat,R: set_Pr1261947904930325089at_nat,X: nat,Y3: nat] :
( ( equiv_equiv_nat @ A4 @ R )
=> ( ( member_nat @ X @ A4 )
=> ( ( member_nat @ Y3 @ A4 )
=> ( ( ( image_nat_nat @ R @ ( insert_nat @ X @ bot_bot_set_nat ) )
= ( image_nat_nat @ R @ ( insert_nat @ Y3 @ bot_bot_set_nat ) ) )
= ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R ) ) ) ) ) ).
% eq_equiv_class_iff
thf(fact_289_eq__equiv__class__iff,axiom,
! [A4: set_o,R: set_Product_prod_o_o,X: $o,Y3: $o] :
( ( equiv_equiv_o @ A4 @ R )
=> ( ( member_o @ X @ A4 )
=> ( ( member_o @ Y3 @ A4 )
=> ( ( ( image_o_o @ R @ ( insert_o @ X @ bot_bot_set_o ) )
= ( image_o_o @ R @ ( insert_o @ Y3 @ bot_bot_set_o ) ) )
= ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y3 ) @ R ) ) ) ) ) ).
% eq_equiv_class_iff
thf(fact_290_equiv__class__eq,axiom,
! [A4: set_nat,R: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
( ( equiv_equiv_nat @ A4 @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
=> ( ( image_nat_nat @ R @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( image_nat_nat @ R @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ) ).
% equiv_class_eq
thf(fact_291_equiv__class__eq,axiom,
! [A4: set_o,R: set_Product_prod_o_o,A: $o,B: $o] :
( ( equiv_equiv_o @ A4 @ R )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R )
=> ( ( image_o_o @ R @ ( insert_o @ A @ bot_bot_set_o ) )
= ( image_o_o @ R @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ) ).
% equiv_class_eq
thf(fact_292_eq__equiv__class,axiom,
! [R: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat,B: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ( ( image_2484338115069272851at_nat @ R @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
= ( image_2484338115069272851at_nat @ R @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) )
=> ( ( equiv_5407799809235660215at_nat @ A4 @ R )
=> ( ( member8440522571783428010at_nat @ B @ A4 )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R ) ) ) ) ).
% eq_equiv_class
thf(fact_293_eq__equiv__class,axiom,
! [R: set_Pr1261947904930325089at_nat,A: nat,B: nat,A4: set_nat] :
( ( ( image_nat_nat @ R @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( image_nat_nat @ R @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
=> ( ( equiv_equiv_nat @ A4 @ R )
=> ( ( member_nat @ B @ A4 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R ) ) ) ) ).
% eq_equiv_class
thf(fact_294_eq__equiv__class,axiom,
! [R: set_Product_prod_o_o,A: $o,B: $o,A4: set_o] :
( ( ( image_o_o @ R @ ( insert_o @ A @ bot_bot_set_o ) )
= ( image_o_o @ R @ ( insert_o @ B @ bot_bot_set_o ) ) )
=> ( ( equiv_equiv_o @ A4 @ R )
=> ( ( member_o @ B @ A4 )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R ) ) ) ) ).
% eq_equiv_class
thf(fact_295_equiv__class__self,axiom,
! [A4: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat] :
( ( equiv_5407799809235660215at_nat @ A4 @ R )
=> ( ( member8440522571783428010at_nat @ A @ A4 )
=> ( member8440522571783428010at_nat @ A @ ( image_2484338115069272851at_nat @ R @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ) ) ).
% equiv_class_self
thf(fact_296_equiv__class__self,axiom,
! [A4: set_nat,R: set_Pr1261947904930325089at_nat,A: nat] :
( ( equiv_equiv_nat @ A4 @ R )
=> ( ( member_nat @ A @ A4 )
=> ( member_nat @ A @ ( image_nat_nat @ R @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) ).
% equiv_class_self
thf(fact_297_equiv__class__self,axiom,
! [A4: set_o,R: set_Product_prod_o_o,A: $o] :
( ( equiv_equiv_o @ A4 @ R )
=> ( ( member_o @ A @ A4 )
=> ( member_o @ A @ ( image_o_o @ R @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) ).
% equiv_class_self
thf(fact_298_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P2: nat > nat > $o,X: nat,Y3: nat,A: product_prod_nat_nat] :
( ( P2 @ X @ Y3 )
=> ( ( A
= ( product_Pair_nat_nat @ X @ Y3 ) )
=> ( P2 @ ( product_fst_nat_nat @ A ) @ ( product_snd_nat_nat @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_299_conjI__realizer,axiom,
! [P2: nat > $o,P: nat,Q: nat > $o,Q2: nat] :
( ( P2 @ P )
=> ( ( Q @ Q2 )
=> ( ( P2 @ ( product_fst_nat_nat @ ( product_Pair_nat_nat @ P @ Q2 ) ) )
& ( Q @ ( product_snd_nat_nat @ ( product_Pair_nat_nat @ P @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_300_exI__realizer,axiom,
! [P2: nat > nat > $o,Y3: nat,X: nat] :
( ( P2 @ Y3 @ X )
=> ( P2 @ ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X @ Y3 ) ) @ ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X @ Y3 ) ) ) ) ).
% exI_realizer
thf(fact_301_case__prod__Pair__iden,axiom,
! [P: product_prod_nat_nat] :
( ( produc2626176000494625587at_nat @ product_Pair_nat_nat @ P )
= P ) ).
% case_prod_Pair_iden
thf(fact_302_the__elem__eq,axiom,
! [X: nat] :
( ( the_elem_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
= X ) ).
% the_elem_eq
thf(fact_303_the__elem__eq,axiom,
! [X: $o] :
( ( the_elem_o @ ( insert_o @ X @ bot_bot_set_o ) )
= X ) ).
% the_elem_eq
thf(fact_304_Collect__empty__eq__bot,axiom,
! [P2: nat > $o] :
( ( ( collect_nat @ P2 )
= bot_bot_set_nat )
= ( P2 = bot_bot_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_305_Collect__empty__eq__bot,axiom,
! [P2: $o > $o] :
( ( ( collect_o @ P2 )
= bot_bot_set_o )
= ( P2 = bot_bot_o_o ) ) ).
% Collect_empty_eq_bot
thf(fact_306_is__singletonI,axiom,
! [X: nat] : ( is_singleton_nat @ ( insert_nat @ X @ bot_bot_set_nat ) ) ).
% is_singletonI
thf(fact_307_is__singletonI,axiom,
! [X: $o] : ( is_singleton_o @ ( insert_o @ X @ bot_bot_set_o ) ) ).
% is_singletonI
thf(fact_308_eq__snd__iff,axiom,
! [B: nat,P: product_prod_nat_nat] :
( ( B
= ( product_snd_nat_nat @ P ) )
= ( ? [A5: nat] :
( P
= ( product_Pair_nat_nat @ A5 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_309_is__singleton__the__elem,axiom,
( is_singleton_nat
= ( ^ [A6: set_nat] :
( A6
= ( insert_nat @ ( the_elem_nat @ A6 ) @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_310_is__singleton__the__elem,axiom,
( is_singleton_o
= ( ^ [A6: set_o] :
( A6
= ( insert_o @ ( the_elem_o @ A6 ) @ bot_bot_set_o ) ) ) ) ).
% is_singleton_the_elem
thf(fact_311_is__singletonI_H,axiom,
! [A4: set_Pr1261947904930325089at_nat] :
( ( A4 != bot_bo2099793752762293965at_nat )
=> ( ! [X3: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A4 )
=> ( ( member8440522571783428010at_nat @ Y @ A4 )
=> ( X3 = Y ) ) )
=> ( is_sin2850979758926227957at_nat @ A4 ) ) ) ).
% is_singletonI'
thf(fact_312_is__singletonI_H,axiom,
! [A4: set_nat] :
( ( A4 != bot_bot_set_nat )
=> ( ! [X3: nat,Y: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ( member_nat @ Y @ A4 )
=> ( X3 = Y ) ) )
=> ( is_singleton_nat @ A4 ) ) ) ).
% is_singletonI'
thf(fact_313_is__singletonI_H,axiom,
! [A4: set_o] :
( ( A4 != bot_bot_set_o )
=> ( ! [X3: $o,Y: $o] :
( ( member_o @ X3 @ A4 )
=> ( ( member_o @ Y @ A4 )
=> ( X3 = Y ) ) )
=> ( is_singleton_o @ A4 ) ) ) ).
% is_singletonI'
thf(fact_314_is__singleton__def,axiom,
( is_singleton_nat
= ( ^ [A6: set_nat] :
? [X4: nat] :
( A6
= ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_def
thf(fact_315_is__singleton__def,axiom,
( is_singleton_o
= ( ^ [A6: set_o] :
? [X4: $o] :
( A6
= ( insert_o @ X4 @ bot_bot_set_o ) ) ) ) ).
% is_singleton_def
thf(fact_316_is__singletonE,axiom,
! [A4: set_nat] :
( ( is_singleton_nat @ A4 )
=> ~ ! [X3: nat] :
( A4
!= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% is_singletonE
thf(fact_317_is__singletonE,axiom,
! [A4: set_o] :
( ( is_singleton_o @ A4 )
=> ~ ! [X3: $o] :
( A4
!= ( insert_o @ X3 @ bot_bot_set_o ) ) ) ).
% is_singletonE
thf(fact_318_the__elem__image__unique,axiom,
! [A4: set_nat,F2: nat > set_nat,X: nat] :
( ( A4 != bot_bot_set_nat )
=> ( ! [Y: nat] :
( ( member_nat @ Y @ A4 )
=> ( ( F2 @ Y )
= ( F2 @ X ) ) )
=> ( ( the_elem_set_nat @ ( image_nat_set_nat @ F2 @ A4 ) )
= ( F2 @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_319_eq__fst__iff,axiom,
! [A: nat,P: product_prod_nat_nat] :
( ( A
= ( product_fst_nat_nat @ P ) )
= ( ? [B4: nat] :
( P
= ( product_Pair_nat_nat @ A @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_320_sndI,axiom,
! [X: product_prod_nat_nat,Y3: nat,Z: nat] :
( ( X
= ( product_Pair_nat_nat @ Y3 @ Z ) )
=> ( ( product_snd_nat_nat @ X )
= Z ) ) ).
% sndI
thf(fact_321_fstI,axiom,
! [X: product_prod_nat_nat,Y3: nat,Z: nat] :
( ( X
= ( product_Pair_nat_nat @ Y3 @ Z ) )
=> ( ( product_fst_nat_nat @ X )
= Y3 ) ) ).
% fstI
thf(fact_322_elim__singleton,axiom,
! [X: product_prod_nat_nat,S: product_prod_nat_nat,X7: product_prod_nat_nat,T: product_prod_nat_nat] :
( ( ( member8440522571783428010at_nat @ X @ ( insert8211810215607154385at_nat @ S @ bot_bo2099793752762293965at_nat ) )
& ( member8440522571783428010at_nat @ X7 @ ( insert8211810215607154385at_nat @ T @ bot_bo2099793752762293965at_nat ) ) )
=> ( ( X = S )
& ( X7 = T ) ) ) ).
% elim_singleton
thf(fact_323_elim__singleton,axiom,
! [X: product_prod_nat_nat,S: product_prod_nat_nat,X7: nat,T: nat] :
( ( ( member8440522571783428010at_nat @ X @ ( insert8211810215607154385at_nat @ S @ bot_bo2099793752762293965at_nat ) )
& ( member_nat @ X7 @ ( insert_nat @ T @ bot_bot_set_nat ) ) )
=> ( ( X = S )
& ( X7 = T ) ) ) ).
% elim_singleton
thf(fact_324_elim__singleton,axiom,
! [X: product_prod_nat_nat,S: product_prod_nat_nat,X7: $o,T: $o] :
( ( ( member8440522571783428010at_nat @ X @ ( insert8211810215607154385at_nat @ S @ bot_bo2099793752762293965at_nat ) )
& ( member_o @ X7 @ ( insert_o @ T @ bot_bot_set_o ) ) )
=> ( ( X = S )
& ( X7 = T ) ) ) ).
% elim_singleton
thf(fact_325_elim__singleton,axiom,
! [X: nat,S: nat,X7: product_prod_nat_nat,T: product_prod_nat_nat] :
( ( ( member_nat @ X @ ( insert_nat @ S @ bot_bot_set_nat ) )
& ( member8440522571783428010at_nat @ X7 @ ( insert8211810215607154385at_nat @ T @ bot_bo2099793752762293965at_nat ) ) )
=> ( ( X = S )
& ( X7 = T ) ) ) ).
% elim_singleton
thf(fact_326_elim__singleton,axiom,
! [X: nat,S: nat,X7: nat,T: nat] :
( ( ( member_nat @ X @ ( insert_nat @ S @ bot_bot_set_nat ) )
& ( member_nat @ X7 @ ( insert_nat @ T @ bot_bot_set_nat ) ) )
=> ( ( X = S )
& ( X7 = T ) ) ) ).
% elim_singleton
thf(fact_327_elim__singleton,axiom,
! [X: nat,S: nat,X7: $o,T: $o] :
( ( ( member_nat @ X @ ( insert_nat @ S @ bot_bot_set_nat ) )
& ( member_o @ X7 @ ( insert_o @ T @ bot_bot_set_o ) ) )
=> ( ( X = S )
& ( X7 = T ) ) ) ).
% elim_singleton
thf(fact_328_elim__singleton,axiom,
! [X: $o,S: $o,X7: product_prod_nat_nat,T: product_prod_nat_nat] :
( ( ( member_o @ X @ ( insert_o @ S @ bot_bot_set_o ) )
& ( member8440522571783428010at_nat @ X7 @ ( insert8211810215607154385at_nat @ T @ bot_bo2099793752762293965at_nat ) ) )
=> ( ( X = S )
& ( X7 = T ) ) ) ).
% elim_singleton
thf(fact_329_elim__singleton,axiom,
! [X: $o,S: $o,X7: nat,T: nat] :
( ( ( member_o @ X @ ( insert_o @ S @ bot_bot_set_o ) )
& ( member_nat @ X7 @ ( insert_nat @ T @ bot_bot_set_nat ) ) )
=> ( ( X = S )
& ( X7 = T ) ) ) ).
% elim_singleton
thf(fact_330_elim__singleton,axiom,
! [X: $o,S: $o,X7: $o,T: $o] :
( ( ( member_o @ X @ ( insert_o @ S @ bot_bot_set_o ) )
& ( member_o @ X7 @ ( insert_o @ T @ bot_bot_set_o ) ) )
=> ( ( X = S )
& ( X7 = T ) ) ) ).
% elim_singleton
thf(fact_331_lex__prod__def,axiom,
( lex_prod_nat_nat
= ( ^ [Ra: set_Pr1261947904930325089at_nat,Rb: set_Pr1261947904930325089at_nat] :
( collec7088162979684241874at_nat
@ ( produc6590410687421337004_nat_o
@ ( produc8739625826339149834_nat_o
@ ^ [A5: nat,B4: nat] :
( produc6081775807080527818_nat_o
@ ^ [A7: nat,B8: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A5 @ A7 ) @ Ra )
| ( ( A5 = A7 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ B8 ) @ Rb ) ) ) ) ) ) ) ) ) ).
% lex_prod_def
thf(fact_332_proj__def,axiom,
( equiv_proj_nat_nat
= ( ^ [R3: set_Pr1261947904930325089at_nat,X4: nat] : ( image_nat_nat @ R3 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ).
% proj_def
thf(fact_333_subset__equiv__class,axiom,
! [A4: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat,B: product_prod_nat_nat,A: product_prod_nat_nat] :
( ( equiv_5407799809235660215at_nat @ A4 @ R )
=> ( ( ord_le3146513528884898305at_nat @ ( image_2484338115069272851at_nat @ R @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) @ ( image_2484338115069272851at_nat @ R @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
=> ( ( member8440522571783428010at_nat @ B @ A4 )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R ) ) ) ) ).
% subset_equiv_class
thf(fact_334_subset__equiv__class,axiom,
! [A4: set_nat,R: set_Pr1261947904930325089at_nat,B: nat,A: nat] :
( ( equiv_equiv_nat @ A4 @ R )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ R @ ( insert_nat @ B @ bot_bot_set_nat ) ) @ ( image_nat_nat @ R @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
=> ( ( member_nat @ B @ A4 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R ) ) ) ) ).
% subset_equiv_class
thf(fact_335_subset__equiv__class,axiom,
! [A4: set_o,R: set_Product_prod_o_o,B: $o,A: $o] :
( ( equiv_equiv_o @ A4 @ R )
=> ( ( ord_less_eq_set_o @ ( image_o_o @ R @ ( insert_o @ B @ bot_bot_set_o ) ) @ ( image_o_o @ R @ ( insert_o @ A @ bot_bot_set_o ) ) )
=> ( ( member_o @ B @ A4 )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R ) ) ) ) ).
% subset_equiv_class
thf(fact_336_equiv__class__subset,axiom,
! [A4: set_nat,R: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
( ( equiv_equiv_nat @ A4 @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ R @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ ( image_nat_nat @ R @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ) ).
% equiv_class_subset
thf(fact_337_equiv__class__subset,axiom,
! [A4: set_o,R: set_Product_prod_o_o,A: $o,B: $o] :
( ( equiv_equiv_o @ A4 @ R )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R )
=> ( ord_less_eq_set_o @ ( image_o_o @ R @ ( insert_o @ A @ bot_bot_set_o ) ) @ ( image_o_o @ R @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ) ).
% equiv_class_subset
thf(fact_338_subsetI,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A4 )
=> ( member8440522571783428010at_nat @ X3 @ B5 ) )
=> ( ord_le3146513528884898305at_nat @ A4 @ B5 ) ) ).
% subsetI
thf(fact_339_subsetI,axiom,
! [A4: set_nat,B5: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( member_nat @ X3 @ B5 ) )
=> ( ord_less_eq_set_nat @ A4 @ B5 ) ) ).
% subsetI
thf(fact_340_subset__empty,axiom,
! [A4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ bot_bot_set_nat )
= ( A4 = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_341_subset__empty,axiom,
! [A4: set_o] :
( ( ord_less_eq_set_o @ A4 @ bot_bot_set_o )
= ( A4 = bot_bot_set_o ) ) ).
% subset_empty
thf(fact_342_empty__subsetI,axiom,
! [A4: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A4 ) ).
% empty_subsetI
thf(fact_343_empty__subsetI,axiom,
! [A4: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A4 ) ).
% empty_subsetI
thf(fact_344_insert__subset,axiom,
! [X: $o,A4: set_o,B5: set_o] :
( ( ord_less_eq_set_o @ ( insert_o @ X @ A4 ) @ B5 )
= ( ( member_o @ X @ B5 )
& ( ord_less_eq_set_o @ A4 @ B5 ) ) ) ).
% insert_subset
thf(fact_345_insert__subset,axiom,
! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ X @ A4 ) @ B5 )
= ( ( member8440522571783428010at_nat @ X @ B5 )
& ( ord_le3146513528884898305at_nat @ A4 @ B5 ) ) ) ).
% insert_subset
thf(fact_346_insert__subset,axiom,
! [X: nat,A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A4 ) @ B5 )
= ( ( member_nat @ X @ B5 )
& ( ord_less_eq_set_nat @ A4 @ B5 ) ) ) ).
% insert_subset
thf(fact_347_singleton__insert__inj__eq,axiom,
! [B: nat,A: nat,A4: set_nat] :
( ( ( insert_nat @ B @ bot_bot_set_nat )
= ( insert_nat @ A @ A4 ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_348_singleton__insert__inj__eq,axiom,
! [B: $o,A: $o,A4: set_o] :
( ( ( insert_o @ B @ bot_bot_set_o )
= ( insert_o @ A @ A4 ) )
= ( ( A = B )
& ( ord_less_eq_set_o @ A4 @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_349_singleton__insert__inj__eq_H,axiom,
! [A: nat,A4: set_nat,B: nat] :
( ( ( insert_nat @ A @ A4 )
= ( insert_nat @ B @ bot_bot_set_nat ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_350_singleton__insert__inj__eq_H,axiom,
! [A: $o,A4: set_o,B: $o] :
( ( ( insert_o @ A @ A4 )
= ( insert_o @ B @ bot_bot_set_o ) )
= ( ( A = B )
& ( ord_less_eq_set_o @ A4 @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_351_lessThan__subset__iff,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X ) @ ( set_ord_lessThan_nat @ Y3 ) )
= ( ord_less_eq_nat @ X @ Y3 ) ) ).
% lessThan_subset_iff
thf(fact_352_in__lex__prod,axiom,
! [A: nat,B: nat,A2: nat,B2: nat,R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( product_Pair_nat_nat @ A2 @ B2 ) ) @ ( lex_prod_nat_nat @ R @ S ) )
= ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A2 ) @ R )
| ( ( A = A2 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ B2 ) @ S ) ) ) ) ).
% in_lex_prod
thf(fact_353_Times__subset__cancel2,axiom,
! [X: nat,C2: set_nat,A4: set_nat,B5: set_nat] :
( ( member_nat @ X @ C2 )
=> ( ( ord_le3146513528884898305at_nat
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : C2 )
@ ( produc457027306803732586at_nat @ B5
@ ^ [Uu: nat] : C2 ) )
= ( ord_less_eq_set_nat @ A4 @ B5 ) ) ) ).
% Times_subset_cancel2
thf(fact_354_Collect__subset,axiom,
! [A4: set_Pr1261947904930325089at_nat,P2: product_prod_nat_nat > $o] :
( ord_le3146513528884898305at_nat
@ ( collec3392354462482085612at_nat
@ ^ [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ A4 )
& ( P2 @ X4 ) ) )
@ A4 ) ).
% Collect_subset
thf(fact_355_Collect__subset,axiom,
! [A4: set_nat,P2: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ( P2 @ X4 ) ) )
@ A4 ) ).
% Collect_subset
thf(fact_356_Image__mono,axiom,
! [R4: set_Pr1261947904930325089at_nat,R: set_Pr1261947904930325089at_nat,A8: set_nat,A4: set_nat] :
( ( ord_le3146513528884898305at_nat @ R4 @ R )
=> ( ( ord_less_eq_set_nat @ A8 @ A4 )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ R4 @ A8 ) @ ( image_nat_nat @ R @ A4 ) ) ) ) ).
% Image_mono
thf(fact_357_subset__iff,axiom,
( ord_le3146513528884898305at_nat
= ( ^ [A6: set_Pr1261947904930325089at_nat,B7: set_Pr1261947904930325089at_nat] :
! [T2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ T2 @ A6 )
=> ( member8440522571783428010at_nat @ T2 @ B7 ) ) ) ) ).
% subset_iff
thf(fact_358_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B7: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A6 )
=> ( member_nat @ T2 @ B7 ) ) ) ) ).
% subset_iff
thf(fact_359_subset__eq,axiom,
( ord_le3146513528884898305at_nat
= ( ^ [A6: set_Pr1261947904930325089at_nat,B7: set_Pr1261947904930325089at_nat] :
! [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ A6 )
=> ( member8440522571783428010at_nat @ X4 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_360_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B7: set_nat] :
! [X4: nat] :
( ( member_nat @ X4 @ A6 )
=> ( member_nat @ X4 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_361_subsetD,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,C: product_prod_nat_nat] :
( ( ord_le3146513528884898305at_nat @ A4 @ B5 )
=> ( ( member8440522571783428010at_nat @ C @ A4 )
=> ( member8440522571783428010at_nat @ C @ B5 ) ) ) ).
% subsetD
thf(fact_362_subsetD,axiom,
! [A4: set_nat,B5: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( member_nat @ C @ A4 )
=> ( member_nat @ C @ B5 ) ) ) ).
% subsetD
thf(fact_363_in__mono,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
( ( ord_le3146513528884898305at_nat @ A4 @ B5 )
=> ( ( member8440522571783428010at_nat @ X @ A4 )
=> ( member8440522571783428010at_nat @ X @ B5 ) ) ) ).
% in_mono
thf(fact_364_in__mono,axiom,
! [A4: set_nat,B5: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( member_nat @ X @ A4 )
=> ( member_nat @ X @ B5 ) ) ) ).
% in_mono
thf(fact_365_Sigma__mono,axiom,
! [A4: set_nat,C2: set_nat,B5: nat > set_nat,D: nat > set_nat] :
( ( ord_less_eq_set_nat @ A4 @ C2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ord_less_eq_set_nat @ ( B5 @ X3 ) @ ( D @ X3 ) ) )
=> ( ord_le3146513528884898305at_nat @ ( produc457027306803732586at_nat @ A4 @ B5 ) @ ( produc457027306803732586at_nat @ C2 @ D ) ) ) ) ).
% Sigma_mono
thf(fact_366_prop__restrict,axiom,
! [X: product_prod_nat_nat,Z3: set_Pr1261947904930325089at_nat,X5: set_Pr1261947904930325089at_nat,P2: product_prod_nat_nat > $o] :
( ( member8440522571783428010at_nat @ X @ Z3 )
=> ( ( ord_le3146513528884898305at_nat @ Z3
@ ( collec3392354462482085612at_nat
@ ^ [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ X5 )
& ( P2 @ X4 ) ) ) )
=> ( P2 @ X ) ) ) ).
% prop_restrict
thf(fact_367_prop__restrict,axiom,
! [X: nat,Z3: set_nat,X5: set_nat,P2: nat > $o] :
( ( member_nat @ X @ Z3 )
=> ( ( ord_less_eq_set_nat @ Z3
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ X5 )
& ( P2 @ X4 ) ) ) )
=> ( P2 @ X ) ) ) ).
% prop_restrict
thf(fact_368_Collect__restrict,axiom,
! [X5: set_Pr1261947904930325089at_nat,P2: product_prod_nat_nat > $o] :
( ord_le3146513528884898305at_nat
@ ( collec3392354462482085612at_nat
@ ^ [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ X5 )
& ( P2 @ X4 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_369_Collect__restrict,axiom,
! [X5: set_nat,P2: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ X5 )
& ( P2 @ X4 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_370_subset__Collect__iff,axiom,
! [B5: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,P2: product_prod_nat_nat > $o] :
( ( ord_le3146513528884898305at_nat @ B5 @ A4 )
=> ( ( ord_le3146513528884898305at_nat @ B5
@ ( collec3392354462482085612at_nat
@ ^ [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ A4 )
& ( P2 @ X4 ) ) ) )
= ( ! [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ B5 )
=> ( P2 @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_371_subset__Collect__iff,axiom,
! [B5: set_nat,A4: set_nat,P2: nat > $o] :
( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( ( ord_less_eq_set_nat @ B5
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ( P2 @ X4 ) ) ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ B5 )
=> ( P2 @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_372_subset__CollectI,axiom,
! [B5: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,Q: product_prod_nat_nat > $o,P2: product_prod_nat_nat > $o] :
( ( ord_le3146513528884898305at_nat @ B5 @ A4 )
=> ( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ B5 )
=> ( ( Q @ X3 )
=> ( P2 @ X3 ) ) )
=> ( ord_le3146513528884898305at_nat
@ ( collec3392354462482085612at_nat
@ ^ [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ B5 )
& ( Q @ X4 ) ) )
@ ( collec3392354462482085612at_nat
@ ^ [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ A4 )
& ( P2 @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_373_subset__CollectI,axiom,
! [B5: set_nat,A4: set_nat,Q: nat > $o,P2: nat > $o] :
( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B5 )
=> ( ( Q @ X3 )
=> ( P2 @ X3 ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ B5 )
& ( Q @ X4 ) ) )
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ( P2 @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_374_bot_Oextremum,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% bot.extremum
thf(fact_375_bot_Oextremum,axiom,
! [A: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A ) ).
% bot.extremum
thf(fact_376_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_377_bot_Oextremum__unique,axiom,
! [A: set_o] :
( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
= ( A = bot_bot_set_o ) ) ).
% bot.extremum_unique
thf(fact_378_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_379_bot_Oextremum__uniqueI,axiom,
! [A: set_o] :
( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
=> ( A = bot_bot_set_o ) ) ).
% bot.extremum_uniqueI
thf(fact_380_subrelI,axiom,
! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ! [X3: nat,Y: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ R )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ S ) )
=> ( ord_le3146513528884898305at_nat @ R @ S ) ) ).
% subrelI
thf(fact_381_image__mono,axiom,
! [A4: set_nat,B5: set_nat,F2: nat > set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F2 @ A4 ) @ ( image_nat_set_nat @ F2 @ B5 ) ) ) ).
% image_mono
thf(fact_382_image__subsetI,axiom,
! [A4: set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A4 )
=> ( member8440522571783428010at_nat @ ( F2 @ X3 ) @ B5 ) )
=> ( ord_le3146513528884898305at_nat @ ( image_5168914502847457605at_nat @ F2 @ A4 ) @ B5 ) ) ).
% image_subsetI
thf(fact_383_image__subsetI,axiom,
! [A4: set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > nat,B5: set_nat] :
( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A4 )
=> ( member_nat @ ( F2 @ X3 ) @ B5 ) )
=> ( ord_less_eq_set_nat @ ( image_2486076414777270412at_nat @ F2 @ A4 ) @ B5 ) ) ).
% image_subsetI
thf(fact_384_image__subsetI,axiom,
! [A4: set_nat,F2: nat > set_nat,B5: set_set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( member_set_nat @ ( F2 @ X3 ) @ B5 ) )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F2 @ A4 ) @ B5 ) ) ).
% image_subsetI
thf(fact_385_image__subsetI,axiom,
! [A4: set_nat,F2: nat > product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( member8440522571783428010at_nat @ ( F2 @ X3 ) @ B5 ) )
=> ( ord_le3146513528884898305at_nat @ ( image_5846123807819985514at_nat @ F2 @ A4 ) @ B5 ) ) ).
% image_subsetI
thf(fact_386_image__subsetI,axiom,
! [A4: set_nat,F2: nat > nat,B5: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( member_nat @ ( F2 @ X3 ) @ B5 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F2 @ A4 ) @ B5 ) ) ).
% image_subsetI
thf(fact_387_subset__imageE,axiom,
! [B5: set_set_nat,F2: nat > set_nat,A4: set_nat] :
( ( ord_le6893508408891458716et_nat @ B5 @ ( image_nat_set_nat @ F2 @ A4 ) )
=> ~ ! [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A4 )
=> ( B5
!= ( image_nat_set_nat @ F2 @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_388_image__subset__iff,axiom,
! [F2: nat > set_nat,A4: set_nat,B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F2 @ A4 ) @ B5 )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( member_set_nat @ ( F2 @ X4 ) @ B5 ) ) ) ) ).
% image_subset_iff
thf(fact_389_subset__image__iff,axiom,
! [B5: set_set_nat,F2: nat > set_nat,A4: set_nat] :
( ( ord_le6893508408891458716et_nat @ B5 @ ( image_nat_set_nat @ F2 @ A4 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A4 )
& ( B5
= ( image_nat_set_nat @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_390_subset__emptyI,axiom,
! [A4: set_Pr1261947904930325089at_nat] :
( ! [X3: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ X3 @ A4 )
=> ( ord_le3146513528884898305at_nat @ A4 @ bot_bo2099793752762293965at_nat ) ) ).
% subset_emptyI
thf(fact_391_subset__emptyI,axiom,
! [A4: set_nat] :
( ! [X3: nat] :
~ ( member_nat @ X3 @ A4 )
=> ( ord_less_eq_set_nat @ A4 @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_392_subset__emptyI,axiom,
! [A4: set_o] :
( ! [X3: $o] :
~ ( member_o @ X3 @ A4 )
=> ( ord_less_eq_set_o @ A4 @ bot_bot_set_o ) ) ).
% subset_emptyI
thf(fact_393_insert__subsetI,axiom,
! [X: $o,A4: set_o,X5: set_o] :
( ( member_o @ X @ A4 )
=> ( ( ord_less_eq_set_o @ X5 @ A4 )
=> ( ord_less_eq_set_o @ ( insert_o @ X @ X5 ) @ A4 ) ) ) ).
% insert_subsetI
thf(fact_394_insert__subsetI,axiom,
! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,X5: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ X @ A4 )
=> ( ( ord_le3146513528884898305at_nat @ X5 @ A4 )
=> ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ X @ X5 ) @ A4 ) ) ) ).
% insert_subsetI
thf(fact_395_insert__subsetI,axiom,
! [X: nat,A4: set_nat,X5: set_nat] :
( ( member_nat @ X @ A4 )
=> ( ( ord_less_eq_set_nat @ X5 @ A4 )
=> ( ord_less_eq_set_nat @ ( insert_nat @ X @ X5 ) @ A4 ) ) ) ).
% insert_subsetI
thf(fact_396_insert__mono,axiom,
! [C2: set_nat,D: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ C2 @ D )
=> ( ord_less_eq_set_nat @ ( insert_nat @ A @ C2 ) @ ( insert_nat @ A @ D ) ) ) ).
% insert_mono
thf(fact_397_insert__mono,axiom,
! [C2: set_o,D: set_o,A: $o] :
( ( ord_less_eq_set_o @ C2 @ D )
=> ( ord_less_eq_set_o @ ( insert_o @ A @ C2 ) @ ( insert_o @ A @ D ) ) ) ).
% insert_mono
thf(fact_398_subset__insert,axiom,
! [X: $o,A4: set_o,B5: set_o] :
( ~ ( member_o @ X @ A4 )
=> ( ( ord_less_eq_set_o @ A4 @ ( insert_o @ X @ B5 ) )
= ( ord_less_eq_set_o @ A4 @ B5 ) ) ) ).
% subset_insert
thf(fact_399_subset__insert,axiom,
! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ~ ( member8440522571783428010at_nat @ X @ A4 )
=> ( ( ord_le3146513528884898305at_nat @ A4 @ ( insert8211810215607154385at_nat @ X @ B5 ) )
= ( ord_le3146513528884898305at_nat @ A4 @ B5 ) ) ) ).
% subset_insert
thf(fact_400_subset__insert,axiom,
! [X: nat,A4: set_nat,B5: set_nat] :
( ~ ( member_nat @ X @ A4 )
=> ( ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ X @ B5 ) )
= ( ord_less_eq_set_nat @ A4 @ B5 ) ) ) ).
% subset_insert
thf(fact_401_subset__insertI,axiom,
! [B5: set_nat,A: nat] : ( ord_less_eq_set_nat @ B5 @ ( insert_nat @ A @ B5 ) ) ).
% subset_insertI
thf(fact_402_subset__insertI,axiom,
! [B5: set_o,A: $o] : ( ord_less_eq_set_o @ B5 @ ( insert_o @ A @ B5 ) ) ).
% subset_insertI
thf(fact_403_subset__insertI2,axiom,
! [A4: set_nat,B5: set_nat,B: nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ B @ B5 ) ) ) ).
% subset_insertI2
thf(fact_404_subset__insertI2,axiom,
! [A4: set_o,B5: set_o,B: $o] :
( ( ord_less_eq_set_o @ A4 @ B5 )
=> ( ord_less_eq_set_o @ A4 @ ( insert_o @ B @ B5 ) ) ) ).
% subset_insertI2
thf(fact_405_times__subset__iff,axiom,
! [A4: set_nat,C2: set_nat,B5: set_nat,D: set_nat] :
( ( ord_le3146513528884898305at_nat
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : C2 )
@ ( produc457027306803732586at_nat @ B5
@ ^ [Uu: nat] : D ) )
= ( ( A4 = bot_bot_set_nat )
| ( C2 = bot_bot_set_nat )
| ( ( ord_less_eq_set_nat @ A4 @ B5 )
& ( ord_less_eq_set_nat @ C2 @ D ) ) ) ) ).
% times_subset_iff
thf(fact_406_times__subset__iff,axiom,
! [A4: set_nat,C2: set_o,B5: set_nat,D: set_o] :
( ( ord_le7519981659162375251_nat_o
@ ( product_Sigma_nat_o @ A4
@ ^ [Uu: nat] : C2 )
@ ( product_Sigma_nat_o @ B5
@ ^ [Uu: nat] : D ) )
= ( ( A4 = bot_bot_set_nat )
| ( C2 = bot_bot_set_o )
| ( ( ord_less_eq_set_nat @ A4 @ B5 )
& ( ord_less_eq_set_o @ C2 @ D ) ) ) ) ).
% times_subset_iff
thf(fact_407_times__subset__iff,axiom,
! [A4: set_o,C2: set_nat,B5: set_o,D: set_nat] :
( ( ord_le6472378536984071597_o_nat
@ ( product_Sigma_o_nat @ A4
@ ^ [Uu: $o] : C2 )
@ ( product_Sigma_o_nat @ B5
@ ^ [Uu: $o] : D ) )
= ( ( A4 = bot_bot_set_o )
| ( C2 = bot_bot_set_nat )
| ( ( ord_less_eq_set_o @ A4 @ B5 )
& ( ord_less_eq_set_nat @ C2 @ D ) ) ) ) ).
% times_subset_iff
thf(fact_408_times__subset__iff,axiom,
! [A4: set_o,C2: set_o,B5: set_o,D: set_o] :
( ( ord_le192195778054999463od_o_o
@ ( product_Sigma_o_o @ A4
@ ^ [Uu: $o] : C2 )
@ ( product_Sigma_o_o @ B5
@ ^ [Uu: $o] : D ) )
= ( ( A4 = bot_bot_set_o )
| ( C2 = bot_bot_set_o )
| ( ( ord_less_eq_set_o @ A4 @ B5 )
& ( ord_less_eq_set_o @ C2 @ D ) ) ) ) ).
% times_subset_iff
thf(fact_409_Image__subset,axiom,
! [R: set_Pr1261947904930325089at_nat,A4: set_nat,B5: set_nat,C2: set_nat] :
( ( ord_le3146513528884898305at_nat @ R
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : B5 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ R @ C2 ) @ B5 ) ) ).
% Image_subset
thf(fact_410_image__Collect__subsetI,axiom,
! [P2: nat > $o,F2: nat > set_nat,B5: set_set_nat] :
( ! [X3: nat] :
( ( P2 @ X3 )
=> ( member_set_nat @ ( F2 @ X3 ) @ B5 ) )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F2 @ ( collect_nat @ P2 ) ) @ B5 ) ) ).
% image_Collect_subsetI
thf(fact_411_subset__singletonD,axiom,
! [A4: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) )
=> ( ( A4 = bot_bot_set_nat )
| ( A4
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_412_subset__singletonD,axiom,
! [A4: set_o,X: $o] :
( ( ord_less_eq_set_o @ A4 @ ( insert_o @ X @ bot_bot_set_o ) )
=> ( ( A4 = bot_bot_set_o )
| ( A4
= ( insert_o @ X @ bot_bot_set_o ) ) ) ) ).
% subset_singletonD
thf(fact_413_subset__singleton__iff,axiom,
! [X5: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ X5 @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( ( X5 = bot_bot_set_nat )
| ( X5
= ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_414_subset__singleton__iff,axiom,
! [X5: set_o,A: $o] :
( ( ord_less_eq_set_o @ X5 @ ( insert_o @ A @ bot_bot_set_o ) )
= ( ( X5 = bot_bot_set_o )
| ( X5
= ( insert_o @ A @ bot_bot_set_o ) ) ) ) ).
% subset_singleton_iff
thf(fact_415_proj__iff,axiom,
! [A4: set_nat,R: set_Pr1261947904930325089at_nat,X: nat,Y3: nat] :
( ( equiv_equiv_nat @ A4 @ R )
=> ( ( ord_less_eq_set_nat @ ( insert_nat @ X @ ( insert_nat @ Y3 @ bot_bot_set_nat ) ) @ A4 )
=> ( ( ( equiv_proj_nat_nat @ R @ X )
= ( equiv_proj_nat_nat @ R @ Y3 ) )
= ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R ) ) ) ) ).
% proj_iff
thf(fact_416_proj__iff,axiom,
! [A4: set_o,R: set_Product_prod_o_o,X: $o,Y3: $o] :
( ( equiv_equiv_o @ A4 @ R )
=> ( ( ord_less_eq_set_o @ ( insert_o @ X @ ( insert_o @ Y3 @ bot_bot_set_o ) ) @ A4 )
=> ( ( ( equiv_proj_o_o @ R @ X )
= ( equiv_proj_o_o @ R @ Y3 ) )
= ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y3 ) @ R ) ) ) ) ).
% proj_iff
thf(fact_417_equiv__type,axiom,
! [A4: set_nat,R: set_Pr1261947904930325089at_nat] :
( ( equiv_equiv_nat @ A4 @ R )
=> ( ord_le3146513528884898305at_nat @ R
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : A4 ) ) ) ).
% equiv_type
thf(fact_418_refines__equiv__class__eq,axiom,
! [R2: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat,A4: set_nat,A: nat] :
( ( ord_le3146513528884898305at_nat @ R2 @ S3 )
=> ( ( equiv_equiv_nat @ A4 @ R2 )
=> ( ( equiv_equiv_nat @ A4 @ S3 )
=> ( ( image_nat_nat @ R2 @ ( image_nat_nat @ S3 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( image_nat_nat @ S3 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) ) ).
% refines_equiv_class_eq
thf(fact_419_refines__equiv__class__eq,axiom,
! [R2: set_Product_prod_o_o,S3: set_Product_prod_o_o,A4: set_o,A: $o] :
( ( ord_le192195778054999463od_o_o @ R2 @ S3 )
=> ( ( equiv_equiv_o @ A4 @ R2 )
=> ( ( equiv_equiv_o @ A4 @ S3 )
=> ( ( image_o_o @ R2 @ ( image_o_o @ S3 @ ( insert_o @ A @ bot_bot_set_o ) ) )
= ( image_o_o @ S3 @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) ) ).
% refines_equiv_class_eq
thf(fact_420_refines__equiv__class__eq2,axiom,
! [R2: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat,A4: set_nat,A: nat] :
( ( ord_le3146513528884898305at_nat @ R2 @ S3 )
=> ( ( equiv_equiv_nat @ A4 @ R2 )
=> ( ( equiv_equiv_nat @ A4 @ S3 )
=> ( ( image_nat_nat @ S3 @ ( image_nat_nat @ R2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( image_nat_nat @ S3 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) ) ).
% refines_equiv_class_eq2
thf(fact_421_refines__equiv__class__eq2,axiom,
! [R2: set_Product_prod_o_o,S3: set_Product_prod_o_o,A4: set_o,A: $o] :
( ( ord_le192195778054999463od_o_o @ R2 @ S3 )
=> ( ( equiv_equiv_o @ A4 @ R2 )
=> ( ( equiv_equiv_o @ A4 @ S3 )
=> ( ( image_o_o @ S3 @ ( image_o_o @ R2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
= ( image_o_o @ S3 @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) ) ).
% refines_equiv_class_eq2
thf(fact_422_subset__fst__snd,axiom,
! [A4: set_Pr1261947904930325089at_nat] :
( ord_le3146513528884898305at_nat @ A4
@ ( produc457027306803732586at_nat @ ( image_2486076414777270412at_nat @ product_fst_nat_nat @ A4 )
@ ^ [Uu: nat] : ( image_2486076414777270412at_nat @ product_snd_nat_nat @ A4 ) ) ) ).
% subset_fst_snd
thf(fact_423_all__subset__image,axiom,
! [F2: nat > set_nat,A4: set_nat,P2: set_set_nat > $o] :
( ( ! [B7: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B7 @ ( image_nat_set_nat @ F2 @ A4 ) )
=> ( P2 @ B7 ) ) )
= ( ! [B7: set_nat] :
( ( ord_less_eq_set_nat @ B7 @ A4 )
=> ( P2 @ ( image_nat_set_nat @ F2 @ B7 ) ) ) ) ) ).
% all_subset_image
thf(fact_424_Gr__incl,axiom,
! [A4: set_nat,F2: nat > set_nat,B5: set_set_nat] :
( ( ord_le3845944159117341623et_nat @ ( bNF_Gr_nat_set_nat @ A4 @ F2 )
@ ( produc8883945523214541856et_nat @ A4
@ ^ [Uu: nat] : B5 ) )
= ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F2 @ A4 ) @ B5 ) ) ).
% Gr_incl
thf(fact_425_Gr__incl,axiom,
! [A4: set_nat,F2: nat > nat,B5: set_nat] :
( ( ord_le3146513528884898305at_nat @ ( bNF_Gr_nat_nat @ A4 @ F2 )
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : B5 ) )
= ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F2 @ A4 ) @ B5 ) ) ).
% Gr_incl
thf(fact_426_subset__singleton__iff__Uniq,axiom,
! [A4: set_Pr1261947904930325089at_nat] :
( ( ? [A5: product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ A4 @ ( insert8211810215607154385at_nat @ A5 @ bot_bo2099793752762293965at_nat ) ) )
= ( uniq_P1720416203664110854at_nat
@ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ A4 ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_427_subset__singleton__iff__Uniq,axiom,
! [A4: set_nat] :
( ( ? [A5: nat] : ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ A5 @ bot_bot_set_nat ) ) )
= ( uniq_nat
@ ^ [X4: nat] : ( member_nat @ X4 @ A4 ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_428_subset__singleton__iff__Uniq,axiom,
! [A4: set_o] :
( ( ? [A5: $o] : ( ord_less_eq_set_o @ A4 @ ( insert_o @ A5 @ bot_bot_set_o ) ) )
= ( uniq_o
@ ^ [X4: $o] : ( member_o @ X4 @ A4 ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_429_in__quotient__imp__in__rel,axiom,
! [A4: set_nat,R: set_Pr1261947904930325089at_nat,X5: set_nat,X: nat,Y3: nat] :
( ( equiv_equiv_nat @ A4 @ R )
=> ( ( member_set_nat @ X5 @ ( equiv_quotient_nat @ A4 @ R ) )
=> ( ( ord_less_eq_set_nat @ ( insert_nat @ X @ ( insert_nat @ Y3 @ bot_bot_set_nat ) ) @ X5 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R ) ) ) ) ).
% in_quotient_imp_in_rel
thf(fact_430_in__quotient__imp__in__rel,axiom,
! [A4: set_o,R: set_Product_prod_o_o,X5: set_o,X: $o,Y3: $o] :
( ( equiv_equiv_o @ A4 @ R )
=> ( ( member_set_o @ X5 @ ( equiv_quotient_o @ A4 @ R ) )
=> ( ( ord_less_eq_set_o @ ( insert_o @ X @ ( insert_o @ Y3 @ bot_bot_set_o ) ) @ X5 )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y3 ) @ R ) ) ) ) ).
% in_quotient_imp_in_rel
thf(fact_431_equiv__class__nondisjoint,axiom,
! [A4: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat,X: product_prod_nat_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
( ( equiv_5407799809235660215at_nat @ A4 @ R )
=> ( ( member8440522571783428010at_nat @ X @ ( inf_in2572325071724192079at_nat @ ( image_2484338115069272851at_nat @ R @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) @ ( image_2484338115069272851at_nat @ R @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) ) )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R ) ) ) ).
% equiv_class_nondisjoint
thf(fact_432_equiv__class__nondisjoint,axiom,
! [A4: set_nat,R: set_Pr1261947904930325089at_nat,X: nat,A: nat,B: nat] :
( ( equiv_equiv_nat @ A4 @ R )
=> ( ( member_nat @ X @ ( inf_inf_set_nat @ ( image_nat_nat @ R @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ ( image_nat_nat @ R @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R ) ) ) ).
% equiv_class_nondisjoint
thf(fact_433_equiv__class__nondisjoint,axiom,
! [A4: set_o,R: set_Product_prod_o_o,X: $o,A: $o,B: $o] :
( ( equiv_equiv_o @ A4 @ R )
=> ( ( member_o @ X @ ( inf_inf_set_o @ ( image_o_o @ R @ ( insert_o @ A @ bot_bot_set_o ) ) @ ( image_o_o @ R @ ( insert_o @ B @ bot_bot_set_o ) ) ) )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R ) ) ) ).
% equiv_class_nondisjoint
thf(fact_434_member__product,axiom,
! [X: product_prod_nat_nat,A4: set_nat,B5: set_nat] :
( ( member8440522571783428010at_nat @ X @ ( produc929334515565554804at_nat @ A4 @ B5 ) )
= ( member8440522571783428010at_nat @ X
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : B5 ) ) ) ).
% member_product
thf(fact_435_Product__Type_Oproduct__def,axiom,
( produc929334515565554804at_nat
= ( ^ [A6: set_nat,B7: set_nat] :
( produc457027306803732586at_nat @ A6
@ ^ [Uu: nat] : B7 ) ) ) ).
% Product_Type.product_def
thf(fact_436_IntI,axiom,
! [C: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ C @ A4 )
=> ( ( member8440522571783428010at_nat @ C @ B5 )
=> ( member8440522571783428010at_nat @ C @ ( inf_in2572325071724192079at_nat @ A4 @ B5 ) ) ) ) ).
% IntI
thf(fact_437_IntI,axiom,
! [C: nat,A4: set_nat,B5: set_nat] :
( ( member_nat @ C @ A4 )
=> ( ( member_nat @ C @ B5 )
=> ( member_nat @ C @ ( inf_inf_set_nat @ A4 @ B5 ) ) ) ) ).
% IntI
thf(fact_438_Int__iff,axiom,
! [C: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ C @ ( inf_in2572325071724192079at_nat @ A4 @ B5 ) )
= ( ( member8440522571783428010at_nat @ C @ A4 )
& ( member8440522571783428010at_nat @ C @ B5 ) ) ) ).
% Int_iff
thf(fact_439_Int__iff,axiom,
! [C: nat,A4: set_nat,B5: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A4 @ B5 ) )
= ( ( member_nat @ C @ A4 )
& ( member_nat @ C @ B5 ) ) ) ).
% Int_iff
thf(fact_440_Int__insert__right__if1,axiom,
! [A: $o,A4: set_o,B5: set_o] :
( ( member_o @ A @ A4 )
=> ( ( inf_inf_set_o @ A4 @ ( insert_o @ A @ B5 ) )
= ( insert_o @ A @ ( inf_inf_set_o @ A4 @ B5 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_441_Int__insert__right__if1,axiom,
! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ A @ A4 )
=> ( ( inf_in2572325071724192079at_nat @ A4 @ ( insert8211810215607154385at_nat @ A @ B5 ) )
= ( insert8211810215607154385at_nat @ A @ ( inf_in2572325071724192079at_nat @ A4 @ B5 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_442_Int__insert__right__if1,axiom,
! [A: nat,A4: set_nat,B5: set_nat] :
( ( member_nat @ A @ A4 )
=> ( ( inf_inf_set_nat @ A4 @ ( insert_nat @ A @ B5 ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A4 @ B5 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_443_Int__insert__right__if0,axiom,
! [A: $o,A4: set_o,B5: set_o] :
( ~ ( member_o @ A @ A4 )
=> ( ( inf_inf_set_o @ A4 @ ( insert_o @ A @ B5 ) )
= ( inf_inf_set_o @ A4 @ B5 ) ) ) ).
% Int_insert_right_if0
thf(fact_444_Int__insert__right__if0,axiom,
! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ~ ( member8440522571783428010at_nat @ A @ A4 )
=> ( ( inf_in2572325071724192079at_nat @ A4 @ ( insert8211810215607154385at_nat @ A @ B5 ) )
= ( inf_in2572325071724192079at_nat @ A4 @ B5 ) ) ) ).
% Int_insert_right_if0
thf(fact_445_Int__insert__right__if0,axiom,
! [A: nat,A4: set_nat,B5: set_nat] :
( ~ ( member_nat @ A @ A4 )
=> ( ( inf_inf_set_nat @ A4 @ ( insert_nat @ A @ B5 ) )
= ( inf_inf_set_nat @ A4 @ B5 ) ) ) ).
% Int_insert_right_if0
thf(fact_446_insert__inter__insert,axiom,
! [A: nat,A4: set_nat,B5: set_nat] :
( ( inf_inf_set_nat @ ( insert_nat @ A @ A4 ) @ ( insert_nat @ A @ B5 ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A4 @ B5 ) ) ) ).
% insert_inter_insert
thf(fact_447_insert__inter__insert,axiom,
! [A: $o,A4: set_o,B5: set_o] :
( ( inf_inf_set_o @ ( insert_o @ A @ A4 ) @ ( insert_o @ A @ B5 ) )
= ( insert_o @ A @ ( inf_inf_set_o @ A4 @ B5 ) ) ) ).
% insert_inter_insert
thf(fact_448_Int__insert__left__if1,axiom,
! [A: $o,C2: set_o,B5: set_o] :
( ( member_o @ A @ C2 )
=> ( ( inf_inf_set_o @ ( insert_o @ A @ B5 ) @ C2 )
= ( insert_o @ A @ ( inf_inf_set_o @ B5 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_449_Int__insert__left__if1,axiom,
! [A: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ A @ C2 )
=> ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ B5 ) @ C2 )
= ( insert8211810215607154385at_nat @ A @ ( inf_in2572325071724192079at_nat @ B5 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_450_Int__insert__left__if1,axiom,
! [A: nat,C2: set_nat,B5: set_nat] :
( ( member_nat @ A @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B5 ) @ C2 )
= ( insert_nat @ A @ ( inf_inf_set_nat @ B5 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_451_Int__insert__left__if0,axiom,
! [A: $o,C2: set_o,B5: set_o] :
( ~ ( member_o @ A @ C2 )
=> ( ( inf_inf_set_o @ ( insert_o @ A @ B5 ) @ C2 )
= ( inf_inf_set_o @ B5 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_452_Int__insert__left__if0,axiom,
! [A: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ~ ( member8440522571783428010at_nat @ A @ C2 )
=> ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ B5 ) @ C2 )
= ( inf_in2572325071724192079at_nat @ B5 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_453_Int__insert__left__if0,axiom,
! [A: nat,C2: set_nat,B5: set_nat] :
( ~ ( member_nat @ A @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B5 ) @ C2 )
= ( inf_inf_set_nat @ B5 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_454_quotient__is__empty2,axiom,
! [A4: set_nat,R: set_Pr1261947904930325089at_nat] :
( ( bot_bot_set_set_nat
= ( equiv_quotient_nat @ A4 @ R ) )
= ( A4 = bot_bot_set_nat ) ) ).
% quotient_is_empty2
thf(fact_455_quotient__is__empty2,axiom,
! [A4: set_o,R: set_Product_prod_o_o] :
( ( bot_bot_set_set_o
= ( equiv_quotient_o @ A4 @ R ) )
= ( A4 = bot_bot_set_o ) ) ).
% quotient_is_empty2
thf(fact_456_quotient__is__empty,axiom,
! [A4: set_nat,R: set_Pr1261947904930325089at_nat] :
( ( ( equiv_quotient_nat @ A4 @ R )
= bot_bot_set_set_nat )
= ( A4 = bot_bot_set_nat ) ) ).
% quotient_is_empty
thf(fact_457_quotient__is__empty,axiom,
! [A4: set_o,R: set_Product_prod_o_o] :
( ( ( equiv_quotient_o @ A4 @ R )
= bot_bot_set_set_o )
= ( A4 = bot_bot_set_o ) ) ).
% quotient_is_empty
thf(fact_458_quotient__empty,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ( equiv_quotient_nat @ bot_bot_set_nat @ R )
= bot_bot_set_set_nat ) ).
% quotient_empty
thf(fact_459_quotient__empty,axiom,
! [R: set_Product_prod_o_o] :
( ( equiv_quotient_o @ bot_bot_set_o @ R )
= bot_bot_set_set_o ) ).
% quotient_empty
thf(fact_460_insert__disjoint_I1_J,axiom,
! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ A4 ) @ B5 )
= bot_bo2099793752762293965at_nat )
= ( ~ ( member8440522571783428010at_nat @ A @ B5 )
& ( ( inf_in2572325071724192079at_nat @ A4 @ B5 )
= bot_bo2099793752762293965at_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_461_insert__disjoint_I1_J,axiom,
! [A: nat,A4: set_nat,B5: set_nat] :
( ( ( inf_inf_set_nat @ ( insert_nat @ A @ A4 ) @ B5 )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B5 )
& ( ( inf_inf_set_nat @ A4 @ B5 )
= bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_462_insert__disjoint_I1_J,axiom,
! [A: $o,A4: set_o,B5: set_o] :
( ( ( inf_inf_set_o @ ( insert_o @ A @ A4 ) @ B5 )
= bot_bot_set_o )
= ( ~ ( member_o @ A @ B5 )
& ( ( inf_inf_set_o @ A4 @ B5 )
= bot_bot_set_o ) ) ) ).
% insert_disjoint(1)
thf(fact_463_insert__disjoint_I2_J,axiom,
! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( bot_bo2099793752762293965at_nat
= ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ A4 ) @ B5 ) )
= ( ~ ( member8440522571783428010at_nat @ A @ B5 )
& ( bot_bo2099793752762293965at_nat
= ( inf_in2572325071724192079at_nat @ A4 @ B5 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_464_insert__disjoint_I2_J,axiom,
! [A: nat,A4: set_nat,B5: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ ( insert_nat @ A @ A4 ) @ B5 ) )
= ( ~ ( member_nat @ A @ B5 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A4 @ B5 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_465_insert__disjoint_I2_J,axiom,
! [A: $o,A4: set_o,B5: set_o] :
( ( bot_bot_set_o
= ( inf_inf_set_o @ ( insert_o @ A @ A4 ) @ B5 ) )
= ( ~ ( member_o @ A @ B5 )
& ( bot_bot_set_o
= ( inf_inf_set_o @ A4 @ B5 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_466_disjoint__insert_I1_J,axiom,
! [B5: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ( ( inf_in2572325071724192079at_nat @ B5 @ ( insert8211810215607154385at_nat @ A @ A4 ) )
= bot_bo2099793752762293965at_nat )
= ( ~ ( member8440522571783428010at_nat @ A @ B5 )
& ( ( inf_in2572325071724192079at_nat @ B5 @ A4 )
= bot_bo2099793752762293965at_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_467_disjoint__insert_I1_J,axiom,
! [B5: set_nat,A: nat,A4: set_nat] :
( ( ( inf_inf_set_nat @ B5 @ ( insert_nat @ A @ A4 ) )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B5 )
& ( ( inf_inf_set_nat @ B5 @ A4 )
= bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_468_disjoint__insert_I1_J,axiom,
! [B5: set_o,A: $o,A4: set_o] :
( ( ( inf_inf_set_o @ B5 @ ( insert_o @ A @ A4 ) )
= bot_bot_set_o )
= ( ~ ( member_o @ A @ B5 )
& ( ( inf_inf_set_o @ B5 @ A4 )
= bot_bot_set_o ) ) ) ).
% disjoint_insert(1)
thf(fact_469_disjoint__insert_I2_J,axiom,
! [A4: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
( ( bot_bo2099793752762293965at_nat
= ( inf_in2572325071724192079at_nat @ A4 @ ( insert8211810215607154385at_nat @ B @ B5 ) ) )
= ( ~ ( member8440522571783428010at_nat @ B @ A4 )
& ( bot_bo2099793752762293965at_nat
= ( inf_in2572325071724192079at_nat @ A4 @ B5 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_470_disjoint__insert_I2_J,axiom,
! [A4: set_nat,B: nat,B5: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ A4 @ ( insert_nat @ B @ B5 ) ) )
= ( ~ ( member_nat @ B @ A4 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A4 @ B5 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_471_disjoint__insert_I2_J,axiom,
! [A4: set_o,B: $o,B5: set_o] :
( ( bot_bot_set_o
= ( inf_inf_set_o @ A4 @ ( insert_o @ B @ B5 ) ) )
= ( ~ ( member_o @ B @ A4 )
& ( bot_bot_set_o
= ( inf_inf_set_o @ A4 @ B5 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_472_Times__Int__distrib1,axiom,
! [A4: set_nat,B5: set_nat,C2: set_nat] :
( ( produc457027306803732586at_nat @ ( inf_inf_set_nat @ A4 @ B5 )
@ ^ [Uu: nat] : C2 )
= ( inf_in2572325071724192079at_nat
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : C2 )
@ ( produc457027306803732586at_nat @ B5
@ ^ [Uu: nat] : C2 ) ) ) ).
% Times_Int_distrib1
thf(fact_473_Sigma__Int__distrib2,axiom,
! [I2: set_nat,A4: nat > set_nat,B5: nat > set_nat] :
( ( produc457027306803732586at_nat @ I2
@ ^ [I: nat] : ( inf_inf_set_nat @ ( A4 @ I ) @ ( B5 @ I ) ) )
= ( inf_in2572325071724192079at_nat @ ( produc457027306803732586at_nat @ I2 @ A4 ) @ ( produc457027306803732586at_nat @ I2 @ B5 ) ) ) ).
% Sigma_Int_distrib2
thf(fact_474_Times__Int__Times,axiom,
! [A4: set_nat,B5: set_nat,C2: set_nat,D: set_nat] :
( ( inf_in2572325071724192079at_nat
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : B5 )
@ ( produc457027306803732586at_nat @ C2
@ ^ [Uu: nat] : D ) )
= ( produc457027306803732586at_nat @ ( inf_inf_set_nat @ A4 @ C2 )
@ ^ [Uu: nat] : ( inf_inf_set_nat @ B5 @ D ) ) ) ).
% Times_Int_Times
thf(fact_475_Sigma__Int__distrib1,axiom,
! [I2: set_nat,J2: set_nat,C2: nat > set_nat] :
( ( produc457027306803732586at_nat @ ( inf_inf_set_nat @ I2 @ J2 ) @ C2 )
= ( inf_in2572325071724192079at_nat @ ( produc457027306803732586at_nat @ I2 @ C2 ) @ ( produc457027306803732586at_nat @ J2 @ C2 ) ) ) ).
% Sigma_Int_distrib1
thf(fact_476_IntE,axiom,
! [C: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ C @ ( inf_in2572325071724192079at_nat @ A4 @ B5 ) )
=> ~ ( ( member8440522571783428010at_nat @ C @ A4 )
=> ~ ( member8440522571783428010at_nat @ C @ B5 ) ) ) ).
% IntE
thf(fact_477_IntE,axiom,
! [C: nat,A4: set_nat,B5: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A4 @ B5 ) )
=> ~ ( ( member_nat @ C @ A4 )
=> ~ ( member_nat @ C @ B5 ) ) ) ).
% IntE
thf(fact_478_IntD1,axiom,
! [C: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ C @ ( inf_in2572325071724192079at_nat @ A4 @ B5 ) )
=> ( member8440522571783428010at_nat @ C @ A4 ) ) ).
% IntD1
thf(fact_479_IntD1,axiom,
! [C: nat,A4: set_nat,B5: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A4 @ B5 ) )
=> ( member_nat @ C @ A4 ) ) ).
% IntD1
thf(fact_480_IntD2,axiom,
! [C: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ C @ ( inf_in2572325071724192079at_nat @ A4 @ B5 ) )
=> ( member8440522571783428010at_nat @ C @ B5 ) ) ).
% IntD2
thf(fact_481_IntD2,axiom,
! [C: nat,A4: set_nat,B5: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A4 @ B5 ) )
=> ( member_nat @ C @ B5 ) ) ).
% IntD2
thf(fact_482_Int__Collect,axiom,
! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,P2: product_prod_nat_nat > $o] :
( ( member8440522571783428010at_nat @ X @ ( inf_in2572325071724192079at_nat @ A4 @ ( collec3392354462482085612at_nat @ P2 ) ) )
= ( ( member8440522571783428010at_nat @ X @ A4 )
& ( P2 @ X ) ) ) ).
% Int_Collect
thf(fact_483_Int__Collect,axiom,
! [X: nat,A4: set_nat,P2: nat > $o] :
( ( member_nat @ X @ ( inf_inf_set_nat @ A4 @ ( collect_nat @ P2 ) ) )
= ( ( member_nat @ X @ A4 )
& ( P2 @ X ) ) ) ).
% Int_Collect
thf(fact_484_Int__def,axiom,
( inf_in2572325071724192079at_nat
= ( ^ [A6: set_Pr1261947904930325089at_nat,B7: set_Pr1261947904930325089at_nat] :
( collec3392354462482085612at_nat
@ ^ [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ A6 )
& ( member8440522571783428010at_nat @ X4 @ B7 ) ) ) ) ) ).
% Int_def
thf(fact_485_Int__def,axiom,
( inf_inf_set_nat
= ( ^ [A6: set_nat,B7: set_nat] :
( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A6 )
& ( member_nat @ X4 @ B7 ) ) ) ) ) ).
% Int_def
thf(fact_486_quotient__disj,axiom,
! [A4: set_nat,R: set_Pr1261947904930325089at_nat,X5: set_nat,Y7: set_nat] :
( ( equiv_equiv_nat @ A4 @ R )
=> ( ( member_set_nat @ X5 @ ( equiv_quotient_nat @ A4 @ R ) )
=> ( ( member_set_nat @ Y7 @ ( equiv_quotient_nat @ A4 @ R ) )
=> ( ( X5 = Y7 )
| ( ( inf_inf_set_nat @ X5 @ Y7 )
= bot_bot_set_nat ) ) ) ) ) ).
% quotient_disj
thf(fact_487_quotient__disj,axiom,
! [A4: set_o,R: set_Product_prod_o_o,X5: set_o,Y7: set_o] :
( ( equiv_equiv_o @ A4 @ R )
=> ( ( member_set_o @ X5 @ ( equiv_quotient_o @ A4 @ R ) )
=> ( ( member_set_o @ Y7 @ ( equiv_quotient_o @ A4 @ R ) )
=> ( ( X5 = Y7 )
| ( ( inf_inf_set_o @ X5 @ Y7 )
= bot_bot_set_o ) ) ) ) ) ).
% quotient_disj
thf(fact_488_disjoint__iff__not__equal,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ( inf_inf_set_nat @ A4 @ B5 )
= bot_bot_set_nat )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ! [Y5: nat] :
( ( member_nat @ Y5 @ B5 )
=> ( X4 != Y5 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_489_disjoint__iff__not__equal,axiom,
! [A4: set_o,B5: set_o] :
( ( ( inf_inf_set_o @ A4 @ B5 )
= bot_bot_set_o )
= ( ! [X4: $o] :
( ( member_o @ X4 @ A4 )
=> ! [Y5: $o] :
( ( member_o @ Y5 @ B5 )
=> ( X4 = (~ Y5) ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_490_Int__empty__right,axiom,
! [A4: set_nat] :
( ( inf_inf_set_nat @ A4 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% Int_empty_right
thf(fact_491_Int__empty__right,axiom,
! [A4: set_o] :
( ( inf_inf_set_o @ A4 @ bot_bot_set_o )
= bot_bot_set_o ) ).
% Int_empty_right
thf(fact_492_Int__empty__left,axiom,
! [B5: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ B5 )
= bot_bot_set_nat ) ).
% Int_empty_left
thf(fact_493_Int__empty__left,axiom,
! [B5: set_o] :
( ( inf_inf_set_o @ bot_bot_set_o @ B5 )
= bot_bot_set_o ) ).
% Int_empty_left
thf(fact_494_disjoint__iff,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( ( inf_in2572325071724192079at_nat @ A4 @ B5 )
= bot_bo2099793752762293965at_nat )
= ( ! [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ A4 )
=> ~ ( member8440522571783428010at_nat @ X4 @ B5 ) ) ) ) ).
% disjoint_iff
thf(fact_495_disjoint__iff,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ( inf_inf_set_nat @ A4 @ B5 )
= bot_bot_set_nat )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ~ ( member_nat @ X4 @ B5 ) ) ) ) ).
% disjoint_iff
thf(fact_496_disjoint__iff,axiom,
! [A4: set_o,B5: set_o] :
( ( ( inf_inf_set_o @ A4 @ B5 )
= bot_bot_set_o )
= ( ! [X4: $o] :
( ( member_o @ X4 @ A4 )
=> ~ ( member_o @ X4 @ B5 ) ) ) ) ).
% disjoint_iff
thf(fact_497_Int__emptyI,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A4 )
=> ~ ( member8440522571783428010at_nat @ X3 @ B5 ) )
=> ( ( inf_in2572325071724192079at_nat @ A4 @ B5 )
= bot_bo2099793752762293965at_nat ) ) ).
% Int_emptyI
thf(fact_498_Int__emptyI,axiom,
! [A4: set_nat,B5: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ~ ( member_nat @ X3 @ B5 ) )
=> ( ( inf_inf_set_nat @ A4 @ B5 )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_499_Int__emptyI,axiom,
! [A4: set_o,B5: set_o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A4 )
=> ~ ( member_o @ X3 @ B5 ) )
=> ( ( inf_inf_set_o @ A4 @ B5 )
= bot_bot_set_o ) ) ).
% Int_emptyI
thf(fact_500_Int__Collect__mono,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,P2: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
( ( ord_le3146513528884898305at_nat @ A4 @ B5 )
=> ( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A4 )
=> ( ( P2 @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A4 @ ( collec3392354462482085612at_nat @ P2 ) ) @ ( inf_in2572325071724192079at_nat @ B5 @ ( collec3392354462482085612at_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_501_Int__Collect__mono,axiom,
! [A4: set_nat,B5: set_nat,P2: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ( P2 @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A4 @ ( collect_nat @ P2 ) ) @ ( inf_inf_set_nat @ B5 @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_502_Int__insert__right,axiom,
! [A: $o,A4: set_o,B5: set_o] :
( ( ( member_o @ A @ A4 )
=> ( ( inf_inf_set_o @ A4 @ ( insert_o @ A @ B5 ) )
= ( insert_o @ A @ ( inf_inf_set_o @ A4 @ B5 ) ) ) )
& ( ~ ( member_o @ A @ A4 )
=> ( ( inf_inf_set_o @ A4 @ ( insert_o @ A @ B5 ) )
= ( inf_inf_set_o @ A4 @ B5 ) ) ) ) ).
% Int_insert_right
thf(fact_503_Int__insert__right,axiom,
! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( ( member8440522571783428010at_nat @ A @ A4 )
=> ( ( inf_in2572325071724192079at_nat @ A4 @ ( insert8211810215607154385at_nat @ A @ B5 ) )
= ( insert8211810215607154385at_nat @ A @ ( inf_in2572325071724192079at_nat @ A4 @ B5 ) ) ) )
& ( ~ ( member8440522571783428010at_nat @ A @ A4 )
=> ( ( inf_in2572325071724192079at_nat @ A4 @ ( insert8211810215607154385at_nat @ A @ B5 ) )
= ( inf_in2572325071724192079at_nat @ A4 @ B5 ) ) ) ) ).
% Int_insert_right
thf(fact_504_Int__insert__right,axiom,
! [A: nat,A4: set_nat,B5: set_nat] :
( ( ( member_nat @ A @ A4 )
=> ( ( inf_inf_set_nat @ A4 @ ( insert_nat @ A @ B5 ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A4 @ B5 ) ) ) )
& ( ~ ( member_nat @ A @ A4 )
=> ( ( inf_inf_set_nat @ A4 @ ( insert_nat @ A @ B5 ) )
= ( inf_inf_set_nat @ A4 @ B5 ) ) ) ) ).
% Int_insert_right
thf(fact_505_Int__insert__left,axiom,
! [A: $o,C2: set_o,B5: set_o] :
( ( ( member_o @ A @ C2 )
=> ( ( inf_inf_set_o @ ( insert_o @ A @ B5 ) @ C2 )
= ( insert_o @ A @ ( inf_inf_set_o @ B5 @ C2 ) ) ) )
& ( ~ ( member_o @ A @ C2 )
=> ( ( inf_inf_set_o @ ( insert_o @ A @ B5 ) @ C2 )
= ( inf_inf_set_o @ B5 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_506_Int__insert__left,axiom,
! [A: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( ( member8440522571783428010at_nat @ A @ C2 )
=> ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ B5 ) @ C2 )
= ( insert8211810215607154385at_nat @ A @ ( inf_in2572325071724192079at_nat @ B5 @ C2 ) ) ) )
& ( ~ ( member8440522571783428010at_nat @ A @ C2 )
=> ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ B5 ) @ C2 )
= ( inf_in2572325071724192079at_nat @ B5 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_507_Int__insert__left,axiom,
! [A: nat,C2: set_nat,B5: set_nat] :
( ( ( member_nat @ A @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B5 ) @ C2 )
= ( insert_nat @ A @ ( inf_inf_set_nat @ B5 @ C2 ) ) ) )
& ( ~ ( member_nat @ A @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B5 ) @ C2 )
= ( inf_inf_set_nat @ B5 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_508_pred__subset__eq2,axiom,
! [R2: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
( ( ord_le2646555220125990790_nat_o
@ ^ [X4: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ R2 )
@ ^ [X4: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ S3 ) )
= ( ord_le3146513528884898305at_nat @ R2 @ S3 ) ) ).
% pred_subset_eq2
thf(fact_509_pred__subset__eq,axiom,
! [R2: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
( ( ord_le704812498762024988_nat_o
@ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ R2 )
@ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ S3 ) )
= ( ord_le3146513528884898305at_nat @ R2 @ S3 ) ) ).
% pred_subset_eq
thf(fact_510_pred__subset__eq,axiom,
! [R2: set_nat,S3: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X4: nat] : ( member_nat @ X4 @ R2 )
@ ^ [X4: nat] : ( member_nat @ X4 @ S3 ) )
= ( ord_less_eq_set_nat @ R2 @ S3 ) ) ).
% pred_subset_eq
thf(fact_511_less__eq__set__def,axiom,
( ord_le3146513528884898305at_nat
= ( ^ [A6: set_Pr1261947904930325089at_nat,B7: set_Pr1261947904930325089at_nat] :
( ord_le704812498762024988_nat_o
@ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ A6 )
@ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ B7 ) ) ) ) ).
% less_eq_set_def
thf(fact_512_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B7: set_nat] :
( ord_less_eq_nat_o
@ ^ [X4: nat] : ( member_nat @ X4 @ A6 )
@ ^ [X4: nat] : ( member_nat @ X4 @ B7 ) ) ) ) ).
% less_eq_set_def
thf(fact_513_proj__image,axiom,
! [R: set_Pr1261947904930325089at_nat,A4: set_nat] :
( ( image_nat_set_nat @ ( equiv_proj_nat_nat @ R ) @ A4 )
= ( equiv_quotient_nat @ A4 @ R ) ) ).
% proj_image
thf(fact_514_proj__preserves,axiom,
! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat] :
( ( member8440522571783428010at_nat @ X @ A4 )
=> ( member2643936169264416010at_nat @ ( equiv_7265450603036132554at_nat @ R @ X ) @ ( equiv_3811336339175339080at_nat @ A4 @ R ) ) ) ).
% proj_preserves
thf(fact_515_proj__preserves,axiom,
! [X: nat,A4: set_nat,R: set_Pr1261947904930325089at_nat] :
( ( member_nat @ X @ A4 )
=> ( member_set_nat @ ( equiv_proj_nat_nat @ R @ X ) @ ( equiv_quotient_nat @ A4 @ R ) ) ) ).
% proj_preserves
thf(fact_516_GrD1,axiom,
! [X: nat,Fx: nat,A4: set_nat,F2: nat > nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Fx ) @ ( bNF_Gr_nat_nat @ A4 @ F2 ) )
=> ( member_nat @ X @ A4 ) ) ).
% GrD1
thf(fact_517_GrD2,axiom,
! [X: nat,Fx: nat,A4: set_nat,F2: nat > nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Fx ) @ ( bNF_Gr_nat_nat @ A4 @ F2 ) )
=> ( ( F2 @ X )
= Fx ) ) ).
% GrD2
thf(fact_518_image__Int__subset,axiom,
! [F2: nat > set_nat,A4: set_nat,B5: set_nat] : ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F2 @ ( inf_inf_set_nat @ A4 @ B5 ) ) @ ( inf_inf_set_set_nat @ ( image_nat_set_nat @ F2 @ A4 ) @ ( image_nat_set_nat @ F2 @ B5 ) ) ) ).
% image_Int_subset
thf(fact_519_Image__Int__subset,axiom,
! [R2: set_Pr1261947904930325089at_nat,A4: set_nat,B5: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ R2 @ ( inf_inf_set_nat @ A4 @ B5 ) ) @ ( inf_inf_set_nat @ ( image_nat_nat @ R2 @ A4 ) @ ( image_nat_nat @ R2 @ B5 ) ) ) ).
% Image_Int_subset
thf(fact_520_quotient__eqI,axiom,
! [A4: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat,X5: set_Pr1261947904930325089at_nat,Y7: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,Y3: product_prod_nat_nat] :
( ( equiv_5407799809235660215at_nat @ A4 @ R )
=> ( ( member2643936169264416010at_nat @ X5 @ ( equiv_3811336339175339080at_nat @ A4 @ R ) )
=> ( ( member2643936169264416010at_nat @ Y7 @ ( equiv_3811336339175339080at_nat @ A4 @ R ) )
=> ( ( member8440522571783428010at_nat @ X @ X5 )
=> ( ( member8440522571783428010at_nat @ Y3 @ Y7 )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ R )
=> ( X5 = Y7 ) ) ) ) ) ) ) ).
% quotient_eqI
thf(fact_521_quotient__eqI,axiom,
! [A4: set_nat,R: set_Pr1261947904930325089at_nat,X5: set_nat,Y7: set_nat,X: nat,Y3: nat] :
( ( equiv_equiv_nat @ A4 @ R )
=> ( ( member_set_nat @ X5 @ ( equiv_quotient_nat @ A4 @ R ) )
=> ( ( member_set_nat @ Y7 @ ( equiv_quotient_nat @ A4 @ R ) )
=> ( ( member_nat @ X @ X5 )
=> ( ( member_nat @ Y3 @ Y7 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R )
=> ( X5 = Y7 ) ) ) ) ) ) ) ).
% quotient_eqI
thf(fact_522_quotient__eq__iff,axiom,
! [A4: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat,X5: set_Pr1261947904930325089at_nat,Y7: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,Y3: product_prod_nat_nat] :
( ( equiv_5407799809235660215at_nat @ A4 @ R )
=> ( ( member2643936169264416010at_nat @ X5 @ ( equiv_3811336339175339080at_nat @ A4 @ R ) )
=> ( ( member2643936169264416010at_nat @ Y7 @ ( equiv_3811336339175339080at_nat @ A4 @ R ) )
=> ( ( member8440522571783428010at_nat @ X @ X5 )
=> ( ( member8440522571783428010at_nat @ Y3 @ Y7 )
=> ( ( X5 = Y7 )
= ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ R ) ) ) ) ) ) ) ).
% quotient_eq_iff
thf(fact_523_quotient__eq__iff,axiom,
! [A4: set_nat,R: set_Pr1261947904930325089at_nat,X5: set_nat,Y7: set_nat,X: nat,Y3: nat] :
( ( equiv_equiv_nat @ A4 @ R )
=> ( ( member_set_nat @ X5 @ ( equiv_quotient_nat @ A4 @ R ) )
=> ( ( member_set_nat @ Y7 @ ( equiv_quotient_nat @ A4 @ R ) )
=> ( ( member_nat @ X @ X5 )
=> ( ( member_nat @ Y3 @ Y7 )
=> ( ( X5 = Y7 )
= ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R ) ) ) ) ) ) ) ).
% quotient_eq_iff
thf(fact_524_in__quotient__imp__closed,axiom,
! [A4: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat,X5: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,Y3: product_prod_nat_nat] :
( ( equiv_5407799809235660215at_nat @ A4 @ R )
=> ( ( member2643936169264416010at_nat @ X5 @ ( equiv_3811336339175339080at_nat @ A4 @ R ) )
=> ( ( member8440522571783428010at_nat @ X @ X5 )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ R )
=> ( member8440522571783428010at_nat @ Y3 @ X5 ) ) ) ) ) ).
% in_quotient_imp_closed
thf(fact_525_in__quotient__imp__closed,axiom,
! [A4: set_nat,R: set_Pr1261947904930325089at_nat,X5: set_nat,X: nat,Y3: nat] :
( ( equiv_equiv_nat @ A4 @ R )
=> ( ( member_set_nat @ X5 @ ( equiv_quotient_nat @ A4 @ R ) )
=> ( ( member_nat @ X @ X5 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R )
=> ( member_nat @ Y3 @ X5 ) ) ) ) ) ).
% in_quotient_imp_closed
thf(fact_526_in__quotient__imp__non__empty,axiom,
! [A4: set_nat,R: set_Pr1261947904930325089at_nat,X5: set_nat] :
( ( equiv_equiv_nat @ A4 @ R )
=> ( ( member_set_nat @ X5 @ ( equiv_quotient_nat @ A4 @ R ) )
=> ( X5 != bot_bot_set_nat ) ) ) ).
% in_quotient_imp_non_empty
thf(fact_527_in__quotient__imp__non__empty,axiom,
! [A4: set_o,R: set_Product_prod_o_o,X5: set_o] :
( ( equiv_equiv_o @ A4 @ R )
=> ( ( member_set_o @ X5 @ ( equiv_quotient_o @ A4 @ R ) )
=> ( X5 != bot_bot_set_o ) ) ) ).
% in_quotient_imp_non_empty
thf(fact_528_in__quotient__imp__subset,axiom,
! [A4: set_nat,R: set_Pr1261947904930325089at_nat,X5: set_nat] :
( ( equiv_equiv_nat @ A4 @ R )
=> ( ( member_set_nat @ X5 @ ( equiv_quotient_nat @ A4 @ R ) )
=> ( ord_less_eq_set_nat @ X5 @ A4 ) ) ) ).
% in_quotient_imp_subset
thf(fact_529_proj__in__iff,axiom,
! [A4: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat,X: product_prod_nat_nat] :
( ( equiv_5407799809235660215at_nat @ A4 @ R )
=> ( ( member2643936169264416010at_nat @ ( equiv_7265450603036132554at_nat @ R @ X ) @ ( equiv_3811336339175339080at_nat @ A4 @ R ) )
= ( member8440522571783428010at_nat @ X @ A4 ) ) ) ).
% proj_in_iff
thf(fact_530_proj__in__iff,axiom,
! [A4: set_nat,R: set_Pr1261947904930325089at_nat,X: nat] :
( ( equiv_equiv_nat @ A4 @ R )
=> ( ( member_set_nat @ ( equiv_proj_nat_nat @ R @ X ) @ ( equiv_quotient_nat @ A4 @ R ) )
= ( member_nat @ X @ A4 ) ) ) ).
% proj_in_iff
thf(fact_531_refines__equiv__image__eq,axiom,
! [R2: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat,A4: set_nat] :
( ( ord_le3146513528884898305at_nat @ R2 @ S3 )
=> ( ( equiv_equiv_nat @ A4 @ R2 )
=> ( ( equiv_equiv_nat @ A4 @ S3 )
=> ( ( image_7916887816326733075et_nat @ ( image_nat_nat @ S3 ) @ ( equiv_quotient_nat @ A4 @ R2 ) )
= ( equiv_quotient_nat @ A4 @ S3 ) ) ) ) ) ).
% refines_equiv_image_eq
thf(fact_532_singleton__quotient,axiom,
! [X: nat,R: set_Pr1261947904930325089at_nat] :
( ( equiv_quotient_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ R )
= ( insert_set_nat @ ( image_nat_nat @ R @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ bot_bot_set_set_nat ) ) ).
% singleton_quotient
thf(fact_533_singleton__quotient,axiom,
! [X: $o,R: set_Product_prod_o_o] :
( ( equiv_quotient_o @ ( insert_o @ X @ bot_bot_set_o ) @ R )
= ( insert_set_o @ ( image_o_o @ R @ ( insert_o @ X @ bot_bot_set_o ) ) @ bot_bot_set_set_o ) ) ).
% singleton_quotient
thf(fact_534_quotientI,axiom,
! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat] :
( ( member8440522571783428010at_nat @ X @ A4 )
=> ( member2643936169264416010at_nat @ ( image_2484338115069272851at_nat @ R @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) @ ( equiv_3811336339175339080at_nat @ A4 @ R ) ) ) ).
% quotientI
thf(fact_535_quotientI,axiom,
! [X: nat,A4: set_nat,R: set_Pr1261947904930325089at_nat] :
( ( member_nat @ X @ A4 )
=> ( member_set_nat @ ( image_nat_nat @ R @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ ( equiv_quotient_nat @ A4 @ R ) ) ) ).
% quotientI
thf(fact_536_quotientI,axiom,
! [X: $o,A4: set_o,R: set_Product_prod_o_o] :
( ( member_o @ X @ A4 )
=> ( member_set_o @ ( image_o_o @ R @ ( insert_o @ X @ bot_bot_set_o ) ) @ ( equiv_quotient_o @ A4 @ R ) ) ) ).
% quotientI
thf(fact_537_quotientE,axiom,
! [X5: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat] :
( ( member2643936169264416010at_nat @ X5 @ ( equiv_3811336339175339080at_nat @ A4 @ R ) )
=> ~ ! [X3: product_prod_nat_nat] :
( ( X5
= ( image_2484338115069272851at_nat @ R @ ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) )
=> ~ ( member8440522571783428010at_nat @ X3 @ A4 ) ) ) ).
% quotientE
thf(fact_538_quotientE,axiom,
! [X5: set_nat,A4: set_nat,R: set_Pr1261947904930325089at_nat] :
( ( member_set_nat @ X5 @ ( equiv_quotient_nat @ A4 @ R ) )
=> ~ ! [X3: nat] :
( ( X5
= ( image_nat_nat @ R @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) )
=> ~ ( member_nat @ X3 @ A4 ) ) ) ).
% quotientE
thf(fact_539_quotientE,axiom,
! [X5: set_o,A4: set_o,R: set_Product_prod_o_o] :
( ( member_set_o @ X5 @ ( equiv_quotient_o @ A4 @ R ) )
=> ~ ! [X3: $o] :
( ( X5
= ( image_o_o @ R @ ( insert_o @ X3 @ bot_bot_set_o ) ) )
=> ~ ( member_o @ X3 @ A4 ) ) ) ).
% quotientE
thf(fact_540_equiv__subset,axiom,
! [B5: set_nat,A4: set_nat,P: set_Pr1261947904930325089at_nat] :
( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( ( equiv_equiv_nat @ A4 @ P )
=> ( equiv_equiv_nat @ B5
@ ( inf_in2572325071724192079at_nat @ P
@ ( produc457027306803732586at_nat @ B5
@ ^ [Uu: nat] : B5 ) ) ) ) ) ).
% equiv_subset
thf(fact_541_eq__equiv__class__iff2,axiom,
! [A4: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat,X: product_prod_nat_nat,Y3: product_prod_nat_nat] :
( ( equiv_5407799809235660215at_nat @ A4 @ R )
=> ( ( member8440522571783428010at_nat @ X @ A4 )
=> ( ( member8440522571783428010at_nat @ Y3 @ A4 )
=> ( ( ( equiv_3811336339175339080at_nat @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) @ R )
= ( equiv_3811336339175339080at_nat @ ( insert8211810215607154385at_nat @ Y3 @ bot_bo2099793752762293965at_nat ) @ R ) )
= ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ R ) ) ) ) ) ).
% eq_equiv_class_iff2
thf(fact_542_eq__equiv__class__iff2,axiom,
! [A4: set_nat,R: set_Pr1261947904930325089at_nat,X: nat,Y3: nat] :
( ( equiv_equiv_nat @ A4 @ R )
=> ( ( member_nat @ X @ A4 )
=> ( ( member_nat @ Y3 @ A4 )
=> ( ( ( equiv_quotient_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ R )
= ( equiv_quotient_nat @ ( insert_nat @ Y3 @ bot_bot_set_nat ) @ R ) )
= ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R ) ) ) ) ) ).
% eq_equiv_class_iff2
thf(fact_543_eq__equiv__class__iff2,axiom,
! [A4: set_o,R: set_Product_prod_o_o,X: $o,Y3: $o] :
( ( equiv_equiv_o @ A4 @ R )
=> ( ( member_o @ X @ A4 )
=> ( ( member_o @ Y3 @ A4 )
=> ( ( ( equiv_quotient_o @ ( insert_o @ X @ bot_bot_set_o ) @ R )
= ( equiv_quotient_o @ ( insert_o @ Y3 @ bot_bot_set_o ) @ R ) )
= ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y3 ) @ R ) ) ) ) ) ).
% eq_equiv_class_iff2
thf(fact_544_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_545_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_o] :
( ( inf_inf_set_o @ X @ bot_bot_set_o )
= bot_bot_set_o ) ).
% boolean_algebra.conj_zero_right
thf(fact_546_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_547_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_o] :
( ( inf_inf_set_o @ bot_bot_set_o @ X )
= bot_bot_set_o ) ).
% boolean_algebra.conj_zero_left
thf(fact_548_inf__bot__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% inf_bot_right
thf(fact_549_inf__bot__right,axiom,
! [X: set_o] :
( ( inf_inf_set_o @ X @ bot_bot_set_o )
= bot_bot_set_o ) ).
% inf_bot_right
thf(fact_550_inf__bot__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
= bot_bot_set_nat ) ).
% inf_bot_left
thf(fact_551_inf__bot__left,axiom,
! [X: set_o] :
( ( inf_inf_set_o @ bot_bot_set_o @ X )
= bot_bot_set_o ) ).
% inf_bot_left
thf(fact_552_Restr__subset,axiom,
! [A4: set_nat,B5: set_nat,R: set_Pr1261947904930325089at_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( inf_in2572325071724192079at_nat
@ ( inf_in2572325071724192079at_nat @ R
@ ( produc457027306803732586at_nat @ B5
@ ^ [Uu: nat] : B5 ) )
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : A4 ) )
= ( inf_in2572325071724192079at_nat @ R
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : A4 ) ) ) ) ).
% Restr_subset
thf(fact_553_image__split__eq__Sigma,axiom,
! [F2: nat > set_nat,G: nat > set_nat,A4: set_nat] :
( ( image_3168736909456693974et_nat
@ ^ [X4: nat] : ( produc4532415448927165861et_nat @ ( F2 @ X4 ) @ ( G @ X4 ) )
@ A4 )
= ( produc105401717395143638et_nat @ ( image_nat_set_nat @ F2 @ A4 )
@ ^ [X4: set_nat] : ( image_nat_set_nat @ G @ ( inf_inf_set_nat @ ( vimage_nat_set_nat @ F2 @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) ) @ A4 ) ) ) ) ).
% image_split_eq_Sigma
thf(fact_554_image__split__eq__Sigma,axiom,
! [F2: nat > nat,G: nat > set_nat,A4: set_nat] :
( ( image_8316717911487899424et_nat
@ ^ [X4: nat] : ( produc4207506657711014383et_nat @ ( F2 @ X4 ) @ ( G @ X4 ) )
@ A4 )
= ( produc8883945523214541856et_nat @ ( image_nat_nat2 @ F2 @ A4 )
@ ^ [X4: nat] : ( image_nat_set_nat @ G @ ( inf_inf_set_nat @ ( vimage_nat_nat @ F2 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) @ A4 ) ) ) ) ).
% image_split_eq_Sigma
thf(fact_555_image__split__eq__Sigma,axiom,
! [F2: nat > $o,G: nat > set_nat,A4: set_nat] :
( ( image_7578179957167012260et_nat
@ ^ [X4: nat] : ( produc818605102413008839et_nat @ ( F2 @ X4 ) @ ( G @ X4 ) )
@ A4 )
= ( produc7056857006710681430et_nat @ ( image_nat_o2 @ F2 @ A4 )
@ ^ [X4: $o] : ( image_nat_set_nat @ G @ ( inf_inf_set_nat @ ( vimage_nat_o @ F2 @ ( insert_o @ X4 @ bot_bot_set_o ) ) @ A4 ) ) ) ) ).
% image_split_eq_Sigma
thf(fact_556_vimage__eq,axiom,
! [A: product_prod_nat_nat,F2: product_prod_nat_nat > product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ A @ ( vimage2449269961533847803at_nat @ F2 @ B5 ) )
= ( member8440522571783428010at_nat @ ( F2 @ A ) @ B5 ) ) ).
% vimage_eq
thf(fact_557_vimage__eq,axiom,
! [A: product_prod_nat_nat,F2: product_prod_nat_nat > nat,B5: set_nat] :
( ( member8440522571783428010at_nat @ A @ ( vimage4653281326611754070at_nat @ F2 @ B5 ) )
= ( member_nat @ ( F2 @ A ) @ B5 ) ) ).
% vimage_eq
thf(fact_558_vimage__eq,axiom,
! [A: nat,F2: nat > product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
( ( member_nat @ A @ ( vimage8013328719654469172at_nat @ F2 @ B5 ) )
= ( member8440522571783428010at_nat @ ( F2 @ A ) @ B5 ) ) ).
% vimage_eq
thf(fact_559_vimage__eq,axiom,
! [A: nat,F2: nat > nat,B5: set_nat] :
( ( member_nat @ A @ ( vimage_nat_nat @ F2 @ B5 ) )
= ( member_nat @ ( F2 @ A ) @ B5 ) ) ).
% vimage_eq
thf(fact_560_vimageI,axiom,
! [F2: product_prod_nat_nat > product_prod_nat_nat,A: product_prod_nat_nat,B: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
( ( ( F2 @ A )
= B )
=> ( ( member8440522571783428010at_nat @ B @ B5 )
=> ( member8440522571783428010at_nat @ A @ ( vimage2449269961533847803at_nat @ F2 @ B5 ) ) ) ) ).
% vimageI
thf(fact_561_vimageI,axiom,
! [F2: nat > product_prod_nat_nat,A: nat,B: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
( ( ( F2 @ A )
= B )
=> ( ( member8440522571783428010at_nat @ B @ B5 )
=> ( member_nat @ A @ ( vimage8013328719654469172at_nat @ F2 @ B5 ) ) ) ) ).
% vimageI
thf(fact_562_vimageI,axiom,
! [F2: product_prod_nat_nat > nat,A: product_prod_nat_nat,B: nat,B5: set_nat] :
( ( ( F2 @ A )
= B )
=> ( ( member_nat @ B @ B5 )
=> ( member8440522571783428010at_nat @ A @ ( vimage4653281326611754070at_nat @ F2 @ B5 ) ) ) ) ).
% vimageI
thf(fact_563_vimageI,axiom,
! [F2: nat > nat,A: nat,B: nat,B5: set_nat] :
( ( ( F2 @ A )
= B )
=> ( ( member_nat @ B @ B5 )
=> ( member_nat @ A @ ( vimage_nat_nat @ F2 @ B5 ) ) ) ) ).
% vimageI
thf(fact_564_vimage__empty,axiom,
! [F2: nat > nat] :
( ( vimage_nat_nat @ F2 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% vimage_empty
thf(fact_565_vimage__empty,axiom,
! [F2: $o > nat] :
( ( vimage_o_nat @ F2 @ bot_bot_set_nat )
= bot_bot_set_o ) ).
% vimage_empty
thf(fact_566_vimage__empty,axiom,
! [F2: nat > $o] :
( ( vimage_nat_o @ F2 @ bot_bot_set_o )
= bot_bot_set_nat ) ).
% vimage_empty
thf(fact_567_vimage__empty,axiom,
! [F2: $o > $o] :
( ( vimage_o_o @ F2 @ bot_bot_set_o )
= bot_bot_set_o ) ).
% vimage_empty
thf(fact_568_vimageI2,axiom,
! [F2: product_prod_nat_nat > product_prod_nat_nat,A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( F2 @ A ) @ A4 )
=> ( member8440522571783428010at_nat @ A @ ( vimage2449269961533847803at_nat @ F2 @ A4 ) ) ) ).
% vimageI2
thf(fact_569_vimageI2,axiom,
! [F2: nat > product_prod_nat_nat,A: nat,A4: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( F2 @ A ) @ A4 )
=> ( member_nat @ A @ ( vimage8013328719654469172at_nat @ F2 @ A4 ) ) ) ).
% vimageI2
thf(fact_570_vimageI2,axiom,
! [F2: product_prod_nat_nat > nat,A: product_prod_nat_nat,A4: set_nat] :
( ( member_nat @ ( F2 @ A ) @ A4 )
=> ( member8440522571783428010at_nat @ A @ ( vimage4653281326611754070at_nat @ F2 @ A4 ) ) ) ).
% vimageI2
thf(fact_571_vimageI2,axiom,
! [F2: nat > nat,A: nat,A4: set_nat] :
( ( member_nat @ ( F2 @ A ) @ A4 )
=> ( member_nat @ A @ ( vimage_nat_nat @ F2 @ A4 ) ) ) ).
% vimageI2
thf(fact_572_vimageE,axiom,
! [A: product_prod_nat_nat,F2: product_prod_nat_nat > product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ A @ ( vimage2449269961533847803at_nat @ F2 @ B5 ) )
=> ( member8440522571783428010at_nat @ ( F2 @ A ) @ B5 ) ) ).
% vimageE
thf(fact_573_vimageE,axiom,
! [A: product_prod_nat_nat,F2: product_prod_nat_nat > nat,B5: set_nat] :
( ( member8440522571783428010at_nat @ A @ ( vimage4653281326611754070at_nat @ F2 @ B5 ) )
=> ( member_nat @ ( F2 @ A ) @ B5 ) ) ).
% vimageE
thf(fact_574_vimageE,axiom,
! [A: nat,F2: nat > product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
( ( member_nat @ A @ ( vimage8013328719654469172at_nat @ F2 @ B5 ) )
=> ( member8440522571783428010at_nat @ ( F2 @ A ) @ B5 ) ) ).
% vimageE
thf(fact_575_vimageE,axiom,
! [A: nat,F2: nat > nat,B5: set_nat] :
( ( member_nat @ A @ ( vimage_nat_nat @ F2 @ B5 ) )
=> ( member_nat @ ( F2 @ A ) @ B5 ) ) ).
% vimageE
thf(fact_576_vimageD,axiom,
! [A: product_prod_nat_nat,F2: product_prod_nat_nat > product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ A @ ( vimage2449269961533847803at_nat @ F2 @ A4 ) )
=> ( member8440522571783428010at_nat @ ( F2 @ A ) @ A4 ) ) ).
% vimageD
thf(fact_577_vimageD,axiom,
! [A: product_prod_nat_nat,F2: product_prod_nat_nat > nat,A4: set_nat] :
( ( member8440522571783428010at_nat @ A @ ( vimage4653281326611754070at_nat @ F2 @ A4 ) )
=> ( member_nat @ ( F2 @ A ) @ A4 ) ) ).
% vimageD
thf(fact_578_vimageD,axiom,
! [A: nat,F2: nat > product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ( member_nat @ A @ ( vimage8013328719654469172at_nat @ F2 @ A4 ) )
=> ( member8440522571783428010at_nat @ ( F2 @ A ) @ A4 ) ) ).
% vimageD
thf(fact_579_vimageD,axiom,
! [A: nat,F2: nat > nat,A4: set_nat] :
( ( member_nat @ A @ ( vimage_nat_nat @ F2 @ A4 ) )
=> ( member_nat @ ( F2 @ A ) @ A4 ) ) ).
% vimageD
thf(fact_580_inf__Int__eq2,axiom,
! [R2: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
( ( inf_inf_nat_nat_o
@ ^ [X4: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ R2 )
@ ^ [X4: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ S3 ) )
= ( ^ [X4: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ ( inf_in2572325071724192079at_nat @ R2 @ S3 ) ) ) ) ).
% inf_Int_eq2
thf(fact_581_inf__set__def,axiom,
( inf_in2572325071724192079at_nat
= ( ^ [A6: set_Pr1261947904930325089at_nat,B7: set_Pr1261947904930325089at_nat] :
( collec3392354462482085612at_nat
@ ( inf_in5163264567034779214_nat_o
@ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ A6 )
@ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ B7 ) ) ) ) ) ).
% inf_set_def
thf(fact_582_inf__set__def,axiom,
( inf_inf_set_nat
= ( ^ [A6: set_nat,B7: set_nat] :
( collect_nat
@ ( inf_inf_nat_o
@ ^ [X4: nat] : ( member_nat @ X4 @ A6 )
@ ^ [X4: nat] : ( member_nat @ X4 @ B7 ) ) ) ) ) ).
% inf_set_def
thf(fact_583_inf__Int__eq,axiom,
! [R2: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
( ( inf_in5163264567034779214_nat_o
@ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ R2 )
@ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ S3 ) )
= ( ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ ( inf_in2572325071724192079at_nat @ R2 @ S3 ) ) ) ) ).
% inf_Int_eq
thf(fact_584_inf__Int__eq,axiom,
! [R2: set_nat,S3: set_nat] :
( ( inf_inf_nat_o
@ ^ [X4: nat] : ( member_nat @ X4 @ R2 )
@ ^ [X4: nat] : ( member_nat @ X4 @ S3 ) )
= ( ^ [X4: nat] : ( member_nat @ X4 @ ( inf_inf_set_nat @ R2 @ S3 ) ) ) ) ).
% inf_Int_eq
thf(fact_585_image__vimage__subset,axiom,
! [F2: nat > set_nat,A4: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F2 @ ( vimage_nat_set_nat @ F2 @ A4 ) ) @ A4 ) ).
% image_vimage_subset
thf(fact_586_image__subset__iff__subset__vimage,axiom,
! [F2: nat > set_nat,A4: set_nat,B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F2 @ A4 ) @ B5 )
= ( ord_less_eq_set_nat @ A4 @ ( vimage_nat_set_nat @ F2 @ B5 ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_587_vimage__singleton__eq,axiom,
! [A: product_prod_nat_nat,F2: product_prod_nat_nat > nat,B: nat] :
( ( member8440522571783428010at_nat @ A @ ( vimage4653281326611754070at_nat @ F2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
= ( ( F2 @ A )
= B ) ) ).
% vimage_singleton_eq
thf(fact_588_vimage__singleton__eq,axiom,
! [A: nat,F2: nat > nat,B: nat] :
( ( member_nat @ A @ ( vimage_nat_nat @ F2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
= ( ( F2 @ A )
= B ) ) ).
% vimage_singleton_eq
thf(fact_589_vimage__singleton__eq,axiom,
! [A: product_prod_nat_nat,F2: product_prod_nat_nat > $o,B: $o] :
( ( member8440522571783428010at_nat @ A @ ( vimage2444243245536687506_nat_o @ F2 @ ( insert_o @ B @ bot_bot_set_o ) ) )
= ( ( F2 @ A )
= B ) ) ).
% vimage_singleton_eq
thf(fact_590_vimage__singleton__eq,axiom,
! [A: nat,F2: nat > $o,B: $o] :
( ( member_nat @ A @ ( vimage_nat_o @ F2 @ ( insert_o @ B @ bot_bot_set_o ) ) )
= ( ( F2 @ A )
= B ) ) ).
% vimage_singleton_eq
thf(fact_591_Pair__vimage__Sigma,axiom,
! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > set_nat] :
( ( ( member8440522571783428010at_nat @ X @ A4 )
=> ( ( vimage3798356988490952717at_nat @ ( produc6350711070570205562at_nat @ X ) @ ( produc7672662199629908489at_nat @ A4 @ F2 ) )
= ( F2 @ X ) ) )
& ( ~ ( member8440522571783428010at_nat @ X @ A4 )
=> ( ( vimage3798356988490952717at_nat @ ( produc6350711070570205562at_nat @ X ) @ ( produc7672662199629908489at_nat @ A4 @ F2 ) )
= bot_bot_set_nat ) ) ) ).
% Pair_vimage_Sigma
thf(fact_592_Pair__vimage__Sigma,axiom,
! [X: nat,A4: set_nat,F2: nat > set_nat] :
( ( ( member_nat @ X @ A4 )
=> ( ( vimage8013328719654469172at_nat @ ( product_Pair_nat_nat @ X ) @ ( produc457027306803732586at_nat @ A4 @ F2 ) )
= ( F2 @ X ) ) )
& ( ~ ( member_nat @ X @ A4 )
=> ( ( vimage8013328719654469172at_nat @ ( product_Pair_nat_nat @ X ) @ ( produc457027306803732586at_nat @ A4 @ F2 ) )
= bot_bot_set_nat ) ) ) ).
% Pair_vimage_Sigma
thf(fact_593_Pair__vimage__Sigma,axiom,
! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > set_o] :
( ( ( member8440522571783428010at_nat @ X @ A4 )
=> ( ( vimage473746520521726369_nat_o @ ( produc855456346497908590_nat_o @ X ) @ ( produc203973861852599583_nat_o @ A4 @ F2 ) )
= ( F2 @ X ) ) )
& ( ~ ( member8440522571783428010at_nat @ X @ A4 )
=> ( ( vimage473746520521726369_nat_o @ ( produc855456346497908590_nat_o @ X ) @ ( produc203973861852599583_nat_o @ A4 @ F2 ) )
= bot_bot_set_o ) ) ) ).
% Pair_vimage_Sigma
thf(fact_594_Pair__vimage__Sigma,axiom,
! [X: nat,A4: set_nat,F2: nat > set_o] :
( ( ( member_nat @ X @ A4 )
=> ( ( vimage6125388955539849434_nat_o @ ( product_Pair_nat_o @ X ) @ ( product_Sigma_nat_o @ A4 @ F2 ) )
= ( F2 @ X ) ) )
& ( ~ ( member_nat @ X @ A4 )
=> ( ( vimage6125388955539849434_nat_o @ ( product_Pair_nat_o @ X ) @ ( product_Sigma_nat_o @ A4 @ F2 ) )
= bot_bot_set_o ) ) ) ).
% Pair_vimage_Sigma
thf(fact_595_quotient__def,axiom,
( equiv_quotient_nat
= ( ^ [A6: set_nat,R3: set_Pr1261947904930325089at_nat] :
( comple548664676211718543et_nat
@ ( image_2194112158459175443et_nat
@ ^ [X4: nat] : ( insert_set_nat @ ( image_nat_nat @ R3 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) @ bot_bot_set_set_nat )
@ A6 ) ) ) ) ).
% quotient_def
thf(fact_596_quotient__def,axiom,
( equiv_quotient_o
= ( ^ [A6: set_o,R3: set_Product_prod_o_o] :
( comple4436988014476444997_set_o
@ ( image_o_set_set_o
@ ^ [X4: $o] : ( insert_set_o @ ( image_o_o @ R3 @ ( insert_o @ X4 @ bot_bot_set_o ) ) @ bot_bot_set_set_o )
@ A6 ) ) ) ) ).
% quotient_def
thf(fact_597_image__Fpow__mono,axiom,
! [F2: nat > set_nat,A4: set_nat,B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F2 @ A4 ) @ B5 )
=> ( ord_le9131159989063066194et_nat @ ( image_6725021117256019401et_nat @ ( image_nat_set_nat @ F2 ) @ ( finite_Fpow_nat @ A4 ) ) @ ( finite_Fpow_set_nat @ B5 ) ) ) ).
% image_Fpow_mono
thf(fact_598_Sigma__Union,axiom,
! [X5: set_set_nat,B5: nat > set_nat] :
( ( produc457027306803732586at_nat @ ( comple7399068483239264473et_nat @ X5 ) @ B5 )
= ( comple5685304695842803022at_nat
@ ( image_6521575952113991828at_nat
@ ^ [A6: set_nat] : ( produc457027306803732586at_nat @ A6 @ B5 )
@ X5 ) ) ) ).
% Sigma_Union
thf(fact_599_UN__Times__distrib,axiom,
! [E: nat > set_nat,F3: nat > set_nat,A4: set_nat,B5: set_nat] :
( ( comple5685304695842803022at_nat
@ ( image_5423882813909314213at_nat
@ ( produc8197505143624133779at_nat
@ ^ [A5: nat,B4: nat] :
( produc457027306803732586at_nat @ ( E @ A5 )
@ ^ [Uu: nat] : ( F3 @ B4 ) ) )
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : B5 ) ) )
= ( produc457027306803732586at_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ E @ A4 ) )
@ ^ [Uu: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F3 @ B5 ) ) ) ) ).
% UN_Times_distrib
thf(fact_600_UN__Image,axiom,
! [X5: nat > set_Pr1261947904930325089at_nat,I2: set_nat,S3: set_nat] :
( ( image_nat_nat @ ( comple5685304695842803022at_nat @ ( image_7178329752028323786at_nat @ X5 @ I2 ) ) @ S3 )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I: nat] : ( image_nat_nat @ ( X5 @ I ) @ S3 )
@ I2 ) ) ) ).
% UN_Image
thf(fact_601_UN__constant__eq,axiom,
! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > set_nat,C: set_nat] :
( ( member8440522571783428010at_nat @ A @ A4 )
=> ( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A4 )
=> ( ( F2 @ X3 )
= C ) )
=> ( ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F2 @ A4 ) )
= C ) ) ) ).
% UN_constant_eq
thf(fact_602_UN__constant__eq,axiom,
! [A: nat,A4: set_nat,F2: nat > set_nat,C: set_nat] :
( ( member_nat @ A @ A4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ( F2 @ X3 )
= C ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ A4 ) )
= C ) ) ) ).
% UN_constant_eq
thf(fact_603_empty__in__Fpow,axiom,
! [A4: set_nat] : ( member_set_nat @ bot_bot_set_nat @ ( finite_Fpow_nat @ A4 ) ) ).
% empty_in_Fpow
thf(fact_604_empty__in__Fpow,axiom,
! [A4: set_o] : ( member_set_o @ bot_bot_set_o @ ( finite_Fpow_o @ A4 ) ) ).
% empty_in_Fpow
thf(fact_605_Union__quotient,axiom,
! [A4: set_nat,R: set_Pr1261947904930325089at_nat] :
( ( equiv_equiv_nat @ A4 @ R )
=> ( ( comple7399068483239264473et_nat @ ( equiv_quotient_nat @ A4 @ R ) )
= A4 ) ) ).
% Union_quotient
thf(fact_606_Image__UN,axiom,
! [R: set_Pr1261947904930325089at_nat,B5: nat > set_nat,A4: set_nat] :
( ( image_nat_nat @ R @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ A4 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X4: nat] : ( image_nat_nat @ R @ ( B5 @ X4 ) )
@ A4 ) ) ) ).
% Image_UN
thf(fact_607_insert__partition,axiom,
! [X: set_o,F3: set_set_o] :
( ~ ( member_set_o @ X @ F3 )
=> ( ! [X3: set_o] :
( ( member_set_o @ X3 @ ( insert_set_o @ X @ F3 ) )
=> ! [Xa: set_o] :
( ( member_set_o @ Xa @ ( insert_set_o @ X @ F3 ) )
=> ( ( X3 != Xa )
=> ( ( inf_inf_set_o @ X3 @ Xa )
= bot_bot_set_o ) ) ) )
=> ( ( inf_inf_set_o @ X @ ( comple90263536869209701_set_o @ F3 ) )
= bot_bot_set_o ) ) ) ).
% insert_partition
thf(fact_608_insert__partition,axiom,
! [X: set_nat,F3: set_set_nat] :
( ~ ( member_set_nat @ X @ F3 )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ ( insert_set_nat @ X @ F3 ) )
=> ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ ( insert_set_nat @ X @ F3 ) )
=> ( ( X3 != Xa )
=> ( ( inf_inf_set_nat @ X3 @ Xa )
= bot_bot_set_nat ) ) ) )
=> ( ( inf_inf_set_nat @ X @ ( comple7399068483239264473et_nat @ F3 ) )
= bot_bot_set_nat ) ) ) ).
% insert_partition
thf(fact_609_snd__fst__flip,axiom,
( product_snd_nat_nat
= ( comp_P1567445206330693457at_nat @ product_fst_nat_nat
@ ( produc2626176000494625587at_nat
@ ^ [X4: nat,Y5: nat] : ( product_Pair_nat_nat @ Y5 @ X4 ) ) ) ) ).
% snd_fst_flip
thf(fact_610_fst__snd__flip,axiom,
( product_fst_nat_nat
= ( comp_P1567445206330693457at_nat @ product_snd_nat_nat
@ ( produc2626176000494625587at_nat
@ ^ [X4: nat,Y5: nat] : ( product_Pair_nat_nat @ Y5 @ X4 ) ) ) ) ).
% fst_snd_flip
thf(fact_611_Image__eq__UN,axiom,
( image_nat_nat
= ( ^ [R3: set_Pr1261947904930325089at_nat,B7: set_nat] :
( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y5: nat] : ( image_nat_nat @ R3 @ ( insert_nat @ Y5 @ bot_bot_set_nat ) )
@ B7 ) ) ) ) ).
% Image_eq_UN
thf(fact_612_Image__eq__UN,axiom,
( image_o_nat
= ( ^ [R3: set_Pr2101469702781467981_o_nat,B7: set_o] :
( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [Y5: $o] : ( image_o_nat @ R3 @ ( insert_o @ Y5 @ bot_bot_set_o ) )
@ B7 ) ) ) ) ).
% Image_eq_UN
thf(fact_613_Sigma__Image,axiom,
! [A4: set_nat,B5: nat > set_nat,X5: set_nat] :
( ( image_nat_nat @ ( produc457027306803732586at_nat @ A4 @ B5 ) @ X5 )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ ( inf_inf_set_nat @ X5 @ A4 ) ) ) ) ).
% Sigma_Image
thf(fact_614_snd__image__Sigma,axiom,
! [A4: set_nat,B5: nat > set_nat] :
( ( image_2486076414777270412at_nat @ product_snd_nat_nat @ ( produc457027306803732586at_nat @ A4 @ B5 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ A4 ) ) ) ).
% snd_image_Sigma
thf(fact_615_UN__singleton,axiom,
! [A4: set_o] :
( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X4: $o] : ( insert_o @ X4 @ bot_bot_set_o )
@ A4 ) )
= A4 ) ).
% UN_singleton
thf(fact_616_UN__singleton,axiom,
! [A4: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X4: nat] : ( insert_nat @ X4 @ bot_bot_set_nat )
@ A4 ) )
= A4 ) ).
% UN_singleton
thf(fact_617_UN__simps_I1_J,axiom,
! [C2: set_nat,A: $o,B5: nat > set_o] :
( ( ( C2 = bot_bot_set_nat )
=> ( ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [X4: nat] : ( insert_o @ A @ ( B5 @ X4 ) )
@ C2 ) )
= bot_bot_set_o ) )
& ( ( C2 != bot_bot_set_nat )
=> ( ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [X4: nat] : ( insert_o @ A @ ( B5 @ X4 ) )
@ C2 ) )
= ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B5 @ C2 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_618_UN__simps_I1_J,axiom,
! [C2: set_o,A: $o,B5: $o > set_o] :
( ( ( C2 = bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X4: $o] : ( insert_o @ A @ ( B5 @ X4 ) )
@ C2 ) )
= bot_bot_set_o ) )
& ( ( C2 != bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X4: $o] : ( insert_o @ A @ ( B5 @ X4 ) )
@ C2 ) )
= ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B5 @ C2 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_619_UN__simps_I1_J,axiom,
! [C2: set_nat,A: nat,B5: nat > set_nat] :
( ( ( C2 = bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X4: nat] : ( insert_nat @ A @ ( B5 @ X4 ) )
@ C2 ) )
= bot_bot_set_nat ) )
& ( ( C2 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X4: nat] : ( insert_nat @ A @ ( B5 @ X4 ) )
@ C2 ) )
= ( insert_nat @ A @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ C2 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_620_UN__simps_I1_J,axiom,
! [C2: set_o,A: nat,B5: $o > set_nat] :
( ( ( C2 = bot_bot_set_o )
=> ( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [X4: $o] : ( insert_nat @ A @ ( B5 @ X4 ) )
@ C2 ) )
= bot_bot_set_nat ) )
& ( ( C2 != bot_bot_set_o )
=> ( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [X4: $o] : ( insert_nat @ A @ ( B5 @ X4 ) )
@ C2 ) )
= ( insert_nat @ A @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B5 @ C2 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_621_UN__constant,axiom,
! [A4: set_nat,C: set_o] :
( ( ( A4 = bot_bot_set_nat )
=> ( ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [Y5: nat] : C
@ A4 ) )
= bot_bot_set_o ) )
& ( ( A4 != bot_bot_set_nat )
=> ( ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [Y5: nat] : C
@ A4 ) )
= C ) ) ) ).
% UN_constant
thf(fact_622_UN__constant,axiom,
! [A4: set_o,C: set_o] :
( ( ( A4 = bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [Y5: $o] : C
@ A4 ) )
= bot_bot_set_o ) )
& ( ( A4 != bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [Y5: $o] : C
@ A4 ) )
= C ) ) ) ).
% UN_constant
thf(fact_623_UN__constant,axiom,
! [A4: set_nat,C: set_nat] :
( ( ( A4 = bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y5: nat] : C
@ A4 ) )
= bot_bot_set_nat ) )
& ( ( A4 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y5: nat] : C
@ A4 ) )
= C ) ) ) ).
% UN_constant
thf(fact_624_UN__constant,axiom,
! [A4: set_o,C: set_nat] :
( ( ( A4 = bot_bot_set_o )
=> ( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [Y5: $o] : C
@ A4 ) )
= bot_bot_set_nat ) )
& ( ( A4 != bot_bot_set_o )
=> ( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [Y5: $o] : C
@ A4 ) )
= C ) ) ) ).
% UN_constant
thf(fact_625_cSUP__const,axiom,
! [A4: set_nat,C: set_nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X4: nat] : C
@ A4 ) )
= C ) ) ).
% cSUP_const
thf(fact_626_cSUP__const,axiom,
! [A4: set_o,C: set_nat] :
( ( A4 != bot_bot_set_o )
=> ( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [X4: $o] : C
@ A4 ) )
= C ) ) ).
% cSUP_const
thf(fact_627_SUP__const,axiom,
! [A4: set_nat,F2: set_nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I: nat] : F2
@ A4 ) )
= F2 ) ) ).
% SUP_const
thf(fact_628_SUP__const,axiom,
! [A4: set_o,F2: set_nat] :
( ( A4 != bot_bot_set_o )
=> ( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [I: $o] : F2
@ A4 ) )
= F2 ) ) ).
% SUP_const
thf(fact_629_Sup__bot__conv_I1_J,axiom,
! [A4: set_set_o] :
( ( ( comple90263536869209701_set_o @ A4 )
= bot_bot_set_o )
= ( ! [X4: set_o] :
( ( member_set_o @ X4 @ A4 )
=> ( X4 = bot_bot_set_o ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_630_Sup__bot__conv_I1_J,axiom,
! [A4: set_set_nat] :
( ( ( comple7399068483239264473et_nat @ A4 )
= bot_bot_set_nat )
= ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A4 )
=> ( X4 = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_631_Sup__bot__conv_I2_J,axiom,
! [A4: set_set_o] :
( ( bot_bot_set_o
= ( comple90263536869209701_set_o @ A4 ) )
= ( ! [X4: set_o] :
( ( member_set_o @ X4 @ A4 )
=> ( X4 = bot_bot_set_o ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_632_Sup__bot__conv_I2_J,axiom,
! [A4: set_set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ A4 ) )
= ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A4 )
=> ( X4 = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_633_SUP__identity__eq,axiom,
! [A4: set_set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [X4: set_nat] : X4
@ A4 ) )
= ( comple7399068483239264473et_nat @ A4 ) ) ).
% SUP_identity_eq
thf(fact_634_UN__iff,axiom,
! [B: nat,B5: nat > set_nat,A4: set_nat] :
( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ A4 ) ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ( member_nat @ B @ ( B5 @ X4 ) ) ) ) ) ).
% UN_iff
thf(fact_635_UN__I,axiom,
! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,B5: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ A @ A4 )
=> ( ( member8440522571783428010at_nat @ B @ ( B5 @ A ) )
=> ( member8440522571783428010at_nat @ B @ ( comple5685304695842803022at_nat @ ( image_5423882813909314213at_nat @ B5 @ A4 ) ) ) ) ) ).
% UN_I
thf(fact_636_UN__I,axiom,
! [A: nat,A4: set_nat,B: product_prod_nat_nat,B5: nat > set_Pr1261947904930325089at_nat] :
( ( member_nat @ A @ A4 )
=> ( ( member8440522571783428010at_nat @ B @ ( B5 @ A ) )
=> ( member8440522571783428010at_nat @ B @ ( comple5685304695842803022at_nat @ ( image_7178329752028323786at_nat @ B5 @ A4 ) ) ) ) ) ).
% UN_I
thf(fact_637_UN__I,axiom,
! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B: nat,B5: product_prod_nat_nat > set_nat] :
( ( member8440522571783428010at_nat @ A @ A4 )
=> ( ( member_nat @ B @ ( B5 @ A ) )
=> ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ B5 @ A4 ) ) ) ) ) ).
% UN_I
thf(fact_638_UN__I,axiom,
! [A: nat,A4: set_nat,B: nat,B5: nat > set_nat] :
( ( member_nat @ A @ A4 )
=> ( ( member_nat @ B @ ( B5 @ A ) )
=> ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ A4 ) ) ) ) ) ).
% UN_I
thf(fact_639_Sup__empty,axiom,
( ( comple90263536869209701_set_o @ bot_bot_set_set_o )
= bot_bot_set_o ) ).
% Sup_empty
thf(fact_640_Sup__empty,axiom,
( ( complete_Sup_Sup_o @ bot_bot_set_o )
= bot_bot_o ) ).
% Sup_empty
thf(fact_641_Sup__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% Sup_empty
thf(fact_642_cSup__singleton,axiom,
! [X: nat] :
( ( complete_Sup_Sup_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
= X ) ).
% cSup_singleton
thf(fact_643_cSup__singleton,axiom,
! [X: $o] :
( ( complete_Sup_Sup_o @ ( insert_o @ X @ bot_bot_set_o ) )
= X ) ).
% cSup_singleton
thf(fact_644_cSup__singleton,axiom,
! [X: set_nat] :
( ( comple7399068483239264473et_nat @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
= X ) ).
% cSup_singleton
thf(fact_645_SUP__bot__conv_I2_J,axiom,
! [B5: nat > set_nat,A4: set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ A4 ) ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ( B5 @ X4 )
= bot_bot_set_nat ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_646_SUP__bot__conv_I1_J,axiom,
! [B5: nat > set_nat,A4: set_nat] :
( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ A4 ) )
= bot_bot_set_nat )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ( B5 @ X4 )
= bot_bot_set_nat ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_647_SUP__bot,axiom,
! [A4: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X4: nat] : bot_bot_set_nat
@ A4 ) )
= bot_bot_set_nat ) ).
% SUP_bot
thf(fact_648_Sup__SUP__eq,axiom,
( comple1901448594430626575_nat_o
= ( ^ [S4: set_Pr5582243495563764594_nat_o,X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ ( comple5685304695842803022at_nat @ ( image_7124889717316225246at_nat @ collec3392354462482085612at_nat @ S4 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_649_Sup__SUP__eq,axiom,
( comple8317665133742190828_nat_o
= ( ^ [S4: set_nat_o,X4: nat] : ( member_nat @ X4 @ ( comple7399068483239264473et_nat @ ( image_nat_o_set_nat @ collect_nat @ S4 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_650_Sup__set__def,axiom,
( comple5685304695842803022at_nat
= ( ^ [A6: set_se7855581050983116737at_nat] :
( collec3392354462482085612at_nat
@ ^ [X4: product_prod_nat_nat] : ( complete_Sup_Sup_o @ ( image_7457375456213313148_nat_o @ ( member8440522571783428010at_nat @ X4 ) @ A6 ) ) ) ) ) ).
% Sup_set_def
thf(fact_651_Sup__set__def,axiom,
( comple7399068483239264473et_nat
= ( ^ [A6: set_set_nat] :
( collect_nat
@ ^ [X4: nat] : ( complete_Sup_Sup_o @ ( image_set_nat_o @ ( member_nat @ X4 ) @ A6 ) ) ) ) ) ).
% Sup_set_def
thf(fact_652_SUP__Sup__eq2,axiom,
! [S3: set_se7855581050983116737at_nat] :
( ( comple3592611370556534995_nat_o
@ ( image_6923511907955291850_nat_o
@ ^ [I: set_Pr1261947904930325089at_nat,X4: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ I )
@ S3 ) )
= ( ^ [X4: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ ( comple5685304695842803022at_nat @ S3 ) ) ) ) ).
% SUP_Sup_eq2
thf(fact_653_Sup__SUP__eq2,axiom,
( comple3592611370556534995_nat_o
= ( ^ [S4: set_nat_nat_o,X4: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ ( comple5685304695842803022at_nat @ ( image_7124889717316225246at_nat @ collec3392354462482085612at_nat @ ( image_7429393840292777309_nat_o @ produc6081775807080527818_nat_o @ S4 ) ) ) ) ) ) ).
% Sup_SUP_eq2
thf(fact_654_SUP__Sup__eq,axiom,
! [S3: set_se7855581050983116737at_nat] :
( ( comple1901448594430626575_nat_o
@ ( image_5253903537071781336_nat_o
@ ^ [I: set_Pr1261947904930325089at_nat,X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ I )
@ S3 ) )
= ( ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ ( comple5685304695842803022at_nat @ S3 ) ) ) ) ).
% SUP_Sup_eq
thf(fact_655_SUP__Sup__eq,axiom,
! [S3: set_set_nat] :
( ( comple8317665133742190828_nat_o
@ ( image_set_nat_nat_o
@ ^ [I: set_nat,X4: nat] : ( member_nat @ X4 @ I )
@ S3 ) )
= ( ^ [X4: nat] : ( member_nat @ X4 @ ( comple7399068483239264473et_nat @ S3 ) ) ) ) ).
% SUP_Sup_eq
thf(fact_656_SUP__UN__eq,axiom,
! [R: nat > set_nat,S3: set_nat] :
( ( comple8317665133742190828_nat_o
@ ( image_nat_nat_o
@ ^ [I: nat,X4: nat] : ( member_nat @ X4 @ ( R @ I ) )
@ S3 ) )
= ( ^ [X4: nat] : ( member_nat @ X4 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ R @ S3 ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_657_Inf_OINF__cong,axiom,
! [A4: set_nat,B5: set_nat,C2: nat > set_nat,D: nat > set_nat,Inf: set_set_nat > set_nat] :
( ( A4 = B5 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B5 )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_nat_set_nat @ C2 @ A4 ) )
= ( Inf @ ( image_nat_set_nat @ D @ B5 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_658_Sup_OSUP__cong,axiom,
! [A4: set_nat,B5: set_nat,C2: nat > set_nat,D: nat > set_nat,Sup: set_set_nat > set_nat] :
( ( A4 = B5 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B5 )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_nat_set_nat @ C2 @ A4 ) )
= ( Sup @ ( image_nat_set_nat @ D @ B5 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_659_Sigma__def,axiom,
( produc457027306803732586at_nat
= ( ^ [A6: set_nat,B7: nat > set_nat] :
( comple5685304695842803022at_nat
@ ( image_7178329752028323786at_nat
@ ^ [X4: nat] :
( comple5685304695842803022at_nat
@ ( image_7178329752028323786at_nat
@ ^ [Y5: nat] : ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ bot_bo2099793752762293965at_nat )
@ ( B7 @ X4 ) ) )
@ A6 ) ) ) ) ).
% Sigma_def
thf(fact_660_equiv__proj,axiom,
! [A4: set_nat,R2: set_Pr1261947904930325089at_nat,Z: product_prod_nat_nat] :
( ( equiv_equiv_nat @ A4 @ R2 )
=> ( ( member8440522571783428010at_nat @ Z @ R2 )
=> ( ( comp_n8456906485034819078at_nat @ ( equiv_proj_nat_nat @ R2 ) @ product_fst_nat_nat @ Z )
= ( comp_n8456906485034819078at_nat @ ( equiv_proj_nat_nat @ R2 ) @ product_snd_nat_nat @ Z ) ) ) ) ).
% equiv_proj
thf(fact_661_SUP__cong,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,C2: product_prod_nat_nat > set_nat,D: product_prod_nat_nat > set_nat] :
( ( A4 = B5 )
=> ( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ B5 )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ C2 @ A4 ) )
= ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ D @ B5 ) ) ) ) ) ).
% SUP_cong
thf(fact_662_SUP__cong,axiom,
! [A4: set_nat,B5: set_nat,C2: nat > set_nat,D: nat > set_nat] :
( ( A4 = B5 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B5 )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C2 @ A4 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ D @ B5 ) ) ) ) ) ).
% SUP_cong
thf(fact_663_Inf_OINF__image,axiom,
! [Inf: set_set_nat > set_nat,G: set_nat > set_nat,F2: nat > set_nat,A4: set_nat] :
( ( Inf @ ( image_7916887816326733075et_nat @ G @ ( image_nat_set_nat @ F2 @ A4 ) ) )
= ( Inf @ ( image_nat_set_nat @ ( comp_s3433241188411525313at_nat @ G @ F2 ) @ A4 ) ) ) ).
% Inf.INF_image
thf(fact_664_Inf_OINF__image,axiom,
! [Inf: set_set_nat > set_nat,G: nat > set_nat,F2: nat > nat,A4: set_nat] :
( ( Inf @ ( image_nat_set_nat @ G @ ( image_nat_nat2 @ F2 @ A4 ) ) )
= ( Inf @ ( image_nat_set_nat @ ( comp_nat_set_nat_nat @ G @ F2 ) @ A4 ) ) ) ).
% Inf.INF_image
thf(fact_665_Sup_OSUP__image,axiom,
! [Sup: set_set_nat > set_nat,G: set_nat > set_nat,F2: nat > set_nat,A4: set_nat] :
( ( Sup @ ( image_7916887816326733075et_nat @ G @ ( image_nat_set_nat @ F2 @ A4 ) ) )
= ( Sup @ ( image_nat_set_nat @ ( comp_s3433241188411525313at_nat @ G @ F2 ) @ A4 ) ) ) ).
% Sup.SUP_image
thf(fact_666_Sup_OSUP__image,axiom,
! [Sup: set_set_nat > set_nat,G: nat > set_nat,F2: nat > nat,A4: set_nat] :
( ( Sup @ ( image_nat_set_nat @ G @ ( image_nat_nat2 @ F2 @ A4 ) ) )
= ( Sup @ ( image_nat_set_nat @ ( comp_nat_set_nat_nat @ G @ F2 ) @ A4 ) ) ) ).
% Sup.SUP_image
thf(fact_667_Union__empty__conv,axiom,
! [A4: set_set_o] :
( ( ( comple90263536869209701_set_o @ A4 )
= bot_bot_set_o )
= ( ! [X4: set_o] :
( ( member_set_o @ X4 @ A4 )
=> ( X4 = bot_bot_set_o ) ) ) ) ).
% Union_empty_conv
thf(fact_668_Union__empty__conv,axiom,
! [A4: set_set_nat] :
( ( ( comple7399068483239264473et_nat @ A4 )
= bot_bot_set_nat )
= ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A4 )
=> ( X4 = bot_bot_set_nat ) ) ) ) ).
% Union_empty_conv
thf(fact_669_empty__Union__conv,axiom,
! [A4: set_set_o] :
( ( bot_bot_set_o
= ( comple90263536869209701_set_o @ A4 ) )
= ( ! [X4: set_o] :
( ( member_set_o @ X4 @ A4 )
=> ( X4 = bot_bot_set_o ) ) ) ) ).
% empty_Union_conv
thf(fact_670_empty__Union__conv,axiom,
! [A4: set_set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ A4 ) )
= ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A4 )
=> ( X4 = bot_bot_set_nat ) ) ) ) ).
% empty_Union_conv
thf(fact_671_SUP__commute,axiom,
! [F2: nat > nat > set_nat,B5: set_nat,A4: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ ( F2 @ I ) @ B5 ) )
@ A4 ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [J: nat] :
( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I: nat] : ( F2 @ I @ J )
@ A4 ) )
@ B5 ) ) ) ).
% SUP_commute
thf(fact_672_UN__UN__flatten,axiom,
! [C2: nat > set_nat,B5: nat > set_nat,A4: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ A4 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y5: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C2 @ ( B5 @ Y5 ) ) )
@ A4 ) ) ) ).
% UN_UN_flatten
thf(fact_673_UN__E,axiom,
! [B: product_prod_nat_nat,B5: product_prod_nat_nat > set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ B @ ( comple5685304695842803022at_nat @ ( image_5423882813909314213at_nat @ B5 @ A4 ) ) )
=> ~ ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A4 )
=> ~ ( member8440522571783428010at_nat @ B @ ( B5 @ X3 ) ) ) ) ).
% UN_E
thf(fact_674_UN__E,axiom,
! [B: product_prod_nat_nat,B5: nat > set_Pr1261947904930325089at_nat,A4: set_nat] :
( ( member8440522571783428010at_nat @ B @ ( comple5685304695842803022at_nat @ ( image_7178329752028323786at_nat @ B5 @ A4 ) ) )
=> ~ ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ~ ( member8440522571783428010at_nat @ B @ ( B5 @ X3 ) ) ) ) ).
% UN_E
thf(fact_675_UN__E,axiom,
! [B: nat,B5: product_prod_nat_nat > set_nat,A4: set_Pr1261947904930325089at_nat] :
( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ B5 @ A4 ) ) )
=> ~ ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A4 )
=> ~ ( member_nat @ B @ ( B5 @ X3 ) ) ) ) ).
% UN_E
thf(fact_676_UN__E,axiom,
! [B: nat,B5: nat > set_nat,A4: set_nat] :
( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ A4 ) ) )
=> ~ ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ~ ( member_nat @ B @ ( B5 @ X3 ) ) ) ) ).
% UN_E
thf(fact_677_UN__extend__simps_I9_J,axiom,
! [C2: nat > set_nat,B5: nat > set_nat,A4: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X4: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C2 @ ( B5 @ X4 ) ) )
@ A4 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ A4 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_678_SUP__eq,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > set_nat,G: product_prod_nat_nat > set_nat] :
( ! [I3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I3 @ A4 )
=> ? [X6: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X6 @ B5 )
& ( ord_less_eq_set_nat @ ( F2 @ I3 ) @ ( G @ X6 ) ) ) )
=> ( ! [J3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ J3 @ B5 )
=> ? [X6: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X6 @ A4 )
& ( ord_less_eq_set_nat @ ( G @ J3 ) @ ( F2 @ X6 ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F2 @ A4 ) )
= ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ G @ B5 ) ) ) ) ) ).
% SUP_eq
thf(fact_679_SUP__eq,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_nat,F2: product_prod_nat_nat > set_nat,G: nat > set_nat] :
( ! [I3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I3 @ A4 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B5 )
& ( ord_less_eq_set_nat @ ( F2 @ I3 ) @ ( G @ X6 ) ) ) )
=> ( ! [J3: nat] :
( ( member_nat @ J3 @ B5 )
=> ? [X6: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X6 @ A4 )
& ( ord_less_eq_set_nat @ ( G @ J3 ) @ ( F2 @ X6 ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F2 @ A4 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B5 ) ) ) ) ) ).
% SUP_eq
thf(fact_680_SUP__eq,axiom,
! [A4: set_nat,B5: set_Pr1261947904930325089at_nat,F2: nat > set_nat,G: product_prod_nat_nat > set_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A4 )
=> ? [X6: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X6 @ B5 )
& ( ord_less_eq_set_nat @ ( F2 @ I3 ) @ ( G @ X6 ) ) ) )
=> ( ! [J3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ J3 @ B5 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A4 )
& ( ord_less_eq_set_nat @ ( G @ J3 ) @ ( F2 @ X6 ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ A4 ) )
= ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ G @ B5 ) ) ) ) ) ).
% SUP_eq
thf(fact_681_SUP__eq,axiom,
! [A4: set_nat,B5: set_nat,F2: nat > set_nat,G: nat > set_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A4 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B5 )
& ( ord_less_eq_set_nat @ ( F2 @ I3 ) @ ( G @ X6 ) ) ) )
=> ( ! [J3: nat] :
( ( member_nat @ J3 @ B5 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A4 )
& ( ord_less_eq_set_nat @ ( G @ J3 ) @ ( F2 @ X6 ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ A4 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B5 ) ) ) ) ) ).
% SUP_eq
thf(fact_682_less__eq__Sup,axiom,
! [A4: set_o,U2: $o] :
( ! [V: $o] :
( ( member_o @ V @ A4 )
=> ( ord_less_eq_o @ U2 @ V ) )
=> ( ( A4 != bot_bot_set_o )
=> ( ord_less_eq_o @ U2 @ ( complete_Sup_Sup_o @ A4 ) ) ) ) ).
% less_eq_Sup
thf(fact_683_less__eq__Sup,axiom,
! [A4: set_set_nat,U2: set_nat] :
( ! [V: set_nat] :
( ( member_set_nat @ V @ A4 )
=> ( ord_less_eq_set_nat @ U2 @ V ) )
=> ( ( A4 != bot_bot_set_set_nat )
=> ( ord_less_eq_set_nat @ U2 @ ( comple7399068483239264473et_nat @ A4 ) ) ) ) ).
% less_eq_Sup
thf(fact_684_cSup__least,axiom,
! [X5: set_nat,Z: nat] :
( ( X5 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X5 )
=> ( ord_less_eq_nat @ X3 @ Z ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ X5 ) @ Z ) ) ) ).
% cSup_least
thf(fact_685_cSup__least,axiom,
! [X5: set_o,Z: $o] :
( ( X5 != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ X5 )
=> ( ord_less_eq_o @ X3 @ Z ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ X5 ) @ Z ) ) ) ).
% cSup_least
thf(fact_686_cSup__least,axiom,
! [X5: set_set_nat,Z: set_nat] :
( ( X5 != bot_bot_set_set_nat )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ X5 )
=> ( ord_less_eq_set_nat @ X3 @ Z ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ X5 ) @ Z ) ) ) ).
% cSup_least
thf(fact_687_cSup__eq__non__empty,axiom,
! [X5: set_nat,A: nat] :
( ( X5 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X5 )
=> ( ord_less_eq_nat @ X3 @ A ) )
=> ( ! [Y: nat] :
( ! [X6: nat] :
( ( member_nat @ X6 @ X5 )
=> ( ord_less_eq_nat @ X6 @ Y ) )
=> ( ord_less_eq_nat @ A @ Y ) )
=> ( ( complete_Sup_Sup_nat @ X5 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_688_cSup__eq__non__empty,axiom,
! [X5: set_o,A: $o] :
( ( X5 != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ X5 )
=> ( ord_less_eq_o @ X3 @ A ) )
=> ( ! [Y: $o] :
( ! [X6: $o] :
( ( member_o @ X6 @ X5 )
=> ( ord_less_eq_o @ X6 @ Y ) )
=> ( ord_less_eq_o @ A @ Y ) )
=> ( ( complete_Sup_Sup_o @ X5 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_689_cSup__eq__non__empty,axiom,
! [X5: set_set_nat,A: set_nat] :
( ( X5 != bot_bot_set_set_nat )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ X5 )
=> ( ord_less_eq_set_nat @ X3 @ A ) )
=> ( ! [Y: set_nat] :
( ! [X6: set_nat] :
( ( member_set_nat @ X6 @ X5 )
=> ( ord_less_eq_set_nat @ X6 @ Y ) )
=> ( ord_less_eq_set_nat @ A @ Y ) )
=> ( ( comple7399068483239264473et_nat @ X5 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_690_SUP__eq__const,axiom,
! [I2: set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > set_nat,X: set_nat] :
( ( I2 != bot_bo2099793752762293965at_nat )
=> ( ! [I3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I3 @ I2 )
=> ( ( F2 @ I3 )
= X ) )
=> ( ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F2 @ I2 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_691_SUP__eq__const,axiom,
! [I2: set_nat,F2: nat > set_nat,X: set_nat] :
( ( I2 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I2 )
=> ( ( F2 @ I3 )
= X ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ I2 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_692_SUP__eq__const,axiom,
! [I2: set_o,F2: $o > set_nat,X: set_nat] :
( ( I2 != bot_bot_set_o )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I2 )
=> ( ( F2 @ I3 )
= X ) )
=> ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F2 @ I2 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_693_SUP__image,axiom,
! [G: set_nat > set_nat,F2: nat > set_nat,A4: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ G @ ( image_nat_set_nat @ F2 @ A4 ) ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ ( comp_s3433241188411525313at_nat @ G @ F2 ) @ A4 ) ) ) ).
% SUP_image
thf(fact_694_SUP__image,axiom,
! [G: nat > set_nat,F2: nat > nat,A4: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ ( image_nat_nat2 @ F2 @ A4 ) ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ ( comp_nat_set_nat_nat @ G @ F2 ) @ A4 ) ) ) ).
% SUP_image
thf(fact_695_Union__disjoint,axiom,
! [C2: set_set_o,A4: set_o] :
( ( ( inf_inf_set_o @ ( comple90263536869209701_set_o @ C2 ) @ A4 )
= bot_bot_set_o )
= ( ! [X4: set_o] :
( ( member_set_o @ X4 @ C2 )
=> ( ( inf_inf_set_o @ X4 @ A4 )
= bot_bot_set_o ) ) ) ) ).
% Union_disjoint
thf(fact_696_Union__disjoint,axiom,
! [C2: set_set_nat,A4: set_nat] :
( ( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ C2 ) @ A4 )
= bot_bot_set_nat )
= ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ C2 )
=> ( ( inf_inf_set_nat @ X4 @ A4 )
= bot_bot_set_nat ) ) ) ) ).
% Union_disjoint
thf(fact_697_Union__empty,axiom,
( ( comple90263536869209701_set_o @ bot_bot_set_set_o )
= bot_bot_set_o ) ).
% Union_empty
thf(fact_698_Union__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% Union_empty
thf(fact_699_SUP__eqI,axiom,
! [A4: set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > set_nat,X: set_nat] :
( ! [I3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I3 @ A4 )
=> ( ord_less_eq_set_nat @ ( F2 @ I3 ) @ X ) )
=> ( ! [Y: set_nat] :
( ! [I4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I4 @ A4 )
=> ( ord_less_eq_set_nat @ ( F2 @ I4 ) @ Y ) )
=> ( ord_less_eq_set_nat @ X @ Y ) )
=> ( ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F2 @ A4 ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_700_SUP__eqI,axiom,
! [A4: set_nat,F2: nat > set_nat,X: set_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A4 )
=> ( ord_less_eq_set_nat @ ( F2 @ I3 ) @ X ) )
=> ( ! [Y: set_nat] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A4 )
=> ( ord_less_eq_set_nat @ ( F2 @ I4 ) @ Y ) )
=> ( ord_less_eq_set_nat @ X @ Y ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ A4 ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_701_SUP__mono,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_nat,F2: product_prod_nat_nat > set_nat,G: nat > set_nat] :
( ! [N3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ N3 @ A4 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B5 )
& ( ord_less_eq_set_nat @ ( F2 @ N3 ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F2 @ A4 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B5 ) ) ) ) ).
% SUP_mono
thf(fact_702_SUP__mono,axiom,
! [A4: set_nat,B5: set_nat,F2: nat > set_nat,G: nat > set_nat] :
( ! [N3: nat] :
( ( member_nat @ N3 @ A4 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B5 )
& ( ord_less_eq_set_nat @ ( F2 @ N3 ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ A4 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B5 ) ) ) ) ).
% SUP_mono
thf(fact_703_SUP__least,axiom,
! [A4: set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > set_nat,U2: set_nat] :
( ! [I3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I3 @ A4 )
=> ( ord_less_eq_set_nat @ ( F2 @ I3 ) @ U2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F2 @ A4 ) ) @ U2 ) ) ).
% SUP_least
thf(fact_704_SUP__least,axiom,
! [A4: set_nat,F2: nat > set_nat,U2: set_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A4 )
=> ( ord_less_eq_set_nat @ ( F2 @ I3 ) @ U2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ A4 ) ) @ U2 ) ) ).
% SUP_least
thf(fact_705_SUP__mono_H,axiom,
! [F2: nat > set_nat,G: nat > set_nat,A4: set_nat] :
( ! [X3: nat] : ( ord_less_eq_set_nat @ ( F2 @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ A4 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ A4 ) ) ) ) ).
% SUP_mono'
thf(fact_706_SUP__upper,axiom,
! [I5: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > set_nat] :
( ( member8440522571783428010at_nat @ I5 @ A4 )
=> ( ord_less_eq_set_nat @ ( F2 @ I5 ) @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F2 @ A4 ) ) ) ) ).
% SUP_upper
thf(fact_707_SUP__upper,axiom,
! [I5: nat,A4: set_nat,F2: nat > set_nat] :
( ( member_nat @ I5 @ A4 )
=> ( ord_less_eq_set_nat @ ( F2 @ I5 ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ A4 ) ) ) ) ).
% SUP_upper
thf(fact_708_SUP__le__iff,axiom,
! [F2: nat > set_nat,A4: set_nat,U2: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ A4 ) ) @ U2 )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_set_nat @ ( F2 @ X4 ) @ U2 ) ) ) ) ).
% SUP_le_iff
thf(fact_709_SUP__upper2,axiom,
! [I5: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,U2: set_nat,F2: product_prod_nat_nat > set_nat] :
( ( member8440522571783428010at_nat @ I5 @ A4 )
=> ( ( ord_less_eq_set_nat @ U2 @ ( F2 @ I5 ) )
=> ( ord_less_eq_set_nat @ U2 @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F2 @ A4 ) ) ) ) ) ).
% SUP_upper2
thf(fact_710_SUP__upper2,axiom,
! [I5: nat,A4: set_nat,U2: set_nat,F2: nat > set_nat] :
( ( member_nat @ I5 @ A4 )
=> ( ( ord_less_eq_set_nat @ U2 @ ( F2 @ I5 ) )
=> ( ord_less_eq_set_nat @ U2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ A4 ) ) ) ) ) ).
% SUP_upper2
thf(fact_711_image__UN,axiom,
! [F2: nat > set_nat,B5: nat > set_nat,A4: set_nat] :
( ( image_nat_set_nat @ F2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ A4 ) ) )
= ( comple548664676211718543et_nat
@ ( image_2194112158459175443et_nat
@ ^ [X4: nat] : ( image_nat_set_nat @ F2 @ ( B5 @ X4 ) )
@ A4 ) ) ) ).
% image_UN
thf(fact_712_image__UN,axiom,
! [F2: nat > nat,B5: nat > set_nat,A4: set_nat] :
( ( image_nat_nat2 @ F2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ A4 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X4: nat] : ( image_nat_nat2 @ F2 @ ( B5 @ X4 ) )
@ A4 ) ) ) ).
% image_UN
thf(fact_713_UN__extend__simps_I10_J,axiom,
! [B5: set_nat > set_nat,F2: nat > set_nat,A4: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [A5: nat] : ( B5 @ ( F2 @ A5 ) )
@ A4 ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B5 @ ( image_nat_set_nat @ F2 @ A4 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_714_UN__extend__simps_I10_J,axiom,
! [B5: nat > set_nat,F2: nat > nat,A4: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [A5: nat] : ( B5 @ ( F2 @ A5 ) )
@ A4 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ ( image_nat_nat2 @ F2 @ A4 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_715_UNION__empty__conv_I2_J,axiom,
! [B5: nat > set_nat,A4: set_nat] :
( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ A4 ) )
= bot_bot_set_nat )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ( B5 @ X4 )
= bot_bot_set_nat ) ) ) ) ).
% UNION_empty_conv(2)
thf(fact_716_UNION__empty__conv_I1_J,axiom,
! [B5: nat > set_nat,A4: set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ A4 ) ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ( B5 @ X4 )
= bot_bot_set_nat ) ) ) ) ).
% UNION_empty_conv(1)
thf(fact_717_UN__empty,axiom,
! [B5: nat > set_o] :
( ( comple90263536869209701_set_o @ ( image_nat_set_o @ B5 @ bot_bot_set_nat ) )
= bot_bot_set_o ) ).
% UN_empty
thf(fact_718_UN__empty,axiom,
! [B5: $o > set_o] :
( ( comple90263536869209701_set_o @ ( image_o_set_o @ B5 @ bot_bot_set_o ) )
= bot_bot_set_o ) ).
% UN_empty
thf(fact_719_UN__empty,axiom,
! [B5: nat > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ bot_bot_set_nat ) )
= bot_bot_set_nat ) ).
% UN_empty
thf(fact_720_UN__empty,axiom,
! [B5: $o > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B5 @ bot_bot_set_o ) )
= bot_bot_set_nat ) ).
% UN_empty
thf(fact_721_UN__empty2,axiom,
! [A4: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X4: nat] : bot_bot_set_nat
@ A4 ) )
= bot_bot_set_nat ) ).
% UN_empty2
thf(fact_722_UN__subset__iff,axiom,
! [A4: nat > set_nat,I2: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A4 @ I2 ) ) @ B5 )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ I2 )
=> ( ord_less_eq_set_nat @ ( A4 @ X4 ) @ B5 ) ) ) ) ).
% UN_subset_iff
thf(fact_723_UN__upper,axiom,
! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_nat] :
( ( member8440522571783428010at_nat @ A @ A4 )
=> ( ord_less_eq_set_nat @ ( B5 @ A ) @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ B5 @ A4 ) ) ) ) ).
% UN_upper
thf(fact_724_UN__upper,axiom,
! [A: nat,A4: set_nat,B5: nat > set_nat] :
( ( member_nat @ A @ A4 )
=> ( ord_less_eq_set_nat @ ( B5 @ A ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ A4 ) ) ) ) ).
% UN_upper
thf(fact_725_UN__least,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_nat,C2: set_nat] :
( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A4 )
=> ( ord_less_eq_set_nat @ ( B5 @ X3 ) @ C2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ B5 @ A4 ) ) @ C2 ) ) ).
% UN_least
thf(fact_726_UN__least,axiom,
! [A4: set_nat,B5: nat > set_nat,C2: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ord_less_eq_set_nat @ ( B5 @ X3 ) @ C2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ A4 ) ) @ C2 ) ) ).
% UN_least
thf(fact_727_UN__mono,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > set_nat,G: product_prod_nat_nat > set_nat] :
( ( ord_le3146513528884898305at_nat @ A4 @ B5 )
=> ( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A4 )
=> ( ord_less_eq_set_nat @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F2 @ A4 ) ) @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ G @ B5 ) ) ) ) ) ).
% UN_mono
thf(fact_728_UN__mono,axiom,
! [A4: set_nat,B5: set_nat,F2: nat > set_nat,G: nat > set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ord_less_eq_set_nat @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ A4 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B5 ) ) ) ) ) ).
% UN_mono
thf(fact_729_UN__insert__distrib,axiom,
! [U2: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,A: $o,B5: product_prod_nat_nat > set_o] :
( ( member8440522571783428010at_nat @ U2 @ A4 )
=> ( ( comple90263536869209701_set_o
@ ( image_501298686397505212_set_o
@ ^ [X4: product_prod_nat_nat] : ( insert_o @ A @ ( B5 @ X4 ) )
@ A4 ) )
= ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_501298686397505212_set_o @ B5 @ A4 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_730_UN__insert__distrib,axiom,
! [U2: nat,A4: set_nat,A: $o,B5: nat > set_o] :
( ( member_nat @ U2 @ A4 )
=> ( ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [X4: nat] : ( insert_o @ A @ ( B5 @ X4 ) )
@ A4 ) )
= ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B5 @ A4 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_731_UN__insert__distrib,axiom,
! [U2: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,A: nat,B5: product_prod_nat_nat > set_nat] :
( ( member8440522571783428010at_nat @ U2 @ A4 )
=> ( ( comple7399068483239264473et_nat
@ ( image_15824709712370754et_nat
@ ^ [X4: product_prod_nat_nat] : ( insert_nat @ A @ ( B5 @ X4 ) )
@ A4 ) )
= ( insert_nat @ A @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ B5 @ A4 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_732_UN__insert__distrib,axiom,
! [U2: nat,A4: set_nat,A: nat,B5: nat > set_nat] :
( ( member_nat @ U2 @ A4 )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X4: nat] : ( insert_nat @ A @ ( B5 @ X4 ) )
@ A4 ) )
= ( insert_nat @ A @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ A4 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_733_Int__UN__distrib2,axiom,
! [A4: nat > set_nat,I2: set_nat,B5: nat > set_nat,J2: set_nat] :
( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A4 @ I2 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ J2 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I: nat] :
( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [J: nat] : ( inf_inf_set_nat @ ( A4 @ I ) @ ( B5 @ J ) )
@ J2 ) )
@ I2 ) ) ) ).
% Int_UN_distrib2
thf(fact_734_Int__UN__distrib,axiom,
! [B5: set_nat,A4: nat > set_nat,I2: set_nat] :
( ( inf_inf_set_nat @ B5 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A4 @ I2 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I: nat] : ( inf_inf_set_nat @ B5 @ ( A4 @ I ) )
@ I2 ) ) ) ).
% Int_UN_distrib
thf(fact_735_UN__extend__simps_I4_J,axiom,
! [A4: nat > set_nat,C2: set_nat,B5: set_nat] :
( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A4 @ C2 ) ) @ B5 )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X4: nat] : ( inf_inf_set_nat @ ( A4 @ X4 ) @ B5 )
@ C2 ) ) ) ).
% UN_extend_simps(4)
thf(fact_736_UN__extend__simps_I5_J,axiom,
! [A4: set_nat,B5: nat > set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ A4 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ C2 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X4: nat] : ( inf_inf_set_nat @ A4 @ ( B5 @ X4 ) )
@ C2 ) ) ) ).
% UN_extend_simps(5)
thf(fact_737_image__Union,axiom,
! [F2: nat > set_nat,S3: set_set_nat] :
( ( image_nat_set_nat @ F2 @ ( comple7399068483239264473et_nat @ S3 ) )
= ( comple548664676211718543et_nat @ ( image_6725021117256019401et_nat @ ( image_nat_set_nat @ F2 ) @ S3 ) ) ) ).
% image_Union
thf(fact_738_image__Union,axiom,
! [F2: nat > nat,S3: set_set_nat] :
( ( image_nat_nat2 @ F2 @ ( comple7399068483239264473et_nat @ S3 ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat2 @ F2 ) @ S3 ) ) ) ).
% image_Union
thf(fact_739_vimage__UN,axiom,
! [F2: nat > nat,B5: nat > set_nat,A4: set_nat] :
( ( vimage_nat_nat @ F2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ A4 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X4: nat] : ( vimage_nat_nat @ F2 @ ( B5 @ X4 ) )
@ A4 ) ) ) ).
% vimage_UN
thf(fact_740_Int__Union2,axiom,
! [B5: set_set_nat,A4: set_nat] :
( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ B5 ) @ A4 )
= ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [C3: set_nat] : ( inf_inf_set_nat @ C3 @ A4 )
@ B5 ) ) ) ).
% Int_Union2
thf(fact_741_Int__Union,axiom,
! [A4: set_nat,B5: set_set_nat] :
( ( inf_inf_set_nat @ A4 @ ( comple7399068483239264473et_nat @ B5 ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ ( inf_inf_set_nat @ A4 ) @ B5 ) ) ) ).
% Int_Union
thf(fact_742_UN__extend__simps_I8_J,axiom,
! [B5: nat > set_nat,A4: set_set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [Y5: set_nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ Y5 ) )
@ A4 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ ( comple7399068483239264473et_nat @ A4 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_743_vimage__Union,axiom,
! [F2: nat > nat,A4: set_set_nat] :
( ( vimage_nat_nat @ F2 @ ( comple7399068483239264473et_nat @ A4 ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ ( vimage_nat_nat @ F2 ) @ A4 ) ) ) ).
% vimage_Union
thf(fact_744_SUP__eq__iff,axiom,
! [I2: set_Pr1261947904930325089at_nat,C: set_nat,F2: product_prod_nat_nat > set_nat] :
( ( I2 != bot_bo2099793752762293965at_nat )
=> ( ! [I3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I3 @ I2 )
=> ( ord_less_eq_set_nat @ C @ ( F2 @ I3 ) ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F2 @ I2 ) )
= C )
= ( ! [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ I2 )
=> ( ( F2 @ X4 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_745_SUP__eq__iff,axiom,
! [I2: set_nat,C: set_nat,F2: nat > set_nat] :
( ( I2 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I2 )
=> ( ord_less_eq_set_nat @ C @ ( F2 @ I3 ) ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ I2 ) )
= C )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ I2 )
=> ( ( F2 @ X4 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_746_SUP__eq__iff,axiom,
! [I2: set_o,C: set_nat,F2: $o > set_nat] :
( ( I2 != bot_bot_set_o )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I2 )
=> ( ord_less_eq_set_nat @ C @ ( F2 @ I3 ) ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F2 @ I2 ) )
= C )
= ( ! [X4: $o] :
( ( member_o @ X4 @ I2 )
=> ( ( F2 @ X4 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_747_cSUP__least,axiom,
! [A4: set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > set_nat,M: set_nat] :
( ( A4 != bot_bo2099793752762293965at_nat )
=> ( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A4 )
=> ( ord_less_eq_set_nat @ ( F2 @ X3 ) @ M ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F2 @ A4 ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_748_cSUP__least,axiom,
! [A4: set_nat,F2: nat > set_nat,M: set_nat] :
( ( A4 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ord_less_eq_set_nat @ ( F2 @ X3 ) @ M ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ A4 ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_749_cSUP__least,axiom,
! [A4: set_o,F2: $o > set_nat,M: set_nat] :
( ( A4 != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A4 )
=> ( ord_less_eq_set_nat @ ( F2 @ X3 ) @ M ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F2 @ A4 ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_750_SUP__subset__mono,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > set_nat,G: product_prod_nat_nat > set_nat] :
( ( ord_le3146513528884898305at_nat @ A4 @ B5 )
=> ( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A4 )
=> ( ord_less_eq_set_nat @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F2 @ A4 ) ) @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ G @ B5 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_751_SUP__subset__mono,axiom,
! [A4: set_nat,B5: set_nat,F2: nat > set_nat,G: nat > set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ord_less_eq_set_nat @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ A4 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B5 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_752_SUP__empty,axiom,
! [F2: nat > set_o] :
( ( comple90263536869209701_set_o @ ( image_nat_set_o @ F2 @ bot_bot_set_nat ) )
= bot_bot_set_o ) ).
% SUP_empty
thf(fact_753_SUP__empty,axiom,
! [F2: $o > set_o] :
( ( comple90263536869209701_set_o @ ( image_o_set_o @ F2 @ bot_bot_set_o ) )
= bot_bot_set_o ) ).
% SUP_empty
thf(fact_754_SUP__empty,axiom,
! [F2: nat > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ bot_bot_set_nat ) )
= bot_bot_set_nat ) ).
% SUP_empty
thf(fact_755_SUP__empty,axiom,
! [F2: $o > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F2 @ bot_bot_set_o ) )
= bot_bot_set_nat ) ).
% SUP_empty
thf(fact_756_SUP__constant,axiom,
! [A4: set_nat,C: set_o] :
( ( ( A4 = bot_bot_set_nat )
=> ( ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [Y5: nat] : C
@ A4 ) )
= bot_bot_set_o ) )
& ( ( A4 != bot_bot_set_nat )
=> ( ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [Y5: nat] : C
@ A4 ) )
= C ) ) ) ).
% SUP_constant
thf(fact_757_SUP__constant,axiom,
! [A4: set_o,C: set_o] :
( ( ( A4 = bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [Y5: $o] : C
@ A4 ) )
= bot_bot_set_o ) )
& ( ( A4 != bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [Y5: $o] : C
@ A4 ) )
= C ) ) ) ).
% SUP_constant
thf(fact_758_SUP__constant,axiom,
! [A4: set_nat,C: set_nat] :
( ( ( A4 = bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y5: nat] : C
@ A4 ) )
= bot_bot_set_nat ) )
& ( ( A4 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y5: nat] : C
@ A4 ) )
= C ) ) ) ).
% SUP_constant
thf(fact_759_SUP__constant,axiom,
! [A4: set_o,C: set_nat] :
( ( ( A4 = bot_bot_set_o )
=> ( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [Y5: $o] : C
@ A4 ) )
= bot_bot_set_nat ) )
& ( ( A4 != bot_bot_set_o )
=> ( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [Y5: $o] : C
@ A4 ) )
= C ) ) ) ).
% SUP_constant
thf(fact_760_UN__extend__simps_I1_J,axiom,
! [C2: set_nat,A: $o,B5: nat > set_o] :
( ( ( C2 = bot_bot_set_nat )
=> ( ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B5 @ C2 ) ) )
= ( insert_o @ A @ bot_bot_set_o ) ) )
& ( ( C2 != bot_bot_set_nat )
=> ( ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B5 @ C2 ) ) )
= ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [X4: nat] : ( insert_o @ A @ ( B5 @ X4 ) )
@ C2 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_761_UN__extend__simps_I1_J,axiom,
! [C2: set_o,A: $o,B5: $o > set_o] :
( ( ( C2 = bot_bot_set_o )
=> ( ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B5 @ C2 ) ) )
= ( insert_o @ A @ bot_bot_set_o ) ) )
& ( ( C2 != bot_bot_set_o )
=> ( ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B5 @ C2 ) ) )
= ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X4: $o] : ( insert_o @ A @ ( B5 @ X4 ) )
@ C2 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_762_UN__extend__simps_I1_J,axiom,
! [C2: set_nat,A: nat,B5: nat > set_nat] :
( ( ( C2 = bot_bot_set_nat )
=> ( ( insert_nat @ A @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ C2 ) ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) )
& ( ( C2 != bot_bot_set_nat )
=> ( ( insert_nat @ A @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ C2 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X4: nat] : ( insert_nat @ A @ ( B5 @ X4 ) )
@ C2 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_763_UN__extend__simps_I1_J,axiom,
! [C2: set_o,A: nat,B5: $o > set_nat] :
( ( ( C2 = bot_bot_set_o )
=> ( ( insert_nat @ A @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B5 @ C2 ) ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) )
& ( ( C2 != bot_bot_set_o )
=> ( ( insert_nat @ A @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B5 @ C2 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [X4: $o] : ( insert_nat @ A @ ( B5 @ X4 ) )
@ C2 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_764_SUP__UNION,axiom,
! [F2: nat > set_nat,G: nat > set_nat,A4: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ A4 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y5: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ ( G @ Y5 ) ) )
@ A4 ) ) ) ).
% SUP_UNION
thf(fact_765_UNION__singleton__eq__range,axiom,
! [F2: nat > set_nat,A4: set_nat] :
( ( comple548664676211718543et_nat
@ ( image_2194112158459175443et_nat
@ ^ [X4: nat] : ( insert_set_nat @ ( F2 @ X4 ) @ bot_bot_set_set_nat )
@ A4 ) )
= ( image_nat_set_nat @ F2 @ A4 ) ) ).
% UNION_singleton_eq_range
thf(fact_766_UNION__singleton__eq__range,axiom,
! [F2: nat > nat,A4: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X4: nat] : ( insert_nat @ ( F2 @ X4 ) @ bot_bot_set_nat )
@ A4 ) )
= ( image_nat_nat2 @ F2 @ A4 ) ) ).
% UNION_singleton_eq_range
thf(fact_767_vimage__eq__UN,axiom,
( vimage_nat_nat
= ( ^ [F4: nat > nat,B7: set_nat] :
( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y5: nat] : ( vimage_nat_nat @ F4 @ ( insert_nat @ Y5 @ bot_bot_set_nat ) )
@ B7 ) ) ) ) ).
% vimage_eq_UN
thf(fact_768_vimage__eq__UN,axiom,
( vimage_nat_o
= ( ^ [F4: nat > $o,B7: set_o] :
( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [Y5: $o] : ( vimage_nat_o @ F4 @ ( insert_o @ Y5 @ bot_bot_set_o ) )
@ B7 ) ) ) ) ).
% vimage_eq_UN
thf(fact_769_ccpo__Sup__singleton,axiom,
! [X: $o] :
( ( complete_Sup_Sup_o @ ( insert_o @ X @ bot_bot_set_o ) )
= X ) ).
% ccpo_Sup_singleton
thf(fact_770_ccpo__Sup__singleton,axiom,
! [X: set_nat] :
( ( comple7399068483239264473et_nat @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
= X ) ).
% ccpo_Sup_singleton
thf(fact_771_Union__natural,axiom,
! [F2: nat > set_nat] :
( ( comp_s5357857527062720591et_nat @ comple548664676211718543et_nat @ ( image_6725021117256019401et_nat @ ( image_nat_set_nat @ F2 ) ) )
= ( comp_s1376360633444582883et_nat @ ( image_nat_set_nat @ F2 ) @ comple7399068483239264473et_nat ) ) ).
% Union_natural
thf(fact_772_Union__natural,axiom,
! [F2: nat > nat] :
( ( comp_s174380336271864291et_nat @ comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat2 @ F2 ) ) )
= ( comp_s6993074788030935341et_nat @ ( image_nat_nat2 @ F2 ) @ comple7399068483239264473et_nat ) ) ).
% Union_natural
thf(fact_773_SUP__inf,axiom,
! [F2: nat > set_nat,B5: set_nat,A: set_nat] :
( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ B5 ) ) @ A )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [B4: nat] : ( inf_inf_set_nat @ ( F2 @ B4 ) @ A )
@ B5 ) ) ) ).
% SUP_inf
thf(fact_774_Sup__inf,axiom,
! [B5: set_set_nat,A: set_nat] :
( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ B5 ) @ A )
= ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [B4: set_nat] : ( inf_inf_set_nat @ B4 @ A )
@ B5 ) ) ) ).
% Sup_inf
thf(fact_775_Sup__inf__eq__bot__iff,axiom,
! [B5: set_set_o,A: set_o] :
( ( ( inf_inf_set_o @ ( comple90263536869209701_set_o @ B5 ) @ A )
= bot_bot_set_o )
= ( ! [X4: set_o] :
( ( member_set_o @ X4 @ B5 )
=> ( ( inf_inf_set_o @ X4 @ A )
= bot_bot_set_o ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_776_Sup__inf__eq__bot__iff,axiom,
! [B5: set_set_nat,A: set_nat] :
( ( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ B5 ) @ A )
= bot_bot_set_nat )
= ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ B5 )
=> ( ( inf_inf_set_nat @ X4 @ A )
= bot_bot_set_nat ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_777_inf__Sup,axiom,
! [A: set_nat,B5: set_set_nat] :
( ( inf_inf_set_nat @ A @ ( comple7399068483239264473et_nat @ B5 ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ ( inf_inf_set_nat @ A ) @ B5 ) ) ) ).
% inf_Sup
thf(fact_778_SUP__inf__distrib2,axiom,
! [F2: nat > set_nat,A4: set_nat,G: nat > set_nat,B5: set_nat] :
( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ A4 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B5 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [A5: nat] :
( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [B4: nat] : ( inf_inf_set_nat @ ( F2 @ A5 ) @ ( G @ B4 ) )
@ B5 ) )
@ A4 ) ) ) ).
% SUP_inf_distrib2
thf(fact_779_inf__SUP,axiom,
! [A: set_nat,F2: nat > set_nat,B5: set_nat] :
( ( inf_inf_set_nat @ A @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ B5 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [B4: nat] : ( inf_inf_set_nat @ A @ ( F2 @ B4 ) )
@ B5 ) ) ) ).
% inf_SUP
thf(fact_780_comp__image,axiom,
! [F2: set_nat > set_nat,G: nat > set_nat] :
( ( comp_s8515515581831355875et_nat @ ( image_7916887816326733075et_nat @ F2 ) @ ( image_nat_set_nat @ G ) )
= ( image_nat_set_nat @ ( comp_s3433241188411525313at_nat @ F2 @ G ) ) ) ).
% comp_image
thf(fact_781_comp__image,axiom,
! [F2: nat > set_nat,G: nat > nat] :
( ( comp_s6110837996735651117et_nat @ ( image_nat_set_nat @ F2 ) @ ( image_nat_nat2 @ G ) )
= ( image_nat_set_nat @ ( comp_nat_set_nat_nat @ F2 @ G ) ) ) ).
% comp_image
thf(fact_782_UN__equiv__class2,axiom,
! [A1: set_Pr1261947904930325089at_nat,R1: set_Pr8693737435421807431at_nat,A22: set_Pr1261947904930325089at_nat,R22: set_Pr8693737435421807431at_nat,F2: product_prod_nat_nat > product_prod_nat_nat > set_nat,A12: product_prod_nat_nat,A23: product_prod_nat_nat] :
( ( equiv_5407799809235660215at_nat @ A1 @ R1 )
=> ( ( equiv_5407799809235660215at_nat @ A22 @ R22 )
=> ( ( equiv_4637926981076332931et_nat @ R1 @ R22 @ F2 )
=> ( ( member8440522571783428010at_nat @ A12 @ A1 )
=> ( ( member8440522571783428010at_nat @ A23 @ A22 )
=> ( ( comple7399068483239264473et_nat
@ ( image_15824709712370754et_nat
@ ^ [X12: product_prod_nat_nat] : ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ ( F2 @ X12 ) @ ( image_2484338115069272851at_nat @ R22 @ ( insert8211810215607154385at_nat @ A23 @ bot_bo2099793752762293965at_nat ) ) ) )
@ ( image_2484338115069272851at_nat @ R1 @ ( insert8211810215607154385at_nat @ A12 @ bot_bo2099793752762293965at_nat ) ) ) )
= ( F2 @ A12 @ A23 ) ) ) ) ) ) ) ).
% UN_equiv_class2
thf(fact_783_UN__equiv__class2,axiom,
! [A1: set_nat,R1: set_Pr1261947904930325089at_nat,A22: set_Pr1261947904930325089at_nat,R22: set_Pr8693737435421807431at_nat,F2: nat > product_prod_nat_nat > set_nat,A12: nat,A23: product_prod_nat_nat] :
( ( equiv_equiv_nat @ A1 @ R1 )
=> ( ( equiv_5407799809235660215at_nat @ A22 @ R22 )
=> ( ( equiv_2960386318186113850et_nat @ R1 @ R22 @ F2 )
=> ( ( member_nat @ A12 @ A1 )
=> ( ( member8440522571783428010at_nat @ A23 @ A22 )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X12: nat] : ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ ( F2 @ X12 ) @ ( image_2484338115069272851at_nat @ R22 @ ( insert8211810215607154385at_nat @ A23 @ bot_bo2099793752762293965at_nat ) ) ) )
@ ( image_nat_nat @ R1 @ ( insert_nat @ A12 @ bot_bot_set_nat ) ) ) )
= ( F2 @ A12 @ A23 ) ) ) ) ) ) ) ).
% UN_equiv_class2
thf(fact_784_UN__equiv__class2,axiom,
! [A1: set_o,R1: set_Product_prod_o_o,A22: set_Pr1261947904930325089at_nat,R22: set_Pr8693737435421807431at_nat,F2: $o > product_prod_nat_nat > set_nat,A12: $o,A23: product_prod_nat_nat] :
( ( equiv_equiv_o @ A1 @ R1 )
=> ( ( equiv_5407799809235660215at_nat @ A22 @ R22 )
=> ( ( equiv_4406527554178595628et_nat @ R1 @ R22 @ F2 )
=> ( ( member_o @ A12 @ A1 )
=> ( ( member8440522571783428010at_nat @ A23 @ A22 )
=> ( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [X12: $o] : ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ ( F2 @ X12 ) @ ( image_2484338115069272851at_nat @ R22 @ ( insert8211810215607154385at_nat @ A23 @ bot_bo2099793752762293965at_nat ) ) ) )
@ ( image_o_o @ R1 @ ( insert_o @ A12 @ bot_bot_set_o ) ) ) )
= ( F2 @ A12 @ A23 ) ) ) ) ) ) ) ).
% UN_equiv_class2
thf(fact_785_UN__equiv__class2,axiom,
! [A1: set_Pr1261947904930325089at_nat,R1: set_Pr8693737435421807431at_nat,A22: set_nat,R22: set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > nat > set_nat,A12: product_prod_nat_nat,A23: nat] :
( ( equiv_5407799809235660215at_nat @ A1 @ R1 )
=> ( ( equiv_equiv_nat @ A22 @ R22 )
=> ( ( equiv_6930685768344799580et_nat @ R1 @ R22 @ F2 )
=> ( ( member8440522571783428010at_nat @ A12 @ A1 )
=> ( ( member_nat @ A23 @ A22 )
=> ( ( comple7399068483239264473et_nat
@ ( image_15824709712370754et_nat
@ ^ [X12: product_prod_nat_nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ ( F2 @ X12 ) @ ( image_nat_nat @ R22 @ ( insert_nat @ A23 @ bot_bot_set_nat ) ) ) )
@ ( image_2484338115069272851at_nat @ R1 @ ( insert8211810215607154385at_nat @ A12 @ bot_bo2099793752762293965at_nat ) ) ) )
= ( F2 @ A12 @ A23 ) ) ) ) ) ) ) ).
% UN_equiv_class2
thf(fact_786_UN__equiv__class2,axiom,
! [A1: set_nat,R1: set_Pr1261947904930325089at_nat,A22: set_nat,R22: set_Pr1261947904930325089at_nat,F2: nat > nat > set_nat,A12: nat,A23: nat] :
( ( equiv_equiv_nat @ A1 @ R1 )
=> ( ( equiv_equiv_nat @ A22 @ R22 )
=> ( ( equiv_4377714518549004389et_nat @ R1 @ R22 @ F2 )
=> ( ( member_nat @ A12 @ A1 )
=> ( ( member_nat @ A23 @ A22 )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X12: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ ( F2 @ X12 ) @ ( image_nat_nat @ R22 @ ( insert_nat @ A23 @ bot_bot_set_nat ) ) ) )
@ ( image_nat_nat @ R1 @ ( insert_nat @ A12 @ bot_bot_set_nat ) ) ) )
= ( F2 @ A12 @ A23 ) ) ) ) ) ) ) ).
% UN_equiv_class2
thf(fact_787_UN__equiv__class2,axiom,
! [A1: set_o,R1: set_Product_prod_o_o,A22: set_nat,R22: set_Pr1261947904930325089at_nat,F2: $o > nat > set_nat,A12: $o,A23: nat] :
( ( equiv_equiv_o @ A1 @ R1 )
=> ( ( equiv_equiv_nat @ A22 @ R22 )
=> ( ( equiv_8341594750169188083et_nat @ R1 @ R22 @ F2 )
=> ( ( member_o @ A12 @ A1 )
=> ( ( member_nat @ A23 @ A22 )
=> ( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [X12: $o] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ ( F2 @ X12 ) @ ( image_nat_nat @ R22 @ ( insert_nat @ A23 @ bot_bot_set_nat ) ) ) )
@ ( image_o_o @ R1 @ ( insert_o @ A12 @ bot_bot_set_o ) ) ) )
= ( F2 @ A12 @ A23 ) ) ) ) ) ) ) ).
% UN_equiv_class2
thf(fact_788_UN__equiv__class2,axiom,
! [A1: set_Pr1261947904930325089at_nat,R1: set_Pr8693737435421807431at_nat,A22: set_o,R22: set_Product_prod_o_o,F2: product_prod_nat_nat > $o > set_nat,A12: product_prod_nat_nat,A23: $o] :
( ( equiv_5407799809235660215at_nat @ A1 @ R1 )
=> ( ( equiv_equiv_o @ A22 @ R22 )
=> ( ( equiv_1136281463066641178et_nat @ R1 @ R22 @ F2 )
=> ( ( member8440522571783428010at_nat @ A12 @ A1 )
=> ( ( member_o @ A23 @ A22 )
=> ( ( comple7399068483239264473et_nat
@ ( image_15824709712370754et_nat
@ ^ [X12: product_prod_nat_nat] : ( comple7399068483239264473et_nat @ ( image_o_set_nat @ ( F2 @ X12 ) @ ( image_o_o @ R22 @ ( insert_o @ A23 @ bot_bot_set_o ) ) ) )
@ ( image_2484338115069272851at_nat @ R1 @ ( insert8211810215607154385at_nat @ A12 @ bot_bo2099793752762293965at_nat ) ) ) )
= ( F2 @ A12 @ A23 ) ) ) ) ) ) ) ).
% UN_equiv_class2
thf(fact_789_UN__equiv__class2,axiom,
! [A1: set_nat,R1: set_Pr1261947904930325089at_nat,A22: set_o,R22: set_Product_prod_o_o,F2: nat > $o > set_nat,A12: nat,A23: $o] :
( ( equiv_equiv_nat @ A1 @ R1 )
=> ( ( equiv_equiv_o @ A22 @ R22 )
=> ( ( equiv_9005645190137227217et_nat @ R1 @ R22 @ F2 )
=> ( ( member_nat @ A12 @ A1 )
=> ( ( member_o @ A23 @ A22 )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X12: nat] : ( comple7399068483239264473et_nat @ ( image_o_set_nat @ ( F2 @ X12 ) @ ( image_o_o @ R22 @ ( insert_o @ A23 @ bot_bot_set_o ) ) ) )
@ ( image_nat_nat @ R1 @ ( insert_nat @ A12 @ bot_bot_set_nat ) ) ) )
= ( F2 @ A12 @ A23 ) ) ) ) ) ) ) ).
% UN_equiv_class2
thf(fact_790_UN__equiv__class2,axiom,
! [A1: set_o,R1: set_Product_prod_o_o,A22: set_o,R22: set_Product_prod_o_o,F2: $o > $o > set_nat,A12: $o,A23: $o] :
( ( equiv_equiv_o @ A1 @ R1 )
=> ( ( equiv_equiv_o @ A22 @ R22 )
=> ( ( equiv_2613626581038891459et_nat @ R1 @ R22 @ F2 )
=> ( ( member_o @ A12 @ A1 )
=> ( ( member_o @ A23 @ A22 )
=> ( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [X12: $o] : ( comple7399068483239264473et_nat @ ( image_o_set_nat @ ( F2 @ X12 ) @ ( image_o_o @ R22 @ ( insert_o @ A23 @ bot_bot_set_o ) ) ) )
@ ( image_o_o @ R1 @ ( insert_o @ A12 @ bot_bot_set_o ) ) ) )
= ( F2 @ A12 @ A23 ) ) ) ) ) ) ) ).
% UN_equiv_class2
thf(fact_791_UN__equiv__class,axiom,
! [A4: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat,F2: product_prod_nat_nat > set_nat,A: product_prod_nat_nat] :
( ( equiv_5407799809235660215at_nat @ A4 @ R )
=> ( ( equiv_7243916823904213361et_nat @ R @ F2 )
=> ( ( member8440522571783428010at_nat @ A @ A4 )
=> ( ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F2 @ ( image_2484338115069272851at_nat @ R @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) )
= ( F2 @ A ) ) ) ) ) ).
% UN_equiv_class
thf(fact_792_UN__equiv__class,axiom,
! [A4: set_nat,R: set_Pr1261947904930325089at_nat,F2: nat > set_nat,A: nat] :
( ( equiv_equiv_nat @ A4 @ R )
=> ( ( equiv_4704879099140956910et_nat @ R @ F2 )
=> ( ( member_nat @ A @ A4 )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ ( image_nat_nat @ R @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) )
= ( F2 @ A ) ) ) ) ) ).
% UN_equiv_class
thf(fact_793_UN__equiv__class,axiom,
! [A4: set_o,R: set_Product_prod_o_o,F2: $o > set_nat,A: $o] :
( ( equiv_equiv_o @ A4 @ R )
=> ( ( equiv_7035509024949518600et_nat @ R @ F2 )
=> ( ( member_o @ A @ A4 )
=> ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F2 @ ( image_o_o @ R @ ( insert_o @ A @ bot_bot_set_o ) ) ) )
= ( F2 @ A ) ) ) ) ) ).
% UN_equiv_class
thf(fact_794_UN__equiv__class__inject,axiom,
! [A4: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat,F2: product_prod_nat_nat > set_nat,X5: set_Pr1261947904930325089at_nat,Y7: set_Pr1261947904930325089at_nat] :
( ( equiv_5407799809235660215at_nat @ A4 @ R )
=> ( ( equiv_7243916823904213361et_nat @ R @ F2 )
=> ( ( ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F2 @ X5 ) )
= ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F2 @ Y7 ) ) )
=> ( ( member2643936169264416010at_nat @ X5 @ ( equiv_3811336339175339080at_nat @ A4 @ R ) )
=> ( ( member2643936169264416010at_nat @ Y7 @ ( equiv_3811336339175339080at_nat @ A4 @ R ) )
=> ( ! [X3: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A4 )
=> ( ( member8440522571783428010at_nat @ Y @ A4 )
=> ( ( ( F2 @ X3 )
= ( F2 @ Y ) )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ R ) ) ) )
=> ( X5 = Y7 ) ) ) ) ) ) ) ).
% UN_equiv_class_inject
thf(fact_795_UN__equiv__class__inject,axiom,
! [A4: set_nat,R: set_Pr1261947904930325089at_nat,F2: nat > set_nat,X5: set_nat,Y7: set_nat] :
( ( equiv_equiv_nat @ A4 @ R )
=> ( ( equiv_4704879099140956910et_nat @ R @ F2 )
=> ( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ X5 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ Y7 ) ) )
=> ( ( member_set_nat @ X5 @ ( equiv_quotient_nat @ A4 @ R ) )
=> ( ( member_set_nat @ Y7 @ ( equiv_quotient_nat @ A4 @ R ) )
=> ( ! [X3: nat,Y: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ( member_nat @ Y @ A4 )
=> ( ( ( F2 @ X3 )
= ( F2 @ Y ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ R ) ) ) )
=> ( X5 = Y7 ) ) ) ) ) ) ) ).
% UN_equiv_class_inject
thf(fact_796_congruent2__implies__congruent__UN,axiom,
! [A1: set_nat,R1: set_Pr1261947904930325089at_nat,A22: set_Pr1261947904930325089at_nat,R22: set_Pr8693737435421807431at_nat,F2: nat > product_prod_nat_nat > set_nat,A: product_prod_nat_nat] :
( ( equiv_equiv_nat @ A1 @ R1 )
=> ( ( equiv_5407799809235660215at_nat @ A22 @ R22 )
=> ( ( equiv_2960386318186113850et_nat @ R1 @ R22 @ F2 )
=> ( ( member8440522571783428010at_nat @ A @ A22 )
=> ( equiv_4704879099140956910et_nat @ R1
@ ^ [X12: nat] : ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ ( F2 @ X12 ) @ ( image_2484338115069272851at_nat @ R22 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ) ) ) ) ) ) ).
% congruent2_implies_congruent_UN
thf(fact_797_congruent2__implies__congruent__UN,axiom,
! [A1: set_nat,R1: set_Pr1261947904930325089at_nat,A22: set_nat,R22: set_Pr1261947904930325089at_nat,F2: nat > nat > set_nat,A: nat] :
( ( equiv_equiv_nat @ A1 @ R1 )
=> ( ( equiv_equiv_nat @ A22 @ R22 )
=> ( ( equiv_4377714518549004389et_nat @ R1 @ R22 @ F2 )
=> ( ( member_nat @ A @ A22 )
=> ( equiv_4704879099140956910et_nat @ R1
@ ^ [X12: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ ( F2 @ X12 ) @ ( image_nat_nat @ R22 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% congruent2_implies_congruent_UN
thf(fact_798_congruent2__implies__congruent__UN,axiom,
! [A1: set_nat,R1: set_Pr1261947904930325089at_nat,A22: set_o,R22: set_Product_prod_o_o,F2: nat > $o > set_nat,A: $o] :
( ( equiv_equiv_nat @ A1 @ R1 )
=> ( ( equiv_equiv_o @ A22 @ R22 )
=> ( ( equiv_9005645190137227217et_nat @ R1 @ R22 @ F2 )
=> ( ( member_o @ A @ A22 )
=> ( equiv_4704879099140956910et_nat @ R1
@ ^ [X12: nat] : ( comple7399068483239264473et_nat @ ( image_o_set_nat @ ( F2 @ X12 ) @ ( image_o_o @ R22 @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) ) ) ) ) ).
% congruent2_implies_congruent_UN
thf(fact_799_UN__equiv__class__type,axiom,
! [A4: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat,F2: product_prod_nat_nat > set_nat,X5: set_Pr1261947904930325089at_nat,B5: set_set_nat] :
( ( equiv_5407799809235660215at_nat @ A4 @ R )
=> ( ( equiv_7243916823904213361et_nat @ R @ F2 )
=> ( ( member2643936169264416010at_nat @ X5 @ ( equiv_3811336339175339080at_nat @ A4 @ R ) )
=> ( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A4 )
=> ( member_set_nat @ ( F2 @ X3 ) @ B5 ) )
=> ( member_set_nat @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F2 @ X5 ) ) @ B5 ) ) ) ) ) ).
% UN_equiv_class_type
thf(fact_800_UN__equiv__class__type,axiom,
! [A4: set_nat,R: set_Pr1261947904930325089at_nat,F2: nat > set_nat,X5: set_nat,B5: set_set_nat] :
( ( equiv_equiv_nat @ A4 @ R )
=> ( ( equiv_4704879099140956910et_nat @ R @ F2 )
=> ( ( member_set_nat @ X5 @ ( equiv_quotient_nat @ A4 @ R ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( member_set_nat @ ( F2 @ X3 ) @ B5 ) )
=> ( member_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ X5 ) ) @ B5 ) ) ) ) ) ).
% UN_equiv_class_type
thf(fact_801_UN__equiv__class__type2,axiom,
! [A1: set_Pr1261947904930325089at_nat,R1: set_Pr8693737435421807431at_nat,A22: set_Pr1261947904930325089at_nat,R22: set_Pr8693737435421807431at_nat,F2: product_prod_nat_nat > product_prod_nat_nat > set_nat,X13: set_Pr1261947904930325089at_nat,X22: set_Pr1261947904930325089at_nat,B5: set_set_nat] :
( ( equiv_5407799809235660215at_nat @ A1 @ R1 )
=> ( ( equiv_5407799809235660215at_nat @ A22 @ R22 )
=> ( ( equiv_4637926981076332931et_nat @ R1 @ R22 @ F2 )
=> ( ( member2643936169264416010at_nat @ X13 @ ( equiv_3811336339175339080at_nat @ A1 @ R1 ) )
=> ( ( member2643936169264416010at_nat @ X22 @ ( equiv_3811336339175339080at_nat @ A22 @ R22 ) )
=> ( ! [X14: product_prod_nat_nat,X23: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X14 @ A1 )
=> ( ( member8440522571783428010at_nat @ X23 @ A22 )
=> ( member_set_nat @ ( F2 @ X14 @ X23 ) @ B5 ) ) )
=> ( member_set_nat
@ ( comple7399068483239264473et_nat
@ ( image_15824709712370754et_nat
@ ^ [X12: product_prod_nat_nat] : ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ ( F2 @ X12 ) @ X22 ) )
@ X13 ) )
@ B5 ) ) ) ) ) ) ) ).
% UN_equiv_class_type2
thf(fact_802_UN__equiv__class__type2,axiom,
! [A1: set_Pr1261947904930325089at_nat,R1: set_Pr8693737435421807431at_nat,A22: set_nat,R22: set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > nat > set_nat,X13: set_Pr1261947904930325089at_nat,X22: set_nat,B5: set_set_nat] :
( ( equiv_5407799809235660215at_nat @ A1 @ R1 )
=> ( ( equiv_equiv_nat @ A22 @ R22 )
=> ( ( equiv_6930685768344799580et_nat @ R1 @ R22 @ F2 )
=> ( ( member2643936169264416010at_nat @ X13 @ ( equiv_3811336339175339080at_nat @ A1 @ R1 ) )
=> ( ( member_set_nat @ X22 @ ( equiv_quotient_nat @ A22 @ R22 ) )
=> ( ! [X14: product_prod_nat_nat,X23: nat] :
( ( member8440522571783428010at_nat @ X14 @ A1 )
=> ( ( member_nat @ X23 @ A22 )
=> ( member_set_nat @ ( F2 @ X14 @ X23 ) @ B5 ) ) )
=> ( member_set_nat
@ ( comple7399068483239264473et_nat
@ ( image_15824709712370754et_nat
@ ^ [X12: product_prod_nat_nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ ( F2 @ X12 ) @ X22 ) )
@ X13 ) )
@ B5 ) ) ) ) ) ) ) ).
% UN_equiv_class_type2
thf(fact_803_UN__equiv__class__type2,axiom,
! [A1: set_nat,R1: set_Pr1261947904930325089at_nat,A22: set_Pr1261947904930325089at_nat,R22: set_Pr8693737435421807431at_nat,F2: nat > product_prod_nat_nat > set_nat,X13: set_nat,X22: set_Pr1261947904930325089at_nat,B5: set_set_nat] :
( ( equiv_equiv_nat @ A1 @ R1 )
=> ( ( equiv_5407799809235660215at_nat @ A22 @ R22 )
=> ( ( equiv_2960386318186113850et_nat @ R1 @ R22 @ F2 )
=> ( ( member_set_nat @ X13 @ ( equiv_quotient_nat @ A1 @ R1 ) )
=> ( ( member2643936169264416010at_nat @ X22 @ ( equiv_3811336339175339080at_nat @ A22 @ R22 ) )
=> ( ! [X14: nat,X23: product_prod_nat_nat] :
( ( member_nat @ X14 @ A1 )
=> ( ( member8440522571783428010at_nat @ X23 @ A22 )
=> ( member_set_nat @ ( F2 @ X14 @ X23 ) @ B5 ) ) )
=> ( member_set_nat
@ ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X12: nat] : ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ ( F2 @ X12 ) @ X22 ) )
@ X13 ) )
@ B5 ) ) ) ) ) ) ) ).
% UN_equiv_class_type2
thf(fact_804_UN__equiv__class__type2,axiom,
! [A1: set_nat,R1: set_Pr1261947904930325089at_nat,A22: set_nat,R22: set_Pr1261947904930325089at_nat,F2: nat > nat > set_nat,X13: set_nat,X22: set_nat,B5: set_set_nat] :
( ( equiv_equiv_nat @ A1 @ R1 )
=> ( ( equiv_equiv_nat @ A22 @ R22 )
=> ( ( equiv_4377714518549004389et_nat @ R1 @ R22 @ F2 )
=> ( ( member_set_nat @ X13 @ ( equiv_quotient_nat @ A1 @ R1 ) )
=> ( ( member_set_nat @ X22 @ ( equiv_quotient_nat @ A22 @ R22 ) )
=> ( ! [X14: nat,X23: nat] :
( ( member_nat @ X14 @ A1 )
=> ( ( member_nat @ X23 @ A22 )
=> ( member_set_nat @ ( F2 @ X14 @ X23 ) @ B5 ) ) )
=> ( member_set_nat
@ ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X12: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ ( F2 @ X12 ) @ X22 ) )
@ X13 ) )
@ B5 ) ) ) ) ) ) ) ).
% UN_equiv_class_type2
thf(fact_805_univ__preserves,axiom,
! [A4: set_nat,R: set_Pr1261947904930325089at_nat,F2: nat > product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
( ( equiv_equiv_nat @ A4 @ R )
=> ( ( equiv_3178832884793977753at_nat @ R @ F2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( member8440522571783428010at_nat @ ( F2 @ X3 ) @ B5 ) )
=> ! [X6: set_nat] :
( ( member_set_nat @ X6 @ ( equiv_quotient_nat @ A4 @ R ) )
=> ( member8440522571783428010at_nat @ ( bNF_Gr881239092033360007at_nat @ F2 @ X6 ) @ B5 ) ) ) ) ) ).
% univ_preserves
thf(fact_806_univ__preserves,axiom,
! [A4: set_nat,R: set_Pr1261947904930325089at_nat,F2: nat > nat,B5: set_nat] :
( ( equiv_equiv_nat @ A4 @ R )
=> ( ( equiv_5105297490605233976at_nat @ R @ F2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( member_nat @ ( F2 @ X3 ) @ B5 ) )
=> ! [X6: set_nat] :
( ( member_set_nat @ X6 @ ( equiv_quotient_nat @ A4 @ R ) )
=> ( member_nat @ ( bNF_Gr5483022400647640010at_nat @ F2 @ X6 ) @ B5 ) ) ) ) ) ).
% univ_preserves
thf(fact_807_Id__on__def,axiom,
( id_on_nat
= ( ^ [A6: set_nat] :
( comple5685304695842803022at_nat
@ ( image_7178329752028323786at_nat
@ ^ [X4: nat] : ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X4 @ X4 ) @ bot_bo2099793752762293965at_nat )
@ A6 ) ) ) ) ).
% Id_on_def
thf(fact_808_image__comp,axiom,
! [F2: set_nat > set_nat,G: nat > set_nat,R: set_nat] :
( ( image_7916887816326733075et_nat @ F2 @ ( image_nat_set_nat @ G @ R ) )
= ( image_nat_set_nat @ ( comp_s3433241188411525313at_nat @ F2 @ G ) @ R ) ) ).
% image_comp
thf(fact_809_image__comp,axiom,
! [F2: nat > set_nat,G: nat > nat,R: set_nat] :
( ( image_nat_set_nat @ F2 @ ( image_nat_nat2 @ G @ R ) )
= ( image_nat_set_nat @ ( comp_nat_set_nat_nat @ F2 @ G ) @ R ) ) ).
% image_comp
thf(fact_810_Id__onI,axiom,
! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ A @ A4 )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ A ) @ ( id_on_2554058798563519774at_nat @ A4 ) ) ) ).
% Id_onI
thf(fact_811_Id__onI,axiom,
! [A: nat,A4: set_nat] :
( ( member_nat @ A @ A4 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ ( id_on_nat @ A4 ) ) ) ).
% Id_onI
thf(fact_812_Id__on__empty,axiom,
( ( id_on_nat @ bot_bot_set_nat )
= bot_bo2099793752762293965at_nat ) ).
% Id_on_empty
thf(fact_813_Id__on__empty,axiom,
( ( id_on_o @ bot_bot_set_o )
= bot_bo7073875226086086771od_o_o ) ).
% Id_on_empty
thf(fact_814_Image__Id__on,axiom,
! [A4: set_nat,B5: set_nat] :
( ( image_nat_nat @ ( id_on_nat @ A4 ) @ B5 )
= ( inf_inf_set_nat @ A4 @ B5 ) ) ).
% Image_Id_on
thf(fact_815_Id__onE,axiom,
! [C: produc859450856879609959at_nat,A4: set_Pr1261947904930325089at_nat] :
( ( member8206827879206165904at_nat @ C @ ( id_on_2554058798563519774at_nat @ A4 ) )
=> ~ ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A4 )
=> ( C
!= ( produc6161850002892822231at_nat @ X3 @ X3 ) ) ) ) ).
% Id_onE
thf(fact_816_Id__onE,axiom,
! [C: product_prod_nat_nat,A4: set_nat] :
( ( member8440522571783428010at_nat @ C @ ( id_on_nat @ A4 ) )
=> ~ ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( C
!= ( product_Pair_nat_nat @ X3 @ X3 ) ) ) ) ).
% Id_onE
thf(fact_817_Id__on__eqI,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ( A = B )
=> ( ( member8440522571783428010at_nat @ A @ A4 )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( id_on_2554058798563519774at_nat @ A4 ) ) ) ) ).
% Id_on_eqI
thf(fact_818_Id__on__eqI,axiom,
! [A: nat,B: nat,A4: set_nat] :
( ( A = B )
=> ( ( member_nat @ A @ A4 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( id_on_nat @ A4 ) ) ) ) ).
% Id_on_eqI
thf(fact_819_Id__on__iff,axiom,
! [X: product_prod_nat_nat,Y3: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ ( id_on_2554058798563519774at_nat @ A4 ) )
= ( ( X = Y3 )
& ( member8440522571783428010at_nat @ X @ A4 ) ) ) ).
% Id_on_iff
thf(fact_820_Id__on__iff,axiom,
! [X: nat,Y3: nat,A4: set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ ( id_on_nat @ A4 ) )
= ( ( X = Y3 )
& ( member_nat @ X @ A4 ) ) ) ).
% Id_on_iff
thf(fact_821_Id__on__subset__Times,axiom,
! [A4: set_nat] :
( ord_le3146513528884898305at_nat @ ( id_on_nat @ A4 )
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : A4 ) ) ).
% Id_on_subset_Times
thf(fact_822_image__eq__imp__comp,axiom,
! [F2: nat > set_nat,A4: set_nat,G: nat > set_nat,B5: set_nat,H: set_nat > set_nat] :
( ( ( image_nat_set_nat @ F2 @ A4 )
= ( image_nat_set_nat @ G @ B5 ) )
=> ( ( image_nat_set_nat @ ( comp_s3433241188411525313at_nat @ H @ F2 ) @ A4 )
= ( image_nat_set_nat @ ( comp_s3433241188411525313at_nat @ H @ G ) @ B5 ) ) ) ).
% image_eq_imp_comp
thf(fact_823_relImage__proj,axiom,
! [A4: set_nat,R2: set_Pr1261947904930325089at_nat] :
( ( equiv_equiv_nat @ A4 @ R2 )
=> ( ord_le4402255868550298733et_nat @ ( bNF_Gr553682781689516860et_nat @ R2 @ ( equiv_proj_nat_nat @ R2 ) ) @ ( id_on_set_nat @ ( equiv_quotient_nat @ A4 @ R2 ) ) ) ) ).
% relImage_proj
thf(fact_824_UN__Pow__subset,axiom,
! [B5: nat > set_nat,A4: set_nat] :
( ord_le6893508408891458716et_nat
@ ( comple548664676211718543et_nat
@ ( image_2194112158459175443et_nat
@ ^ [X4: nat] : ( pow_nat @ ( B5 @ X4 ) )
@ A4 ) )
@ ( pow_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ A4 ) ) ) ) ).
% UN_Pow_subset
thf(fact_825_Pow__singleton__iff,axiom,
! [X5: set_nat,Y7: set_nat] :
( ( ( pow_nat @ X5 )
= ( insert_set_nat @ Y7 @ bot_bot_set_set_nat ) )
= ( ( X5 = bot_bot_set_nat )
& ( Y7 = bot_bot_set_nat ) ) ) ).
% Pow_singleton_iff
thf(fact_826_Pow__singleton__iff,axiom,
! [X5: set_o,Y7: set_o] :
( ( ( pow_o @ X5 )
= ( insert_set_o @ Y7 @ bot_bot_set_set_o ) )
= ( ( X5 = bot_bot_set_o )
& ( Y7 = bot_bot_set_o ) ) ) ).
% Pow_singleton_iff
thf(fact_827_Pow__empty,axiom,
( ( pow_nat @ bot_bot_set_nat )
= ( insert_set_nat @ bot_bot_set_nat @ bot_bot_set_set_nat ) ) ).
% Pow_empty
thf(fact_828_Pow__empty,axiom,
( ( pow_o @ bot_bot_set_o )
= ( insert_set_o @ bot_bot_set_o @ bot_bot_set_set_o ) ) ).
% Pow_empty
thf(fact_829_Pow__bottom,axiom,
! [B5: set_nat] : ( member_set_nat @ bot_bot_set_nat @ ( pow_nat @ B5 ) ) ).
% Pow_bottom
thf(fact_830_Pow__bottom,axiom,
! [B5: set_o] : ( member_set_o @ bot_bot_set_o @ ( pow_o @ B5 ) ) ).
% Pow_bottom
thf(fact_831_Cantors__paradox,axiom,
! [A4: set_nat] :
~ ? [F5: nat > set_nat] :
( ( image_nat_set_nat @ F5 @ A4 )
= ( pow_nat @ A4 ) ) ).
% Cantors_paradox
thf(fact_832_image__Pow__surj,axiom,
! [F2: nat > set_nat,A4: set_nat,B5: set_set_nat] :
( ( ( image_nat_set_nat @ F2 @ A4 )
= B5 )
=> ( ( image_6725021117256019401et_nat @ ( image_nat_set_nat @ F2 ) @ ( pow_nat @ A4 ) )
= ( pow_set_nat @ B5 ) ) ) ).
% image_Pow_surj
thf(fact_833_image__Pow__mono,axiom,
! [F2: nat > set_nat,A4: set_nat,B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F2 @ A4 ) @ B5 )
=> ( ord_le9131159989063066194et_nat @ ( image_6725021117256019401et_nat @ ( image_nat_set_nat @ F2 ) @ ( pow_nat @ A4 ) ) @ ( pow_set_nat @ B5 ) ) ) ).
% image_Pow_mono
thf(fact_834_relImage__relInvImage,axiom,
! [R2: set_Pr5488025237498180813et_nat,F2: nat > set_nat,A4: set_nat] :
( ( ord_le4402255868550298733et_nat @ R2
@ ( produc105401717395143638et_nat @ ( image_nat_set_nat @ F2 @ A4 )
@ ^ [Uu: set_nat] : ( image_nat_set_nat @ F2 @ A4 ) ) )
=> ( ( bNF_Gr553682781689516860et_nat @ ( bNF_Gr7866296307673166617et_nat @ A4 @ R2 @ F2 ) @ F2 )
= R2 ) ) ).
% relImage_relInvImage
thf(fact_835_refl__on__singleton,axiom,
! [X: nat] : ( refl_on_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X @ X ) @ bot_bo2099793752762293965at_nat ) ) ).
% refl_on_singleton
thf(fact_836_refl__on__singleton,axiom,
! [X: $o] : ( refl_on_o @ ( insert_o @ X @ bot_bot_set_o ) @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ X @ X ) @ bot_bo7073875226086086771od_o_o ) ) ).
% refl_on_singleton
thf(fact_837_refl__onD2,axiom,
! [A4: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat,X: product_prod_nat_nat,Y3: product_prod_nat_nat] :
( ( refl_o7122697186039111552at_nat @ A4 @ R )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ R )
=> ( member8440522571783428010at_nat @ Y3 @ A4 ) ) ) ).
% refl_onD2
thf(fact_838_refl__onD2,axiom,
! [A4: set_nat,R: set_Pr1261947904930325089at_nat,X: nat,Y3: nat] :
( ( refl_on_nat @ A4 @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R )
=> ( member_nat @ Y3 @ A4 ) ) ) ).
% refl_onD2
thf(fact_839_refl__onD1,axiom,
! [A4: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat,X: product_prod_nat_nat,Y3: product_prod_nat_nat] :
( ( refl_o7122697186039111552at_nat @ A4 @ R )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ R )
=> ( member8440522571783428010at_nat @ X @ A4 ) ) ) ).
% refl_onD1
thf(fact_840_refl__onD1,axiom,
! [A4: set_nat,R: set_Pr1261947904930325089at_nat,X: nat,Y3: nat] :
( ( refl_on_nat @ A4 @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R )
=> ( member_nat @ X @ A4 ) ) ) ).
% refl_onD1
thf(fact_841_refl__onD,axiom,
! [A4: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat] :
( ( refl_o7122697186039111552at_nat @ A4 @ R )
=> ( ( member8440522571783428010at_nat @ A @ A4 )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ A ) @ R ) ) ) ).
% refl_onD
thf(fact_842_refl__onD,axiom,
! [A4: set_nat,R: set_Pr1261947904930325089at_nat,A: nat] :
( ( refl_on_nat @ A4 @ R )
=> ( ( member_nat @ A @ A4 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ R ) ) ) ).
% refl_onD
thf(fact_843_refl__on__empty,axiom,
refl_on_nat @ bot_bot_set_nat @ bot_bo2099793752762293965at_nat ).
% refl_on_empty
thf(fact_844_refl__on__empty,axiom,
refl_on_o @ bot_bot_set_o @ bot_bo7073875226086086771od_o_o ).
% refl_on_empty
thf(fact_845_refl__on__def,axiom,
( refl_on_nat
= ( ^ [A6: set_nat,R3: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ R3
@ ( produc457027306803732586at_nat @ A6
@ ^ [Uu: nat] : A6 ) )
& ! [X4: nat] :
( ( member_nat @ X4 @ A6 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ X4 ) @ R3 ) ) ) ) ) ).
% refl_on_def
thf(fact_846_refl__onI,axiom,
! [R: set_Pr8693737435421807431at_nat,A4: set_Pr1261947904930325089at_nat] :
( ( ord_le3000389064537975527at_nat @ R
@ ( produc2761391749766926216at_nat @ A4
@ ^ [Uu: product_prod_nat_nat] : A4 ) )
=> ( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A4 )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ X3 ) @ R ) )
=> ( refl_o7122697186039111552at_nat @ A4 @ R ) ) ) ).
% refl_onI
thf(fact_847_refl__onI,axiom,
! [R: set_Pr1261947904930325089at_nat,A4: set_nat] :
( ( ord_le3146513528884898305at_nat @ R
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : A4 ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ R ) )
=> ( refl_on_nat @ A4 @ R ) ) ) ).
% refl_onI
thf(fact_848_refl__on__UNION,axiom,
! [S3: set_nat,A4: nat > set_nat,R: nat > set_Pr1261947904930325089at_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ S3 )
=> ( refl_on_nat @ ( A4 @ X3 ) @ ( R @ X3 ) ) )
=> ( refl_on_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A4 @ S3 ) ) @ ( comple5685304695842803022at_nat @ ( image_7178329752028323786at_nat @ R @ S3 ) ) ) ) ).
% refl_on_UNION
thf(fact_849_refl__on__domain,axiom,
! [A4: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
( ( refl_o7122697186039111552at_nat @ A4 @ R )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R )
=> ( ( member8440522571783428010at_nat @ A @ A4 )
& ( member8440522571783428010at_nat @ B @ A4 ) ) ) ) ).
% refl_on_domain
thf(fact_850_refl__on__domain,axiom,
! [A4: set_nat,R: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
( ( refl_on_nat @ A4 @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
=> ( ( member_nat @ A @ A4 )
& ( member_nat @ B @ A4 ) ) ) ) ).
% refl_on_domain
thf(fact_851_insert__Times__insert,axiom,
! [A: nat,A4: set_nat,B: $o,B5: set_o] :
( ( product_Sigma_nat_o @ ( insert_nat @ A @ A4 )
@ ^ [Uu: nat] : ( insert_o @ B @ B5 ) )
= ( insert2233377226759017069_nat_o @ ( product_Pair_nat_o @ A @ B )
@ ( sup_su503371503184139039_nat_o
@ ( product_Sigma_nat_o @ A4
@ ^ [Uu: nat] : ( insert_o @ B @ B5 ) )
@ ( product_Sigma_nat_o @ ( insert_nat @ A @ A4 )
@ ^ [Uu: nat] : B5 ) ) ) ) ).
% insert_Times_insert
thf(fact_852_insert__Times__insert,axiom,
! [A: $o,A4: set_o,B: nat,B5: set_nat] :
( ( product_Sigma_o_nat @ ( insert_o @ A @ A4 )
@ ^ [Uu: $o] : ( insert_nat @ B @ B5 ) )
= ( insert7948214739558299847_o_nat @ ( product_Pair_o_nat @ A @ B )
@ ( sup_su8679140417860611193_o_nat
@ ( product_Sigma_o_nat @ A4
@ ^ [Uu: $o] : ( insert_nat @ B @ B5 ) )
@ ( product_Sigma_o_nat @ ( insert_o @ A @ A4 )
@ ^ [Uu: $o] : B5 ) ) ) ) ).
% insert_Times_insert
thf(fact_853_insert__Times__insert,axiom,
! [A: $o,A4: set_o,B: $o,B5: set_o] :
( ( product_Sigma_o_o @ ( insert_o @ A @ A4 )
@ ^ [Uu: $o] : ( insert_o @ B @ B5 ) )
= ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ A @ B )
@ ( sup_su5769328420594410459od_o_o
@ ( product_Sigma_o_o @ A4
@ ^ [Uu: $o] : ( insert_o @ B @ B5 ) )
@ ( product_Sigma_o_o @ ( insert_o @ A @ A4 )
@ ^ [Uu: $o] : B5 ) ) ) ) ).
% insert_Times_insert
thf(fact_854_insert__Times__insert,axiom,
! [A: nat,A4: set_nat,B: nat,B5: set_nat] :
( ( produc457027306803732586at_nat @ ( insert_nat @ A @ A4 )
@ ^ [Uu: nat] : ( insert_nat @ B @ B5 ) )
= ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B )
@ ( sup_su6327502436637775413at_nat
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : ( insert_nat @ B @ B5 ) )
@ ( produc457027306803732586at_nat @ ( insert_nat @ A @ A4 )
@ ^ [Uu: nat] : B5 ) ) ) ) ).
% insert_Times_insert
thf(fact_855_cSUP__UNION,axiom,
! [A4: set_nat,B5: nat > set_o,F2: $o > nat] :
( ( A4 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ( B5 @ X3 )
!= bot_bot_set_o ) )
=> ( ( condit2214826472909112428ve_nat
@ ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X4: nat] : ( image_o_nat2 @ F2 @ ( B5 @ X4 ) )
@ A4 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_o_nat2 @ F2 @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B5 @ A4 ) ) ) )
= ( complete_Sup_Sup_nat
@ ( image_nat_nat2
@ ^ [X4: nat] : ( complete_Sup_Sup_nat @ ( image_o_nat2 @ F2 @ ( B5 @ X4 ) ) )
@ A4 ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_856_cSUP__UNION,axiom,
! [A4: set_o,B5: $o > set_o,F2: $o > nat] :
( ( A4 != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A4 )
=> ( ( B5 @ X3 )
!= bot_bot_set_o ) )
=> ( ( condit2214826472909112428ve_nat
@ ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [X4: $o] : ( image_o_nat2 @ F2 @ ( B5 @ X4 ) )
@ A4 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_o_nat2 @ F2 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B5 @ A4 ) ) ) )
= ( complete_Sup_Sup_nat
@ ( image_o_nat2
@ ^ [X4: $o] : ( complete_Sup_Sup_nat @ ( image_o_nat2 @ F2 @ ( B5 @ X4 ) ) )
@ A4 ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_857_cSUP__UNION,axiom,
! [A4: set_nat,B5: nat > set_nat,F2: nat > nat] :
( ( A4 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ( B5 @ X3 )
!= bot_bot_set_nat ) )
=> ( ( condit2214826472909112428ve_nat
@ ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X4: nat] : ( image_nat_nat2 @ F2 @ ( B5 @ X4 ) )
@ A4 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ A4 ) ) ) )
= ( complete_Sup_Sup_nat
@ ( image_nat_nat2
@ ^ [X4: nat] : ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F2 @ ( B5 @ X4 ) ) )
@ A4 ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_858_cSUP__UNION,axiom,
! [A4: set_o,B5: $o > set_nat,F2: nat > nat] :
( ( A4 != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A4 )
=> ( ( B5 @ X3 )
!= bot_bot_set_nat ) )
=> ( ( condit2214826472909112428ve_nat
@ ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [X4: $o] : ( image_nat_nat2 @ F2 @ ( B5 @ X4 ) )
@ A4 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F2 @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B5 @ A4 ) ) ) )
= ( complete_Sup_Sup_nat
@ ( image_o_nat2
@ ^ [X4: $o] : ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F2 @ ( B5 @ X4 ) ) )
@ A4 ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_859_cSUP__UNION,axiom,
! [A4: set_nat,B5: nat > set_o,F2: $o > set_nat] :
( ( A4 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ( B5 @ X3 )
!= bot_bot_set_o ) )
=> ( ( condit5477540289124974626et_nat
@ ( comple548664676211718543et_nat
@ ( image_2194112158459175443et_nat
@ ^ [X4: nat] : ( image_o_set_nat @ F2 @ ( B5 @ X4 ) )
@ A4 ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F2 @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B5 @ A4 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X4: nat] : ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F2 @ ( B5 @ X4 ) ) )
@ A4 ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_860_cSUP__UNION,axiom,
! [A4: set_o,B5: $o > set_o,F2: $o > set_nat] :
( ( A4 != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A4 )
=> ( ( B5 @ X3 )
!= bot_bot_set_o ) )
=> ( ( condit5477540289124974626et_nat
@ ( comple548664676211718543et_nat
@ ( image_o_set_set_nat
@ ^ [X4: $o] : ( image_o_set_nat @ F2 @ ( B5 @ X4 ) )
@ A4 ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F2 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B5 @ A4 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [X4: $o] : ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F2 @ ( B5 @ X4 ) ) )
@ A4 ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_861_cSUP__UNION,axiom,
! [A4: set_nat,B5: nat > set_nat,F2: nat > set_nat] :
( ( A4 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ( B5 @ X3 )
!= bot_bot_set_nat ) )
=> ( ( condit5477540289124974626et_nat
@ ( comple548664676211718543et_nat
@ ( image_2194112158459175443et_nat
@ ^ [X4: nat] : ( image_nat_set_nat @ F2 @ ( B5 @ X4 ) )
@ A4 ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ A4 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X4: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ ( B5 @ X4 ) ) )
@ A4 ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_862_cSUP__UNION,axiom,
! [A4: set_o,B5: $o > set_nat,F2: nat > set_nat] :
( ( A4 != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A4 )
=> ( ( B5 @ X3 )
!= bot_bot_set_nat ) )
=> ( ( condit5477540289124974626et_nat
@ ( comple548664676211718543et_nat
@ ( image_o_set_set_nat
@ ^ [X4: $o] : ( image_nat_set_nat @ F2 @ ( B5 @ X4 ) )
@ A4 ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B5 @ A4 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [X4: $o] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ ( B5 @ X4 ) ) )
@ A4 ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_863_cSUP__UNION,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_o,F2: $o > nat] :
( ( A4 != bot_bo2099793752762293965at_nat )
=> ( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A4 )
=> ( ( B5 @ X3 )
!= bot_bot_set_o ) )
=> ( ( condit2214826472909112428ve_nat
@ ( comple7399068483239264473et_nat
@ ( image_15824709712370754et_nat
@ ^ [X4: product_prod_nat_nat] : ( image_o_nat2 @ F2 @ ( B5 @ X4 ) )
@ A4 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_o_nat2 @ F2 @ ( comple90263536869209701_set_o @ ( image_501298686397505212_set_o @ B5 @ A4 ) ) ) )
= ( complete_Sup_Sup_nat
@ ( image_2486076414777270412at_nat
@ ^ [X4: product_prod_nat_nat] : ( complete_Sup_Sup_nat @ ( image_o_nat2 @ F2 @ ( B5 @ X4 ) ) )
@ A4 ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_864_cSUP__UNION,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_nat,F2: nat > nat] :
( ( A4 != bot_bo2099793752762293965at_nat )
=> ( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A4 )
=> ( ( B5 @ X3 )
!= bot_bot_set_nat ) )
=> ( ( condit2214826472909112428ve_nat
@ ( comple7399068483239264473et_nat
@ ( image_15824709712370754et_nat
@ ^ [X4: product_prod_nat_nat] : ( image_nat_nat2 @ F2 @ ( B5 @ X4 ) )
@ A4 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F2 @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ B5 @ A4 ) ) ) )
= ( complete_Sup_Sup_nat
@ ( image_2486076414777270412at_nat
@ ^ [X4: product_prod_nat_nat] : ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F2 @ ( B5 @ X4 ) ) )
@ A4 ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_865_Ex__inj__on__UNION__Sigma,axiom,
! [A4: nat > set_nat,I2: set_nat] :
? [F: nat > product_prod_nat_nat] :
( ( inj_on5538052773655684606at_nat @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A4 @ I2 ) ) )
& ( ord_le3146513528884898305at_nat @ ( image_5846123807819985514at_nat @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A4 @ I2 ) ) ) @ ( produc457027306803732586at_nat @ I2 @ A4 ) ) ) ).
% Ex_inj_on_UNION_Sigma
thf(fact_866_Un__iff,axiom,
! [C: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ C @ ( sup_su6327502436637775413at_nat @ A4 @ B5 ) )
= ( ( member8440522571783428010at_nat @ C @ A4 )
| ( member8440522571783428010at_nat @ C @ B5 ) ) ) ).
% Un_iff
thf(fact_867_Un__iff,axiom,
! [C: nat,A4: set_nat,B5: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A4 @ B5 ) )
= ( ( member_nat @ C @ A4 )
| ( member_nat @ C @ B5 ) ) ) ).
% Un_iff
thf(fact_868_UnCI,axiom,
! [C: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
( ( ~ ( member8440522571783428010at_nat @ C @ B5 )
=> ( member8440522571783428010at_nat @ C @ A4 ) )
=> ( member8440522571783428010at_nat @ C @ ( sup_su6327502436637775413at_nat @ A4 @ B5 ) ) ) ).
% UnCI
thf(fact_869_UnCI,axiom,
! [C: nat,B5: set_nat,A4: set_nat] :
( ( ~ ( member_nat @ C @ B5 )
=> ( member_nat @ C @ A4 ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A4 @ B5 ) ) ) ).
% UnCI
thf(fact_870_sup__bot_Oright__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% sup_bot.right_neutral
thf(fact_871_sup__bot_Oright__neutral,axiom,
! [A: set_o] :
( ( sup_sup_set_o @ A @ bot_bot_set_o )
= A ) ).
% sup_bot.right_neutral
thf(fact_872_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ A @ B ) )
= ( ( A = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_873_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_o,B: set_o] :
( ( bot_bot_set_o
= ( sup_sup_set_o @ A @ B ) )
= ( ( A = bot_bot_set_o )
& ( B = bot_bot_set_o ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_874_sup__bot_Oleft__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_875_sup__bot_Oleft__neutral,axiom,
! [A: set_o] :
( ( sup_sup_set_o @ bot_bot_set_o @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_876_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( sup_sup_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ( A = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_877_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_o,B: set_o] :
( ( ( sup_sup_set_o @ A @ B )
= bot_bot_set_o )
= ( ( A = bot_bot_set_o )
& ( B = bot_bot_set_o ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_878_sup__eq__bot__iff,axiom,
! [X: set_nat,Y3: set_nat] :
( ( ( sup_sup_set_nat @ X @ Y3 )
= bot_bot_set_nat )
= ( ( X = bot_bot_set_nat )
& ( Y3 = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_879_sup__eq__bot__iff,axiom,
! [X: set_o,Y3: set_o] :
( ( ( sup_sup_set_o @ X @ Y3 )
= bot_bot_set_o )
= ( ( X = bot_bot_set_o )
& ( Y3 = bot_bot_set_o ) ) ) ).
% sup_eq_bot_iff
thf(fact_880_bot__eq__sup__iff,axiom,
! [X: set_nat,Y3: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X @ Y3 ) )
= ( ( X = bot_bot_set_nat )
& ( Y3 = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_881_bot__eq__sup__iff,axiom,
! [X: set_o,Y3: set_o] :
( ( bot_bot_set_o
= ( sup_sup_set_o @ X @ Y3 ) )
= ( ( X = bot_bot_set_o )
& ( Y3 = bot_bot_set_o ) ) ) ).
% bot_eq_sup_iff
thf(fact_882_sup__bot__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
= X ) ).
% sup_bot_right
thf(fact_883_sup__bot__right,axiom,
! [X: set_o] :
( ( sup_sup_set_o @ X @ bot_bot_set_o )
= X ) ).
% sup_bot_right
thf(fact_884_sup__bot__left,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X )
= X ) ).
% sup_bot_left
thf(fact_885_sup__bot__left,axiom,
! [X: set_o] :
( ( sup_sup_set_o @ bot_bot_set_o @ X )
= X ) ).
% sup_bot_left
thf(fact_886_Un__empty,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ( sup_sup_set_nat @ A4 @ B5 )
= bot_bot_set_nat )
= ( ( A4 = bot_bot_set_nat )
& ( B5 = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_887_Un__empty,axiom,
! [A4: set_o,B5: set_o] :
( ( ( sup_sup_set_o @ A4 @ B5 )
= bot_bot_set_o )
= ( ( A4 = bot_bot_set_o )
& ( B5 = bot_bot_set_o ) ) ) ).
% Un_empty
thf(fact_888_Un__insert__right,axiom,
! [A4: set_nat,A: nat,B5: set_nat] :
( ( sup_sup_set_nat @ A4 @ ( insert_nat @ A @ B5 ) )
= ( insert_nat @ A @ ( sup_sup_set_nat @ A4 @ B5 ) ) ) ).
% Un_insert_right
thf(fact_889_Un__insert__right,axiom,
! [A4: set_o,A: $o,B5: set_o] :
( ( sup_sup_set_o @ A4 @ ( insert_o @ A @ B5 ) )
= ( insert_o @ A @ ( sup_sup_set_o @ A4 @ B5 ) ) ) ).
% Un_insert_right
thf(fact_890_Un__insert__left,axiom,
! [A: nat,B5: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( insert_nat @ A @ B5 ) @ C2 )
= ( insert_nat @ A @ ( sup_sup_set_nat @ B5 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_891_Un__insert__left,axiom,
! [A: $o,B5: set_o,C2: set_o] :
( ( sup_sup_set_o @ ( insert_o @ A @ B5 ) @ C2 )
= ( insert_o @ A @ ( sup_sup_set_o @ B5 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_892_bdd__above__empty,axiom,
condit2214826472909112428ve_nat @ bot_bot_set_nat ).
% bdd_above_empty
thf(fact_893_bdd__above__empty,axiom,
condit5488710616941104124bove_o @ bot_bot_set_o ).
% bdd_above_empty
thf(fact_894_bdd__above__insert,axiom,
! [A: nat,A4: set_nat] :
( ( condit2214826472909112428ve_nat @ ( insert_nat @ A @ A4 ) )
= ( condit2214826472909112428ve_nat @ A4 ) ) ).
% bdd_above_insert
thf(fact_895_bdd__above__insert,axiom,
! [A: $o,A4: set_o] :
( ( condit5488710616941104124bove_o @ ( insert_o @ A @ A4 ) )
= ( condit5488710616941104124bove_o @ A4 ) ) ).
% bdd_above_insert
thf(fact_896_bdd__above__Iio,axiom,
! [B: nat] : ( condit2214826472909112428ve_nat @ ( set_ord_lessThan_nat @ B ) ) ).
% bdd_above_Iio
thf(fact_897_Sup__insert,axiom,
! [A: $o,A4: set_o] :
( ( complete_Sup_Sup_o @ ( insert_o @ A @ A4 ) )
= ( sup_sup_o @ A @ ( complete_Sup_Sup_o @ A4 ) ) ) ).
% Sup_insert
thf(fact_898_Sup__insert,axiom,
! [A: set_nat,A4: set_set_nat] :
( ( comple7399068483239264473et_nat @ ( insert_set_nat @ A @ A4 ) )
= ( sup_sup_set_nat @ A @ ( comple7399068483239264473et_nat @ A4 ) ) ) ).
% Sup_insert
thf(fact_899_if__image__distrib,axiom,
! [P2: nat > $o,F2: nat > set_nat,G: nat > set_nat,S3: set_nat] :
( ( image_nat_set_nat
@ ^ [X4: nat] : ( if_set_nat @ ( P2 @ X4 ) @ ( F2 @ X4 ) @ ( G @ X4 ) )
@ S3 )
= ( sup_sup_set_set_nat @ ( image_nat_set_nat @ F2 @ ( inf_inf_set_nat @ S3 @ ( collect_nat @ P2 ) ) )
@ ( image_nat_set_nat @ G
@ ( inf_inf_set_nat @ S3
@ ( collect_nat
@ ^ [X4: nat] :
~ ( P2 @ X4 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_900_UN__Un,axiom,
! [M: nat > set_nat,A4: set_nat,B5: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M @ ( sup_sup_set_nat @ A4 @ B5 ) ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M @ A4 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M @ B5 ) ) ) ) ).
% UN_Un
thf(fact_901_bdd__above__image__sup,axiom,
! [F2: nat > set_nat,G: nat > set_nat,A4: set_nat] :
( ( condit5477540289124974626et_nat
@ ( image_nat_set_nat
@ ^ [X4: nat] : ( sup_sup_set_nat @ ( F2 @ X4 ) @ ( G @ X4 ) )
@ A4 ) )
= ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ F2 @ A4 ) )
& ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ G @ A4 ) ) ) ) ).
% bdd_above_image_sup
thf(fact_902_UN__simps_I2_J,axiom,
! [C2: set_nat,A4: nat > set_o,B5: set_o] :
( ( ( C2 = bot_bot_set_nat )
=> ( ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [X4: nat] : ( sup_sup_set_o @ ( A4 @ X4 ) @ B5 )
@ C2 ) )
= bot_bot_set_o ) )
& ( ( C2 != bot_bot_set_nat )
=> ( ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [X4: nat] : ( sup_sup_set_o @ ( A4 @ X4 ) @ B5 )
@ C2 ) )
= ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ A4 @ C2 ) ) @ B5 ) ) ) ) ).
% UN_simps(2)
thf(fact_903_UN__simps_I2_J,axiom,
! [C2: set_o,A4: $o > set_o,B5: set_o] :
( ( ( C2 = bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X4: $o] : ( sup_sup_set_o @ ( A4 @ X4 ) @ B5 )
@ C2 ) )
= bot_bot_set_o ) )
& ( ( C2 != bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X4: $o] : ( sup_sup_set_o @ ( A4 @ X4 ) @ B5 )
@ C2 ) )
= ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ A4 @ C2 ) ) @ B5 ) ) ) ) ).
% UN_simps(2)
thf(fact_904_UN__simps_I2_J,axiom,
! [C2: set_nat,A4: nat > set_nat,B5: set_nat] :
( ( ( C2 = bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X4: nat] : ( sup_sup_set_nat @ ( A4 @ X4 ) @ B5 )
@ C2 ) )
= bot_bot_set_nat ) )
& ( ( C2 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X4: nat] : ( sup_sup_set_nat @ ( A4 @ X4 ) @ B5 )
@ C2 ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A4 @ C2 ) ) @ B5 ) ) ) ) ).
% UN_simps(2)
thf(fact_905_UN__simps_I2_J,axiom,
! [C2: set_o,A4: $o > set_nat,B5: set_nat] :
( ( ( C2 = bot_bot_set_o )
=> ( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [X4: $o] : ( sup_sup_set_nat @ ( A4 @ X4 ) @ B5 )
@ C2 ) )
= bot_bot_set_nat ) )
& ( ( C2 != bot_bot_set_o )
=> ( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [X4: $o] : ( sup_sup_set_nat @ ( A4 @ X4 ) @ B5 )
@ C2 ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A4 @ C2 ) ) @ B5 ) ) ) ) ).
% UN_simps(2)
thf(fact_906_UN__simps_I3_J,axiom,
! [C2: set_nat,A4: set_o,B5: nat > set_o] :
( ( ( C2 = bot_bot_set_nat )
=> ( ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [X4: nat] : ( sup_sup_set_o @ A4 @ ( B5 @ X4 ) )
@ C2 ) )
= bot_bot_set_o ) )
& ( ( C2 != bot_bot_set_nat )
=> ( ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [X4: nat] : ( sup_sup_set_o @ A4 @ ( B5 @ X4 ) )
@ C2 ) )
= ( sup_sup_set_o @ A4 @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B5 @ C2 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_907_UN__simps_I3_J,axiom,
! [C2: set_o,A4: set_o,B5: $o > set_o] :
( ( ( C2 = bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X4: $o] : ( sup_sup_set_o @ A4 @ ( B5 @ X4 ) )
@ C2 ) )
= bot_bot_set_o ) )
& ( ( C2 != bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X4: $o] : ( sup_sup_set_o @ A4 @ ( B5 @ X4 ) )
@ C2 ) )
= ( sup_sup_set_o @ A4 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B5 @ C2 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_908_UN__simps_I3_J,axiom,
! [C2: set_nat,A4: set_nat,B5: nat > set_nat] :
( ( ( C2 = bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X4: nat] : ( sup_sup_set_nat @ A4 @ ( B5 @ X4 ) )
@ C2 ) )
= bot_bot_set_nat ) )
& ( ( C2 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X4: nat] : ( sup_sup_set_nat @ A4 @ ( B5 @ X4 ) )
@ C2 ) )
= ( sup_sup_set_nat @ A4 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ C2 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_909_UN__simps_I3_J,axiom,
! [C2: set_o,A4: set_nat,B5: $o > set_nat] :
( ( ( C2 = bot_bot_set_o )
=> ( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [X4: $o] : ( sup_sup_set_nat @ A4 @ ( B5 @ X4 ) )
@ C2 ) )
= bot_bot_set_nat ) )
& ( ( C2 != bot_bot_set_o )
=> ( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [X4: $o] : ( sup_sup_set_nat @ A4 @ ( B5 @ X4 ) )
@ C2 ) )
= ( sup_sup_set_nat @ A4 @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B5 @ C2 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_910_UN__insert,axiom,
! [B5: nat > set_nat,A: nat,A4: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ ( insert_nat @ A @ A4 ) ) )
= ( sup_sup_set_nat @ ( B5 @ A ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ A4 ) ) ) ) ).
% UN_insert
thf(fact_911_UN__insert,axiom,
! [B5: $o > set_nat,A: $o,A4: set_o] :
( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B5 @ ( insert_o @ A @ A4 ) ) )
= ( sup_sup_set_nat @ ( B5 @ A ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B5 @ A4 ) ) ) ) ).
% UN_insert
thf(fact_912_inj__on__Un__image__eq__iff,axiom,
! [F2: nat > set_nat,A4: set_nat,B5: set_nat] :
( ( inj_on_nat_set_nat @ F2 @ ( sup_sup_set_nat @ A4 @ B5 ) )
=> ( ( ( image_nat_set_nat @ F2 @ A4 )
= ( image_nat_set_nat @ F2 @ B5 ) )
= ( A4 = B5 ) ) ) ).
% inj_on_Un_image_eq_iff
thf(fact_913_inj__on__image__Pow,axiom,
! [F2: nat > set_nat,A4: set_nat] :
( ( inj_on_nat_set_nat @ F2 @ A4 )
=> ( inj_on2776966659131765557et_nat @ ( image_nat_set_nat @ F2 ) @ ( pow_nat @ A4 ) ) ) ).
% inj_on_image_Pow
thf(fact_914_inj__on__image__Fpow,axiom,
! [F2: nat > set_nat,A4: set_nat] :
( ( inj_on_nat_set_nat @ F2 @ A4 )
=> ( inj_on2776966659131765557et_nat @ ( image_nat_set_nat @ F2 ) @ ( finite_Fpow_nat @ A4 ) ) ) ).
% inj_on_image_Fpow
thf(fact_915_cSUP__union,axiom,
! [A4: set_nat,F2: nat > set_nat,B5: set_nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ F2 @ A4 ) )
=> ( ( B5 != bot_bot_set_nat )
=> ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ F2 @ B5 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ ( sup_sup_set_nat @ A4 @ B5 ) ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ A4 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ B5 ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_916_cSUP__union,axiom,
! [A4: set_o,F2: $o > set_nat,B5: set_o] :
( ( A4 != bot_bot_set_o )
=> ( ( condit5477540289124974626et_nat @ ( image_o_set_nat @ F2 @ A4 ) )
=> ( ( B5 != bot_bot_set_o )
=> ( ( condit5477540289124974626et_nat @ ( image_o_set_nat @ F2 @ B5 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F2 @ ( sup_sup_set_o @ A4 @ B5 ) ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F2 @ A4 ) ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F2 @ B5 ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_917_SUP__union,axiom,
! [M: nat > set_nat,A4: set_nat,B5: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M @ ( sup_sup_set_nat @ A4 @ B5 ) ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M @ A4 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M @ B5 ) ) ) ) ).
% SUP_union
thf(fact_918_inj__on__image,axiom,
! [F2: nat > set_nat,A4: set_set_nat] :
( ( inj_on_nat_set_nat @ F2 @ ( comple7399068483239264473et_nat @ A4 ) )
=> ( inj_on2776966659131765557et_nat @ ( image_nat_set_nat @ F2 ) @ A4 ) ) ).
% inj_on_image
thf(fact_919_image__Un,axiom,
! [F2: nat > set_nat,A4: set_nat,B5: set_nat] :
( ( image_nat_set_nat @ F2 @ ( sup_sup_set_nat @ A4 @ B5 ) )
= ( sup_sup_set_set_nat @ ( image_nat_set_nat @ F2 @ A4 ) @ ( image_nat_set_nat @ F2 @ B5 ) ) ) ).
% image_Un
thf(fact_920_inj__on__disjoint__Un,axiom,
! [F2: nat > set_nat,A4: set_nat,G: nat > set_nat,B5: set_nat] :
( ( inj_on_nat_set_nat @ F2 @ A4 )
=> ( ( inj_on_nat_set_nat @ G @ B5 )
=> ( ( ( inf_inf_set_set_nat @ ( image_nat_set_nat @ F2 @ A4 ) @ ( image_nat_set_nat @ G @ B5 ) )
= bot_bot_set_set_nat )
=> ( inj_on_nat_set_nat
@ ^ [X4: nat] : ( if_set_nat @ ( member_nat @ X4 @ A4 ) @ ( F2 @ X4 ) @ ( G @ X4 ) )
@ ( sup_sup_set_nat @ A4 @ B5 ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_921_inj__on__disjoint__Un,axiom,
! [F2: product_prod_nat_nat > nat,A4: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > nat,B5: set_Pr1261947904930325089at_nat] :
( ( inj_on2178005380612969504at_nat @ F2 @ A4 )
=> ( ( inj_on2178005380612969504at_nat @ G @ B5 )
=> ( ( ( inf_inf_set_nat @ ( image_2486076414777270412at_nat @ F2 @ A4 ) @ ( image_2486076414777270412at_nat @ G @ B5 ) )
= bot_bot_set_nat )
=> ( inj_on2178005380612969504at_nat
@ ^ [X4: product_prod_nat_nat] : ( if_nat @ ( member8440522571783428010at_nat @ X4 @ A4 ) @ ( F2 @ X4 ) @ ( G @ X4 ) )
@ ( sup_su6327502436637775413at_nat @ A4 @ B5 ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_922_inj__on__disjoint__Un,axiom,
! [F2: nat > nat,A4: set_nat,G: nat > nat,B5: set_nat] :
( ( inj_on_nat_nat @ F2 @ A4 )
=> ( ( inj_on_nat_nat @ G @ B5 )
=> ( ( ( inf_inf_set_nat @ ( image_nat_nat2 @ F2 @ A4 ) @ ( image_nat_nat2 @ G @ B5 ) )
= bot_bot_set_nat )
=> ( inj_on_nat_nat
@ ^ [X4: nat] : ( if_nat @ ( member_nat @ X4 @ A4 ) @ ( F2 @ X4 ) @ ( G @ X4 ) )
@ ( sup_sup_set_nat @ A4 @ B5 ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_923_inj__on__disjoint__Un,axiom,
! [F2: product_prod_nat_nat > $o,A4: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > $o,B5: set_Pr1261947904930325089at_nat] :
( ( inj_on3734678552557954888_nat_o @ F2 @ A4 )
=> ( ( inj_on3734678552557954888_nat_o @ G @ B5 )
=> ( ( ( inf_inf_set_o @ ( image_3693632289388996572_nat_o @ F2 @ A4 ) @ ( image_3693632289388996572_nat_o @ G @ B5 ) )
= bot_bot_set_o )
=> ( inj_on3734678552557954888_nat_o
@ ^ [X4: product_prod_nat_nat] :
( ( ( member8440522571783428010at_nat @ X4 @ A4 )
=> ( F2 @ X4 ) )
& ( ~ ( member8440522571783428010at_nat @ X4 @ A4 )
=> ( G @ X4 ) ) )
@ ( sup_su6327502436637775413at_nat @ A4 @ B5 ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_924_inj__on__disjoint__Un,axiom,
! [F2: nat > $o,A4: set_nat,G: nat > $o,B5: set_nat] :
( ( inj_on_nat_o @ F2 @ A4 )
=> ( ( inj_on_nat_o @ G @ B5 )
=> ( ( ( inf_inf_set_o @ ( image_nat_o2 @ F2 @ A4 ) @ ( image_nat_o2 @ G @ B5 ) )
= bot_bot_set_o )
=> ( inj_on_nat_o
@ ^ [X4: nat] :
( ( ( member_nat @ X4 @ A4 )
=> ( F2 @ X4 ) )
& ( ~ ( member_nat @ X4 @ A4 )
=> ( G @ X4 ) ) )
@ ( sup_sup_set_nat @ A4 @ B5 ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_925_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
! [A4: set_nat,F2: nat > set_nat,G: nat > set_nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ F2 @ A4 ) )
=> ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ G @ A4 ) )
=> ( ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ A4 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ A4 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [A5: nat] : ( sup_sup_set_nat @ ( F2 @ A5 ) @ ( G @ A5 ) )
@ A4 ) ) ) ) ) ) ).
% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_926_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
! [A4: set_o,F2: $o > set_nat,G: $o > set_nat] :
( ( A4 != bot_bot_set_o )
=> ( ( condit5477540289124974626et_nat @ ( image_o_set_nat @ F2 @ A4 ) )
=> ( ( condit5477540289124974626et_nat @ ( image_o_set_nat @ G @ A4 ) )
=> ( ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F2 @ A4 ) ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ G @ A4 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [A5: $o] : ( sup_sup_set_nat @ ( F2 @ A5 ) @ ( G @ A5 ) )
@ A4 ) ) ) ) ) ) ).
% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_927_insert__def,axiom,
( insert_nat
= ( ^ [A5: nat] :
( sup_sup_set_nat
@ ( collect_nat
@ ^ [X4: nat] : ( X4 = A5 ) ) ) ) ) ).
% insert_def
thf(fact_928_insert__def,axiom,
( insert_o
= ( ^ [A5: $o] :
( sup_sup_set_o
@ ( collect_o
@ ^ [X4: $o] : ( X4 = A5 ) ) ) ) ) ).
% insert_def
thf(fact_929_cSup__union__distrib,axiom,
! [A4: set_nat,B5: set_nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ A4 )
=> ( ( B5 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ B5 )
=> ( ( complete_Sup_Sup_nat @ ( sup_sup_set_nat @ A4 @ B5 ) )
= ( sup_sup_nat @ ( complete_Sup_Sup_nat @ A4 ) @ ( complete_Sup_Sup_nat @ B5 ) ) ) ) ) ) ) ).
% cSup_union_distrib
thf(fact_930_cSup__union__distrib,axiom,
! [A4: set_o,B5: set_o] :
( ( A4 != bot_bot_set_o )
=> ( ( condit5488710616941104124bove_o @ A4 )
=> ( ( B5 != bot_bot_set_o )
=> ( ( condit5488710616941104124bove_o @ B5 )
=> ( ( complete_Sup_Sup_o @ ( sup_sup_set_o @ A4 @ B5 ) )
= ( sup_sup_o @ ( complete_Sup_Sup_o @ A4 ) @ ( complete_Sup_Sup_o @ B5 ) ) ) ) ) ) ) ).
% cSup_union_distrib
thf(fact_931_cSup__union__distrib,axiom,
! [A4: set_set_nat,B5: set_set_nat] :
( ( A4 != bot_bot_set_set_nat )
=> ( ( condit5477540289124974626et_nat @ A4 )
=> ( ( B5 != bot_bot_set_set_nat )
=> ( ( condit5477540289124974626et_nat @ B5 )
=> ( ( comple7399068483239264473et_nat @ ( sup_sup_set_set_nat @ A4 @ B5 ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ A4 ) @ ( comple7399068483239264473et_nat @ B5 ) ) ) ) ) ) ) ).
% cSup_union_distrib
thf(fact_932_Image__Un,axiom,
! [R2: set_Pr1261947904930325089at_nat,A4: set_nat,B5: set_nat] :
( ( image_nat_nat @ R2 @ ( sup_sup_set_nat @ A4 @ B5 ) )
= ( sup_sup_set_nat @ ( image_nat_nat @ R2 @ A4 ) @ ( image_nat_nat @ R2 @ B5 ) ) ) ).
% Image_Un
thf(fact_933_inj__on__convol__ident,axiom,
! [F2: nat > nat,X5: set_nat] :
( inj_on5538052773655684606at_nat
@ ^ [X4: nat] : ( product_Pair_nat_nat @ X4 @ ( F2 @ X4 ) )
@ X5 ) ).
% inj_on_convol_ident
thf(fact_934_cSup__insert__If,axiom,
! [X5: set_nat,A: nat] :
( ( condit2214826472909112428ve_nat @ X5 )
=> ( ( ( X5 = bot_bot_set_nat )
=> ( ( complete_Sup_Sup_nat @ ( insert_nat @ A @ X5 ) )
= A ) )
& ( ( X5 != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_nat @ ( insert_nat @ A @ X5 ) )
= ( sup_sup_nat @ A @ ( complete_Sup_Sup_nat @ X5 ) ) ) ) ) ) ).
% cSup_insert_If
thf(fact_935_cSup__insert__If,axiom,
! [X5: set_o,A: $o] :
( ( condit5488710616941104124bove_o @ X5 )
=> ( ( complete_Sup_Sup_o @ ( insert_o @ A @ X5 ) )
= ( ( ( X5 = bot_bot_set_o )
=> A )
& ( ( X5 != bot_bot_set_o )
=> ( sup_sup_o @ A @ ( complete_Sup_Sup_o @ X5 ) ) ) ) ) ) ).
% cSup_insert_If
thf(fact_936_cSup__insert__If,axiom,
! [X5: set_set_nat,A: set_nat] :
( ( condit5477540289124974626et_nat @ X5 )
=> ( ( ( X5 = bot_bot_set_set_nat )
=> ( ( comple7399068483239264473et_nat @ ( insert_set_nat @ A @ X5 ) )
= A ) )
& ( ( X5 != bot_bot_set_set_nat )
=> ( ( comple7399068483239264473et_nat @ ( insert_set_nat @ A @ X5 ) )
= ( sup_sup_set_nat @ A @ ( comple7399068483239264473et_nat @ X5 ) ) ) ) ) ) ).
% cSup_insert_If
thf(fact_937_cSup__insert,axiom,
! [X5: set_nat,A: nat] :
( ( X5 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ X5 )
=> ( ( complete_Sup_Sup_nat @ ( insert_nat @ A @ X5 ) )
= ( sup_sup_nat @ A @ ( complete_Sup_Sup_nat @ X5 ) ) ) ) ) ).
% cSup_insert
thf(fact_938_cSup__insert,axiom,
! [X5: set_o,A: $o] :
( ( X5 != bot_bot_set_o )
=> ( ( condit5488710616941104124bove_o @ X5 )
=> ( ( complete_Sup_Sup_o @ ( insert_o @ A @ X5 ) )
= ( sup_sup_o @ A @ ( complete_Sup_Sup_o @ X5 ) ) ) ) ) ).
% cSup_insert
thf(fact_939_cSup__insert,axiom,
! [X5: set_set_nat,A: set_nat] :
( ( X5 != bot_bot_set_set_nat )
=> ( ( condit5477540289124974626et_nat @ X5 )
=> ( ( comple7399068483239264473et_nat @ ( insert_set_nat @ A @ X5 ) )
= ( sup_sup_set_nat @ A @ ( comple7399068483239264473et_nat @ X5 ) ) ) ) ) ).
% cSup_insert
thf(fact_940_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_941_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_o] :
( ( sup_sup_set_o @ X @ bot_bot_set_o )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_942_Un__empty__right,axiom,
! [A4: set_nat] :
( ( sup_sup_set_nat @ A4 @ bot_bot_set_nat )
= A4 ) ).
% Un_empty_right
thf(fact_943_Un__empty__right,axiom,
! [A4: set_o] :
( ( sup_sup_set_o @ A4 @ bot_bot_set_o )
= A4 ) ).
% Un_empty_right
thf(fact_944_Un__empty__left,axiom,
! [B5: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B5 )
= B5 ) ).
% Un_empty_left
thf(fact_945_Un__empty__left,axiom,
! [B5: set_o] :
( ( sup_sup_set_o @ bot_bot_set_o @ B5 )
= B5 ) ).
% Un_empty_left
thf(fact_946_Un__def,axiom,
( sup_su6327502436637775413at_nat
= ( ^ [A6: set_Pr1261947904930325089at_nat,B7: set_Pr1261947904930325089at_nat] :
( collec3392354462482085612at_nat
@ ^ [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ A6 )
| ( member8440522571783428010at_nat @ X4 @ B7 ) ) ) ) ) ).
% Un_def
thf(fact_947_Un__def,axiom,
( sup_sup_set_nat
= ( ^ [A6: set_nat,B7: set_nat] :
( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A6 )
| ( member_nat @ X4 @ B7 ) ) ) ) ) ).
% Un_def
thf(fact_948_Un__Image,axiom,
! [R2: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat,A4: set_nat] :
( ( image_nat_nat @ ( sup_su6327502436637775413at_nat @ R2 @ S3 ) @ A4 )
= ( sup_sup_set_nat @ ( image_nat_nat @ R2 @ A4 ) @ ( image_nat_nat @ S3 @ A4 ) ) ) ).
% Un_Image
thf(fact_949_UnI2,axiom,
! [C: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ C @ B5 )
=> ( member8440522571783428010at_nat @ C @ ( sup_su6327502436637775413at_nat @ A4 @ B5 ) ) ) ).
% UnI2
thf(fact_950_UnI2,axiom,
! [C: nat,B5: set_nat,A4: set_nat] :
( ( member_nat @ C @ B5 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A4 @ B5 ) ) ) ).
% UnI2
thf(fact_951_UnI1,axiom,
! [C: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ C @ A4 )
=> ( member8440522571783428010at_nat @ C @ ( sup_su6327502436637775413at_nat @ A4 @ B5 ) ) ) ).
% UnI1
thf(fact_952_UnI1,axiom,
! [C: nat,A4: set_nat,B5: set_nat] :
( ( member_nat @ C @ A4 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A4 @ B5 ) ) ) ).
% UnI1
thf(fact_953_UnE,axiom,
! [C: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ C @ ( sup_su6327502436637775413at_nat @ A4 @ B5 ) )
=> ( ~ ( member8440522571783428010at_nat @ C @ A4 )
=> ( member8440522571783428010at_nat @ C @ B5 ) ) ) ).
% UnE
thf(fact_954_UnE,axiom,
! [C: nat,A4: set_nat,B5: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A4 @ B5 ) )
=> ( ~ ( member_nat @ C @ A4 )
=> ( member_nat @ C @ B5 ) ) ) ).
% UnE
thf(fact_955_Sigma__Un__distrib1,axiom,
! [I2: set_nat,J2: set_nat,C2: nat > set_nat] :
( ( produc457027306803732586at_nat @ ( sup_sup_set_nat @ I2 @ J2 ) @ C2 )
= ( sup_su6327502436637775413at_nat @ ( produc457027306803732586at_nat @ I2 @ C2 ) @ ( produc457027306803732586at_nat @ J2 @ C2 ) ) ) ).
% Sigma_Un_distrib1
thf(fact_956_Sigma__Un__distrib2,axiom,
! [I2: set_nat,A4: nat > set_nat,B5: nat > set_nat] :
( ( produc457027306803732586at_nat @ I2
@ ^ [I: nat] : ( sup_sup_set_nat @ ( A4 @ I ) @ ( B5 @ I ) ) )
= ( sup_su6327502436637775413at_nat @ ( produc457027306803732586at_nat @ I2 @ A4 ) @ ( produc457027306803732586at_nat @ I2 @ B5 ) ) ) ).
% Sigma_Un_distrib2
thf(fact_957_Times__Un__distrib1,axiom,
! [A4: set_nat,B5: set_nat,C2: set_nat] :
( ( produc457027306803732586at_nat @ ( sup_sup_set_nat @ A4 @ B5 )
@ ^ [Uu: nat] : C2 )
= ( sup_su6327502436637775413at_nat
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : C2 )
@ ( produc457027306803732586at_nat @ B5
@ ^ [Uu: nat] : C2 ) ) ) ).
% Times_Un_distrib1
thf(fact_958_cSup__inter__less__eq,axiom,
! [A4: set_nat,B5: set_nat] :
( ( condit2214826472909112428ve_nat @ A4 )
=> ( ( condit2214826472909112428ve_nat @ B5 )
=> ( ( ( inf_inf_set_nat @ A4 @ B5 )
!= bot_bot_set_nat )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( inf_inf_set_nat @ A4 @ B5 ) ) @ ( sup_sup_nat @ ( complete_Sup_Sup_nat @ A4 ) @ ( complete_Sup_Sup_nat @ B5 ) ) ) ) ) ) ).
% cSup_inter_less_eq
thf(fact_959_cSup__inter__less__eq,axiom,
! [A4: set_o,B5: set_o] :
( ( condit5488710616941104124bove_o @ A4 )
=> ( ( condit5488710616941104124bove_o @ B5 )
=> ( ( ( inf_inf_set_o @ A4 @ B5 )
!= bot_bot_set_o )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( inf_inf_set_o @ A4 @ B5 ) ) @ ( sup_sup_o @ ( complete_Sup_Sup_o @ A4 ) @ ( complete_Sup_Sup_o @ B5 ) ) ) ) ) ) ).
% cSup_inter_less_eq
thf(fact_960_cSup__inter__less__eq,axiom,
! [A4: set_set_nat,B5: set_set_nat] :
( ( condit5477540289124974626et_nat @ A4 )
=> ( ( condit5477540289124974626et_nat @ B5 )
=> ( ( ( inf_inf_set_set_nat @ A4 @ B5 )
!= bot_bot_set_set_nat )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( inf_inf_set_set_nat @ A4 @ B5 ) ) @ ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ A4 ) @ ( comple7399068483239264473et_nat @ B5 ) ) ) ) ) ) ).
% cSup_inter_less_eq
thf(fact_961_cSUP__insert,axiom,
! [A4: set_nat,F2: nat > set_nat,A: nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ F2 @ A4 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ ( insert_nat @ A @ A4 ) ) )
= ( sup_sup_set_nat @ ( F2 @ A ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ A4 ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_962_cSUP__insert,axiom,
! [A4: set_o,F2: $o > set_nat,A: $o] :
( ( A4 != bot_bot_set_o )
=> ( ( condit5477540289124974626et_nat @ ( image_o_set_nat @ F2 @ A4 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F2 @ ( insert_o @ A @ A4 ) ) )
= ( sup_sup_set_nat @ ( F2 @ A ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F2 @ A4 ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_963_subset__image__inj,axiom,
! [S3: set_set_nat,F2: nat > set_nat,T3: set_nat] :
( ( ord_le6893508408891458716et_nat @ S3 @ ( image_nat_set_nat @ F2 @ T3 ) )
= ( ? [U3: set_nat] :
( ( ord_less_eq_set_nat @ U3 @ T3 )
& ( inj_on_nat_set_nat @ F2 @ U3 )
& ( S3
= ( image_nat_set_nat @ F2 @ U3 ) ) ) ) ) ).
% subset_image_inj
thf(fact_964_inj__on__image__mem__iff,axiom,
! [F2: product_prod_nat_nat > product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ( inj_on8969904277767023793at_nat @ F2 @ B5 )
=> ( ( member8440522571783428010at_nat @ A @ B5 )
=> ( ( ord_le3146513528884898305at_nat @ A4 @ B5 )
=> ( ( member8440522571783428010at_nat @ ( F2 @ A ) @ ( image_5168914502847457605at_nat @ F2 @ A4 ) )
= ( member8440522571783428010at_nat @ A @ A4 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_965_inj__on__image__mem__iff,axiom,
! [F2: product_prod_nat_nat > nat,B5: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ( inj_on2178005380612969504at_nat @ F2 @ B5 )
=> ( ( member8440522571783428010at_nat @ A @ B5 )
=> ( ( ord_le3146513528884898305at_nat @ A4 @ B5 )
=> ( ( member_nat @ ( F2 @ A ) @ ( image_2486076414777270412at_nat @ F2 @ A4 ) )
= ( member8440522571783428010at_nat @ A @ A4 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_966_inj__on__image__mem__iff,axiom,
! [F2: nat > set_nat,B5: set_nat,A: nat,A4: set_nat] :
( ( inj_on_nat_set_nat @ F2 @ B5 )
=> ( ( member_nat @ A @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( member_set_nat @ ( F2 @ A ) @ ( image_nat_set_nat @ F2 @ A4 ) )
= ( member_nat @ A @ A4 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_967_inj__on__image__mem__iff,axiom,
! [F2: nat > product_prod_nat_nat,B5: set_nat,A: nat,A4: set_nat] :
( ( inj_on5538052773655684606at_nat @ F2 @ B5 )
=> ( ( member_nat @ A @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( member8440522571783428010at_nat @ ( F2 @ A ) @ ( image_5846123807819985514at_nat @ F2 @ A4 ) )
= ( member_nat @ A @ A4 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_968_inj__on__image__mem__iff,axiom,
! [F2: nat > nat,B5: set_nat,A: nat,A4: set_nat] :
( ( inj_on_nat_nat @ F2 @ B5 )
=> ( ( member_nat @ A @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( member_nat @ ( F2 @ A ) @ ( image_nat_nat2 @ F2 @ A4 ) )
= ( member_nat @ A @ A4 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_969_inj__on__image__eq__iff,axiom,
! [F2: nat > set_nat,C2: set_nat,A4: set_nat,B5: set_nat] :
( ( inj_on_nat_set_nat @ F2 @ C2 )
=> ( ( ord_less_eq_set_nat @ A4 @ C2 )
=> ( ( ord_less_eq_set_nat @ B5 @ C2 )
=> ( ( ( image_nat_set_nat @ F2 @ A4 )
= ( image_nat_set_nat @ F2 @ B5 ) )
= ( A4 = B5 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_970_inj__img__insertE,axiom,
! [F2: $o > $o,A4: set_o,X: $o,B5: set_o] :
( ( inj_on_o_o @ F2 @ A4 )
=> ( ~ ( member_o @ X @ B5 )
=> ( ( ( insert_o @ X @ B5 )
= ( image_o_o2 @ F2 @ A4 ) )
=> ~ ! [X8: $o,A9: set_o] :
( ~ ( member_o @ X8 @ A9 )
=> ( ( A4
= ( insert_o @ X8 @ A9 ) )
=> ( ( X
= ( F2 @ X8 ) )
=> ( B5
!= ( image_o_o2 @ F2 @ A9 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_971_inj__img__insertE,axiom,
! [F2: product_prod_nat_nat > $o,A4: set_Pr1261947904930325089at_nat,X: $o,B5: set_o] :
( ( inj_on3734678552557954888_nat_o @ F2 @ A4 )
=> ( ~ ( member_o @ X @ B5 )
=> ( ( ( insert_o @ X @ B5 )
= ( image_3693632289388996572_nat_o @ F2 @ A4 ) )
=> ~ ! [X8: product_prod_nat_nat,A9: set_Pr1261947904930325089at_nat] :
( ~ ( member8440522571783428010at_nat @ X8 @ A9 )
=> ( ( A4
= ( insert8211810215607154385at_nat @ X8 @ A9 ) )
=> ( ( X
= ( F2 @ X8 ) )
=> ( B5
!= ( image_3693632289388996572_nat_o @ F2 @ A9 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_972_inj__img__insertE,axiom,
! [F2: nat > set_nat,A4: set_nat,X: set_nat,B5: set_set_nat] :
( ( inj_on_nat_set_nat @ F2 @ A4 )
=> ( ~ ( member_set_nat @ X @ B5 )
=> ( ( ( insert_set_nat @ X @ B5 )
= ( image_nat_set_nat @ F2 @ A4 ) )
=> ~ ! [X8: nat,A9: set_nat] :
( ~ ( member_nat @ X8 @ A9 )
=> ( ( A4
= ( insert_nat @ X8 @ A9 ) )
=> ( ( X
= ( F2 @ X8 ) )
=> ( B5
!= ( image_nat_set_nat @ F2 @ A9 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_973_inj__img__insertE,axiom,
! [F2: nat > $o,A4: set_nat,X: $o,B5: set_o] :
( ( inj_on_nat_o @ F2 @ A4 )
=> ( ~ ( member_o @ X @ B5 )
=> ( ( ( insert_o @ X @ B5 )
= ( image_nat_o2 @ F2 @ A4 ) )
=> ~ ! [X8: nat,A9: set_nat] :
( ~ ( member_nat @ X8 @ A9 )
=> ( ( A4
= ( insert_nat @ X8 @ A9 ) )
=> ( ( X
= ( F2 @ X8 ) )
=> ( B5
!= ( image_nat_o2 @ F2 @ A9 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_974_inj__img__insertE,axiom,
! [F2: $o > product_prod_nat_nat,A4: set_o,X: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
( ( inj_on3896976348050468698at_nat @ F2 @ A4 )
=> ( ~ ( member8440522571783428010at_nat @ X @ B5 )
=> ( ( ( insert8211810215607154385at_nat @ X @ B5 )
= ( image_3855930084881510382at_nat @ F2 @ A4 ) )
=> ~ ! [X8: $o,A9: set_o] :
( ~ ( member_o @ X8 @ A9 )
=> ( ( A4
= ( insert_o @ X8 @ A9 ) )
=> ( ( X
= ( F2 @ X8 ) )
=> ( B5
!= ( image_3855930084881510382at_nat @ F2 @ A9 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_975_inj__img__insertE,axiom,
! [F2: product_prod_nat_nat > product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
( ( inj_on8969904277767023793at_nat @ F2 @ A4 )
=> ( ~ ( member8440522571783428010at_nat @ X @ B5 )
=> ( ( ( insert8211810215607154385at_nat @ X @ B5 )
= ( image_5168914502847457605at_nat @ F2 @ A4 ) )
=> ~ ! [X8: product_prod_nat_nat,A9: set_Pr1261947904930325089at_nat] :
( ~ ( member8440522571783428010at_nat @ X8 @ A9 )
=> ( ( A4
= ( insert8211810215607154385at_nat @ X8 @ A9 ) )
=> ( ( X
= ( F2 @ X8 ) )
=> ( B5
!= ( image_5168914502847457605at_nat @ F2 @ A9 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_976_inj__img__insertE,axiom,
! [F2: nat > product_prod_nat_nat,A4: set_nat,X: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
( ( inj_on5538052773655684606at_nat @ F2 @ A4 )
=> ( ~ ( member8440522571783428010at_nat @ X @ B5 )
=> ( ( ( insert8211810215607154385at_nat @ X @ B5 )
= ( image_5846123807819985514at_nat @ F2 @ A4 ) )
=> ~ ! [X8: nat,A9: set_nat] :
( ~ ( member_nat @ X8 @ A9 )
=> ( ( A4
= ( insert_nat @ X8 @ A9 ) )
=> ( ( X
= ( F2 @ X8 ) )
=> ( B5
!= ( image_5846123807819985514at_nat @ F2 @ A9 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_977_inj__img__insertE,axiom,
! [F2: $o > nat,A4: set_o,X: nat,B5: set_nat] :
( ( inj_on_o_nat @ F2 @ A4 )
=> ( ~ ( member_nat @ X @ B5 )
=> ( ( ( insert_nat @ X @ B5 )
= ( image_o_nat2 @ F2 @ A4 ) )
=> ~ ! [X8: $o,A9: set_o] :
( ~ ( member_o @ X8 @ A9 )
=> ( ( A4
= ( insert_o @ X8 @ A9 ) )
=> ( ( X
= ( F2 @ X8 ) )
=> ( B5
!= ( image_o_nat2 @ F2 @ A9 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_978_inj__img__insertE,axiom,
! [F2: product_prod_nat_nat > nat,A4: set_Pr1261947904930325089at_nat,X: nat,B5: set_nat] :
( ( inj_on2178005380612969504at_nat @ F2 @ A4 )
=> ( ~ ( member_nat @ X @ B5 )
=> ( ( ( insert_nat @ X @ B5 )
= ( image_2486076414777270412at_nat @ F2 @ A4 ) )
=> ~ ! [X8: product_prod_nat_nat,A9: set_Pr1261947904930325089at_nat] :
( ~ ( member8440522571783428010at_nat @ X8 @ A9 )
=> ( ( A4
= ( insert8211810215607154385at_nat @ X8 @ A9 ) )
=> ( ( X
= ( F2 @ X8 ) )
=> ( B5
!= ( image_2486076414777270412at_nat @ F2 @ A9 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_979_inj__img__insertE,axiom,
! [F2: nat > nat,A4: set_nat,X: nat,B5: set_nat] :
( ( inj_on_nat_nat @ F2 @ A4 )
=> ( ~ ( member_nat @ X @ B5 )
=> ( ( ( insert_nat @ X @ B5 )
= ( image_nat_nat2 @ F2 @ A4 ) )
=> ~ ! [X8: nat,A9: set_nat] :
( ~ ( member_nat @ X8 @ A9 )
=> ( ( A4
= ( insert_nat @ X8 @ A9 ) )
=> ( ( X
= ( F2 @ X8 ) )
=> ( B5
!= ( image_nat_nat2 @ F2 @ A9 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_980_bdd__above_OI2,axiom,
! [A4: set_nat,F2: nat > set_nat,M: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ord_less_eq_set_nat @ ( F2 @ X3 ) @ M ) )
=> ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ F2 @ A4 ) ) ) ).
% bdd_above.I2
thf(fact_981_insert__is__Un,axiom,
( insert_nat
= ( ^ [A5: nat] : ( sup_sup_set_nat @ ( insert_nat @ A5 @ bot_bot_set_nat ) ) ) ) ).
% insert_is_Un
thf(fact_982_insert__is__Un,axiom,
( insert_o
= ( ^ [A5: $o] : ( sup_sup_set_o @ ( insert_o @ A5 @ bot_bot_set_o ) ) ) ) ).
% insert_is_Un
thf(fact_983_Un__singleton__iff,axiom,
! [A4: set_nat,B5: set_nat,X: nat] :
( ( ( sup_sup_set_nat @ A4 @ B5 )
= ( insert_nat @ X @ bot_bot_set_nat ) )
= ( ( ( A4 = bot_bot_set_nat )
& ( B5
= ( insert_nat @ X @ bot_bot_set_nat ) ) )
| ( ( A4
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B5 = bot_bot_set_nat ) )
| ( ( A4
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B5
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_984_Un__singleton__iff,axiom,
! [A4: set_o,B5: set_o,X: $o] :
( ( ( sup_sup_set_o @ A4 @ B5 )
= ( insert_o @ X @ bot_bot_set_o ) )
= ( ( ( A4 = bot_bot_set_o )
& ( B5
= ( insert_o @ X @ bot_bot_set_o ) ) )
| ( ( A4
= ( insert_o @ X @ bot_bot_set_o ) )
& ( B5 = bot_bot_set_o ) )
| ( ( A4
= ( insert_o @ X @ bot_bot_set_o ) )
& ( B5
= ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_985_singleton__Un__iff,axiom,
! [X: nat,A4: set_nat,B5: set_nat] :
( ( ( insert_nat @ X @ bot_bot_set_nat )
= ( sup_sup_set_nat @ A4 @ B5 ) )
= ( ( ( A4 = bot_bot_set_nat )
& ( B5
= ( insert_nat @ X @ bot_bot_set_nat ) ) )
| ( ( A4
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B5 = bot_bot_set_nat ) )
| ( ( A4
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B5
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_986_singleton__Un__iff,axiom,
! [X: $o,A4: set_o,B5: set_o] :
( ( ( insert_o @ X @ bot_bot_set_o )
= ( sup_sup_set_o @ A4 @ B5 ) )
= ( ( ( A4 = bot_bot_set_o )
& ( B5
= ( insert_o @ X @ bot_bot_set_o ) ) )
| ( ( A4
= ( insert_o @ X @ bot_bot_set_o ) )
& ( B5 = bot_bot_set_o ) )
| ( ( A4
= ( insert_o @ X @ bot_bot_set_o ) )
& ( B5
= ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_987_complete__lattice__class_OSUP__sup__distrib,axiom,
! [F2: nat > set_nat,A4: set_nat,G: nat > set_nat] :
( ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ A4 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ A4 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [A5: nat] : ( sup_sup_set_nat @ ( F2 @ A5 ) @ ( G @ A5 ) )
@ A4 ) ) ) ).
% complete_lattice_class.SUP_sup_distrib
thf(fact_988_SUP__absorb,axiom,
! [K: product_prod_nat_nat,I2: set_Pr1261947904930325089at_nat,A4: product_prod_nat_nat > set_nat] :
( ( member8440522571783428010at_nat @ K @ I2 )
=> ( ( sup_sup_set_nat @ ( A4 @ K ) @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ A4 @ I2 ) ) )
= ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ A4 @ I2 ) ) ) ) ).
% SUP_absorb
thf(fact_989_SUP__absorb,axiom,
! [K: nat,I2: set_nat,A4: nat > set_nat] :
( ( member_nat @ K @ I2 )
=> ( ( sup_sup_set_nat @ ( A4 @ K ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A4 @ I2 ) ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A4 @ I2 ) ) ) ) ).
% SUP_absorb
thf(fact_990_swap__inj__on,axiom,
! [A4: set_Pr1261947904930325089at_nat] :
( inj_on8969904277767023793at_nat
@ ( produc2626176000494625587at_nat
@ ^ [I: nat,J: nat] : ( product_Pair_nat_nat @ J @ I ) )
@ A4 ) ).
% swap_inj_on
thf(fact_991_Un__Union__image,axiom,
! [A4: nat > set_nat,B5: nat > set_nat,C2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X4: nat] : ( sup_sup_set_nat @ ( A4 @ X4 ) @ ( B5 @ X4 ) )
@ C2 ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A4 @ C2 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ C2 ) ) ) ) ).
% Un_Union_image
thf(fact_992_UN__Un__distrib,axiom,
! [A4: nat > set_nat,B5: nat > set_nat,I2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I: nat] : ( sup_sup_set_nat @ ( A4 @ I ) @ ( B5 @ I ) )
@ I2 ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A4 @ I2 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ I2 ) ) ) ) ).
% UN_Un_distrib
thf(fact_993_UN__absorb,axiom,
! [K: product_prod_nat_nat,I2: set_Pr1261947904930325089at_nat,A4: product_prod_nat_nat > set_nat] :
( ( member8440522571783428010at_nat @ K @ I2 )
=> ( ( sup_sup_set_nat @ ( A4 @ K ) @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ A4 @ I2 ) ) )
= ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ A4 @ I2 ) ) ) ) ).
% UN_absorb
thf(fact_994_UN__absorb,axiom,
! [K: nat,I2: set_nat,A4: nat > set_nat] :
( ( member_nat @ K @ I2 )
=> ( ( sup_sup_set_nat @ ( A4 @ K ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A4 @ I2 ) ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A4 @ I2 ) ) ) ) ).
% UN_absorb
thf(fact_995_inj__on__iff__surj,axiom,
! [A4: set_set_nat,A8: set_nat] :
( ( A4 != bot_bot_set_set_nat )
=> ( ( ? [F4: set_nat > nat] :
( ( inj_on_set_nat_nat @ F4 @ A4 )
& ( ord_less_eq_set_nat @ ( image_set_nat_nat @ F4 @ A4 ) @ A8 ) ) )
= ( ? [G2: nat > set_nat] :
( ( image_nat_set_nat @ G2 @ A8 )
= A4 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_996_inj__on__iff__surj,axiom,
! [A4: set_nat,A8: set_set_nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( ? [F4: nat > set_nat] :
( ( inj_on_nat_set_nat @ F4 @ A4 )
& ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F4 @ A4 ) @ A8 ) ) )
= ( ? [G2: set_nat > nat] :
( ( image_set_nat_nat @ G2 @ A8 )
= A4 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_997_inj__on__image__Int,axiom,
! [F2: nat > set_nat,C2: set_nat,A4: set_nat,B5: set_nat] :
( ( inj_on_nat_set_nat @ F2 @ C2 )
=> ( ( ord_less_eq_set_nat @ A4 @ C2 )
=> ( ( ord_less_eq_set_nat @ B5 @ C2 )
=> ( ( image_nat_set_nat @ F2 @ ( inf_inf_set_nat @ A4 @ B5 ) )
= ( inf_inf_set_set_nat @ ( image_nat_set_nat @ F2 @ A4 ) @ ( image_nat_set_nat @ F2 @ B5 ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_998_cSUP__upper2,axiom,
! [F2: product_prod_nat_nat > set_nat,A4: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,U2: set_nat] :
( ( condit5477540289124974626et_nat @ ( image_15824709712370754et_nat @ F2 @ A4 ) )
=> ( ( member8440522571783428010at_nat @ X @ A4 )
=> ( ( ord_less_eq_set_nat @ U2 @ ( F2 @ X ) )
=> ( ord_less_eq_set_nat @ U2 @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F2 @ A4 ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_999_cSUP__upper2,axiom,
! [F2: nat > set_nat,A4: set_nat,X: nat,U2: set_nat] :
( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ F2 @ A4 ) )
=> ( ( member_nat @ X @ A4 )
=> ( ( ord_less_eq_set_nat @ U2 @ ( F2 @ X ) )
=> ( ord_less_eq_set_nat @ U2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ A4 ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_1000_cSUP__upper,axiom,
! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > set_nat] :
( ( member8440522571783428010at_nat @ X @ A4 )
=> ( ( condit5477540289124974626et_nat @ ( image_15824709712370754et_nat @ F2 @ A4 ) )
=> ( ord_less_eq_set_nat @ ( F2 @ X ) @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F2 @ A4 ) ) ) ) ) ).
% cSUP_upper
thf(fact_1001_cSUP__upper,axiom,
! [X: nat,A4: set_nat,F2: nat > set_nat] :
( ( member_nat @ X @ A4 )
=> ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ F2 @ A4 ) )
=> ( ord_less_eq_set_nat @ ( F2 @ X ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ A4 ) ) ) ) ) ).
% cSUP_upper
thf(fact_1002_cSup__le__iff,axiom,
! [S3: set_nat,A: nat] :
( ( S3 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ S3 )
=> ( ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ S3 ) @ A )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ S3 )
=> ( ord_less_eq_nat @ X4 @ A ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_1003_cSup__le__iff,axiom,
! [S3: set_o,A: $o] :
( ( S3 != bot_bot_set_o )
=> ( ( condit5488710616941104124bove_o @ S3 )
=> ( ( ord_less_eq_o @ ( complete_Sup_Sup_o @ S3 ) @ A )
= ( ! [X4: $o] :
( ( member_o @ X4 @ S3 )
=> ( ord_less_eq_o @ X4 @ A ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_1004_cSup__le__iff,axiom,
! [S3: set_set_nat,A: set_nat] :
( ( S3 != bot_bot_set_set_nat )
=> ( ( condit5477540289124974626et_nat @ S3 )
=> ( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ S3 ) @ A )
= ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ S3 )
=> ( ord_less_eq_set_nat @ X4 @ A ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_1005_cSup__mono,axiom,
! [B5: set_nat,A4: set_nat] :
( ( B5 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ A4 )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ B5 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A4 )
& ( ord_less_eq_nat @ B3 @ X6 ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ B5 ) @ ( complete_Sup_Sup_nat @ A4 ) ) ) ) ) ).
% cSup_mono
thf(fact_1006_cSup__mono,axiom,
! [B5: set_o,A4: set_o] :
( ( B5 != bot_bot_set_o )
=> ( ( condit5488710616941104124bove_o @ A4 )
=> ( ! [B3: $o] :
( ( member_o @ B3 @ B5 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A4 )
& ( ord_less_eq_o @ B3 @ X6 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ B5 ) @ ( complete_Sup_Sup_o @ A4 ) ) ) ) ) ).
% cSup_mono
thf(fact_1007_cSup__mono,axiom,
! [B5: set_set_nat,A4: set_set_nat] :
( ( B5 != bot_bot_set_set_nat )
=> ( ( condit5477540289124974626et_nat @ A4 )
=> ( ! [B3: set_nat] :
( ( member_set_nat @ B3 @ B5 )
=> ? [X6: set_nat] :
( ( member_set_nat @ X6 @ A4 )
& ( ord_less_eq_set_nat @ B3 @ X6 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ B5 ) @ ( comple7399068483239264473et_nat @ A4 ) ) ) ) ) ).
% cSup_mono
thf(fact_1008_Union__image__empty,axiom,
! [A4: set_nat,F2: nat > set_nat] :
( ( sup_sup_set_nat @ A4 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ bot_bot_set_nat ) ) )
= A4 ) ).
% Union_image_empty
thf(fact_1009_Union__image__empty,axiom,
! [A4: set_nat,F2: $o > set_nat] :
( ( sup_sup_set_nat @ A4 @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F2 @ bot_bot_set_o ) ) )
= A4 ) ).
% Union_image_empty
thf(fact_1010_Union__image__insert,axiom,
! [F2: nat > set_nat,A: nat,B5: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ ( insert_nat @ A @ B5 ) ) )
= ( sup_sup_set_nat @ ( F2 @ A ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ B5 ) ) ) ) ).
% Union_image_insert
thf(fact_1011_Union__image__insert,axiom,
! [F2: $o > set_nat,A: $o,B5: set_o] :
( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F2 @ ( insert_o @ A @ B5 ) ) )
= ( sup_sup_set_nat @ ( F2 @ A ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F2 @ B5 ) ) ) ) ).
% Union_image_insert
thf(fact_1012_SUP__insert,axiom,
! [F2: nat > set_nat,A: nat,A4: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ ( insert_nat @ A @ A4 ) ) )
= ( sup_sup_set_nat @ ( F2 @ A ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ A4 ) ) ) ) ).
% SUP_insert
thf(fact_1013_SUP__insert,axiom,
! [F2: $o > set_nat,A: $o,A4: set_o] :
( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F2 @ ( insert_o @ A @ A4 ) ) )
= ( sup_sup_set_nat @ ( F2 @ A ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F2 @ A4 ) ) ) ) ).
% SUP_insert
thf(fact_1014_UN__extend__simps_I2_J,axiom,
! [C2: set_nat,A4: nat > set_nat,B5: set_nat] :
( ( ( C2 = bot_bot_set_nat )
=> ( ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A4 @ C2 ) ) @ B5 )
= B5 ) )
& ( ( C2 != bot_bot_set_nat )
=> ( ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A4 @ C2 ) ) @ B5 )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X4: nat] : ( sup_sup_set_nat @ ( A4 @ X4 ) @ B5 )
@ C2 ) ) ) ) ) ).
% UN_extend_simps(2)
thf(fact_1015_UN__extend__simps_I2_J,axiom,
! [C2: set_o,A4: $o > set_nat,B5: set_nat] :
( ( ( C2 = bot_bot_set_o )
=> ( ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A4 @ C2 ) ) @ B5 )
= B5 ) )
& ( ( C2 != bot_bot_set_o )
=> ( ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A4 @ C2 ) ) @ B5 )
= ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [X4: $o] : ( sup_sup_set_nat @ ( A4 @ X4 ) @ B5 )
@ C2 ) ) ) ) ) ).
% UN_extend_simps(2)
thf(fact_1016_UN__extend__simps_I3_J,axiom,
! [C2: set_nat,A4: set_nat,B5: nat > set_nat] :
( ( ( C2 = bot_bot_set_nat )
=> ( ( sup_sup_set_nat @ A4 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ C2 ) ) )
= A4 ) )
& ( ( C2 != bot_bot_set_nat )
=> ( ( sup_sup_set_nat @ A4 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ C2 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X4: nat] : ( sup_sup_set_nat @ A4 @ ( B5 @ X4 ) )
@ C2 ) ) ) ) ) ).
% UN_extend_simps(3)
thf(fact_1017_UN__extend__simps_I3_J,axiom,
! [C2: set_o,A4: set_nat,B5: $o > set_nat] :
( ( ( C2 = bot_bot_set_o )
=> ( ( sup_sup_set_nat @ A4 @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B5 @ C2 ) ) )
= A4 ) )
& ( ( C2 != bot_bot_set_o )
=> ( ( sup_sup_set_nat @ A4 @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B5 @ C2 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [X4: $o] : ( sup_sup_set_nat @ A4 @ ( B5 @ X4 ) )
@ C2 ) ) ) ) ) ).
% UN_extend_simps(3)
thf(fact_1018_cSUP__le__iff,axiom,
! [A4: set_nat,F2: nat > set_nat,U2: set_nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ F2 @ A4 ) )
=> ( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ A4 ) ) @ U2 )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_set_nat @ ( F2 @ X4 ) @ U2 ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_1019_cSUP__le__iff,axiom,
! [A4: set_o,F2: $o > set_nat,U2: set_nat] :
( ( A4 != bot_bot_set_o )
=> ( ( condit5477540289124974626et_nat @ ( image_o_set_nat @ F2 @ A4 ) )
=> ( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F2 @ A4 ) ) @ U2 )
= ( ! [X4: $o] :
( ( member_o @ X4 @ A4 )
=> ( ord_less_eq_set_nat @ ( F2 @ X4 ) @ U2 ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_1020_cSUP__mono,axiom,
! [A4: set_Pr1261947904930325089at_nat,G: nat > set_nat,B5: set_nat,F2: product_prod_nat_nat > set_nat] :
( ( A4 != bot_bo2099793752762293965at_nat )
=> ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ G @ B5 ) )
=> ( ! [N3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ N3 @ A4 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B5 )
& ( ord_less_eq_set_nat @ ( F2 @ N3 ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F2 @ A4 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B5 ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_1021_cSUP__mono,axiom,
! [A4: set_nat,G: nat > set_nat,B5: set_nat,F2: nat > set_nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ G @ B5 ) )
=> ( ! [N3: nat] :
( ( member_nat @ N3 @ A4 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B5 )
& ( ord_less_eq_set_nat @ ( F2 @ N3 ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ A4 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B5 ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_1022_cSUP__mono,axiom,
! [A4: set_o,G: nat > set_nat,B5: set_nat,F2: $o > set_nat] :
( ( A4 != bot_bot_set_o )
=> ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ G @ B5 ) )
=> ( ! [N3: $o] :
( ( member_o @ N3 @ A4 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B5 )
& ( ord_less_eq_set_nat @ ( F2 @ N3 ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F2 @ A4 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B5 ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_1023_cSup__subset__mono,axiom,
! [A4: set_nat,B5: set_nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ A4 ) @ ( complete_Sup_Sup_nat @ B5 ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_1024_cSup__subset__mono,axiom,
! [A4: set_o,B5: set_o] :
( ( A4 != bot_bot_set_o )
=> ( ( condit5488710616941104124bove_o @ B5 )
=> ( ( ord_less_eq_set_o @ A4 @ B5 )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A4 ) @ ( complete_Sup_Sup_o @ B5 ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_1025_cSup__subset__mono,axiom,
! [A4: set_set_nat,B5: set_set_nat] :
( ( A4 != bot_bot_set_set_nat )
=> ( ( condit5477540289124974626et_nat @ B5 )
=> ( ( ord_le6893508408891458716et_nat @ A4 @ B5 )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A4 ) @ ( comple7399068483239264473et_nat @ B5 ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_1026_cSUP__subset__mono,axiom,
! [A4: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > set_nat,B5: set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > set_nat] :
( ( A4 != bot_bo2099793752762293965at_nat )
=> ( ( condit5477540289124974626et_nat @ ( image_15824709712370754et_nat @ G @ B5 ) )
=> ( ( ord_le3146513528884898305at_nat @ A4 @ B5 )
=> ( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A4 )
=> ( ord_less_eq_set_nat @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F2 @ A4 ) ) @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ G @ B5 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_1027_cSUP__subset__mono,axiom,
! [A4: set_nat,G: nat > set_nat,B5: set_nat,F2: nat > set_nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ G @ B5 ) )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ord_less_eq_set_nat @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ A4 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B5 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_1028_cSUP__subset__mono,axiom,
! [A4: set_o,G: $o > set_nat,B5: set_o,F2: $o > set_nat] :
( ( A4 != bot_bot_set_o )
=> ( ( condit5477540289124974626et_nat @ ( image_o_set_nat @ G @ B5 ) )
=> ( ( ord_less_eq_set_o @ A4 @ B5 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A4 )
=> ( ord_less_eq_set_nat @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F2 @ A4 ) ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ G @ B5 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_1029_injectivity__image__union,axiom,
! [F3: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,P2: set_se7855581050983116737at_nat,G3: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,P4: set_se7855581050983116737at_nat,A4: set_Pr1261947904930325089at_nat] :
( ( ( image_3684629450409544005at_nat
@ ^ [X9: set_Pr1261947904930325089at_nat] : ( sup_su6327502436637775413at_nat @ X9 @ ( F3 @ X9 ) )
@ P2 )
= ( image_3684629450409544005at_nat
@ ^ [X9: set_Pr1261947904930325089at_nat] : ( sup_su6327502436637775413at_nat @ X9 @ ( G3 @ X9 ) )
@ P4 ) )
=> ( ! [X3: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ X3 @ P2 )
=> ( ord_le3146513528884898305at_nat @ X3 @ A4 ) )
=> ( ! [X3: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ X3 @ P4 )
=> ( ord_le3146513528884898305at_nat @ X3 @ A4 ) )
=> ( ! [X3: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ X3 @ P2 )
=> ! [Xa: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Xa @ ( F3 @ X3 ) )
=> ~ ( member8440522571783428010at_nat @ Xa @ A4 ) ) )
=> ( ! [X3: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ X3 @ P4 )
=> ! [Xa: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Xa @ ( G3 @ X3 ) )
=> ~ ( member8440522571783428010at_nat @ Xa @ A4 ) ) )
=> ( P2 = P4 ) ) ) ) ) ) ).
% injectivity_image_union
thf(fact_1030_injectivity__image__union,axiom,
! [F3: set_nat > set_nat,P2: set_set_nat,G3: set_nat > set_nat,P4: set_set_nat,A4: set_nat] :
( ( ( image_7916887816326733075et_nat
@ ^ [X9: set_nat] : ( sup_sup_set_nat @ X9 @ ( F3 @ X9 ) )
@ P2 )
= ( image_7916887816326733075et_nat
@ ^ [X9: set_nat] : ( sup_sup_set_nat @ X9 @ ( G3 @ X9 ) )
@ P4 ) )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ P2 )
=> ( ord_less_eq_set_nat @ X3 @ A4 ) )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ P4 )
=> ( ord_less_eq_set_nat @ X3 @ A4 ) )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ P2 )
=> ! [Xa: nat] :
( ( member_nat @ Xa @ ( F3 @ X3 ) )
=> ~ ( member_nat @ Xa @ A4 ) ) )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ P4 )
=> ! [Xa: nat] :
( ( member_nat @ Xa @ ( G3 @ X3 ) )
=> ~ ( member_nat @ Xa @ A4 ) ) )
=> ( P2 = P4 ) ) ) ) ) ) ).
% injectivity_image_union
thf(fact_1031_injectivity__union,axiom,
! [A4: set_nat,B5: set_nat,C2: set_nat,D: set_nat,P2: set_nat > $o,Q: set_nat > $o] :
( ( ( sup_sup_set_nat @ A4 @ B5 )
= ( sup_sup_set_nat @ C2 @ D ) )
=> ( ( P2 @ A4 )
=> ( ( P2 @ C2 )
=> ( ( Q @ B5 )
=> ( ( Q @ D )
=> ( ! [S5: set_nat,T4: set_nat] :
( ( P2 @ S5 )
=> ( ( Q @ T4 )
=> ( ( inf_inf_set_nat @ S5 @ T4 )
= bot_bot_set_nat ) ) )
=> ( ( A4 = C2 )
& ( B5 = D ) ) ) ) ) ) ) ) ).
% injectivity_union
thf(fact_1032_injectivity__union,axiom,
! [A4: set_o,B5: set_o,C2: set_o,D: set_o,P2: set_o > $o,Q: set_o > $o] :
( ( ( sup_sup_set_o @ A4 @ B5 )
= ( sup_sup_set_o @ C2 @ D ) )
=> ( ( P2 @ A4 )
=> ( ( P2 @ C2 )
=> ( ( Q @ B5 )
=> ( ( Q @ D )
=> ( ! [S5: set_o,T4: set_o] :
( ( P2 @ S5 )
=> ( ( Q @ T4 )
=> ( ( inf_inf_set_o @ S5 @ T4 )
= bot_bot_set_o ) ) )
=> ( ( A4 = C2 )
& ( B5 = D ) ) ) ) ) ) ) ) ).
% injectivity_union
thf(fact_1033_inj__on__impl__inj__on__image,axiom,
! [F2: nat > set_nat,A4: set_nat,X5: set_set_nat] :
( ( inj_on_nat_set_nat @ F2 @ A4 )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ X5 )
=> ( ord_less_eq_set_nat @ X3 @ A4 ) )
=> ( inj_on2776966659131765557et_nat @ ( image_nat_set_nat @ F2 ) @ X5 ) ) ) ).
% inj_on_impl_inj_on_image
thf(fact_1034_sup__set__def,axiom,
( sup_su6327502436637775413at_nat
= ( ^ [A6: set_Pr1261947904930325089at_nat,B7: set_Pr1261947904930325089at_nat] :
( collec3392354462482085612at_nat
@ ( sup_su798857527126471912_nat_o
@ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ A6 )
@ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ B7 ) ) ) ) ) ).
% sup_set_def
thf(fact_1035_sup__set__def,axiom,
( sup_sup_set_nat
= ( ^ [A6: set_nat,B7: set_nat] :
( collect_nat
@ ( sup_sup_nat_o
@ ^ [X4: nat] : ( member_nat @ X4 @ A6 )
@ ^ [X4: nat] : ( member_nat @ X4 @ B7 ) ) ) ) ) ).
% sup_set_def
thf(fact_1036_sup__Un__eq,axiom,
! [R2: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
( ( sup_su798857527126471912_nat_o
@ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ R2 )
@ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ S3 ) )
= ( ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ ( sup_su6327502436637775413at_nat @ R2 @ S3 ) ) ) ) ).
% sup_Un_eq
thf(fact_1037_sup__Un__eq,axiom,
! [R2: set_nat,S3: set_nat] :
( ( sup_sup_nat_o
@ ^ [X4: nat] : ( member_nat @ X4 @ R2 )
@ ^ [X4: nat] : ( member_nat @ X4 @ S3 ) )
= ( ^ [X4: nat] : ( member_nat @ X4 @ ( sup_sup_set_nat @ R2 @ S3 ) ) ) ) ).
% sup_Un_eq
thf(fact_1038_inj__singleton,axiom,
! [A4: set_nat] :
( inj_on_nat_set_nat
@ ^ [X4: nat] : ( insert_nat @ X4 @ bot_bot_set_nat )
@ A4 ) ).
% inj_singleton
thf(fact_1039_inj__singleton,axiom,
! [A4: set_o] :
( inj_on_o_set_o
@ ^ [X4: $o] : ( insert_o @ X4 @ bot_bot_set_o )
@ A4 ) ).
% inj_singleton
thf(fact_1040_sup__Un__eq2,axiom,
! [R2: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
( ( sup_sup_nat_nat_o
@ ^ [X4: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ R2 )
@ ^ [X4: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ S3 ) )
= ( ^ [X4: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ ( sup_su6327502436637775413at_nat @ R2 @ S3 ) ) ) ) ).
% sup_Un_eq2
thf(fact_1041_injectivity__image,axiom,
! [F2: nat > set_nat,A4: set_nat,G: nat > set_nat,Invert: set_nat > nat] :
( ( ( image_nat_set_nat @ F2 @ A4 )
= ( image_nat_set_nat @ G @ A4 ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ( ( Invert @ ( F2 @ X3 ) )
= X3 )
& ( ( Invert @ ( G @ X3 ) )
= X3 ) ) )
=> ! [X6: nat] :
( ( member_nat @ X6 @ A4 )
=> ( ( F2 @ X6 )
= ( G @ X6 ) ) ) ) ) ).
% injectivity_image
thf(fact_1042_Pow__insert,axiom,
! [A: nat,A4: set_nat] :
( ( pow_nat @ ( insert_nat @ A @ A4 ) )
= ( sup_sup_set_set_nat @ ( pow_nat @ A4 ) @ ( image_7916887816326733075et_nat @ ( insert_nat @ A ) @ ( pow_nat @ A4 ) ) ) ) ).
% Pow_insert
thf(fact_1043_Pow__insert,axiom,
! [A: $o,A4: set_o] :
( ( pow_o @ ( insert_o @ A @ A4 ) )
= ( sup_sup_set_set_o @ ( pow_o @ A4 ) @ ( image_set_o_set_o @ ( insert_o @ A ) @ ( pow_o @ A4 ) ) ) ) ).
% Pow_insert
thf(fact_1044_inj__on__vimage__singleton,axiom,
! [F2: product_prod_nat_nat > nat,A4: set_Pr1261947904930325089at_nat,A: nat] :
( ( inj_on2178005380612969504at_nat @ F2 @ A4 )
=> ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ ( vimage4653281326611754070at_nat @ F2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ A4 )
@ ( insert8211810215607154385at_nat
@ ( the_Pr7557018466319803784at_nat
@ ^ [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ A4 )
& ( ( F2 @ X4 )
= A ) ) )
@ bot_bo2099793752762293965at_nat ) ) ) ).
% inj_on_vimage_singleton
thf(fact_1045_inj__on__vimage__singleton,axiom,
! [F2: nat > nat,A4: set_nat,A: nat] :
( ( inj_on_nat_nat @ F2 @ A4 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ ( vimage_nat_nat @ F2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ A4 )
@ ( insert_nat
@ ( the_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ( ( F2 @ X4 )
= A ) ) )
@ bot_bot_set_nat ) ) ) ).
% inj_on_vimage_singleton
thf(fact_1046_inj__on__vimage__singleton,axiom,
! [F2: $o > nat,A4: set_o,A: nat] :
( ( inj_on_o_nat @ F2 @ A4 )
=> ( ord_less_eq_set_o @ ( inf_inf_set_o @ ( vimage_o_nat @ F2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ A4 )
@ ( insert_o
@ ( the_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ A4 )
& ( ( F2 @ X4 )
= A ) ) )
@ bot_bot_set_o ) ) ) ).
% inj_on_vimage_singleton
thf(fact_1047_inj__on__vimage__singleton,axiom,
! [F2: product_prod_nat_nat > $o,A4: set_Pr1261947904930325089at_nat,A: $o] :
( ( inj_on3734678552557954888_nat_o @ F2 @ A4 )
=> ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ ( vimage2444243245536687506_nat_o @ F2 @ ( insert_o @ A @ bot_bot_set_o ) ) @ A4 )
@ ( insert8211810215607154385at_nat
@ ( the_Pr7557018466319803784at_nat
@ ^ [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ A4 )
& ( ( F2 @ X4 )
= A ) ) )
@ bot_bo2099793752762293965at_nat ) ) ) ).
% inj_on_vimage_singleton
thf(fact_1048_inj__on__vimage__singleton,axiom,
! [F2: nat > $o,A4: set_nat,A: $o] :
( ( inj_on_nat_o @ F2 @ A4 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ ( vimage_nat_o @ F2 @ ( insert_o @ A @ bot_bot_set_o ) ) @ A4 )
@ ( insert_nat
@ ( the_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ( ( F2 @ X4 )
= A ) ) )
@ bot_bot_set_nat ) ) ) ).
% inj_on_vimage_singleton
thf(fact_1049_inj__on__vimage__singleton,axiom,
! [F2: $o > $o,A4: set_o,A: $o] :
( ( inj_on_o_o @ F2 @ A4 )
=> ( ord_less_eq_set_o @ ( inf_inf_set_o @ ( vimage_o_o @ F2 @ ( insert_o @ A @ bot_bot_set_o ) ) @ A4 )
@ ( insert_o
@ ( the_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ A4 )
& ( ( F2 @ X4 )
= A ) ) )
@ bot_bot_set_o ) ) ) ).
% inj_on_vimage_singleton
thf(fact_1050_linear__order__on__singleton,axiom,
! [X: nat] : ( order_4473980167227706203on_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X @ X ) @ bot_bo2099793752762293965at_nat ) ) ).
% linear_order_on_singleton
thf(fact_1051_linear__order__on__singleton,axiom,
! [X: $o] : ( order_6238756238976269133r_on_o @ ( insert_o @ X @ bot_bot_set_o ) @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ X @ X ) @ bot_bo7073875226086086771od_o_o ) ) ).
% linear_order_on_singleton
thf(fact_1052_The__split__eq,axiom,
! [X: nat,Y3: nat] :
( ( the_Pr7557018466319803784at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [X10: nat,Y8: nat] :
( ( X = X10 )
& ( Y3 = Y8 ) ) ) )
= ( product_Pair_nat_nat @ X @ Y3 ) ) ).
% The_split_eq
thf(fact_1053_lnear__order__on__empty,axiom,
order_4473980167227706203on_nat @ bot_bot_set_nat @ bot_bo2099793752762293965at_nat ).
% lnear_order_on_empty
thf(fact_1054_lnear__order__on__empty,axiom,
order_6238756238976269133r_on_o @ bot_bot_set_o @ bot_bo7073875226086086771od_o_o ).
% lnear_order_on_empty
thf(fact_1055_the__elem__def,axiom,
( the_elem_nat
= ( ^ [X9: set_nat] :
( the_nat
@ ^ [X4: nat] :
( X9
= ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) ).
% the_elem_def
thf(fact_1056_the__elem__def,axiom,
( the_elem_o
= ( ^ [X9: set_o] :
( the_o
@ ^ [X4: $o] :
( X9
= ( insert_o @ X4 @ bot_bot_set_o ) ) ) ) ) ).
% the_elem_def
thf(fact_1057_linear__order__on__Restr,axiom,
! [A4: set_nat,R: set_Pr1261947904930325089at_nat,X: nat] :
( ( order_4473980167227706203on_nat @ A4 @ R )
=> ( order_4473980167227706203on_nat @ ( inf_inf_set_nat @ A4 @ ( order_above_nat @ R @ X ) )
@ ( inf_in2572325071724192079at_nat @ R
@ ( produc457027306803732586at_nat @ ( order_above_nat @ R @ X )
@ ^ [Uu: nat] : ( order_above_nat @ R @ X ) ) ) ) ) ).
% linear_order_on_Restr
thf(fact_1058_above__def,axiom,
( order_above_nat
= ( ^ [R3: set_Pr1261947904930325089at_nat,A5: nat] :
( collect_nat
@ ^ [B4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A5 @ B4 ) @ R3 ) ) ) ) ).
% above_def
thf(fact_1059_inj__vimage__singleton,axiom,
! [F2: nat > nat,A: nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
=> ( ord_less_eq_set_nat @ ( vimage_nat_nat @ F2 @ ( insert_nat @ A @ bot_bot_set_nat ) )
@ ( insert_nat
@ ( the_nat
@ ^ [X4: nat] :
( ( F2 @ X4 )
= A ) )
@ bot_bot_set_nat ) ) ) ).
% inj_vimage_singleton
thf(fact_1060_inj__vimage__singleton,axiom,
! [F2: nat > $o,A: $o] :
( ( inj_on_nat_o @ F2 @ top_top_set_nat )
=> ( ord_less_eq_set_nat @ ( vimage_nat_o @ F2 @ ( insert_o @ A @ bot_bot_set_o ) )
@ ( insert_nat
@ ( the_nat
@ ^ [X4: nat] :
( ( F2 @ X4 )
= A ) )
@ bot_bot_set_nat ) ) ) ).
% inj_vimage_singleton
thf(fact_1061_inj__vimage__singleton,axiom,
! [F2: $o > nat,A: nat] :
( ( inj_on_o_nat @ F2 @ top_top_set_o )
=> ( ord_less_eq_set_o @ ( vimage_o_nat @ F2 @ ( insert_nat @ A @ bot_bot_set_nat ) )
@ ( insert_o
@ ( the_o
@ ^ [X4: $o] :
( ( F2 @ X4 )
= A ) )
@ bot_bot_set_o ) ) ) ).
% inj_vimage_singleton
thf(fact_1062_inj__vimage__singleton,axiom,
! [F2: $o > $o,A: $o] :
( ( inj_on_o_o @ F2 @ top_top_set_o )
=> ( ord_less_eq_set_o @ ( vimage_o_o @ F2 @ ( insert_o @ A @ bot_bot_set_o ) )
@ ( insert_o
@ ( the_o
@ ^ [X4: $o] :
( ( F2 @ X4 )
= A ) )
@ bot_bot_set_o ) ) ) ).
% inj_vimage_singleton
thf(fact_1063_UNIV__I,axiom,
! [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ top_to4669805908274784177at_nat ) ).
% UNIV_I
thf(fact_1064_UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% UNIV_I
thf(fact_1065_UNIV__I,axiom,
! [X: $o] : ( member_o @ X @ top_top_set_o ) ).
% UNIV_I
thf(fact_1066_Int__UNIV,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ( inf_inf_set_nat @ A4 @ B5 )
= top_top_set_nat )
= ( ( A4 = top_top_set_nat )
& ( B5 = top_top_set_nat ) ) ) ).
% Int_UNIV
thf(fact_1067_Int__UNIV,axiom,
! [A4: set_o,B5: set_o] :
( ( ( inf_inf_set_o @ A4 @ B5 )
= top_top_set_o )
= ( ( A4 = top_top_set_o )
& ( B5 = top_top_set_o ) ) ) ).
% Int_UNIV
thf(fact_1068_vimage__UNIV,axiom,
! [F2: nat > nat] :
( ( vimage_nat_nat @ F2 @ top_top_set_nat )
= top_top_set_nat ) ).
% vimage_UNIV
thf(fact_1069_vimage__UNIV,axiom,
! [F2: $o > nat] :
( ( vimage_o_nat @ F2 @ top_top_set_nat )
= top_top_set_o ) ).
% vimage_UNIV
thf(fact_1070_vimage__UNIV,axiom,
! [F2: nat > $o] :
( ( vimage_nat_o @ F2 @ top_top_set_o )
= top_top_set_nat ) ).
% vimage_UNIV
thf(fact_1071_vimage__UNIV,axiom,
! [F2: $o > $o] :
( ( vimage_o_o @ F2 @ top_top_set_o )
= top_top_set_o ) ).
% vimage_UNIV
thf(fact_1072_Pow__UNIV,axiom,
( ( pow_nat @ top_top_set_nat )
= top_top_set_set_nat ) ).
% Pow_UNIV
thf(fact_1073_Pow__UNIV,axiom,
( ( pow_o @ top_top_set_o )
= top_top_set_set_o ) ).
% Pow_UNIV
thf(fact_1074_Collect__const,axiom,
! [P2: $o] :
( ( P2
=> ( ( collect_nat
@ ^ [S6: nat] : P2 )
= top_top_set_nat ) )
& ( ~ P2
=> ( ( collect_nat
@ ^ [S6: nat] : P2 )
= bot_bot_set_nat ) ) ) ).
% Collect_const
thf(fact_1075_Collect__const,axiom,
! [P2: $o] :
( ( P2
=> ( ( collect_o
@ ^ [S6: $o] : P2 )
= top_top_set_o ) )
& ( ~ P2
=> ( ( collect_o
@ ^ [S6: $o] : P2 )
= bot_bot_set_o ) ) ) ).
% Collect_const
thf(fact_1076_UNIV__Times__UNIV,axiom,
( ( produc457027306803732586at_nat @ top_top_set_nat
@ ^ [Uu: nat] : top_top_set_nat )
= top_to4669805908274784177at_nat ) ).
% UNIV_Times_UNIV
thf(fact_1077_UNIV__Times__UNIV,axiom,
( ( product_Sigma_nat_o @ top_top_set_nat
@ ^ [Uu: nat] : top_top_set_o )
= top_to8070287629520841379_nat_o ) ).
% UNIV_Times_UNIV
thf(fact_1078_UNIV__Times__UNIV,axiom,
( ( product_Sigma_o_nat @ top_top_set_o
@ ^ [Uu: $o] : top_top_set_nat )
= top_to7022684507342537725_o_nat ) ).
% UNIV_Times_UNIV
thf(fact_1079_UNIV__Times__UNIV,axiom,
( ( product_Sigma_o_o @ top_top_set_o
@ ^ [Uu: $o] : top_top_set_o )
= top_to7721136755696657239od_o_o ) ).
% UNIV_Times_UNIV
thf(fact_1080_vimage__const,axiom,
! [C: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ( ( member8440522571783428010at_nat @ C @ A4 )
=> ( ( vimage8013328719654469172at_nat
@ ^ [X4: nat] : C
@ A4 )
= top_top_set_nat ) )
& ( ~ ( member8440522571783428010at_nat @ C @ A4 )
=> ( ( vimage8013328719654469172at_nat
@ ^ [X4: nat] : C
@ A4 )
= bot_bot_set_nat ) ) ) ).
% vimage_const
thf(fact_1081_vimage__const,axiom,
! [C: nat,A4: set_nat] :
( ( ( member_nat @ C @ A4 )
=> ( ( vimage_nat_nat
@ ^ [X4: nat] : C
@ A4 )
= top_top_set_nat ) )
& ( ~ ( member_nat @ C @ A4 )
=> ( ( vimage_nat_nat
@ ^ [X4: nat] : C
@ A4 )
= bot_bot_set_nat ) ) ) ).
% vimage_const
thf(fact_1082_vimage__const,axiom,
! [C: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ( ( member8440522571783428010at_nat @ C @ A4 )
=> ( ( vimage2606541041029201316at_nat
@ ^ [X4: $o] : C
@ A4 )
= top_top_set_o ) )
& ( ~ ( member8440522571783428010at_nat @ C @ A4 )
=> ( ( vimage2606541041029201316at_nat
@ ^ [X4: $o] : C
@ A4 )
= bot_bot_set_o ) ) ) ).
% vimage_const
thf(fact_1083_vimage__const,axiom,
! [C: nat,A4: set_nat] :
( ( ( member_nat @ C @ A4 )
=> ( ( vimage_o_nat
@ ^ [X4: $o] : C
@ A4 )
= top_top_set_o ) )
& ( ~ ( member_nat @ C @ A4 )
=> ( ( vimage_o_nat
@ ^ [X4: $o] : C
@ A4 )
= bot_bot_set_o ) ) ) ).
% vimage_const
thf(fact_1084_image__vimage__eq,axiom,
! [F2: nat > set_nat,A4: set_set_nat] :
( ( image_nat_set_nat @ F2 @ ( vimage_nat_set_nat @ F2 @ A4 ) )
= ( inf_inf_set_set_nat @ A4 @ ( image_nat_set_nat @ F2 @ top_top_set_nat ) ) ) ).
% image_vimage_eq
thf(fact_1085_range__constant,axiom,
! [X: set_nat] :
( ( image_nat_set_nat
@ ^ [Uu: nat] : X
@ top_top_set_nat )
= ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ).
% range_constant
thf(fact_1086_range__constant,axiom,
! [X: nat] :
( ( image_nat_nat2
@ ^ [Uu: nat] : X
@ top_top_set_nat )
= ( insert_nat @ X @ bot_bot_set_nat ) ) ).
% range_constant
thf(fact_1087_range__constant,axiom,
! [X: $o] :
( ( image_nat_o2
@ ^ [Uu: nat] : X
@ top_top_set_nat )
= ( insert_o @ X @ bot_bot_set_o ) ) ).
% range_constant
thf(fact_1088_range__constant,axiom,
! [X: nat] :
( ( image_o_nat2
@ ^ [Uu: $o] : X
@ top_top_set_o )
= ( insert_nat @ X @ bot_bot_set_nat ) ) ).
% range_constant
thf(fact_1089_range__constant,axiom,
! [X: $o] :
( ( image_o_o2
@ ^ [Uu: $o] : X
@ top_top_set_o )
= ( insert_o @ X @ bot_bot_set_o ) ) ).
% range_constant
thf(fact_1090_UN__lessThan__UNIV,axiom,
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_lessThan_nat @ top_top_set_nat ) )
= top_top_set_nat ) ).
% UN_lessThan_UNIV
thf(fact_1091_surjD,axiom,
! [F2: nat > set_nat,Y3: set_nat] :
( ( ( image_nat_set_nat @ F2 @ top_top_set_nat )
= top_top_set_set_nat )
=> ? [X3: nat] :
( Y3
= ( F2 @ X3 ) ) ) ).
% surjD
thf(fact_1092_surjD,axiom,
! [F2: nat > nat,Y3: nat] :
( ( ( image_nat_nat2 @ F2 @ top_top_set_nat )
= top_top_set_nat )
=> ? [X3: nat] :
( Y3
= ( F2 @ X3 ) ) ) ).
% surjD
thf(fact_1093_surjD,axiom,
! [F2: nat > $o,Y3: $o] :
( ( ( image_nat_o2 @ F2 @ top_top_set_nat )
= top_top_set_o )
=> ? [X3: nat] :
( Y3
= ( F2 @ X3 ) ) ) ).
% surjD
thf(fact_1094_surjD,axiom,
! [F2: $o > nat,Y3: nat] :
( ( ( image_o_nat2 @ F2 @ top_top_set_o )
= top_top_set_nat )
=> ? [X3: $o] :
( Y3
= ( F2 @ X3 ) ) ) ).
% surjD
thf(fact_1095_surjD,axiom,
! [F2: $o > $o,Y3: $o] :
( ( ( image_o_o2 @ F2 @ top_top_set_o )
= top_top_set_o )
=> ? [X3: $o] :
( Y3
= ( F2 @ X3 ) ) ) ).
% surjD
thf(fact_1096_surjE,axiom,
! [F2: nat > set_nat,Y3: set_nat] :
( ( ( image_nat_set_nat @ F2 @ top_top_set_nat )
= top_top_set_set_nat )
=> ~ ! [X3: nat] :
( Y3
!= ( F2 @ X3 ) ) ) ).
% surjE
thf(fact_1097_surjE,axiom,
! [F2: nat > nat,Y3: nat] :
( ( ( image_nat_nat2 @ F2 @ top_top_set_nat )
= top_top_set_nat )
=> ~ ! [X3: nat] :
( Y3
!= ( F2 @ X3 ) ) ) ).
% surjE
thf(fact_1098_surjE,axiom,
! [F2: nat > $o,Y3: $o] :
( ( ( image_nat_o2 @ F2 @ top_top_set_nat )
= top_top_set_o )
=> ~ ! [X3: nat] :
( Y3
= ( ~ ( F2 @ X3 ) ) ) ) ).
% surjE
thf(fact_1099_surjE,axiom,
! [F2: $o > nat,Y3: nat] :
( ( ( image_o_nat2 @ F2 @ top_top_set_o )
= top_top_set_nat )
=> ~ ! [X3: $o] :
( Y3
!= ( F2 @ X3 ) ) ) ).
% surjE
thf(fact_1100_surjE,axiom,
! [F2: $o > $o,Y3: $o] :
( ( ( image_o_o2 @ F2 @ top_top_set_o )
= top_top_set_o )
=> ~ ! [X3: $o] :
( Y3
= ( ~ ( F2 @ X3 ) ) ) ) ).
% surjE
thf(fact_1101_surjI,axiom,
! [G: nat > set_nat,F2: set_nat > nat] :
( ! [X3: set_nat] :
( ( G @ ( F2 @ X3 ) )
= X3 )
=> ( ( image_nat_set_nat @ G @ top_top_set_nat )
= top_top_set_set_nat ) ) ).
% surjI
thf(fact_1102_surjI,axiom,
! [G: nat > nat,F2: nat > nat] :
( ! [X3: nat] :
( ( G @ ( F2 @ X3 ) )
= X3 )
=> ( ( image_nat_nat2 @ G @ top_top_set_nat )
= top_top_set_nat ) ) ).
% surjI
thf(fact_1103_surjI,axiom,
! [G: nat > $o,F2: $o > nat] :
( ! [X3: $o] :
( ( G @ ( F2 @ X3 ) )
= X3 )
=> ( ( image_nat_o2 @ G @ top_top_set_nat )
= top_top_set_o ) ) ).
% surjI
thf(fact_1104_surjI,axiom,
! [G: $o > nat,F2: nat > $o] :
( ! [X3: nat] :
( ( G @ ( F2 @ X3 ) )
= X3 )
=> ( ( image_o_nat2 @ G @ top_top_set_o )
= top_top_set_nat ) ) ).
% surjI
thf(fact_1105_surjI,axiom,
! [G: $o > $o,F2: $o > $o] :
( ! [X3: $o] :
( ( G @ ( F2 @ X3 ) )
= X3 )
=> ( ( image_o_o2 @ G @ top_top_set_o )
= top_top_set_o ) ) ).
% surjI
thf(fact_1106_surj__def,axiom,
! [F2: nat > set_nat] :
( ( ( image_nat_set_nat @ F2 @ top_top_set_nat )
= top_top_set_set_nat )
= ( ! [Y5: set_nat] :
? [X4: nat] :
( Y5
= ( F2 @ X4 ) ) ) ) ).
% surj_def
thf(fact_1107_surj__def,axiom,
! [F2: nat > nat] :
( ( ( image_nat_nat2 @ F2 @ top_top_set_nat )
= top_top_set_nat )
= ( ! [Y5: nat] :
? [X4: nat] :
( Y5
= ( F2 @ X4 ) ) ) ) ).
% surj_def
thf(fact_1108_surj__def,axiom,
! [F2: nat > $o] :
( ( ( image_nat_o2 @ F2 @ top_top_set_nat )
= top_top_set_o )
= ( ! [Y5: $o] :
? [X4: nat] :
( Y5
= ( F2 @ X4 ) ) ) ) ).
% surj_def
thf(fact_1109_surj__def,axiom,
! [F2: $o > nat] :
( ( ( image_o_nat2 @ F2 @ top_top_set_o )
= top_top_set_nat )
= ( ! [Y5: nat] :
? [X4: $o] :
( Y5
= ( F2 @ X4 ) ) ) ) ).
% surj_def
thf(fact_1110_surj__def,axiom,
! [F2: $o > $o] :
( ( ( image_o_o2 @ F2 @ top_top_set_o )
= top_top_set_o )
= ( ! [Y5: $o] :
? [X4: $o] :
( Y5
= ( F2 @ X4 ) ) ) ) ).
% surj_def
thf(fact_1111_range__eqI,axiom,
! [B: set_nat,F2: nat > set_nat,X: nat] :
( ( B
= ( F2 @ X ) )
=> ( member_set_nat @ B @ ( image_nat_set_nat @ F2 @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_1112_range__eqI,axiom,
! [B: product_prod_nat_nat,F2: nat > product_prod_nat_nat,X: nat] :
( ( B
= ( F2 @ X ) )
=> ( member8440522571783428010at_nat @ B @ ( image_5846123807819985514at_nat @ F2 @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_1113_range__eqI,axiom,
! [B: nat,F2: nat > nat,X: nat] :
( ( B
= ( F2 @ X ) )
=> ( member_nat @ B @ ( image_nat_nat2 @ F2 @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_1114_range__eqI,axiom,
! [B: product_prod_nat_nat,F2: $o > product_prod_nat_nat,X: $o] :
( ( B
= ( F2 @ X ) )
=> ( member8440522571783428010at_nat @ B @ ( image_3855930084881510382at_nat @ F2 @ top_top_set_o ) ) ) ).
% range_eqI
thf(fact_1115_range__eqI,axiom,
! [B: nat,F2: $o > nat,X: $o] :
( ( B
= ( F2 @ X ) )
=> ( member_nat @ B @ ( image_o_nat2 @ F2 @ top_top_set_o ) ) ) ).
% range_eqI
thf(fact_1116_rangeI,axiom,
! [F2: nat > set_nat,X: nat] : ( member_set_nat @ ( F2 @ X ) @ ( image_nat_set_nat @ F2 @ top_top_set_nat ) ) ).
% rangeI
thf(fact_1117_rangeI,axiom,
! [F2: nat > product_prod_nat_nat,X: nat] : ( member8440522571783428010at_nat @ ( F2 @ X ) @ ( image_5846123807819985514at_nat @ F2 @ top_top_set_nat ) ) ).
% rangeI
thf(fact_1118_rangeI,axiom,
! [F2: nat > nat,X: nat] : ( member_nat @ ( F2 @ X ) @ ( image_nat_nat2 @ F2 @ top_top_set_nat ) ) ).
% rangeI
thf(fact_1119_rangeI,axiom,
! [F2: $o > product_prod_nat_nat,X: $o] : ( member8440522571783428010at_nat @ ( F2 @ X ) @ ( image_3855930084881510382at_nat @ F2 @ top_top_set_o ) ) ).
% rangeI
thf(fact_1120_rangeI,axiom,
! [F2: $o > nat,X: $o] : ( member_nat @ ( F2 @ X ) @ ( image_o_nat2 @ F2 @ top_top_set_o ) ) ).
% rangeI
thf(fact_1121_Int__UNIV__right,axiom,
! [A4: set_nat] :
( ( inf_inf_set_nat @ A4 @ top_top_set_nat )
= A4 ) ).
% Int_UNIV_right
thf(fact_1122_Int__UNIV__right,axiom,
! [A4: set_o] :
( ( inf_inf_set_o @ A4 @ top_top_set_o )
= A4 ) ).
% Int_UNIV_right
thf(fact_1123_Int__UNIV__left,axiom,
! [B5: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ B5 )
= B5 ) ).
% Int_UNIV_left
thf(fact_1124_Int__UNIV__left,axiom,
! [B5: set_o] :
( ( inf_inf_set_o @ top_top_set_o @ B5 )
= B5 ) ).
% Int_UNIV_left
thf(fact_1125_range__composition,axiom,
! [F2: set_nat > set_nat,G: nat > set_nat] :
( ( image_nat_set_nat
@ ^ [X4: nat] : ( F2 @ ( G @ X4 ) )
@ top_top_set_nat )
= ( image_7916887816326733075et_nat @ F2 @ ( image_nat_set_nat @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_1126_range__composition,axiom,
! [F2: nat > set_nat,G: nat > nat] :
( ( image_nat_set_nat
@ ^ [X4: nat] : ( F2 @ ( G @ X4 ) )
@ top_top_set_nat )
= ( image_nat_set_nat @ F2 @ ( image_nat_nat2 @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_1127_range__composition,axiom,
! [F2: nat > set_nat,G: $o > nat] :
( ( image_o_set_nat
@ ^ [X4: $o] : ( F2 @ ( G @ X4 ) )
@ top_top_set_o )
= ( image_nat_set_nat @ F2 @ ( image_o_nat2 @ G @ top_top_set_o ) ) ) ).
% range_composition
thf(fact_1128_rangeE,axiom,
! [B: set_nat,F2: nat > set_nat] :
( ( member_set_nat @ B @ ( image_nat_set_nat @ F2 @ top_top_set_nat ) )
=> ~ ! [X3: nat] :
( B
!= ( F2 @ X3 ) ) ) ).
% rangeE
thf(fact_1129_rangeE,axiom,
! [B: product_prod_nat_nat,F2: nat > product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ B @ ( image_5846123807819985514at_nat @ F2 @ top_top_set_nat ) )
=> ~ ! [X3: nat] :
( B
!= ( F2 @ X3 ) ) ) ).
% rangeE
thf(fact_1130_rangeE,axiom,
! [B: nat,F2: nat > nat] :
( ( member_nat @ B @ ( image_nat_nat2 @ F2 @ top_top_set_nat ) )
=> ~ ! [X3: nat] :
( B
!= ( F2 @ X3 ) ) ) ).
% rangeE
thf(fact_1131_rangeE,axiom,
! [B: product_prod_nat_nat,F2: $o > product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ B @ ( image_3855930084881510382at_nat @ F2 @ top_top_set_o ) )
=> ~ ! [X3: $o] :
( B
!= ( F2 @ X3 ) ) ) ).
% rangeE
thf(fact_1132_rangeE,axiom,
! [B: nat,F2: $o > nat] :
( ( member_nat @ B @ ( image_o_nat2 @ F2 @ top_top_set_o ) )
=> ~ ! [X3: $o] :
( B
!= ( F2 @ X3 ) ) ) ).
% rangeE
thf(fact_1133_empty__not__UNIV,axiom,
bot_bot_set_nat != top_top_set_nat ).
% empty_not_UNIV
thf(fact_1134_empty__not__UNIV,axiom,
bot_bot_set_o != top_top_set_o ).
% empty_not_UNIV
thf(fact_1135_insert__UNIV,axiom,
! [X: nat] :
( ( insert_nat @ X @ top_top_set_nat )
= top_top_set_nat ) ).
% insert_UNIV
thf(fact_1136_insert__UNIV,axiom,
! [X: $o] :
( ( insert_o @ X @ top_top_set_o )
= top_top_set_o ) ).
% insert_UNIV
thf(fact_1137_UNIV__witness,axiom,
? [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ top_to4669805908274784177at_nat ) ).
% UNIV_witness
thf(fact_1138_UNIV__witness,axiom,
? [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_1139_UNIV__witness,axiom,
? [X3: $o] : ( member_o @ X3 @ top_top_set_o ) ).
% UNIV_witness
thf(fact_1140_UNIV__eq__I,axiom,
! [A4: set_Pr1261947904930325089at_nat] :
( ! [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ A4 )
=> ( top_to4669805908274784177at_nat = A4 ) ) ).
% UNIV_eq_I
thf(fact_1141_UNIV__eq__I,axiom,
! [A4: set_nat] :
( ! [X3: nat] : ( member_nat @ X3 @ A4 )
=> ( top_top_set_nat = A4 ) ) ).
% UNIV_eq_I
thf(fact_1142_UNIV__eq__I,axiom,
! [A4: set_o] :
( ! [X3: $o] : ( member_o @ X3 @ A4 )
=> ( top_top_set_o = A4 ) ) ).
% UNIV_eq_I
thf(fact_1143_UNIV__def,axiom,
( top_top_set_nat
= ( collect_nat
@ ^ [X4: nat] : $true ) ) ).
% UNIV_def
thf(fact_1144_UNIV__def,axiom,
( top_top_set_o
= ( collect_o
@ ^ [X4: $o] : $true ) ) ).
% UNIV_def
thf(fact_1145_subset__UNIV,axiom,
! [A4: set_nat] : ( ord_less_eq_set_nat @ A4 @ top_top_set_nat ) ).
% subset_UNIV
thf(fact_1146_subset__UNIV,axiom,
! [A4: set_o] : ( ord_less_eq_set_o @ A4 @ top_top_set_o ) ).
% subset_UNIV
thf(fact_1147_Un__UNIV__left,axiom,
! [B5: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ B5 )
= top_top_set_nat ) ).
% Un_UNIV_left
thf(fact_1148_Un__UNIV__left,axiom,
! [B5: set_o] :
( ( sup_sup_set_o @ top_top_set_o @ B5 )
= top_top_set_o ) ).
% Un_UNIV_left
thf(fact_1149_Un__UNIV__right,axiom,
! [A4: set_nat] :
( ( sup_sup_set_nat @ A4 @ top_top_set_nat )
= top_top_set_nat ) ).
% Un_UNIV_right
thf(fact_1150_Un__UNIV__right,axiom,
! [A4: set_o] :
( ( sup_sup_set_o @ A4 @ top_top_set_o )
= top_top_set_o ) ).
% Un_UNIV_right
thf(fact_1151_range__subsetD,axiom,
! [F2: nat > set_nat,B5: set_set_nat,I5: nat] :
( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F2 @ top_top_set_nat ) @ B5 )
=> ( member_set_nat @ ( F2 @ I5 ) @ B5 ) ) ).
% range_subsetD
thf(fact_1152_range__subsetD,axiom,
! [F2: nat > product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat,I5: nat] :
( ( ord_le3146513528884898305at_nat @ ( image_5846123807819985514at_nat @ F2 @ top_top_set_nat ) @ B5 )
=> ( member8440522571783428010at_nat @ ( F2 @ I5 ) @ B5 ) ) ).
% range_subsetD
thf(fact_1153_range__subsetD,axiom,
! [F2: nat > nat,B5: set_nat,I5: nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F2 @ top_top_set_nat ) @ B5 )
=> ( member_nat @ ( F2 @ I5 ) @ B5 ) ) ).
% range_subsetD
thf(fact_1154_range__subsetD,axiom,
! [F2: $o > product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat,I5: $o] :
( ( ord_le3146513528884898305at_nat @ ( image_3855930084881510382at_nat @ F2 @ top_top_set_o ) @ B5 )
=> ( member8440522571783428010at_nat @ ( F2 @ I5 ) @ B5 ) ) ).
% range_subsetD
thf(fact_1155_range__subsetD,axiom,
! [F2: $o > nat,B5: set_nat,I5: $o] :
( ( ord_less_eq_set_nat @ ( image_o_nat2 @ F2 @ top_top_set_o ) @ B5 )
=> ( member_nat @ ( F2 @ I5 ) @ B5 ) ) ).
% range_subsetD
thf(fact_1156_comp__surj,axiom,
! [F2: nat > nat,G: nat > nat] :
( ( ( image_nat_nat2 @ F2 @ top_top_set_nat )
= top_top_set_nat )
=> ( ( ( image_nat_nat2 @ G @ top_top_set_nat )
= top_top_set_nat )
=> ( ( image_nat_nat2 @ ( comp_nat_nat_nat @ G @ F2 ) @ top_top_set_nat )
= top_top_set_nat ) ) ) ).
% comp_surj
thf(fact_1157_comp__surj,axiom,
! [F2: nat > nat,G: nat > $o] :
( ( ( image_nat_nat2 @ F2 @ top_top_set_nat )
= top_top_set_nat )
=> ( ( ( image_nat_o2 @ G @ top_top_set_nat )
= top_top_set_o )
=> ( ( image_nat_o2 @ ( comp_nat_o_nat @ G @ F2 ) @ top_top_set_nat )
= top_top_set_o ) ) ) ).
% comp_surj
thf(fact_1158_comp__surj,axiom,
! [F2: nat > $o,G: $o > nat] :
( ( ( image_nat_o2 @ F2 @ top_top_set_nat )
= top_top_set_o )
=> ( ( ( image_o_nat2 @ G @ top_top_set_o )
= top_top_set_nat )
=> ( ( image_nat_nat2 @ ( comp_o_nat_nat @ G @ F2 ) @ top_top_set_nat )
= top_top_set_nat ) ) ) ).
% comp_surj
thf(fact_1159_comp__surj,axiom,
! [F2: nat > $o,G: $o > $o] :
( ( ( image_nat_o2 @ F2 @ top_top_set_nat )
= top_top_set_o )
=> ( ( ( image_o_o2 @ G @ top_top_set_o )
= top_top_set_o )
=> ( ( image_nat_o2 @ ( comp_o_o_nat @ G @ F2 ) @ top_top_set_nat )
= top_top_set_o ) ) ) ).
% comp_surj
thf(fact_1160_comp__surj,axiom,
! [F2: $o > nat,G: nat > nat] :
( ( ( image_o_nat2 @ F2 @ top_top_set_o )
= top_top_set_nat )
=> ( ( ( image_nat_nat2 @ G @ top_top_set_nat )
= top_top_set_nat )
=> ( ( image_o_nat2 @ ( comp_nat_nat_o @ G @ F2 ) @ top_top_set_o )
= top_top_set_nat ) ) ) ).
% comp_surj
thf(fact_1161_comp__surj,axiom,
! [F2: $o > nat,G: nat > $o] :
( ( ( image_o_nat2 @ F2 @ top_top_set_o )
= top_top_set_nat )
=> ( ( ( image_nat_o2 @ G @ top_top_set_nat )
= top_top_set_o )
=> ( ( image_o_o2 @ ( comp_nat_o_o @ G @ F2 ) @ top_top_set_o )
= top_top_set_o ) ) ) ).
% comp_surj
thf(fact_1162_comp__surj,axiom,
! [F2: $o > $o,G: $o > nat] :
( ( ( image_o_o2 @ F2 @ top_top_set_o )
= top_top_set_o )
=> ( ( ( image_o_nat2 @ G @ top_top_set_o )
= top_top_set_nat )
=> ( ( image_o_nat2 @ ( comp_o_nat_o @ G @ F2 ) @ top_top_set_o )
= top_top_set_nat ) ) ) ).
% comp_surj
thf(fact_1163_comp__surj,axiom,
! [F2: $o > $o,G: $o > $o] :
( ( ( image_o_o2 @ F2 @ top_top_set_o )
= top_top_set_o )
=> ( ( ( image_o_o2 @ G @ top_top_set_o )
= top_top_set_o )
=> ( ( image_o_o2 @ ( comp_o_o_o @ G @ F2 ) @ top_top_set_o )
= top_top_set_o ) ) ) ).
% comp_surj
thf(fact_1164_comp__surj,axiom,
! [F2: nat > set_nat,G: set_nat > nat] :
( ( ( image_nat_set_nat @ F2 @ top_top_set_nat )
= top_top_set_set_nat )
=> ( ( ( image_set_nat_nat @ G @ top_top_set_set_nat )
= top_top_set_nat )
=> ( ( image_nat_nat2 @ ( comp_set_nat_nat_nat @ G @ F2 ) @ top_top_set_nat )
= top_top_set_nat ) ) ) ).
% comp_surj
thf(fact_1165_comp__surj,axiom,
! [F2: nat > set_nat,G: set_nat > $o] :
( ( ( image_nat_set_nat @ F2 @ top_top_set_nat )
= top_top_set_set_nat )
=> ( ( ( image_set_nat_o @ G @ top_top_set_set_nat )
= top_top_set_o )
=> ( ( image_nat_o2 @ ( comp_set_nat_o_nat @ G @ F2 ) @ top_top_set_nat )
= top_top_set_o ) ) ) ).
% comp_surj
thf(fact_1166_inj__image__mem__iff,axiom,
! [F2: product_prod_nat_nat > product_prod_nat_nat,A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ( inj_on8969904277767023793at_nat @ F2 @ top_to4669805908274784177at_nat )
=> ( ( member8440522571783428010at_nat @ ( F2 @ A ) @ ( image_5168914502847457605at_nat @ F2 @ A4 ) )
= ( member8440522571783428010at_nat @ A @ A4 ) ) ) ).
% inj_image_mem_iff
thf(fact_1167_inj__image__mem__iff,axiom,
! [F2: product_prod_nat_nat > nat,A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ( inj_on2178005380612969504at_nat @ F2 @ top_to4669805908274784177at_nat )
=> ( ( member_nat @ ( F2 @ A ) @ ( image_2486076414777270412at_nat @ F2 @ A4 ) )
= ( member8440522571783428010at_nat @ A @ A4 ) ) ) ).
% inj_image_mem_iff
thf(fact_1168_inj__image__mem__iff,axiom,
! [F2: nat > set_nat,A: nat,A4: set_nat] :
( ( inj_on_nat_set_nat @ F2 @ top_top_set_nat )
=> ( ( member_set_nat @ ( F2 @ A ) @ ( image_nat_set_nat @ F2 @ A4 ) )
= ( member_nat @ A @ A4 ) ) ) ).
% inj_image_mem_iff
thf(fact_1169_inj__image__mem__iff,axiom,
! [F2: nat > product_prod_nat_nat,A: nat,A4: set_nat] :
( ( inj_on5538052773655684606at_nat @ F2 @ top_top_set_nat )
=> ( ( member8440522571783428010at_nat @ ( F2 @ A ) @ ( image_5846123807819985514at_nat @ F2 @ A4 ) )
= ( member_nat @ A @ A4 ) ) ) ).
% inj_image_mem_iff
thf(fact_1170_inj__image__mem__iff,axiom,
! [F2: nat > nat,A: nat,A4: set_nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
=> ( ( member_nat @ ( F2 @ A ) @ ( image_nat_nat2 @ F2 @ A4 ) )
= ( member_nat @ A @ A4 ) ) ) ).
% inj_image_mem_iff
thf(fact_1171_inj__image__mem__iff,axiom,
! [F2: $o > product_prod_nat_nat,A: $o,A4: set_o] :
( ( inj_on3896976348050468698at_nat @ F2 @ top_top_set_o )
=> ( ( member8440522571783428010at_nat @ ( F2 @ A ) @ ( image_3855930084881510382at_nat @ F2 @ A4 ) )
= ( member_o @ A @ A4 ) ) ) ).
% inj_image_mem_iff
thf(fact_1172_inj__image__mem__iff,axiom,
! [F2: $o > nat,A: $o,A4: set_o] :
( ( inj_on_o_nat @ F2 @ top_top_set_o )
=> ( ( member_nat @ ( F2 @ A ) @ ( image_o_nat2 @ F2 @ A4 ) )
= ( member_o @ A @ A4 ) ) ) ).
% inj_image_mem_iff
thf(fact_1173_inj__image__eq__iff,axiom,
! [F2: nat > set_nat,A4: set_nat,B5: set_nat] :
( ( inj_on_nat_set_nat @ F2 @ top_top_set_nat )
=> ( ( ( image_nat_set_nat @ F2 @ A4 )
= ( image_nat_set_nat @ F2 @ B5 ) )
= ( A4 = B5 ) ) ) ).
% inj_image_eq_iff
thf(fact_1174_range__ex1__eq,axiom,
! [F2: nat > set_nat,B: set_nat] :
( ( inj_on_nat_set_nat @ F2 @ top_top_set_nat )
=> ( ( member_set_nat @ B @ ( image_nat_set_nat @ F2 @ top_top_set_nat ) )
= ( ? [X4: nat] :
( ( B
= ( F2 @ X4 ) )
& ! [Y5: nat] :
( ( B
= ( F2 @ Y5 ) )
=> ( Y5 = X4 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_1175_range__ex1__eq,axiom,
! [F2: nat > product_prod_nat_nat,B: product_prod_nat_nat] :
( ( inj_on5538052773655684606at_nat @ F2 @ top_top_set_nat )
=> ( ( member8440522571783428010at_nat @ B @ ( image_5846123807819985514at_nat @ F2 @ top_top_set_nat ) )
= ( ? [X4: nat] :
( ( B
= ( F2 @ X4 ) )
& ! [Y5: nat] :
( ( B
= ( F2 @ Y5 ) )
=> ( Y5 = X4 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_1176_range__ex1__eq,axiom,
! [F2: nat > nat,B: nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
=> ( ( member_nat @ B @ ( image_nat_nat2 @ F2 @ top_top_set_nat ) )
= ( ? [X4: nat] :
( ( B
= ( F2 @ X4 ) )
& ! [Y5: nat] :
( ( B
= ( F2 @ Y5 ) )
=> ( Y5 = X4 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_1177_range__ex1__eq,axiom,
! [F2: $o > product_prod_nat_nat,B: product_prod_nat_nat] :
( ( inj_on3896976348050468698at_nat @ F2 @ top_top_set_o )
=> ( ( member8440522571783428010at_nat @ B @ ( image_3855930084881510382at_nat @ F2 @ top_top_set_o ) )
= ( ? [X4: $o] :
( ( B
= ( F2 @ X4 ) )
& ! [Y5: $o] :
( ( B
= ( F2 @ Y5 ) )
=> ( Y5 = X4 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_1178_range__ex1__eq,axiom,
! [F2: $o > nat,B: nat] :
( ( inj_on_o_nat @ F2 @ top_top_set_o )
=> ( ( member_nat @ B @ ( image_o_nat2 @ F2 @ top_top_set_o ) )
= ( ? [X4: $o] :
( ( B
= ( F2 @ X4 ) )
& ! [Y5: $o] :
( ( B
= ( F2 @ Y5 ) )
=> ( Y5 = X4 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_1179_surj__image__vimage__eq,axiom,
! [F2: nat > set_nat,A4: set_set_nat] :
( ( ( image_nat_set_nat @ F2 @ top_top_set_nat )
= top_top_set_set_nat )
=> ( ( image_nat_set_nat @ F2 @ ( vimage_nat_set_nat @ F2 @ A4 ) )
= A4 ) ) ).
% surj_image_vimage_eq
thf(fact_1180_surj__image__vimage__eq,axiom,
! [F2: nat > nat,A4: set_nat] :
( ( ( image_nat_nat2 @ F2 @ top_top_set_nat )
= top_top_set_nat )
=> ( ( image_nat_nat2 @ F2 @ ( vimage_nat_nat @ F2 @ A4 ) )
= A4 ) ) ).
% surj_image_vimage_eq
thf(fact_1181_surj__image__vimage__eq,axiom,
! [F2: nat > $o,A4: set_o] :
( ( ( image_nat_o2 @ F2 @ top_top_set_nat )
= top_top_set_o )
=> ( ( image_nat_o2 @ F2 @ ( vimage_nat_o @ F2 @ A4 ) )
= A4 ) ) ).
% surj_image_vimage_eq
thf(fact_1182_surj__image__vimage__eq,axiom,
! [F2: $o > nat,A4: set_nat] :
( ( ( image_o_nat2 @ F2 @ top_top_set_o )
= top_top_set_nat )
=> ( ( image_o_nat2 @ F2 @ ( vimage_o_nat @ F2 @ A4 ) )
= A4 ) ) ).
% surj_image_vimage_eq
thf(fact_1183_surj__image__vimage__eq,axiom,
! [F2: $o > $o,A4: set_o] :
( ( ( image_o_o2 @ F2 @ top_top_set_o )
= top_top_set_o )
=> ( ( image_o_o2 @ F2 @ ( vimage_o_o @ F2 @ A4 ) )
= A4 ) ) ).
% surj_image_vimage_eq
thf(fact_1184_reflD,axiom,
! [R: set_Pr1261947904930325089at_nat,A: nat] :
( ( refl_on_nat @ top_top_set_nat @ R )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ R ) ) ).
% reflD
thf(fact_1185_reflD,axiom,
! [R: set_Product_prod_o_o,A: $o] :
( ( refl_on_o @ top_top_set_o @ R )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ A ) @ R ) ) ).
% reflD
thf(fact_1186_reflI,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ! [X3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ R )
=> ( refl_on_nat @ top_top_set_nat @ R ) ) ).
% reflI
thf(fact_1187_reflI,axiom,
! [R: set_Product_prod_o_o] :
( ! [X3: $o] : ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ X3 ) @ R )
=> ( refl_on_o @ top_top_set_o @ R ) ) ).
% reflI
thf(fact_1188_SUP__UNIV__bool__expand,axiom,
! [A4: $o > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A4 @ top_top_set_o ) )
= ( sup_sup_set_nat @ ( A4 @ $true ) @ ( A4 @ $false ) ) ) ).
% SUP_UNIV_bool_expand
thf(fact_1189_Un__eq__UN,axiom,
( sup_sup_set_nat
= ( ^ [A6: set_nat,B7: set_nat] :
( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [B4: $o] : ( if_set_nat @ B4 @ A6 @ B7 )
@ top_top_set_o ) ) ) ) ).
% Un_eq_UN
thf(fact_1190_UN__bool__eq,axiom,
! [A4: $o > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A4 @ top_top_set_o ) )
= ( sup_sup_set_nat @ ( A4 @ $true ) @ ( A4 @ $false ) ) ) ).
% UN_bool_eq
thf(fact_1191_vimage__fst,axiom,
! [A4: set_nat] :
( ( vimage4653281326611754070at_nat @ product_fst_nat_nat @ A4 )
= ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : top_top_set_nat ) ) ).
% vimage_fst
thf(fact_1192_vimage__snd,axiom,
! [A4: set_nat] :
( ( vimage4653281326611754070at_nat @ product_snd_nat_nat @ A4 )
= ( produc457027306803732586at_nat @ top_top_set_nat
@ ^ [Uu: nat] : A4 ) ) ).
% vimage_snd
thf(fact_1193_boolean__algebra_Ocomplement__unique,axiom,
! [A: set_nat,X: set_nat,Y3: set_nat] :
( ( ( inf_inf_set_nat @ A @ X )
= bot_bot_set_nat )
=> ( ( ( sup_sup_set_nat @ A @ X )
= top_top_set_nat )
=> ( ( ( inf_inf_set_nat @ A @ Y3 )
= bot_bot_set_nat )
=> ( ( ( sup_sup_set_nat @ A @ Y3 )
= top_top_set_nat )
=> ( X = Y3 ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_1194_boolean__algebra_Ocomplement__unique,axiom,
! [A: set_o,X: set_o,Y3: set_o] :
( ( ( inf_inf_set_o @ A @ X )
= bot_bot_set_o )
=> ( ( ( sup_sup_set_o @ A @ X )
= top_top_set_o )
=> ( ( ( inf_inf_set_o @ A @ Y3 )
= bot_bot_set_o )
=> ( ( ( sup_sup_set_o @ A @ Y3 )
= top_top_set_o )
=> ( X = Y3 ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_1195_range__eq__singletonD,axiom,
! [F2: nat > set_nat,A: set_nat,X: nat] :
( ( ( image_nat_set_nat @ F2 @ top_top_set_nat )
= ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
=> ( ( F2 @ X )
= A ) ) ).
% range_eq_singletonD
thf(fact_1196_range__eq__singletonD,axiom,
! [F2: nat > nat,A: nat,X: nat] :
( ( ( image_nat_nat2 @ F2 @ top_top_set_nat )
= ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( ( F2 @ X )
= A ) ) ).
% range_eq_singletonD
thf(fact_1197_range__eq__singletonD,axiom,
! [F2: nat > $o,A: $o,X: nat] :
( ( ( image_nat_o2 @ F2 @ top_top_set_nat )
= ( insert_o @ A @ bot_bot_set_o ) )
=> ( ( F2 @ X )
= A ) ) ).
% range_eq_singletonD
thf(fact_1198_range__eq__singletonD,axiom,
! [F2: $o > nat,A: nat,X: $o] :
( ( ( image_o_nat2 @ F2 @ top_top_set_o )
= ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( ( F2 @ X )
= A ) ) ).
% range_eq_singletonD
thf(fact_1199_range__eq__singletonD,axiom,
! [F2: $o > $o,A: $o,X: $o] :
( ( ( image_o_o2 @ F2 @ top_top_set_o )
= ( insert_o @ A @ bot_bot_set_o ) )
=> ( ( F2 @ X )
= A ) ) ).
% range_eq_singletonD
thf(fact_1200_inj__image__subset__iff,axiom,
! [F2: nat > set_nat,A4: set_nat,B5: set_nat] :
( ( inj_on_nat_set_nat @ F2 @ top_top_set_nat )
=> ( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F2 @ A4 ) @ ( image_nat_set_nat @ F2 @ B5 ) )
= ( ord_less_eq_set_nat @ A4 @ B5 ) ) ) ).
% inj_image_subset_iff
thf(fact_1201_image__Int,axiom,
! [F2: nat > set_nat,A4: set_nat,B5: set_nat] :
( ( inj_on_nat_set_nat @ F2 @ top_top_set_nat )
=> ( ( image_nat_set_nat @ F2 @ ( inf_inf_set_nat @ A4 @ B5 ) )
= ( inf_inf_set_set_nat @ ( image_nat_set_nat @ F2 @ A4 ) @ ( image_nat_set_nat @ F2 @ B5 ) ) ) ) ).
% image_Int
thf(fact_1202_surj__vimage__empty,axiom,
! [F2: nat > set_nat,A4: set_set_nat] :
( ( ( image_nat_set_nat @ F2 @ top_top_set_nat )
= top_top_set_set_nat )
=> ( ( ( vimage_nat_set_nat @ F2 @ A4 )
= bot_bot_set_nat )
= ( A4 = bot_bot_set_set_nat ) ) ) ).
% surj_vimage_empty
thf(fact_1203_surj__vimage__empty,axiom,
! [F2: nat > nat,A4: set_nat] :
( ( ( image_nat_nat2 @ F2 @ top_top_set_nat )
= top_top_set_nat )
=> ( ( ( vimage_nat_nat @ F2 @ A4 )
= bot_bot_set_nat )
= ( A4 = bot_bot_set_nat ) ) ) ).
% surj_vimage_empty
thf(fact_1204_surj__vimage__empty,axiom,
! [F2: nat > $o,A4: set_o] :
( ( ( image_nat_o2 @ F2 @ top_top_set_nat )
= top_top_set_o )
=> ( ( ( vimage_nat_o @ F2 @ A4 )
= bot_bot_set_nat )
= ( A4 = bot_bot_set_o ) ) ) ).
% surj_vimage_empty
thf(fact_1205_surj__vimage__empty,axiom,
! [F2: $o > nat,A4: set_nat] :
( ( ( image_o_nat2 @ F2 @ top_top_set_o )
= top_top_set_nat )
=> ( ( ( vimage_o_nat @ F2 @ A4 )
= bot_bot_set_o )
= ( A4 = bot_bot_set_nat ) ) ) ).
% surj_vimage_empty
thf(fact_1206_surj__vimage__empty,axiom,
! [F2: $o > $o,A4: set_o] :
( ( ( image_o_o2 @ F2 @ top_top_set_o )
= top_top_set_o )
=> ( ( ( vimage_o_o @ F2 @ A4 )
= bot_bot_set_o )
= ( A4 = bot_bot_set_o ) ) ) ).
% surj_vimage_empty
thf(fact_1207_vimage__subsetD,axiom,
! [F2: nat > set_nat,B5: set_set_nat,A4: set_nat] :
( ( ( image_nat_set_nat @ F2 @ top_top_set_nat )
= top_top_set_set_nat )
=> ( ( ord_less_eq_set_nat @ ( vimage_nat_set_nat @ F2 @ B5 ) @ A4 )
=> ( ord_le6893508408891458716et_nat @ B5 @ ( image_nat_set_nat @ F2 @ A4 ) ) ) ) ).
% vimage_subsetD
thf(fact_1208_vimage__subsetD,axiom,
! [F2: nat > nat,B5: set_nat,A4: set_nat] :
( ( ( image_nat_nat2 @ F2 @ top_top_set_nat )
= top_top_set_nat )
=> ( ( ord_less_eq_set_nat @ ( vimage_nat_nat @ F2 @ B5 ) @ A4 )
=> ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat2 @ F2 @ A4 ) ) ) ) ).
% vimage_subsetD
thf(fact_1209_vimage__subsetD,axiom,
! [F2: nat > $o,B5: set_o,A4: set_nat] :
( ( ( image_nat_o2 @ F2 @ top_top_set_nat )
= top_top_set_o )
=> ( ( ord_less_eq_set_nat @ ( vimage_nat_o @ F2 @ B5 ) @ A4 )
=> ( ord_less_eq_set_o @ B5 @ ( image_nat_o2 @ F2 @ A4 ) ) ) ) ).
% vimage_subsetD
thf(fact_1210_vimage__subsetD,axiom,
! [F2: $o > nat,B5: set_nat,A4: set_o] :
( ( ( image_o_nat2 @ F2 @ top_top_set_o )
= top_top_set_nat )
=> ( ( ord_less_eq_set_o @ ( vimage_o_nat @ F2 @ B5 ) @ A4 )
=> ( ord_less_eq_set_nat @ B5 @ ( image_o_nat2 @ F2 @ A4 ) ) ) ) ).
% vimage_subsetD
thf(fact_1211_vimage__subsetD,axiom,
! [F2: $o > $o,B5: set_o,A4: set_o] :
( ( ( image_o_o2 @ F2 @ top_top_set_o )
= top_top_set_o )
=> ( ( ord_less_eq_set_o @ ( vimage_o_o @ F2 @ B5 ) @ A4 )
=> ( ord_less_eq_set_o @ B5 @ ( image_o_o2 @ F2 @ A4 ) ) ) ) ).
% vimage_subsetD
thf(fact_1212_inj__vimage__image__eq,axiom,
! [F2: nat > set_nat,A4: set_nat] :
( ( inj_on_nat_set_nat @ F2 @ top_top_set_nat )
=> ( ( vimage_nat_set_nat @ F2 @ ( image_nat_set_nat @ F2 @ A4 ) )
= A4 ) ) ).
% inj_vimage_image_eq
thf(fact_1213_vimage__subsetI,axiom,
! [F2: nat > set_nat,B5: set_set_nat,A4: set_nat] :
( ( inj_on_nat_set_nat @ F2 @ top_top_set_nat )
=> ( ( ord_le6893508408891458716et_nat @ B5 @ ( image_nat_set_nat @ F2 @ A4 ) )
=> ( ord_less_eq_set_nat @ ( vimage_nat_set_nat @ F2 @ B5 ) @ A4 ) ) ) ).
% vimage_subsetI
thf(fact_1214_fun_Omap__ident__strong,axiom,
! [T: nat > set_nat,F2: set_nat > set_nat] :
( ! [Z4: set_nat] :
( ( member_set_nat @ Z4 @ ( image_nat_set_nat @ T @ top_top_set_nat ) )
=> ( ( F2 @ Z4 )
= Z4 ) )
=> ( ( comp_s3433241188411525313at_nat @ F2 @ T )
= T ) ) ).
% fun.map_ident_strong
thf(fact_1215_fun_Omap__ident__strong,axiom,
! [T: nat > product_prod_nat_nat,F2: product_prod_nat_nat > product_prod_nat_nat] :
( ! [Z4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Z4 @ ( image_5846123807819985514at_nat @ T @ top_top_set_nat ) )
=> ( ( F2 @ Z4 )
= Z4 ) )
=> ( ( comp_P7430769850142754163at_nat @ F2 @ T )
= T ) ) ).
% fun.map_ident_strong
thf(fact_1216_fun_Omap__ident__strong,axiom,
! [T: nat > nat,F2: nat > nat] :
( ! [Z4: nat] :
( ( member_nat @ Z4 @ ( image_nat_nat2 @ T @ top_top_set_nat ) )
=> ( ( F2 @ Z4 )
= Z4 ) )
=> ( ( comp_nat_nat_nat @ F2 @ T )
= T ) ) ).
% fun.map_ident_strong
thf(fact_1217_fun_Omap__ident__strong,axiom,
! [T: $o > product_prod_nat_nat,F2: product_prod_nat_nat > product_prod_nat_nat] :
( ! [Z4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Z4 @ ( image_3855930084881510382at_nat @ T @ top_top_set_o ) )
=> ( ( F2 @ Z4 )
= Z4 ) )
=> ( ( comp_P6182352318297508661_nat_o @ F2 @ T )
= T ) ) ).
% fun.map_ident_strong
thf(fact_1218_fun_Omap__ident__strong,axiom,
! [T: $o > nat,F2: nat > nat] :
( ! [Z4: nat] :
( ( member_nat @ Z4 @ ( image_o_nat2 @ T @ top_top_set_o ) )
=> ( ( F2 @ Z4 )
= Z4 ) )
=> ( ( comp_nat_nat_o @ F2 @ T )
= T ) ) ).
% fun.map_ident_strong
thf(fact_1219_UNIV__bool,axiom,
( top_top_set_o
= ( insert_o @ $false @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).
% UNIV_bool
thf(fact_1220_top__empty__eq,axiom,
( top_to3137496036531963500_nat_o
= ( ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ top_to4669805908274784177at_nat ) ) ) ).
% top_empty_eq
thf(fact_1221_top__empty__eq,axiom,
( top_top_nat_o
= ( ^ [X4: nat] : ( member_nat @ X4 @ top_top_set_nat ) ) ) ).
% top_empty_eq
thf(fact_1222_top__empty__eq,axiom,
( top_top_o_o
= ( ^ [X4: $o] : ( member_o @ X4 @ top_top_set_o ) ) ) ).
% top_empty_eq
thf(fact_1223_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_1224_top__set__def,axiom,
( top_top_set_o
= ( collect_o @ top_top_o_o ) ) ).
% top_set_def
thf(fact_1225_top__empty__eq2,axiom,
( top_top_nat_nat_o
= ( ^ [X4: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ top_to4669805908274784177at_nat ) ) ) ).
% top_empty_eq2
thf(fact_1226_fun_Oset__map,axiom,
! [F2: set_nat > set_nat,V2: nat > set_nat] :
( ( image_nat_set_nat @ ( comp_s3433241188411525313at_nat @ F2 @ V2 ) @ top_top_set_nat )
= ( image_7916887816326733075et_nat @ F2 @ ( image_nat_set_nat @ V2 @ top_top_set_nat ) ) ) ).
% fun.set_map
thf(fact_1227_fun_Oset__map,axiom,
! [F2: nat > set_nat,V2: nat > nat] :
( ( image_nat_set_nat @ ( comp_nat_set_nat_nat @ F2 @ V2 ) @ top_top_set_nat )
= ( image_nat_set_nat @ F2 @ ( image_nat_nat2 @ V2 @ top_top_set_nat ) ) ) ).
% fun.set_map
thf(fact_1228_fun_Oset__map,axiom,
! [F2: nat > set_nat,V2: $o > nat] :
( ( image_o_set_nat @ ( comp_nat_set_nat_o @ F2 @ V2 ) @ top_top_set_o )
= ( image_nat_set_nat @ F2 @ ( image_o_nat2 @ V2 @ top_top_set_o ) ) ) ).
% fun.set_map
thf(fact_1229_sorted__list__of__set_Oinj__on,axiom,
( inj_on_nat_nat
@ ^ [X4: nat] : X4
@ top_top_set_nat ) ).
% sorted_list_of_set.inj_on
thf(fact_1230_sorted__list__of__set_Oinj__on,axiom,
( inj_on_o_o
@ ^ [X4: $o] : X4
@ top_top_set_o ) ).
% sorted_list_of_set.inj_on
thf(fact_1231_fun_Orel__cong,axiom,
! [X: nat > nat,Ya: nat > nat,Y3: nat > nat,Xa2: nat > nat,R2: nat > nat > $o,Ra2: nat > nat > $o] :
( ( X = Ya )
=> ( ( Y3 = Xa2 )
=> ( ! [Z4: nat,Yb: nat] :
( ( member_nat @ Z4 @ ( image_nat_nat2 @ Ya @ top_top_set_nat ) )
=> ( ( member_nat @ Yb @ ( image_nat_nat2 @ Xa2 @ top_top_set_nat ) )
=> ( ( R2 @ Z4 @ Yb )
= ( Ra2 @ Z4 @ Yb ) ) ) )
=> ( ( bNF_re5653821019739307937at_nat
@ ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 )
@ R2
@ X
@ Y3 )
= ( bNF_re5653821019739307937at_nat
@ ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 )
@ Ra2
@ Ya
@ Xa2 ) ) ) ) ) ).
% fun.rel_cong
thf(fact_1232_fun_Orel__cong,axiom,
! [X: $o > nat,Ya: $o > nat,Y3: $o > nat,Xa2: $o > nat,R2: nat > nat > $o,Ra2: nat > nat > $o] :
( ( X = Ya )
=> ( ( Y3 = Xa2 )
=> ( ! [Z4: nat,Yb: nat] :
( ( member_nat @ Z4 @ ( image_o_nat2 @ Ya @ top_top_set_o ) )
=> ( ( member_nat @ Yb @ ( image_o_nat2 @ Xa2 @ top_top_set_o ) )
=> ( ( R2 @ Z4 @ Yb )
= ( Ra2 @ Z4 @ Yb ) ) ) )
=> ( ( bNF_re552426215088010623at_nat
@ ^ [Y6: $o,Z2: $o] : ( Y6 = Z2 )
@ R2
@ X
@ Y3 )
= ( bNF_re552426215088010623at_nat
@ ^ [Y6: $o,Z2: $o] : ( Y6 = Z2 )
@ Ra2
@ Ya
@ Xa2 ) ) ) ) ) ).
% fun.rel_cong
thf(fact_1233_fun_Orel__cong,axiom,
! [X: nat > set_nat,Ya: nat > set_nat,Y3: nat > nat,Xa2: nat > nat,R2: set_nat > nat > $o,Ra2: set_nat > nat > $o] :
( ( X = Ya )
=> ( ( Y3 = Xa2 )
=> ( ! [Z4: set_nat,Yb: nat] :
( ( member_set_nat @ Z4 @ ( image_nat_set_nat @ Ya @ top_top_set_nat ) )
=> ( ( member_nat @ Yb @ ( image_nat_nat2 @ Xa2 @ top_top_set_nat ) )
=> ( ( R2 @ Z4 @ Yb )
= ( Ra2 @ Z4 @ Yb ) ) ) )
=> ( ( bNF_re4574843485292204887at_nat
@ ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 )
@ R2
@ X
@ Y3 )
= ( bNF_re4574843485292204887at_nat
@ ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 )
@ Ra2
@ Ya
@ Xa2 ) ) ) ) ) ).
% fun.rel_cong
thf(fact_1234_fun_Orel__cong,axiom,
! [X: nat > nat,Ya: nat > nat,Y3: nat > set_nat,Xa2: nat > set_nat,R2: nat > set_nat > $o,Ra2: nat > set_nat > $o] :
( ( X = Ya )
=> ( ( Y3 = Xa2 )
=> ( ! [Z4: nat,Yb: set_nat] :
( ( member_nat @ Z4 @ ( image_nat_nat2 @ Ya @ top_top_set_nat ) )
=> ( ( member_set_nat @ Yb @ ( image_nat_set_nat @ Xa2 @ top_top_set_nat ) )
=> ( ( R2 @ Z4 @ Yb )
= ( Ra2 @ Z4 @ Yb ) ) ) )
=> ( ( bNF_re8140478389947574103et_nat
@ ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 )
@ R2
@ X
@ Y3 )
= ( bNF_re8140478389947574103et_nat
@ ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 )
@ Ra2
@ Ya
@ Xa2 ) ) ) ) ) ).
% fun.rel_cong
thf(fact_1235_fun_Orel__cong,axiom,
! [X: nat > set_nat,Ya: nat > set_nat,Y3: nat > set_nat,Xa2: nat > set_nat,R2: set_nat > set_nat > $o,Ra2: set_nat > set_nat > $o] :
( ( X = Ya )
=> ( ( Y3 = Xa2 )
=> ( ! [Z4: set_nat,Yb: set_nat] :
( ( member_set_nat @ Z4 @ ( image_nat_set_nat @ Ya @ top_top_set_nat ) )
=> ( ( member_set_nat @ Yb @ ( image_nat_set_nat @ Xa2 @ top_top_set_nat ) )
=> ( ( R2 @ Z4 @ Yb )
= ( Ra2 @ Z4 @ Yb ) ) ) )
=> ( ( bNF_re4683002380766094093et_nat
@ ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 )
@ R2
@ X
@ Y3 )
= ( bNF_re4683002380766094093et_nat
@ ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 )
@ Ra2
@ Ya
@ Xa2 ) ) ) ) ) ).
% fun.rel_cong
thf(fact_1236_fun_Orel__cong,axiom,
! [X: nat > product_prod_nat_nat,Ya: nat > product_prod_nat_nat,Y3: nat > nat,Xa2: nat > nat,R2: product_prod_nat_nat > nat > $o,Ra2: product_prod_nat_nat > nat > $o] :
( ( X = Ya )
=> ( ( Y3 = Xa2 )
=> ( ! [Z4: product_prod_nat_nat,Yb: nat] :
( ( member8440522571783428010at_nat @ Z4 @ ( image_5846123807819985514at_nat @ Ya @ top_top_set_nat ) )
=> ( ( member_nat @ Yb @ ( image_nat_nat2 @ Xa2 @ top_top_set_nat ) )
=> ( ( R2 @ Z4 @ Yb )
= ( Ra2 @ Z4 @ Yb ) ) ) )
=> ( ( bNF_re6832768993106356754at_nat
@ ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 )
@ R2
@ X
@ Y3 )
= ( bNF_re6832768993106356754at_nat
@ ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 )
@ Ra2
@ Ya
@ Xa2 ) ) ) ) ) ).
% fun.rel_cong
thf(fact_1237_fun_Orel__cong,axiom,
! [X: nat > nat,Ya: nat > nat,Y3: nat > product_prod_nat_nat,Xa2: nat > product_prod_nat_nat,R2: nat > product_prod_nat_nat > $o,Ra2: nat > product_prod_nat_nat > $o] :
( ( X = Ya )
=> ( ( Y3 = Xa2 )
=> ( ! [Z4: nat,Yb: product_prod_nat_nat] :
( ( member_nat @ Z4 @ ( image_nat_nat2 @ Ya @ top_top_set_nat ) )
=> ( ( member8440522571783428010at_nat @ Yb @ ( image_5846123807819985514at_nat @ Xa2 @ top_top_set_nat ) )
=> ( ( R2 @ Z4 @ Yb )
= ( Ra2 @ Z4 @ Yb ) ) ) )
=> ( ( bNF_re969444349294296048at_nat
@ ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 )
@ R2
@ X
@ Y3 )
= ( bNF_re969444349294296048at_nat
@ ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 )
@ Ra2
@ Ya
@ Xa2 ) ) ) ) ) ).
% fun.rel_cong
thf(fact_1238_fun_Orel__cong,axiom,
! [X: $o > product_prod_nat_nat,Ya: $o > product_prod_nat_nat,Y3: $o > nat,Xa2: $o > nat,R2: product_prod_nat_nat > nat > $o,Ra2: product_prod_nat_nat > nat > $o] :
( ( X = Ya )
=> ( ( Y3 = Xa2 )
=> ( ! [Z4: product_prod_nat_nat,Yb: nat] :
( ( member8440522571783428010at_nat @ Z4 @ ( image_3855930084881510382at_nat @ Ya @ top_top_set_o ) )
=> ( ( member_nat @ Yb @ ( image_o_nat2 @ Xa2 @ top_top_set_o ) )
=> ( ( R2 @ Z4 @ Yb )
= ( Ra2 @ Z4 @ Yb ) ) ) )
=> ( ( bNF_re5458109625844885172at_nat
@ ^ [Y6: $o,Z2: $o] : ( Y6 = Z2 )
@ R2
@ X
@ Y3 )
= ( bNF_re5458109625844885172at_nat
@ ^ [Y6: $o,Z2: $o] : ( Y6 = Z2 )
@ Ra2
@ Ya
@ Xa2 ) ) ) ) ) ).
% fun.rel_cong
thf(fact_1239_fun_Orel__cong,axiom,
! [X: $o > nat,Ya: $o > nat,Y3: $o > product_prod_nat_nat,Xa2: $o > product_prod_nat_nat,R2: nat > product_prod_nat_nat > $o,Ra2: nat > product_prod_nat_nat > $o] :
( ( X = Ya )
=> ( ( Y3 = Xa2 )
=> ( ! [Z4: nat,Yb: product_prod_nat_nat] :
( ( member_nat @ Z4 @ ( image_o_nat2 @ Ya @ top_top_set_o ) )
=> ( ( member8440522571783428010at_nat @ Yb @ ( image_3855930084881510382at_nat @ Xa2 @ top_top_set_o ) )
=> ( ( R2 @ Z4 @ Yb )
= ( Ra2 @ Z4 @ Yb ) ) ) )
=> ( ( bNF_re8818157018887600274at_nat
@ ^ [Y6: $o,Z2: $o] : ( Y6 = Z2 )
@ R2
@ X
@ Y3 )
= ( bNF_re8818157018887600274at_nat
@ ^ [Y6: $o,Z2: $o] : ( Y6 = Z2 )
@ Ra2
@ Ya
@ Xa2 ) ) ) ) ) ).
% fun.rel_cong
thf(fact_1240_fun_Orel__cong,axiom,
! [X: nat > set_nat,Ya: nat > set_nat,Y3: nat > product_prod_nat_nat,Xa2: nat > product_prod_nat_nat,R2: set_nat > product_prod_nat_nat > $o,Ra2: set_nat > product_prod_nat_nat > $o] :
( ( X = Ya )
=> ( ( Y3 = Xa2 )
=> ( ! [Z4: set_nat,Yb: product_prod_nat_nat] :
( ( member_set_nat @ Z4 @ ( image_nat_set_nat @ Ya @ top_top_set_nat ) )
=> ( ( member8440522571783428010at_nat @ Yb @ ( image_5846123807819985514at_nat @ Xa2 @ top_top_set_nat ) )
=> ( ( R2 @ Z4 @ Yb )
= ( Ra2 @ Z4 @ Yb ) ) ) )
=> ( ( bNF_re7960066483578023098at_nat
@ ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 )
@ R2
@ X
@ Y3 )
= ( bNF_re7960066483578023098at_nat
@ ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 )
@ Ra2
@ Ya
@ Xa2 ) ) ) ) ) ).
% fun.rel_cong
thf(fact_1241_fun_Orel__mono__strong,axiom,
! [R2: nat > nat > $o,X: nat > nat,Y3: nat > nat,Ra2: nat > nat > $o] :
( ( bNF_re5653821019739307937at_nat
@ ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 )
@ R2
@ X
@ Y3 )
=> ( ! [Z4: nat,Yb: nat] :
( ( member_nat @ Z4 @ ( image_nat_nat2 @ X @ top_top_set_nat ) )
=> ( ( member_nat @ Yb @ ( image_nat_nat2 @ Y3 @ top_top_set_nat ) )
=> ( ( R2 @ Z4 @ Yb )
=> ( Ra2 @ Z4 @ Yb ) ) ) )
=> ( bNF_re5653821019739307937at_nat
@ ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 )
@ Ra2
@ X
@ Y3 ) ) ) ).
% fun.rel_mono_strong
thf(fact_1242_fun_Orel__mono__strong,axiom,
! [R2: nat > nat > $o,X: $o > nat,Y3: $o > nat,Ra2: nat > nat > $o] :
( ( bNF_re552426215088010623at_nat
@ ^ [Y6: $o,Z2: $o] : ( Y6 = Z2 )
@ R2
@ X
@ Y3 )
=> ( ! [Z4: nat,Yb: nat] :
( ( member_nat @ Z4 @ ( image_o_nat2 @ X @ top_top_set_o ) )
=> ( ( member_nat @ Yb @ ( image_o_nat2 @ Y3 @ top_top_set_o ) )
=> ( ( R2 @ Z4 @ Yb )
=> ( Ra2 @ Z4 @ Yb ) ) ) )
=> ( bNF_re552426215088010623at_nat
@ ^ [Y6: $o,Z2: $o] : ( Y6 = Z2 )
@ Ra2
@ X
@ Y3 ) ) ) ).
% fun.rel_mono_strong
thf(fact_1243_fun_Orel__mono__strong,axiom,
! [R2: set_nat > nat > $o,X: nat > set_nat,Y3: nat > nat,Ra2: set_nat > nat > $o] :
( ( bNF_re4574843485292204887at_nat
@ ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 )
@ R2
@ X
@ Y3 )
=> ( ! [Z4: set_nat,Yb: nat] :
( ( member_set_nat @ Z4 @ ( image_nat_set_nat @ X @ top_top_set_nat ) )
=> ( ( member_nat @ Yb @ ( image_nat_nat2 @ Y3 @ top_top_set_nat ) )
=> ( ( R2 @ Z4 @ Yb )
=> ( Ra2 @ Z4 @ Yb ) ) ) )
=> ( bNF_re4574843485292204887at_nat
@ ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 )
@ Ra2
@ X
@ Y3 ) ) ) ).
% fun.rel_mono_strong
thf(fact_1244_fun_Orel__mono__strong,axiom,
! [R2: nat > set_nat > $o,X: nat > nat,Y3: nat > set_nat,Ra2: nat > set_nat > $o] :
( ( bNF_re8140478389947574103et_nat
@ ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 )
@ R2
@ X
@ Y3 )
=> ( ! [Z4: nat,Yb: set_nat] :
( ( member_nat @ Z4 @ ( image_nat_nat2 @ X @ top_top_set_nat ) )
=> ( ( member_set_nat @ Yb @ ( image_nat_set_nat @ Y3 @ top_top_set_nat ) )
=> ( ( R2 @ Z4 @ Yb )
=> ( Ra2 @ Z4 @ Yb ) ) ) )
=> ( bNF_re8140478389947574103et_nat
@ ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 )
@ Ra2
@ X
@ Y3 ) ) ) ).
% fun.rel_mono_strong
thf(fact_1245_fun_Orel__mono__strong,axiom,
! [R2: set_nat > set_nat > $o,X: nat > set_nat,Y3: nat > set_nat,Ra2: set_nat > set_nat > $o] :
( ( bNF_re4683002380766094093et_nat
@ ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 )
@ R2
@ X
@ Y3 )
=> ( ! [Z4: set_nat,Yb: set_nat] :
( ( member_set_nat @ Z4 @ ( image_nat_set_nat @ X @ top_top_set_nat ) )
=> ( ( member_set_nat @ Yb @ ( image_nat_set_nat @ Y3 @ top_top_set_nat ) )
=> ( ( R2 @ Z4 @ Yb )
=> ( Ra2 @ Z4 @ Yb ) ) ) )
=> ( bNF_re4683002380766094093et_nat
@ ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 )
@ Ra2
@ X
@ Y3 ) ) ) ).
% fun.rel_mono_strong
thf(fact_1246_fun_Orel__mono__strong,axiom,
! [R2: product_prod_nat_nat > nat > $o,X: nat > product_prod_nat_nat,Y3: nat > nat,Ra2: product_prod_nat_nat > nat > $o] :
( ( bNF_re6832768993106356754at_nat
@ ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 )
@ R2
@ X
@ Y3 )
=> ( ! [Z4: product_prod_nat_nat,Yb: nat] :
( ( member8440522571783428010at_nat @ Z4 @ ( image_5846123807819985514at_nat @ X @ top_top_set_nat ) )
=> ( ( member_nat @ Yb @ ( image_nat_nat2 @ Y3 @ top_top_set_nat ) )
=> ( ( R2 @ Z4 @ Yb )
=> ( Ra2 @ Z4 @ Yb ) ) ) )
=> ( bNF_re6832768993106356754at_nat
@ ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 )
@ Ra2
@ X
@ Y3 ) ) ) ).
% fun.rel_mono_strong
thf(fact_1247_fun_Orel__mono__strong,axiom,
! [R2: nat > product_prod_nat_nat > $o,X: nat > nat,Y3: nat > product_prod_nat_nat,Ra2: nat > product_prod_nat_nat > $o] :
( ( bNF_re969444349294296048at_nat
@ ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 )
@ R2
@ X
@ Y3 )
=> ( ! [Z4: nat,Yb: product_prod_nat_nat] :
( ( member_nat @ Z4 @ ( image_nat_nat2 @ X @ top_top_set_nat ) )
=> ( ( member8440522571783428010at_nat @ Yb @ ( image_5846123807819985514at_nat @ Y3 @ top_top_set_nat ) )
=> ( ( R2 @ Z4 @ Yb )
=> ( Ra2 @ Z4 @ Yb ) ) ) )
=> ( bNF_re969444349294296048at_nat
@ ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 )
@ Ra2
@ X
@ Y3 ) ) ) ).
% fun.rel_mono_strong
thf(fact_1248_fun_Orel__mono__strong,axiom,
! [R2: product_prod_nat_nat > nat > $o,X: $o > product_prod_nat_nat,Y3: $o > nat,Ra2: product_prod_nat_nat > nat > $o] :
( ( bNF_re5458109625844885172at_nat
@ ^ [Y6: $o,Z2: $o] : ( Y6 = Z2 )
@ R2
@ X
@ Y3 )
=> ( ! [Z4: product_prod_nat_nat,Yb: nat] :
( ( member8440522571783428010at_nat @ Z4 @ ( image_3855930084881510382at_nat @ X @ top_top_set_o ) )
=> ( ( member_nat @ Yb @ ( image_o_nat2 @ Y3 @ top_top_set_o ) )
=> ( ( R2 @ Z4 @ Yb )
=> ( Ra2 @ Z4 @ Yb ) ) ) )
=> ( bNF_re5458109625844885172at_nat
@ ^ [Y6: $o,Z2: $o] : ( Y6 = Z2 )
@ Ra2
@ X
@ Y3 ) ) ) ).
% fun.rel_mono_strong
thf(fact_1249_fun_Orel__mono__strong,axiom,
! [R2: nat > product_prod_nat_nat > $o,X: $o > nat,Y3: $o > product_prod_nat_nat,Ra2: nat > product_prod_nat_nat > $o] :
( ( bNF_re8818157018887600274at_nat
@ ^ [Y6: $o,Z2: $o] : ( Y6 = Z2 )
@ R2
@ X
@ Y3 )
=> ( ! [Z4: nat,Yb: product_prod_nat_nat] :
( ( member_nat @ Z4 @ ( image_o_nat2 @ X @ top_top_set_o ) )
=> ( ( member8440522571783428010at_nat @ Yb @ ( image_3855930084881510382at_nat @ Y3 @ top_top_set_o ) )
=> ( ( R2 @ Z4 @ Yb )
=> ( Ra2 @ Z4 @ Yb ) ) ) )
=> ( bNF_re8818157018887600274at_nat
@ ^ [Y6: $o,Z2: $o] : ( Y6 = Z2 )
@ Ra2
@ X
@ Y3 ) ) ) ).
% fun.rel_mono_strong
thf(fact_1250_fun_Orel__mono__strong,axiom,
! [R2: set_nat > product_prod_nat_nat > $o,X: nat > set_nat,Y3: nat > product_prod_nat_nat,Ra2: set_nat > product_prod_nat_nat > $o] :
( ( bNF_re7960066483578023098at_nat
@ ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 )
@ R2
@ X
@ Y3 )
=> ( ! [Z4: set_nat,Yb: product_prod_nat_nat] :
( ( member_set_nat @ Z4 @ ( image_nat_set_nat @ X @ top_top_set_nat ) )
=> ( ( member8440522571783428010at_nat @ Yb @ ( image_5846123807819985514at_nat @ Y3 @ top_top_set_nat ) )
=> ( ( R2 @ Z4 @ Yb )
=> ( Ra2 @ Z4 @ Yb ) ) ) )
=> ( bNF_re7960066483578023098at_nat
@ ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 )
@ Ra2
@ X
@ Y3 ) ) ) ).
% fun.rel_mono_strong
thf(fact_1251_fun_Orel__refl__strong,axiom,
! [X: nat > set_nat,Ra2: set_nat > set_nat > $o] :
( ! [Z4: set_nat] :
( ( member_set_nat @ Z4 @ ( image_nat_set_nat @ X @ top_top_set_nat ) )
=> ( Ra2 @ Z4 @ Z4 ) )
=> ( bNF_re4683002380766094093et_nat
@ ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 )
@ Ra2
@ X
@ X ) ) ).
% fun.rel_refl_strong
thf(fact_1252_fun_Orel__refl__strong,axiom,
! [X: nat > product_prod_nat_nat,Ra2: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ! [Z4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Z4 @ ( image_5846123807819985514at_nat @ X @ top_top_set_nat ) )
=> ( Ra2 @ Z4 @ Z4 ) )
=> ( bNF_re901410630616637503at_nat
@ ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 )
@ Ra2
@ X
@ X ) ) ).
% fun.rel_refl_strong
thf(fact_1253_fun_Orel__refl__strong,axiom,
! [X: nat > nat,Ra2: nat > nat > $o] :
( ! [Z4: nat] :
( ( member_nat @ Z4 @ ( image_nat_nat2 @ X @ top_top_set_nat ) )
=> ( Ra2 @ Z4 @ Z4 ) )
=> ( bNF_re5653821019739307937at_nat
@ ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 )
@ Ra2
@ X
@ X ) ) ).
% fun.rel_refl_strong
thf(fact_1254_fun_Orel__refl__strong,axiom,
! [X: $o > product_prod_nat_nat,Ra2: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ! [Z4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Z4 @ ( image_3855930084881510382at_nat @ X @ top_top_set_o ) )
=> ( Ra2 @ Z4 @ Z4 ) )
=> ( bNF_re45780121106632989at_nat
@ ^ [Y6: $o,Z2: $o] : ( Y6 = Z2 )
@ Ra2
@ X
@ X ) ) ).
% fun.rel_refl_strong
thf(fact_1255_fun_Orel__refl__strong,axiom,
! [X: $o > nat,Ra2: nat > nat > $o] :
( ! [Z4: nat] :
( ( member_nat @ Z4 @ ( image_o_nat2 @ X @ top_top_set_o ) )
=> ( Ra2 @ Z4 @ Z4 ) )
=> ( bNF_re552426215088010623at_nat
@ ^ [Y6: $o,Z2: $o] : ( Y6 = Z2 )
@ Ra2
@ X
@ X ) ) ).
% fun.rel_refl_strong
thf(fact_1256_predicate2__transferD,axiom,
! [R12: nat > nat > $o,R23: nat > nat > $o,P2: nat > nat > $o,Q: nat > nat > $o,A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
( ( bNF_re578469030762574527_nat_o @ R12
@ ( bNF_re4705727531993890431at_o_o @ R23
@ ^ [Y6: $o,Z2: $o] : ( Y6 = Z2 ) )
@ P2
@ Q )
=> ( ( member8440522571783428010at_nat @ A @ A4 )
=> ( ( member8440522571783428010at_nat @ B @ B5 )
=> ( ( ord_le3146513528884898305at_nat @ A4 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R12 ) ) )
=> ( ( ord_le3146513528884898305at_nat @ B5 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R23 ) ) )
=> ( ( P2 @ ( product_fst_nat_nat @ A ) @ ( product_fst_nat_nat @ B ) )
= ( Q @ ( product_snd_nat_nat @ A ) @ ( product_snd_nat_nat @ B ) ) ) ) ) ) ) ) ).
% predicate2_transferD
thf(fact_1257_Field__insert,axiom,
! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat] :
( ( field_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R ) )
= ( sup_sup_set_nat @ ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) @ ( field_nat @ R ) ) ) ).
% Field_insert
thf(fact_1258_Field__insert,axiom,
! [A: $o,B: $o,R: set_Product_prod_o_o] :
( ( field_o @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ A @ B ) @ R ) )
= ( sup_sup_set_o @ ( insert_o @ A @ ( insert_o @ B @ bot_bot_set_o ) ) @ ( field_o @ R ) ) ) ).
% Field_insert
thf(fact_1259_Field__square,axiom,
! [X: set_nat] :
( ( field_nat
@ ( produc457027306803732586at_nat @ X
@ ^ [Uu: nat] : X ) )
= X ) ).
% Field_square
thf(fact_1260_Field__empty,axiom,
( ( field_nat @ bot_bo2099793752762293965at_nat )
= bot_bot_set_nat ) ).
% Field_empty
thf(fact_1261_Field__empty,axiom,
( ( field_o @ bot_bo7073875226086086771od_o_o )
= bot_bot_set_o ) ).
% Field_empty
thf(fact_1262_Field__Union,axiom,
! [R2: set_se7855581050983116737at_nat] :
( ( field_nat @ ( comple5685304695842803022at_nat @ R2 ) )
= ( comple7399068483239264473et_nat @ ( image_5853224409075402146et_nat @ field_nat @ R2 ) ) ) ).
% Field_Union
thf(fact_1263_FieldI2,axiom,
! [I5: product_prod_nat_nat,J4: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ I5 @ J4 ) @ R2 )
=> ( member8440522571783428010at_nat @ J4 @ ( field_4706345496283612622at_nat @ R2 ) ) ) ).
% FieldI2
thf(fact_1264_FieldI2,axiom,
! [I5: nat,J4: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I5 @ J4 ) @ R2 )
=> ( member_nat @ J4 @ ( field_nat @ R2 ) ) ) ).
% FieldI2
thf(fact_1265_FieldI1,axiom,
! [I5: product_prod_nat_nat,J4: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ I5 @ J4 ) @ R2 )
=> ( member8440522571783428010at_nat @ I5 @ ( field_4706345496283612622at_nat @ R2 ) ) ) ).
% FieldI1
thf(fact_1266_FieldI1,axiom,
! [I5: nat,J4: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I5 @ J4 ) @ R2 )
=> ( member_nat @ I5 @ ( field_nat @ R2 ) ) ) ).
% FieldI1
thf(fact_1267_Restr__Field,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ R
@ ( produc457027306803732586at_nat @ ( field_nat @ R )
@ ^ [Uu: nat] : ( field_nat @ R ) ) )
= R ) ).
% Restr_Field
thf(fact_1268_Field__Restr__subset,axiom,
! [R: set_Pr1261947904930325089at_nat,A4: set_nat] :
( ord_less_eq_set_nat
@ ( field_nat
@ ( inf_in2572325071724192079at_nat @ R
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : A4 ) ) )
@ A4 ) ).
% Field_Restr_subset
thf(fact_1269_Refl__Restr,axiom,
! [R: set_Pr1261947904930325089at_nat,A4: set_nat] :
( ( refl_on_nat @ ( field_nat @ R ) @ R )
=> ( refl_on_nat
@ ( field_nat
@ ( inf_in2572325071724192079at_nat @ R
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : A4 ) ) )
@ ( inf_in2572325071724192079at_nat @ R
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : A4 ) ) ) ) ).
% Refl_Restr
thf(fact_1270_Linear__order__Restr,axiom,
! [R: set_Pr1261947904930325089at_nat,A4: set_nat] :
( ( order_4473980167227706203on_nat @ ( field_nat @ R ) @ R )
=> ( order_4473980167227706203on_nat
@ ( field_nat
@ ( inf_in2572325071724192079at_nat @ R
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : A4 ) ) )
@ ( inf_in2572325071724192079at_nat @ R
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : A4 ) ) ) ) ).
% Linear_order_Restr
thf(fact_1271_Refl__Field__Restr2,axiom,
! [R: set_Pr1261947904930325089at_nat,A4: set_nat] :
( ( refl_on_nat @ ( field_nat @ R ) @ R )
=> ( ( ord_less_eq_set_nat @ A4 @ ( field_nat @ R ) )
=> ( ( field_nat
@ ( inf_in2572325071724192079at_nat @ R
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : A4 ) ) )
= A4 ) ) ) ).
% Refl_Field_Restr2
thf(fact_1272_Refl__Field__Restr,axiom,
! [R: set_Pr1261947904930325089at_nat,A4: set_nat] :
( ( refl_on_nat @ ( field_nat @ R ) @ R )
=> ( ( field_nat
@ ( inf_in2572325071724192079at_nat @ R
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : A4 ) ) )
= ( inf_inf_set_nat @ ( field_nat @ R ) @ A4 ) ) ) ).
% Refl_Field_Restr
thf(fact_1273_Inf__nat__def1,axiom,
! [K2: set_nat] :
( ( K2 != bot_bot_set_nat )
=> ( member_nat @ ( complete_Inf_Inf_nat @ K2 ) @ K2 ) ) ).
% Inf_nat_def1
% Helper facts (5)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y3: nat] :
( ( if_nat @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y3: nat] :
( ( if_nat @ $true @ X @ Y3 )
= X ) ).
thf(help_If_3_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X: set_nat,Y3: set_nat] :
( ( if_set_nat @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X: set_nat,Y3: set_nat] :
( ( if_set_nat @ $true @ X @ Y3 )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ u @ v ) @ ( equiva50036288466687814et_nat @ r ) )
= ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ u @ v ) @ p ) ) ).
%------------------------------------------------------------------------------