TPTP Problem File: SLH0555^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : FOL_Seq_Calc2/0018_EPathHintikka/prob_00145_005744__13038662_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1641 ( 641 unt; 366 typ; 0 def)
% Number of atoms : 3712 (1487 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 11349 ( 313 ~; 20 |; 253 &;9286 @)
% ( 0 <=>;1477 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 6 avg)
% Number of types : 75 ( 74 usr)
% Number of type conns : 1636 (1636 >; 0 *; 0 +; 0 <<)
% Number of symbols : 295 ( 292 usr; 44 con; 0-2 aty)
% Number of variables : 4219 ( 501 ^;3521 !; 197 ?;4219 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 08:45:11.099
%------------------------------------------------------------------------------
% Could-be-implicit typings (74)
thf(ty_n_t__Stream__Ostream_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_Eo_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J_J,type,
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thf(ty_n_t__Stream__Ostream_It__Stream__Ostream_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__SeCaV__Ofm_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
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thf(ty_n_t__Set__Oset_It__Stream__Ostream_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J_J,type,
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produc5731340206564399485m_rule: $tType ).
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list_P2774625669004309958m_rule: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
set_Pr1822751329126368876m_rule: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Prover__Orule_Mt__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
produc2353945479044264792ist_fm: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
produc340336539035504054m_rule: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J_J,type,
set_se6159873173319230773ist_fm: $tType ).
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stream4408948924543953275ist_fm: $tType ).
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set_Pr5202636777678657877ist_fm: $tType ).
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set_Pr7964886650082825447_fm_fm: $tType ).
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Rat__Orat_Mt__Rat__Orat_J_J,type,
set_Pr8928021450653196913at_rat: $tType ).
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set_Pr6084635751276098665at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Rat__Orat_J_J,type,
set_Pr4105333604307423337at_rat: $tType ).
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set_Pr1261947904930325089at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__List__Olist_It__SeCaV__Ofm_J_J_J,type,
set_set_list_fm: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__SeCaV__Ofm_M_Eo_J_J,type,
set_Pr2813250270173872364d_fm_o: $tType ).
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set_Pr3863124289995447138d_o_fm: $tType ).
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set_Pr2399638329824935915_rat_o: $tType ).
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set_Pr3149072824959771635_nat_o: $tType ).
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set_Pr4944855402158566229_o_rat: $tType ).
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set_Pr2101469702781467981_o_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__SeCaV__Ofm_J_J_J,type,
set_set_set_fm: $tType ).
thf(ty_n_t__Stream__Ostream_It__Stream__Ostream_It__SeCaV__Ofm_J_J,type,
stream_stream_fm: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__SeCaV__Ofm_Mt__SeCaV__Ofm_J,type,
product_prod_fm_fm: $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__Stream__Ostream_It__SeCaV__Ofm_J_J,type,
set_stream_fm: $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__Stream__Ostream_It__Stream__Ostream_I_Eo_J_J,type,
stream_stream_o: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__SeCaV__Otm_J_J,type,
set_list_tm: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__SeCaV__Ofm_J_J,type,
set_list_fm: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__SeCaV__Ofm_M_Eo_J,type,
product_prod_fm_o: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__SeCaV__Ofm_J,type,
product_prod_o_fm: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__SeCaV__Ofm_J_J,type,
list_set_fm: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__SeCaV__Ofm_J_J,type,
set_set_fm: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
set_set_rat: $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__SeCaV__Ofm_M_Eo_J_J,type,
set_fm_o: $tType ).
thf(ty_n_t__Set__Oset_It__Stream__Ostream_I_Eo_J_J,type,
set_stream_o: $tType ).
thf(ty_n_t__Stream__Ostream_It__Prover__Orule_J,type,
stream_rule: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
product_prod_o_o: $tType ).
thf(ty_n_t__Stream__Ostream_It__SeCaV__Ofm_J,type,
stream_fm: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
set_set_o: $tType ).
thf(ty_n_t__Stream__Ostream_It__Rat__Orat_J,type,
stream_rat: $tType ).
thf(ty_n_t__Set__Oset_It__Prover__Orule_J,type,
set_rule: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_Eo_M_Eo_J_J,type,
set_o_o: $tType ).
thf(ty_n_t__List__Olist_It__SeCaV__Otm_J,type,
list_tm: $tType ).
thf(ty_n_t__List__Olist_It__SeCaV__Ofm_J,type,
list_fm: $tType ).
thf(ty_n_t__Set__Oset_It__SeCaV__Ofm_J,type,
set_fm: $tType ).
thf(ty_n_t__Set__Oset_It__Rat__Orat_J,type,
set_rat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Stream__Ostream_I_Eo_J,type,
stream_o: $tType ).
thf(ty_n_t__List__Olist_I_Eo_J,type,
list_o: $tType ).
thf(ty_n_t__Set__Oset_I_Eo_J,type,
set_o: $tType ).
thf(ty_n_t__Prover__Orule,type,
rule: $tType ).
thf(ty_n_t__SeCaV__Ofm,type,
fm: $tType ).
thf(ty_n_t__Rat__Orat,type,
rat: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (292)
thf(sy_c_Complete__Lattices_OInf__class_OInf_001_Eo,type,
complete_Inf_Inf_o: set_o > $o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_I_Eo_J,type,
comple3063163877087187839_set_o: set_set_o > set_o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Nat__Onat_J,type,
comple7806235888213564991et_nat: set_set_nat > set_nat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
comple1222484001442006309m_rule: set_se4309406418413213730m_rule > set_Pr1822751329126368876m_rule ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Rat__Orat_J,type,
comple4298007329820168263et_rat: set_set_rat > set_rat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__SeCaV__Ofm_J,type,
comple1024113957266099930set_fm: set_set_fm > set_fm ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_I_Eo_M_Eo_J,type,
complete_Sup_Sup_o_o: set_o_o > $o > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_M_Eo_J,type,
comple3597286212584221062rule_o: set_Pr3586676963299879929rule_o > produc340336539035504054m_rule > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__SeCaV__Ofm_M_Eo_J,type,
comple3443612784043876893p_fm_o: set_fm_o > fm > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_Eo,type,
complete_Sup_Sup_o: set_o > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_I_Eo_J,type,
comple90263536869209701_set_o: set_set_o > set_o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__List__Olist_It__SeCaV__Ofm_J_J,type,
comple8784269564784259782ist_fm: set_set_list_fm > set_list_fm ).
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__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
comple4923826267547576514ist_fm: set_se6159873173319230773ist_fm > set_Pr5202636777678657877ist_fm ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
comple5648871819318716351m_rule: set_se4309406418413213730m_rule > set_Pr1822751329126368876m_rule ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Rat__Orat_J,type,
comple3890839924845867745et_rat: set_set_rat > set_rat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__SeCaV__Ofm_J,type,
comple2134933779557159616set_fm: set_set_fm > set_fm ).
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__SeCaV__Ofm_J_J,type,
comple4326961365146087328set_fm: set_set_set_fm > set_set_fm ).
thf(sy_c_Countable_Ofrom__nat_001_Eo,type,
from_nat_o: nat > $o ).
thf(sy_c_Countable_Ofrom__nat_001t__Nat__Onat,type,
from_nat_nat: nat > nat ).
thf(sy_c_Countable_Ofrom__nat_001t__Rat__Orat,type,
from_nat_rat: nat > rat ).
thf(sy_c_Countable_Onat__to__rat__surj,type,
nat_to_rat_surj: nat > rat ).
thf(sy_c_EPathHintikka_Opseq,type,
pseq: produc340336539035504054m_rule > list_fm ).
thf(sy_c_EPathHintikka_Otree__fms,type,
tree_fms: stream2709947120125613254m_rule > set_fm ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_List_Obind_001_Eo_001_Eo,type,
bind_o_o: list_o > ( $o > list_o ) > list_o ).
thf(sy_c_List_Obind_001_Eo_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
bind_o2471881556210961415m_rule: list_o > ( $o > list_P2774625669004309958m_rule ) > list_P2774625669004309958m_rule ).
thf(sy_c_List_Obind_001_Eo_001t__SeCaV__Ofm,type,
bind_o_fm: list_o > ( $o > list_fm ) > list_fm ).
thf(sy_c_List_Obind_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_001t__SeCaV__Ofm,type,
bind_P95098981060663284ule_fm: list_P2774625669004309958m_rule > ( produc340336539035504054m_rule > list_fm ) > list_fm ).
thf(sy_c_List_Obind_001t__SeCaV__Ofm_001_Eo,type,
bind_fm_o: list_fm > ( fm > list_o ) > list_o ).
thf(sy_c_List_Obind_001t__SeCaV__Ofm_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
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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__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
member5802698409304902403m_rule: set_Pr1822751329126368876m_rule > set_se4309406418413213730m_rule > $o ).
thf(sy_c_member_001t__Set__Oset_It__Rat__Orat_J,type,
member_set_rat: set_rat > set_set_rat > $o ).
thf(sy_c_member_001t__Set__Oset_It__SeCaV__Ofm_J,type,
member_set_fm: set_fm > set_set_fm > $o ).
thf(sy_v_p,type,
p: fm ).
thf(sy_v_steps,type,
steps: stream2709947120125613254m_rule ).
% Relevant facts (1273)
thf(fact_0__092_060open_062i_OR_Asteps_A_061_Arange_A_I_I_B_B_J_Asteps_J_092_060close_062,axiom,
( ( sset_P4484857331586881186m_rule @ steps )
= ( image_8361736131336225677m_rule @ ( snth_P6679518042731451922m_rule @ steps ) @ top_top_set_nat ) ) ).
% \<open>i.R steps = range ((!!) steps)\<close>
thf(fact_1_UN__I,axiom,
! [A: set_fm,A2: set_set_fm,B: fm,B2: set_fm > set_fm] :
( ( member_set_fm @ A @ A2 )
=> ( ( member_fm @ B @ ( B2 @ A ) )
=> ( member_fm @ B @ ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_2_UN__I,axiom,
! [A: fm,A2: set_fm,B: produc340336539035504054m_rule,B2: fm > set_Pr1822751329126368876m_rule] :
( ( member_fm @ A @ A2 )
=> ( ( member7231649785386036813m_rule @ B @ ( B2 @ A ) )
=> ( member7231649785386036813m_rule @ B @ ( comple5648871819318716351m_rule @ ( image_7929583154892365210m_rule @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_3_UN__I,axiom,
! [A: fm,A2: set_fm,B: $o,B2: fm > set_o] :
( ( member_fm @ A @ A2 )
=> ( ( member_o @ B @ ( B2 @ A ) )
=> ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_fm_set_o @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_4_UN__I,axiom,
! [A: produc340336539035504054m_rule,A2: set_Pr1822751329126368876m_rule,B: produc340336539035504054m_rule,B2: produc340336539035504054m_rule > set_Pr1822751329126368876m_rule] :
( ( member7231649785386036813m_rule @ A @ A2 )
=> ( ( member7231649785386036813m_rule @ B @ ( B2 @ A ) )
=> ( member7231649785386036813m_rule @ B @ ( comple5648871819318716351m_rule @ ( image_2683796897779739177m_rule @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_5_UN__I,axiom,
! [A: produc340336539035504054m_rule,A2: set_Pr1822751329126368876m_rule,B: $o,B2: produc340336539035504054m_rule > set_o] :
( ( member7231649785386036813m_rule @ A @ A2 )
=> ( ( member_o @ B @ ( B2 @ A ) )
=> ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_2025595867862115707_set_o @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_6_UN__I,axiom,
! [A: $o,A2: set_o,B: produc340336539035504054m_rule,B2: $o > set_Pr1822751329126368876m_rule] :
( ( member_o @ A @ A2 )
=> ( ( member7231649785386036813m_rule @ B @ ( B2 @ A ) )
=> ( member7231649785386036813m_rule @ B @ ( comple5648871819318716351m_rule @ ( image_6392605097815222207m_rule @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_7_UN__I,axiom,
! [A: $o,A2: set_o,B: $o,B2: $o > set_o] :
( ( member_o @ A @ A2 )
=> ( ( member_o @ B @ ( B2 @ A ) )
=> ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_8_UN__I,axiom,
! [A: fm,A2: set_fm,B: fm,B2: fm > set_fm] :
( ( member_fm @ A @ A2 )
=> ( ( member_fm @ B @ ( B2 @ A ) )
=> ( member_fm @ B @ ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_9_UN__I,axiom,
! [A: produc340336539035504054m_rule,A2: set_Pr1822751329126368876m_rule,B: fm,B2: produc340336539035504054m_rule > set_fm] :
( ( member7231649785386036813m_rule @ A @ A2 )
=> ( ( member_fm @ B @ ( B2 @ A ) )
=> ( member_fm @ B @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_10_UN__I,axiom,
! [A: $o,A2: set_o,B: fm,B2: $o > set_fm] :
( ( member_o @ A @ A2 )
=> ( ( member_fm @ B @ ( B2 @ A ) )
=> ( member_fm @ B @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_11_UN__iff,axiom,
! [B: fm,B2: $o > set_fm,A2: set_o] :
( ( member_fm @ B @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ B2 @ A2 ) ) )
= ( ? [X: $o] :
( ( member_o @ X @ A2 )
& ( member_fm @ B @ ( B2 @ X ) ) ) ) ) ).
% UN_iff
thf(fact_12_UN__iff,axiom,
! [B: fm,B2: set_fm > set_fm,A2: set_set_fm] :
( ( member_fm @ B @ ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ B2 @ A2 ) ) )
= ( ? [X: set_fm] :
( ( member_set_fm @ X @ A2 )
& ( member_fm @ B @ ( B2 @ X ) ) ) ) ) ).
% UN_iff
thf(fact_13_UN__iff,axiom,
! [B: $o,B2: $o > set_o,A2: set_o] :
( ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
= ( ? [X: $o] :
( ( member_o @ X @ A2 )
& ( member_o @ B @ ( B2 @ X ) ) ) ) ) ).
% UN_iff
thf(fact_14_UN__iff,axiom,
! [B: produc340336539035504054m_rule,B2: $o > set_Pr1822751329126368876m_rule,A2: set_o] :
( ( member7231649785386036813m_rule @ B @ ( comple5648871819318716351m_rule @ ( image_6392605097815222207m_rule @ B2 @ A2 ) ) )
= ( ? [X: $o] :
( ( member_o @ X @ A2 )
& ( member7231649785386036813m_rule @ B @ ( B2 @ X ) ) ) ) ) ).
% UN_iff
thf(fact_15_UN__iff,axiom,
! [B: fm,B2: produc340336539035504054m_rule > set_fm,A2: set_Pr1822751329126368876m_rule] :
( ( member_fm @ B @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B2 @ A2 ) ) )
= ( ? [X: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X @ A2 )
& ( member_fm @ B @ ( B2 @ X ) ) ) ) ) ).
% UN_iff
thf(fact_16_SUP__identity__eq,axiom,
! [A2: set_set_o] :
( ( comple90263536869209701_set_o
@ ( image_set_o_set_o
@ ^ [X: set_o] : X
@ A2 ) )
= ( comple90263536869209701_set_o @ A2 ) ) ).
% SUP_identity_eq
thf(fact_17_SUP__identity__eq,axiom,
! [A2: set_se4309406418413213730m_rule] :
( ( comple5648871819318716351m_rule
@ ( image_7670284417601411807m_rule
@ ^ [X: set_Pr1822751329126368876m_rule] : X
@ A2 ) )
= ( comple5648871819318716351m_rule @ A2 ) ) ).
% SUP_identity_eq
thf(fact_18_SUP__identity__eq,axiom,
! [A2: set_set_fm] :
( ( comple2134933779557159616set_fm
@ ( image_set_fm_set_fm
@ ^ [X: set_fm] : X
@ A2 ) )
= ( comple2134933779557159616set_fm @ A2 ) ) ).
% SUP_identity_eq
thf(fact_19_SUP__identity__eq,axiom,
! [A2: set_o] :
( ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [X: $o] : X
@ A2 ) )
= ( complete_Sup_Sup_o @ A2 ) ) ).
% SUP_identity_eq
thf(fact_20_UN__ball__bex__simps_I4_J,axiom,
! [B2: $o > set_fm,A2: set_o,P: fm > $o] :
( ( ? [X: fm] :
( ( member_fm @ X @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ B2 @ A2 ) ) )
& ( P @ X ) ) )
= ( ? [X: $o] :
( ( member_o @ X @ A2 )
& ? [Y: fm] :
( ( member_fm @ Y @ ( B2 @ X ) )
& ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_21_UN__ball__bex__simps_I4_J,axiom,
! [B2: set_fm > set_fm,A2: set_set_fm,P: fm > $o] :
( ( ? [X: fm] :
( ( member_fm @ X @ ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ B2 @ A2 ) ) )
& ( P @ X ) ) )
= ( ? [X: set_fm] :
( ( member_set_fm @ X @ A2 )
& ? [Y: fm] :
( ( member_fm @ Y @ ( B2 @ X ) )
& ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_22_UN__ball__bex__simps_I4_J,axiom,
! [B2: $o > set_o,A2: set_o,P: $o > $o] :
( ( ? [X: $o] :
( ( member_o @ X @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
& ( P @ X ) ) )
= ( ? [X: $o] :
( ( member_o @ X @ A2 )
& ? [Y: $o] :
( ( member_o @ Y @ ( B2 @ X ) )
& ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_23_UN__ball__bex__simps_I4_J,axiom,
! [B2: $o > set_Pr1822751329126368876m_rule,A2: set_o,P: produc340336539035504054m_rule > $o] :
( ( ? [X: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X @ ( comple5648871819318716351m_rule @ ( image_6392605097815222207m_rule @ B2 @ A2 ) ) )
& ( P @ X ) ) )
= ( ? [X: $o] :
( ( member_o @ X @ A2 )
& ? [Y: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ Y @ ( B2 @ X ) )
& ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_24_UN__ball__bex__simps_I4_J,axiom,
! [B2: produc340336539035504054m_rule > set_fm,A2: set_Pr1822751329126368876m_rule,P: fm > $o] :
( ( ? [X: fm] :
( ( member_fm @ X @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B2 @ A2 ) ) )
& ( P @ X ) ) )
= ( ? [X: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X @ A2 )
& ? [Y: fm] :
( ( member_fm @ Y @ ( B2 @ X ) )
& ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_25_UN__ball__bex__simps_I2_J,axiom,
! [B2: $o > set_fm,A2: set_o,P: fm > $o] :
( ( ! [X: fm] :
( ( member_fm @ X @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ B2 @ A2 ) ) )
=> ( P @ X ) ) )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ! [Y: fm] :
( ( member_fm @ Y @ ( B2 @ X ) )
=> ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_26_UN__ball__bex__simps_I2_J,axiom,
! [B2: set_fm > set_fm,A2: set_set_fm,P: fm > $o] :
( ( ! [X: fm] :
( ( member_fm @ X @ ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ B2 @ A2 ) ) )
=> ( P @ X ) ) )
= ( ! [X: set_fm] :
( ( member_set_fm @ X @ A2 )
=> ! [Y: fm] :
( ( member_fm @ Y @ ( B2 @ X ) )
=> ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_27_UN__ball__bex__simps_I2_J,axiom,
! [B2: $o > set_o,A2: set_o,P: $o > $o] :
( ( ! [X: $o] :
( ( member_o @ X @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
=> ( P @ X ) ) )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ! [Y: $o] :
( ( member_o @ Y @ ( B2 @ X ) )
=> ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_28_UN__ball__bex__simps_I2_J,axiom,
! [B2: $o > set_Pr1822751329126368876m_rule,A2: set_o,P: produc340336539035504054m_rule > $o] :
( ( ! [X: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X @ ( comple5648871819318716351m_rule @ ( image_6392605097815222207m_rule @ B2 @ A2 ) ) )
=> ( P @ X ) ) )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ! [Y: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ Y @ ( B2 @ X ) )
=> ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_29_UN__ball__bex__simps_I2_J,axiom,
! [B2: produc340336539035504054m_rule > set_fm,A2: set_Pr1822751329126368876m_rule,P: fm > $o] :
( ( ! [X: fm] :
( ( member_fm @ X @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B2 @ A2 ) ) )
=> ( P @ X ) ) )
= ( ! [X: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X @ A2 )
=> ! [Y: fm] :
( ( member_fm @ Y @ ( B2 @ X ) )
=> ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_30_bex__UN,axiom,
! [B2: $o > set_fm,A2: set_o,P: fm > $o] :
( ( ? [X: fm] :
( ( member_fm @ X @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ B2 @ A2 ) ) )
& ( P @ X ) ) )
= ( ? [X: $o] :
( ( member_o @ X @ A2 )
& ? [Y: fm] :
( ( member_fm @ Y @ ( B2 @ X ) )
& ( P @ Y ) ) ) ) ) ).
% bex_UN
thf(fact_31_bex__UN,axiom,
! [B2: set_fm > set_fm,A2: set_set_fm,P: fm > $o] :
( ( ? [X: fm] :
( ( member_fm @ X @ ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ B2 @ A2 ) ) )
& ( P @ X ) ) )
= ( ? [X: set_fm] :
( ( member_set_fm @ X @ A2 )
& ? [Y: fm] :
( ( member_fm @ Y @ ( B2 @ X ) )
& ( P @ Y ) ) ) ) ) ).
% bex_UN
thf(fact_32_bex__UN,axiom,
! [B2: $o > set_o,A2: set_o,P: $o > $o] :
( ( ? [X: $o] :
( ( member_o @ X @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
& ( P @ X ) ) )
= ( ? [X: $o] :
( ( member_o @ X @ A2 )
& ? [Y: $o] :
( ( member_o @ Y @ ( B2 @ X ) )
& ( P @ Y ) ) ) ) ) ).
% bex_UN
thf(fact_33_bex__UN,axiom,
! [B2: $o > set_Pr1822751329126368876m_rule,A2: set_o,P: produc340336539035504054m_rule > $o] :
( ( ? [X: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X @ ( comple5648871819318716351m_rule @ ( image_6392605097815222207m_rule @ B2 @ A2 ) ) )
& ( P @ X ) ) )
= ( ? [X: $o] :
( ( member_o @ X @ A2 )
& ? [Y: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ Y @ ( B2 @ X ) )
& ( P @ Y ) ) ) ) ) ).
% bex_UN
thf(fact_34_bex__UN,axiom,
! [B2: produc340336539035504054m_rule > set_fm,A2: set_Pr1822751329126368876m_rule,P: fm > $o] :
( ( ? [X: fm] :
( ( member_fm @ X @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B2 @ A2 ) ) )
& ( P @ X ) ) )
= ( ? [X: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X @ A2 )
& ? [Y: fm] :
( ( member_fm @ Y @ ( B2 @ X ) )
& ( P @ Y ) ) ) ) ) ).
% bex_UN
thf(fact_35_ball__UN,axiom,
! [B2: $o > set_fm,A2: set_o,P: fm > $o] :
( ( ! [X: fm] :
( ( member_fm @ X @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ B2 @ A2 ) ) )
=> ( P @ X ) ) )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ! [Y: fm] :
( ( member_fm @ Y @ ( B2 @ X ) )
=> ( P @ Y ) ) ) ) ) ).
% ball_UN
thf(fact_36_ball__UN,axiom,
! [B2: set_fm > set_fm,A2: set_set_fm,P: fm > $o] :
( ( ! [X: fm] :
( ( member_fm @ X @ ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ B2 @ A2 ) ) )
=> ( P @ X ) ) )
= ( ! [X: set_fm] :
( ( member_set_fm @ X @ A2 )
=> ! [Y: fm] :
( ( member_fm @ Y @ ( B2 @ X ) )
=> ( P @ Y ) ) ) ) ) ).
% ball_UN
thf(fact_37_ball__UN,axiom,
! [B2: $o > set_o,A2: set_o,P: $o > $o] :
( ( ! [X: $o] :
( ( member_o @ X @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
=> ( P @ X ) ) )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ! [Y: $o] :
( ( member_o @ Y @ ( B2 @ X ) )
=> ( P @ Y ) ) ) ) ) ).
% ball_UN
thf(fact_38_ball__UN,axiom,
! [B2: $o > set_Pr1822751329126368876m_rule,A2: set_o,P: produc340336539035504054m_rule > $o] :
( ( ! [X: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X @ ( comple5648871819318716351m_rule @ ( image_6392605097815222207m_rule @ B2 @ A2 ) ) )
=> ( P @ X ) ) )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ! [Y: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ Y @ ( B2 @ X ) )
=> ( P @ Y ) ) ) ) ) ).
% ball_UN
thf(fact_39_ball__UN,axiom,
! [B2: produc340336539035504054m_rule > set_fm,A2: set_Pr1822751329126368876m_rule,P: fm > $o] :
( ( ! [X: fm] :
( ( member_fm @ X @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B2 @ A2 ) ) )
=> ( P @ X ) ) )
= ( ! [X: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X @ A2 )
=> ! [Y: fm] :
( ( member_fm @ Y @ ( B2 @ X ) )
=> ( P @ Y ) ) ) ) ) ).
% ball_UN
thf(fact_40_SUP__UNION,axiom,
! [F: fm > $o,G: produc340336539035504054m_rule > set_fm,A2: set_Pr1822751329126368876m_rule] :
( ( complete_Sup_Sup_o @ ( image_fm_o @ F @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ G @ A2 ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_8969478372357302939rule_o
@ ^ [Y: produc340336539035504054m_rule] : ( complete_Sup_Sup_o @ ( image_fm_o @ F @ ( G @ Y ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_41_SUP__UNION,axiom,
! [F: fm > set_fm,G: produc340336539035504054m_rule > set_fm,A2: set_Pr1822751329126368876m_rule] :
( ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ F @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ G @ A2 ) ) ) )
= ( comple2134933779557159616set_fm
@ ( image_377115208147135190set_fm
@ ^ [Y: produc340336539035504054m_rule] : ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ F @ ( G @ Y ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_42_SUP__UNION,axiom,
! [F: produc340336539035504054m_rule > set_fm,G: produc340336539035504054m_rule > set_Pr1822751329126368876m_rule,A2: set_Pr1822751329126368876m_rule] :
( ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ F @ ( comple5648871819318716351m_rule @ ( image_2683796897779739177m_rule @ G @ A2 ) ) ) )
= ( comple2134933779557159616set_fm
@ ( image_377115208147135190set_fm
@ ^ [Y: produc340336539035504054m_rule] : ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ F @ ( G @ Y ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_43_SUP__UNION,axiom,
! [F: fm > $o,G: $o > set_fm,A2: set_o] :
( ( complete_Sup_Sup_o @ ( image_fm_o @ F @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ G @ A2 ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [Y: $o] : ( complete_Sup_Sup_o @ ( image_fm_o @ F @ ( G @ Y ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_44_SUP__UNION,axiom,
! [F: $o > $o,G: $o > set_o,A2: set_o] :
( ( complete_Sup_Sup_o @ ( image_o_o @ F @ ( comple90263536869209701_set_o @ ( image_o_set_o @ G @ A2 ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [Y: $o] : ( complete_Sup_Sup_o @ ( image_o_o @ F @ ( G @ Y ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_45_SUP__UNION,axiom,
! [F: fm > set_fm,G: $o > set_fm,A2: set_o] :
( ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ F @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ G @ A2 ) ) ) )
= ( comple2134933779557159616set_fm
@ ( image_o_set_fm
@ ^ [Y: $o] : ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ F @ ( G @ Y ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_46_SUP__UNION,axiom,
! [F: $o > set_fm,G: $o > set_o,A2: set_o] :
( ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ ( comple90263536869209701_set_o @ ( image_o_set_o @ G @ A2 ) ) ) )
= ( comple2134933779557159616set_fm
@ ( image_o_set_fm
@ ^ [Y: $o] : ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ ( G @ Y ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_47_SUP__UNION,axiom,
! [F: fm > $o,G: set_fm > set_fm,A2: set_set_fm] :
( ( complete_Sup_Sup_o @ ( image_fm_o @ F @ ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ G @ A2 ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_set_fm_o
@ ^ [Y: set_fm] : ( complete_Sup_Sup_o @ ( image_fm_o @ F @ ( G @ Y ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_48_SUP__UNION,axiom,
! [F: fm > set_o,G: $o > set_fm,A2: set_o] :
( ( comple90263536869209701_set_o @ ( image_fm_set_o @ F @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ G @ A2 ) ) ) )
= ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [Y: $o] : ( comple90263536869209701_set_o @ ( image_fm_set_o @ F @ ( G @ Y ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_49_SUP__UNION,axiom,
! [F: $o > set_o,G: $o > set_o,A2: set_o] :
( ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ ( comple90263536869209701_set_o @ ( image_o_set_o @ G @ A2 ) ) ) )
= ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [Y: $o] : ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ ( G @ Y ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_50_tree__fms__def,axiom,
( tree_fms
= ( ^ [Steps: stream2709947120125613254m_rule] :
( comple2134933779557159616set_fm
@ ( image_377115208147135190set_fm
@ ^ [Ss: produc340336539035504054m_rule] : ( set_fm2 @ ( pseq @ Ss ) )
@ ( sset_P4484857331586881186m_rule @ Steps ) ) ) ) ) ).
% tree_fms_def
thf(fact_51_UN__extend__simps_I10_J,axiom,
! [B2: rat > set_fm,F: nat > rat,A2: set_nat] :
( ( comple2134933779557159616set_fm
@ ( image_nat_set_fm
@ ^ [A3: nat] : ( B2 @ ( F @ A3 ) )
@ A2 ) )
= ( comple2134933779557159616set_fm @ ( image_rat_set_fm @ B2 @ ( image_nat_rat @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_52_UN__extend__simps_I10_J,axiom,
! [B2: $o > set_fm,F: $o > $o,A2: set_o] :
( ( comple2134933779557159616set_fm
@ ( image_o_set_fm
@ ^ [A3: $o] : ( B2 @ ( F @ A3 ) )
@ A2 ) )
= ( comple2134933779557159616set_fm @ ( image_o_set_fm @ B2 @ ( image_o_o @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_53_UN__extend__simps_I10_J,axiom,
! [B2: rat > set_o,F: nat > rat,A2: set_nat] :
( ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [A3: nat] : ( B2 @ ( F @ A3 ) )
@ A2 ) )
= ( comple90263536869209701_set_o @ ( image_rat_set_o @ B2 @ ( image_nat_rat @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_54_UN__extend__simps_I10_J,axiom,
! [B2: $o > set_o,F: $o > $o,A2: set_o] :
( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [A3: $o] : ( B2 @ ( F @ A3 ) )
@ A2 ) )
= ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ ( image_o_o @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_55_UN__extend__simps_I10_J,axiom,
! [B2: set_o > set_fm,F: $o > set_o,A2: set_o] :
( ( comple2134933779557159616set_fm
@ ( image_o_set_fm
@ ^ [A3: $o] : ( B2 @ ( F @ A3 ) )
@ A2 ) )
= ( comple2134933779557159616set_fm @ ( image_set_o_set_fm @ B2 @ ( image_o_set_o @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_56_UN__extend__simps_I10_J,axiom,
! [B2: set_fm > set_fm,F: $o > set_fm,A2: set_o] :
( ( comple2134933779557159616set_fm
@ ( image_o_set_fm
@ ^ [A3: $o] : ( B2 @ ( F @ A3 ) )
@ A2 ) )
= ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ B2 @ ( image_o_set_fm @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_57_UN__extend__simps_I10_J,axiom,
! [B2: $o > set_fm,F: set_fm > $o,A2: set_set_fm] :
( ( comple2134933779557159616set_fm
@ ( image_set_fm_set_fm
@ ^ [A3: set_fm] : ( B2 @ ( F @ A3 ) )
@ A2 ) )
= ( comple2134933779557159616set_fm @ ( image_o_set_fm @ B2 @ ( image_set_fm_o @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_58_UN__extend__simps_I10_J,axiom,
! [B2: set_fm > set_o,F: $o > set_fm,A2: set_o] :
( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [A3: $o] : ( B2 @ ( F @ A3 ) )
@ A2 ) )
= ( comple90263536869209701_set_o @ ( image_set_fm_set_o @ B2 @ ( image_o_set_fm @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_59_UN__extend__simps_I10_J,axiom,
! [B2: set_o > set_o,F: $o > set_o,A2: set_o] :
( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [A3: $o] : ( B2 @ ( F @ A3 ) )
@ A2 ) )
= ( comple90263536869209701_set_o @ ( image_set_o_set_o @ B2 @ ( image_o_set_o @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_60_UN__extend__simps_I10_J,axiom,
! [B2: set_fm > set_fm,F: set_fm > set_fm,A2: set_set_fm] :
( ( comple2134933779557159616set_fm
@ ( image_set_fm_set_fm
@ ^ [A3: set_fm] : ( B2 @ ( F @ A3 ) )
@ A2 ) )
= ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ B2 @ ( image_set_fm_set_fm @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_61_image__UN,axiom,
! [F: fm > fm,B2: $o > set_fm,A2: set_o] :
( ( image_fm_fm @ F @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ B2 @ A2 ) ) )
= ( comple2134933779557159616set_fm
@ ( image_o_set_fm
@ ^ [X: $o] : ( image_fm_fm @ F @ ( B2 @ X ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_62_image__UN,axiom,
! [F: fm > $o,B2: $o > set_fm,A2: set_o] :
( ( image_fm_o @ F @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ B2 @ A2 ) ) )
= ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( image_fm_o @ F @ ( B2 @ X ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_63_image__UN,axiom,
! [F: $o > fm,B2: $o > set_o,A2: set_o] :
( ( image_o_fm @ F @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
= ( comple2134933779557159616set_fm
@ ( image_o_set_fm
@ ^ [X: $o] : ( image_o_fm @ F @ ( B2 @ X ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_64_image__UN,axiom,
! [F: $o > $o,B2: $o > set_o,A2: set_o] :
( ( image_o_o @ F @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
= ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( image_o_o @ F @ ( B2 @ X ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_65_image__UN,axiom,
! [F: fm > fm,B2: set_fm > set_fm,A2: set_set_fm] :
( ( image_fm_fm @ F @ ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ B2 @ A2 ) ) )
= ( comple2134933779557159616set_fm
@ ( image_set_fm_set_fm
@ ^ [X: set_fm] : ( image_fm_fm @ F @ ( B2 @ X ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_66_image__UN,axiom,
! [F: fm > $o,B2: set_fm > set_fm,A2: set_set_fm] :
( ( image_fm_o @ F @ ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ B2 @ A2 ) ) )
= ( comple90263536869209701_set_o
@ ( image_set_fm_set_o
@ ^ [X: set_fm] : ( image_fm_o @ F @ ( B2 @ X ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_67_image__UN,axiom,
! [F: $o > set_fm,B2: $o > set_o,A2: set_o] :
( ( image_o_set_fm @ F @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
= ( comple4326961365146087328set_fm
@ ( image_o_set_set_fm
@ ^ [X: $o] : ( image_o_set_fm @ F @ ( B2 @ X ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_68_image__UN,axiom,
! [F: $o > set_o,B2: $o > set_o,A2: set_o] :
( ( image_o_set_o @ F @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
= ( comple4436988014476444997_set_o
@ ( image_o_set_set_o
@ ^ [X: $o] : ( image_o_set_o @ F @ ( B2 @ X ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_69_image__UN,axiom,
! [F: $o > fm,B2: set_fm > set_o,A2: set_set_fm] :
( ( image_o_fm @ F @ ( comple90263536869209701_set_o @ ( image_set_fm_set_o @ B2 @ A2 ) ) )
= ( comple2134933779557159616set_fm
@ ( image_set_fm_set_fm
@ ^ [X: set_fm] : ( image_o_fm @ F @ ( B2 @ X ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_70_image__UN,axiom,
! [F: nat > produc340336539035504054m_rule,B2: $o > set_nat,A2: set_o] :
( ( image_8361736131336225677m_rule @ F @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ A2 ) ) )
= ( comple5648871819318716351m_rule
@ ( image_6392605097815222207m_rule
@ ^ [X: $o] : ( image_8361736131336225677m_rule @ F @ ( B2 @ X ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_71_Union__iff,axiom,
! [A2: fm,C: set_set_fm] :
( ( member_fm @ A2 @ ( comple2134933779557159616set_fm @ C ) )
= ( ? [X: set_fm] :
( ( member_set_fm @ X @ C )
& ( member_fm @ A2 @ X ) ) ) ) ).
% Union_iff
thf(fact_72_Union__iff,axiom,
! [A2: $o,C: set_set_o] :
( ( member_o @ A2 @ ( comple90263536869209701_set_o @ C ) )
= ( ? [X: set_o] :
( ( member_set_o @ X @ C )
& ( member_o @ A2 @ X ) ) ) ) ).
% Union_iff
thf(fact_73_Union__iff,axiom,
! [A2: produc340336539035504054m_rule,C: set_se4309406418413213730m_rule] :
( ( member7231649785386036813m_rule @ A2 @ ( comple5648871819318716351m_rule @ C ) )
= ( ? [X: set_Pr1822751329126368876m_rule] :
( ( member5802698409304902403m_rule @ X @ C )
& ( member7231649785386036813m_rule @ A2 @ X ) ) ) ) ).
% Union_iff
thf(fact_74_UnionI,axiom,
! [X2: set_fm,C: set_set_fm,A2: fm] :
( ( member_set_fm @ X2 @ C )
=> ( ( member_fm @ A2 @ X2 )
=> ( member_fm @ A2 @ ( comple2134933779557159616set_fm @ C ) ) ) ) ).
% UnionI
thf(fact_75_UnionI,axiom,
! [X2: set_o,C: set_set_o,A2: $o] :
( ( member_set_o @ X2 @ C )
=> ( ( member_o @ A2 @ X2 )
=> ( member_o @ A2 @ ( comple90263536869209701_set_o @ C ) ) ) ) ).
% UnionI
thf(fact_76_UnionI,axiom,
! [X2: set_Pr1822751329126368876m_rule,C: set_se4309406418413213730m_rule,A2: produc340336539035504054m_rule] :
( ( member5802698409304902403m_rule @ X2 @ C )
=> ( ( member7231649785386036813m_rule @ A2 @ X2 )
=> ( member7231649785386036813m_rule @ A2 @ ( comple5648871819318716351m_rule @ C ) ) ) ) ).
% UnionI
thf(fact_77_UN__ball__bex__simps_I1_J,axiom,
! [A2: set_set_fm,P: fm > $o] :
( ( ! [X: fm] :
( ( member_fm @ X @ ( comple2134933779557159616set_fm @ A2 ) )
=> ( P @ X ) ) )
= ( ! [X: set_fm] :
( ( member_set_fm @ X @ A2 )
=> ! [Y: fm] :
( ( member_fm @ Y @ X )
=> ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_78_UN__ball__bex__simps_I1_J,axiom,
! [A2: set_set_o,P: $o > $o] :
( ( ! [X: $o] :
( ( member_o @ X @ ( comple90263536869209701_set_o @ A2 ) )
=> ( P @ X ) ) )
= ( ! [X: set_o] :
( ( member_set_o @ X @ A2 )
=> ! [Y: $o] :
( ( member_o @ Y @ X )
=> ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_79_UN__ball__bex__simps_I1_J,axiom,
! [A2: set_se4309406418413213730m_rule,P: produc340336539035504054m_rule > $o] :
( ( ! [X: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X @ ( comple5648871819318716351m_rule @ A2 ) )
=> ( P @ X ) ) )
= ( ! [X: set_Pr1822751329126368876m_rule] :
( ( member5802698409304902403m_rule @ X @ A2 )
=> ! [Y: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ Y @ X )
=> ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_80_UN__ball__bex__simps_I3_J,axiom,
! [A2: set_set_fm,P: fm > $o] :
( ( ? [X: fm] :
( ( member_fm @ X @ ( comple2134933779557159616set_fm @ A2 ) )
& ( P @ X ) ) )
= ( ? [X: set_fm] :
( ( member_set_fm @ X @ A2 )
& ? [Y: fm] :
( ( member_fm @ Y @ X )
& ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_81_UN__ball__bex__simps_I3_J,axiom,
! [A2: set_set_o,P: $o > $o] :
( ( ? [X: $o] :
( ( member_o @ X @ ( comple90263536869209701_set_o @ A2 ) )
& ( P @ X ) ) )
= ( ? [X: set_o] :
( ( member_set_o @ X @ A2 )
& ? [Y: $o] :
( ( member_o @ Y @ X )
& ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_82_UN__ball__bex__simps_I3_J,axiom,
! [A2: set_se4309406418413213730m_rule,P: produc340336539035504054m_rule > $o] :
( ( ? [X: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X @ ( comple5648871819318716351m_rule @ A2 ) )
& ( P @ X ) ) )
= ( ? [X: set_Pr1822751329126368876m_rule] :
( ( member5802698409304902403m_rule @ X @ A2 )
& ? [Y: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ Y @ X )
& ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_83_Sup__UNIV,axiom,
( ( comple7399068483239264473et_nat @ top_top_set_set_nat )
= top_top_set_nat ) ).
% Sup_UNIV
thf(fact_84_Sup__UNIV,axiom,
( ( comple3890839924845867745et_rat @ top_top_set_set_rat )
= top_top_set_rat ) ).
% Sup_UNIV
thf(fact_85_Sup__UNIV,axiom,
( ( comple2134933779557159616set_fm @ top_top_set_set_fm )
= top_top_set_fm ) ).
% Sup_UNIV
thf(fact_86_Sup__UNIV,axiom,
( ( complete_Sup_Sup_o @ top_top_set_o )
= top_top_o ) ).
% Sup_UNIV
thf(fact_87_Sup__UNIV,axiom,
( ( comple90263536869209701_set_o @ top_top_set_set_o )
= top_top_set_o ) ).
% Sup_UNIV
thf(fact_88_Sup__UNIV,axiom,
( ( comple5648871819318716351m_rule @ top_to7036587647274415826m_rule )
= top_to8442024021002298652m_rule ) ).
% Sup_UNIV
thf(fact_89_Sup__set__def,axiom,
( comple2134933779557159616set_fm
= ( ^ [A4: set_set_fm] :
( collect_fm
@ ^ [X: fm] : ( complete_Sup_Sup_o @ ( image_set_fm_o @ ( member_fm @ X ) @ A4 ) ) ) ) ) ).
% Sup_set_def
thf(fact_90_Sup__set__def,axiom,
( comple90263536869209701_set_o
= ( ^ [A4: set_set_o] :
( collect_o
@ ^ [X: $o] : ( complete_Sup_Sup_o @ ( image_set_o_o @ ( member_o @ X ) @ A4 ) ) ) ) ) ).
% Sup_set_def
thf(fact_91_Sup__set__def,axiom,
( comple5648871819318716351m_rule
= ( ^ [A4: set_se4309406418413213730m_rule] :
( collec1009276759030335883m_rule
@ ^ [X: produc340336539035504054m_rule] : ( complete_Sup_Sup_o @ ( image_4506595978750863461rule_o @ ( member7231649785386036813m_rule @ X ) @ A4 ) ) ) ) ) ).
% Sup_set_def
thf(fact_92_Union__UNIV,axiom,
( ( comple7399068483239264473et_nat @ top_top_set_set_nat )
= top_top_set_nat ) ).
% Union_UNIV
thf(fact_93_Union__UNIV,axiom,
( ( comple3890839924845867745et_rat @ top_top_set_set_rat )
= top_top_set_rat ) ).
% Union_UNIV
thf(fact_94_Union__UNIV,axiom,
( ( comple2134933779557159616set_fm @ top_top_set_set_fm )
= top_top_set_fm ) ).
% Union_UNIV
thf(fact_95_Union__UNIV,axiom,
( ( comple90263536869209701_set_o @ top_top_set_set_o )
= top_top_set_o ) ).
% Union_UNIV
thf(fact_96_Union__UNIV,axiom,
( ( comple5648871819318716351m_rule @ top_to7036587647274415826m_rule )
= top_to8442024021002298652m_rule ) ).
% Union_UNIV
thf(fact_97_Sup_OSUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C: nat > produc340336539035504054m_rule,D: nat > produc340336539035504054m_rule,Sup: set_Pr1822751329126368876m_rule > produc340336539035504054m_rule] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_8361736131336225677m_rule @ C @ A2 ) )
= ( Sup @ ( image_8361736131336225677m_rule @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_98_Sup_OSUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C: nat > rat,D: nat > rat,Sup: set_rat > rat] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_nat_rat @ C @ A2 ) )
= ( Sup @ ( image_nat_rat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_99_Sup_OSUP__cong,axiom,
! [A2: set_set_fm,B2: set_set_fm,C: set_fm > set_fm,D: set_fm > set_fm,Sup: set_set_fm > set_fm] :
( ( A2 = B2 )
=> ( ! [X3: set_fm] :
( ( member_set_fm @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_set_fm_set_fm @ C @ A2 ) )
= ( Sup @ ( image_set_fm_set_fm @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_100_Sup_OSUP__cong,axiom,
! [A2: set_Pr1822751329126368876m_rule,B2: set_Pr1822751329126368876m_rule,C: produc340336539035504054m_rule > set_fm,D: produc340336539035504054m_rule > set_fm,Sup: set_set_fm > set_fm] :
( ( A2 = B2 )
=> ( ! [X3: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_377115208147135190set_fm @ C @ A2 ) )
= ( Sup @ ( image_377115208147135190set_fm @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_101_Sup_OSUP__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > $o,D: $o > $o,Sup: set_o > $o] :
( ( A2 = B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_o_o @ C @ A2 ) )
= ( Sup @ ( image_o_o @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_102_Sup_OSUP__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > set_Pr1822751329126368876m_rule,D: $o > set_Pr1822751329126368876m_rule,Sup: set_se4309406418413213730m_rule > set_Pr1822751329126368876m_rule] :
( ( A2 = B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_6392605097815222207m_rule @ C @ A2 ) )
= ( Sup @ ( image_6392605097815222207m_rule @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_103_Sup_OSUP__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > set_fm,D: $o > set_fm,Sup: set_set_fm > set_fm] :
( ( A2 = B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_o_set_fm @ C @ A2 ) )
= ( Sup @ ( image_o_set_fm @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_104_Sup_OSUP__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > set_o,D: $o > set_o,Sup: set_set_o > set_o] :
( ( A2 = B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_o_set_o @ C @ A2 ) )
= ( Sup @ ( image_o_set_o @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_105_Inf_OINF__cong,axiom,
! [A2: set_nat,B2: set_nat,C: nat > produc340336539035504054m_rule,D: nat > produc340336539035504054m_rule,Inf: set_Pr1822751329126368876m_rule > produc340336539035504054m_rule] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_8361736131336225677m_rule @ C @ A2 ) )
= ( Inf @ ( image_8361736131336225677m_rule @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_106_Inf_OINF__cong,axiom,
! [A2: set_nat,B2: set_nat,C: nat > rat,D: nat > rat,Inf: set_rat > rat] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_nat_rat @ C @ A2 ) )
= ( Inf @ ( image_nat_rat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_107_Inf_OINF__cong,axiom,
! [A2: set_set_fm,B2: set_set_fm,C: set_fm > set_fm,D: set_fm > set_fm,Inf: set_set_fm > set_fm] :
( ( A2 = B2 )
=> ( ! [X3: set_fm] :
( ( member_set_fm @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_set_fm_set_fm @ C @ A2 ) )
= ( Inf @ ( image_set_fm_set_fm @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_108_Inf_OINF__cong,axiom,
! [A2: set_Pr1822751329126368876m_rule,B2: set_Pr1822751329126368876m_rule,C: produc340336539035504054m_rule > set_fm,D: produc340336539035504054m_rule > set_fm,Inf: set_set_fm > set_fm] :
( ( A2 = B2 )
=> ( ! [X3: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_377115208147135190set_fm @ C @ A2 ) )
= ( Inf @ ( image_377115208147135190set_fm @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_109_Inf_OINF__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > $o,D: $o > $o,Inf: set_o > $o] :
( ( A2 = B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_o_o @ C @ A2 ) )
= ( Inf @ ( image_o_o @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_110_Inf_OINF__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > set_Pr1822751329126368876m_rule,D: $o > set_Pr1822751329126368876m_rule,Inf: set_se4309406418413213730m_rule > set_Pr1822751329126368876m_rule] :
( ( A2 = B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_6392605097815222207m_rule @ C @ A2 ) )
= ( Inf @ ( image_6392605097815222207m_rule @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_111_Inf_OINF__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > set_fm,D: $o > set_fm,Inf: set_set_fm > set_fm] :
( ( A2 = B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_o_set_fm @ C @ A2 ) )
= ( Inf @ ( image_o_set_fm @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_112_Inf_OINF__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > set_o,D: $o > set_o,Inf: set_set_o > set_o] :
( ( A2 = B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_o_set_o @ C @ A2 ) )
= ( Inf @ ( image_o_set_o @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_113_UnionE,axiom,
! [A2: fm,C: set_set_fm] :
( ( member_fm @ A2 @ ( comple2134933779557159616set_fm @ C ) )
=> ~ ! [X4: set_fm] :
( ( member_fm @ A2 @ X4 )
=> ~ ( member_set_fm @ X4 @ C ) ) ) ).
% UnionE
thf(fact_114_UnionE,axiom,
! [A2: $o,C: set_set_o] :
( ( member_o @ A2 @ ( comple90263536869209701_set_o @ C ) )
=> ~ ! [X4: set_o] :
( ( member_o @ A2 @ X4 )
=> ~ ( member_set_o @ X4 @ C ) ) ) ).
% UnionE
thf(fact_115_UnionE,axiom,
! [A2: produc340336539035504054m_rule,C: set_se4309406418413213730m_rule] :
( ( member7231649785386036813m_rule @ A2 @ ( comple5648871819318716351m_rule @ C ) )
=> ~ ! [X4: set_Pr1822751329126368876m_rule] :
( ( member7231649785386036813m_rule @ A2 @ X4 )
=> ~ ( member5802698409304902403m_rule @ X4 @ C ) ) ) ).
% UnionE
thf(fact_116_pseq__in__tree__fms,axiom,
! [X5: produc340336539035504054m_rule,Steps2: stream2709947120125613254m_rule,P2: fm] :
( ( member7231649785386036813m_rule @ X5 @ ( sset_P4484857331586881186m_rule @ Steps2 ) )
=> ( ( member_fm @ P2 @ ( set_fm2 @ ( pseq @ X5 ) ) )
=> ( member_fm @ P2 @ ( tree_fms @ Steps2 ) ) ) ) ).
% pseq_in_tree_fms
thf(fact_117_Sup_OSUP__identity__eq,axiom,
! [Sup: set_o > $o,A2: set_o] :
( ( Sup
@ ( image_o_o
@ ^ [X: $o] : X
@ A2 ) )
= ( Sup @ A2 ) ) ).
% Sup.SUP_identity_eq
thf(fact_118_Sup_OSUP__identity__eq,axiom,
! [Sup: set_set_fm > set_fm,A2: set_set_fm] :
( ( Sup
@ ( image_set_fm_set_fm
@ ^ [X: set_fm] : X
@ A2 ) )
= ( Sup @ A2 ) ) ).
% Sup.SUP_identity_eq
thf(fact_119_Inf_OINF__identity__eq,axiom,
! [Inf: set_o > $o,A2: set_o] :
( ( Inf
@ ( image_o_o
@ ^ [X: $o] : X
@ A2 ) )
= ( Inf @ A2 ) ) ).
% Inf.INF_identity_eq
thf(fact_120_Inf_OINF__identity__eq,axiom,
! [Inf: set_set_fm > set_fm,A2: set_set_fm] :
( ( Inf
@ ( image_set_fm_set_fm
@ ^ [X: set_fm] : X
@ A2 ) )
= ( Inf @ A2 ) ) ).
% Inf.INF_identity_eq
thf(fact_121_SUP__cong,axiom,
! [A2: set_fm,B2: set_fm,C: fm > $o,D: fm > $o] :
( ( A2 = B2 )
=> ( ! [X3: fm] :
( ( member_fm @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_fm_o @ C @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_fm_o @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_122_SUP__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > $o,D: $o > $o] :
( ( A2 = B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ C @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_123_SUP__cong,axiom,
! [A2: set_fm,B2: set_fm,C: fm > set_fm,D: fm > set_fm] :
( ( A2 = B2 )
=> ( ! [X3: fm] :
( ( member_fm @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ C @ A2 ) )
= ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_124_SUP__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > set_fm,D: $o > set_fm] :
( ( A2 = B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( comple2134933779557159616set_fm @ ( image_o_set_fm @ C @ A2 ) )
= ( comple2134933779557159616set_fm @ ( image_o_set_fm @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_125_SUP__cong,axiom,
! [A2: set_fm,B2: set_fm,C: fm > set_o,D: fm > set_o] :
( ( A2 = B2 )
=> ( ! [X3: fm] :
( ( member_fm @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( comple90263536869209701_set_o @ ( image_fm_set_o @ C @ A2 ) )
= ( comple90263536869209701_set_o @ ( image_fm_set_o @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_126_SUP__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > set_o,D: $o > set_o] :
( ( A2 = B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( comple90263536869209701_set_o @ ( image_o_set_o @ C @ A2 ) )
= ( comple90263536869209701_set_o @ ( image_o_set_o @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_127_SUP__cong,axiom,
! [A2: set_set_fm,B2: set_set_fm,C: set_fm > set_fm,D: set_fm > set_fm] :
( ( A2 = B2 )
=> ( ! [X3: set_fm] :
( ( member_set_fm @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ C @ A2 ) )
= ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_128_SUP__cong,axiom,
! [A2: set_Pr1822751329126368876m_rule,B2: set_Pr1822751329126368876m_rule,C: produc340336539035504054m_rule > $o,D: produc340336539035504054m_rule > $o] :
( ( A2 = B2 )
=> ( ! [X3: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_8969478372357302939rule_o @ C @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_8969478372357302939rule_o @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_129_SUP__cong,axiom,
! [A2: set_Pr1822751329126368876m_rule,B2: set_Pr1822751329126368876m_rule,C: produc340336539035504054m_rule > set_fm,D: produc340336539035504054m_rule > set_fm] :
( ( A2 = B2 )
=> ( ! [X3: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ C @ A2 ) )
= ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_130_SUP__cong,axiom,
! [A2: set_Pr1822751329126368876m_rule,B2: set_Pr1822751329126368876m_rule,C: produc340336539035504054m_rule > set_o,D: produc340336539035504054m_rule > set_o] :
( ( A2 = B2 )
=> ( ! [X3: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( comple90263536869209701_set_o @ ( image_2025595867862115707_set_o @ C @ A2 ) )
= ( comple90263536869209701_set_o @ ( image_2025595867862115707_set_o @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_131_SUP__commute,axiom,
! [F: $o > $o > $o,B2: set_o,A2: set_o] :
( ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [I: $o] : ( complete_Sup_Sup_o @ ( image_o_o @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [J: $o] :
( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [I: $o] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_132_SUP__commute,axiom,
! [F: $o > $o > set_fm,B2: set_o,A2: set_o] :
( ( comple2134933779557159616set_fm
@ ( image_o_set_fm
@ ^ [I: $o] : ( comple2134933779557159616set_fm @ ( image_o_set_fm @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple2134933779557159616set_fm
@ ( image_o_set_fm
@ ^ [J: $o] :
( comple2134933779557159616set_fm
@ ( image_o_set_fm
@ ^ [I: $o] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_133_SUP__commute,axiom,
! [F: $o > $o > set_o,B2: set_o,A2: set_o] :
( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [I: $o] : ( comple90263536869209701_set_o @ ( image_o_set_o @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [J: $o] :
( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [I: $o] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_134_SUP__commute,axiom,
! [F: $o > set_fm > set_fm,B2: set_set_fm,A2: set_o] :
( ( comple2134933779557159616set_fm
@ ( image_o_set_fm
@ ^ [I: $o] : ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple2134933779557159616set_fm
@ ( image_set_fm_set_fm
@ ^ [J: set_fm] :
( comple2134933779557159616set_fm
@ ( image_o_set_fm
@ ^ [I: $o] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_135_SUP__commute,axiom,
! [F: set_fm > $o > set_fm,B2: set_o,A2: set_set_fm] :
( ( comple2134933779557159616set_fm
@ ( image_set_fm_set_fm
@ ^ [I: set_fm] : ( comple2134933779557159616set_fm @ ( image_o_set_fm @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple2134933779557159616set_fm
@ ( image_o_set_fm
@ ^ [J: $o] :
( comple2134933779557159616set_fm
@ ( image_set_fm_set_fm
@ ^ [I: set_fm] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_136_SUP__commute,axiom,
! [F: set_fm > set_fm > set_fm,B2: set_set_fm,A2: set_set_fm] :
( ( comple2134933779557159616set_fm
@ ( image_set_fm_set_fm
@ ^ [I: set_fm] : ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple2134933779557159616set_fm
@ ( image_set_fm_set_fm
@ ^ [J: set_fm] :
( comple2134933779557159616set_fm
@ ( image_set_fm_set_fm
@ ^ [I: set_fm] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_137_SUP__commute,axiom,
! [F: produc340336539035504054m_rule > $o > set_fm,B2: set_o,A2: set_Pr1822751329126368876m_rule] :
( ( comple2134933779557159616set_fm
@ ( image_377115208147135190set_fm
@ ^ [I: produc340336539035504054m_rule] : ( comple2134933779557159616set_fm @ ( image_o_set_fm @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple2134933779557159616set_fm
@ ( image_o_set_fm
@ ^ [J: $o] :
( comple2134933779557159616set_fm
@ ( image_377115208147135190set_fm
@ ^ [I: produc340336539035504054m_rule] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_138_SUP__commute,axiom,
! [F: $o > produc340336539035504054m_rule > set_fm,B2: set_Pr1822751329126368876m_rule,A2: set_o] :
( ( comple2134933779557159616set_fm
@ ( image_o_set_fm
@ ^ [I: $o] : ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple2134933779557159616set_fm
@ ( image_377115208147135190set_fm
@ ^ [J: produc340336539035504054m_rule] :
( comple2134933779557159616set_fm
@ ( image_o_set_fm
@ ^ [I: $o] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_139_SUP__commute,axiom,
! [F: $o > $o > set_Pr1822751329126368876m_rule,B2: set_o,A2: set_o] :
( ( comple5648871819318716351m_rule
@ ( image_6392605097815222207m_rule
@ ^ [I: $o] : ( comple5648871819318716351m_rule @ ( image_6392605097815222207m_rule @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple5648871819318716351m_rule
@ ( image_6392605097815222207m_rule
@ ^ [J: $o] :
( comple5648871819318716351m_rule
@ ( image_6392605097815222207m_rule
@ ^ [I: $o] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_140_SUP__commute,axiom,
! [F: produc340336539035504054m_rule > set_fm > set_fm,B2: set_set_fm,A2: set_Pr1822751329126368876m_rule] :
( ( comple2134933779557159616set_fm
@ ( image_377115208147135190set_fm
@ ^ [I: produc340336539035504054m_rule] : ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple2134933779557159616set_fm
@ ( image_set_fm_set_fm
@ ^ [J: set_fm] :
( comple2134933779557159616set_fm
@ ( image_377115208147135190set_fm
@ ^ [I: produc340336539035504054m_rule] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_141_image__Union,axiom,
! [F: nat > rat,S: set_set_nat] :
( ( image_nat_rat @ F @ ( comple7399068483239264473et_nat @ S ) )
= ( comple3890839924845867745et_rat @ ( image_4408659257933336347et_rat @ ( image_nat_rat @ F ) @ S ) ) ) ).
% image_Union
thf(fact_142_image__Union,axiom,
! [F: fm > fm,S: set_set_fm] :
( ( image_fm_fm @ F @ ( comple2134933779557159616set_fm @ S ) )
= ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ ( image_fm_fm @ F ) @ S ) ) ) ).
% image_Union
thf(fact_143_image__Union,axiom,
! [F: fm > $o,S: set_set_fm] :
( ( image_fm_o @ F @ ( comple2134933779557159616set_fm @ S ) )
= ( comple90263536869209701_set_o @ ( image_set_fm_set_o @ ( image_fm_o @ F ) @ S ) ) ) ).
% image_Union
thf(fact_144_image__Union,axiom,
! [F: $o > fm,S: set_set_o] :
( ( image_o_fm @ F @ ( comple90263536869209701_set_o @ S ) )
= ( comple2134933779557159616set_fm @ ( image_set_o_set_fm @ ( image_o_fm @ F ) @ S ) ) ) ).
% image_Union
thf(fact_145_image__Union,axiom,
! [F: $o > $o,S: set_set_o] :
( ( image_o_o @ F @ ( comple90263536869209701_set_o @ S ) )
= ( comple90263536869209701_set_o @ ( image_set_o_set_o @ ( image_o_o @ F ) @ S ) ) ) ).
% image_Union
thf(fact_146_image__Union,axiom,
! [F: $o > set_fm,S: set_set_o] :
( ( image_o_set_fm @ F @ ( comple90263536869209701_set_o @ S ) )
= ( comple4326961365146087328set_fm @ ( image_7161930260996802752set_fm @ ( image_o_set_fm @ F ) @ S ) ) ) ).
% image_Union
thf(fact_147_image__Union,axiom,
! [F: $o > set_o,S: set_set_o] :
( ( image_o_set_o @ F @ ( comple90263536869209701_set_o @ S ) )
= ( comple4436988014476444997_set_o @ ( image_5023573440332574309_set_o @ ( image_o_set_o @ F ) @ S ) ) ) ).
% image_Union
thf(fact_148_image__Union,axiom,
! [F: set_fm > set_fm,S: set_set_set_fm] :
( ( image_set_fm_set_fm @ F @ ( comple4326961365146087328set_fm @ S ) )
= ( comple4326961365146087328set_fm @ ( image_7100568860784662213set_fm @ ( image_set_fm_set_fm @ F ) @ S ) ) ) ).
% image_Union
thf(fact_149_image__Union,axiom,
! [F: nat > produc340336539035504054m_rule,S: set_set_nat] :
( ( image_8361736131336225677m_rule @ F @ ( comple7399068483239264473et_nat @ S ) )
= ( comple5648871819318716351m_rule @ ( image_6203217489821156345m_rule @ ( image_8361736131336225677m_rule @ F ) @ S ) ) ) ).
% image_Union
thf(fact_150_image__Union,axiom,
! [F: fm > produc340336539035504054m_rule,S: set_set_fm] :
( ( image_6079799016930204900m_rule @ F @ ( comple2134933779557159616set_fm @ S ) )
= ( comple5648871819318716351m_rule @ ( image_2204476524374002298m_rule @ ( image_6079799016930204900m_rule @ F ) @ S ) ) ) ).
% image_Union
thf(fact_151_UN__UN__flatten,axiom,
! [C: fm > set_fm,B2: $o > set_fm,A2: set_o] :
( ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ C @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ B2 @ A2 ) ) ) )
= ( comple2134933779557159616set_fm
@ ( image_o_set_fm
@ ^ [Y: $o] : ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ C @ ( B2 @ Y ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_152_UN__UN__flatten,axiom,
! [C: $o > set_fm,B2: $o > set_o,A2: set_o] :
( ( comple2134933779557159616set_fm @ ( image_o_set_fm @ C @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) ) )
= ( comple2134933779557159616set_fm
@ ( image_o_set_fm
@ ^ [Y: $o] : ( comple2134933779557159616set_fm @ ( image_o_set_fm @ C @ ( B2 @ Y ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_153_UN__UN__flatten,axiom,
! [C: fm > set_o,B2: $o > set_fm,A2: set_o] :
( ( comple90263536869209701_set_o @ ( image_fm_set_o @ C @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ B2 @ A2 ) ) ) )
= ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [Y: $o] : ( comple90263536869209701_set_o @ ( image_fm_set_o @ C @ ( B2 @ Y ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_154_UN__UN__flatten,axiom,
! [C: $o > set_o,B2: $o > set_o,A2: set_o] :
( ( comple90263536869209701_set_o @ ( image_o_set_o @ C @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) ) )
= ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [Y: $o] : ( comple90263536869209701_set_o @ ( image_o_set_o @ C @ ( B2 @ Y ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_155_UN__UN__flatten,axiom,
! [C: set_fm > set_fm,B2: $o > set_set_fm,A2: set_o] :
( ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ C @ ( comple4326961365146087328set_fm @ ( image_o_set_set_fm @ B2 @ A2 ) ) ) )
= ( comple2134933779557159616set_fm
@ ( image_o_set_fm
@ ^ [Y: $o] : ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ C @ ( B2 @ Y ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_156_UN__UN__flatten,axiom,
! [C: fm > set_fm,B2: set_fm > set_fm,A2: set_set_fm] :
( ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ C @ ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ B2 @ A2 ) ) ) )
= ( comple2134933779557159616set_fm
@ ( image_set_fm_set_fm
@ ^ [Y: set_fm] : ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ C @ ( B2 @ Y ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_157_UN__UN__flatten,axiom,
! [C: $o > set_fm,B2: set_fm > set_o,A2: set_set_fm] :
( ( comple2134933779557159616set_fm @ ( image_o_set_fm @ C @ ( comple90263536869209701_set_o @ ( image_set_fm_set_o @ B2 @ A2 ) ) ) )
= ( comple2134933779557159616set_fm
@ ( image_set_fm_set_fm
@ ^ [Y: set_fm] : ( comple2134933779557159616set_fm @ ( image_o_set_fm @ C @ ( B2 @ Y ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_158_UN__UN__flatten,axiom,
! [C: fm > set_o,B2: set_fm > set_fm,A2: set_set_fm] :
( ( comple90263536869209701_set_o @ ( image_fm_set_o @ C @ ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ B2 @ A2 ) ) ) )
= ( comple90263536869209701_set_o
@ ( image_set_fm_set_o
@ ^ [Y: set_fm] : ( comple90263536869209701_set_o @ ( image_fm_set_o @ C @ ( B2 @ Y ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_159_UN__UN__flatten,axiom,
! [C: set_fm > set_fm,B2: set_fm > set_set_fm,A2: set_set_fm] :
( ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ C @ ( comple4326961365146087328set_fm @ ( image_2051799731705857957set_fm @ B2 @ A2 ) ) ) )
= ( comple2134933779557159616set_fm
@ ( image_set_fm_set_fm
@ ^ [Y: set_fm] : ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ C @ ( B2 @ Y ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_160_UN__UN__flatten,axiom,
! [C: fm > set_fm,B2: produc340336539035504054m_rule > set_fm,A2: set_Pr1822751329126368876m_rule] :
( ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ C @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B2 @ A2 ) ) ) )
= ( comple2134933779557159616set_fm
@ ( image_377115208147135190set_fm
@ ^ [Y: produc340336539035504054m_rule] : ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ C @ ( B2 @ Y ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_161_UN__E,axiom,
! [B: fm,B2: set_fm > set_fm,A2: set_set_fm] :
( ( member_fm @ B @ ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ B2 @ A2 ) ) )
=> ~ ! [X3: set_fm] :
( ( member_set_fm @ X3 @ A2 )
=> ~ ( member_fm @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_162_UN__E,axiom,
! [B: fm,B2: fm > set_fm,A2: set_fm] :
( ( member_fm @ B @ ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ B2 @ A2 ) ) )
=> ~ ! [X3: fm] :
( ( member_fm @ X3 @ A2 )
=> ~ ( member_fm @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_163_UN__E,axiom,
! [B: fm,B2: produc340336539035504054m_rule > set_fm,A2: set_Pr1822751329126368876m_rule] :
( ( member_fm @ B @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B2 @ A2 ) ) )
=> ~ ! [X3: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X3 @ A2 )
=> ~ ( member_fm @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_164_UN__E,axiom,
! [B: fm,B2: $o > set_fm,A2: set_o] :
( ( member_fm @ B @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ B2 @ A2 ) ) )
=> ~ ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ~ ( member_fm @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_165_UN__E,axiom,
! [B: $o,B2: fm > set_o,A2: set_fm] :
( ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_fm_set_o @ B2 @ A2 ) ) )
=> ~ ! [X3: fm] :
( ( member_fm @ X3 @ A2 )
=> ~ ( member_o @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_166_UN__E,axiom,
! [B: $o,B2: produc340336539035504054m_rule > set_o,A2: set_Pr1822751329126368876m_rule] :
( ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_2025595867862115707_set_o @ B2 @ A2 ) ) )
=> ~ ! [X3: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X3 @ A2 )
=> ~ ( member_o @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_167_UN__E,axiom,
! [B: $o,B2: $o > set_o,A2: set_o] :
( ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
=> ~ ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ~ ( member_o @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_168_UN__E,axiom,
! [B: produc340336539035504054m_rule,B2: fm > set_Pr1822751329126368876m_rule,A2: set_fm] :
( ( member7231649785386036813m_rule @ B @ ( comple5648871819318716351m_rule @ ( image_7929583154892365210m_rule @ B2 @ A2 ) ) )
=> ~ ! [X3: fm] :
( ( member_fm @ X3 @ A2 )
=> ~ ( member7231649785386036813m_rule @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_169_UN__E,axiom,
! [B: produc340336539035504054m_rule,B2: produc340336539035504054m_rule > set_Pr1822751329126368876m_rule,A2: set_Pr1822751329126368876m_rule] :
( ( member7231649785386036813m_rule @ B @ ( comple5648871819318716351m_rule @ ( image_2683796897779739177m_rule @ B2 @ A2 ) ) )
=> ~ ! [X3: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X3 @ A2 )
=> ~ ( member7231649785386036813m_rule @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_170_UN__E,axiom,
! [B: produc340336539035504054m_rule,B2: $o > set_Pr1822751329126368876m_rule,A2: set_o] :
( ( member7231649785386036813m_rule @ B @ ( comple5648871819318716351m_rule @ ( image_6392605097815222207m_rule @ B2 @ A2 ) ) )
=> ~ ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ~ ( member7231649785386036813m_rule @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_171_UN__extend__simps_I8_J,axiom,
! [B2: set_fm > set_fm,A2: set_set_set_fm] :
( ( comple2134933779557159616set_fm
@ ( image_2606594240855772645set_fm
@ ^ [Y: set_set_fm] : ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ B2 @ Y ) )
@ A2 ) )
= ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ B2 @ ( comple4326961365146087328set_fm @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_172_UN__extend__simps_I8_J,axiom,
! [B2: fm > set_fm,A2: set_set_fm] :
( ( comple2134933779557159616set_fm
@ ( image_set_fm_set_fm
@ ^ [Y: set_fm] : ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ B2 @ Y ) )
@ A2 ) )
= ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ B2 @ ( comple2134933779557159616set_fm @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_173_UN__extend__simps_I8_J,axiom,
! [B2: $o > set_fm,A2: set_set_o] :
( ( comple2134933779557159616set_fm
@ ( image_set_o_set_fm
@ ^ [Y: set_o] : ( comple2134933779557159616set_fm @ ( image_o_set_fm @ B2 @ Y ) )
@ A2 ) )
= ( comple2134933779557159616set_fm @ ( image_o_set_fm @ B2 @ ( comple90263536869209701_set_o @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_174_UN__extend__simps_I8_J,axiom,
! [B2: produc340336539035504054m_rule > set_fm,A2: set_se4309406418413213730m_rule] :
( ( comple2134933779557159616set_fm
@ ( image_338663757272164256set_fm
@ ^ [Y: set_Pr1822751329126368876m_rule] : ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B2 @ Y ) )
@ A2 ) )
= ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B2 @ ( comple5648871819318716351m_rule @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_175_UN__extend__simps_I8_J,axiom,
! [B2: fm > set_o,A2: set_set_fm] :
( ( comple90263536869209701_set_o
@ ( image_set_fm_set_o
@ ^ [Y: set_fm] : ( comple90263536869209701_set_o @ ( image_fm_set_o @ B2 @ Y ) )
@ A2 ) )
= ( comple90263536869209701_set_o @ ( image_fm_set_o @ B2 @ ( comple2134933779557159616set_fm @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_176_UN__extend__simps_I8_J,axiom,
! [B2: $o > set_o,A2: set_set_o] :
( ( comple90263536869209701_set_o
@ ( image_set_o_set_o
@ ^ [Y: set_o] : ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ Y ) )
@ A2 ) )
= ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ ( comple90263536869209701_set_o @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_177_UN__extend__simps_I8_J,axiom,
! [B2: produc340336539035504054m_rule > set_o,A2: set_se4309406418413213730m_rule] :
( ( comple90263536869209701_set_o
@ ( image_1016064580196875077_set_o
@ ^ [Y: set_Pr1822751329126368876m_rule] : ( comple90263536869209701_set_o @ ( image_2025595867862115707_set_o @ B2 @ Y ) )
@ A2 ) )
= ( comple90263536869209701_set_o @ ( image_2025595867862115707_set_o @ B2 @ ( comple5648871819318716351m_rule @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_178_UN__extend__simps_I8_J,axiom,
! [B2: fm > set_Pr1822751329126368876m_rule,A2: set_set_fm] :
( ( comple5648871819318716351m_rule
@ ( image_2204476524374002298m_rule
@ ^ [Y: set_fm] : ( comple5648871819318716351m_rule @ ( image_7929583154892365210m_rule @ B2 @ Y ) )
@ A2 ) )
= ( comple5648871819318716351m_rule @ ( image_7929583154892365210m_rule @ B2 @ ( comple2134933779557159616set_fm @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_179_UN__extend__simps_I8_J,axiom,
! [B2: $o > set_Pr1822751329126368876m_rule,A2: set_set_o] :
( ( comple5648871819318716351m_rule
@ ( image_6429356636794881695m_rule
@ ^ [Y: set_o] : ( comple5648871819318716351m_rule @ ( image_6392605097815222207m_rule @ B2 @ Y ) )
@ A2 ) )
= ( comple5648871819318716351m_rule @ ( image_6392605097815222207m_rule @ B2 @ ( comple90263536869209701_set_o @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_180_UN__extend__simps_I8_J,axiom,
! [B2: produc340336539035504054m_rule > set_Pr1822751329126368876m_rule,A2: set_se4309406418413213730m_rule] :
( ( comple5648871819318716351m_rule
@ ( image_7670284417601411807m_rule
@ ^ [Y: set_Pr1822751329126368876m_rule] : ( comple5648871819318716351m_rule @ ( image_2683796897779739177m_rule @ B2 @ Y ) )
@ A2 ) )
= ( comple5648871819318716351m_rule @ ( image_2683796897779739177m_rule @ B2 @ ( comple5648871819318716351m_rule @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_181_UN__extend__simps_I9_J,axiom,
! [C: fm > set_fm,B2: $o > set_fm,A2: set_o] :
( ( comple2134933779557159616set_fm
@ ( image_o_set_fm
@ ^ [X: $o] : ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ C @ ( B2 @ X ) ) )
@ A2 ) )
= ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ C @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_182_UN__extend__simps_I9_J,axiom,
! [C: $o > set_fm,B2: $o > set_o,A2: set_o] :
( ( comple2134933779557159616set_fm
@ ( image_o_set_fm
@ ^ [X: $o] : ( comple2134933779557159616set_fm @ ( image_o_set_fm @ C @ ( B2 @ X ) ) )
@ A2 ) )
= ( comple2134933779557159616set_fm @ ( image_o_set_fm @ C @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_183_UN__extend__simps_I9_J,axiom,
! [C: fm > set_o,B2: $o > set_fm,A2: set_o] :
( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( comple90263536869209701_set_o @ ( image_fm_set_o @ C @ ( B2 @ X ) ) )
@ A2 ) )
= ( comple90263536869209701_set_o @ ( image_fm_set_o @ C @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_184_UN__extend__simps_I9_J,axiom,
! [C: $o > set_o,B2: $o > set_o,A2: set_o] :
( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( comple90263536869209701_set_o @ ( image_o_set_o @ C @ ( B2 @ X ) ) )
@ A2 ) )
= ( comple90263536869209701_set_o @ ( image_o_set_o @ C @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_185_UN__extend__simps_I9_J,axiom,
! [C: set_fm > set_fm,B2: $o > set_set_fm,A2: set_o] :
( ( comple2134933779557159616set_fm
@ ( image_o_set_fm
@ ^ [X: $o] : ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ C @ ( B2 @ X ) ) )
@ A2 ) )
= ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ C @ ( comple4326961365146087328set_fm @ ( image_o_set_set_fm @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_186_UN__extend__simps_I9_J,axiom,
! [C: fm > set_fm,B2: set_fm > set_fm,A2: set_set_fm] :
( ( comple2134933779557159616set_fm
@ ( image_set_fm_set_fm
@ ^ [X: set_fm] : ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ C @ ( B2 @ X ) ) )
@ A2 ) )
= ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ C @ ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_187_UN__extend__simps_I9_J,axiom,
! [C: $o > set_fm,B2: set_fm > set_o,A2: set_set_fm] :
( ( comple2134933779557159616set_fm
@ ( image_set_fm_set_fm
@ ^ [X: set_fm] : ( comple2134933779557159616set_fm @ ( image_o_set_fm @ C @ ( B2 @ X ) ) )
@ A2 ) )
= ( comple2134933779557159616set_fm @ ( image_o_set_fm @ C @ ( comple90263536869209701_set_o @ ( image_set_fm_set_o @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_188_UN__extend__simps_I9_J,axiom,
! [C: fm > set_o,B2: set_fm > set_fm,A2: set_set_fm] :
( ( comple90263536869209701_set_o
@ ( image_set_fm_set_o
@ ^ [X: set_fm] : ( comple90263536869209701_set_o @ ( image_fm_set_o @ C @ ( B2 @ X ) ) )
@ A2 ) )
= ( comple90263536869209701_set_o @ ( image_fm_set_o @ C @ ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_189_UN__extend__simps_I9_J,axiom,
! [C: set_fm > set_fm,B2: set_fm > set_set_fm,A2: set_set_fm] :
( ( comple2134933779557159616set_fm
@ ( image_set_fm_set_fm
@ ^ [X: set_fm] : ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ C @ ( B2 @ X ) ) )
@ A2 ) )
= ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ C @ ( comple4326961365146087328set_fm @ ( image_2051799731705857957set_fm @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_190_UN__extend__simps_I9_J,axiom,
! [C: fm > set_fm,B2: produc340336539035504054m_rule > set_fm,A2: set_Pr1822751329126368876m_rule] :
( ( comple2134933779557159616set_fm
@ ( image_377115208147135190set_fm
@ ^ [X: produc340336539035504054m_rule] : ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ C @ ( B2 @ X ) ) )
@ A2 ) )
= ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ C @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_191_pseq__def,axiom,
( pseq
= ( ^ [Z: produc340336539035504054m_rule] : ( produc3242453064496107283ist_fm @ ( produc993771546023551940m_rule @ Z ) ) ) ) ).
% pseq_def
thf(fact_192_range__snd,axiom,
( ( image_2429618640364022857ist_fm @ produc3242453064496107283ist_fm @ top_to1730629564055774885ist_fm )
= top_top_set_list_fm ) ).
% range_snd
thf(fact_193_range__fst,axiom,
( ( image_8684537122991555832ist_fm @ produc993771546023551940m_rule @ top_to8442024021002298652m_rule )
= top_to1730629564055774885ist_fm ) ).
% range_fst
thf(fact_194_sset__range,axiom,
( sset_rat
= ( ^ [S2: stream_rat] : ( image_nat_rat @ ( snth_rat @ S2 ) @ top_top_set_nat ) ) ) ).
% sset_range
thf(fact_195_sset__range,axiom,
( sset_P4484857331586881186m_rule
= ( ^ [S2: stream2709947120125613254m_rule] : ( image_8361736131336225677m_rule @ ( snth_P6679518042731451922m_rule @ S2 ) @ top_top_set_nat ) ) ) ).
% sset_range
thf(fact_196_image__ident,axiom,
! [Y2: set_o] :
( ( image_o_o
@ ^ [X: $o] : X
@ Y2 )
= Y2 ) ).
% image_ident
thf(fact_197_image__ident,axiom,
! [Y2: set_set_fm] :
( ( image_set_fm_set_fm
@ ^ [X: set_fm] : X
@ Y2 )
= Y2 ) ).
% image_ident
thf(fact_198_image__eqI,axiom,
! [B: rat,F: nat > rat,X5: nat,A2: set_nat] :
( ( B
= ( F @ X5 ) )
=> ( ( member_nat @ X5 @ A2 )
=> ( member_rat @ B @ ( image_nat_rat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_199_image__eqI,axiom,
! [B: fm,F: fm > fm,X5: fm,A2: set_fm] :
( ( B
= ( F @ X5 ) )
=> ( ( member_fm @ X5 @ A2 )
=> ( member_fm @ B @ ( image_fm_fm @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_200_image__eqI,axiom,
! [B: $o,F: fm > $o,X5: fm,A2: set_fm] :
( ( B
= ( F @ X5 ) )
=> ( ( member_fm @ X5 @ A2 )
=> ( member_o @ B @ ( image_fm_o @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_201_image__eqI,axiom,
! [B: fm,F: $o > fm,X5: $o,A2: set_o] :
( ( B
= ( F @ X5 ) )
=> ( ( member_o @ X5 @ A2 )
=> ( member_fm @ B @ ( image_o_fm @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_202_image__eqI,axiom,
! [B: $o,F: $o > $o,X5: $o,A2: set_o] :
( ( B
= ( F @ X5 ) )
=> ( ( member_o @ X5 @ A2 )
=> ( member_o @ B @ ( image_o_o @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_203_image__eqI,axiom,
! [B: set_fm,F: $o > set_fm,X5: $o,A2: set_o] :
( ( B
= ( F @ X5 ) )
=> ( ( member_o @ X5 @ A2 )
=> ( member_set_fm @ B @ ( image_o_set_fm @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_204_image__eqI,axiom,
! [B: set_o,F: $o > set_o,X5: $o,A2: set_o] :
( ( B
= ( F @ X5 ) )
=> ( ( member_o @ X5 @ A2 )
=> ( member_set_o @ B @ ( image_o_set_o @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_205_image__eqI,axiom,
! [B: set_fm,F: set_fm > set_fm,X5: set_fm,A2: set_set_fm] :
( ( B
= ( F @ X5 ) )
=> ( ( member_set_fm @ X5 @ A2 )
=> ( member_set_fm @ B @ ( image_set_fm_set_fm @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_206_image__eqI,axiom,
! [B: produc340336539035504054m_rule,F: nat > produc340336539035504054m_rule,X5: nat,A2: set_nat] :
( ( B
= ( F @ X5 ) )
=> ( ( member_nat @ X5 @ A2 )
=> ( member7231649785386036813m_rule @ B @ ( image_8361736131336225677m_rule @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_207_image__eqI,axiom,
! [B: produc340336539035504054m_rule,F: fm > produc340336539035504054m_rule,X5: fm,A2: set_fm] :
( ( B
= ( F @ X5 ) )
=> ( ( member_fm @ X5 @ A2 )
=> ( member7231649785386036813m_rule @ B @ ( image_6079799016930204900m_rule @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_208_SUP__UN__eq,axiom,
! [R: produc340336539035504054m_rule > set_fm,S: set_Pr1822751329126368876m_rule] :
( ( comple3443612784043876893p_fm_o
@ ( image_5627174782981912967e_fm_o
@ ^ [I: produc340336539035504054m_rule,X: fm] : ( member_fm @ X @ ( R @ I ) )
@ S ) )
= ( ^ [X: fm] : ( member_fm @ X @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ R @ S ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_209_SUP__UN__eq,axiom,
! [R: $o > set_fm,S: set_o] :
( ( comple3443612784043876893p_fm_o
@ ( image_o_fm_o
@ ^ [I: $o,X: fm] : ( member_fm @ X @ ( R @ I ) )
@ S ) )
= ( ^ [X: fm] : ( member_fm @ X @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ R @ S ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_210_SUP__UN__eq,axiom,
! [R: set_fm > set_fm,S: set_set_fm] :
( ( comple3443612784043876893p_fm_o
@ ( image_set_fm_fm_o
@ ^ [I: set_fm,X: fm] : ( member_fm @ X @ ( R @ I ) )
@ S ) )
= ( ^ [X: fm] : ( member_fm @ X @ ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ R @ S ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_211_SUP__UN__eq,axiom,
! [R: $o > set_o,S: set_o] :
( ( complete_Sup_Sup_o_o
@ ( image_o_o_o
@ ^ [I: $o,X: $o] : ( member_o @ X @ ( R @ I ) )
@ S ) )
= ( ^ [X: $o] : ( member_o @ X @ ( comple90263536869209701_set_o @ ( image_o_set_o @ R @ S ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_212_SUP__UN__eq,axiom,
! [R: $o > set_Pr1822751329126368876m_rule,S: set_o] :
( ( comple3597286212584221062rule_o
@ ( image_7960912856160037254rule_o
@ ^ [I: $o,X: produc340336539035504054m_rule] : ( member7231649785386036813m_rule @ X @ ( R @ I ) )
@ S ) )
= ( ^ [X: produc340336539035504054m_rule] : ( member7231649785386036813m_rule @ X @ ( comple5648871819318716351m_rule @ ( image_6392605097815222207m_rule @ R @ S ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_213_UN__constant__eq,axiom,
! [A: set_fm,A2: set_set_fm,F: set_fm > set_fm,C2: set_fm] :
( ( member_set_fm @ A @ A2 )
=> ( ! [X3: set_fm] :
( ( member_set_fm @ X3 @ A2 )
=> ( ( F @ X3 )
= C2 ) )
=> ( ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ F @ A2 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_214_UN__constant__eq,axiom,
! [A: fm,A2: set_fm,F: fm > set_fm,C2: set_fm] :
( ( member_fm @ A @ A2 )
=> ( ! [X3: fm] :
( ( member_fm @ X3 @ A2 )
=> ( ( F @ X3 )
= C2 ) )
=> ( ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ F @ A2 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_215_UN__constant__eq,axiom,
! [A: produc340336539035504054m_rule,A2: set_Pr1822751329126368876m_rule,F: produc340336539035504054m_rule > set_fm,C2: set_fm] :
( ( member7231649785386036813m_rule @ A @ A2 )
=> ( ! [X3: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X3 @ A2 )
=> ( ( F @ X3 )
= C2 ) )
=> ( ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ F @ A2 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_216_UN__constant__eq,axiom,
! [A: $o,A2: set_o,F: $o > set_fm,C2: set_fm] :
( ( member_o @ A @ A2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ( F @ X3 )
= C2 ) )
=> ( ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ A2 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_217_UN__constant__eq,axiom,
! [A: fm,A2: set_fm,F: fm > set_o,C2: set_o] :
( ( member_fm @ A @ A2 )
=> ( ! [X3: fm] :
( ( member_fm @ X3 @ A2 )
=> ( ( F @ X3 )
= C2 ) )
=> ( ( comple90263536869209701_set_o @ ( image_fm_set_o @ F @ A2 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_218_UN__constant__eq,axiom,
! [A: produc340336539035504054m_rule,A2: set_Pr1822751329126368876m_rule,F: produc340336539035504054m_rule > set_o,C2: set_o] :
( ( member7231649785386036813m_rule @ A @ A2 )
=> ( ! [X3: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X3 @ A2 )
=> ( ( F @ X3 )
= C2 ) )
=> ( ( comple90263536869209701_set_o @ ( image_2025595867862115707_set_o @ F @ A2 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_219_UN__constant__eq,axiom,
! [A: $o,A2: set_o,F: $o > set_o,C2: set_o] :
( ( member_o @ A @ A2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ( F @ X3 )
= C2 ) )
=> ( ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ A2 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_220_UN__constant__eq,axiom,
! [A: fm,A2: set_fm,F: fm > set_Pr1822751329126368876m_rule,C2: set_Pr1822751329126368876m_rule] :
( ( member_fm @ A @ A2 )
=> ( ! [X3: fm] :
( ( member_fm @ X3 @ A2 )
=> ( ( F @ X3 )
= C2 ) )
=> ( ( comple5648871819318716351m_rule @ ( image_7929583154892365210m_rule @ F @ A2 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_221_UN__constant__eq,axiom,
! [A: produc340336539035504054m_rule,A2: set_Pr1822751329126368876m_rule,F: produc340336539035504054m_rule > set_Pr1822751329126368876m_rule,C2: set_Pr1822751329126368876m_rule] :
( ( member7231649785386036813m_rule @ A @ A2 )
=> ( ! [X3: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X3 @ A2 )
=> ( ( F @ X3 )
= C2 ) )
=> ( ( comple5648871819318716351m_rule @ ( image_2683796897779739177m_rule @ F @ A2 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_222_UN__constant__eq,axiom,
! [A: $o,A2: set_o,F: $o > set_Pr1822751329126368876m_rule,C2: set_Pr1822751329126368876m_rule] :
( ( member_o @ A @ A2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ( F @ X3 )
= C2 ) )
=> ( ( comple5648871819318716351m_rule @ ( image_6392605097815222207m_rule @ F @ A2 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_223_UNIV__I,axiom,
! [X5: fm] : ( member_fm @ X5 @ top_top_set_fm ) ).
% UNIV_I
thf(fact_224_UNIV__I,axiom,
! [X5: produc340336539035504054m_rule] : ( member7231649785386036813m_rule @ X5 @ top_to8442024021002298652m_rule ) ).
% UNIV_I
thf(fact_225_UNIV__I,axiom,
! [X5: nat] : ( member_nat @ X5 @ top_top_set_nat ) ).
% UNIV_I
thf(fact_226_UNIV__I,axiom,
! [X5: rat] : ( member_rat @ X5 @ top_top_set_rat ) ).
% UNIV_I
thf(fact_227_UNIV__I,axiom,
! [X5: $o] : ( member_o @ X5 @ top_top_set_o ) ).
% UNIV_I
thf(fact_228_mem__Collect__eq,axiom,
! [A: fm,P: fm > $o] :
( ( member_fm @ A @ ( collect_fm @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_229_mem__Collect__eq,axiom,
! [A: produc340336539035504054m_rule,P: produc340336539035504054m_rule > $o] :
( ( member7231649785386036813m_rule @ A @ ( collec1009276759030335883m_rule @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_230_mem__Collect__eq,axiom,
! [A: $o,P: $o > $o] :
( ( member_o @ A @ ( collect_o @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_231_Collect__mem__eq,axiom,
! [A2: set_fm] :
( ( collect_fm
@ ^ [X: fm] : ( member_fm @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_232_Collect__mem__eq,axiom,
! [A2: set_Pr1822751329126368876m_rule] :
( ( collec1009276759030335883m_rule
@ ^ [X: produc340336539035504054m_rule] : ( member7231649785386036813m_rule @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_233_Collect__mem__eq,axiom,
! [A2: set_o] :
( ( collect_o
@ ^ [X: $o] : ( member_o @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_234_iso__tuple__UNIV__I,axiom,
! [X5: fm] : ( member_fm @ X5 @ top_top_set_fm ) ).
% iso_tuple_UNIV_I
thf(fact_235_iso__tuple__UNIV__I,axiom,
! [X5: produc340336539035504054m_rule] : ( member7231649785386036813m_rule @ X5 @ top_to8442024021002298652m_rule ) ).
% iso_tuple_UNIV_I
thf(fact_236_iso__tuple__UNIV__I,axiom,
! [X5: nat] : ( member_nat @ X5 @ top_top_set_nat ) ).
% iso_tuple_UNIV_I
thf(fact_237_iso__tuple__UNIV__I,axiom,
! [X5: rat] : ( member_rat @ X5 @ top_top_set_rat ) ).
% iso_tuple_UNIV_I
thf(fact_238_iso__tuple__UNIV__I,axiom,
! [X5: $o] : ( member_o @ X5 @ top_top_set_o ) ).
% iso_tuple_UNIV_I
thf(fact_239_range__composition,axiom,
! [F: $o > $o,G: nat > $o] :
( ( image_nat_o
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ top_top_set_nat )
= ( image_o_o @ F @ ( image_nat_o @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_240_range__composition,axiom,
! [F: rat > rat,G: nat > rat] :
( ( image_nat_rat
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ top_top_set_nat )
= ( image_rat_rat @ F @ ( image_nat_rat @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_241_range__composition,axiom,
! [F: nat > rat,G: nat > nat] :
( ( image_nat_rat
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ top_top_set_nat )
= ( image_nat_rat @ F @ ( image_nat_nat @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_242_range__composition,axiom,
! [F: nat > rat,G: rat > nat] :
( ( image_rat_rat
@ ^ [X: rat] : ( F @ ( G @ X ) )
@ top_top_set_rat )
= ( image_nat_rat @ F @ ( image_rat_nat @ G @ top_top_set_rat ) ) ) ).
% range_composition
thf(fact_243_range__composition,axiom,
! [F: $o > $o,G: rat > $o] :
( ( image_rat_o
@ ^ [X: rat] : ( F @ ( G @ X ) )
@ top_top_set_rat )
= ( image_o_o @ F @ ( image_rat_o @ G @ top_top_set_rat ) ) ) ).
% range_composition
thf(fact_244_range__composition,axiom,
! [F: nat > rat,G: $o > nat] :
( ( image_o_rat
@ ^ [X: $o] : ( F @ ( G @ X ) )
@ top_top_set_o )
= ( image_nat_rat @ F @ ( image_o_nat @ G @ top_top_set_o ) ) ) ).
% range_composition
thf(fact_245_range__composition,axiom,
! [F: $o > $o,G: $o > $o] :
( ( image_o_o
@ ^ [X: $o] : ( F @ ( G @ X ) )
@ top_top_set_o )
= ( image_o_o @ F @ ( image_o_o @ G @ top_top_set_o ) ) ) ).
% range_composition
thf(fact_246_range__composition,axiom,
! [F: $o > set_fm,G: nat > $o] :
( ( image_nat_set_fm
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ top_top_set_nat )
= ( image_o_set_fm @ F @ ( image_nat_o @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_247_range__composition,axiom,
! [F: $o > set_o,G: nat > $o] :
( ( image_nat_set_o
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ top_top_set_nat )
= ( image_o_set_o @ F @ ( image_nat_o @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_248_range__composition,axiom,
! [F: $o > set_fm,G: rat > $o] :
( ( image_rat_set_fm
@ ^ [X: rat] : ( F @ ( G @ X ) )
@ top_top_set_rat )
= ( image_o_set_fm @ F @ ( image_rat_o @ G @ top_top_set_rat ) ) ) ).
% range_composition
thf(fact_249_Sup__bool__def,axiom,
( complete_Sup_Sup_o
= ( member_o @ $true ) ) ).
% Sup_bool_def
thf(fact_250_top__empty__eq,axiom,
( top_top_fm_o
= ( ^ [X: fm] : ( member_fm @ X @ top_top_set_fm ) ) ) ).
% top_empty_eq
thf(fact_251_top__empty__eq,axiom,
( top_to4689891595912939113rule_o
= ( ^ [X: produc340336539035504054m_rule] : ( member7231649785386036813m_rule @ X @ top_to8442024021002298652m_rule ) ) ) ).
% top_empty_eq
thf(fact_252_top__empty__eq,axiom,
( top_top_nat_o
= ( ^ [X: nat] : ( member_nat @ X @ top_top_set_nat ) ) ) ).
% top_empty_eq
thf(fact_253_top__empty__eq,axiom,
( top_top_rat_o
= ( ^ [X: rat] : ( member_rat @ X @ top_top_set_rat ) ) ) ).
% top_empty_eq
thf(fact_254_top__empty__eq,axiom,
( top_top_o_o
= ( ^ [X: $o] : ( member_o @ X @ top_top_set_o ) ) ) ).
% top_empty_eq
thf(fact_255_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_256_top__set__def,axiom,
( top_top_set_rat
= ( collect_rat @ top_top_rat_o ) ) ).
% top_set_def
thf(fact_257_top__set__def,axiom,
( top_top_set_o
= ( collect_o @ top_top_o_o ) ) ).
% top_set_def
thf(fact_258_UNIV__witness,axiom,
? [X3: fm] : ( member_fm @ X3 @ top_top_set_fm ) ).
% UNIV_witness
thf(fact_259_UNIV__witness,axiom,
? [X3: produc340336539035504054m_rule] : ( member7231649785386036813m_rule @ X3 @ top_to8442024021002298652m_rule ) ).
% UNIV_witness
thf(fact_260_UNIV__witness,axiom,
? [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_261_UNIV__witness,axiom,
? [X3: rat] : ( member_rat @ X3 @ top_top_set_rat ) ).
% UNIV_witness
thf(fact_262_UNIV__witness,axiom,
? [X3: $o] : ( member_o @ X3 @ top_top_set_o ) ).
% UNIV_witness
thf(fact_263_UNIV__eq__I,axiom,
! [A2: set_fm] :
( ! [X3: fm] : ( member_fm @ X3 @ A2 )
=> ( top_top_set_fm = A2 ) ) ).
% UNIV_eq_I
thf(fact_264_UNIV__eq__I,axiom,
! [A2: set_Pr1822751329126368876m_rule] :
( ! [X3: produc340336539035504054m_rule] : ( member7231649785386036813m_rule @ X3 @ A2 )
=> ( top_to8442024021002298652m_rule = A2 ) ) ).
% UNIV_eq_I
thf(fact_265_UNIV__eq__I,axiom,
! [A2: set_nat] :
( ! [X3: nat] : ( member_nat @ X3 @ A2 )
=> ( top_top_set_nat = A2 ) ) ).
% UNIV_eq_I
thf(fact_266_UNIV__eq__I,axiom,
! [A2: set_rat] :
( ! [X3: rat] : ( member_rat @ X3 @ A2 )
=> ( top_top_set_rat = A2 ) ) ).
% UNIV_eq_I
thf(fact_267_UNIV__eq__I,axiom,
! [A2: set_o] :
( ! [X3: $o] : ( member_o @ X3 @ A2 )
=> ( top_top_set_o = A2 ) ) ).
% UNIV_eq_I
thf(fact_268_imageI,axiom,
! [X5: nat,A2: set_nat,F: nat > rat] :
( ( member_nat @ X5 @ A2 )
=> ( member_rat @ ( F @ X5 ) @ ( image_nat_rat @ F @ A2 ) ) ) ).
% imageI
thf(fact_269_imageI,axiom,
! [X5: fm,A2: set_fm,F: fm > fm] :
( ( member_fm @ X5 @ A2 )
=> ( member_fm @ ( F @ X5 ) @ ( image_fm_fm @ F @ A2 ) ) ) ).
% imageI
thf(fact_270_imageI,axiom,
! [X5: fm,A2: set_fm,F: fm > $o] :
( ( member_fm @ X5 @ A2 )
=> ( member_o @ ( F @ X5 ) @ ( image_fm_o @ F @ A2 ) ) ) ).
% imageI
thf(fact_271_imageI,axiom,
! [X5: $o,A2: set_o,F: $o > fm] :
( ( member_o @ X5 @ A2 )
=> ( member_fm @ ( F @ X5 ) @ ( image_o_fm @ F @ A2 ) ) ) ).
% imageI
thf(fact_272_imageI,axiom,
! [X5: $o,A2: set_o,F: $o > $o] :
( ( member_o @ X5 @ A2 )
=> ( member_o @ ( F @ X5 ) @ ( image_o_o @ F @ A2 ) ) ) ).
% imageI
thf(fact_273_imageI,axiom,
! [X5: $o,A2: set_o,F: $o > set_fm] :
( ( member_o @ X5 @ A2 )
=> ( member_set_fm @ ( F @ X5 ) @ ( image_o_set_fm @ F @ A2 ) ) ) ).
% imageI
thf(fact_274_imageI,axiom,
! [X5: $o,A2: set_o,F: $o > set_o] :
( ( member_o @ X5 @ A2 )
=> ( member_set_o @ ( F @ X5 ) @ ( image_o_set_o @ F @ A2 ) ) ) ).
% imageI
thf(fact_275_imageI,axiom,
! [X5: set_fm,A2: set_set_fm,F: set_fm > set_fm] :
( ( member_set_fm @ X5 @ A2 )
=> ( member_set_fm @ ( F @ X5 ) @ ( image_set_fm_set_fm @ F @ A2 ) ) ) ).
% imageI
thf(fact_276_imageI,axiom,
! [X5: nat,A2: set_nat,F: nat > produc340336539035504054m_rule] :
( ( member_nat @ X5 @ A2 )
=> ( member7231649785386036813m_rule @ ( F @ X5 ) @ ( image_8361736131336225677m_rule @ F @ A2 ) ) ) ).
% imageI
thf(fact_277_imageI,axiom,
! [X5: fm,A2: set_fm,F: fm > produc340336539035504054m_rule] :
( ( member_fm @ X5 @ A2 )
=> ( member7231649785386036813m_rule @ ( F @ X5 ) @ ( image_6079799016930204900m_rule @ F @ A2 ) ) ) ).
% imageI
thf(fact_278_image__iff,axiom,
! [Z2: set_fm,F: produc340336539035504054m_rule > set_fm,A2: set_Pr1822751329126368876m_rule] :
( ( member_set_fm @ Z2 @ ( image_377115208147135190set_fm @ F @ A2 ) )
= ( ? [X: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X @ A2 )
& ( Z2
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_279_image__iff,axiom,
! [Z2: rat,F: nat > rat,A2: set_nat] :
( ( member_rat @ Z2 @ ( image_nat_rat @ F @ A2 ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A2 )
& ( Z2
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_280_image__iff,axiom,
! [Z2: set_Pr1822751329126368876m_rule,F: $o > set_Pr1822751329126368876m_rule,A2: set_o] :
( ( member5802698409304902403m_rule @ Z2 @ ( image_6392605097815222207m_rule @ F @ A2 ) )
= ( ? [X: $o] :
( ( member_o @ X @ A2 )
& ( Z2
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_281_image__iff,axiom,
! [Z2: set_fm,F: $o > set_fm,A2: set_o] :
( ( member_set_fm @ Z2 @ ( image_o_set_fm @ F @ A2 ) )
= ( ? [X: $o] :
( ( member_o @ X @ A2 )
& ( Z2
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_282_image__iff,axiom,
! [Z2: set_o,F: $o > set_o,A2: set_o] :
( ( member_set_o @ Z2 @ ( image_o_set_o @ F @ A2 ) )
= ( ? [X: $o] :
( ( member_o @ X @ A2 )
& ( Z2
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_283_image__iff,axiom,
! [Z2: set_fm,F: set_fm > set_fm,A2: set_set_fm] :
( ( member_set_fm @ Z2 @ ( image_set_fm_set_fm @ F @ A2 ) )
= ( ? [X: set_fm] :
( ( member_set_fm @ X @ A2 )
& ( Z2
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_284_image__iff,axiom,
! [Z2: produc340336539035504054m_rule,F: nat > produc340336539035504054m_rule,A2: set_nat] :
( ( member7231649785386036813m_rule @ Z2 @ ( image_8361736131336225677m_rule @ F @ A2 ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A2 )
& ( Z2
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_285_image__iff,axiom,
! [Z2: $o,F: $o > $o,A2: set_o] :
( ( member_o @ Z2 @ ( image_o_o @ F @ A2 ) )
= ( ? [X: $o] :
( ( member_o @ X @ A2 )
& ( Z2
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_286_bex__imageD,axiom,
! [F: produc340336539035504054m_rule > set_fm,A2: set_Pr1822751329126368876m_rule,P: set_fm > $o] :
( ? [X6: set_fm] :
( ( member_set_fm @ X6 @ ( image_377115208147135190set_fm @ F @ A2 ) )
& ( P @ X6 ) )
=> ? [X3: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_287_bex__imageD,axiom,
! [F: nat > produc340336539035504054m_rule,A2: set_nat,P: produc340336539035504054m_rule > $o] :
( ? [X6: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X6 @ ( image_8361736131336225677m_rule @ F @ A2 ) )
& ( P @ X6 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_288_bex__imageD,axiom,
! [F: nat > rat,A2: set_nat,P: rat > $o] :
( ? [X6: rat] :
( ( member_rat @ X6 @ ( image_nat_rat @ F @ A2 ) )
& ( P @ X6 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_289_bex__imageD,axiom,
! [F: $o > $o,A2: set_o,P: $o > $o] :
( ? [X6: $o] :
( ( member_o @ X6 @ ( image_o_o @ F @ A2 ) )
& ( P @ X6 ) )
=> ? [X3: $o] :
( ( member_o @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_290_bex__imageD,axiom,
! [F: $o > set_Pr1822751329126368876m_rule,A2: set_o,P: set_Pr1822751329126368876m_rule > $o] :
( ? [X6: set_Pr1822751329126368876m_rule] :
( ( member5802698409304902403m_rule @ X6 @ ( image_6392605097815222207m_rule @ F @ A2 ) )
& ( P @ X6 ) )
=> ? [X3: $o] :
( ( member_o @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_291_bex__imageD,axiom,
! [F: $o > set_fm,A2: set_o,P: set_fm > $o] :
( ? [X6: set_fm] :
( ( member_set_fm @ X6 @ ( image_o_set_fm @ F @ A2 ) )
& ( P @ X6 ) )
=> ? [X3: $o] :
( ( member_o @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_292_bex__imageD,axiom,
! [F: $o > set_o,A2: set_o,P: set_o > $o] :
( ? [X6: set_o] :
( ( member_set_o @ X6 @ ( image_o_set_o @ F @ A2 ) )
& ( P @ X6 ) )
=> ? [X3: $o] :
( ( member_o @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_293_bex__imageD,axiom,
! [F: set_fm > set_fm,A2: set_set_fm,P: set_fm > $o] :
( ? [X6: set_fm] :
( ( member_set_fm @ X6 @ ( image_set_fm_set_fm @ F @ A2 ) )
& ( P @ X6 ) )
=> ? [X3: set_fm] :
( ( member_set_fm @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_294_image__cong,axiom,
! [M: set_nat,N: set_nat,F: nat > produc340336539035504054m_rule,G: nat > produc340336539035504054m_rule] :
( ( M = N )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_8361736131336225677m_rule @ F @ M )
= ( image_8361736131336225677m_rule @ G @ N ) ) ) ) ).
% image_cong
thf(fact_295_image__cong,axiom,
! [M: set_nat,N: set_nat,F: nat > rat,G: nat > rat] :
( ( M = N )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_nat_rat @ F @ M )
= ( image_nat_rat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_296_image__cong,axiom,
! [M: set_set_fm,N: set_set_fm,F: set_fm > set_fm,G: set_fm > set_fm] :
( ( M = N )
=> ( ! [X3: set_fm] :
( ( member_set_fm @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_set_fm_set_fm @ F @ M )
= ( image_set_fm_set_fm @ G @ N ) ) ) ) ).
% image_cong
thf(fact_297_image__cong,axiom,
! [M: set_Pr1822751329126368876m_rule,N: set_Pr1822751329126368876m_rule,F: produc340336539035504054m_rule > set_fm,G: produc340336539035504054m_rule > set_fm] :
( ( M = N )
=> ( ! [X3: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_377115208147135190set_fm @ F @ M )
= ( image_377115208147135190set_fm @ G @ N ) ) ) ) ).
% image_cong
thf(fact_298_image__cong,axiom,
! [M: set_o,N: set_o,F: $o > $o,G: $o > $o] :
( ( M = N )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_o_o @ F @ M )
= ( image_o_o @ G @ N ) ) ) ) ).
% image_cong
thf(fact_299_image__cong,axiom,
! [M: set_o,N: set_o,F: $o > set_Pr1822751329126368876m_rule,G: $o > set_Pr1822751329126368876m_rule] :
( ( M = N )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_6392605097815222207m_rule @ F @ M )
= ( image_6392605097815222207m_rule @ G @ N ) ) ) ) ).
% image_cong
thf(fact_300_image__cong,axiom,
! [M: set_o,N: set_o,F: $o > set_fm,G: $o > set_fm] :
( ( M = N )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_o_set_fm @ F @ M )
= ( image_o_set_fm @ G @ N ) ) ) ) ).
% image_cong
thf(fact_301_image__cong,axiom,
! [M: set_o,N: set_o,F: $o > set_o,G: $o > set_o] :
( ( M = N )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_o_set_o @ F @ M )
= ( image_o_set_o @ G @ N ) ) ) ) ).
% image_cong
thf(fact_302_ball__imageD,axiom,
! [F: produc340336539035504054m_rule > set_fm,A2: set_Pr1822751329126368876m_rule,P: set_fm > $o] :
( ! [X3: set_fm] :
( ( member_set_fm @ X3 @ ( image_377115208147135190set_fm @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X6: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X6 @ A2 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_303_ball__imageD,axiom,
! [F: nat > produc340336539035504054m_rule,A2: set_nat,P: produc340336539035504054m_rule > $o] :
( ! [X3: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X3 @ ( image_8361736131336225677m_rule @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X6: nat] :
( ( member_nat @ X6 @ A2 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_304_ball__imageD,axiom,
! [F: nat > rat,A2: set_nat,P: rat > $o] :
( ! [X3: rat] :
( ( member_rat @ X3 @ ( image_nat_rat @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X6: nat] :
( ( member_nat @ X6 @ A2 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_305_ball__imageD,axiom,
! [F: $o > $o,A2: set_o,P: $o > $o] :
( ! [X3: $o] :
( ( member_o @ X3 @ ( image_o_o @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X6: $o] :
( ( member_o @ X6 @ A2 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_306_ball__imageD,axiom,
! [F: $o > set_Pr1822751329126368876m_rule,A2: set_o,P: set_Pr1822751329126368876m_rule > $o] :
( ! [X3: set_Pr1822751329126368876m_rule] :
( ( member5802698409304902403m_rule @ X3 @ ( image_6392605097815222207m_rule @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X6: $o] :
( ( member_o @ X6 @ A2 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_307_ball__imageD,axiom,
! [F: $o > set_fm,A2: set_o,P: set_fm > $o] :
( ! [X3: set_fm] :
( ( member_set_fm @ X3 @ ( image_o_set_fm @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X6: $o] :
( ( member_o @ X6 @ A2 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_308_ball__imageD,axiom,
! [F: $o > set_o,A2: set_o,P: set_o > $o] :
( ! [X3: set_o] :
( ( member_set_o @ X3 @ ( image_o_set_o @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X6: $o] :
( ( member_o @ X6 @ A2 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_309_ball__imageD,axiom,
! [F: set_fm > set_fm,A2: set_set_fm,P: set_fm > $o] :
( ! [X3: set_fm] :
( ( member_set_fm @ X3 @ ( image_set_fm_set_fm @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X6: set_fm] :
( ( member_set_fm @ X6 @ A2 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_310_rev__image__eqI,axiom,
! [X5: nat,A2: set_nat,B: rat,F: nat > rat] :
( ( member_nat @ X5 @ A2 )
=> ( ( B
= ( F @ X5 ) )
=> ( member_rat @ B @ ( image_nat_rat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_311_rev__image__eqI,axiom,
! [X5: fm,A2: set_fm,B: fm,F: fm > fm] :
( ( member_fm @ X5 @ A2 )
=> ( ( B
= ( F @ X5 ) )
=> ( member_fm @ B @ ( image_fm_fm @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_312_rev__image__eqI,axiom,
! [X5: fm,A2: set_fm,B: $o,F: fm > $o] :
( ( member_fm @ X5 @ A2 )
=> ( ( B
= ( F @ X5 ) )
=> ( member_o @ B @ ( image_fm_o @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_313_rev__image__eqI,axiom,
! [X5: $o,A2: set_o,B: fm,F: $o > fm] :
( ( member_o @ X5 @ A2 )
=> ( ( B
= ( F @ X5 ) )
=> ( member_fm @ B @ ( image_o_fm @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_314_rev__image__eqI,axiom,
! [X5: $o,A2: set_o,B: $o,F: $o > $o] :
( ( member_o @ X5 @ A2 )
=> ( ( B
= ( F @ X5 ) )
=> ( member_o @ B @ ( image_o_o @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_315_rev__image__eqI,axiom,
! [X5: $o,A2: set_o,B: set_fm,F: $o > set_fm] :
( ( member_o @ X5 @ A2 )
=> ( ( B
= ( F @ X5 ) )
=> ( member_set_fm @ B @ ( image_o_set_fm @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_316_rev__image__eqI,axiom,
! [X5: $o,A2: set_o,B: set_o,F: $o > set_o] :
( ( member_o @ X5 @ A2 )
=> ( ( B
= ( F @ X5 ) )
=> ( member_set_o @ B @ ( image_o_set_o @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_317_rev__image__eqI,axiom,
! [X5: set_fm,A2: set_set_fm,B: set_fm,F: set_fm > set_fm] :
( ( member_set_fm @ X5 @ A2 )
=> ( ( B
= ( F @ X5 ) )
=> ( member_set_fm @ B @ ( image_set_fm_set_fm @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_318_rev__image__eqI,axiom,
! [X5: nat,A2: set_nat,B: produc340336539035504054m_rule,F: nat > produc340336539035504054m_rule] :
( ( member_nat @ X5 @ A2 )
=> ( ( B
= ( F @ X5 ) )
=> ( member7231649785386036813m_rule @ B @ ( image_8361736131336225677m_rule @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_319_rev__image__eqI,axiom,
! [X5: fm,A2: set_fm,B: produc340336539035504054m_rule,F: fm > produc340336539035504054m_rule] :
( ( member_fm @ X5 @ A2 )
=> ( ( B
= ( F @ X5 ) )
=> ( member7231649785386036813m_rule @ B @ ( image_6079799016930204900m_rule @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_320_Sup__SUP__eq,axiom,
( comple3443612784043876893p_fm_o
= ( ^ [S3: set_fm_o,X: fm] : ( member_fm @ X @ ( comple2134933779557159616set_fm @ ( image_fm_o_set_fm @ collect_fm @ S3 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_321_Sup__SUP__eq,axiom,
( complete_Sup_Sup_o_o
= ( ^ [S3: set_o_o,X: $o] : ( member_o @ X @ ( comple90263536869209701_set_o @ ( image_o_o_set_o @ collect_o @ S3 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_322_Sup__SUP__eq,axiom,
( comple3597286212584221062rule_o
= ( ^ [S3: set_Pr3586676963299879929rule_o,X: produc340336539035504054m_rule] : ( member7231649785386036813m_rule @ X @ ( comple5648871819318716351m_rule @ ( image_3007582100768677440m_rule @ collec1009276759030335883m_rule @ S3 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_323_UNIV__def,axiom,
( top_top_set_nat
= ( collect_nat
@ ^ [X: nat] : $true ) ) ).
% UNIV_def
thf(fact_324_UNIV__def,axiom,
( top_top_set_rat
= ( collect_rat
@ ^ [X: rat] : $true ) ) ).
% UNIV_def
thf(fact_325_UNIV__def,axiom,
( top_top_set_o
= ( collect_o
@ ^ [X: $o] : $true ) ) ).
% UNIV_def
thf(fact_326_imageE,axiom,
! [B: rat,F: nat > rat,A2: set_nat] :
( ( member_rat @ B @ ( image_nat_rat @ F @ A2 ) )
=> ~ ! [X3: nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_327_imageE,axiom,
! [B: fm,F: fm > fm,A2: set_fm] :
( ( member_fm @ B @ ( image_fm_fm @ F @ A2 ) )
=> ~ ! [X3: fm] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_fm @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_328_imageE,axiom,
! [B: fm,F: $o > fm,A2: set_o] :
( ( member_fm @ B @ ( image_o_fm @ F @ A2 ) )
=> ~ ! [X3: $o] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_o @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_329_imageE,axiom,
! [B: $o,F: fm > $o,A2: set_fm] :
( ( member_o @ B @ ( image_fm_o @ F @ A2 ) )
=> ~ ! [X3: fm] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_fm @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_330_imageE,axiom,
! [B: $o,F: $o > $o,A2: set_o] :
( ( member_o @ B @ ( image_o_o @ F @ A2 ) )
=> ~ ! [X3: $o] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_o @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_331_imageE,axiom,
! [B: set_fm,F: $o > set_fm,A2: set_o] :
( ( member_set_fm @ B @ ( image_o_set_fm @ F @ A2 ) )
=> ~ ! [X3: $o] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_o @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_332_imageE,axiom,
! [B: set_o,F: $o > set_o,A2: set_o] :
( ( member_set_o @ B @ ( image_o_set_o @ F @ A2 ) )
=> ~ ! [X3: $o] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_o @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_333_imageE,axiom,
! [B: set_fm,F: set_fm > set_fm,A2: set_set_fm] :
( ( member_set_fm @ B @ ( image_set_fm_set_fm @ F @ A2 ) )
=> ~ ! [X3: set_fm] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_set_fm @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_334_imageE,axiom,
! [B: fm,F: produc340336539035504054m_rule > fm,A2: set_Pr1822751329126368876m_rule] :
( ( member_fm @ B @ ( image_2303374963288105462ule_fm @ F @ A2 ) )
=> ~ ! [X3: produc340336539035504054m_rule] :
( ( B
= ( F @ X3 ) )
=> ~ ( member7231649785386036813m_rule @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_335_imageE,axiom,
! [B: produc340336539035504054m_rule,F: nat > produc340336539035504054m_rule,A2: set_nat] :
( ( member7231649785386036813m_rule @ B @ ( image_8361736131336225677m_rule @ F @ A2 ) )
=> ~ ! [X3: nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_336_image__image,axiom,
! [F: rat > rat,G: nat > rat,A2: set_nat] :
( ( image_rat_rat @ F @ ( image_nat_rat @ G @ A2 ) )
= ( image_nat_rat
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_337_image__image,axiom,
! [F: nat > rat,G: nat > nat,A2: set_nat] :
( ( image_nat_rat @ F @ ( image_nat_nat @ G @ A2 ) )
= ( image_nat_rat
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_338_image__image,axiom,
! [F: $o > $o,G: $o > $o,A2: set_o] :
( ( image_o_o @ F @ ( image_o_o @ G @ A2 ) )
= ( image_o_o
@ ^ [X: $o] : ( F @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_339_image__image,axiom,
! [F: set_fm > $o,G: $o > set_fm,A2: set_o] :
( ( image_set_fm_o @ F @ ( image_o_set_fm @ G @ A2 ) )
= ( image_o_o
@ ^ [X: $o] : ( F @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_340_image__image,axiom,
! [F: set_o > $o,G: $o > set_o,A2: set_o] :
( ( image_set_o_o @ F @ ( image_o_set_o @ G @ A2 ) )
= ( image_o_o
@ ^ [X: $o] : ( F @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_341_image__image,axiom,
! [F: $o > set_fm,G: $o > $o,A2: set_o] :
( ( image_o_set_fm @ F @ ( image_o_o @ G @ A2 ) )
= ( image_o_set_fm
@ ^ [X: $o] : ( F @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_342_image__image,axiom,
! [F: $o > set_o,G: $o > $o,A2: set_o] :
( ( image_o_set_o @ F @ ( image_o_o @ G @ A2 ) )
= ( image_o_set_o
@ ^ [X: $o] : ( F @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_343_image__image,axiom,
! [F: set_fm > set_o,G: $o > set_fm,A2: set_o] :
( ( image_set_fm_set_o @ F @ ( image_o_set_fm @ G @ A2 ) )
= ( image_o_set_o
@ ^ [X: $o] : ( F @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_344_image__image,axiom,
! [F: set_o > set_fm,G: $o > set_o,A2: set_o] :
( ( image_set_o_set_fm @ F @ ( image_o_set_o @ G @ A2 ) )
= ( image_o_set_fm
@ ^ [X: $o] : ( F @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_345_image__image,axiom,
! [F: set_o > set_o,G: $o > set_o,A2: set_o] :
( ( image_set_o_set_o @ F @ ( image_o_set_o @ G @ A2 ) )
= ( image_o_set_o
@ ^ [X: $o] : ( F @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_346_Compr__image__eq,axiom,
! [F: nat > rat,A2: set_nat,P: rat > $o] :
( ( collect_rat
@ ^ [X: rat] :
( ( member_rat @ X @ ( image_nat_rat @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_nat_rat @ F
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_347_Compr__image__eq,axiom,
! [F: fm > fm,A2: set_fm,P: fm > $o] :
( ( collect_fm
@ ^ [X: fm] :
( ( member_fm @ X @ ( image_fm_fm @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_fm_fm @ F
@ ( collect_fm
@ ^ [X: fm] :
( ( member_fm @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_348_Compr__image__eq,axiom,
! [F: $o > fm,A2: set_o,P: fm > $o] :
( ( collect_fm
@ ^ [X: fm] :
( ( member_fm @ X @ ( image_o_fm @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_o_fm @ F
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_349_Compr__image__eq,axiom,
! [F: fm > $o,A2: set_fm,P: $o > $o] :
( ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( image_fm_o @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_fm_o @ F
@ ( collect_fm
@ ^ [X: fm] :
( ( member_fm @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_350_Compr__image__eq,axiom,
! [F: $o > $o,A2: set_o,P: $o > $o] :
( ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( image_o_o @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_o_o @ F
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_351_Compr__image__eq,axiom,
! [F: $o > set_fm,A2: set_o,P: set_fm > $o] :
( ( collect_set_fm
@ ^ [X: set_fm] :
( ( member_set_fm @ X @ ( image_o_set_fm @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_o_set_fm @ F
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_352_Compr__image__eq,axiom,
! [F: $o > set_o,A2: set_o,P: set_o > $o] :
( ( collect_set_o
@ ^ [X: set_o] :
( ( member_set_o @ X @ ( image_o_set_o @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_o_set_o @ F
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_353_Compr__image__eq,axiom,
! [F: set_fm > set_fm,A2: set_set_fm,P: set_fm > $o] :
( ( collect_set_fm
@ ^ [X: set_fm] :
( ( member_set_fm @ X @ ( image_set_fm_set_fm @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_set_fm_set_fm @ F
@ ( collect_set_fm
@ ^ [X: set_fm] :
( ( member_set_fm @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_354_Compr__image__eq,axiom,
! [F: produc340336539035504054m_rule > fm,A2: set_Pr1822751329126368876m_rule,P: fm > $o] :
( ( collect_fm
@ ^ [X: fm] :
( ( member_fm @ X @ ( image_2303374963288105462ule_fm @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_2303374963288105462ule_fm @ F
@ ( collec1009276759030335883m_rule
@ ^ [X: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_355_Compr__image__eq,axiom,
! [F: nat > produc340336539035504054m_rule,A2: set_nat,P: produc340336539035504054m_rule > $o] :
( ( collec1009276759030335883m_rule
@ ^ [X: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X @ ( image_8361736131336225677m_rule @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_8361736131336225677m_rule @ F
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_356_SUP__Sup__eq,axiom,
! [S: set_set_fm] :
( ( comple3443612784043876893p_fm_o
@ ( image_set_fm_fm_o
@ ^ [I: set_fm,X: fm] : ( member_fm @ X @ I )
@ S ) )
= ( ^ [X: fm] : ( member_fm @ X @ ( comple2134933779557159616set_fm @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_357_SUP__Sup__eq,axiom,
! [S: set_set_o] :
( ( complete_Sup_Sup_o_o
@ ( image_set_o_o_o
@ ^ [I: set_o,X: $o] : ( member_o @ X @ I )
@ S ) )
= ( ^ [X: $o] : ( member_o @ X @ ( comple90263536869209701_set_o @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_358_SUP__Sup__eq,axiom,
! [S: set_se4309406418413213730m_rule] :
( ( comple3597286212584221062rule_o
@ ( image_3805248563727310438rule_o
@ ^ [I: set_Pr1822751329126368876m_rule,X: produc340336539035504054m_rule] : ( member7231649785386036813m_rule @ X @ I )
@ S ) )
= ( ^ [X: produc340336539035504054m_rule] : ( member7231649785386036813m_rule @ X @ ( comple5648871819318716351m_rule @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_359_rangeI,axiom,
! [F: nat > rat,X5: nat] : ( member_rat @ ( F @ X5 ) @ ( image_nat_rat @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_360_rangeI,axiom,
! [F: nat > fm,X5: nat] : ( member_fm @ ( F @ X5 ) @ ( image_nat_fm @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_361_rangeI,axiom,
! [F: nat > $o,X5: nat] : ( member_o @ ( F @ X5 ) @ ( image_nat_o @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_362_rangeI,axiom,
! [F: rat > fm,X5: rat] : ( member_fm @ ( F @ X5 ) @ ( image_rat_fm @ F @ top_top_set_rat ) ) ).
% rangeI
thf(fact_363_rangeI,axiom,
! [F: rat > $o,X5: rat] : ( member_o @ ( F @ X5 ) @ ( image_rat_o @ F @ top_top_set_rat ) ) ).
% rangeI
thf(fact_364_rangeI,axiom,
! [F: $o > fm,X5: $o] : ( member_fm @ ( F @ X5 ) @ ( image_o_fm @ F @ top_top_set_o ) ) ).
% rangeI
thf(fact_365_rangeI,axiom,
! [F: $o > $o,X5: $o] : ( member_o @ ( F @ X5 ) @ ( image_o_o @ F @ top_top_set_o ) ) ).
% rangeI
thf(fact_366_rangeI,axiom,
! [F: $o > set_fm,X5: $o] : ( member_set_fm @ ( F @ X5 ) @ ( image_o_set_fm @ F @ top_top_set_o ) ) ).
% rangeI
thf(fact_367_rangeI,axiom,
! [F: $o > set_o,X5: $o] : ( member_set_o @ ( F @ X5 ) @ ( image_o_set_o @ F @ top_top_set_o ) ) ).
% rangeI
thf(fact_368_rangeI,axiom,
! [F: set_fm > set_fm,X5: set_fm] : ( member_set_fm @ ( F @ X5 ) @ ( image_set_fm_set_fm @ F @ top_top_set_set_fm ) ) ).
% rangeI
thf(fact_369_range__eqI,axiom,
! [B: rat,F: nat > rat,X5: nat] :
( ( B
= ( F @ X5 ) )
=> ( member_rat @ B @ ( image_nat_rat @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_370_range__eqI,axiom,
! [B: fm,F: nat > fm,X5: nat] :
( ( B
= ( F @ X5 ) )
=> ( member_fm @ B @ ( image_nat_fm @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_371_range__eqI,axiom,
! [B: $o,F: nat > $o,X5: nat] :
( ( B
= ( F @ X5 ) )
=> ( member_o @ B @ ( image_nat_o @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_372_range__eqI,axiom,
! [B: fm,F: rat > fm,X5: rat] :
( ( B
= ( F @ X5 ) )
=> ( member_fm @ B @ ( image_rat_fm @ F @ top_top_set_rat ) ) ) ).
% range_eqI
thf(fact_373_range__eqI,axiom,
! [B: $o,F: rat > $o,X5: rat] :
( ( B
= ( F @ X5 ) )
=> ( member_o @ B @ ( image_rat_o @ F @ top_top_set_rat ) ) ) ).
% range_eqI
thf(fact_374_range__eqI,axiom,
! [B: fm,F: $o > fm,X5: $o] :
( ( B
= ( F @ X5 ) )
=> ( member_fm @ B @ ( image_o_fm @ F @ top_top_set_o ) ) ) ).
% range_eqI
thf(fact_375_range__eqI,axiom,
! [B: $o,F: $o > $o,X5: $o] :
( ( B
= ( F @ X5 ) )
=> ( member_o @ B @ ( image_o_o @ F @ top_top_set_o ) ) ) ).
% range_eqI
thf(fact_376_range__eqI,axiom,
! [B: set_fm,F: $o > set_fm,X5: $o] :
( ( B
= ( F @ X5 ) )
=> ( member_set_fm @ B @ ( image_o_set_fm @ F @ top_top_set_o ) ) ) ).
% range_eqI
thf(fact_377_range__eqI,axiom,
! [B: set_o,F: $o > set_o,X5: $o] :
( ( B
= ( F @ X5 ) )
=> ( member_set_o @ B @ ( image_o_set_o @ F @ top_top_set_o ) ) ) ).
% range_eqI
thf(fact_378_range__eqI,axiom,
! [B: set_fm,F: set_fm > set_fm,X5: set_fm] :
( ( B
= ( F @ X5 ) )
=> ( member_set_fm @ B @ ( image_set_fm_set_fm @ F @ top_top_set_set_fm ) ) ) ).
% range_eqI
thf(fact_379_snth__sset,axiom,
! [S4: stream_fm,N2: nat] : ( member_fm @ ( snth_fm @ S4 @ N2 ) @ ( sset_fm @ S4 ) ) ).
% snth_sset
thf(fact_380_snth__sset,axiom,
! [S4: stream_o,N2: nat] : ( member_o @ ( snth_o @ S4 @ N2 ) @ ( sset_o @ S4 ) ) ).
% snth_sset
thf(fact_381_snth__sset,axiom,
! [S4: stream2709947120125613254m_rule,N2: nat] : ( member7231649785386036813m_rule @ ( snth_P6679518042731451922m_rule @ S4 @ N2 ) @ ( sset_P4484857331586881186m_rule @ S4 ) ) ).
% snth_sset
thf(fact_382_rangeE,axiom,
! [B: rat,F: nat > rat] :
( ( member_rat @ B @ ( image_nat_rat @ F @ top_top_set_nat ) )
=> ~ ! [X3: nat] :
( B
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_383_rangeE,axiom,
! [B: fm,F: nat > fm] :
( ( member_fm @ B @ ( image_nat_fm @ F @ top_top_set_nat ) )
=> ~ ! [X3: nat] :
( B
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_384_rangeE,axiom,
! [B: $o,F: nat > $o] :
( ( member_o @ B @ ( image_nat_o @ F @ top_top_set_nat ) )
=> ~ ! [X3: nat] :
( B
= ( ~ ( F @ X3 ) ) ) ) ).
% rangeE
thf(fact_385_rangeE,axiom,
! [B: fm,F: rat > fm] :
( ( member_fm @ B @ ( image_rat_fm @ F @ top_top_set_rat ) )
=> ~ ! [X3: rat] :
( B
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_386_rangeE,axiom,
! [B: $o,F: rat > $o] :
( ( member_o @ B @ ( image_rat_o @ F @ top_top_set_rat ) )
=> ~ ! [X3: rat] :
( B
= ( ~ ( F @ X3 ) ) ) ) ).
% rangeE
thf(fact_387_rangeE,axiom,
! [B: fm,F: $o > fm] :
( ( member_fm @ B @ ( image_o_fm @ F @ top_top_set_o ) )
=> ~ ! [X3: $o] :
( B
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_388_rangeE,axiom,
! [B: $o,F: $o > $o] :
( ( member_o @ B @ ( image_o_o @ F @ top_top_set_o ) )
=> ~ ! [X3: $o] :
( B
= ( ~ ( F @ X3 ) ) ) ) ).
% rangeE
thf(fact_389_rangeE,axiom,
! [B: set_fm,F: $o > set_fm] :
( ( member_set_fm @ B @ ( image_o_set_fm @ F @ top_top_set_o ) )
=> ~ ! [X3: $o] :
( B
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_390_rangeE,axiom,
! [B: set_o,F: $o > set_o] :
( ( member_set_o @ B @ ( image_o_set_o @ F @ top_top_set_o ) )
=> ~ ! [X3: $o] :
( B
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_391_rangeE,axiom,
! [B: set_fm,F: set_fm > set_fm] :
( ( member_set_fm @ B @ ( image_set_fm_set_fm @ F @ top_top_set_set_fm ) )
=> ~ ! [X3: set_fm] :
( B
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_392_snth__sset__smerge,axiom,
! [Ss2: stream_stream_fm,N2: nat,M2: nat] : ( member_fm @ ( snth_fm @ ( snth_stream_fm @ Ss2 @ N2 ) @ M2 ) @ ( sset_fm @ ( smerge_fm @ Ss2 ) ) ) ).
% snth_sset_smerge
thf(fact_393_snth__sset__smerge,axiom,
! [Ss2: stream_stream_o,N2: nat,M2: nat] : ( member_o @ ( snth_o @ ( snth_stream_o @ Ss2 @ N2 ) @ M2 ) @ ( sset_o @ ( smerge_o @ Ss2 ) ) ) ).
% snth_sset_smerge
thf(fact_394_snth__sset__smerge,axiom,
! [Ss2: stream3752074346242807894m_rule,N2: nat,M2: nat] : ( member7231649785386036813m_rule @ ( snth_P6679518042731451922m_rule @ ( snth_s6182113952396108578m_rule @ Ss2 @ N2 ) @ M2 ) @ ( sset_P4484857331586881186m_rule @ ( smerge193809993764105000m_rule @ Ss2 ) ) ) ).
% snth_sset_smerge
thf(fact_395_prod_Oexpand,axiom,
! [Prod: produc340336539035504054m_rule,Prod2: produc340336539035504054m_rule] :
( ( ( ( produc993771546023551940m_rule @ Prod )
= ( produc993771546023551940m_rule @ Prod2 ) )
& ( ( produc6514711208036230658m_rule @ Prod )
= ( produc6514711208036230658m_rule @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_396_prod_Oexpand,axiom,
! [Prod: produc6018962875968178549ist_fm,Prod2: produc6018962875968178549ist_fm] :
( ( ( ( produc4874987732838623697ist_fm @ Prod )
= ( produc4874987732838623697ist_fm @ Prod2 ) )
& ( ( produc3242453064496107283ist_fm @ Prod )
= ( produc3242453064496107283ist_fm @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_397_prod__eqI,axiom,
! [P2: produc340336539035504054m_rule,Q: produc340336539035504054m_rule] :
( ( ( produc993771546023551940m_rule @ P2 )
= ( produc993771546023551940m_rule @ Q ) )
=> ( ( ( produc6514711208036230658m_rule @ P2 )
= ( produc6514711208036230658m_rule @ Q ) )
=> ( P2 = Q ) ) ) ).
% prod_eqI
thf(fact_398_prod__eqI,axiom,
! [P2: produc6018962875968178549ist_fm,Q: produc6018962875968178549ist_fm] :
( ( ( produc4874987732838623697ist_fm @ P2 )
= ( produc4874987732838623697ist_fm @ Q ) )
=> ( ( ( produc3242453064496107283ist_fm @ P2 )
= ( produc3242453064496107283ist_fm @ Q ) )
=> ( P2 = Q ) ) ) ).
% prod_eqI
thf(fact_399_exE__realizer_H,axiom,
! [P: rule > produc6018962875968178549ist_fm > $o,P2: produc340336539035504054m_rule] :
( ( P @ ( produc6514711208036230658m_rule @ P2 ) @ ( produc993771546023551940m_rule @ P2 ) )
=> ~ ! [X3: produc6018962875968178549ist_fm,Y3: rule] :
~ ( P @ Y3 @ X3 ) ) ).
% exE_realizer'
thf(fact_400_exE__realizer_H,axiom,
! [P: list_fm > list_tm > $o,P2: produc6018962875968178549ist_fm] :
( ( P @ ( produc3242453064496107283ist_fm @ P2 ) @ ( produc4874987732838623697ist_fm @ P2 ) )
=> ~ ! [X3: list_tm,Y3: list_fm] :
~ ( P @ Y3 @ X3 ) ) ).
% exE_realizer'
thf(fact_401_prod__eq__iff,axiom,
( ( ^ [Y4: produc340336539035504054m_rule,Z3: produc340336539035504054m_rule] : ( Y4 = Z3 ) )
= ( ^ [S2: produc340336539035504054m_rule,T: produc340336539035504054m_rule] :
( ( ( produc993771546023551940m_rule @ S2 )
= ( produc993771546023551940m_rule @ T ) )
& ( ( produc6514711208036230658m_rule @ S2 )
= ( produc6514711208036230658m_rule @ T ) ) ) ) ) ).
% prod_eq_iff
thf(fact_402_prod__eq__iff,axiom,
( ( ^ [Y4: produc6018962875968178549ist_fm,Z3: produc6018962875968178549ist_fm] : ( Y4 = Z3 ) )
= ( ^ [S2: produc6018962875968178549ist_fm,T: produc6018962875968178549ist_fm] :
( ( ( produc4874987732838623697ist_fm @ S2 )
= ( produc4874987732838623697ist_fm @ T ) )
& ( ( produc3242453064496107283ist_fm @ S2 )
= ( produc3242453064496107283ist_fm @ T ) ) ) ) ) ).
% prod_eq_iff
thf(fact_403_sset__smerge,axiom,
! [Ss2: stream_stream_fm] :
( ( sset_fm @ ( smerge_fm @ Ss2 ) )
= ( comple2134933779557159616set_fm @ ( image_7285378133382788895set_fm @ sset_fm @ ( sset_stream_fm @ Ss2 ) ) ) ) ).
% sset_smerge
thf(fact_404_sset__smerge,axiom,
! [Ss2: stream_stream_o] :
( ( sset_o @ ( smerge_o @ Ss2 ) )
= ( comple90263536869209701_set_o @ ( image_stream_o_set_o @ sset_o @ ( sset_stream_o @ Ss2 ) ) ) ) ).
% sset_smerge
thf(fact_405_sset__smerge,axiom,
! [Ss2: stream3752074346242807894m_rule] :
( ( sset_P4484857331586881186m_rule @ ( smerge193809993764105000m_rule @ Ss2 ) )
= ( comple5648871819318716351m_rule @ ( image_3045026045495828537m_rule @ sset_P4484857331586881186m_rule @ ( sset_s1746085050419710898m_rule @ Ss2 ) ) ) ) ).
% sset_smerge
thf(fact_406_surjD,axiom,
! [F: nat > nat,Y5: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ? [X3: nat] :
( Y5
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_407_surjD,axiom,
! [F: nat > rat,Y5: rat] :
( ( ( image_nat_rat @ F @ top_top_set_nat )
= top_top_set_rat )
=> ? [X3: nat] :
( Y5
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_408_surjD,axiom,
! [F: nat > $o,Y5: $o] :
( ( ( image_nat_o @ F @ top_top_set_nat )
= top_top_set_o )
=> ? [X3: nat] :
( Y5
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_409_surjD,axiom,
! [F: rat > nat,Y5: nat] :
( ( ( image_rat_nat @ F @ top_top_set_rat )
= top_top_set_nat )
=> ? [X3: rat] :
( Y5
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_410_surjD,axiom,
! [F: rat > rat,Y5: rat] :
( ( ( image_rat_rat @ F @ top_top_set_rat )
= top_top_set_rat )
=> ? [X3: rat] :
( Y5
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_411_surjD,axiom,
! [F: rat > $o,Y5: $o] :
( ( ( image_rat_o @ F @ top_top_set_rat )
= top_top_set_o )
=> ? [X3: rat] :
( Y5
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_412_surjD,axiom,
! [F: $o > nat,Y5: nat] :
( ( ( image_o_nat @ F @ top_top_set_o )
= top_top_set_nat )
=> ? [X3: $o] :
( Y5
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_413_surjD,axiom,
! [F: $o > rat,Y5: rat] :
( ( ( image_o_rat @ F @ top_top_set_o )
= top_top_set_rat )
=> ? [X3: $o] :
( Y5
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_414_surjD,axiom,
! [F: $o > $o,Y5: $o] :
( ( ( image_o_o @ F @ top_top_set_o )
= top_top_set_o )
=> ? [X3: $o] :
( Y5
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_415_surjD,axiom,
! [F: $o > set_fm,Y5: set_fm] :
( ( ( image_o_set_fm @ F @ top_top_set_o )
= top_top_set_set_fm )
=> ? [X3: $o] :
( Y5
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_416_surjE,axiom,
! [F: nat > nat,Y5: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ~ ! [X3: nat] :
( Y5
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_417_surjE,axiom,
! [F: nat > rat,Y5: rat] :
( ( ( image_nat_rat @ F @ top_top_set_nat )
= top_top_set_rat )
=> ~ ! [X3: nat] :
( Y5
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_418_surjE,axiom,
! [F: nat > $o,Y5: $o] :
( ( ( image_nat_o @ F @ top_top_set_nat )
= top_top_set_o )
=> ~ ! [X3: nat] :
( Y5
= ( ~ ( F @ X3 ) ) ) ) ).
% surjE
thf(fact_419_surjE,axiom,
! [F: rat > nat,Y5: nat] :
( ( ( image_rat_nat @ F @ top_top_set_rat )
= top_top_set_nat )
=> ~ ! [X3: rat] :
( Y5
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_420_surjE,axiom,
! [F: rat > rat,Y5: rat] :
( ( ( image_rat_rat @ F @ top_top_set_rat )
= top_top_set_rat )
=> ~ ! [X3: rat] :
( Y5
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_421_surjE,axiom,
! [F: rat > $o,Y5: $o] :
( ( ( image_rat_o @ F @ top_top_set_rat )
= top_top_set_o )
=> ~ ! [X3: rat] :
( Y5
= ( ~ ( F @ X3 ) ) ) ) ).
% surjE
thf(fact_422_surjE,axiom,
! [F: $o > nat,Y5: nat] :
( ( ( image_o_nat @ F @ top_top_set_o )
= top_top_set_nat )
=> ~ ! [X3: $o] :
( Y5
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_423_surjE,axiom,
! [F: $o > rat,Y5: rat] :
( ( ( image_o_rat @ F @ top_top_set_o )
= top_top_set_rat )
=> ~ ! [X3: $o] :
( Y5
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_424_surjE,axiom,
! [F: $o > $o,Y5: $o] :
( ( ( image_o_o @ F @ top_top_set_o )
= top_top_set_o )
=> ~ ! [X3: $o] :
( Y5
= ( ~ ( F @ X3 ) ) ) ) ).
% surjE
thf(fact_425_surjE,axiom,
! [F: $o > set_fm,Y5: set_fm] :
( ( ( image_o_set_fm @ F @ top_top_set_o )
= top_top_set_set_fm )
=> ~ ! [X3: $o] :
( Y5
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_426_surjI,axiom,
! [G: nat > nat,F: nat > nat] :
( ! [X3: nat] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_nat_nat @ G @ top_top_set_nat )
= top_top_set_nat ) ) ).
% surjI
thf(fact_427_surjI,axiom,
! [G: nat > rat,F: rat > nat] :
( ! [X3: rat] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_nat_rat @ G @ top_top_set_nat )
= top_top_set_rat ) ) ).
% surjI
thf(fact_428_surjI,axiom,
! [G: nat > $o,F: $o > nat] :
( ! [X3: $o] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_nat_o @ G @ top_top_set_nat )
= top_top_set_o ) ) ).
% surjI
thf(fact_429_surjI,axiom,
! [G: rat > nat,F: nat > rat] :
( ! [X3: nat] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_rat_nat @ G @ top_top_set_rat )
= top_top_set_nat ) ) ).
% surjI
thf(fact_430_surjI,axiom,
! [G: rat > rat,F: rat > rat] :
( ! [X3: rat] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_rat_rat @ G @ top_top_set_rat )
= top_top_set_rat ) ) ).
% surjI
thf(fact_431_surjI,axiom,
! [G: rat > $o,F: $o > rat] :
( ! [X3: $o] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_rat_o @ G @ top_top_set_rat )
= top_top_set_o ) ) ).
% surjI
thf(fact_432_surjI,axiom,
! [G: $o > nat,F: nat > $o] :
( ! [X3: nat] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_o_nat @ G @ top_top_set_o )
= top_top_set_nat ) ) ).
% surjI
thf(fact_433_surjI,axiom,
! [G: $o > rat,F: rat > $o] :
( ! [X3: rat] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_o_rat @ G @ top_top_set_o )
= top_top_set_rat ) ) ).
% surjI
thf(fact_434_surjI,axiom,
! [G: $o > $o,F: $o > $o] :
( ! [X3: $o] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_o_o @ G @ top_top_set_o )
= top_top_set_o ) ) ).
% surjI
thf(fact_435_surjI,axiom,
! [G: $o > set_fm,F: set_fm > $o] :
( ! [X3: set_fm] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_o_set_fm @ G @ top_top_set_o )
= top_top_set_set_fm ) ) ).
% surjI
thf(fact_436_surj__def,axiom,
! [F: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
= ( ! [Y: nat] :
? [X: nat] :
( Y
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_437_surj__def,axiom,
! [F: nat > rat] :
( ( ( image_nat_rat @ F @ top_top_set_nat )
= top_top_set_rat )
= ( ! [Y: rat] :
? [X: nat] :
( Y
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_438_surj__def,axiom,
! [F: nat > $o] :
( ( ( image_nat_o @ F @ top_top_set_nat )
= top_top_set_o )
= ( ! [Y: $o] :
? [X: nat] :
( Y
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_439_surj__def,axiom,
! [F: rat > nat] :
( ( ( image_rat_nat @ F @ top_top_set_rat )
= top_top_set_nat )
= ( ! [Y: nat] :
? [X: rat] :
( Y
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_440_surj__def,axiom,
! [F: rat > rat] :
( ( ( image_rat_rat @ F @ top_top_set_rat )
= top_top_set_rat )
= ( ! [Y: rat] :
? [X: rat] :
( Y
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_441_surj__def,axiom,
! [F: rat > $o] :
( ( ( image_rat_o @ F @ top_top_set_rat )
= top_top_set_o )
= ( ! [Y: $o] :
? [X: rat] :
( Y
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_442_surj__def,axiom,
! [F: $o > nat] :
( ( ( image_o_nat @ F @ top_top_set_o )
= top_top_set_nat )
= ( ! [Y: nat] :
? [X: $o] :
( Y
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_443_surj__def,axiom,
! [F: $o > rat] :
( ( ( image_o_rat @ F @ top_top_set_o )
= top_top_set_rat )
= ( ! [Y: rat] :
? [X: $o] :
( Y
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_444_surj__def,axiom,
! [F: $o > $o] :
( ( ( image_o_o @ F @ top_top_set_o )
= top_top_set_o )
= ( ! [Y: $o] :
? [X: $o] :
( Y
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_445_surj__def,axiom,
! [F: $o > set_fm] :
( ( ( image_o_set_fm @ F @ top_top_set_o )
= top_top_set_set_fm )
= ( ! [Y: set_fm] :
? [X: $o] :
( Y
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_446_surj__from__nat,axiom,
( ( image_nat_nat @ from_nat_nat @ top_top_set_nat )
= top_top_set_nat ) ).
% surj_from_nat
thf(fact_447_surj__from__nat,axiom,
( ( image_nat_rat @ from_nat_rat @ top_top_set_nat )
= top_top_set_rat ) ).
% surj_from_nat
thf(fact_448_surj__from__nat,axiom,
( ( image_nat_o @ from_nat_o @ top_top_set_nat )
= top_top_set_o ) ).
% surj_from_nat
thf(fact_449_rat__denum,axiom,
? [F2: nat > rat] :
( ( image_nat_rat @ F2 @ top_top_set_nat )
= top_top_set_rat ) ).
% rat_denum
thf(fact_450_SUP__bool__eq,axiom,
( ( ^ [A4: set_o,F3: $o > $o] : ( complete_Sup_Sup_o @ ( image_o_o @ F3 @ A4 ) ) )
= bex_o ) ).
% SUP_bool_eq
thf(fact_451_snd__swap,axiom,
! [X5: produc340336539035504054m_rule] :
( ( produc9041034967185247836ist_fm @ ( produc3121145899861168912m_rule @ X5 ) )
= ( produc993771546023551940m_rule @ X5 ) ) ).
% snd_swap
thf(fact_452_snd__swap,axiom,
! [X5: produc3002719820330532825ist_tm] :
( ( produc3242453064496107283ist_fm @ ( produc3209917318656576745ist_tm @ X5 ) )
= ( produc2105211372612122933ist_tm @ X5 ) ) ).
% snd_swap
thf(fact_453_fst__swap,axiom,
! [X5: produc2353945479044264792ist_fm] :
( ( produc993771546023551940m_rule @ ( produc5647469659010186090ist_fm @ X5 ) )
= ( produc9041034967185247836ist_fm @ X5 ) ) ).
% fst_swap
thf(fact_454_fst__swap,axiom,
! [X5: produc6018962875968178549ist_fm] :
( ( produc2105211372612122933ist_tm @ ( produc5979693678883077509ist_fm @ X5 ) )
= ( produc3242453064496107283ist_fm @ X5 ) ) ).
% fst_swap
thf(fact_455_snd__image__Sigma,axiom,
! [A2: set_list_tm,B2: list_tm > set_list_fm] :
( ( image_2429618640364022857ist_fm @ produc3242453064496107283ist_fm @ ( produc3940501721629090966ist_fm @ A2 @ B2 ) )
= ( comple8784269564784259782ist_fm @ ( image_1622534648157636019ist_fm @ B2 @ A2 ) ) ) ).
% snd_image_Sigma
thf(fact_456_snd__image__Sigma,axiom,
! [A2: set_Pr1822751329126368876m_rule,B2: produc340336539035504054m_rule > set_fm] :
( ( image_2218355382418627706_fm_fm @ produc2872832660204944694ule_fm @ ( produc4849014715371551731ule_fm @ A2 @ B2 ) )
= ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B2 @ A2 ) ) ) ).
% snd_image_Sigma
thf(fact_457_snd__image__Sigma,axiom,
! [A2: set_o,B2: $o > set_fm] :
( ( image_5810579989281242486_fm_fm @ product_snd_o_fm @ ( product_Sigma_o_fm @ A2 @ B2 ) )
= ( comple2134933779557159616set_fm @ ( image_o_set_fm @ B2 @ A2 ) ) ) ).
% snd_image_Sigma
thf(fact_458_snd__image__Sigma,axiom,
! [A2: set_set_fm,B2: set_fm > set_fm] :
( ( image_7407043975336642353_fm_fm @ produc1578770745001813669_fm_fm @ ( produc5628978988381973288_fm_fm @ A2 @ B2 ) )
= ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ B2 @ A2 ) ) ) ).
% snd_image_Sigma
thf(fact_459_snd__image__Sigma,axiom,
! [A2: set_o,B2: $o > set_o] :
( ( image_7896445794123959606_o_o_o @ product_snd_o_o @ ( product_Sigma_o_o @ A2 @ B2 ) )
= ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) ) ).
% snd_image_Sigma
thf(fact_460_snd__image__Sigma,axiom,
! [A2: set_o,B2: $o > set_Pr1822751329126368876m_rule] :
( ( image_1310218892822311482m_rule @ produc7147932500738176585m_rule @ ( produc7949347099473444230m_rule @ A2 @ B2 ) )
= ( comple5648871819318716351m_rule @ ( image_6392605097815222207m_rule @ B2 @ A2 ) ) ) ).
% snd_image_Sigma
thf(fact_461_Stream_Osmember__def,axiom,
( smember_fm
= ( ^ [X: fm,S2: stream_fm] : ( member_fm @ X @ ( sset_fm @ S2 ) ) ) ) ).
% Stream.smember_def
thf(fact_462_Stream_Osmember__def,axiom,
( smember_o
= ( ^ [X: $o,S2: stream_o] : ( member_o @ X @ ( sset_o @ S2 ) ) ) ) ).
% Stream.smember_def
thf(fact_463_Stream_Osmember__def,axiom,
( smembe4439892024482649336m_rule
= ( ^ [X: produc340336539035504054m_rule,S2: stream2709947120125613254m_rule] : ( member7231649785386036813m_rule @ X @ ( sset_P4484857331586881186m_rule @ S2 ) ) ) ) ).
% Stream.smember_def
thf(fact_464_True__in__image__Bex,axiom,
! [P: $o > $o,A2: set_o] :
( ( member_o @ $true @ ( image_o_o @ P @ A2 ) )
= ( ? [X: $o] :
( ( member_o @ X @ A2 )
& ( P @ X ) ) ) ) ).
% True_in_image_Bex
thf(fact_465_UNIV__Times__UNIV,axiom,
( ( produc457027306803732586at_nat @ top_top_set_nat
@ ^ [Uu: nat] : top_top_set_nat )
= top_to4669805908274784177at_nat ) ).
% UNIV_Times_UNIV
thf(fact_466_UNIV__Times__UNIV,axiom,
( ( produc9045269283572012658at_rat @ top_top_set_nat
@ ^ [Uu: nat] : top_top_set_rat )
= top_to7513191607651882425at_rat ) ).
% UNIV_Times_UNIV
thf(fact_467_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_468_UNIV__Times__UNIV,axiom,
( ( produc140472662868866674at_nat @ top_top_set_rat
@ ^ [Uu: rat] : top_top_set_nat )
= top_to269121717765781945at_nat ) ).
% UNIV_Times_UNIV
thf(fact_469_UNIV__Times__UNIV,axiom,
( ( produc8728714639637146746at_rat @ top_top_set_rat
@ ^ [Uu: rat] : top_top_set_rat )
= top_to3112507417142880193at_rat ) ).
% UNIV_Times_UNIV
thf(fact_470_UNIV__Times__UNIV,axiom,
( ( product_Sigma_rat_o @ top_top_set_rat
@ ^ [Uu: rat] : top_top_set_o )
= top_to7320853134386005659_rat_o ) ).
% UNIV_Times_UNIV
thf(fact_471_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_472_UNIV__Times__UNIV,axiom,
( ( product_Sigma_o_rat @ top_top_set_o
@ ^ [Uu: $o] : top_top_set_rat )
= top_to642698169864860165_o_rat ) ).
% UNIV_Times_UNIV
thf(fact_473_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_474_gt__ex,axiom,
! [X5: nat] :
? [X_1: nat] : ( ord_less_nat @ X5 @ X_1 ) ).
% gt_ex
thf(fact_475_less__imp__neq,axiom,
! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ( X5 != Y5 ) ) ).
% less_imp_neq
thf(fact_476_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_477_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_478_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_479_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y6: nat] :
( ( ord_less_nat @ Y6 @ X3 )
=> ( P @ Y6 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_480_antisym__conv3,axiom,
! [Y5: nat,X5: nat] :
( ~ ( ord_less_nat @ Y5 @ X5 )
=> ( ( ~ ( ord_less_nat @ X5 @ Y5 ) )
= ( X5 = Y5 ) ) ) ).
% antisym_conv3
thf(fact_481_linorder__cases,axiom,
! [X5: nat,Y5: nat] :
( ~ ( ord_less_nat @ X5 @ Y5 )
=> ( ( X5 != Y5 )
=> ( ord_less_nat @ Y5 @ X5 ) ) ) ).
% linorder_cases
thf(fact_482_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_483_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_484_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X7: nat] : ( P3 @ X7 ) )
= ( ^ [P4: nat > $o] :
? [N3: nat] :
( ( P4 @ N3 )
& ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ~ ( P4 @ M3 ) ) ) ) ) ).
% exists_least_iff
thf(fact_485_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B3: nat] :
( ( ord_less_nat @ A5 @ B3 )
=> ( P @ A5 @ B3 ) )
=> ( ! [A5: nat] : ( P @ A5 @ A5 )
=> ( ! [A5: nat,B3: nat] :
( ( P @ B3 @ A5 )
=> ( P @ A5 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_486_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_487_not__less__iff__gr__or__eq,axiom,
! [X5: nat,Y5: nat] :
( ( ~ ( ord_less_nat @ X5 @ Y5 ) )
= ( ( ord_less_nat @ Y5 @ X5 )
| ( X5 = Y5 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_488_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_489_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_490_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_491_Bex__def,axiom,
( bex_fm
= ( ^ [A4: set_fm,P4: fm > $o] :
? [X: fm] :
( ( member_fm @ X @ A4 )
& ( P4 @ X ) ) ) ) ).
% Bex_def
thf(fact_492_Bex__def,axiom,
( bex_Pr954714274454064726m_rule
= ( ^ [A4: set_Pr1822751329126368876m_rule,P4: produc340336539035504054m_rule > $o] :
? [X: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X @ A4 )
& ( P4 @ X ) ) ) ) ).
% Bex_def
thf(fact_493_Bex__def,axiom,
( bex_o
= ( ^ [A4: set_o,P4: $o > $o] :
? [X: $o] :
( ( member_o @ X @ A4 )
& ( P4 @ X ) ) ) ) ).
% Bex_def
thf(fact_494_linorder__neqE,axiom,
! [X5: nat,Y5: nat] :
( ( X5 != Y5 )
=> ( ~ ( ord_less_nat @ X5 @ Y5 )
=> ( ord_less_nat @ Y5 @ X5 ) ) ) ).
% linorder_neqE
thf(fact_495_order__less__asym,axiom,
! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ~ ( ord_less_nat @ Y5 @ X5 ) ) ).
% order_less_asym
thf(fact_496_linorder__neq__iff,axiom,
! [X5: nat,Y5: nat] :
( ( X5 != Y5 )
= ( ( ord_less_nat @ X5 @ Y5 )
| ( ord_less_nat @ Y5 @ X5 ) ) ) ).
% linorder_neq_iff
thf(fact_497_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_498_order__less__trans,axiom,
! [X5: nat,Y5: nat,Z2: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ( ( ord_less_nat @ Y5 @ Z2 )
=> ( ord_less_nat @ X5 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_499_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_500_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_501_order__less__irrefl,axiom,
! [X5: nat] :
~ ( ord_less_nat @ X5 @ X5 ) ).
% order_less_irrefl
thf(fact_502_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_503_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_504_order__less__not__sym,axiom,
! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ~ ( ord_less_nat @ Y5 @ X5 ) ) ).
% order_less_not_sym
thf(fact_505_order__less__imp__triv,axiom,
! [X5: nat,Y5: nat,P: $o] :
( ( ord_less_nat @ X5 @ Y5 )
=> ( ( ord_less_nat @ Y5 @ X5 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_506_linorder__less__linear,axiom,
! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
| ( X5 = Y5 )
| ( ord_less_nat @ Y5 @ X5 ) ) ).
% linorder_less_linear
thf(fact_507_order__less__imp__not__eq,axiom,
! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ( X5 != Y5 ) ) ).
% order_less_imp_not_eq
thf(fact_508_order__less__imp__not__eq2,axiom,
! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ( Y5 != X5 ) ) ).
% order_less_imp_not_eq2
thf(fact_509_order__less__imp__not__less,axiom,
! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ~ ( ord_less_nat @ Y5 @ X5 ) ) ).
% order_less_imp_not_less
thf(fact_510_top_Onot__eq__extremum,axiom,
! [A: set_nat] :
( ( A != top_top_set_nat )
= ( ord_less_set_nat @ A @ top_top_set_nat ) ) ).
% top.not_eq_extremum
thf(fact_511_top_Onot__eq__extremum,axiom,
! [A: set_rat] :
( ( A != top_top_set_rat )
= ( ord_less_set_rat @ A @ top_top_set_rat ) ) ).
% top.not_eq_extremum
thf(fact_512_top_Onot__eq__extremum,axiom,
! [A: set_o] :
( ( A != top_top_set_o )
= ( ord_less_set_o @ A @ top_top_set_o ) ) ).
% top.not_eq_extremum
thf(fact_513_top_Oextremum__strict,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ top_top_set_nat @ A ) ).
% top.extremum_strict
thf(fact_514_top_Oextremum__strict,axiom,
! [A: set_rat] :
~ ( ord_less_set_rat @ top_top_set_rat @ A ) ).
% top.extremum_strict
thf(fact_515_top_Oextremum__strict,axiom,
! [A: set_o] :
~ ( ord_less_set_o @ top_top_set_o @ A ) ).
% top.extremum_strict
thf(fact_516_image__def,axiom,
( image_377115208147135190set_fm
= ( ^ [F3: produc340336539035504054m_rule > set_fm,A4: set_Pr1822751329126368876m_rule] :
( collect_set_fm
@ ^ [Y: set_fm] :
? [X: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X @ A4 )
& ( Y
= ( F3 @ X ) ) ) ) ) ) ).
% image_def
thf(fact_517_image__def,axiom,
( image_8361736131336225677m_rule
= ( ^ [F3: nat > produc340336539035504054m_rule,A4: set_nat] :
( collec1009276759030335883m_rule
@ ^ [Y: produc340336539035504054m_rule] :
? [X: nat] :
( ( member_nat @ X @ A4 )
& ( Y
= ( F3 @ X ) ) ) ) ) ) ).
% image_def
thf(fact_518_image__def,axiom,
( image_nat_rat
= ( ^ [F3: nat > rat,A4: set_nat] :
( collect_rat
@ ^ [Y: rat] :
? [X: nat] :
( ( member_nat @ X @ A4 )
& ( Y
= ( F3 @ X ) ) ) ) ) ) ).
% image_def
thf(fact_519_image__def,axiom,
( image_o_o
= ( ^ [F3: $o > $o,A4: set_o] :
( collect_o
@ ^ [Y: $o] :
? [X: $o] :
( ( member_o @ X @ A4 )
& ( Y
= ( F3 @ X ) ) ) ) ) ) ).
% image_def
thf(fact_520_image__def,axiom,
( image_6392605097815222207m_rule
= ( ^ [F3: $o > set_Pr1822751329126368876m_rule,A4: set_o] :
( collec7794190411160120897m_rule
@ ^ [Y: set_Pr1822751329126368876m_rule] :
? [X: $o] :
( ( member_o @ X @ A4 )
& ( Y
= ( F3 @ X ) ) ) ) ) ) ).
% image_def
thf(fact_521_image__def,axiom,
( image_o_set_fm
= ( ^ [F3: $o > set_fm,A4: set_o] :
( collect_set_fm
@ ^ [Y: set_fm] :
? [X: $o] :
( ( member_o @ X @ A4 )
& ( Y
= ( F3 @ X ) ) ) ) ) ) ).
% image_def
thf(fact_522_image__def,axiom,
( image_o_set_o
= ( ^ [F3: $o > set_o,A4: set_o] :
( collect_set_o
@ ^ [Y: set_o] :
? [X: $o] :
( ( member_o @ X @ A4 )
& ( Y
= ( F3 @ X ) ) ) ) ) ) ).
% image_def
thf(fact_523_image__def,axiom,
( image_set_fm_set_fm
= ( ^ [F3: set_fm > set_fm,A4: set_set_fm] :
( collect_set_fm
@ ^ [Y: set_fm] :
? [X: set_fm] :
( ( member_set_fm @ X @ A4 )
& ( Y
= ( F3 @ X ) ) ) ) ) ) ).
% image_def
thf(fact_524_Union__eq,axiom,
( comple2134933779557159616set_fm
= ( ^ [A4: set_set_fm] :
( collect_fm
@ ^ [X: fm] :
? [Y: set_fm] :
( ( member_set_fm @ Y @ A4 )
& ( member_fm @ X @ Y ) ) ) ) ) ).
% Union_eq
thf(fact_525_Union__eq,axiom,
( comple90263536869209701_set_o
= ( ^ [A4: set_set_o] :
( collect_o
@ ^ [X: $o] :
? [Y: set_o] :
( ( member_set_o @ Y @ A4 )
& ( member_o @ X @ Y ) ) ) ) ) ).
% Union_eq
thf(fact_526_Union__eq,axiom,
( comple5648871819318716351m_rule
= ( ^ [A4: set_se4309406418413213730m_rule] :
( collec1009276759030335883m_rule
@ ^ [X: produc340336539035504054m_rule] :
? [Y: set_Pr1822751329126368876m_rule] :
( ( member5802698409304902403m_rule @ Y @ A4 )
& ( member7231649785386036813m_rule @ X @ Y ) ) ) ) ) ).
% Union_eq
thf(fact_527_Sigma__Union,axiom,
! [X2: set_se6159873173319230773ist_fm,B2: produc6018962875968178549ist_fm > set_rule] :
( ( produc4664469595690820543m_rule @ ( comple4923826267547576514ist_fm @ X2 ) @ B2 )
= ( comple5648871819318716351m_rule
@ ( image_6480592836540255356m_rule
@ ^ [A4: set_Pr5202636777678657877ist_fm] : ( produc4664469595690820543m_rule @ A4 @ B2 )
@ X2 ) ) ) ).
% Sigma_Union
thf(fact_528_UNION__eq,axiom,
! [B2: produc340336539035504054m_rule > set_fm,A2: set_Pr1822751329126368876m_rule] :
( ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B2 @ A2 ) )
= ( collect_fm
@ ^ [Y: fm] :
? [X: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X @ A2 )
& ( member_fm @ Y @ ( B2 @ X ) ) ) ) ) ).
% UNION_eq
thf(fact_529_UNION__eq,axiom,
! [B2: $o > set_fm,A2: set_o] :
( ( comple2134933779557159616set_fm @ ( image_o_set_fm @ B2 @ A2 ) )
= ( collect_fm
@ ^ [Y: fm] :
? [X: $o] :
( ( member_o @ X @ A2 )
& ( member_fm @ Y @ ( B2 @ X ) ) ) ) ) ).
% UNION_eq
thf(fact_530_UNION__eq,axiom,
! [B2: set_fm > set_fm,A2: set_set_fm] :
( ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ B2 @ A2 ) )
= ( collect_fm
@ ^ [Y: fm] :
? [X: set_fm] :
( ( member_set_fm @ X @ A2 )
& ( member_fm @ Y @ ( B2 @ X ) ) ) ) ) ).
% UNION_eq
thf(fact_531_UNION__eq,axiom,
! [B2: $o > set_o,A2: set_o] :
( ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) )
= ( collect_o
@ ^ [Y: $o] :
? [X: $o] :
( ( member_o @ X @ A2 )
& ( member_o @ Y @ ( B2 @ X ) ) ) ) ) ).
% UNION_eq
thf(fact_532_UNION__eq,axiom,
! [B2: $o > set_Pr1822751329126368876m_rule,A2: set_o] :
( ( comple5648871819318716351m_rule @ ( image_6392605097815222207m_rule @ B2 @ A2 ) )
= ( collec1009276759030335883m_rule
@ ^ [Y: produc340336539035504054m_rule] :
? [X: $o] :
( ( member_o @ X @ A2 )
& ( member7231649785386036813m_rule @ Y @ ( B2 @ X ) ) ) ) ) ).
% UNION_eq
thf(fact_533_Collect__bex__eq,axiom,
! [A2: set_Pr1822751329126368876m_rule,P: fm > produc340336539035504054m_rule > $o] :
( ( collect_fm
@ ^ [X: fm] :
? [Y: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ Y @ A2 )
& ( P @ X @ Y ) ) )
= ( comple2134933779557159616set_fm
@ ( image_377115208147135190set_fm
@ ^ [Y: produc340336539035504054m_rule] :
( collect_fm
@ ^ [X: fm] : ( P @ X @ Y ) )
@ A2 ) ) ) ).
% Collect_bex_eq
thf(fact_534_Collect__bex__eq,axiom,
! [A2: set_o,P: fm > $o > $o] :
( ( collect_fm
@ ^ [X: fm] :
? [Y: $o] :
( ( member_o @ Y @ A2 )
& ( P @ X @ Y ) ) )
= ( comple2134933779557159616set_fm
@ ( image_o_set_fm
@ ^ [Y: $o] :
( collect_fm
@ ^ [X: fm] : ( P @ X @ Y ) )
@ A2 ) ) ) ).
% Collect_bex_eq
thf(fact_535_Collect__bex__eq,axiom,
! [A2: set_set_fm,P: fm > set_fm > $o] :
( ( collect_fm
@ ^ [X: fm] :
? [Y: set_fm] :
( ( member_set_fm @ Y @ A2 )
& ( P @ X @ Y ) ) )
= ( comple2134933779557159616set_fm
@ ( image_set_fm_set_fm
@ ^ [Y: set_fm] :
( collect_fm
@ ^ [X: fm] : ( P @ X @ Y ) )
@ A2 ) ) ) ).
% Collect_bex_eq
thf(fact_536_Collect__bex__eq,axiom,
! [A2: set_o,P: $o > $o > $o] :
( ( collect_o
@ ^ [X: $o] :
? [Y: $o] :
( ( member_o @ Y @ A2 )
& ( P @ X @ Y ) ) )
= ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [Y: $o] :
( collect_o
@ ^ [X: $o] : ( P @ X @ Y ) )
@ A2 ) ) ) ).
% Collect_bex_eq
thf(fact_537_Collect__bex__eq,axiom,
! [A2: set_o,P: produc340336539035504054m_rule > $o > $o] :
( ( collec1009276759030335883m_rule
@ ^ [X: produc340336539035504054m_rule] :
? [Y: $o] :
( ( member_o @ Y @ A2 )
& ( P @ X @ Y ) ) )
= ( comple5648871819318716351m_rule
@ ( image_6392605097815222207m_rule
@ ^ [Y: $o] :
( collec1009276759030335883m_rule
@ ^ [X: produc340336539035504054m_rule] : ( P @ X @ Y ) )
@ A2 ) ) ) ).
% Collect_bex_eq
thf(fact_538_mem__Times__iff,axiom,
! [X5: product_prod_fm_fm,A2: set_fm,B2: set_fm] :
( ( member7780952600467998736_fm_fm @ X5
@ ( product_Sigma_fm_fm @ A2
@ ^ [Uu: fm] : B2 ) )
= ( ( member_fm @ ( product_fst_fm_fm @ X5 ) @ A2 )
& ( member_fm @ ( product_snd_fm_fm @ X5 ) @ B2 ) ) ) ).
% mem_Times_iff
thf(fact_539_mem__Times__iff,axiom,
! [X5: product_prod_fm_o,A2: set_fm,B2: set_o] :
( ( member1731323461681951669d_fm_o @ X5
@ ( product_Sigma_fm_o @ A2
@ ^ [Uu: fm] : B2 ) )
= ( ( member_fm @ ( product_fst_fm_o @ X5 ) @ A2 )
& ( member_o @ ( product_snd_fm_o @ X5 ) @ B2 ) ) ) ).
% mem_Times_iff
thf(fact_540_mem__Times__iff,axiom,
! [X5: product_prod_o_fm,A2: set_o,B2: set_fm] :
( ( member6824368184199080235d_o_fm @ X5
@ ( product_Sigma_o_fm @ A2
@ ^ [Uu: $o] : B2 ) )
= ( ( member_o @ ( product_fst_o_fm @ X5 ) @ A2 )
& ( member_fm @ ( product_snd_o_fm @ X5 ) @ B2 ) ) ) ).
% mem_Times_iff
thf(fact_541_mem__Times__iff,axiom,
! [X5: product_prod_o_o,A2: set_o,B2: set_o] :
( ( member7466972457876170832od_o_o @ X5
@ ( product_Sigma_o_o @ A2
@ ^ [Uu: $o] : B2 ) )
= ( ( member_o @ ( product_fst_o_o @ X5 ) @ A2 )
& ( member_o @ ( product_snd_o_o @ X5 ) @ B2 ) ) ) ).
% mem_Times_iff
thf(fact_542_mem__Times__iff,axiom,
! [X5: produc6018962875968178549ist_fm,A2: set_list_tm,B2: set_list_fm] :
( ( member4699826688122452638ist_fm @ X5
@ ( produc3940501721629090966ist_fm @ A2
@ ^ [Uu: list_tm] : B2 ) )
= ( ( member_list_tm @ ( produc4874987732838623697ist_fm @ X5 ) @ A2 )
& ( member_list_fm @ ( produc3242453064496107283ist_fm @ X5 ) @ B2 ) ) ) ).
% mem_Times_iff
thf(fact_543_mem__Times__iff,axiom,
! [X5: produc340336539035504054m_rule,A2: set_Pr5202636777678657877ist_fm,B2: set_rule] :
( ( member7231649785386036813m_rule @ X5
@ ( produc4664469595690820543m_rule @ A2
@ ^ [Uu: produc6018962875968178549ist_fm] : B2 ) )
= ( ( member4699826688122452638ist_fm @ ( produc993771546023551940m_rule @ X5 ) @ A2 )
& ( member_rule @ ( produc6514711208036230658m_rule @ X5 ) @ B2 ) ) ) ).
% mem_Times_iff
thf(fact_544_mem__Times__iff,axiom,
! [X5: produc7032816097177688792m_rule,A2: set_fm,B2: set_Pr1822751329126368876m_rule] :
( ( member1076108276309761647m_rule @ X5
@ ( produc8625438769013651169m_rule @ A2
@ ^ [Uu: fm] : B2 ) )
= ( ( member_fm @ ( produc758635412052336614m_rule @ X5 ) @ A2 )
& ( member7231649785386036813m_rule @ ( produc6649256713847044132m_rule @ X5 ) @ B2 ) ) ) ).
% mem_Times_iff
thf(fact_545_mem__Times__iff,axiom,
! [X5: produc5590745105550754866ule_fm,A2: set_Pr1822751329126368876m_rule,B2: set_fm] :
( ( member8857409321537603529ule_fm @ X5
@ ( produc4849014715371551731ule_fm @ A2
@ ^ [Uu: produc340336539035504054m_rule] : B2 ) )
= ( ( member7231649785386036813m_rule @ ( produc6205583395265012984ule_fm @ X5 ) @ A2 )
& ( member_fm @ ( produc2872832660204944694ule_fm @ X5 ) @ B2 ) ) ) ).
% mem_Times_iff
thf(fact_546_mem__Times__iff,axiom,
! [X5: produc8024173992080876759rule_o,A2: set_Pr1822751329126368876m_rule,B2: set_o] :
( ( member5462180432759943662rule_o @ X5
@ ( produc5570680280418397464rule_o @ A2
@ ^ [Uu: produc340336539035504054m_rule] : B2 ) )
= ( ( member7231649785386036813m_rule @ ( produc7946234125759730333rule_o @ X5 ) @ A2 )
& ( member_o @ ( produc4769265681683129819rule_o @ X5 ) @ B2 ) ) ) ).
% mem_Times_iff
thf(fact_547_mem__Times__iff,axiom,
! [X5: produc5731340206564399485m_rule,A2: set_o,B2: set_Pr1822751329126368876m_rule] :
( ( member3169346647243466388m_rule @ X5
@ ( produc7949347099473444230m_rule @ A2
@ ^ [Uu: $o] : B2 ) )
= ( ( member_o @ ( produc1101528907960001291m_rule @ X5 ) @ A2 )
& ( member7231649785386036813m_rule @ ( produc7147932500738176585m_rule @ X5 ) @ B2 ) ) ) ).
% mem_Times_iff
thf(fact_548_SUP__lessD,axiom,
! [F: fm > $o,A2: set_fm,Y5: $o,I2: fm] :
( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_fm_o @ F @ A2 ) ) @ Y5 )
=> ( ( member_fm @ I2 @ A2 )
=> ( ord_less_o @ ( F @ I2 ) @ Y5 ) ) ) ).
% SUP_lessD
thf(fact_549_SUP__lessD,axiom,
! [F: $o > $o,A2: set_o,Y5: $o,I2: $o] :
( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) ) @ Y5 )
=> ( ( member_o @ I2 @ A2 )
=> ( ord_less_o @ ( F @ I2 ) @ Y5 ) ) ) ).
% SUP_lessD
thf(fact_550_SUP__lessD,axiom,
! [F: fm > set_fm,A2: set_fm,Y5: set_fm,I2: fm] :
( ( ord_less_set_fm @ ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ F @ A2 ) ) @ Y5 )
=> ( ( member_fm @ I2 @ A2 )
=> ( ord_less_set_fm @ ( F @ I2 ) @ Y5 ) ) ) ).
% SUP_lessD
thf(fact_551_SUP__lessD,axiom,
! [F: $o > set_fm,A2: set_o,Y5: set_fm,I2: $o] :
( ( ord_less_set_fm @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ A2 ) ) @ Y5 )
=> ( ( member_o @ I2 @ A2 )
=> ( ord_less_set_fm @ ( F @ I2 ) @ Y5 ) ) ) ).
% SUP_lessD
thf(fact_552_SUP__lessD,axiom,
! [F: fm > set_o,A2: set_fm,Y5: set_o,I2: fm] :
( ( ord_less_set_o @ ( comple90263536869209701_set_o @ ( image_fm_set_o @ F @ A2 ) ) @ Y5 )
=> ( ( member_fm @ I2 @ A2 )
=> ( ord_less_set_o @ ( F @ I2 ) @ Y5 ) ) ) ).
% SUP_lessD
thf(fact_553_SUP__lessD,axiom,
! [F: $o > set_o,A2: set_o,Y5: set_o,I2: $o] :
( ( ord_less_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ A2 ) ) @ Y5 )
=> ( ( member_o @ I2 @ A2 )
=> ( ord_less_set_o @ ( F @ I2 ) @ Y5 ) ) ) ).
% SUP_lessD
thf(fact_554_SUP__lessD,axiom,
! [F: set_fm > set_fm,A2: set_set_fm,Y5: set_fm,I2: set_fm] :
( ( ord_less_set_fm @ ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ F @ A2 ) ) @ Y5 )
=> ( ( member_set_fm @ I2 @ A2 )
=> ( ord_less_set_fm @ ( F @ I2 ) @ Y5 ) ) ) ).
% SUP_lessD
thf(fact_555_SUP__lessD,axiom,
! [F: produc340336539035504054m_rule > $o,A2: set_Pr1822751329126368876m_rule,Y5: $o,I2: produc340336539035504054m_rule] :
( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_8969478372357302939rule_o @ F @ A2 ) ) @ Y5 )
=> ( ( member7231649785386036813m_rule @ I2 @ A2 )
=> ( ord_less_o @ ( F @ I2 ) @ Y5 ) ) ) ).
% SUP_lessD
thf(fact_556_SUP__lessD,axiom,
! [F: produc340336539035504054m_rule > set_fm,A2: set_Pr1822751329126368876m_rule,Y5: set_fm,I2: produc340336539035504054m_rule] :
( ( ord_less_set_fm @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ F @ A2 ) ) @ Y5 )
=> ( ( member7231649785386036813m_rule @ I2 @ A2 )
=> ( ord_less_set_fm @ ( F @ I2 ) @ Y5 ) ) ) ).
% SUP_lessD
thf(fact_557_SUP__lessD,axiom,
! [F: produc340336539035504054m_rule > set_o,A2: set_Pr1822751329126368876m_rule,Y5: set_o,I2: produc340336539035504054m_rule] :
( ( ord_less_set_o @ ( comple90263536869209701_set_o @ ( image_2025595867862115707_set_o @ F @ A2 ) ) @ Y5 )
=> ( ( member7231649785386036813m_rule @ I2 @ A2 )
=> ( ord_less_set_o @ ( F @ I2 ) @ Y5 ) ) ) ).
% SUP_lessD
thf(fact_558_sset__sproduct,axiom,
! [S1: stream2709947120125613254m_rule,S22: stream2709947120125613254m_rule] :
( ( sset_P3768201806174902753m_rule @ ( sprodu7455803463837168214m_rule @ S1 @ S22 ) )
= ( produc1849974139216714806m_rule @ ( sset_P4484857331586881186m_rule @ S1 )
@ ^ [Uu: produc340336539035504054m_rule] : ( sset_P4484857331586881186m_rule @ S22 ) ) ) ).
% sset_sproduct
thf(fact_559_sset__sproduct,axiom,
! [S1: stream4408948924543953275ist_fm,S22: stream_rule] :
( ( sset_P4484857331586881186m_rule @ ( sprodu2217682051239556511m_rule @ S1 @ S22 ) )
= ( produc4664469595690820543m_rule @ ( sset_P5379989128061332361ist_fm @ S1 )
@ ^ [Uu: produc6018962875968178549ist_fm] : ( sset_rule @ S22 ) ) ) ).
% sset_sproduct
thf(fact_560_surj__nat__to__rat__surj,axiom,
( ( image_nat_rat @ nat_to_rat_surj @ top_top_set_nat )
= top_top_set_rat ) ).
% surj_nat_to_rat_surj
thf(fact_561_member__product,axiom,
! [X5: produc340336539035504054m_rule,A2: set_Pr5202636777678657877ist_fm,B2: set_rule] :
( ( member7231649785386036813m_rule @ X5 @ ( produc8488632834096016437m_rule @ A2 @ B2 ) )
= ( member7231649785386036813m_rule @ X5
@ ( produc4664469595690820543m_rule @ A2
@ ^ [Uu: produc6018962875968178549ist_fm] : B2 ) ) ) ).
% member_product
thf(fact_562_subset__fst__snd,axiom,
! [A2: set_Pr1822751329126368876m_rule] :
( ord_le6390412330253371084m_rule @ A2
@ ( produc4664469595690820543m_rule @ ( image_8684537122991555832ist_fm @ produc993771546023551940m_rule @ A2 )
@ ^ [Uu: produc6018962875968178549ist_fm] : ( image_4399899049665951567e_rule @ produc6514711208036230658m_rule @ A2 ) ) ) ).
% subset_fst_snd
thf(fact_563_subset__fst__snd,axiom,
! [A2: set_Pr5202636777678657877ist_fm] :
( ord_le1771420097867575541ist_fm @ A2
@ ( produc3940501721629090966ist_fm @ ( image_2433570665449658043ist_tm @ produc4874987732838623697ist_fm @ A2 )
@ ^ [Uu: list_tm] : ( image_2429618640364022857ist_fm @ produc3242453064496107283ist_fm @ A2 ) ) ) ).
% subset_fst_snd
thf(fact_564_snd__image__times,axiom,
! [A2: set_list_tm,B2: set_list_fm] :
( ( ( A2 = bot_bot_set_list_tm )
=> ( ( image_2429618640364022857ist_fm @ produc3242453064496107283ist_fm
@ ( produc3940501721629090966ist_fm @ A2
@ ^ [Uu: list_tm] : B2 ) )
= bot_bot_set_list_fm ) )
& ( ( A2 != bot_bot_set_list_tm )
=> ( ( image_2429618640364022857ist_fm @ produc3242453064496107283ist_fm
@ ( produc3940501721629090966ist_fm @ A2
@ ^ [Uu: list_tm] : B2 ) )
= B2 ) ) ) ).
% snd_image_times
thf(fact_565_snd__image__times,axiom,
! [A2: set_o,B2: set_o] :
( ( ( A2 = bot_bot_set_o )
=> ( ( image_7896445794123959606_o_o_o @ product_snd_o_o
@ ( product_Sigma_o_o @ A2
@ ^ [Uu: $o] : B2 ) )
= bot_bot_set_o ) )
& ( ( A2 != bot_bot_set_o )
=> ( ( image_7896445794123959606_o_o_o @ product_snd_o_o
@ ( product_Sigma_o_o @ A2
@ ^ [Uu: $o] : B2 ) )
= B2 ) ) ) ).
% snd_image_times
thf(fact_566_fst__image__times,axiom,
! [B2: set_rule,A2: set_Pr5202636777678657877ist_fm] :
( ( ( B2 = bot_bot_set_rule )
=> ( ( image_8684537122991555832ist_fm @ produc993771546023551940m_rule
@ ( produc4664469595690820543m_rule @ A2
@ ^ [Uu: produc6018962875968178549ist_fm] : B2 ) )
= bot_bo1798318783834493889ist_fm ) )
& ( ( B2 != bot_bot_set_rule )
=> ( ( image_8684537122991555832ist_fm @ produc993771546023551940m_rule
@ ( produc4664469595690820543m_rule @ A2
@ ^ [Uu: produc6018962875968178549ist_fm] : B2 ) )
= A2 ) ) ) ).
% fst_image_times
thf(fact_567_fst__image__times,axiom,
! [B2: set_o,A2: set_o] :
( ( ( B2 = bot_bot_set_o )
=> ( ( image_7896445794123959606_o_o_o @ product_fst_o_o
@ ( product_Sigma_o_o @ A2
@ ^ [Uu: $o] : B2 ) )
= bot_bot_set_o ) )
& ( ( B2 != bot_bot_set_o )
=> ( ( image_7896445794123959606_o_o_o @ product_fst_o_o
@ ( product_Sigma_o_o @ A2
@ ^ [Uu: $o] : B2 ) )
= A2 ) ) ) ).
% fst_image_times
thf(fact_568_set__list__bind,axiom,
! [Xs: list_P2774625669004309958m_rule,F: produc340336539035504054m_rule > list_fm] :
( ( set_fm2 @ ( bind_P95098981060663284ule_fm @ Xs @ F ) )
= ( comple2134933779557159616set_fm
@ ( image_377115208147135190set_fm
@ ^ [X: produc340336539035504054m_rule] : ( set_fm2 @ ( F @ X ) )
@ ( set_Pr4534715572506550497m_rule @ Xs ) ) ) ) ).
% set_list_bind
thf(fact_569_set__list__bind,axiom,
! [Xs: list_o,F: $o > list_fm] :
( ( set_fm2 @ ( bind_o_fm @ Xs @ F ) )
= ( comple2134933779557159616set_fm
@ ( image_o_set_fm
@ ^ [X: $o] : ( set_fm2 @ ( F @ X ) )
@ ( set_o2 @ Xs ) ) ) ) ).
% set_list_bind
thf(fact_570_set__list__bind,axiom,
! [Xs: list_set_fm,F: set_fm > list_fm] :
( ( set_fm2 @ ( bind_set_fm_fm @ Xs @ F ) )
= ( comple2134933779557159616set_fm
@ ( image_set_fm_set_fm
@ ^ [X: set_fm] : ( set_fm2 @ ( F @ X ) )
@ ( set_set_fm2 @ Xs ) ) ) ) ).
% set_list_bind
thf(fact_571_set__list__bind,axiom,
! [Xs: list_fm,F: fm > list_fm] :
( ( set_fm2 @ ( bind_fm_fm @ Xs @ F ) )
= ( comple2134933779557159616set_fm
@ ( image_fm_set_fm
@ ^ [X: fm] : ( set_fm2 @ ( F @ X ) )
@ ( set_fm2 @ Xs ) ) ) ) ).
% set_list_bind
thf(fact_572_set__list__bind,axiom,
! [Xs: list_o,F: $o > list_o] :
( ( set_o2 @ ( bind_o_o @ Xs @ F ) )
= ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( set_o2 @ ( F @ X ) )
@ ( set_o2 @ Xs ) ) ) ) ).
% set_list_bind
thf(fact_573_set__list__bind,axiom,
! [Xs: list_fm,F: fm > list_o] :
( ( set_o2 @ ( bind_fm_o @ Xs @ F ) )
= ( comple90263536869209701_set_o
@ ( image_fm_set_o
@ ^ [X: fm] : ( set_o2 @ ( F @ X ) )
@ ( set_fm2 @ Xs ) ) ) ) ).
% set_list_bind
thf(fact_574_set__list__bind,axiom,
! [Xs: list_o,F: $o > list_P2774625669004309958m_rule] :
( ( set_Pr4534715572506550497m_rule @ ( bind_o2471881556210961415m_rule @ Xs @ F ) )
= ( comple5648871819318716351m_rule
@ ( image_6392605097815222207m_rule
@ ^ [X: $o] : ( set_Pr4534715572506550497m_rule @ ( F @ X ) )
@ ( set_o2 @ Xs ) ) ) ) ).
% set_list_bind
thf(fact_575_set__list__bind,axiom,
! [Xs: list_fm,F: fm > list_P2774625669004309958m_rule] :
( ( set_Pr4534715572506550497m_rule @ ( bind_f3871523034702762722m_rule @ Xs @ F ) )
= ( comple5648871819318716351m_rule
@ ( image_7929583154892365210m_rule
@ ^ [X: fm] : ( set_Pr4534715572506550497m_rule @ ( F @ X ) )
@ ( set_fm2 @ Xs ) ) ) ) ).
% set_list_bind
thf(fact_576_fst__image__Sigma,axiom,
! [A2: set_Pr5202636777678657877ist_fm,B2: produc6018962875968178549ist_fm > set_rule] :
( ( image_8684537122991555832ist_fm @ produc993771546023551940m_rule @ ( produc4664469595690820543m_rule @ A2 @ B2 ) )
= ( collec5845330005437105888ist_fm
@ ^ [X: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ X @ A2 )
& ( ( B2 @ X )
!= bot_bot_set_rule ) ) ) ) ).
% fst_image_Sigma
thf(fact_577_fst__image__Sigma,axiom,
! [A2: set_fm,B2: fm > set_o] :
( ( image_2170215323712322924m_o_fm @ product_fst_fm_o @ ( product_Sigma_fm_o @ A2 @ B2 ) )
= ( collect_fm
@ ^ [X: fm] :
( ( member_fm @ X @ A2 )
& ( ( B2 @ X )
!= bot_bot_set_o ) ) ) ) ).
% fst_image_Sigma
thf(fact_578_fst__image__Sigma,axiom,
! [A2: set_Pr1822751329126368876m_rule,B2: produc340336539035504054m_rule > set_o] :
( ( image_2710498626411009428m_rule @ produc7946234125759730333rule_o @ ( produc5570680280418397464rule_o @ A2 @ B2 ) )
= ( collec1009276759030335883m_rule
@ ^ [X: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X @ A2 )
& ( ( B2 @ X )
!= bot_bot_set_o ) ) ) ) ).
% fst_image_Sigma
thf(fact_579_fst__image__Sigma,axiom,
! [A2: set_o,B2: $o > set_o] :
( ( image_7896445794123959606_o_o_o @ product_fst_o_o @ ( product_Sigma_o_o @ A2 @ B2 ) )
= ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A2 )
& ( ( B2 @ X )
!= bot_bot_set_o ) ) ) ) ).
% fst_image_Sigma
thf(fact_580_INT__extend__simps_I8_J,axiom,
! [B2: set_fm > set_fm,A2: set_set_set_fm] :
( ( comple1024113957266099930set_fm
@ ( image_2606594240855772645set_fm
@ ^ [Y: set_set_fm] : ( comple1024113957266099930set_fm @ ( image_set_fm_set_fm @ B2 @ Y ) )
@ A2 ) )
= ( comple1024113957266099930set_fm @ ( image_set_fm_set_fm @ B2 @ ( comple4326961365146087328set_fm @ A2 ) ) ) ) ).
% INT_extend_simps(8)
thf(fact_581_INT__extend__simps_I8_J,axiom,
! [B2: fm > set_fm,A2: set_set_fm] :
( ( comple1024113957266099930set_fm
@ ( image_set_fm_set_fm
@ ^ [Y: set_fm] : ( comple1024113957266099930set_fm @ ( image_fm_set_fm @ B2 @ Y ) )
@ A2 ) )
= ( comple1024113957266099930set_fm @ ( image_fm_set_fm @ B2 @ ( comple2134933779557159616set_fm @ A2 ) ) ) ) ).
% INT_extend_simps(8)
thf(fact_582_INT__extend__simps_I8_J,axiom,
! [B2: $o > set_Pr1822751329126368876m_rule,A2: set_set_o] :
( ( comple1222484001442006309m_rule
@ ( image_6429356636794881695m_rule
@ ^ [Y: set_o] : ( comple1222484001442006309m_rule @ ( image_6392605097815222207m_rule @ B2 @ Y ) )
@ A2 ) )
= ( comple1222484001442006309m_rule @ ( image_6392605097815222207m_rule @ B2 @ ( comple90263536869209701_set_o @ A2 ) ) ) ) ).
% INT_extend_simps(8)
thf(fact_583_INT__extend__simps_I8_J,axiom,
! [B2: $o > set_fm,A2: set_set_o] :
( ( comple1024113957266099930set_fm
@ ( image_set_o_set_fm
@ ^ [Y: set_o] : ( comple1024113957266099930set_fm @ ( image_o_set_fm @ B2 @ Y ) )
@ A2 ) )
= ( comple1024113957266099930set_fm @ ( image_o_set_fm @ B2 @ ( comple90263536869209701_set_o @ A2 ) ) ) ) ).
% INT_extend_simps(8)
thf(fact_584_INT__extend__simps_I8_J,axiom,
! [B2: $o > set_o,A2: set_set_o] :
( ( comple3063163877087187839_set_o
@ ( image_set_o_set_o
@ ^ [Y: set_o] : ( comple3063163877087187839_set_o @ ( image_o_set_o @ B2 @ Y ) )
@ A2 ) )
= ( comple3063163877087187839_set_o @ ( image_o_set_o @ B2 @ ( comple90263536869209701_set_o @ A2 ) ) ) ) ).
% INT_extend_simps(8)
thf(fact_585_INT__extend__simps_I8_J,axiom,
! [B2: produc340336539035504054m_rule > set_fm,A2: set_se4309406418413213730m_rule] :
( ( comple1024113957266099930set_fm
@ ( image_338663757272164256set_fm
@ ^ [Y: set_Pr1822751329126368876m_rule] : ( comple1024113957266099930set_fm @ ( image_377115208147135190set_fm @ B2 @ Y ) )
@ A2 ) )
= ( comple1024113957266099930set_fm @ ( image_377115208147135190set_fm @ B2 @ ( comple5648871819318716351m_rule @ A2 ) ) ) ) ).
% INT_extend_simps(8)
thf(fact_586_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_587_order__refl,axiom,
! [X5: nat] : ( ord_less_eq_nat @ X5 @ X5 ) ).
% order_refl
thf(fact_588_empty__Collect__eq,axiom,
! [P: $o > $o] :
( ( bot_bot_set_o
= ( collect_o @ P ) )
= ( ! [X: $o] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_589_Collect__empty__eq,axiom,
! [P: $o > $o] :
( ( ( collect_o @ P )
= bot_bot_set_o )
= ( ! [X: $o] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_590_all__not__in__conv,axiom,
! [A2: set_fm] :
( ( ! [X: fm] :
~ ( member_fm @ X @ A2 ) )
= ( A2 = bot_bot_set_fm ) ) ).
% all_not_in_conv
thf(fact_591_all__not__in__conv,axiom,
! [A2: set_Pr1822751329126368876m_rule] :
( ( ! [X: produc340336539035504054m_rule] :
~ ( member7231649785386036813m_rule @ X @ A2 ) )
= ( A2 = bot_bo3426410157013319936m_rule ) ) ).
% all_not_in_conv
thf(fact_592_all__not__in__conv,axiom,
! [A2: set_o] :
( ( ! [X: $o] :
~ ( member_o @ X @ A2 ) )
= ( A2 = bot_bot_set_o ) ) ).
% all_not_in_conv
thf(fact_593_empty__iff,axiom,
! [C2: fm] :
~ ( member_fm @ C2 @ bot_bot_set_fm ) ).
% empty_iff
thf(fact_594_empty__iff,axiom,
! [C2: produc340336539035504054m_rule] :
~ ( member7231649785386036813m_rule @ C2 @ bot_bo3426410157013319936m_rule ) ).
% empty_iff
thf(fact_595_empty__iff,axiom,
! [C2: $o] :
~ ( member_o @ C2 @ bot_bot_set_o ) ).
% empty_iff
thf(fact_596_subsetI,axiom,
! [A2: set_fm,B2: set_fm] :
( ! [X3: fm] :
( ( member_fm @ X3 @ A2 )
=> ( member_fm @ X3 @ B2 ) )
=> ( ord_less_eq_set_fm @ A2 @ B2 ) ) ).
% subsetI
thf(fact_597_subsetI,axiom,
! [A2: set_Pr1822751329126368876m_rule,B2: set_Pr1822751329126368876m_rule] :
( ! [X3: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X3 @ A2 )
=> ( member7231649785386036813m_rule @ X3 @ B2 ) )
=> ( ord_le6390412330253371084m_rule @ A2 @ B2 ) ) ).
% subsetI
thf(fact_598_subsetI,axiom,
! [A2: set_o,B2: set_o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( member_o @ X3 @ B2 ) )
=> ( ord_less_eq_set_o @ A2 @ B2 ) ) ).
% subsetI
thf(fact_599_Inter__iff,axiom,
! [A2: fm,C: set_set_fm] :
( ( member_fm @ A2 @ ( comple1024113957266099930set_fm @ C ) )
= ( ! [X: set_fm] :
( ( member_set_fm @ X @ C )
=> ( member_fm @ A2 @ X ) ) ) ) ).
% Inter_iff
thf(fact_600_Inter__iff,axiom,
! [A2: produc340336539035504054m_rule,C: set_se4309406418413213730m_rule] :
( ( member7231649785386036813m_rule @ A2 @ ( comple1222484001442006309m_rule @ C ) )
= ( ! [X: set_Pr1822751329126368876m_rule] :
( ( member5802698409304902403m_rule @ X @ C )
=> ( member7231649785386036813m_rule @ A2 @ X ) ) ) ) ).
% Inter_iff
thf(fact_601_Inter__iff,axiom,
! [A2: $o,C: set_set_o] :
( ( member_o @ A2 @ ( comple3063163877087187839_set_o @ C ) )
= ( ! [X: set_o] :
( ( member_set_o @ X @ C )
=> ( member_o @ A2 @ X ) ) ) ) ).
% Inter_iff
thf(fact_602_InterI,axiom,
! [C: set_set_fm,A2: fm] :
( ! [X4: set_fm] :
( ( member_set_fm @ X4 @ C )
=> ( member_fm @ A2 @ X4 ) )
=> ( member_fm @ A2 @ ( comple1024113957266099930set_fm @ C ) ) ) ).
% InterI
thf(fact_603_InterI,axiom,
! [C: set_se4309406418413213730m_rule,A2: produc340336539035504054m_rule] :
( ! [X4: set_Pr1822751329126368876m_rule] :
( ( member5802698409304902403m_rule @ X4 @ C )
=> ( member7231649785386036813m_rule @ A2 @ X4 ) )
=> ( member7231649785386036813m_rule @ A2 @ ( comple1222484001442006309m_rule @ C ) ) ) ).
% InterI
thf(fact_604_InterI,axiom,
! [C: set_set_o,A2: $o] :
( ! [X4: set_o] :
( ( member_set_o @ X4 @ C )
=> ( member_o @ A2 @ X4 ) )
=> ( member_o @ A2 @ ( comple3063163877087187839_set_o @ C ) ) ) ).
% InterI
thf(fact_605_image__is__empty,axiom,
! [F: produc340336539035504054m_rule > set_fm,A2: set_Pr1822751329126368876m_rule] :
( ( ( image_377115208147135190set_fm @ F @ A2 )
= bot_bot_set_set_fm )
= ( A2 = bot_bo3426410157013319936m_rule ) ) ).
% image_is_empty
thf(fact_606_image__is__empty,axiom,
! [F: nat > produc340336539035504054m_rule,A2: set_nat] :
( ( ( image_8361736131336225677m_rule @ F @ A2 )
= bot_bo3426410157013319936m_rule )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_607_image__is__empty,axiom,
! [F: nat > rat,A2: set_nat] :
( ( ( image_nat_rat @ F @ A2 )
= bot_bot_set_rat )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_608_image__is__empty,axiom,
! [F: set_fm > set_fm,A2: set_set_fm] :
( ( ( image_set_fm_set_fm @ F @ A2 )
= bot_bot_set_set_fm )
= ( A2 = bot_bot_set_set_fm ) ) ).
% image_is_empty
thf(fact_609_image__is__empty,axiom,
! [F: $o > set_Pr1822751329126368876m_rule,A2: set_o] :
( ( ( image_6392605097815222207m_rule @ F @ A2 )
= bot_bo8650941071013544630m_rule )
= ( A2 = bot_bot_set_o ) ) ).
% image_is_empty
thf(fact_610_image__is__empty,axiom,
! [F: $o > set_fm,A2: set_o] :
( ( ( image_o_set_fm @ F @ A2 )
= bot_bot_set_set_fm )
= ( A2 = bot_bot_set_o ) ) ).
% image_is_empty
thf(fact_611_image__is__empty,axiom,
! [F: $o > set_o,A2: set_o] :
( ( ( image_o_set_o @ F @ A2 )
= bot_bot_set_set_o )
= ( A2 = bot_bot_set_o ) ) ).
% image_is_empty
thf(fact_612_image__is__empty,axiom,
! [F: $o > $o,A2: set_o] :
( ( ( image_o_o @ F @ A2 )
= bot_bot_set_o )
= ( A2 = bot_bot_set_o ) ) ).
% image_is_empty
thf(fact_613_empty__is__image,axiom,
! [F: produc340336539035504054m_rule > set_fm,A2: set_Pr1822751329126368876m_rule] :
( ( bot_bot_set_set_fm
= ( image_377115208147135190set_fm @ F @ A2 ) )
= ( A2 = bot_bo3426410157013319936m_rule ) ) ).
% empty_is_image
thf(fact_614_empty__is__image,axiom,
! [F: nat > produc340336539035504054m_rule,A2: set_nat] :
( ( bot_bo3426410157013319936m_rule
= ( image_8361736131336225677m_rule @ F @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_615_empty__is__image,axiom,
! [F: nat > rat,A2: set_nat] :
( ( bot_bot_set_rat
= ( image_nat_rat @ F @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_616_empty__is__image,axiom,
! [F: set_fm > set_fm,A2: set_set_fm] :
( ( bot_bot_set_set_fm
= ( image_set_fm_set_fm @ F @ A2 ) )
= ( A2 = bot_bot_set_set_fm ) ) ).
% empty_is_image
thf(fact_617_empty__is__image,axiom,
! [F: $o > set_Pr1822751329126368876m_rule,A2: set_o] :
( ( bot_bo8650941071013544630m_rule
= ( image_6392605097815222207m_rule @ F @ A2 ) )
= ( A2 = bot_bot_set_o ) ) ).
% empty_is_image
thf(fact_618_empty__is__image,axiom,
! [F: $o > set_fm,A2: set_o] :
( ( bot_bot_set_set_fm
= ( image_o_set_fm @ F @ A2 ) )
= ( A2 = bot_bot_set_o ) ) ).
% empty_is_image
thf(fact_619_empty__is__image,axiom,
! [F: $o > set_o,A2: set_o] :
( ( bot_bot_set_set_o
= ( image_o_set_o @ F @ A2 ) )
= ( A2 = bot_bot_set_o ) ) ).
% empty_is_image
thf(fact_620_empty__is__image,axiom,
! [F: $o > $o,A2: set_o] :
( ( bot_bot_set_o
= ( image_o_o @ F @ A2 ) )
= ( A2 = bot_bot_set_o ) ) ).
% empty_is_image
thf(fact_621_image__empty,axiom,
! [F: produc340336539035504054m_rule > set_fm] :
( ( image_377115208147135190set_fm @ F @ bot_bo3426410157013319936m_rule )
= bot_bot_set_set_fm ) ).
% image_empty
thf(fact_622_image__empty,axiom,
! [F: nat > produc340336539035504054m_rule] :
( ( image_8361736131336225677m_rule @ F @ bot_bot_set_nat )
= bot_bo3426410157013319936m_rule ) ).
% image_empty
thf(fact_623_image__empty,axiom,
! [F: nat > rat] :
( ( image_nat_rat @ F @ bot_bot_set_nat )
= bot_bot_set_rat ) ).
% image_empty
thf(fact_624_image__empty,axiom,
! [F: set_fm > set_fm] :
( ( image_set_fm_set_fm @ F @ bot_bot_set_set_fm )
= bot_bot_set_set_fm ) ).
% image_empty
thf(fact_625_image__empty,axiom,
! [F: $o > set_Pr1822751329126368876m_rule] :
( ( image_6392605097815222207m_rule @ F @ bot_bot_set_o )
= bot_bo8650941071013544630m_rule ) ).
% image_empty
thf(fact_626_image__empty,axiom,
! [F: $o > set_fm] :
( ( image_o_set_fm @ F @ bot_bot_set_o )
= bot_bot_set_set_fm ) ).
% image_empty
thf(fact_627_image__empty,axiom,
! [F: $o > set_o] :
( ( image_o_set_o @ F @ bot_bot_set_o )
= bot_bot_set_set_o ) ).
% image_empty
thf(fact_628_image__empty,axiom,
! [F: $o > $o] :
( ( image_o_o @ F @ bot_bot_set_o )
= bot_bot_set_o ) ).
% image_empty
thf(fact_629_empty__subsetI,axiom,
! [A2: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A2 ) ).
% empty_subsetI
thf(fact_630_subset__empty,axiom,
! [A2: set_o] :
( ( ord_less_eq_set_o @ A2 @ bot_bot_set_o )
= ( A2 = bot_bot_set_o ) ) ).
% subset_empty
thf(fact_631_Sup__bot__conv_I1_J,axiom,
! [A2: set_set_fm] :
( ( ( comple2134933779557159616set_fm @ A2 )
= bot_bot_set_fm )
= ( ! [X: set_fm] :
( ( member_set_fm @ X @ A2 )
=> ( X = bot_bot_set_fm ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_632_Sup__bot__conv_I1_J,axiom,
! [A2: set_o] :
( ( ( complete_Sup_Sup_o @ A2 )
= bot_bot_o )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( X = bot_bot_o ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_633_Sup__bot__conv_I1_J,axiom,
! [A2: set_set_o] :
( ( ( comple90263536869209701_set_o @ A2 )
= bot_bot_set_o )
= ( ! [X: set_o] :
( ( member_set_o @ X @ A2 )
=> ( X = bot_bot_set_o ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_634_Sup__bot__conv_I1_J,axiom,
! [A2: set_se4309406418413213730m_rule] :
( ( ( comple5648871819318716351m_rule @ A2 )
= bot_bo3426410157013319936m_rule )
= ( ! [X: set_Pr1822751329126368876m_rule] :
( ( member5802698409304902403m_rule @ X @ A2 )
=> ( X = bot_bo3426410157013319936m_rule ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_635_Sup__bot__conv_I2_J,axiom,
! [A2: set_set_fm] :
( ( bot_bot_set_fm
= ( comple2134933779557159616set_fm @ A2 ) )
= ( ! [X: set_fm] :
( ( member_set_fm @ X @ A2 )
=> ( X = bot_bot_set_fm ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_636_Sup__bot__conv_I2_J,axiom,
! [A2: set_o] :
( ( bot_bot_o
= ( complete_Sup_Sup_o @ A2 ) )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( X = bot_bot_o ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_637_Sup__bot__conv_I2_J,axiom,
! [A2: set_set_o] :
( ( bot_bot_set_o
= ( comple90263536869209701_set_o @ A2 ) )
= ( ! [X: set_o] :
( ( member_set_o @ X @ A2 )
=> ( X = bot_bot_set_o ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_638_Sup__bot__conv_I2_J,axiom,
! [A2: set_se4309406418413213730m_rule] :
( ( bot_bo3426410157013319936m_rule
= ( comple5648871819318716351m_rule @ A2 ) )
= ( ! [X: set_Pr1822751329126368876m_rule] :
( ( member5802698409304902403m_rule @ X @ A2 )
=> ( X = bot_bo3426410157013319936m_rule ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_639_Inf__top__conv_I2_J,axiom,
! [A2: set_set_nat] :
( ( top_top_set_nat
= ( comple7806235888213564991et_nat @ A2 ) )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( X = top_top_set_nat ) ) ) ) ).
% Inf_top_conv(2)
thf(fact_640_Inf__top__conv_I2_J,axiom,
! [A2: set_set_rat] :
( ( top_top_set_rat
= ( comple4298007329820168263et_rat @ A2 ) )
= ( ! [X: set_rat] :
( ( member_set_rat @ X @ A2 )
=> ( X = top_top_set_rat ) ) ) ) ).
% Inf_top_conv(2)
thf(fact_641_Inf__top__conv_I2_J,axiom,
! [A2: set_set_o] :
( ( top_top_set_o
= ( comple3063163877087187839_set_o @ A2 ) )
= ( ! [X: set_o] :
( ( member_set_o @ X @ A2 )
=> ( X = top_top_set_o ) ) ) ) ).
% Inf_top_conv(2)
thf(fact_642_Inf__top__conv_I2_J,axiom,
! [A2: set_o] :
( ( top_top_o
= ( complete_Inf_Inf_o @ A2 ) )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( X = top_top_o ) ) ) ) ).
% Inf_top_conv(2)
thf(fact_643_Inf__top__conv_I1_J,axiom,
! [A2: set_set_nat] :
( ( ( comple7806235888213564991et_nat @ A2 )
= top_top_set_nat )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( X = top_top_set_nat ) ) ) ) ).
% Inf_top_conv(1)
thf(fact_644_Inf__top__conv_I1_J,axiom,
! [A2: set_set_rat] :
( ( ( comple4298007329820168263et_rat @ A2 )
= top_top_set_rat )
= ( ! [X: set_rat] :
( ( member_set_rat @ X @ A2 )
=> ( X = top_top_set_rat ) ) ) ) ).
% Inf_top_conv(1)
thf(fact_645_Inf__top__conv_I1_J,axiom,
! [A2: set_set_o] :
( ( ( comple3063163877087187839_set_o @ A2 )
= top_top_set_o )
= ( ! [X: set_o] :
( ( member_set_o @ X @ A2 )
=> ( X = top_top_set_o ) ) ) ) ).
% Inf_top_conv(1)
thf(fact_646_Inf__top__conv_I1_J,axiom,
! [A2: set_o] :
( ( ( complete_Inf_Inf_o @ A2 )
= top_top_o )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( X = top_top_o ) ) ) ) ).
% Inf_top_conv(1)
thf(fact_647_bex__empty,axiom,
! [P: $o > $o] :
~ ? [X6: $o] :
( ( member_o @ X6 @ bot_bot_set_o )
& ( P @ X6 ) ) ).
% bex_empty
thf(fact_648_Inter__UNIV__conv_I2_J,axiom,
! [A2: set_set_nat] :
( ( top_top_set_nat
= ( comple7806235888213564991et_nat @ A2 ) )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( X = top_top_set_nat ) ) ) ) ).
% Inter_UNIV_conv(2)
thf(fact_649_Inter__UNIV__conv_I2_J,axiom,
! [A2: set_set_rat] :
( ( top_top_set_rat
= ( comple4298007329820168263et_rat @ A2 ) )
= ( ! [X: set_rat] :
( ( member_set_rat @ X @ A2 )
=> ( X = top_top_set_rat ) ) ) ) ).
% Inter_UNIV_conv(2)
thf(fact_650_Inter__UNIV__conv_I2_J,axiom,
! [A2: set_set_o] :
( ( top_top_set_o
= ( comple3063163877087187839_set_o @ A2 ) )
= ( ! [X: set_o] :
( ( member_set_o @ X @ A2 )
=> ( X = top_top_set_o ) ) ) ) ).
% Inter_UNIV_conv(2)
thf(fact_651_Inter__UNIV__conv_I1_J,axiom,
! [A2: set_set_nat] :
( ( ( comple7806235888213564991et_nat @ A2 )
= top_top_set_nat )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( X = top_top_set_nat ) ) ) ) ).
% Inter_UNIV_conv(1)
thf(fact_652_Inter__UNIV__conv_I1_J,axiom,
! [A2: set_set_rat] :
( ( ( comple4298007329820168263et_rat @ A2 )
= top_top_set_rat )
= ( ! [X: set_rat] :
( ( member_set_rat @ X @ A2 )
=> ( X = top_top_set_rat ) ) ) ) ).
% Inter_UNIV_conv(1)
thf(fact_653_Inter__UNIV__conv_I1_J,axiom,
! [A2: set_set_o] :
( ( ( comple3063163877087187839_set_o @ A2 )
= top_top_set_o )
= ( ! [X: set_o] :
( ( member_set_o @ X @ A2 )
=> ( X = top_top_set_o ) ) ) ) ).
% Inter_UNIV_conv(1)
thf(fact_654_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collect_nat
@ ^ [S2: nat] : P )
= top_top_set_nat ) )
& ( ~ P
=> ( ( collect_nat
@ ^ [S2: nat] : P )
= bot_bot_set_nat ) ) ) ).
% Collect_const
thf(fact_655_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collect_rat
@ ^ [S2: rat] : P )
= top_top_set_rat ) )
& ( ~ P
=> ( ( collect_rat
@ ^ [S2: rat] : P )
= bot_bot_set_rat ) ) ) ).
% Collect_const
thf(fact_656_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collect_o
@ ^ [S2: $o] : P )
= top_top_set_o ) )
& ( ~ P
=> ( ( collect_o
@ ^ [S2: $o] : P )
= bot_bot_set_o ) ) ) ).
% Collect_const
thf(fact_657_INF__identity__eq,axiom,
! [A2: set_set_fm] :
( ( comple1024113957266099930set_fm
@ ( image_set_fm_set_fm
@ ^ [X: set_fm] : X
@ A2 ) )
= ( comple1024113957266099930set_fm @ A2 ) ) ).
% INF_identity_eq
thf(fact_658_INF__identity__eq,axiom,
! [A2: set_o] :
( ( complete_Inf_Inf_o
@ ( image_o_o
@ ^ [X: $o] : X
@ A2 ) )
= ( complete_Inf_Inf_o @ A2 ) ) ).
% INF_identity_eq
thf(fact_659_Times__empty,axiom,
! [A2: set_o,B2: set_o] :
( ( ( product_Sigma_o_o @ A2
@ ^ [Uu: $o] : B2 )
= bot_bo7073875226086086771od_o_o )
= ( ( A2 = bot_bot_set_o )
| ( B2 = bot_bot_set_o ) ) ) ).
% Times_empty
thf(fact_660_INT__iff,axiom,
! [B: fm,B2: produc340336539035504054m_rule > set_fm,A2: set_Pr1822751329126368876m_rule] :
( ( member_fm @ B @ ( comple1024113957266099930set_fm @ ( image_377115208147135190set_fm @ B2 @ A2 ) ) )
= ( ! [X: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X @ A2 )
=> ( member_fm @ B @ ( B2 @ X ) ) ) ) ) ).
% INT_iff
thf(fact_661_INT__iff,axiom,
! [B: fm,B2: $o > set_fm,A2: set_o] :
( ( member_fm @ B @ ( comple1024113957266099930set_fm @ ( image_o_set_fm @ B2 @ A2 ) ) )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( member_fm @ B @ ( B2 @ X ) ) ) ) ) ).
% INT_iff
thf(fact_662_INT__iff,axiom,
! [B: fm,B2: set_fm > set_fm,A2: set_set_fm] :
( ( member_fm @ B @ ( comple1024113957266099930set_fm @ ( image_set_fm_set_fm @ B2 @ A2 ) ) )
= ( ! [X: set_fm] :
( ( member_set_fm @ X @ A2 )
=> ( member_fm @ B @ ( B2 @ X ) ) ) ) ) ).
% INT_iff
thf(fact_663_INT__iff,axiom,
! [B: produc340336539035504054m_rule,B2: $o > set_Pr1822751329126368876m_rule,A2: set_o] :
( ( member7231649785386036813m_rule @ B @ ( comple1222484001442006309m_rule @ ( image_6392605097815222207m_rule @ B2 @ A2 ) ) )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( member7231649785386036813m_rule @ B @ ( B2 @ X ) ) ) ) ) ).
% INT_iff
thf(fact_664_INT__iff,axiom,
! [B: $o,B2: $o > set_o,A2: set_o] :
( ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( member_o @ B @ ( B2 @ X ) ) ) ) ) ).
% INT_iff
thf(fact_665_INT__I,axiom,
! [A2: set_set_fm,B: fm,B2: set_fm > set_fm] :
( ! [X3: set_fm] :
( ( member_set_fm @ X3 @ A2 )
=> ( member_fm @ B @ ( B2 @ X3 ) ) )
=> ( member_fm @ B @ ( comple1024113957266099930set_fm @ ( image_set_fm_set_fm @ B2 @ A2 ) ) ) ) ).
% INT_I
thf(fact_666_INT__I,axiom,
! [A2: set_fm,B: fm,B2: fm > set_fm] :
( ! [X3: fm] :
( ( member_fm @ X3 @ A2 )
=> ( member_fm @ B @ ( B2 @ X3 ) ) )
=> ( member_fm @ B @ ( comple1024113957266099930set_fm @ ( image_fm_set_fm @ B2 @ A2 ) ) ) ) ).
% INT_I
thf(fact_667_INT__I,axiom,
! [A2: set_fm,B: produc340336539035504054m_rule,B2: fm > set_Pr1822751329126368876m_rule] :
( ! [X3: fm] :
( ( member_fm @ X3 @ A2 )
=> ( member7231649785386036813m_rule @ B @ ( B2 @ X3 ) ) )
=> ( member7231649785386036813m_rule @ B @ ( comple1222484001442006309m_rule @ ( image_7929583154892365210m_rule @ B2 @ A2 ) ) ) ) ).
% INT_I
thf(fact_668_INT__I,axiom,
! [A2: set_fm,B: $o,B2: fm > set_o] :
( ! [X3: fm] :
( ( member_fm @ X3 @ A2 )
=> ( member_o @ B @ ( B2 @ X3 ) ) )
=> ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_fm_set_o @ B2 @ A2 ) ) ) ) ).
% INT_I
thf(fact_669_INT__I,axiom,
! [A2: set_Pr1822751329126368876m_rule,B: fm,B2: produc340336539035504054m_rule > set_fm] :
( ! [X3: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X3 @ A2 )
=> ( member_fm @ B @ ( B2 @ X3 ) ) )
=> ( member_fm @ B @ ( comple1024113957266099930set_fm @ ( image_377115208147135190set_fm @ B2 @ A2 ) ) ) ) ).
% INT_I
thf(fact_670_INT__I,axiom,
! [A2: set_Pr1822751329126368876m_rule,B: produc340336539035504054m_rule,B2: produc340336539035504054m_rule > set_Pr1822751329126368876m_rule] :
( ! [X3: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X3 @ A2 )
=> ( member7231649785386036813m_rule @ B @ ( B2 @ X3 ) ) )
=> ( member7231649785386036813m_rule @ B @ ( comple1222484001442006309m_rule @ ( image_2683796897779739177m_rule @ B2 @ A2 ) ) ) ) ).
% INT_I
thf(fact_671_INT__I,axiom,
! [A2: set_Pr1822751329126368876m_rule,B: $o,B2: produc340336539035504054m_rule > set_o] :
( ! [X3: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X3 @ A2 )
=> ( member_o @ B @ ( B2 @ X3 ) ) )
=> ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_2025595867862115707_set_o @ B2 @ A2 ) ) ) ) ).
% INT_I
thf(fact_672_INT__I,axiom,
! [A2: set_o,B: fm,B2: $o > set_fm] :
( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( member_fm @ B @ ( B2 @ X3 ) ) )
=> ( member_fm @ B @ ( comple1024113957266099930set_fm @ ( image_o_set_fm @ B2 @ A2 ) ) ) ) ).
% INT_I
thf(fact_673_INT__I,axiom,
! [A2: set_o,B: produc340336539035504054m_rule,B2: $o > set_Pr1822751329126368876m_rule] :
( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( member7231649785386036813m_rule @ B @ ( B2 @ X3 ) ) )
=> ( member7231649785386036813m_rule @ B @ ( comple1222484001442006309m_rule @ ( image_6392605097815222207m_rule @ B2 @ A2 ) ) ) ) ).
% INT_I
thf(fact_674_INT__I,axiom,
! [A2: set_o,B: $o,B2: $o > set_o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( member_o @ B @ ( B2 @ X3 ) ) )
=> ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ B2 @ A2 ) ) ) ) ).
% INT_I
thf(fact_675_Sup__empty,axiom,
( ( comple2134933779557159616set_fm @ bot_bot_set_set_fm )
= bot_bot_set_fm ) ).
% Sup_empty
thf(fact_676_Sup__empty,axiom,
( ( complete_Sup_Sup_o @ bot_bot_set_o )
= bot_bot_o ) ).
% Sup_empty
thf(fact_677_Sup__empty,axiom,
( ( comple90263536869209701_set_o @ bot_bot_set_set_o )
= bot_bot_set_o ) ).
% Sup_empty
thf(fact_678_Sup__empty,axiom,
( ( comple5648871819318716351m_rule @ bot_bo8650941071013544630m_rule )
= bot_bo3426410157013319936m_rule ) ).
% Sup_empty
thf(fact_679_Inf__UNIV,axiom,
( ( comple3063163877087187839_set_o @ top_top_set_set_o )
= bot_bot_set_o ) ).
% Inf_UNIV
thf(fact_680_Inf__UNIV,axiom,
( ( complete_Inf_Inf_o @ top_top_set_o )
= bot_bot_o ) ).
% Inf_UNIV
thf(fact_681_Inf__empty,axiom,
( ( comple7806235888213564991et_nat @ bot_bot_set_set_nat )
= top_top_set_nat ) ).
% Inf_empty
thf(fact_682_Inf__empty,axiom,
( ( comple4298007329820168263et_rat @ bot_bot_set_set_rat )
= top_top_set_rat ) ).
% Inf_empty
thf(fact_683_Inf__empty,axiom,
( ( comple3063163877087187839_set_o @ bot_bot_set_set_o )
= top_top_set_o ) ).
% Inf_empty
thf(fact_684_Inf__empty,axiom,
( ( complete_Inf_Inf_o @ bot_bot_set_o )
= top_top_o ) ).
% Inf_empty
thf(fact_685_SUP__bot,axiom,
! [A2: set_Pr1822751329126368876m_rule] :
( ( comple2134933779557159616set_fm
@ ( image_377115208147135190set_fm
@ ^ [X: produc340336539035504054m_rule] : bot_bot_set_fm
@ A2 ) )
= bot_bot_set_fm ) ).
% SUP_bot
thf(fact_686_SUP__bot,axiom,
! [A2: set_o] :
( ( comple2134933779557159616set_fm
@ ( image_o_set_fm
@ ^ [X: $o] : bot_bot_set_fm
@ A2 ) )
= bot_bot_set_fm ) ).
% SUP_bot
thf(fact_687_SUP__bot,axiom,
! [A2: set_set_fm] :
( ( comple2134933779557159616set_fm
@ ( image_set_fm_set_fm
@ ^ [X: set_fm] : bot_bot_set_fm
@ A2 ) )
= bot_bot_set_fm ) ).
% SUP_bot
thf(fact_688_SUP__bot,axiom,
! [A2: set_o] :
( ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [X: $o] : bot_bot_o
@ A2 ) )
= bot_bot_o ) ).
% SUP_bot
thf(fact_689_SUP__bot,axiom,
! [A2: set_o] :
( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X: $o] : bot_bot_set_o
@ A2 ) )
= bot_bot_set_o ) ).
% SUP_bot
thf(fact_690_SUP__bot,axiom,
! [A2: set_o] :
( ( comple5648871819318716351m_rule
@ ( image_6392605097815222207m_rule
@ ^ [X: $o] : bot_bo3426410157013319936m_rule
@ A2 ) )
= bot_bo3426410157013319936m_rule ) ).
% SUP_bot
thf(fact_691_SUP__bot__conv_I1_J,axiom,
! [B2: produc340336539035504054m_rule > set_fm,A2: set_Pr1822751329126368876m_rule] :
( ( ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B2 @ A2 ) )
= bot_bot_set_fm )
= ( ! [X: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X @ A2 )
=> ( ( B2 @ X )
= bot_bot_set_fm ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_692_SUP__bot__conv_I1_J,axiom,
! [B2: $o > set_fm,A2: set_o] :
( ( ( comple2134933779557159616set_fm @ ( image_o_set_fm @ B2 @ A2 ) )
= bot_bot_set_fm )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ( B2 @ X )
= bot_bot_set_fm ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_693_SUP__bot__conv_I1_J,axiom,
! [B2: set_fm > set_fm,A2: set_set_fm] :
( ( ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ B2 @ A2 ) )
= bot_bot_set_fm )
= ( ! [X: set_fm] :
( ( member_set_fm @ X @ A2 )
=> ( ( B2 @ X )
= bot_bot_set_fm ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_694_SUP__bot__conv_I1_J,axiom,
! [B2: $o > $o,A2: set_o] :
( ( ( complete_Sup_Sup_o @ ( image_o_o @ B2 @ A2 ) )
= bot_bot_o )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ( B2 @ X )
= bot_bot_o ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_695_SUP__bot__conv_I1_J,axiom,
! [B2: $o > set_o,A2: set_o] :
( ( ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) )
= bot_bot_set_o )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ( B2 @ X )
= bot_bot_set_o ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_696_SUP__bot__conv_I1_J,axiom,
! [B2: $o > set_Pr1822751329126368876m_rule,A2: set_o] :
( ( ( comple5648871819318716351m_rule @ ( image_6392605097815222207m_rule @ B2 @ A2 ) )
= bot_bo3426410157013319936m_rule )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ( B2 @ X )
= bot_bo3426410157013319936m_rule ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_697_SUP__bot__conv_I2_J,axiom,
! [B2: produc340336539035504054m_rule > set_fm,A2: set_Pr1822751329126368876m_rule] :
( ( bot_bot_set_fm
= ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B2 @ A2 ) ) )
= ( ! [X: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X @ A2 )
=> ( ( B2 @ X )
= bot_bot_set_fm ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_698_SUP__bot__conv_I2_J,axiom,
! [B2: $o > set_fm,A2: set_o] :
( ( bot_bot_set_fm
= ( comple2134933779557159616set_fm @ ( image_o_set_fm @ B2 @ A2 ) ) )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ( B2 @ X )
= bot_bot_set_fm ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_699_SUP__bot__conv_I2_J,axiom,
! [B2: set_fm > set_fm,A2: set_set_fm] :
( ( bot_bot_set_fm
= ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ B2 @ A2 ) ) )
= ( ! [X: set_fm] :
( ( member_set_fm @ X @ A2 )
=> ( ( B2 @ X )
= bot_bot_set_fm ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_700_SUP__bot__conv_I2_J,axiom,
! [B2: $o > $o,A2: set_o] :
( ( bot_bot_o
= ( complete_Sup_Sup_o @ ( image_o_o @ B2 @ A2 ) ) )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ( B2 @ X )
= bot_bot_o ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_701_SUP__bot__conv_I2_J,axiom,
! [B2: $o > set_o,A2: set_o] :
( ( bot_bot_set_o
= ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ( B2 @ X )
= bot_bot_set_o ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_702_SUP__bot__conv_I2_J,axiom,
! [B2: $o > set_Pr1822751329126368876m_rule,A2: set_o] :
( ( bot_bo3426410157013319936m_rule
= ( comple5648871819318716351m_rule @ ( image_6392605097815222207m_rule @ B2 @ A2 ) ) )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ( B2 @ X )
= bot_bo3426410157013319936m_rule ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_703_INF__top,axiom,
! [A2: set_Pr1822751329126368876m_rule] :
( ( comple1024113957266099930set_fm
@ ( image_377115208147135190set_fm
@ ^ [X: produc340336539035504054m_rule] : top_top_set_fm
@ A2 ) )
= top_top_set_fm ) ).
% INF_top
thf(fact_704_INF__top,axiom,
! [A2: set_o] :
( ( comple1222484001442006309m_rule
@ ( image_6392605097815222207m_rule
@ ^ [X: $o] : top_to8442024021002298652m_rule
@ A2 ) )
= top_to8442024021002298652m_rule ) ).
% INF_top
thf(fact_705_INF__top,axiom,
! [A2: set_o] :
( ( comple1024113957266099930set_fm
@ ( image_o_set_fm
@ ^ [X: $o] : top_top_set_fm
@ A2 ) )
= top_top_set_fm ) ).
% INF_top
thf(fact_706_INF__top,axiom,
! [A2: set_set_fm] :
( ( comple1024113957266099930set_fm
@ ( image_set_fm_set_fm
@ ^ [X: set_fm] : top_top_set_fm
@ A2 ) )
= top_top_set_fm ) ).
% INF_top
thf(fact_707_INF__top,axiom,
! [A2: set_o] :
( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [X: $o] : top_top_set_o
@ A2 ) )
= top_top_set_o ) ).
% INF_top
thf(fact_708_INF__top,axiom,
! [A2: set_o] :
( ( complete_Inf_Inf_o
@ ( image_o_o
@ ^ [X: $o] : top_top_o
@ A2 ) )
= top_top_o ) ).
% INF_top
thf(fact_709_INF__top__conv_I1_J,axiom,
! [B2: produc340336539035504054m_rule > set_fm,A2: set_Pr1822751329126368876m_rule] :
( ( ( comple1024113957266099930set_fm @ ( image_377115208147135190set_fm @ B2 @ A2 ) )
= top_top_set_fm )
= ( ! [X: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X @ A2 )
=> ( ( B2 @ X )
= top_top_set_fm ) ) ) ) ).
% INF_top_conv(1)
thf(fact_710_INF__top__conv_I1_J,axiom,
! [B2: $o > set_Pr1822751329126368876m_rule,A2: set_o] :
( ( ( comple1222484001442006309m_rule @ ( image_6392605097815222207m_rule @ B2 @ A2 ) )
= top_to8442024021002298652m_rule )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ( B2 @ X )
= top_to8442024021002298652m_rule ) ) ) ) ).
% INF_top_conv(1)
thf(fact_711_INF__top__conv_I1_J,axiom,
! [B2: $o > set_fm,A2: set_o] :
( ( ( comple1024113957266099930set_fm @ ( image_o_set_fm @ B2 @ A2 ) )
= top_top_set_fm )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ( B2 @ X )
= top_top_set_fm ) ) ) ) ).
% INF_top_conv(1)
thf(fact_712_INF__top__conv_I1_J,axiom,
! [B2: set_fm > set_fm,A2: set_set_fm] :
( ( ( comple1024113957266099930set_fm @ ( image_set_fm_set_fm @ B2 @ A2 ) )
= top_top_set_fm )
= ( ! [X: set_fm] :
( ( member_set_fm @ X @ A2 )
=> ( ( B2 @ X )
= top_top_set_fm ) ) ) ) ).
% INF_top_conv(1)
thf(fact_713_INF__top__conv_I1_J,axiom,
! [B2: $o > set_o,A2: set_o] :
( ( ( comple3063163877087187839_set_o @ ( image_o_set_o @ B2 @ A2 ) )
= top_top_set_o )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ( B2 @ X )
= top_top_set_o ) ) ) ) ).
% INF_top_conv(1)
thf(fact_714_INF__top__conv_I1_J,axiom,
! [B2: $o > $o,A2: set_o] :
( ( ( complete_Inf_Inf_o @ ( image_o_o @ B2 @ A2 ) )
= top_top_o )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ( B2 @ X )
= top_top_o ) ) ) ) ).
% INF_top_conv(1)
thf(fact_715_INF__top__conv_I2_J,axiom,
! [B2: produc340336539035504054m_rule > set_fm,A2: set_Pr1822751329126368876m_rule] :
( ( top_top_set_fm
= ( comple1024113957266099930set_fm @ ( image_377115208147135190set_fm @ B2 @ A2 ) ) )
= ( ! [X: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X @ A2 )
=> ( ( B2 @ X )
= top_top_set_fm ) ) ) ) ).
% INF_top_conv(2)
thf(fact_716_INF__top__conv_I2_J,axiom,
! [B2: $o > set_Pr1822751329126368876m_rule,A2: set_o] :
( ( top_to8442024021002298652m_rule
= ( comple1222484001442006309m_rule @ ( image_6392605097815222207m_rule @ B2 @ A2 ) ) )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ( B2 @ X )
= top_to8442024021002298652m_rule ) ) ) ) ).
% INF_top_conv(2)
thf(fact_717_INF__top__conv_I2_J,axiom,
! [B2: $o > set_fm,A2: set_o] :
( ( top_top_set_fm
= ( comple1024113957266099930set_fm @ ( image_o_set_fm @ B2 @ A2 ) ) )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ( B2 @ X )
= top_top_set_fm ) ) ) ) ).
% INF_top_conv(2)
thf(fact_718_INF__top__conv_I2_J,axiom,
! [B2: set_fm > set_fm,A2: set_set_fm] :
( ( top_top_set_fm
= ( comple1024113957266099930set_fm @ ( image_set_fm_set_fm @ B2 @ A2 ) ) )
= ( ! [X: set_fm] :
( ( member_set_fm @ X @ A2 )
=> ( ( B2 @ X )
= top_top_set_fm ) ) ) ) ).
% INF_top_conv(2)
thf(fact_719_INF__top__conv_I2_J,axiom,
! [B2: $o > set_o,A2: set_o] :
( ( top_top_set_o
= ( comple3063163877087187839_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ( B2 @ X )
= top_top_set_o ) ) ) ) ).
% INF_top_conv(2)
thf(fact_720_INF__top__conv_I2_J,axiom,
! [B2: $o > $o,A2: set_o] :
( ( top_top_o
= ( complete_Inf_Inf_o @ ( image_o_o @ B2 @ A2 ) ) )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ( B2 @ X )
= top_top_o ) ) ) ) ).
% INF_top_conv(2)
thf(fact_721_SUP__const,axiom,
! [A2: set_Pr1822751329126368876m_rule,F: set_fm] :
( ( A2 != bot_bo3426410157013319936m_rule )
=> ( ( comple2134933779557159616set_fm
@ ( image_377115208147135190set_fm
@ ^ [I: produc340336539035504054m_rule] : F
@ A2 ) )
= F ) ) ).
% SUP_const
thf(fact_722_SUP__const,axiom,
! [A2: set_set_fm,F: set_fm] :
( ( A2 != bot_bot_set_set_fm )
=> ( ( comple2134933779557159616set_fm
@ ( image_set_fm_set_fm
@ ^ [I: set_fm] : F
@ A2 ) )
= F ) ) ).
% SUP_const
thf(fact_723_SUP__const,axiom,
! [A2: set_o,F: set_fm] :
( ( A2 != bot_bot_set_o )
=> ( ( comple2134933779557159616set_fm
@ ( image_o_set_fm
@ ^ [I: $o] : F
@ A2 ) )
= F ) ) ).
% SUP_const
thf(fact_724_SUP__const,axiom,
! [A2: set_o,F: $o] :
( ( A2 != bot_bot_set_o )
=> ( ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [I: $o] : F
@ A2 ) )
= F ) ) ).
% SUP_const
thf(fact_725_SUP__const,axiom,
! [A2: set_o,F: set_o] :
( ( A2 != bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [I: $o] : F
@ A2 ) )
= F ) ) ).
% SUP_const
thf(fact_726_SUP__const,axiom,
! [A2: set_o,F: set_Pr1822751329126368876m_rule] :
( ( A2 != bot_bot_set_o )
=> ( ( comple5648871819318716351m_rule
@ ( image_6392605097815222207m_rule
@ ^ [I: $o] : F
@ A2 ) )
= F ) ) ).
% SUP_const
thf(fact_727_INF__const,axiom,
! [A2: set_Pr1822751329126368876m_rule,F: set_fm] :
( ( A2 != bot_bo3426410157013319936m_rule )
=> ( ( comple1024113957266099930set_fm
@ ( image_377115208147135190set_fm
@ ^ [I: produc340336539035504054m_rule] : F
@ A2 ) )
= F ) ) ).
% INF_const
thf(fact_728_INF__const,axiom,
! [A2: set_set_fm,F: set_fm] :
( ( A2 != bot_bot_set_set_fm )
=> ( ( comple1024113957266099930set_fm
@ ( image_set_fm_set_fm
@ ^ [I: set_fm] : F
@ A2 ) )
= F ) ) ).
% INF_const
thf(fact_729_INF__const,axiom,
! [A2: set_o,F: set_Pr1822751329126368876m_rule] :
( ( A2 != bot_bot_set_o )
=> ( ( comple1222484001442006309m_rule
@ ( image_6392605097815222207m_rule
@ ^ [I: $o] : F
@ A2 ) )
= F ) ) ).
% INF_const
thf(fact_730_INF__const,axiom,
! [A2: set_o,F: set_fm] :
( ( A2 != bot_bot_set_o )
=> ( ( comple1024113957266099930set_fm
@ ( image_o_set_fm
@ ^ [I: $o] : F
@ A2 ) )
= F ) ) ).
% INF_const
thf(fact_731_INF__const,axiom,
! [A2: set_o,F: set_o] :
( ( A2 != bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [I: $o] : F
@ A2 ) )
= F ) ) ).
% INF_const
thf(fact_732_INF__const,axiom,
! [A2: set_o,F: $o] :
( ( A2 != bot_bot_set_o )
=> ( ( complete_Inf_Inf_o
@ ( image_o_o
@ ^ [I: $o] : F
@ A2 ) )
= F ) ) ).
% INF_const
thf(fact_733_UN__constant,axiom,
! [A2: set_Pr1822751329126368876m_rule,C2: set_fm] :
( ( ( A2 = bot_bo3426410157013319936m_rule )
=> ( ( comple2134933779557159616set_fm
@ ( image_377115208147135190set_fm
@ ^ [Y: produc340336539035504054m_rule] : C2
@ A2 ) )
= bot_bot_set_fm ) )
& ( ( A2 != bot_bo3426410157013319936m_rule )
=> ( ( comple2134933779557159616set_fm
@ ( image_377115208147135190set_fm
@ ^ [Y: produc340336539035504054m_rule] : C2
@ A2 ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_734_UN__constant,axiom,
! [A2: set_set_fm,C2: set_fm] :
( ( ( A2 = bot_bot_set_set_fm )
=> ( ( comple2134933779557159616set_fm
@ ( image_set_fm_set_fm
@ ^ [Y: set_fm] : C2
@ A2 ) )
= bot_bot_set_fm ) )
& ( ( A2 != bot_bot_set_set_fm )
=> ( ( comple2134933779557159616set_fm
@ ( image_set_fm_set_fm
@ ^ [Y: set_fm] : C2
@ A2 ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_735_UN__constant,axiom,
! [A2: set_o,C2: set_fm] :
( ( ( A2 = bot_bot_set_o )
=> ( ( comple2134933779557159616set_fm
@ ( image_o_set_fm
@ ^ [Y: $o] : C2
@ A2 ) )
= bot_bot_set_fm ) )
& ( ( A2 != bot_bot_set_o )
=> ( ( comple2134933779557159616set_fm
@ ( image_o_set_fm
@ ^ [Y: $o] : C2
@ A2 ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_736_UN__constant,axiom,
! [A2: set_o,C2: set_o] :
( ( ( A2 = bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [Y: $o] : C2
@ A2 ) )
= bot_bot_set_o ) )
& ( ( A2 != bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [Y: $o] : C2
@ A2 ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_737_UN__constant,axiom,
! [A2: set_o,C2: set_Pr1822751329126368876m_rule] :
( ( ( A2 = bot_bot_set_o )
=> ( ( comple5648871819318716351m_rule
@ ( image_6392605097815222207m_rule
@ ^ [Y: $o] : C2
@ A2 ) )
= bot_bo3426410157013319936m_rule ) )
& ( ( A2 != bot_bot_set_o )
=> ( ( comple5648871819318716351m_rule
@ ( image_6392605097815222207m_rule
@ ^ [Y: $o] : C2
@ A2 ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_738_INT__constant,axiom,
! [A2: set_Pr1822751329126368876m_rule,C2: set_fm] :
( ( ( A2 = bot_bo3426410157013319936m_rule )
=> ( ( comple1024113957266099930set_fm
@ ( image_377115208147135190set_fm
@ ^ [Y: produc340336539035504054m_rule] : C2
@ A2 ) )
= top_top_set_fm ) )
& ( ( A2 != bot_bo3426410157013319936m_rule )
=> ( ( comple1024113957266099930set_fm
@ ( image_377115208147135190set_fm
@ ^ [Y: produc340336539035504054m_rule] : C2
@ A2 ) )
= C2 ) ) ) ).
% INT_constant
thf(fact_739_INT__constant,axiom,
! [A2: set_set_fm,C2: set_fm] :
( ( ( A2 = bot_bot_set_set_fm )
=> ( ( comple1024113957266099930set_fm
@ ( image_set_fm_set_fm
@ ^ [Y: set_fm] : C2
@ A2 ) )
= top_top_set_fm ) )
& ( ( A2 != bot_bot_set_set_fm )
=> ( ( comple1024113957266099930set_fm
@ ( image_set_fm_set_fm
@ ^ [Y: set_fm] : C2
@ A2 ) )
= C2 ) ) ) ).
% INT_constant
thf(fact_740_INT__constant,axiom,
! [A2: set_o,C2: set_Pr1822751329126368876m_rule] :
( ( ( A2 = bot_bot_set_o )
=> ( ( comple1222484001442006309m_rule
@ ( image_6392605097815222207m_rule
@ ^ [Y: $o] : C2
@ A2 ) )
= top_to8442024021002298652m_rule ) )
& ( ( A2 != bot_bot_set_o )
=> ( ( comple1222484001442006309m_rule
@ ( image_6392605097815222207m_rule
@ ^ [Y: $o] : C2
@ A2 ) )
= C2 ) ) ) ).
% INT_constant
thf(fact_741_INT__constant,axiom,
! [A2: set_o,C2: set_fm] :
( ( ( A2 = bot_bot_set_o )
=> ( ( comple1024113957266099930set_fm
@ ( image_o_set_fm
@ ^ [Y: $o] : C2
@ A2 ) )
= top_top_set_fm ) )
& ( ( A2 != bot_bot_set_o )
=> ( ( comple1024113957266099930set_fm
@ ( image_o_set_fm
@ ^ [Y: $o] : C2
@ A2 ) )
= C2 ) ) ) ).
% INT_constant
thf(fact_742_INT__constant,axiom,
! [A2: set_o,C2: set_nat] :
( ( ( A2 = bot_bot_set_o )
=> ( ( comple7806235888213564991et_nat
@ ( image_o_set_nat
@ ^ [Y: $o] : C2
@ A2 ) )
= top_top_set_nat ) )
& ( ( A2 != bot_bot_set_o )
=> ( ( comple7806235888213564991et_nat
@ ( image_o_set_nat
@ ^ [Y: $o] : C2
@ A2 ) )
= C2 ) ) ) ).
% INT_constant
thf(fact_743_INT__constant,axiom,
! [A2: set_o,C2: set_rat] :
( ( ( A2 = bot_bot_set_o )
=> ( ( comple4298007329820168263et_rat
@ ( image_o_set_rat
@ ^ [Y: $o] : C2
@ A2 ) )
= top_top_set_rat ) )
& ( ( A2 != bot_bot_set_o )
=> ( ( comple4298007329820168263et_rat
@ ( image_o_set_rat
@ ^ [Y: $o] : C2
@ A2 ) )
= C2 ) ) ) ).
% INT_constant
thf(fact_744_INT__constant,axiom,
! [A2: set_o,C2: set_o] :
( ( ( A2 = bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [Y: $o] : C2
@ A2 ) )
= top_top_set_o ) )
& ( ( A2 != bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [Y: $o] : C2
@ A2 ) )
= C2 ) ) ) ).
% INT_constant
thf(fact_745_INF__eq__iff,axiom,
! [I3: set_set_fm,F: set_fm > set_fm,C2: set_fm] :
( ( I3 != bot_bot_set_set_fm )
=> ( ! [I4: set_fm] :
( ( member_set_fm @ I4 @ I3 )
=> ( ord_less_eq_set_fm @ ( F @ I4 ) @ C2 ) )
=> ( ( ( comple1024113957266099930set_fm @ ( image_set_fm_set_fm @ F @ I3 ) )
= C2 )
= ( ! [X: set_fm] :
( ( member_set_fm @ X @ I3 )
=> ( ( F @ X )
= C2 ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_746_INF__eq__iff,axiom,
! [I3: set_Pr1822751329126368876m_rule,F: produc340336539035504054m_rule > set_fm,C2: set_fm] :
( ( I3 != bot_bo3426410157013319936m_rule )
=> ( ! [I4: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ I4 @ I3 )
=> ( ord_less_eq_set_fm @ ( F @ I4 ) @ C2 ) )
=> ( ( ( comple1024113957266099930set_fm @ ( image_377115208147135190set_fm @ F @ I3 ) )
= C2 )
= ( ! [X: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X @ I3 )
=> ( ( F @ X )
= C2 ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_747_INF__eq__iff,axiom,
! [I3: set_o,F: $o > set_Pr1822751329126368876m_rule,C2: set_Pr1822751329126368876m_rule] :
( ( I3 != bot_bot_set_o )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ I3 )
=> ( ord_le6390412330253371084m_rule @ ( F @ I4 ) @ C2 ) )
=> ( ( ( comple1222484001442006309m_rule @ ( image_6392605097815222207m_rule @ F @ I3 ) )
= C2 )
= ( ! [X: $o] :
( ( member_o @ X @ I3 )
=> ( ( F @ X )
= C2 ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_748_INF__eq__iff,axiom,
! [I3: set_o,F: $o > set_fm,C2: set_fm] :
( ( I3 != bot_bot_set_o )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ I3 )
=> ( ord_less_eq_set_fm @ ( F @ I4 ) @ C2 ) )
=> ( ( ( comple1024113957266099930set_fm @ ( image_o_set_fm @ F @ I3 ) )
= C2 )
= ( ! [X: $o] :
( ( member_o @ X @ I3 )
=> ( ( F @ X )
= C2 ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_749_INF__eq__iff,axiom,
! [I3: set_o,F: $o > set_o,C2: set_o] :
( ( I3 != bot_bot_set_o )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ I3 )
=> ( ord_less_eq_set_o @ ( F @ I4 ) @ C2 ) )
=> ( ( ( comple3063163877087187839_set_o @ ( image_o_set_o @ F @ I3 ) )
= C2 )
= ( ! [X: $o] :
( ( member_o @ X @ I3 )
=> ( ( F @ X )
= C2 ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_750_INF__eq__iff,axiom,
! [I3: set_fm,F: fm > $o,C2: $o] :
( ( I3 != bot_bot_set_fm )
=> ( ! [I4: fm] :
( ( member_fm @ I4 @ I3 )
=> ( ord_less_eq_o @ ( F @ I4 ) @ C2 ) )
=> ( ( ( complete_Inf_Inf_o @ ( image_fm_o @ F @ I3 ) )
= C2 )
= ( ! [X: fm] :
( ( member_fm @ X @ I3 )
=> ( ( F @ X )
= C2 ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_751_INF__eq__iff,axiom,
! [I3: set_Pr1822751329126368876m_rule,F: produc340336539035504054m_rule > $o,C2: $o] :
( ( I3 != bot_bo3426410157013319936m_rule )
=> ( ! [I4: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ I4 @ I3 )
=> ( ord_less_eq_o @ ( F @ I4 ) @ C2 ) )
=> ( ( ( complete_Inf_Inf_o @ ( image_8969478372357302939rule_o @ F @ I3 ) )
= C2 )
= ( ! [X: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X @ I3 )
=> ( ( F @ X )
= C2 ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_752_INF__eq__iff,axiom,
! [I3: set_o,F: $o > $o,C2: $o] :
( ( I3 != bot_bot_set_o )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ I3 )
=> ( ord_less_eq_o @ ( F @ I4 ) @ C2 ) )
=> ( ( ( complete_Inf_Inf_o @ ( image_o_o @ F @ I3 ) )
= C2 )
= ( ! [X: $o] :
( ( member_o @ X @ I3 )
=> ( ( F @ X )
= C2 ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_753_INF__superset__mono,axiom,
! [B2: set_set_fm,A2: set_set_fm,F: set_fm > set_fm,G: set_fm > set_fm] :
( ( ord_le5844446314808584147set_fm @ B2 @ A2 )
=> ( ! [X3: set_fm] :
( ( member_set_fm @ X3 @ B2 )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_fm @ ( comple1024113957266099930set_fm @ ( image_set_fm_set_fm @ F @ A2 ) ) @ ( comple1024113957266099930set_fm @ ( image_set_fm_set_fm @ G @ B2 ) ) ) ) ) ).
% INF_superset_mono
thf(fact_754_INF__superset__mono,axiom,
! [B2: set_Pr1822751329126368876m_rule,A2: set_Pr1822751329126368876m_rule,F: produc340336539035504054m_rule > set_fm,G: produc340336539035504054m_rule > set_fm] :
( ( ord_le6390412330253371084m_rule @ B2 @ A2 )
=> ( ! [X3: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X3 @ B2 )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_fm @ ( comple1024113957266099930set_fm @ ( image_377115208147135190set_fm @ F @ A2 ) ) @ ( comple1024113957266099930set_fm @ ( image_377115208147135190set_fm @ G @ B2 ) ) ) ) ) ).
% INF_superset_mono
thf(fact_755_INF__superset__mono,axiom,
! [B2: set_o,A2: set_o,F: $o > set_Pr1822751329126368876m_rule,G: $o > set_Pr1822751329126368876m_rule] :
( ( ord_less_eq_set_o @ B2 @ A2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B2 )
=> ( ord_le6390412330253371084m_rule @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_le6390412330253371084m_rule @ ( comple1222484001442006309m_rule @ ( image_6392605097815222207m_rule @ F @ A2 ) ) @ ( comple1222484001442006309m_rule @ ( image_6392605097815222207m_rule @ G @ B2 ) ) ) ) ) ).
% INF_superset_mono
thf(fact_756_INF__superset__mono,axiom,
! [B2: set_o,A2: set_o,F: $o > set_fm,G: $o > set_fm] :
( ( ord_less_eq_set_o @ B2 @ A2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B2 )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_fm @ ( comple1024113957266099930set_fm @ ( image_o_set_fm @ F @ A2 ) ) @ ( comple1024113957266099930set_fm @ ( image_o_set_fm @ G @ B2 ) ) ) ) ) ).
% INF_superset_mono
thf(fact_757_INF__superset__mono,axiom,
! [B2: set_o,A2: set_o,F: $o > set_o,G: $o > set_o] :
( ( ord_less_eq_set_o @ B2 @ A2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B2 )
=> ( ord_less_eq_set_o @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_o @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ F @ A2 ) ) @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ G @ B2 ) ) ) ) ) ).
% INF_superset_mono
thf(fact_758_INF__superset__mono,axiom,
! [B2: set_fm,A2: set_fm,F: fm > $o,G: fm > $o] :
( ( ord_less_eq_set_fm @ B2 @ A2 )
=> ( ! [X3: fm] :
( ( member_fm @ X3 @ B2 )
=> ( ord_less_eq_o @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_fm_o @ F @ A2 ) ) @ ( complete_Inf_Inf_o @ ( image_fm_o @ G @ B2 ) ) ) ) ) ).
% INF_superset_mono
thf(fact_759_INF__superset__mono,axiom,
! [B2: set_Pr1822751329126368876m_rule,A2: set_Pr1822751329126368876m_rule,F: produc340336539035504054m_rule > $o,G: produc340336539035504054m_rule > $o] :
( ( ord_le6390412330253371084m_rule @ B2 @ A2 )
=> ( ! [X3: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X3 @ B2 )
=> ( ord_less_eq_o @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_8969478372357302939rule_o @ F @ A2 ) ) @ ( complete_Inf_Inf_o @ ( image_8969478372357302939rule_o @ G @ B2 ) ) ) ) ) ).
% INF_superset_mono
thf(fact_760_INF__superset__mono,axiom,
! [B2: set_o,A2: set_o,F: $o > $o,G: $o > $o] :
( ( ord_less_eq_set_o @ B2 @ A2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B2 )
=> ( ord_less_eq_o @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) ) @ ( complete_Inf_Inf_o @ ( image_o_o @ G @ B2 ) ) ) ) ) ).
% INF_superset_mono
thf(fact_761_times__subset__iff,axiom,
! [A2: set_o,C: set_o,B2: set_o,D: set_o] :
( ( ord_le192195778054999463od_o_o
@ ( product_Sigma_o_o @ A2
@ ^ [Uu: $o] : C )
@ ( product_Sigma_o_o @ B2
@ ^ [Uu: $o] : D ) )
= ( ( A2 = bot_bot_set_o )
| ( C = bot_bot_set_o )
| ( ( ord_less_eq_set_o @ A2 @ B2 )
& ( ord_less_eq_set_o @ C @ D ) ) ) ) ).
% times_subset_iff
thf(fact_762_INF__eq,axiom,
! [A2: set_fm,B2: set_fm,G: fm > $o,F: fm > $o] :
( ! [I4: fm] :
( ( member_fm @ I4 @ A2 )
=> ? [X6: fm] :
( ( member_fm @ X6 @ B2 )
& ( ord_less_eq_o @ ( G @ X6 ) @ ( F @ I4 ) ) ) )
=> ( ! [J2: fm] :
( ( member_fm @ J2 @ B2 )
=> ? [X6: fm] :
( ( member_fm @ X6 @ A2 )
& ( ord_less_eq_o @ ( F @ X6 ) @ ( G @ J2 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_fm_o @ F @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_fm_o @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_763_INF__eq,axiom,
! [A2: set_fm,B2: set_o,G: $o > $o,F: fm > $o] :
( ! [I4: fm] :
( ( member_fm @ I4 @ A2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ B2 )
& ( ord_less_eq_o @ ( G @ X6 ) @ ( F @ I4 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B2 )
=> ? [X6: fm] :
( ( member_fm @ X6 @ A2 )
& ( ord_less_eq_o @ ( F @ X6 ) @ ( G @ J2 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_fm_o @ F @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_o_o @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_764_INF__eq,axiom,
! [A2: set_o,B2: set_fm,G: fm > $o,F: $o > $o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ? [X6: fm] :
( ( member_fm @ X6 @ B2 )
& ( ord_less_eq_o @ ( G @ X6 ) @ ( F @ I4 ) ) ) )
=> ( ! [J2: fm] :
( ( member_fm @ J2 @ B2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A2 )
& ( ord_less_eq_o @ ( F @ X6 ) @ ( G @ J2 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_fm_o @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_765_INF__eq,axiom,
! [A2: set_o,B2: set_o,G: $o > $o,F: $o > $o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ B2 )
& ( ord_less_eq_o @ ( G @ X6 ) @ ( F @ I4 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A2 )
& ( ord_less_eq_o @ ( F @ X6 ) @ ( G @ J2 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_o_o @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_766_INF__eq,axiom,
! [A2: set_fm,B2: set_o,G: $o > set_fm,F: fm > set_fm] :
( ! [I4: fm] :
( ( member_fm @ I4 @ A2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ B2 )
& ( ord_less_eq_set_fm @ ( G @ X6 ) @ ( F @ I4 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B2 )
=> ? [X6: fm] :
( ( member_fm @ X6 @ A2 )
& ( ord_less_eq_set_fm @ ( F @ X6 ) @ ( G @ J2 ) ) ) )
=> ( ( comple1024113957266099930set_fm @ ( image_fm_set_fm @ F @ A2 ) )
= ( comple1024113957266099930set_fm @ ( image_o_set_fm @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_767_INF__eq,axiom,
! [A2: set_fm,B2: set_o,G: $o > set_o,F: fm > set_o] :
( ! [I4: fm] :
( ( member_fm @ I4 @ A2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ B2 )
& ( ord_less_eq_set_o @ ( G @ X6 ) @ ( F @ I4 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B2 )
=> ? [X6: fm] :
( ( member_fm @ X6 @ A2 )
& ( ord_less_eq_set_o @ ( F @ X6 ) @ ( G @ J2 ) ) ) )
=> ( ( comple3063163877087187839_set_o @ ( image_fm_set_o @ F @ A2 ) )
= ( comple3063163877087187839_set_o @ ( image_o_set_o @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_768_INF__eq,axiom,
! [A2: set_o,B2: set_fm,G: fm > set_fm,F: $o > set_fm] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ? [X6: fm] :
( ( member_fm @ X6 @ B2 )
& ( ord_less_eq_set_fm @ ( G @ X6 ) @ ( F @ I4 ) ) ) )
=> ( ! [J2: fm] :
( ( member_fm @ J2 @ B2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A2 )
& ( ord_less_eq_set_fm @ ( F @ X6 ) @ ( G @ J2 ) ) ) )
=> ( ( comple1024113957266099930set_fm @ ( image_o_set_fm @ F @ A2 ) )
= ( comple1024113957266099930set_fm @ ( image_fm_set_fm @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_769_INF__eq,axiom,
! [A2: set_o,B2: set_fm,G: fm > set_o,F: $o > set_o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ? [X6: fm] :
( ( member_fm @ X6 @ B2 )
& ( ord_less_eq_set_o @ ( G @ X6 ) @ ( F @ I4 ) ) ) )
=> ( ! [J2: fm] :
( ( member_fm @ J2 @ B2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A2 )
& ( ord_less_eq_set_o @ ( F @ X6 ) @ ( G @ J2 ) ) ) )
=> ( ( comple3063163877087187839_set_o @ ( image_o_set_o @ F @ A2 ) )
= ( comple3063163877087187839_set_o @ ( image_fm_set_o @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_770_INF__eq,axiom,
! [A2: set_o,B2: set_o,G: $o > set_fm,F: $o > set_fm] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ B2 )
& ( ord_less_eq_set_fm @ ( G @ X6 ) @ ( F @ I4 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A2 )
& ( ord_less_eq_set_fm @ ( F @ X6 ) @ ( G @ J2 ) ) ) )
=> ( ( comple1024113957266099930set_fm @ ( image_o_set_fm @ F @ A2 ) )
= ( comple1024113957266099930set_fm @ ( image_o_set_fm @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_771_INF__eq,axiom,
! [A2: set_o,B2: set_o,G: $o > set_o,F: $o > set_o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ B2 )
& ( ord_less_eq_set_o @ ( G @ X6 ) @ ( F @ I4 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A2 )
& ( ord_less_eq_set_o @ ( F @ X6 ) @ ( G @ J2 ) ) ) )
=> ( ( comple3063163877087187839_set_o @ ( image_o_set_o @ F @ A2 ) )
= ( comple3063163877087187839_set_o @ ( image_o_set_o @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_772_INF__eq__const,axiom,
! [I3: set_set_fm,F: set_fm > set_fm,X5: set_fm] :
( ( I3 != bot_bot_set_set_fm )
=> ( ! [I4: set_fm] :
( ( member_set_fm @ I4 @ I3 )
=> ( ( F @ I4 )
= X5 ) )
=> ( ( comple1024113957266099930set_fm @ ( image_set_fm_set_fm @ F @ I3 ) )
= X5 ) ) ) ).
% INF_eq_const
thf(fact_773_INF__eq__const,axiom,
! [I3: set_Pr1822751329126368876m_rule,F: produc340336539035504054m_rule > set_fm,X5: set_fm] :
( ( I3 != bot_bo3426410157013319936m_rule )
=> ( ! [I4: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ I4 @ I3 )
=> ( ( F @ I4 )
= X5 ) )
=> ( ( comple1024113957266099930set_fm @ ( image_377115208147135190set_fm @ F @ I3 ) )
= X5 ) ) ) ).
% INF_eq_const
thf(fact_774_INF__eq__const,axiom,
! [I3: set_o,F: $o > set_Pr1822751329126368876m_rule,X5: set_Pr1822751329126368876m_rule] :
( ( I3 != bot_bot_set_o )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ I3 )
=> ( ( F @ I4 )
= X5 ) )
=> ( ( comple1222484001442006309m_rule @ ( image_6392605097815222207m_rule @ F @ I3 ) )
= X5 ) ) ) ).
% INF_eq_const
thf(fact_775_INF__eq__const,axiom,
! [I3: set_o,F: $o > set_fm,X5: set_fm] :
( ( I3 != bot_bot_set_o )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ I3 )
=> ( ( F @ I4 )
= X5 ) )
=> ( ( comple1024113957266099930set_fm @ ( image_o_set_fm @ F @ I3 ) )
= X5 ) ) ) ).
% INF_eq_const
thf(fact_776_INF__eq__const,axiom,
! [I3: set_o,F: $o > set_o,X5: set_o] :
( ( I3 != bot_bot_set_o )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ I3 )
=> ( ( F @ I4 )
= X5 ) )
=> ( ( comple3063163877087187839_set_o @ ( image_o_set_o @ F @ I3 ) )
= X5 ) ) ) ).
% INF_eq_const
thf(fact_777_INF__eq__const,axiom,
! [I3: set_fm,F: fm > $o,X5: $o] :
( ( I3 != bot_bot_set_fm )
=> ( ! [I4: fm] :
( ( member_fm @ I4 @ I3 )
=> ( ( F @ I4 )
= X5 ) )
=> ( ( complete_Inf_Inf_o @ ( image_fm_o @ F @ I3 ) )
= X5 ) ) ) ).
% INF_eq_const
thf(fact_778_INF__eq__const,axiom,
! [I3: set_Pr1822751329126368876m_rule,F: produc340336539035504054m_rule > $o,X5: $o] :
( ( I3 != bot_bo3426410157013319936m_rule )
=> ( ! [I4: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ I4 @ I3 )
=> ( ( F @ I4 )
= X5 ) )
=> ( ( complete_Inf_Inf_o @ ( image_8969478372357302939rule_o @ F @ I3 ) )
= X5 ) ) ) ).
% INF_eq_const
thf(fact_779_INF__eq__const,axiom,
! [I3: set_o,F: $o > $o,X5: $o] :
( ( I3 != bot_bot_set_o )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ I3 )
=> ( ( F @ I4 )
= X5 ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ F @ I3 ) )
= X5 ) ) ) ).
% INF_eq_const
thf(fact_780_Inter__empty,axiom,
( ( comple7806235888213564991et_nat @ bot_bot_set_set_nat )
= top_top_set_nat ) ).
% Inter_empty
thf(fact_781_Inter__empty,axiom,
( ( comple4298007329820168263et_rat @ bot_bot_set_set_rat )
= top_top_set_rat ) ).
% Inter_empty
thf(fact_782_Inter__empty,axiom,
( ( comple3063163877087187839_set_o @ bot_bot_set_set_o )
= top_top_set_o ) ).
% Inter_empty
thf(fact_783_not__psubset__empty,axiom,
! [A2: set_o] :
~ ( ord_less_set_o @ A2 @ bot_bot_set_o ) ).
% not_psubset_empty
thf(fact_784_Sup__subset__mono,axiom,
! [A2: set_set_fm,B2: set_set_fm] :
( ( ord_le5844446314808584147set_fm @ A2 @ B2 )
=> ( ord_less_eq_set_fm @ ( comple2134933779557159616set_fm @ A2 ) @ ( comple2134933779557159616set_fm @ B2 ) ) ) ).
% Sup_subset_mono
thf(fact_785_Sup__subset__mono,axiom,
! [A2: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A2 ) @ ( complete_Sup_Sup_o @ B2 ) ) ) ).
% Sup_subset_mono
thf(fact_786_Sup__subset__mono,axiom,
! [A2: set_set_o,B2: set_set_o] :
( ( ord_le4374716579403074808_set_o @ A2 @ B2 )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ A2 ) @ ( comple90263536869209701_set_o @ B2 ) ) ) ).
% Sup_subset_mono
thf(fact_787_Sup__subset__mono,axiom,
! [A2: set_se4309406418413213730m_rule,B2: set_se4309406418413213730m_rule] :
( ( ord_le4271096684612802690m_rule @ A2 @ B2 )
=> ( ord_le6390412330253371084m_rule @ ( comple5648871819318716351m_rule @ A2 ) @ ( comple5648871819318716351m_rule @ B2 ) ) ) ).
% Sup_subset_mono
thf(fact_788_less__eq__Sup,axiom,
! [A2: set_set_fm,U: set_fm] :
( ! [V: set_fm] :
( ( member_set_fm @ V @ A2 )
=> ( ord_less_eq_set_fm @ U @ V ) )
=> ( ( A2 != bot_bot_set_set_fm )
=> ( ord_less_eq_set_fm @ U @ ( comple2134933779557159616set_fm @ A2 ) ) ) ) ).
% less_eq_Sup
thf(fact_789_less__eq__Sup,axiom,
! [A2: set_o,U: $o] :
( ! [V: $o] :
( ( member_o @ V @ A2 )
=> ( ord_less_eq_o @ U @ V ) )
=> ( ( A2 != bot_bot_set_o )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ A2 ) ) ) ) ).
% less_eq_Sup
thf(fact_790_less__eq__Sup,axiom,
! [A2: set_set_o,U: set_o] :
( ! [V: set_o] :
( ( member_set_o @ V @ A2 )
=> ( ord_less_eq_set_o @ U @ V ) )
=> ( ( A2 != bot_bot_set_set_o )
=> ( ord_less_eq_set_o @ U @ ( comple90263536869209701_set_o @ A2 ) ) ) ) ).
% less_eq_Sup
thf(fact_791_less__eq__Sup,axiom,
! [A2: set_se4309406418413213730m_rule,U: set_Pr1822751329126368876m_rule] :
( ! [V: set_Pr1822751329126368876m_rule] :
( ( member5802698409304902403m_rule @ V @ A2 )
=> ( ord_le6390412330253371084m_rule @ U @ V ) )
=> ( ( A2 != bot_bo8650941071013544630m_rule )
=> ( ord_le6390412330253371084m_rule @ U @ ( comple5648871819318716351m_rule @ A2 ) ) ) ) ).
% less_eq_Sup
thf(fact_792_Inf__le__Sup,axiom,
! [A2: set_set_fm] :
( ( A2 != bot_bot_set_set_fm )
=> ( ord_less_eq_set_fm @ ( comple1024113957266099930set_fm @ A2 ) @ ( comple2134933779557159616set_fm @ A2 ) ) ) ).
% Inf_le_Sup
thf(fact_793_Inf__le__Sup,axiom,
! [A2: set_o] :
( ( A2 != bot_bot_set_o )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A2 ) @ ( complete_Sup_Sup_o @ A2 ) ) ) ).
% Inf_le_Sup
thf(fact_794_Inf__le__Sup,axiom,
! [A2: set_set_o] :
( ( A2 != bot_bot_set_set_o )
=> ( ord_less_eq_set_o @ ( comple3063163877087187839_set_o @ A2 ) @ ( comple90263536869209701_set_o @ A2 ) ) ) ).
% Inf_le_Sup
thf(fact_795_Inf__le__Sup,axiom,
! [A2: set_se4309406418413213730m_rule] :
( ( A2 != bot_bo8650941071013544630m_rule )
=> ( ord_le6390412330253371084m_rule @ ( comple1222484001442006309m_rule @ A2 ) @ ( comple5648871819318716351m_rule @ A2 ) ) ) ).
% Inf_le_Sup
thf(fact_796_order__antisym__conv,axiom,
! [Y5: nat,X5: nat] :
( ( ord_less_eq_nat @ Y5 @ X5 )
=> ( ( ord_less_eq_nat @ X5 @ Y5 )
= ( X5 = Y5 ) ) ) ).
% order_antisym_conv
thf(fact_797_linorder__le__cases,axiom,
! [X5: nat,Y5: nat] :
( ~ ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X5 ) ) ).
% linorder_le_cases
thf(fact_798_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_799_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_800_linorder__linear,axiom,
! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
| ( ord_less_eq_nat @ Y5 @ X5 ) ) ).
% linorder_linear
thf(fact_801_order__eq__refl,axiom,
! [X5: nat,Y5: nat] :
( ( X5 = Y5 )
=> ( ord_less_eq_nat @ X5 @ Y5 ) ) ).
% order_eq_refl
thf(fact_802_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_803_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_804_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A3: nat,B4: nat] :
( ( ord_less_eq_nat @ A3 @ B4 )
& ( ord_less_eq_nat @ B4 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_805_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_806_subset__iff,axiom,
( ord_less_eq_set_fm
= ( ^ [A4: set_fm,B5: set_fm] :
! [T: fm] :
( ( member_fm @ T @ A4 )
=> ( member_fm @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_807_subset__iff,axiom,
( ord_le6390412330253371084m_rule
= ( ^ [A4: set_Pr1822751329126368876m_rule,B5: set_Pr1822751329126368876m_rule] :
! [T: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ T @ A4 )
=> ( member7231649785386036813m_rule @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_808_subset__iff,axiom,
( ord_less_eq_set_o
= ( ^ [A4: set_o,B5: set_o] :
! [T: $o] :
( ( member_o @ T @ A4 )
=> ( member_o @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_809_ex__in__conv,axiom,
! [A2: set_fm] :
( ( ? [X: fm] : ( member_fm @ X @ A2 ) )
= ( A2 != bot_bot_set_fm ) ) ).
% ex_in_conv
thf(fact_810_ex__in__conv,axiom,
! [A2: set_Pr1822751329126368876m_rule] :
( ( ? [X: produc340336539035504054m_rule] : ( member7231649785386036813m_rule @ X @ A2 ) )
= ( A2 != bot_bo3426410157013319936m_rule ) ) ).
% ex_in_conv
thf(fact_811_ex__in__conv,axiom,
! [A2: set_o] :
( ( ? [X: $o] : ( member_o @ X @ A2 ) )
= ( A2 != bot_bot_set_o ) ) ).
% ex_in_conv
thf(fact_812_subset__eq,axiom,
( ord_less_eq_set_fm
= ( ^ [A4: set_fm,B5: set_fm] :
! [X: fm] :
( ( member_fm @ X @ A4 )
=> ( member_fm @ X @ B5 ) ) ) ) ).
% subset_eq
thf(fact_813_subset__eq,axiom,
( ord_le6390412330253371084m_rule
= ( ^ [A4: set_Pr1822751329126368876m_rule,B5: set_Pr1822751329126368876m_rule] :
! [X: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X @ A4 )
=> ( member7231649785386036813m_rule @ X @ B5 ) ) ) ) ).
% subset_eq
thf(fact_814_subset__eq,axiom,
( ord_less_eq_set_o
= ( ^ [A4: set_o,B5: set_o] :
! [X: $o] :
( ( member_o @ X @ A4 )
=> ( member_o @ X @ B5 ) ) ) ) ).
% subset_eq
thf(fact_815_equals0I,axiom,
! [A2: set_fm] :
( ! [Y3: fm] :
~ ( member_fm @ Y3 @ A2 )
=> ( A2 = bot_bot_set_fm ) ) ).
% equals0I
thf(fact_816_equals0I,axiom,
! [A2: set_Pr1822751329126368876m_rule] :
( ! [Y3: produc340336539035504054m_rule] :
~ ( member7231649785386036813m_rule @ Y3 @ A2 )
=> ( A2 = bot_bo3426410157013319936m_rule ) ) ).
% equals0I
thf(fact_817_equals0I,axiom,
! [A2: set_o] :
( ! [Y3: $o] :
~ ( member_o @ Y3 @ A2 )
=> ( A2 = bot_bot_set_o ) ) ).
% equals0I
thf(fact_818_equals0D,axiom,
! [A2: set_fm,A: fm] :
( ( A2 = bot_bot_set_fm )
=> ~ ( member_fm @ A @ A2 ) ) ).
% equals0D
thf(fact_819_equals0D,axiom,
! [A2: set_Pr1822751329126368876m_rule,A: produc340336539035504054m_rule] :
( ( A2 = bot_bo3426410157013319936m_rule )
=> ~ ( member7231649785386036813m_rule @ A @ A2 ) ) ).
% equals0D
thf(fact_820_equals0D,axiom,
! [A2: set_o,A: $o] :
( ( A2 = bot_bot_set_o )
=> ~ ( member_o @ A @ A2 ) ) ).
% equals0D
thf(fact_821_subsetD,axiom,
! [A2: set_fm,B2: set_fm,C2: fm] :
( ( ord_less_eq_set_fm @ A2 @ B2 )
=> ( ( member_fm @ C2 @ A2 )
=> ( member_fm @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_822_subsetD,axiom,
! [A2: set_Pr1822751329126368876m_rule,B2: set_Pr1822751329126368876m_rule,C2: produc340336539035504054m_rule] :
( ( ord_le6390412330253371084m_rule @ A2 @ B2 )
=> ( ( member7231649785386036813m_rule @ C2 @ A2 )
=> ( member7231649785386036813m_rule @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_823_subsetD,axiom,
! [A2: set_o,B2: set_o,C2: $o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ( member_o @ C2 @ A2 )
=> ( member_o @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_824_in__mono,axiom,
! [A2: set_fm,B2: set_fm,X5: fm] :
( ( ord_less_eq_set_fm @ A2 @ B2 )
=> ( ( member_fm @ X5 @ A2 )
=> ( member_fm @ X5 @ B2 ) ) ) ).
% in_mono
thf(fact_825_in__mono,axiom,
! [A2: set_Pr1822751329126368876m_rule,B2: set_Pr1822751329126368876m_rule,X5: produc340336539035504054m_rule] :
( ( ord_le6390412330253371084m_rule @ A2 @ B2 )
=> ( ( member7231649785386036813m_rule @ X5 @ A2 )
=> ( member7231649785386036813m_rule @ X5 @ B2 ) ) ) ).
% in_mono
thf(fact_826_in__mono,axiom,
! [A2: set_o,B2: set_o,X5: $o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ( member_o @ X5 @ A2 )
=> ( member_o @ X5 @ B2 ) ) ) ).
% in_mono
thf(fact_827_emptyE,axiom,
! [A: fm] :
~ ( member_fm @ A @ bot_bot_set_fm ) ).
% emptyE
thf(fact_828_emptyE,axiom,
! [A: produc340336539035504054m_rule] :
~ ( member7231649785386036813m_rule @ A @ bot_bo3426410157013319936m_rule ) ).
% emptyE
thf(fact_829_emptyE,axiom,
! [A: $o] :
~ ( member_o @ A @ bot_bot_set_o ) ).
% emptyE
thf(fact_830_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_831_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_832_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_833_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_834_dual__order_Otrans,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_835_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_836_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A3: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A3 )
& ( ord_less_eq_nat @ A3 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_837_bot_Oextremum,axiom,
! [A: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A ) ).
% bot.extremum
thf(fact_838_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_839_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B3: nat] :
( ( ord_less_eq_nat @ A5 @ B3 )
=> ( P @ A5 @ B3 ) )
=> ( ! [A5: nat,B3: nat] :
( ( P @ B3 @ A5 )
=> ( P @ A5 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_840_order__trans,axiom,
! [X5: nat,Y5: nat,Z2: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ( ord_less_eq_nat @ Y5 @ Z2 )
=> ( ord_less_eq_nat @ X5 @ Z2 ) ) ) ).
% order_trans
thf(fact_841_order_Otrans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% order.trans
thf(fact_842_order__antisym,axiom,
! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ( ord_less_eq_nat @ Y5 @ X5 )
=> ( X5 = Y5 ) ) ) ).
% order_antisym
thf(fact_843_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_844_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_845_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_846_le__cases3,axiom,
! [X5: nat,Y5: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X5 @ Y5 )
=> ~ ( ord_less_eq_nat @ Y5 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y5 @ X5 )
=> ~ ( ord_less_eq_nat @ X5 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X5 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y5 ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y5 )
=> ~ ( ord_less_eq_nat @ Y5 @ X5 ) )
=> ( ( ( ord_less_eq_nat @ Y5 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X5 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X5 )
=> ~ ( ord_less_eq_nat @ X5 @ Y5 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_847_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_848_InterE,axiom,
! [A2: fm,C: set_set_fm,X2: set_fm] :
( ( member_fm @ A2 @ ( comple1024113957266099930set_fm @ C ) )
=> ( ( member_set_fm @ X2 @ C )
=> ( member_fm @ A2 @ X2 ) ) ) ).
% InterE
thf(fact_849_InterE,axiom,
! [A2: produc340336539035504054m_rule,C: set_se4309406418413213730m_rule,X2: set_Pr1822751329126368876m_rule] :
( ( member7231649785386036813m_rule @ A2 @ ( comple1222484001442006309m_rule @ C ) )
=> ( ( member5802698409304902403m_rule @ X2 @ C )
=> ( member7231649785386036813m_rule @ A2 @ X2 ) ) ) ).
% InterE
thf(fact_850_InterE,axiom,
! [A2: $o,C: set_set_o,X2: set_o] :
( ( member_o @ A2 @ ( comple3063163877087187839_set_o @ C ) )
=> ( ( member_set_o @ X2 @ C )
=> ( member_o @ A2 @ X2 ) ) ) ).
% InterE
thf(fact_851_InterD,axiom,
! [A2: fm,C: set_set_fm,X2: set_fm] :
( ( member_fm @ A2 @ ( comple1024113957266099930set_fm @ C ) )
=> ( ( member_set_fm @ X2 @ C )
=> ( member_fm @ A2 @ X2 ) ) ) ).
% InterD
thf(fact_852_InterD,axiom,
! [A2: produc340336539035504054m_rule,C: set_se4309406418413213730m_rule,X2: set_Pr1822751329126368876m_rule] :
( ( member7231649785386036813m_rule @ A2 @ ( comple1222484001442006309m_rule @ C ) )
=> ( ( member5802698409304902403m_rule @ X2 @ C )
=> ( member7231649785386036813m_rule @ A2 @ X2 ) ) ) ).
% InterD
thf(fact_853_InterD,axiom,
! [A2: $o,C: set_set_o,X2: set_o] :
( ( member_o @ A2 @ ( comple3063163877087187839_set_o @ C ) )
=> ( ( member_set_o @ X2 @ C )
=> ( member_o @ A2 @ X2 ) ) ) ).
% InterD
thf(fact_854_Inf__superset__mono,axiom,
! [B2: set_o,A2: set_o] :
( ( ord_less_eq_set_o @ B2 @ A2 )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A2 ) @ ( complete_Inf_Inf_o @ B2 ) ) ) ).
% Inf_superset_mono
thf(fact_855_Inf__greatest,axiom,
! [A2: set_o,Z2: $o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_less_eq_o @ Z2 @ X3 ) )
=> ( ord_less_eq_o @ Z2 @ ( complete_Inf_Inf_o @ A2 ) ) ) ).
% Inf_greatest
thf(fact_856_Inf__less__eq,axiom,
! [A2: set_o,U: $o] :
( ! [V: $o] :
( ( member_o @ V @ A2 )
=> ( ord_less_eq_o @ V @ U ) )
=> ( ( A2 != bot_bot_set_o )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A2 ) @ U ) ) ) ).
% Inf_less_eq
thf(fact_857_le__Inf__iff,axiom,
! [B: $o,A2: set_o] :
( ( ord_less_eq_o @ B @ ( complete_Inf_Inf_o @ A2 ) )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ord_less_eq_o @ B @ X ) ) ) ) ).
% le_Inf_iff
thf(fact_858_Inf__lower2,axiom,
! [U: $o,A2: set_o,V2: $o] :
( ( member_o @ U @ A2 )
=> ( ( ord_less_eq_o @ U @ V2 )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A2 ) @ V2 ) ) ) ).
% Inf_lower2
thf(fact_859_Inf__lower,axiom,
! [X5: $o,A2: set_o] :
( ( member_o @ X5 @ A2 )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A2 ) @ X5 ) ) ).
% Inf_lower
thf(fact_860_Inf__mono,axiom,
! [B2: set_o,A2: set_o] :
( ! [B3: $o] :
( ( member_o @ B3 @ B2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A2 )
& ( ord_less_eq_o @ X6 @ B3 ) ) )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A2 ) @ ( complete_Inf_Inf_o @ B2 ) ) ) ).
% Inf_mono
thf(fact_861_Inf__eqI,axiom,
! [A2: set_o,X5: $o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ( ord_less_eq_o @ X5 @ I4 ) )
=> ( ! [Y3: $o] :
( ! [I5: $o] :
( ( member_o @ I5 @ A2 )
=> ( ord_less_eq_o @ Y3 @ I5 ) )
=> ( ord_less_eq_o @ Y3 @ X5 ) )
=> ( ( complete_Inf_Inf_o @ A2 )
= X5 ) ) ) ).
% Inf_eqI
thf(fact_862_Union__mono,axiom,
! [A2: set_set_fm,B2: set_set_fm] :
( ( ord_le5844446314808584147set_fm @ A2 @ B2 )
=> ( ord_less_eq_set_fm @ ( comple2134933779557159616set_fm @ A2 ) @ ( comple2134933779557159616set_fm @ B2 ) ) ) ).
% Union_mono
thf(fact_863_Union__mono,axiom,
! [A2: set_set_o,B2: set_set_o] :
( ( ord_le4374716579403074808_set_o @ A2 @ B2 )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ A2 ) @ ( comple90263536869209701_set_o @ B2 ) ) ) ).
% Union_mono
thf(fact_864_Union__mono,axiom,
! [A2: set_se4309406418413213730m_rule,B2: set_se4309406418413213730m_rule] :
( ( ord_le4271096684612802690m_rule @ A2 @ B2 )
=> ( ord_le6390412330253371084m_rule @ ( comple5648871819318716351m_rule @ A2 ) @ ( comple5648871819318716351m_rule @ B2 ) ) ) ).
% Union_mono
thf(fact_865_Union__empty,axiom,
( ( comple2134933779557159616set_fm @ bot_bot_set_set_fm )
= bot_bot_set_fm ) ).
% Union_empty
thf(fact_866_Union__empty,axiom,
( ( comple90263536869209701_set_o @ bot_bot_set_set_o )
= bot_bot_set_o ) ).
% Union_empty
thf(fact_867_Union__empty,axiom,
( ( comple5648871819318716351m_rule @ bot_bo8650941071013544630m_rule )
= bot_bo3426410157013319936m_rule ) ).
% Union_empty
thf(fact_868_INF__le__SUP,axiom,
! [A2: set_Pr1822751329126368876m_rule,F: produc340336539035504054m_rule > set_fm] :
( ( A2 != bot_bo3426410157013319936m_rule )
=> ( ord_less_eq_set_fm @ ( comple1024113957266099930set_fm @ ( image_377115208147135190set_fm @ F @ A2 ) ) @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ F @ A2 ) ) ) ) ).
% INF_le_SUP
thf(fact_869_INF__le__SUP,axiom,
! [A2: set_set_fm,F: set_fm > set_fm] :
( ( A2 != bot_bot_set_set_fm )
=> ( ord_less_eq_set_fm @ ( comple1024113957266099930set_fm @ ( image_set_fm_set_fm @ F @ A2 ) ) @ ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ F @ A2 ) ) ) ) ).
% INF_le_SUP
thf(fact_870_INF__le__SUP,axiom,
! [A2: set_o,F: $o > set_fm] :
( ( A2 != bot_bot_set_o )
=> ( ord_less_eq_set_fm @ ( comple1024113957266099930set_fm @ ( image_o_set_fm @ F @ A2 ) ) @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ A2 ) ) ) ) ).
% INF_le_SUP
thf(fact_871_INF__le__SUP,axiom,
! [A2: set_o,F: $o > $o] :
( ( A2 != bot_bot_set_o )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) ) @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) ) ) ) ).
% INF_le_SUP
thf(fact_872_INF__le__SUP,axiom,
! [A2: set_o,F: $o > set_o] :
( ( A2 != bot_bot_set_o )
=> ( ord_less_eq_set_o @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ F @ A2 ) ) @ ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ A2 ) ) ) ) ).
% INF_le_SUP
thf(fact_873_INF__le__SUP,axiom,
! [A2: set_o,F: $o > set_Pr1822751329126368876m_rule] :
( ( A2 != bot_bot_set_o )
=> ( ord_le6390412330253371084m_rule @ ( comple1222484001442006309m_rule @ ( image_6392605097815222207m_rule @ F @ A2 ) ) @ ( comple5648871819318716351m_rule @ ( image_6392605097815222207m_rule @ F @ A2 ) ) ) ) ).
% INF_le_SUP
thf(fact_874_Collect__subset,axiom,
! [A2: set_fm,P: fm > $o] :
( ord_less_eq_set_fm
@ ( collect_fm
@ ^ [X: fm] :
( ( member_fm @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_875_Collect__subset,axiom,
! [A2: set_Pr1822751329126368876m_rule,P: produc340336539035504054m_rule > $o] :
( ord_le6390412330253371084m_rule
@ ( collec1009276759030335883m_rule
@ ^ [X: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_876_Collect__subset,axiom,
! [A2: set_o,P: $o > $o] :
( ord_less_eq_set_o
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_877_subset__code_I1_J,axiom,
! [Xs: list_P2774625669004309958m_rule,B2: set_Pr1822751329126368876m_rule] :
( ( ord_le6390412330253371084m_rule @ ( set_Pr4534715572506550497m_rule @ Xs ) @ B2 )
= ( ! [X: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X @ ( set_Pr4534715572506550497m_rule @ Xs ) )
=> ( member7231649785386036813m_rule @ X @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_878_subset__code_I1_J,axiom,
! [Xs: list_o,B2: set_o] :
( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ B2 )
= ( ! [X: $o] :
( ( member_o @ X @ ( set_o2 @ Xs ) )
=> ( member_o @ X @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_879_subset__code_I1_J,axiom,
! [Xs: list_fm,B2: set_fm] :
( ( ord_less_eq_set_fm @ ( set_fm2 @ Xs ) @ B2 )
= ( ! [X: fm] :
( ( member_fm @ X @ ( set_fm2 @ Xs ) )
=> ( member_fm @ X @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_880_empty__def,axiom,
( bot_bot_set_o
= ( collect_o
@ ^ [X: $o] : $false ) ) ).
% empty_def
thf(fact_881_Inter__UNIV,axiom,
( ( comple3063163877087187839_set_o @ top_top_set_set_o )
= bot_bot_set_o ) ).
% Inter_UNIV
thf(fact_882_INF__greatest,axiom,
! [A2: set_set_fm,U: set_fm,F: set_fm > set_fm] :
( ! [I4: set_fm] :
( ( member_set_fm @ I4 @ A2 )
=> ( ord_less_eq_set_fm @ U @ ( F @ I4 ) ) )
=> ( ord_less_eq_set_fm @ U @ ( comple1024113957266099930set_fm @ ( image_set_fm_set_fm @ F @ A2 ) ) ) ) ).
% INF_greatest
thf(fact_883_INF__greatest,axiom,
! [A2: set_Pr1822751329126368876m_rule,U: set_fm,F: produc340336539035504054m_rule > set_fm] :
( ! [I4: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ I4 @ A2 )
=> ( ord_less_eq_set_fm @ U @ ( F @ I4 ) ) )
=> ( ord_less_eq_set_fm @ U @ ( comple1024113957266099930set_fm @ ( image_377115208147135190set_fm @ F @ A2 ) ) ) ) ).
% INF_greatest
thf(fact_884_INF__greatest,axiom,
! [A2: set_o,U: set_Pr1822751329126368876m_rule,F: $o > set_Pr1822751329126368876m_rule] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ( ord_le6390412330253371084m_rule @ U @ ( F @ I4 ) ) )
=> ( ord_le6390412330253371084m_rule @ U @ ( comple1222484001442006309m_rule @ ( image_6392605097815222207m_rule @ F @ A2 ) ) ) ) ).
% INF_greatest
thf(fact_885_INF__greatest,axiom,
! [A2: set_o,U: set_fm,F: $o > set_fm] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ( ord_less_eq_set_fm @ U @ ( F @ I4 ) ) )
=> ( ord_less_eq_set_fm @ U @ ( comple1024113957266099930set_fm @ ( image_o_set_fm @ F @ A2 ) ) ) ) ).
% INF_greatest
thf(fact_886_INF__greatest,axiom,
! [A2: set_o,U: set_o,F: $o > set_o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ( ord_less_eq_set_o @ U @ ( F @ I4 ) ) )
=> ( ord_less_eq_set_o @ U @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ F @ A2 ) ) ) ) ).
% INF_greatest
thf(fact_887_INF__greatest,axiom,
! [A2: set_fm,U: $o,F: fm > $o] :
( ! [I4: fm] :
( ( member_fm @ I4 @ A2 )
=> ( ord_less_eq_o @ U @ ( F @ I4 ) ) )
=> ( ord_less_eq_o @ U @ ( complete_Inf_Inf_o @ ( image_fm_o @ F @ A2 ) ) ) ) ).
% INF_greatest
thf(fact_888_INF__greatest,axiom,
! [A2: set_Pr1822751329126368876m_rule,U: $o,F: produc340336539035504054m_rule > $o] :
( ! [I4: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ I4 @ A2 )
=> ( ord_less_eq_o @ U @ ( F @ I4 ) ) )
=> ( ord_less_eq_o @ U @ ( complete_Inf_Inf_o @ ( image_8969478372357302939rule_o @ F @ A2 ) ) ) ) ).
% INF_greatest
thf(fact_889_INF__greatest,axiom,
! [A2: set_o,U: $o,F: $o > $o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ( ord_less_eq_o @ U @ ( F @ I4 ) ) )
=> ( ord_less_eq_o @ U @ ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) ) ) ) ).
% INF_greatest
thf(fact_890_le__INF__iff,axiom,
! [U: set_fm,F: produc340336539035504054m_rule > set_fm,A2: set_Pr1822751329126368876m_rule] :
( ( ord_less_eq_set_fm @ U @ ( comple1024113957266099930set_fm @ ( image_377115208147135190set_fm @ F @ A2 ) ) )
= ( ! [X: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X @ A2 )
=> ( ord_less_eq_set_fm @ U @ ( F @ X ) ) ) ) ) ).
% le_INF_iff
thf(fact_891_le__INF__iff,axiom,
! [U: set_Pr1822751329126368876m_rule,F: $o > set_Pr1822751329126368876m_rule,A2: set_o] :
( ( ord_le6390412330253371084m_rule @ U @ ( comple1222484001442006309m_rule @ ( image_6392605097815222207m_rule @ F @ A2 ) ) )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ord_le6390412330253371084m_rule @ U @ ( F @ X ) ) ) ) ) ).
% le_INF_iff
thf(fact_892_le__INF__iff,axiom,
! [U: set_fm,F: $o > set_fm,A2: set_o] :
( ( ord_less_eq_set_fm @ U @ ( comple1024113957266099930set_fm @ ( image_o_set_fm @ F @ A2 ) ) )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ord_less_eq_set_fm @ U @ ( F @ X ) ) ) ) ) ).
% le_INF_iff
thf(fact_893_le__INF__iff,axiom,
! [U: set_o,F: $o > set_o,A2: set_o] :
( ( ord_less_eq_set_o @ U @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ F @ A2 ) ) )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ord_less_eq_set_o @ U @ ( F @ X ) ) ) ) ) ).
% le_INF_iff
thf(fact_894_le__INF__iff,axiom,
! [U: set_fm,F: set_fm > set_fm,A2: set_set_fm] :
( ( ord_less_eq_set_fm @ U @ ( comple1024113957266099930set_fm @ ( image_set_fm_set_fm @ F @ A2 ) ) )
= ( ! [X: set_fm] :
( ( member_set_fm @ X @ A2 )
=> ( ord_less_eq_set_fm @ U @ ( F @ X ) ) ) ) ) ).
% le_INF_iff
thf(fact_895_le__INF__iff,axiom,
! [U: $o,F: $o > $o,A2: set_o] :
( ( ord_less_eq_o @ U @ ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) ) )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ord_less_eq_o @ U @ ( F @ X ) ) ) ) ) ).
% le_INF_iff
thf(fact_896_INF__lower2,axiom,
! [I2: set_fm,A2: set_set_fm,F: set_fm > set_fm,U: set_fm] :
( ( member_set_fm @ I2 @ A2 )
=> ( ( ord_less_eq_set_fm @ ( F @ I2 ) @ U )
=> ( ord_less_eq_set_fm @ ( comple1024113957266099930set_fm @ ( image_set_fm_set_fm @ F @ A2 ) ) @ U ) ) ) ).
% INF_lower2
thf(fact_897_INF__lower2,axiom,
! [I2: produc340336539035504054m_rule,A2: set_Pr1822751329126368876m_rule,F: produc340336539035504054m_rule > set_fm,U: set_fm] :
( ( member7231649785386036813m_rule @ I2 @ A2 )
=> ( ( ord_less_eq_set_fm @ ( F @ I2 ) @ U )
=> ( ord_less_eq_set_fm @ ( comple1024113957266099930set_fm @ ( image_377115208147135190set_fm @ F @ A2 ) ) @ U ) ) ) ).
% INF_lower2
thf(fact_898_INF__lower2,axiom,
! [I2: $o,A2: set_o,F: $o > set_Pr1822751329126368876m_rule,U: set_Pr1822751329126368876m_rule] :
( ( member_o @ I2 @ A2 )
=> ( ( ord_le6390412330253371084m_rule @ ( F @ I2 ) @ U )
=> ( ord_le6390412330253371084m_rule @ ( comple1222484001442006309m_rule @ ( image_6392605097815222207m_rule @ F @ A2 ) ) @ U ) ) ) ).
% INF_lower2
thf(fact_899_INF__lower2,axiom,
! [I2: $o,A2: set_o,F: $o > set_fm,U: set_fm] :
( ( member_o @ I2 @ A2 )
=> ( ( ord_less_eq_set_fm @ ( F @ I2 ) @ U )
=> ( ord_less_eq_set_fm @ ( comple1024113957266099930set_fm @ ( image_o_set_fm @ F @ A2 ) ) @ U ) ) ) ).
% INF_lower2
thf(fact_900_INF__lower2,axiom,
! [I2: $o,A2: set_o,F: $o > set_o,U: set_o] :
( ( member_o @ I2 @ A2 )
=> ( ( ord_less_eq_set_o @ ( F @ I2 ) @ U )
=> ( ord_less_eq_set_o @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ F @ A2 ) ) @ U ) ) ) ).
% INF_lower2
thf(fact_901_INF__lower2,axiom,
! [I2: fm,A2: set_fm,F: fm > $o,U: $o] :
( ( member_fm @ I2 @ A2 )
=> ( ( ord_less_eq_o @ ( F @ I2 ) @ U )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_fm_o @ F @ A2 ) ) @ U ) ) ) ).
% INF_lower2
thf(fact_902_INF__lower2,axiom,
! [I2: produc340336539035504054m_rule,A2: set_Pr1822751329126368876m_rule,F: produc340336539035504054m_rule > $o,U: $o] :
( ( member7231649785386036813m_rule @ I2 @ A2 )
=> ( ( ord_less_eq_o @ ( F @ I2 ) @ U )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_8969478372357302939rule_o @ F @ A2 ) ) @ U ) ) ) ).
% INF_lower2
thf(fact_903_INF__lower2,axiom,
! [I2: $o,A2: set_o,F: $o > $o,U: $o] :
( ( member_o @ I2 @ A2 )
=> ( ( ord_less_eq_o @ ( F @ I2 ) @ U )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) ) @ U ) ) ) ).
% INF_lower2
thf(fact_904_INF__mono_H,axiom,
! [F: produc340336539035504054m_rule > set_fm,G: produc340336539035504054m_rule > set_fm,A2: set_Pr1822751329126368876m_rule] :
( ! [X3: produc340336539035504054m_rule] : ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq_set_fm @ ( comple1024113957266099930set_fm @ ( image_377115208147135190set_fm @ F @ A2 ) ) @ ( comple1024113957266099930set_fm @ ( image_377115208147135190set_fm @ G @ A2 ) ) ) ) ).
% INF_mono'
thf(fact_905_INF__mono_H,axiom,
! [F: $o > set_Pr1822751329126368876m_rule,G: $o > set_Pr1822751329126368876m_rule,A2: set_o] :
( ! [X3: $o] : ( ord_le6390412330253371084m_rule @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_le6390412330253371084m_rule @ ( comple1222484001442006309m_rule @ ( image_6392605097815222207m_rule @ F @ A2 ) ) @ ( comple1222484001442006309m_rule @ ( image_6392605097815222207m_rule @ G @ A2 ) ) ) ) ).
% INF_mono'
thf(fact_906_INF__mono_H,axiom,
! [F: $o > set_fm,G: $o > set_fm,A2: set_o] :
( ! [X3: $o] : ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq_set_fm @ ( comple1024113957266099930set_fm @ ( image_o_set_fm @ F @ A2 ) ) @ ( comple1024113957266099930set_fm @ ( image_o_set_fm @ G @ A2 ) ) ) ) ).
% INF_mono'
thf(fact_907_INF__mono_H,axiom,
! [F: $o > set_o,G: $o > set_o,A2: set_o] :
( ! [X3: $o] : ( ord_less_eq_set_o @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq_set_o @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ F @ A2 ) ) @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ G @ A2 ) ) ) ) ).
% INF_mono'
thf(fact_908_INF__mono_H,axiom,
! [F: set_fm > set_fm,G: set_fm > set_fm,A2: set_set_fm] :
( ! [X3: set_fm] : ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq_set_fm @ ( comple1024113957266099930set_fm @ ( image_set_fm_set_fm @ F @ A2 ) ) @ ( comple1024113957266099930set_fm @ ( image_set_fm_set_fm @ G @ A2 ) ) ) ) ).
% INF_mono'
thf(fact_909_INF__mono_H,axiom,
! [F: $o > $o,G: $o > $o,A2: set_o] :
( ! [X3: $o] : ( ord_less_eq_o @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) ) @ ( complete_Inf_Inf_o @ ( image_o_o @ G @ A2 ) ) ) ) ).
% INF_mono'
thf(fact_910_INF__lower,axiom,
! [I2: set_fm,A2: set_set_fm,F: set_fm > set_fm] :
( ( member_set_fm @ I2 @ A2 )
=> ( ord_less_eq_set_fm @ ( comple1024113957266099930set_fm @ ( image_set_fm_set_fm @ F @ A2 ) ) @ ( F @ I2 ) ) ) ).
% INF_lower
thf(fact_911_INF__lower,axiom,
! [I2: produc340336539035504054m_rule,A2: set_Pr1822751329126368876m_rule,F: produc340336539035504054m_rule > set_fm] :
( ( member7231649785386036813m_rule @ I2 @ A2 )
=> ( ord_less_eq_set_fm @ ( comple1024113957266099930set_fm @ ( image_377115208147135190set_fm @ F @ A2 ) ) @ ( F @ I2 ) ) ) ).
% INF_lower
thf(fact_912_INF__lower,axiom,
! [I2: $o,A2: set_o,F: $o > set_Pr1822751329126368876m_rule] :
( ( member_o @ I2 @ A2 )
=> ( ord_le6390412330253371084m_rule @ ( comple1222484001442006309m_rule @ ( image_6392605097815222207m_rule @ F @ A2 ) ) @ ( F @ I2 ) ) ) ).
% INF_lower
thf(fact_913_INF__lower,axiom,
! [I2: $o,A2: set_o,F: $o > set_fm] :
( ( member_o @ I2 @ A2 )
=> ( ord_less_eq_set_fm @ ( comple1024113957266099930set_fm @ ( image_o_set_fm @ F @ A2 ) ) @ ( F @ I2 ) ) ) ).
% INF_lower
thf(fact_914_INF__lower,axiom,
! [I2: $o,A2: set_o,F: $o > set_o] :
( ( member_o @ I2 @ A2 )
=> ( ord_less_eq_set_o @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ F @ A2 ) ) @ ( F @ I2 ) ) ) ).
% INF_lower
thf(fact_915_INF__lower,axiom,
! [I2: fm,A2: set_fm,F: fm > $o] :
( ( member_fm @ I2 @ A2 )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_fm_o @ F @ A2 ) ) @ ( F @ I2 ) ) ) ).
% INF_lower
thf(fact_916_INF__lower,axiom,
! [I2: produc340336539035504054m_rule,A2: set_Pr1822751329126368876m_rule,F: produc340336539035504054m_rule > $o] :
( ( member7231649785386036813m_rule @ I2 @ A2 )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_8969478372357302939rule_o @ F @ A2 ) ) @ ( F @ I2 ) ) ) ).
% INF_lower
thf(fact_917_INF__lower,axiom,
! [I2: $o,A2: set_o,F: $o > $o] :
( ( member_o @ I2 @ A2 )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) ) @ ( F @ I2 ) ) ) ).
% INF_lower
thf(fact_918_INF__mono,axiom,
! [B2: set_fm,A2: set_o,F: $o > $o,G: fm > $o] :
( ! [M4: fm] :
( ( member_fm @ M4 @ B2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A2 )
& ( ord_less_eq_o @ ( F @ X6 ) @ ( G @ M4 ) ) ) )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) ) @ ( complete_Inf_Inf_o @ ( image_fm_o @ G @ B2 ) ) ) ) ).
% INF_mono
thf(fact_919_INF__mono,axiom,
! [B2: set_o,A2: set_o,F: $o > $o,G: $o > $o] :
( ! [M4: $o] :
( ( member_o @ M4 @ B2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A2 )
& ( ord_less_eq_o @ ( F @ X6 ) @ ( G @ M4 ) ) ) )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) ) @ ( complete_Inf_Inf_o @ ( image_o_o @ G @ B2 ) ) ) ) ).
% INF_mono
thf(fact_920_INF__mono,axiom,
! [B2: set_fm,A2: set_o,F: $o > set_fm,G: fm > set_fm] :
( ! [M4: fm] :
( ( member_fm @ M4 @ B2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A2 )
& ( ord_less_eq_set_fm @ ( F @ X6 ) @ ( G @ M4 ) ) ) )
=> ( ord_less_eq_set_fm @ ( comple1024113957266099930set_fm @ ( image_o_set_fm @ F @ A2 ) ) @ ( comple1024113957266099930set_fm @ ( image_fm_set_fm @ G @ B2 ) ) ) ) ).
% INF_mono
thf(fact_921_INF__mono,axiom,
! [B2: set_fm,A2: set_o,F: $o > set_o,G: fm > set_o] :
( ! [M4: fm] :
( ( member_fm @ M4 @ B2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A2 )
& ( ord_less_eq_set_o @ ( F @ X6 ) @ ( G @ M4 ) ) ) )
=> ( ord_less_eq_set_o @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ F @ A2 ) ) @ ( comple3063163877087187839_set_o @ ( image_fm_set_o @ G @ B2 ) ) ) ) ).
% INF_mono
thf(fact_922_INF__mono,axiom,
! [B2: set_o,A2: set_o,F: $o > set_fm,G: $o > set_fm] :
( ! [M4: $o] :
( ( member_o @ M4 @ B2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A2 )
& ( ord_less_eq_set_fm @ ( F @ X6 ) @ ( G @ M4 ) ) ) )
=> ( ord_less_eq_set_fm @ ( comple1024113957266099930set_fm @ ( image_o_set_fm @ F @ A2 ) ) @ ( comple1024113957266099930set_fm @ ( image_o_set_fm @ G @ B2 ) ) ) ) ).
% INF_mono
thf(fact_923_INF__mono,axiom,
! [B2: set_o,A2: set_o,F: $o > set_o,G: $o > set_o] :
( ! [M4: $o] :
( ( member_o @ M4 @ B2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A2 )
& ( ord_less_eq_set_o @ ( F @ X6 ) @ ( G @ M4 ) ) ) )
=> ( ord_less_eq_set_o @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ F @ A2 ) ) @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ G @ B2 ) ) ) ) ).
% INF_mono
thf(fact_924_INF__mono,axiom,
! [B2: set_set_fm,A2: set_o,F: $o > set_fm,G: set_fm > set_fm] :
( ! [M4: set_fm] :
( ( member_set_fm @ M4 @ B2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A2 )
& ( ord_less_eq_set_fm @ ( F @ X6 ) @ ( G @ M4 ) ) ) )
=> ( ord_less_eq_set_fm @ ( comple1024113957266099930set_fm @ ( image_o_set_fm @ F @ A2 ) ) @ ( comple1024113957266099930set_fm @ ( image_set_fm_set_fm @ G @ B2 ) ) ) ) ).
% INF_mono
thf(fact_925_INF__mono,axiom,
! [B2: set_fm,A2: set_set_fm,F: set_fm > set_fm,G: fm > set_fm] :
( ! [M4: fm] :
( ( member_fm @ M4 @ B2 )
=> ? [X6: set_fm] :
( ( member_set_fm @ X6 @ A2 )
& ( ord_less_eq_set_fm @ ( F @ X6 ) @ ( G @ M4 ) ) ) )
=> ( ord_less_eq_set_fm @ ( comple1024113957266099930set_fm @ ( image_set_fm_set_fm @ F @ A2 ) ) @ ( comple1024113957266099930set_fm @ ( image_fm_set_fm @ G @ B2 ) ) ) ) ).
% INF_mono
thf(fact_926_INF__mono,axiom,
! [B2: set_o,A2: set_set_fm,F: set_fm > set_fm,G: $o > set_fm] :
( ! [M4: $o] :
( ( member_o @ M4 @ B2 )
=> ? [X6: set_fm] :
( ( member_set_fm @ X6 @ A2 )
& ( ord_less_eq_set_fm @ ( F @ X6 ) @ ( G @ M4 ) ) ) )
=> ( ord_less_eq_set_fm @ ( comple1024113957266099930set_fm @ ( image_set_fm_set_fm @ F @ A2 ) ) @ ( comple1024113957266099930set_fm @ ( image_o_set_fm @ G @ B2 ) ) ) ) ).
% INF_mono
thf(fact_927_INF__mono,axiom,
! [B2: set_set_fm,A2: set_set_fm,F: set_fm > set_fm,G: set_fm > set_fm] :
( ! [M4: set_fm] :
( ( member_set_fm @ M4 @ B2 )
=> ? [X6: set_fm] :
( ( member_set_fm @ X6 @ A2 )
& ( ord_less_eq_set_fm @ ( F @ X6 ) @ ( G @ M4 ) ) ) )
=> ( ord_less_eq_set_fm @ ( comple1024113957266099930set_fm @ ( image_set_fm_set_fm @ F @ A2 ) ) @ ( comple1024113957266099930set_fm @ ( image_set_fm_set_fm @ G @ B2 ) ) ) ) ).
% INF_mono
thf(fact_928_INF__eqI,axiom,
! [A2: set_set_fm,X5: set_fm,F: set_fm > set_fm] :
( ! [I4: set_fm] :
( ( member_set_fm @ I4 @ A2 )
=> ( ord_less_eq_set_fm @ X5 @ ( F @ I4 ) ) )
=> ( ! [Y3: set_fm] :
( ! [I5: set_fm] :
( ( member_set_fm @ I5 @ A2 )
=> ( ord_less_eq_set_fm @ Y3 @ ( F @ I5 ) ) )
=> ( ord_less_eq_set_fm @ Y3 @ X5 ) )
=> ( ( comple1024113957266099930set_fm @ ( image_set_fm_set_fm @ F @ A2 ) )
= X5 ) ) ) ).
% INF_eqI
thf(fact_929_INF__eqI,axiom,
! [A2: set_Pr1822751329126368876m_rule,X5: set_fm,F: produc340336539035504054m_rule > set_fm] :
( ! [I4: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ I4 @ A2 )
=> ( ord_less_eq_set_fm @ X5 @ ( F @ I4 ) ) )
=> ( ! [Y3: set_fm] :
( ! [I5: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ I5 @ A2 )
=> ( ord_less_eq_set_fm @ Y3 @ ( F @ I5 ) ) )
=> ( ord_less_eq_set_fm @ Y3 @ X5 ) )
=> ( ( comple1024113957266099930set_fm @ ( image_377115208147135190set_fm @ F @ A2 ) )
= X5 ) ) ) ).
% INF_eqI
thf(fact_930_INF__eqI,axiom,
! [A2: set_o,X5: set_Pr1822751329126368876m_rule,F: $o > set_Pr1822751329126368876m_rule] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ( ord_le6390412330253371084m_rule @ X5 @ ( F @ I4 ) ) )
=> ( ! [Y3: set_Pr1822751329126368876m_rule] :
( ! [I5: $o] :
( ( member_o @ I5 @ A2 )
=> ( ord_le6390412330253371084m_rule @ Y3 @ ( F @ I5 ) ) )
=> ( ord_le6390412330253371084m_rule @ Y3 @ X5 ) )
=> ( ( comple1222484001442006309m_rule @ ( image_6392605097815222207m_rule @ F @ A2 ) )
= X5 ) ) ) ).
% INF_eqI
thf(fact_931_INF__eqI,axiom,
! [A2: set_o,X5: set_fm,F: $o > set_fm] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ( ord_less_eq_set_fm @ X5 @ ( F @ I4 ) ) )
=> ( ! [Y3: set_fm] :
( ! [I5: $o] :
( ( member_o @ I5 @ A2 )
=> ( ord_less_eq_set_fm @ Y3 @ ( F @ I5 ) ) )
=> ( ord_less_eq_set_fm @ Y3 @ X5 ) )
=> ( ( comple1024113957266099930set_fm @ ( image_o_set_fm @ F @ A2 ) )
= X5 ) ) ) ).
% INF_eqI
thf(fact_932_INF__eqI,axiom,
! [A2: set_o,X5: set_o,F: $o > set_o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ( ord_less_eq_set_o @ X5 @ ( F @ I4 ) ) )
=> ( ! [Y3: set_o] :
( ! [I5: $o] :
( ( member_o @ I5 @ A2 )
=> ( ord_less_eq_set_o @ Y3 @ ( F @ I5 ) ) )
=> ( ord_less_eq_set_o @ Y3 @ X5 ) )
=> ( ( comple3063163877087187839_set_o @ ( image_o_set_o @ F @ A2 ) )
= X5 ) ) ) ).
% INF_eqI
thf(fact_933_INF__eqI,axiom,
! [A2: set_fm,X5: $o,F: fm > $o] :
( ! [I4: fm] :
( ( member_fm @ I4 @ A2 )
=> ( ord_less_eq_o @ X5 @ ( F @ I4 ) ) )
=> ( ! [Y3: $o] :
( ! [I5: fm] :
( ( member_fm @ I5 @ A2 )
=> ( ord_less_eq_o @ Y3 @ ( F @ I5 ) ) )
=> ( ord_less_eq_o @ Y3 @ X5 ) )
=> ( ( complete_Inf_Inf_o @ ( image_fm_o @ F @ A2 ) )
= X5 ) ) ) ).
% INF_eqI
thf(fact_934_INF__eqI,axiom,
! [A2: set_Pr1822751329126368876m_rule,X5: $o,F: produc340336539035504054m_rule > $o] :
( ! [I4: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ I4 @ A2 )
=> ( ord_less_eq_o @ X5 @ ( F @ I4 ) ) )
=> ( ! [Y3: $o] :
( ! [I5: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ I5 @ A2 )
=> ( ord_less_eq_o @ Y3 @ ( F @ I5 ) ) )
=> ( ord_less_eq_o @ Y3 @ X5 ) )
=> ( ( complete_Inf_Inf_o @ ( image_8969478372357302939rule_o @ F @ A2 ) )
= X5 ) ) ) ).
% INF_eqI
thf(fact_935_INF__eqI,axiom,
! [A2: set_o,X5: $o,F: $o > $o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ( ord_less_eq_o @ X5 @ ( F @ I4 ) ) )
=> ( ! [Y3: $o] :
( ! [I5: $o] :
( ( member_o @ I5 @ A2 )
=> ( ord_less_eq_o @ Y3 @ ( F @ I5 ) ) )
=> ( ord_less_eq_o @ Y3 @ X5 ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) )
= X5 ) ) ) ).
% INF_eqI
thf(fact_936_INT__subset__iff,axiom,
! [B2: set_fm,A2: produc340336539035504054m_rule > set_fm,I3: set_Pr1822751329126368876m_rule] :
( ( ord_less_eq_set_fm @ B2 @ ( comple1024113957266099930set_fm @ ( image_377115208147135190set_fm @ A2 @ I3 ) ) )
= ( ! [X: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X @ I3 )
=> ( ord_less_eq_set_fm @ B2 @ ( A2 @ X ) ) ) ) ) ).
% INT_subset_iff
thf(fact_937_INT__subset__iff,axiom,
! [B2: set_Pr1822751329126368876m_rule,A2: $o > set_Pr1822751329126368876m_rule,I3: set_o] :
( ( ord_le6390412330253371084m_rule @ B2 @ ( comple1222484001442006309m_rule @ ( image_6392605097815222207m_rule @ A2 @ I3 ) ) )
= ( ! [X: $o] :
( ( member_o @ X @ I3 )
=> ( ord_le6390412330253371084m_rule @ B2 @ ( A2 @ X ) ) ) ) ) ).
% INT_subset_iff
thf(fact_938_INT__subset__iff,axiom,
! [B2: set_fm,A2: $o > set_fm,I3: set_o] :
( ( ord_less_eq_set_fm @ B2 @ ( comple1024113957266099930set_fm @ ( image_o_set_fm @ A2 @ I3 ) ) )
= ( ! [X: $o] :
( ( member_o @ X @ I3 )
=> ( ord_less_eq_set_fm @ B2 @ ( A2 @ X ) ) ) ) ) ).
% INT_subset_iff
thf(fact_939_INT__subset__iff,axiom,
! [B2: set_o,A2: $o > set_o,I3: set_o] :
( ( ord_less_eq_set_o @ B2 @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ A2 @ I3 ) ) )
= ( ! [X: $o] :
( ( member_o @ X @ I3 )
=> ( ord_less_eq_set_o @ B2 @ ( A2 @ X ) ) ) ) ) ).
% INT_subset_iff
thf(fact_940_INT__subset__iff,axiom,
! [B2: set_fm,A2: set_fm > set_fm,I3: set_set_fm] :
( ( ord_less_eq_set_fm @ B2 @ ( comple1024113957266099930set_fm @ ( image_set_fm_set_fm @ A2 @ I3 ) ) )
= ( ! [X: set_fm] :
( ( member_set_fm @ X @ I3 )
=> ( ord_less_eq_set_fm @ B2 @ ( A2 @ X ) ) ) ) ) ).
% INT_subset_iff
thf(fact_941_INT__anti__mono,axiom,
! [A2: set_set_fm,B2: set_set_fm,F: set_fm > set_fm,G: set_fm > set_fm] :
( ( ord_le5844446314808584147set_fm @ A2 @ B2 )
=> ( ! [X3: set_fm] :
( ( member_set_fm @ X3 @ A2 )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_fm @ ( comple1024113957266099930set_fm @ ( image_set_fm_set_fm @ F @ B2 ) ) @ ( comple1024113957266099930set_fm @ ( image_set_fm_set_fm @ G @ A2 ) ) ) ) ) ).
% INT_anti_mono
thf(fact_942_INT__anti__mono,axiom,
! [A2: set_Pr1822751329126368876m_rule,B2: set_Pr1822751329126368876m_rule,F: produc340336539035504054m_rule > set_fm,G: produc340336539035504054m_rule > set_fm] :
( ( ord_le6390412330253371084m_rule @ A2 @ B2 )
=> ( ! [X3: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X3 @ A2 )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_fm @ ( comple1024113957266099930set_fm @ ( image_377115208147135190set_fm @ F @ B2 ) ) @ ( comple1024113957266099930set_fm @ ( image_377115208147135190set_fm @ G @ A2 ) ) ) ) ) ).
% INT_anti_mono
thf(fact_943_INT__anti__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > set_Pr1822751329126368876m_rule,G: $o > set_Pr1822751329126368876m_rule] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_le6390412330253371084m_rule @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_le6390412330253371084m_rule @ ( comple1222484001442006309m_rule @ ( image_6392605097815222207m_rule @ F @ B2 ) ) @ ( comple1222484001442006309m_rule @ ( image_6392605097815222207m_rule @ G @ A2 ) ) ) ) ) ).
% INT_anti_mono
thf(fact_944_INT__anti__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > set_fm,G: $o > set_fm] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_fm @ ( comple1024113957266099930set_fm @ ( image_o_set_fm @ F @ B2 ) ) @ ( comple1024113957266099930set_fm @ ( image_o_set_fm @ G @ A2 ) ) ) ) ) ).
% INT_anti_mono
thf(fact_945_INT__anti__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > set_o,G: $o > set_o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_less_eq_set_o @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_o @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ F @ B2 ) ) @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ G @ A2 ) ) ) ) ) ).
% INT_anti_mono
thf(fact_946_INT__greatest,axiom,
! [A2: set_set_fm,C: set_fm,B2: set_fm > set_fm] :
( ! [X3: set_fm] :
( ( member_set_fm @ X3 @ A2 )
=> ( ord_less_eq_set_fm @ C @ ( B2 @ X3 ) ) )
=> ( ord_less_eq_set_fm @ C @ ( comple1024113957266099930set_fm @ ( image_set_fm_set_fm @ B2 @ A2 ) ) ) ) ).
% INT_greatest
thf(fact_947_INT__greatest,axiom,
! [A2: set_Pr1822751329126368876m_rule,C: set_fm,B2: produc340336539035504054m_rule > set_fm] :
( ! [X3: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X3 @ A2 )
=> ( ord_less_eq_set_fm @ C @ ( B2 @ X3 ) ) )
=> ( ord_less_eq_set_fm @ C @ ( comple1024113957266099930set_fm @ ( image_377115208147135190set_fm @ B2 @ A2 ) ) ) ) ).
% INT_greatest
thf(fact_948_INT__greatest,axiom,
! [A2: set_o,C: set_Pr1822751329126368876m_rule,B2: $o > set_Pr1822751329126368876m_rule] :
( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_le6390412330253371084m_rule @ C @ ( B2 @ X3 ) ) )
=> ( ord_le6390412330253371084m_rule @ C @ ( comple1222484001442006309m_rule @ ( image_6392605097815222207m_rule @ B2 @ A2 ) ) ) ) ).
% INT_greatest
thf(fact_949_INT__greatest,axiom,
! [A2: set_o,C: set_fm,B2: $o > set_fm] :
( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_less_eq_set_fm @ C @ ( B2 @ X3 ) ) )
=> ( ord_less_eq_set_fm @ C @ ( comple1024113957266099930set_fm @ ( image_o_set_fm @ B2 @ A2 ) ) ) ) ).
% INT_greatest
thf(fact_950_INT__greatest,axiom,
! [A2: set_o,C: set_o,B2: $o > set_o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_less_eq_set_o @ C @ ( B2 @ X3 ) ) )
=> ( ord_less_eq_set_o @ C @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ B2 @ A2 ) ) ) ) ).
% INT_greatest
thf(fact_951_INT__lower,axiom,
! [A: set_fm,A2: set_set_fm,B2: set_fm > set_fm] :
( ( member_set_fm @ A @ A2 )
=> ( ord_less_eq_set_fm @ ( comple1024113957266099930set_fm @ ( image_set_fm_set_fm @ B2 @ A2 ) ) @ ( B2 @ A ) ) ) ).
% INT_lower
thf(fact_952_INT__lower,axiom,
! [A: produc340336539035504054m_rule,A2: set_Pr1822751329126368876m_rule,B2: produc340336539035504054m_rule > set_fm] :
( ( member7231649785386036813m_rule @ A @ A2 )
=> ( ord_less_eq_set_fm @ ( comple1024113957266099930set_fm @ ( image_377115208147135190set_fm @ B2 @ A2 ) ) @ ( B2 @ A ) ) ) ).
% INT_lower
thf(fact_953_INT__lower,axiom,
! [A: $o,A2: set_o,B2: $o > set_Pr1822751329126368876m_rule] :
( ( member_o @ A @ A2 )
=> ( ord_le6390412330253371084m_rule @ ( comple1222484001442006309m_rule @ ( image_6392605097815222207m_rule @ B2 @ A2 ) ) @ ( B2 @ A ) ) ) ).
% INT_lower
thf(fact_954_INT__lower,axiom,
! [A: $o,A2: set_o,B2: $o > set_fm] :
( ( member_o @ A @ A2 )
=> ( ord_less_eq_set_fm @ ( comple1024113957266099930set_fm @ ( image_o_set_fm @ B2 @ A2 ) ) @ ( B2 @ A ) ) ) ).
% INT_lower
thf(fact_955_INT__lower,axiom,
! [A: $o,A2: set_o,B2: $o > set_o] :
( ( member_o @ A @ A2 )
=> ( ord_less_eq_set_o @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ B2 @ A2 ) ) @ ( B2 @ A ) ) ) ).
% INT_lower
thf(fact_956_SUP__eq__iff,axiom,
! [I3: set_fm,C2: $o,F: fm > $o] :
( ( I3 != bot_bot_set_fm )
=> ( ! [I4: fm] :
( ( member_fm @ I4 @ I3 )
=> ( ord_less_eq_o @ C2 @ ( F @ I4 ) ) )
=> ( ( ( complete_Sup_Sup_o @ ( image_fm_o @ F @ I3 ) )
= C2 )
= ( ! [X: fm] :
( ( member_fm @ X @ I3 )
=> ( ( F @ X )
= C2 ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_957_SUP__eq__iff,axiom,
! [I3: set_o,C2: $o,F: $o > $o] :
( ( I3 != bot_bot_set_o )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ I3 )
=> ( ord_less_eq_o @ C2 @ ( F @ I4 ) ) )
=> ( ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ I3 ) )
= C2 )
= ( ! [X: $o] :
( ( member_o @ X @ I3 )
=> ( ( F @ X )
= C2 ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_958_SUP__eq__iff,axiom,
! [I3: set_fm,C2: set_fm,F: fm > set_fm] :
( ( I3 != bot_bot_set_fm )
=> ( ! [I4: fm] :
( ( member_fm @ I4 @ I3 )
=> ( ord_less_eq_set_fm @ C2 @ ( F @ I4 ) ) )
=> ( ( ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ F @ I3 ) )
= C2 )
= ( ! [X: fm] :
( ( member_fm @ X @ I3 )
=> ( ( F @ X )
= C2 ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_959_SUP__eq__iff,axiom,
! [I3: set_o,C2: set_fm,F: $o > set_fm] :
( ( I3 != bot_bot_set_o )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ I3 )
=> ( ord_less_eq_set_fm @ C2 @ ( F @ I4 ) ) )
=> ( ( ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ I3 ) )
= C2 )
= ( ! [X: $o] :
( ( member_o @ X @ I3 )
=> ( ( F @ X )
= C2 ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_960_SUP__eq__iff,axiom,
! [I3: set_fm,C2: set_o,F: fm > set_o] :
( ( I3 != bot_bot_set_fm )
=> ( ! [I4: fm] :
( ( member_fm @ I4 @ I3 )
=> ( ord_less_eq_set_o @ C2 @ ( F @ I4 ) ) )
=> ( ( ( comple90263536869209701_set_o @ ( image_fm_set_o @ F @ I3 ) )
= C2 )
= ( ! [X: fm] :
( ( member_fm @ X @ I3 )
=> ( ( F @ X )
= C2 ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_961_SUP__eq__iff,axiom,
! [I3: set_o,C2: set_o,F: $o > set_o] :
( ( I3 != bot_bot_set_o )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ I3 )
=> ( ord_less_eq_set_o @ C2 @ ( F @ I4 ) ) )
=> ( ( ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ I3 ) )
= C2 )
= ( ! [X: $o] :
( ( member_o @ X @ I3 )
=> ( ( F @ X )
= C2 ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_962_SUP__eq__iff,axiom,
! [I3: set_set_fm,C2: set_fm,F: set_fm > set_fm] :
( ( I3 != bot_bot_set_set_fm )
=> ( ! [I4: set_fm] :
( ( member_set_fm @ I4 @ I3 )
=> ( ord_less_eq_set_fm @ C2 @ ( F @ I4 ) ) )
=> ( ( ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ F @ I3 ) )
= C2 )
= ( ! [X: set_fm] :
( ( member_set_fm @ X @ I3 )
=> ( ( F @ X )
= C2 ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_963_SUP__eq__iff,axiom,
! [I3: set_Pr1822751329126368876m_rule,C2: $o,F: produc340336539035504054m_rule > $o] :
( ( I3 != bot_bo3426410157013319936m_rule )
=> ( ! [I4: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ I4 @ I3 )
=> ( ord_less_eq_o @ C2 @ ( F @ I4 ) ) )
=> ( ( ( complete_Sup_Sup_o @ ( image_8969478372357302939rule_o @ F @ I3 ) )
= C2 )
= ( ! [X: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X @ I3 )
=> ( ( F @ X )
= C2 ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_964_SUP__eq__iff,axiom,
! [I3: set_Pr1822751329126368876m_rule,C2: set_fm,F: produc340336539035504054m_rule > set_fm] :
( ( I3 != bot_bo3426410157013319936m_rule )
=> ( ! [I4: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ I4 @ I3 )
=> ( ord_less_eq_set_fm @ C2 @ ( F @ I4 ) ) )
=> ( ( ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ F @ I3 ) )
= C2 )
= ( ! [X: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X @ I3 )
=> ( ( F @ X )
= C2 ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_965_SUP__eq__iff,axiom,
! [I3: set_Pr1822751329126368876m_rule,C2: set_o,F: produc340336539035504054m_rule > set_o] :
( ( I3 != bot_bo3426410157013319936m_rule )
=> ( ! [I4: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ I4 @ I3 )
=> ( ord_less_eq_set_o @ C2 @ ( F @ I4 ) ) )
=> ( ( ( comple90263536869209701_set_o @ ( image_2025595867862115707_set_o @ F @ I3 ) )
= C2 )
= ( ! [X: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X @ I3 )
=> ( ( F @ X )
= C2 ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_966_bot_Onot__eq__extremum,axiom,
! [A: set_o] :
( ( A != bot_bot_set_o )
= ( ord_less_set_o @ bot_bot_set_o @ A ) ) ).
% bot.not_eq_extremum
thf(fact_967_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_968_bot_Oextremum__strict,axiom,
! [A: set_o] :
~ ( ord_less_set_o @ A @ bot_bot_set_o ) ).
% bot.extremum_strict
thf(fact_969_bot_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_970_INF__cong,axiom,
! [A2: set_set_fm,B2: set_set_fm,C: set_fm > set_fm,D: set_fm > set_fm] :
( ( A2 = B2 )
=> ( ! [X3: set_fm] :
( ( member_set_fm @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( comple1024113957266099930set_fm @ ( image_set_fm_set_fm @ C @ A2 ) )
= ( comple1024113957266099930set_fm @ ( image_set_fm_set_fm @ D @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_971_INF__cong,axiom,
! [A2: set_Pr1822751329126368876m_rule,B2: set_Pr1822751329126368876m_rule,C: produc340336539035504054m_rule > set_fm,D: produc340336539035504054m_rule > set_fm] :
( ( A2 = B2 )
=> ( ! [X3: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( comple1024113957266099930set_fm @ ( image_377115208147135190set_fm @ C @ A2 ) )
= ( comple1024113957266099930set_fm @ ( image_377115208147135190set_fm @ D @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_972_INF__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > set_Pr1822751329126368876m_rule,D: $o > set_Pr1822751329126368876m_rule] :
( ( A2 = B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( comple1222484001442006309m_rule @ ( image_6392605097815222207m_rule @ C @ A2 ) )
= ( comple1222484001442006309m_rule @ ( image_6392605097815222207m_rule @ D @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_973_INF__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > set_fm,D: $o > set_fm] :
( ( A2 = B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( comple1024113957266099930set_fm @ ( image_o_set_fm @ C @ A2 ) )
= ( comple1024113957266099930set_fm @ ( image_o_set_fm @ D @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_974_INF__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > set_o,D: $o > set_o] :
( ( A2 = B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( comple3063163877087187839_set_o @ ( image_o_set_o @ C @ A2 ) )
= ( comple3063163877087187839_set_o @ ( image_o_set_o @ D @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_975_INF__cong,axiom,
! [A2: set_fm,B2: set_fm,C: fm > $o,D: fm > $o] :
( ( A2 = B2 )
=> ( ! [X3: fm] :
( ( member_fm @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_fm_o @ C @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_fm_o @ D @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_976_INF__cong,axiom,
! [A2: set_Pr1822751329126368876m_rule,B2: set_Pr1822751329126368876m_rule,C: produc340336539035504054m_rule > $o,D: produc340336539035504054m_rule > $o] :
( ( A2 = B2 )
=> ( ! [X3: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_8969478372357302939rule_o @ C @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_8969478372357302939rule_o @ D @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_977_INF__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > $o,D: $o > $o] :
( ( A2 = B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ C @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_o_o @ D @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_978_INF__empty,axiom,
! [F: produc340336539035504054m_rule > set_fm] :
( ( comple1024113957266099930set_fm @ ( image_377115208147135190set_fm @ F @ bot_bo3426410157013319936m_rule ) )
= top_top_set_fm ) ).
% INF_empty
thf(fact_979_INF__empty,axiom,
! [F: set_fm > set_fm] :
( ( comple1024113957266099930set_fm @ ( image_set_fm_set_fm @ F @ bot_bot_set_set_fm ) )
= top_top_set_fm ) ).
% INF_empty
thf(fact_980_INF__empty,axiom,
! [F: $o > set_Pr1822751329126368876m_rule] :
( ( comple1222484001442006309m_rule @ ( image_6392605097815222207m_rule @ F @ bot_bot_set_o ) )
= top_to8442024021002298652m_rule ) ).
% INF_empty
thf(fact_981_INF__empty,axiom,
! [F: $o > set_fm] :
( ( comple1024113957266099930set_fm @ ( image_o_set_fm @ F @ bot_bot_set_o ) )
= top_top_set_fm ) ).
% INF_empty
thf(fact_982_INF__empty,axiom,
! [F: $o > set_nat] :
( ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ bot_bot_set_o ) )
= top_top_set_nat ) ).
% INF_empty
thf(fact_983_INF__empty,axiom,
! [F: $o > set_rat] :
( ( comple4298007329820168263et_rat @ ( image_o_set_rat @ F @ bot_bot_set_o ) )
= top_top_set_rat ) ).
% INF_empty
thf(fact_984_INF__empty,axiom,
! [F: $o > set_o] :
( ( comple3063163877087187839_set_o @ ( image_o_set_o @ F @ bot_bot_set_o ) )
= top_top_set_o ) ).
% INF_empty
thf(fact_985_INF__empty,axiom,
! [F: $o > $o] :
( ( complete_Inf_Inf_o @ ( image_o_o @ F @ bot_bot_set_o ) )
= top_top_o ) ).
% INF_empty
thf(fact_986_INF__constant,axiom,
! [A2: set_Pr1822751329126368876m_rule,C2: set_fm] :
( ( ( A2 = bot_bo3426410157013319936m_rule )
=> ( ( comple1024113957266099930set_fm
@ ( image_377115208147135190set_fm
@ ^ [Y: produc340336539035504054m_rule] : C2
@ A2 ) )
= top_top_set_fm ) )
& ( ( A2 != bot_bo3426410157013319936m_rule )
=> ( ( comple1024113957266099930set_fm
@ ( image_377115208147135190set_fm
@ ^ [Y: produc340336539035504054m_rule] : C2
@ A2 ) )
= C2 ) ) ) ).
% INF_constant
thf(fact_987_INF__constant,axiom,
! [A2: set_set_fm,C2: set_fm] :
( ( ( A2 = bot_bot_set_set_fm )
=> ( ( comple1024113957266099930set_fm
@ ( image_set_fm_set_fm
@ ^ [Y: set_fm] : C2
@ A2 ) )
= top_top_set_fm ) )
& ( ( A2 != bot_bot_set_set_fm )
=> ( ( comple1024113957266099930set_fm
@ ( image_set_fm_set_fm
@ ^ [Y: set_fm] : C2
@ A2 ) )
= C2 ) ) ) ).
% INF_constant
thf(fact_988_INF__constant,axiom,
! [A2: set_o,C2: set_Pr1822751329126368876m_rule] :
( ( ( A2 = bot_bot_set_o )
=> ( ( comple1222484001442006309m_rule
@ ( image_6392605097815222207m_rule
@ ^ [Y: $o] : C2
@ A2 ) )
= top_to8442024021002298652m_rule ) )
& ( ( A2 != bot_bot_set_o )
=> ( ( comple1222484001442006309m_rule
@ ( image_6392605097815222207m_rule
@ ^ [Y: $o] : C2
@ A2 ) )
= C2 ) ) ) ).
% INF_constant
thf(fact_989_INF__constant,axiom,
! [A2: set_o,C2: set_fm] :
( ( ( A2 = bot_bot_set_o )
=> ( ( comple1024113957266099930set_fm
@ ( image_o_set_fm
@ ^ [Y: $o] : C2
@ A2 ) )
= top_top_set_fm ) )
& ( ( A2 != bot_bot_set_o )
=> ( ( comple1024113957266099930set_fm
@ ( image_o_set_fm
@ ^ [Y: $o] : C2
@ A2 ) )
= C2 ) ) ) ).
% INF_constant
thf(fact_990_INF__constant,axiom,
! [A2: set_o,C2: set_nat] :
( ( ( A2 = bot_bot_set_o )
=> ( ( comple7806235888213564991et_nat
@ ( image_o_set_nat
@ ^ [Y: $o] : C2
@ A2 ) )
= top_top_set_nat ) )
& ( ( A2 != bot_bot_set_o )
=> ( ( comple7806235888213564991et_nat
@ ( image_o_set_nat
@ ^ [Y: $o] : C2
@ A2 ) )
= C2 ) ) ) ).
% INF_constant
thf(fact_991_INF__constant,axiom,
! [A2: set_o,C2: set_rat] :
( ( ( A2 = bot_bot_set_o )
=> ( ( comple4298007329820168263et_rat
@ ( image_o_set_rat
@ ^ [Y: $o] : C2
@ A2 ) )
= top_top_set_rat ) )
& ( ( A2 != bot_bot_set_o )
=> ( ( comple4298007329820168263et_rat
@ ( image_o_set_rat
@ ^ [Y: $o] : C2
@ A2 ) )
= C2 ) ) ) ).
% INF_constant
thf(fact_992_INF__constant,axiom,
! [A2: set_o,C2: set_o] :
( ( ( A2 = bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [Y: $o] : C2
@ A2 ) )
= top_top_set_o ) )
& ( ( A2 != bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [Y: $o] : C2
@ A2 ) )
= C2 ) ) ) ).
% INF_constant
thf(fact_993_INF__constant,axiom,
! [C2: $o,A2: set_o] :
( ( complete_Inf_Inf_o
@ ( image_o_o
@ ^ [Y: $o] : C2
@ A2 ) )
= ( ( ( A2 = bot_bot_set_o )
=> top_top_o )
& ( ( A2 != bot_bot_set_o )
=> C2 ) ) ) ).
% INF_constant
thf(fact_994_order__le__imp__less__or__eq,axiom,
! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ( ord_less_nat @ X5 @ Y5 )
| ( X5 = Y5 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_995_linorder__le__less__linear,axiom,
! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
| ( ord_less_nat @ Y5 @ X5 ) ) ).
% linorder_le_less_linear
thf(fact_996_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_997_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_998_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_999_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1000_order__less__le__trans,axiom,
! [X5: nat,Y5: nat,Z2: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ( ( ord_less_eq_nat @ Y5 @ Z2 )
=> ( ord_less_nat @ X5 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_1001_order__le__less__trans,axiom,
! [X5: nat,Y5: nat,Z2: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ( ord_less_nat @ Y5 @ Z2 )
=> ( ord_less_nat @ X5 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_1002_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1003_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1004_order__less__imp__le,axiom,
! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ( ord_less_eq_nat @ X5 @ Y5 ) ) ).
% order_less_imp_le
thf(fact_1005_linorder__not__less,axiom,
! [X5: nat,Y5: nat] :
( ( ~ ( ord_less_nat @ X5 @ Y5 ) )
= ( ord_less_eq_nat @ Y5 @ X5 ) ) ).
% linorder_not_less
thf(fact_1006_linorder__not__le,axiom,
! [X5: nat,Y5: nat] :
( ( ~ ( ord_less_eq_nat @ X5 @ Y5 ) )
= ( ord_less_nat @ Y5 @ X5 ) ) ).
% linorder_not_le
thf(fact_1007_order__less__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
& ( X != Y ) ) ) ) ).
% order_less_le
thf(fact_1008_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ) ).
% order_le_less
thf(fact_1009_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1010_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_1011_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B4: nat,A3: nat] :
( ( ord_less_eq_nat @ B4 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1012_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1013_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1014_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B4: nat,A3: nat] :
( ( ord_less_eq_nat @ B4 @ A3 )
& ( A3 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1015_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A3: nat] :
( ( ord_less_nat @ B4 @ A3 )
| ( A3 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1016_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B4: nat] :
( ( ord_less_eq_nat @ A3 @ B4 )
& ~ ( ord_less_eq_nat @ B4 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1017_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_1018_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_1019_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B4: nat] :
( ( ord_less_eq_nat @ A3 @ B4 )
& ( A3 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1020_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B4: nat] :
( ( ord_less_nat @ A3 @ B4 )
| ( A3 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1021_not__le__imp__less,axiom,
! [Y5: nat,X5: nat] :
( ~ ( ord_less_eq_nat @ Y5 @ X5 )
=> ( ord_less_nat @ X5 @ Y5 ) ) ).
% not_le_imp_less
thf(fact_1022_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
& ~ ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% less_le_not_le
thf(fact_1023_antisym__conv2,axiom,
! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ( ~ ( ord_less_nat @ X5 @ Y5 ) )
= ( X5 = Y5 ) ) ) ).
% antisym_conv2
thf(fact_1024_antisym__conv1,axiom,
! [X5: nat,Y5: nat] :
( ~ ( ord_less_nat @ X5 @ Y5 )
=> ( ( ord_less_eq_nat @ X5 @ Y5 )
= ( X5 = Y5 ) ) ) ).
% antisym_conv1
thf(fact_1025_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_1026_leI,axiom,
! [X5: nat,Y5: nat] :
( ~ ( ord_less_nat @ X5 @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X5 ) ) ).
% leI
thf(fact_1027_leD,axiom,
! [Y5: nat,X5: nat] :
( ( ord_less_eq_nat @ Y5 @ X5 )
=> ~ ( ord_less_nat @ X5 @ Y5 ) ) ).
% leD
thf(fact_1028_SUP__constant,axiom,
! [A2: set_Pr1822751329126368876m_rule,C2: set_fm] :
( ( ( A2 = bot_bo3426410157013319936m_rule )
=> ( ( comple2134933779557159616set_fm
@ ( image_377115208147135190set_fm
@ ^ [Y: produc340336539035504054m_rule] : C2
@ A2 ) )
= bot_bot_set_fm ) )
& ( ( A2 != bot_bo3426410157013319936m_rule )
=> ( ( comple2134933779557159616set_fm
@ ( image_377115208147135190set_fm
@ ^ [Y: produc340336539035504054m_rule] : C2
@ A2 ) )
= C2 ) ) ) ).
% SUP_constant
thf(fact_1029_SUP__constant,axiom,
! [A2: set_set_fm,C2: set_fm] :
( ( ( A2 = bot_bot_set_set_fm )
=> ( ( comple2134933779557159616set_fm
@ ( image_set_fm_set_fm
@ ^ [Y: set_fm] : C2
@ A2 ) )
= bot_bot_set_fm ) )
& ( ( A2 != bot_bot_set_set_fm )
=> ( ( comple2134933779557159616set_fm
@ ( image_set_fm_set_fm
@ ^ [Y: set_fm] : C2
@ A2 ) )
= C2 ) ) ) ).
% SUP_constant
thf(fact_1030_SUP__constant,axiom,
! [A2: set_o,C2: set_fm] :
( ( ( A2 = bot_bot_set_o )
=> ( ( comple2134933779557159616set_fm
@ ( image_o_set_fm
@ ^ [Y: $o] : C2
@ A2 ) )
= bot_bot_set_fm ) )
& ( ( A2 != bot_bot_set_o )
=> ( ( comple2134933779557159616set_fm
@ ( image_o_set_fm
@ ^ [Y: $o] : C2
@ A2 ) )
= C2 ) ) ) ).
% SUP_constant
thf(fact_1031_SUP__constant,axiom,
! [C2: $o,A2: set_o] :
( ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [Y: $o] : C2
@ A2 ) )
= ( ( ( A2 = bot_bot_set_o )
=> bot_bot_o )
& ( ( A2 != bot_bot_set_o )
=> C2 ) ) ) ).
% SUP_constant
thf(fact_1032_SUP__constant,axiom,
! [A2: set_o,C2: set_o] :
( ( ( A2 = bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [Y: $o] : C2
@ A2 ) )
= bot_bot_set_o ) )
& ( ( A2 != bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [Y: $o] : C2
@ A2 ) )
= C2 ) ) ) ).
% SUP_constant
thf(fact_1033_SUP__constant,axiom,
! [A2: set_o,C2: set_Pr1822751329126368876m_rule] :
( ( ( A2 = bot_bot_set_o )
=> ( ( comple5648871819318716351m_rule
@ ( image_6392605097815222207m_rule
@ ^ [Y: $o] : C2
@ A2 ) )
= bot_bo3426410157013319936m_rule ) )
& ( ( A2 != bot_bot_set_o )
=> ( ( comple5648871819318716351m_rule
@ ( image_6392605097815222207m_rule
@ ^ [Y: $o] : C2
@ A2 ) )
= C2 ) ) ) ).
% SUP_constant
thf(fact_1034_SUP__empty,axiom,
! [F: produc340336539035504054m_rule > set_fm] :
( ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ F @ bot_bo3426410157013319936m_rule ) )
= bot_bot_set_fm ) ).
% SUP_empty
thf(fact_1035_SUP__empty,axiom,
! [F: set_fm > set_fm] :
( ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ F @ bot_bot_set_set_fm ) )
= bot_bot_set_fm ) ).
% SUP_empty
thf(fact_1036_SUP__empty,axiom,
! [F: $o > set_fm] :
( ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ bot_bot_set_o ) )
= bot_bot_set_fm ) ).
% SUP_empty
thf(fact_1037_SUP__empty,axiom,
! [F: $o > $o] :
( ( complete_Sup_Sup_o @ ( image_o_o @ F @ bot_bot_set_o ) )
= bot_bot_o ) ).
% SUP_empty
thf(fact_1038_SUP__empty,axiom,
! [F: $o > set_o] :
( ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ bot_bot_set_o ) )
= bot_bot_set_o ) ).
% SUP_empty
thf(fact_1039_SUP__empty,axiom,
! [F: $o > set_Pr1822751329126368876m_rule] :
( ( comple5648871819318716351m_rule @ ( image_6392605097815222207m_rule @ F @ bot_bot_set_o ) )
= bot_bo3426410157013319936m_rule ) ).
% SUP_empty
thf(fact_1040_top__greatest,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% top_greatest
thf(fact_1041_top__greatest,axiom,
! [A: set_rat] : ( ord_less_eq_set_rat @ A @ top_top_set_rat ) ).
% top_greatest
thf(fact_1042_top__greatest,axiom,
! [A: set_o] : ( ord_less_eq_set_o @ A @ top_top_set_o ) ).
% top_greatest
thf(fact_1043_top_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A )
= ( A = top_top_set_nat ) ) ).
% top.extremum_unique
thf(fact_1044_top_Oextremum__unique,axiom,
! [A: set_rat] :
( ( ord_less_eq_set_rat @ top_top_set_rat @ A )
= ( A = top_top_set_rat ) ) ).
% top.extremum_unique
thf(fact_1045_top_Oextremum__unique,axiom,
! [A: set_o] :
( ( ord_less_eq_set_o @ top_top_set_o @ A )
= ( A = top_top_set_o ) ) ).
% top.extremum_unique
thf(fact_1046_top_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A )
=> ( A = top_top_set_nat ) ) ).
% top.extremum_uniqueI
thf(fact_1047_top_Oextremum__uniqueI,axiom,
! [A: set_rat] :
( ( ord_less_eq_set_rat @ top_top_set_rat @ A )
=> ( A = top_top_set_rat ) ) ).
% top.extremum_uniqueI
thf(fact_1048_top_Oextremum__uniqueI,axiom,
! [A: set_o] :
( ( ord_less_eq_set_o @ top_top_set_o @ A )
=> ( A = top_top_set_o ) ) ).
% top.extremum_uniqueI
thf(fact_1049_INT__empty,axiom,
! [B2: produc340336539035504054m_rule > set_fm] :
( ( comple1024113957266099930set_fm @ ( image_377115208147135190set_fm @ B2 @ bot_bo3426410157013319936m_rule ) )
= top_top_set_fm ) ).
% INT_empty
thf(fact_1050_INT__empty,axiom,
! [B2: set_fm > set_fm] :
( ( comple1024113957266099930set_fm @ ( image_set_fm_set_fm @ B2 @ bot_bot_set_set_fm ) )
= top_top_set_fm ) ).
% INT_empty
thf(fact_1051_INT__empty,axiom,
! [B2: $o > set_Pr1822751329126368876m_rule] :
( ( comple1222484001442006309m_rule @ ( image_6392605097815222207m_rule @ B2 @ bot_bot_set_o ) )
= top_to8442024021002298652m_rule ) ).
% INT_empty
thf(fact_1052_INT__empty,axiom,
! [B2: $o > set_fm] :
( ( comple1024113957266099930set_fm @ ( image_o_set_fm @ B2 @ bot_bot_set_o ) )
= top_top_set_fm ) ).
% INT_empty
thf(fact_1053_INT__empty,axiom,
! [B2: $o > set_nat] :
( ( comple7806235888213564991et_nat @ ( image_o_set_nat @ B2 @ bot_bot_set_o ) )
= top_top_set_nat ) ).
% INT_empty
thf(fact_1054_INT__empty,axiom,
! [B2: $o > set_rat] :
( ( comple4298007329820168263et_rat @ ( image_o_set_rat @ B2 @ bot_bot_set_o ) )
= top_top_set_rat ) ).
% INT_empty
thf(fact_1055_INT__empty,axiom,
! [B2: $o > set_o] :
( ( comple3063163877087187839_set_o @ ( image_o_set_o @ B2 @ bot_bot_set_o ) )
= top_top_set_o ) ).
% INT_empty
thf(fact_1056_subset__UNIV,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ top_top_set_nat ) ).
% subset_UNIV
thf(fact_1057_subset__UNIV,axiom,
! [A2: set_rat] : ( ord_less_eq_set_rat @ A2 @ top_top_set_rat ) ).
% subset_UNIV
thf(fact_1058_subset__UNIV,axiom,
! [A2: set_o] : ( ord_less_eq_set_o @ A2 @ top_top_set_o ) ).
% subset_UNIV
thf(fact_1059_subset__image__iff,axiom,
! [B2: set_set_fm,F: produc340336539035504054m_rule > set_fm,A2: set_Pr1822751329126368876m_rule] :
( ( ord_le5844446314808584147set_fm @ B2 @ ( image_377115208147135190set_fm @ F @ A2 ) )
= ( ? [AA: set_Pr1822751329126368876m_rule] :
( ( ord_le6390412330253371084m_rule @ AA @ A2 )
& ( B2
= ( image_377115208147135190set_fm @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1060_subset__image__iff,axiom,
! [B2: set_Pr1822751329126368876m_rule,F: nat > produc340336539035504054m_rule,A2: set_nat] :
( ( ord_le6390412330253371084m_rule @ B2 @ ( image_8361736131336225677m_rule @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B2
= ( image_8361736131336225677m_rule @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1061_subset__image__iff,axiom,
! [B2: set_rat,F: nat > rat,A2: set_nat] :
( ( ord_less_eq_set_rat @ B2 @ ( image_nat_rat @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B2
= ( image_nat_rat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1062_subset__image__iff,axiom,
! [B2: set_o,F: $o > $o,A2: set_o] :
( ( ord_less_eq_set_o @ B2 @ ( image_o_o @ F @ A2 ) )
= ( ? [AA: set_o] :
( ( ord_less_eq_set_o @ AA @ A2 )
& ( B2
= ( image_o_o @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1063_subset__image__iff,axiom,
! [B2: set_se4309406418413213730m_rule,F: $o > set_Pr1822751329126368876m_rule,A2: set_o] :
( ( ord_le4271096684612802690m_rule @ B2 @ ( image_6392605097815222207m_rule @ F @ A2 ) )
= ( ? [AA: set_o] :
( ( ord_less_eq_set_o @ AA @ A2 )
& ( B2
= ( image_6392605097815222207m_rule @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1064_subset__image__iff,axiom,
! [B2: set_set_fm,F: $o > set_fm,A2: set_o] :
( ( ord_le5844446314808584147set_fm @ B2 @ ( image_o_set_fm @ F @ A2 ) )
= ( ? [AA: set_o] :
( ( ord_less_eq_set_o @ AA @ A2 )
& ( B2
= ( image_o_set_fm @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1065_subset__image__iff,axiom,
! [B2: set_set_o,F: $o > set_o,A2: set_o] :
( ( ord_le4374716579403074808_set_o @ B2 @ ( image_o_set_o @ F @ A2 ) )
= ( ? [AA: set_o] :
( ( ord_less_eq_set_o @ AA @ A2 )
& ( B2
= ( image_o_set_o @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1066_subset__image__iff,axiom,
! [B2: set_set_fm,F: set_fm > set_fm,A2: set_set_fm] :
( ( ord_le5844446314808584147set_fm @ B2 @ ( image_set_fm_set_fm @ F @ A2 ) )
= ( ? [AA: set_set_fm] :
( ( ord_le5844446314808584147set_fm @ AA @ A2 )
& ( B2
= ( image_set_fm_set_fm @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1067_image__subset__iff,axiom,
! [F: produc340336539035504054m_rule > set_fm,A2: set_Pr1822751329126368876m_rule,B2: set_set_fm] :
( ( ord_le5844446314808584147set_fm @ ( image_377115208147135190set_fm @ F @ A2 ) @ B2 )
= ( ! [X: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X @ A2 )
=> ( member_set_fm @ ( F @ X ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_1068_image__subset__iff,axiom,
! [F: nat > rat,A2: set_nat,B2: set_rat] :
( ( ord_less_eq_set_rat @ ( image_nat_rat @ F @ A2 ) @ B2 )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_rat @ ( F @ X ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_1069_image__subset__iff,axiom,
! [F: $o > set_Pr1822751329126368876m_rule,A2: set_o,B2: set_se4309406418413213730m_rule] :
( ( ord_le4271096684612802690m_rule @ ( image_6392605097815222207m_rule @ F @ A2 ) @ B2 )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( member5802698409304902403m_rule @ ( F @ X ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_1070_image__subset__iff,axiom,
! [F: $o > set_fm,A2: set_o,B2: set_set_fm] :
( ( ord_le5844446314808584147set_fm @ ( image_o_set_fm @ F @ A2 ) @ B2 )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( member_set_fm @ ( F @ X ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_1071_image__subset__iff,axiom,
! [F: $o > set_o,A2: set_o,B2: set_set_o] :
( ( ord_le4374716579403074808_set_o @ ( image_o_set_o @ F @ A2 ) @ B2 )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( member_set_o @ ( F @ X ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_1072_image__subset__iff,axiom,
! [F: set_fm > set_fm,A2: set_set_fm,B2: set_set_fm] :
( ( ord_le5844446314808584147set_fm @ ( image_set_fm_set_fm @ F @ A2 ) @ B2 )
= ( ! [X: set_fm] :
( ( member_set_fm @ X @ A2 )
=> ( member_set_fm @ ( F @ X ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_1073_image__subset__iff,axiom,
! [F: nat > produc340336539035504054m_rule,A2: set_nat,B2: set_Pr1822751329126368876m_rule] :
( ( ord_le6390412330253371084m_rule @ ( image_8361736131336225677m_rule @ F @ A2 ) @ B2 )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member7231649785386036813m_rule @ ( F @ X ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_1074_image__subset__iff,axiom,
! [F: $o > $o,A2: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ ( image_o_o @ F @ A2 ) @ B2 )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( member_o @ ( F @ X ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_1075_subset__imageE,axiom,
! [B2: set_set_fm,F: produc340336539035504054m_rule > set_fm,A2: set_Pr1822751329126368876m_rule] :
( ( ord_le5844446314808584147set_fm @ B2 @ ( image_377115208147135190set_fm @ F @ A2 ) )
=> ~ ! [C3: set_Pr1822751329126368876m_rule] :
( ( ord_le6390412330253371084m_rule @ C3 @ A2 )
=> ( B2
!= ( image_377115208147135190set_fm @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_1076_subset__imageE,axiom,
! [B2: set_Pr1822751329126368876m_rule,F: nat > produc340336539035504054m_rule,A2: set_nat] :
( ( ord_le6390412330253371084m_rule @ B2 @ ( image_8361736131336225677m_rule @ F @ A2 ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A2 )
=> ( B2
!= ( image_8361736131336225677m_rule @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_1077_subset__imageE,axiom,
! [B2: set_rat,F: nat > rat,A2: set_nat] :
( ( ord_less_eq_set_rat @ B2 @ ( image_nat_rat @ F @ A2 ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A2 )
=> ( B2
!= ( image_nat_rat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_1078_subset__imageE,axiom,
! [B2: set_o,F: $o > $o,A2: set_o] :
( ( ord_less_eq_set_o @ B2 @ ( image_o_o @ F @ A2 ) )
=> ~ ! [C3: set_o] :
( ( ord_less_eq_set_o @ C3 @ A2 )
=> ( B2
!= ( image_o_o @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_1079_subset__imageE,axiom,
! [B2: set_se4309406418413213730m_rule,F: $o > set_Pr1822751329126368876m_rule,A2: set_o] :
( ( ord_le4271096684612802690m_rule @ B2 @ ( image_6392605097815222207m_rule @ F @ A2 ) )
=> ~ ! [C3: set_o] :
( ( ord_less_eq_set_o @ C3 @ A2 )
=> ( B2
!= ( image_6392605097815222207m_rule @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_1080_subset__imageE,axiom,
! [B2: set_set_fm,F: $o > set_fm,A2: set_o] :
( ( ord_le5844446314808584147set_fm @ B2 @ ( image_o_set_fm @ F @ A2 ) )
=> ~ ! [C3: set_o] :
( ( ord_less_eq_set_o @ C3 @ A2 )
=> ( B2
!= ( image_o_set_fm @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_1081_subset__imageE,axiom,
! [B2: set_set_o,F: $o > set_o,A2: set_o] :
( ( ord_le4374716579403074808_set_o @ B2 @ ( image_o_set_o @ F @ A2 ) )
=> ~ ! [C3: set_o] :
( ( ord_less_eq_set_o @ C3 @ A2 )
=> ( B2
!= ( image_o_set_o @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_1082_subset__imageE,axiom,
! [B2: set_set_fm,F: set_fm > set_fm,A2: set_set_fm] :
( ( ord_le5844446314808584147set_fm @ B2 @ ( image_set_fm_set_fm @ F @ A2 ) )
=> ~ ! [C3: set_set_fm] :
( ( ord_le5844446314808584147set_fm @ C3 @ A2 )
=> ( B2
!= ( image_set_fm_set_fm @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_1083_image__subsetI,axiom,
! [A2: set_nat,F: nat > rat,B2: set_rat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_rat @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_rat @ ( image_nat_rat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1084_image__subsetI,axiom,
! [A2: set_fm,F: fm > fm,B2: set_fm] :
( ! [X3: fm] :
( ( member_fm @ X3 @ A2 )
=> ( member_fm @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_fm @ ( image_fm_fm @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1085_image__subsetI,axiom,
! [A2: set_fm,F: fm > $o,B2: set_o] :
( ! [X3: fm] :
( ( member_fm @ X3 @ A2 )
=> ( member_o @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_o @ ( image_fm_o @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1086_image__subsetI,axiom,
! [A2: set_o,F: $o > fm,B2: set_fm] :
( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( member_fm @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_fm @ ( image_o_fm @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1087_image__subsetI,axiom,
! [A2: set_o,F: $o > $o,B2: set_o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( member_o @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_o @ ( image_o_o @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1088_image__subsetI,axiom,
! [A2: set_o,F: $o > set_fm,B2: set_set_fm] :
( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( member_set_fm @ ( F @ X3 ) @ B2 ) )
=> ( ord_le5844446314808584147set_fm @ ( image_o_set_fm @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1089_image__subsetI,axiom,
! [A2: set_o,F: $o > set_o,B2: set_set_o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( member_set_o @ ( F @ X3 ) @ B2 ) )
=> ( ord_le4374716579403074808_set_o @ ( image_o_set_o @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1090_image__subsetI,axiom,
! [A2: set_set_fm,F: set_fm > set_fm,B2: set_set_fm] :
( ! [X3: set_fm] :
( ( member_set_fm @ X3 @ A2 )
=> ( member_set_fm @ ( F @ X3 ) @ B2 ) )
=> ( ord_le5844446314808584147set_fm @ ( image_set_fm_set_fm @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1091_image__subsetI,axiom,
! [A2: set_nat,F: nat > produc340336539035504054m_rule,B2: set_Pr1822751329126368876m_rule] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member7231649785386036813m_rule @ ( F @ X3 ) @ B2 ) )
=> ( ord_le6390412330253371084m_rule @ ( image_8361736131336225677m_rule @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1092_image__subsetI,axiom,
! [A2: set_fm,F: fm > produc340336539035504054m_rule,B2: set_Pr1822751329126368876m_rule] :
( ! [X3: fm] :
( ( member_fm @ X3 @ A2 )
=> ( member7231649785386036813m_rule @ ( F @ X3 ) @ B2 ) )
=> ( ord_le6390412330253371084m_rule @ ( image_6079799016930204900m_rule @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1093_image__mono,axiom,
! [A2: set_Pr1822751329126368876m_rule,B2: set_Pr1822751329126368876m_rule,F: produc340336539035504054m_rule > set_fm] :
( ( ord_le6390412330253371084m_rule @ A2 @ B2 )
=> ( ord_le5844446314808584147set_fm @ ( image_377115208147135190set_fm @ F @ A2 ) @ ( image_377115208147135190set_fm @ F @ B2 ) ) ) ).
% image_mono
thf(fact_1094_image__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > produc340336539035504054m_rule] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_le6390412330253371084m_rule @ ( image_8361736131336225677m_rule @ F @ A2 ) @ ( image_8361736131336225677m_rule @ F @ B2 ) ) ) ).
% image_mono
thf(fact_1095_image__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > rat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_rat @ ( image_nat_rat @ F @ A2 ) @ ( image_nat_rat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_1096_image__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > $o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ord_less_eq_set_o @ ( image_o_o @ F @ A2 ) @ ( image_o_o @ F @ B2 ) ) ) ).
% image_mono
thf(fact_1097_image__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > set_Pr1822751329126368876m_rule] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ord_le4271096684612802690m_rule @ ( image_6392605097815222207m_rule @ F @ A2 ) @ ( image_6392605097815222207m_rule @ F @ B2 ) ) ) ).
% image_mono
thf(fact_1098_image__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > set_fm] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ord_le5844446314808584147set_fm @ ( image_o_set_fm @ F @ A2 ) @ ( image_o_set_fm @ F @ B2 ) ) ) ).
% image_mono
thf(fact_1099_image__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > set_o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ord_le4374716579403074808_set_o @ ( image_o_set_o @ F @ A2 ) @ ( image_o_set_o @ F @ B2 ) ) ) ).
% image_mono
thf(fact_1100_image__mono,axiom,
! [A2: set_set_fm,B2: set_set_fm,F: set_fm > set_fm] :
( ( ord_le5844446314808584147set_fm @ A2 @ B2 )
=> ( ord_le5844446314808584147set_fm @ ( image_set_fm_set_fm @ F @ A2 ) @ ( image_set_fm_set_fm @ F @ B2 ) ) ) ).
% image_mono
thf(fact_1101_SUP__subset__mono,axiom,
! [A2: set_fm,B2: set_fm,F: fm > $o,G: fm > $o] :
( ( ord_less_eq_set_fm @ A2 @ B2 )
=> ( ! [X3: fm] :
( ( member_fm @ X3 @ A2 )
=> ( ord_less_eq_o @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_fm_o @ F @ A2 ) ) @ ( complete_Sup_Sup_o @ ( image_fm_o @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1102_SUP__subset__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > $o,G: $o > $o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_less_eq_o @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) ) @ ( complete_Sup_Sup_o @ ( image_o_o @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1103_SUP__subset__mono,axiom,
! [A2: set_fm,B2: set_fm,F: fm > set_fm,G: fm > set_fm] :
( ( ord_less_eq_set_fm @ A2 @ B2 )
=> ( ! [X3: fm] :
( ( member_fm @ X3 @ A2 )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_fm @ ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ F @ A2 ) ) @ ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1104_SUP__subset__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > set_fm,G: $o > set_fm] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_fm @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ A2 ) ) @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1105_SUP__subset__mono,axiom,
! [A2: set_fm,B2: set_fm,F: fm > set_o,G: fm > set_o] :
( ( ord_less_eq_set_fm @ A2 @ B2 )
=> ( ! [X3: fm] :
( ( member_fm @ X3 @ A2 )
=> ( ord_less_eq_set_o @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_fm_set_o @ F @ A2 ) ) @ ( comple90263536869209701_set_o @ ( image_fm_set_o @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1106_SUP__subset__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > set_o,G: $o > set_o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_less_eq_set_o @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ A2 ) ) @ ( comple90263536869209701_set_o @ ( image_o_set_o @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1107_SUP__subset__mono,axiom,
! [A2: set_set_fm,B2: set_set_fm,F: set_fm > set_fm,G: set_fm > set_fm] :
( ( ord_le5844446314808584147set_fm @ A2 @ B2 )
=> ( ! [X3: set_fm] :
( ( member_set_fm @ X3 @ A2 )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_fm @ ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ F @ A2 ) ) @ ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1108_SUP__subset__mono,axiom,
! [A2: set_Pr1822751329126368876m_rule,B2: set_Pr1822751329126368876m_rule,F: produc340336539035504054m_rule > $o,G: produc340336539035504054m_rule > $o] :
( ( ord_le6390412330253371084m_rule @ A2 @ B2 )
=> ( ! [X3: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X3 @ A2 )
=> ( ord_less_eq_o @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_8969478372357302939rule_o @ F @ A2 ) ) @ ( complete_Sup_Sup_o @ ( image_8969478372357302939rule_o @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1109_SUP__subset__mono,axiom,
! [A2: set_Pr1822751329126368876m_rule,B2: set_Pr1822751329126368876m_rule,F: produc340336539035504054m_rule > set_fm,G: produc340336539035504054m_rule > set_fm] :
( ( ord_le6390412330253371084m_rule @ A2 @ B2 )
=> ( ! [X3: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X3 @ A2 )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_fm @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ F @ A2 ) ) @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1110_SUP__subset__mono,axiom,
! [A2: set_Pr1822751329126368876m_rule,B2: set_Pr1822751329126368876m_rule,F: produc340336539035504054m_rule > set_o,G: produc340336539035504054m_rule > set_o] :
( ( ord_le6390412330253371084m_rule @ A2 @ B2 )
=> ( ! [X3: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X3 @ A2 )
=> ( ord_less_eq_set_o @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_2025595867862115707_set_o @ F @ A2 ) ) @ ( comple90263536869209701_set_o @ ( image_2025595867862115707_set_o @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1111_Sup__upper2,axiom,
! [U: set_fm,A2: set_set_fm,V2: set_fm] :
( ( member_set_fm @ U @ A2 )
=> ( ( ord_less_eq_set_fm @ V2 @ U )
=> ( ord_less_eq_set_fm @ V2 @ ( comple2134933779557159616set_fm @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_1112_Sup__upper2,axiom,
! [U: $o,A2: set_o,V2: $o] :
( ( member_o @ U @ A2 )
=> ( ( ord_less_eq_o @ V2 @ U )
=> ( ord_less_eq_o @ V2 @ ( complete_Sup_Sup_o @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_1113_Sup__upper2,axiom,
! [U: set_o,A2: set_set_o,V2: set_o] :
( ( member_set_o @ U @ A2 )
=> ( ( ord_less_eq_set_o @ V2 @ U )
=> ( ord_less_eq_set_o @ V2 @ ( comple90263536869209701_set_o @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_1114_Sup__upper2,axiom,
! [U: set_Pr1822751329126368876m_rule,A2: set_se4309406418413213730m_rule,V2: set_Pr1822751329126368876m_rule] :
( ( member5802698409304902403m_rule @ U @ A2 )
=> ( ( ord_le6390412330253371084m_rule @ V2 @ U )
=> ( ord_le6390412330253371084m_rule @ V2 @ ( comple5648871819318716351m_rule @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_1115_Sup__le__iff,axiom,
! [A2: set_set_fm,B: set_fm] :
( ( ord_less_eq_set_fm @ ( comple2134933779557159616set_fm @ A2 ) @ B )
= ( ! [X: set_fm] :
( ( member_set_fm @ X @ A2 )
=> ( ord_less_eq_set_fm @ X @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_1116_Sup__le__iff,axiom,
! [A2: set_o,B: $o] :
( ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A2 ) @ B )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ord_less_eq_o @ X @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_1117_Sup__le__iff,axiom,
! [A2: set_set_o,B: set_o] :
( ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ A2 ) @ B )
= ( ! [X: set_o] :
( ( member_set_o @ X @ A2 )
=> ( ord_less_eq_set_o @ X @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_1118_Sup__le__iff,axiom,
! [A2: set_se4309406418413213730m_rule,B: set_Pr1822751329126368876m_rule] :
( ( ord_le6390412330253371084m_rule @ ( comple5648871819318716351m_rule @ A2 ) @ B )
= ( ! [X: set_Pr1822751329126368876m_rule] :
( ( member5802698409304902403m_rule @ X @ A2 )
=> ( ord_le6390412330253371084m_rule @ X @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_1119_Sup__upper,axiom,
! [X5: set_fm,A2: set_set_fm] :
( ( member_set_fm @ X5 @ A2 )
=> ( ord_less_eq_set_fm @ X5 @ ( comple2134933779557159616set_fm @ A2 ) ) ) ).
% Sup_upper
thf(fact_1120_Sup__upper,axiom,
! [X5: $o,A2: set_o] :
( ( member_o @ X5 @ A2 )
=> ( ord_less_eq_o @ X5 @ ( complete_Sup_Sup_o @ A2 ) ) ) ).
% Sup_upper
thf(fact_1121_Sup__upper,axiom,
! [X5: set_o,A2: set_set_o] :
( ( member_set_o @ X5 @ A2 )
=> ( ord_less_eq_set_o @ X5 @ ( comple90263536869209701_set_o @ A2 ) ) ) ).
% Sup_upper
thf(fact_1122_Sup__upper,axiom,
! [X5: set_Pr1822751329126368876m_rule,A2: set_se4309406418413213730m_rule] :
( ( member5802698409304902403m_rule @ X5 @ A2 )
=> ( ord_le6390412330253371084m_rule @ X5 @ ( comple5648871819318716351m_rule @ A2 ) ) ) ).
% Sup_upper
thf(fact_1123_Sup__least,axiom,
! [A2: set_set_fm,Z2: set_fm] :
( ! [X3: set_fm] :
( ( member_set_fm @ X3 @ A2 )
=> ( ord_less_eq_set_fm @ X3 @ Z2 ) )
=> ( ord_less_eq_set_fm @ ( comple2134933779557159616set_fm @ A2 ) @ Z2 ) ) ).
% Sup_least
thf(fact_1124_Sup__least,axiom,
! [A2: set_o,Z2: $o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_less_eq_o @ X3 @ Z2 ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A2 ) @ Z2 ) ) ).
% Sup_least
thf(fact_1125_Sup__least,axiom,
! [A2: set_set_o,Z2: set_o] :
( ! [X3: set_o] :
( ( member_set_o @ X3 @ A2 )
=> ( ord_less_eq_set_o @ X3 @ Z2 ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ A2 ) @ Z2 ) ) ).
% Sup_least
thf(fact_1126_Sup__least,axiom,
! [A2: set_se4309406418413213730m_rule,Z2: set_Pr1822751329126368876m_rule] :
( ! [X3: set_Pr1822751329126368876m_rule] :
( ( member5802698409304902403m_rule @ X3 @ A2 )
=> ( ord_le6390412330253371084m_rule @ X3 @ Z2 ) )
=> ( ord_le6390412330253371084m_rule @ ( comple5648871819318716351m_rule @ A2 ) @ Z2 ) ) ).
% Sup_least
thf(fact_1127_Sup__mono,axiom,
! [A2: set_set_fm,B2: set_set_fm] :
( ! [A5: set_fm] :
( ( member_set_fm @ A5 @ A2 )
=> ? [X6: set_fm] :
( ( member_set_fm @ X6 @ B2 )
& ( ord_less_eq_set_fm @ A5 @ X6 ) ) )
=> ( ord_less_eq_set_fm @ ( comple2134933779557159616set_fm @ A2 ) @ ( comple2134933779557159616set_fm @ B2 ) ) ) ).
% Sup_mono
thf(fact_1128_Sup__mono,axiom,
! [A2: set_o,B2: set_o] :
( ! [A5: $o] :
( ( member_o @ A5 @ A2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ B2 )
& ( ord_less_eq_o @ A5 @ X6 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A2 ) @ ( complete_Sup_Sup_o @ B2 ) ) ) ).
% Sup_mono
thf(fact_1129_Sup__mono,axiom,
! [A2: set_set_o,B2: set_set_o] :
( ! [A5: set_o] :
( ( member_set_o @ A5 @ A2 )
=> ? [X6: set_o] :
( ( member_set_o @ X6 @ B2 )
& ( ord_less_eq_set_o @ A5 @ X6 ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ A2 ) @ ( comple90263536869209701_set_o @ B2 ) ) ) ).
% Sup_mono
thf(fact_1130_Sup__mono,axiom,
! [A2: set_se4309406418413213730m_rule,B2: set_se4309406418413213730m_rule] :
( ! [A5: set_Pr1822751329126368876m_rule] :
( ( member5802698409304902403m_rule @ A5 @ A2 )
=> ? [X6: set_Pr1822751329126368876m_rule] :
( ( member5802698409304902403m_rule @ X6 @ B2 )
& ( ord_le6390412330253371084m_rule @ A5 @ X6 ) ) )
=> ( ord_le6390412330253371084m_rule @ ( comple5648871819318716351m_rule @ A2 ) @ ( comple5648871819318716351m_rule @ B2 ) ) ) ).
% Sup_mono
thf(fact_1131_Sup__eqI,axiom,
! [A2: set_set_fm,X5: set_fm] :
( ! [Y3: set_fm] :
( ( member_set_fm @ Y3 @ A2 )
=> ( ord_less_eq_set_fm @ Y3 @ X5 ) )
=> ( ! [Y3: set_fm] :
( ! [Z4: set_fm] :
( ( member_set_fm @ Z4 @ A2 )
=> ( ord_less_eq_set_fm @ Z4 @ Y3 ) )
=> ( ord_less_eq_set_fm @ X5 @ Y3 ) )
=> ( ( comple2134933779557159616set_fm @ A2 )
= X5 ) ) ) ).
% Sup_eqI
thf(fact_1132_Sup__eqI,axiom,
! [A2: set_o,X5: $o] :
( ! [Y3: $o] :
( ( member_o @ Y3 @ A2 )
=> ( ord_less_eq_o @ Y3 @ X5 ) )
=> ( ! [Y3: $o] :
( ! [Z4: $o] :
( ( member_o @ Z4 @ A2 )
=> ( ord_less_eq_o @ Z4 @ Y3 ) )
=> ( ord_less_eq_o @ X5 @ Y3 ) )
=> ( ( complete_Sup_Sup_o @ A2 )
= X5 ) ) ) ).
% Sup_eqI
thf(fact_1133_Sup__eqI,axiom,
! [A2: set_set_o,X5: set_o] :
( ! [Y3: set_o] :
( ( member_set_o @ Y3 @ A2 )
=> ( ord_less_eq_set_o @ Y3 @ X5 ) )
=> ( ! [Y3: set_o] :
( ! [Z4: set_o] :
( ( member_set_o @ Z4 @ A2 )
=> ( ord_less_eq_set_o @ Z4 @ Y3 ) )
=> ( ord_less_eq_set_o @ X5 @ Y3 ) )
=> ( ( comple90263536869209701_set_o @ A2 )
= X5 ) ) ) ).
% Sup_eqI
thf(fact_1134_Sup__eqI,axiom,
! [A2: set_se4309406418413213730m_rule,X5: set_Pr1822751329126368876m_rule] :
( ! [Y3: set_Pr1822751329126368876m_rule] :
( ( member5802698409304902403m_rule @ Y3 @ A2 )
=> ( ord_le6390412330253371084m_rule @ Y3 @ X5 ) )
=> ( ! [Y3: set_Pr1822751329126368876m_rule] :
( ! [Z4: set_Pr1822751329126368876m_rule] :
( ( member5802698409304902403m_rule @ Z4 @ A2 )
=> ( ord_le6390412330253371084m_rule @ Z4 @ Y3 ) )
=> ( ord_le6390412330253371084m_rule @ X5 @ Y3 ) )
=> ( ( comple5648871819318716351m_rule @ A2 )
= X5 ) ) ) ).
% Sup_eqI
thf(fact_1135_empty__not__UNIV,axiom,
bot_bot_set_nat != top_top_set_nat ).
% empty_not_UNIV
thf(fact_1136_empty__not__UNIV,axiom,
bot_bot_set_rat != top_top_set_rat ).
% empty_not_UNIV
thf(fact_1137_empty__not__UNIV,axiom,
bot_bot_set_o != top_top_set_o ).
% empty_not_UNIV
thf(fact_1138_Union__subsetI,axiom,
! [A2: set_set_fm,B2: set_set_fm] :
( ! [X3: set_fm] :
( ( member_set_fm @ X3 @ A2 )
=> ? [Y6: set_fm] :
( ( member_set_fm @ Y6 @ B2 )
& ( ord_less_eq_set_fm @ X3 @ Y6 ) ) )
=> ( ord_less_eq_set_fm @ ( comple2134933779557159616set_fm @ A2 ) @ ( comple2134933779557159616set_fm @ B2 ) ) ) ).
% Union_subsetI
thf(fact_1139_Union__subsetI,axiom,
! [A2: set_set_o,B2: set_set_o] :
( ! [X3: set_o] :
( ( member_set_o @ X3 @ A2 )
=> ? [Y6: set_o] :
( ( member_set_o @ Y6 @ B2 )
& ( ord_less_eq_set_o @ X3 @ Y6 ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ A2 ) @ ( comple90263536869209701_set_o @ B2 ) ) ) ).
% Union_subsetI
thf(fact_1140_Union__subsetI,axiom,
! [A2: set_se4309406418413213730m_rule,B2: set_se4309406418413213730m_rule] :
( ! [X3: set_Pr1822751329126368876m_rule] :
( ( member5802698409304902403m_rule @ X3 @ A2 )
=> ? [Y6: set_Pr1822751329126368876m_rule] :
( ( member5802698409304902403m_rule @ Y6 @ B2 )
& ( ord_le6390412330253371084m_rule @ X3 @ Y6 ) ) )
=> ( ord_le6390412330253371084m_rule @ ( comple5648871819318716351m_rule @ A2 ) @ ( comple5648871819318716351m_rule @ B2 ) ) ) ).
% Union_subsetI
thf(fact_1141_Union__upper,axiom,
! [B2: set_fm,A2: set_set_fm] :
( ( member_set_fm @ B2 @ A2 )
=> ( ord_less_eq_set_fm @ B2 @ ( comple2134933779557159616set_fm @ A2 ) ) ) ).
% Union_upper
thf(fact_1142_Union__upper,axiom,
! [B2: set_o,A2: set_set_o] :
( ( member_set_o @ B2 @ A2 )
=> ( ord_less_eq_set_o @ B2 @ ( comple90263536869209701_set_o @ A2 ) ) ) ).
% Union_upper
thf(fact_1143_Union__upper,axiom,
! [B2: set_Pr1822751329126368876m_rule,A2: set_se4309406418413213730m_rule] :
( ( member5802698409304902403m_rule @ B2 @ A2 )
=> ( ord_le6390412330253371084m_rule @ B2 @ ( comple5648871819318716351m_rule @ A2 ) ) ) ).
% Union_upper
thf(fact_1144_Union__least,axiom,
! [A2: set_set_fm,C: set_fm] :
( ! [X4: set_fm] :
( ( member_set_fm @ X4 @ A2 )
=> ( ord_less_eq_set_fm @ X4 @ C ) )
=> ( ord_less_eq_set_fm @ ( comple2134933779557159616set_fm @ A2 ) @ C ) ) ).
% Union_least
thf(fact_1145_Union__least,axiom,
! [A2: set_set_o,C: set_o] :
( ! [X4: set_o] :
( ( member_set_o @ X4 @ A2 )
=> ( ord_less_eq_set_o @ X4 @ C ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ A2 ) @ C ) ) ).
% Union_least
thf(fact_1146_Union__least,axiom,
! [A2: set_se4309406418413213730m_rule,C: set_Pr1822751329126368876m_rule] :
( ! [X4: set_Pr1822751329126368876m_rule] :
( ( member5802698409304902403m_rule @ X4 @ A2 )
=> ( ord_le6390412330253371084m_rule @ X4 @ C ) )
=> ( ord_le6390412330253371084m_rule @ ( comple5648871819318716351m_rule @ A2 ) @ C ) ) ).
% Union_least
thf(fact_1147_empty__Union__conv,axiom,
! [A2: set_set_fm] :
( ( bot_bot_set_fm
= ( comple2134933779557159616set_fm @ A2 ) )
= ( ! [X: set_fm] :
( ( member_set_fm @ X @ A2 )
=> ( X = bot_bot_set_fm ) ) ) ) ).
% empty_Union_conv
thf(fact_1148_empty__Union__conv,axiom,
! [A2: set_set_o] :
( ( bot_bot_set_o
= ( comple90263536869209701_set_o @ A2 ) )
= ( ! [X: set_o] :
( ( member_set_o @ X @ A2 )
=> ( X = bot_bot_set_o ) ) ) ) ).
% empty_Union_conv
thf(fact_1149_empty__Union__conv,axiom,
! [A2: set_se4309406418413213730m_rule] :
( ( bot_bo3426410157013319936m_rule
= ( comple5648871819318716351m_rule @ A2 ) )
= ( ! [X: set_Pr1822751329126368876m_rule] :
( ( member5802698409304902403m_rule @ X @ A2 )
=> ( X = bot_bo3426410157013319936m_rule ) ) ) ) ).
% empty_Union_conv
thf(fact_1150_Union__empty__conv,axiom,
! [A2: set_set_fm] :
( ( ( comple2134933779557159616set_fm @ A2 )
= bot_bot_set_fm )
= ( ! [X: set_fm] :
( ( member_set_fm @ X @ A2 )
=> ( X = bot_bot_set_fm ) ) ) ) ).
% Union_empty_conv
thf(fact_1151_Union__empty__conv,axiom,
! [A2: set_set_o] :
( ( ( comple90263536869209701_set_o @ A2 )
= bot_bot_set_o )
= ( ! [X: set_o] :
( ( member_set_o @ X @ A2 )
=> ( X = bot_bot_set_o ) ) ) ) ).
% Union_empty_conv
thf(fact_1152_Union__empty__conv,axiom,
! [A2: set_se4309406418413213730m_rule] :
( ( ( comple5648871819318716351m_rule @ A2 )
= bot_bo3426410157013319936m_rule )
= ( ! [X: set_Pr1822751329126368876m_rule] :
( ( member5802698409304902403m_rule @ X @ A2 )
=> ( X = bot_bo3426410157013319936m_rule ) ) ) ) ).
% Union_empty_conv
thf(fact_1153_INF__commute,axiom,
! [F: $o > $o > $o,B2: set_o,A2: set_o] :
( ( complete_Inf_Inf_o
@ ( image_o_o
@ ^ [I: $o] : ( complete_Inf_Inf_o @ ( image_o_o @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( complete_Inf_Inf_o
@ ( image_o_o
@ ^ [J: $o] :
( complete_Inf_Inf_o
@ ( image_o_o
@ ^ [I: $o] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% INF_commute
thf(fact_1154_INF__commute,axiom,
! [F: $o > $o > set_fm,B2: set_o,A2: set_o] :
( ( comple1024113957266099930set_fm
@ ( image_o_set_fm
@ ^ [I: $o] : ( comple1024113957266099930set_fm @ ( image_o_set_fm @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple1024113957266099930set_fm
@ ( image_o_set_fm
@ ^ [J: $o] :
( comple1024113957266099930set_fm
@ ( image_o_set_fm
@ ^ [I: $o] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% INF_commute
thf(fact_1155_INF__commute,axiom,
! [F: $o > $o > set_o,B2: set_o,A2: set_o] :
( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [I: $o] : ( comple3063163877087187839_set_o @ ( image_o_set_o @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [J: $o] :
( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [I: $o] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% INF_commute
thf(fact_1156_INF__commute,axiom,
! [F: $o > set_fm > set_fm,B2: set_set_fm,A2: set_o] :
( ( comple1024113957266099930set_fm
@ ( image_o_set_fm
@ ^ [I: $o] : ( comple1024113957266099930set_fm @ ( image_set_fm_set_fm @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple1024113957266099930set_fm
@ ( image_set_fm_set_fm
@ ^ [J: set_fm] :
( comple1024113957266099930set_fm
@ ( image_o_set_fm
@ ^ [I: $o] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% INF_commute
thf(fact_1157_INF__commute,axiom,
! [F: set_fm > $o > set_fm,B2: set_o,A2: set_set_fm] :
( ( comple1024113957266099930set_fm
@ ( image_set_fm_set_fm
@ ^ [I: set_fm] : ( comple1024113957266099930set_fm @ ( image_o_set_fm @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple1024113957266099930set_fm
@ ( image_o_set_fm
@ ^ [J: $o] :
( comple1024113957266099930set_fm
@ ( image_set_fm_set_fm
@ ^ [I: set_fm] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% INF_commute
thf(fact_1158_INF__commute,axiom,
! [F: set_fm > set_fm > set_fm,B2: set_set_fm,A2: set_set_fm] :
( ( comple1024113957266099930set_fm
@ ( image_set_fm_set_fm
@ ^ [I: set_fm] : ( comple1024113957266099930set_fm @ ( image_set_fm_set_fm @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple1024113957266099930set_fm
@ ( image_set_fm_set_fm
@ ^ [J: set_fm] :
( comple1024113957266099930set_fm
@ ( image_set_fm_set_fm
@ ^ [I: set_fm] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% INF_commute
thf(fact_1159_INF__commute,axiom,
! [F: produc340336539035504054m_rule > $o > set_fm,B2: set_o,A2: set_Pr1822751329126368876m_rule] :
( ( comple1024113957266099930set_fm
@ ( image_377115208147135190set_fm
@ ^ [I: produc340336539035504054m_rule] : ( comple1024113957266099930set_fm @ ( image_o_set_fm @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple1024113957266099930set_fm
@ ( image_o_set_fm
@ ^ [J: $o] :
( comple1024113957266099930set_fm
@ ( image_377115208147135190set_fm
@ ^ [I: produc340336539035504054m_rule] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% INF_commute
thf(fact_1160_INF__commute,axiom,
! [F: $o > $o > set_Pr1822751329126368876m_rule,B2: set_o,A2: set_o] :
( ( comple1222484001442006309m_rule
@ ( image_6392605097815222207m_rule
@ ^ [I: $o] : ( comple1222484001442006309m_rule @ ( image_6392605097815222207m_rule @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple1222484001442006309m_rule
@ ( image_6392605097815222207m_rule
@ ^ [J: $o] :
( comple1222484001442006309m_rule
@ ( image_6392605097815222207m_rule
@ ^ [I: $o] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% INF_commute
thf(fact_1161_INF__commute,axiom,
! [F: $o > produc340336539035504054m_rule > set_fm,B2: set_Pr1822751329126368876m_rule,A2: set_o] :
( ( comple1024113957266099930set_fm
@ ( image_o_set_fm
@ ^ [I: $o] : ( comple1024113957266099930set_fm @ ( image_377115208147135190set_fm @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple1024113957266099930set_fm
@ ( image_377115208147135190set_fm
@ ^ [J: produc340336539035504054m_rule] :
( comple1024113957266099930set_fm
@ ( image_o_set_fm
@ ^ [I: $o] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% INF_commute
thf(fact_1162_INF__commute,axiom,
! [F: produc340336539035504054m_rule > set_fm > set_fm,B2: set_set_fm,A2: set_Pr1822751329126368876m_rule] :
( ( comple1024113957266099930set_fm
@ ( image_377115208147135190set_fm
@ ^ [I: produc340336539035504054m_rule] : ( comple1024113957266099930set_fm @ ( image_set_fm_set_fm @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple1024113957266099930set_fm
@ ( image_set_fm_set_fm
@ ^ [J: set_fm] :
( comple1024113957266099930set_fm
@ ( image_377115208147135190set_fm
@ ^ [I: produc340336539035504054m_rule] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% INF_commute
thf(fact_1163_INT__E,axiom,
! [B: fm,B2: set_fm > set_fm,A2: set_set_fm,A: set_fm] :
( ( member_fm @ B @ ( comple1024113957266099930set_fm @ ( image_set_fm_set_fm @ B2 @ A2 ) ) )
=> ( ~ ( member_fm @ B @ ( B2 @ A ) )
=> ~ ( member_set_fm @ A @ A2 ) ) ) ).
% INT_E
thf(fact_1164_INT__E,axiom,
! [B: fm,B2: fm > set_fm,A2: set_fm,A: fm] :
( ( member_fm @ B @ ( comple1024113957266099930set_fm @ ( image_fm_set_fm @ B2 @ A2 ) ) )
=> ( ~ ( member_fm @ B @ ( B2 @ A ) )
=> ~ ( member_fm @ A @ A2 ) ) ) ).
% INT_E
thf(fact_1165_INT__E,axiom,
! [B: fm,B2: produc340336539035504054m_rule > set_fm,A2: set_Pr1822751329126368876m_rule,A: produc340336539035504054m_rule] :
( ( member_fm @ B @ ( comple1024113957266099930set_fm @ ( image_377115208147135190set_fm @ B2 @ A2 ) ) )
=> ( ~ ( member_fm @ B @ ( B2 @ A ) )
=> ~ ( member7231649785386036813m_rule @ A @ A2 ) ) ) ).
% INT_E
thf(fact_1166_INT__E,axiom,
! [B: fm,B2: $o > set_fm,A2: set_o,A: $o] :
( ( member_fm @ B @ ( comple1024113957266099930set_fm @ ( image_o_set_fm @ B2 @ A2 ) ) )
=> ( ~ ( member_fm @ B @ ( B2 @ A ) )
=> ~ ( member_o @ A @ A2 ) ) ) ).
% INT_E
thf(fact_1167_INT__E,axiom,
! [B: produc340336539035504054m_rule,B2: fm > set_Pr1822751329126368876m_rule,A2: set_fm,A: fm] :
( ( member7231649785386036813m_rule @ B @ ( comple1222484001442006309m_rule @ ( image_7929583154892365210m_rule @ B2 @ A2 ) ) )
=> ( ~ ( member7231649785386036813m_rule @ B @ ( B2 @ A ) )
=> ~ ( member_fm @ A @ A2 ) ) ) ).
% INT_E
thf(fact_1168_INT__E,axiom,
! [B: produc340336539035504054m_rule,B2: produc340336539035504054m_rule > set_Pr1822751329126368876m_rule,A2: set_Pr1822751329126368876m_rule,A: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ B @ ( comple1222484001442006309m_rule @ ( image_2683796897779739177m_rule @ B2 @ A2 ) ) )
=> ( ~ ( member7231649785386036813m_rule @ B @ ( B2 @ A ) )
=> ~ ( member7231649785386036813m_rule @ A @ A2 ) ) ) ).
% INT_E
thf(fact_1169_INT__E,axiom,
! [B: produc340336539035504054m_rule,B2: $o > set_Pr1822751329126368876m_rule,A2: set_o,A: $o] :
( ( member7231649785386036813m_rule @ B @ ( comple1222484001442006309m_rule @ ( image_6392605097815222207m_rule @ B2 @ A2 ) ) )
=> ( ~ ( member7231649785386036813m_rule @ B @ ( B2 @ A ) )
=> ~ ( member_o @ A @ A2 ) ) ) ).
% INT_E
thf(fact_1170_INT__E,axiom,
! [B: $o,B2: fm > set_o,A2: set_fm,A: fm] :
( ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_fm_set_o @ B2 @ A2 ) ) )
=> ( ~ ( member_o @ B @ ( B2 @ A ) )
=> ~ ( member_fm @ A @ A2 ) ) ) ).
% INT_E
thf(fact_1171_INT__E,axiom,
! [B: $o,B2: produc340336539035504054m_rule > set_o,A2: set_Pr1822751329126368876m_rule,A: produc340336539035504054m_rule] :
( ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_2025595867862115707_set_o @ B2 @ A2 ) ) )
=> ( ~ ( member_o @ B @ ( B2 @ A ) )
=> ~ ( member7231649785386036813m_rule @ A @ A2 ) ) ) ).
% INT_E
thf(fact_1172_INT__E,axiom,
! [B: $o,B2: $o > set_o,A2: set_o,A: $o] :
( ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
=> ( ~ ( member_o @ B @ ( B2 @ A ) )
=> ~ ( member_o @ A @ A2 ) ) ) ).
% INT_E
thf(fact_1173_INT__D,axiom,
! [B: fm,B2: set_fm > set_fm,A2: set_set_fm,A: set_fm] :
( ( member_fm @ B @ ( comple1024113957266099930set_fm @ ( image_set_fm_set_fm @ B2 @ A2 ) ) )
=> ( ( member_set_fm @ A @ A2 )
=> ( member_fm @ B @ ( B2 @ A ) ) ) ) ).
% INT_D
thf(fact_1174_INT__D,axiom,
! [B: fm,B2: fm > set_fm,A2: set_fm,A: fm] :
( ( member_fm @ B @ ( comple1024113957266099930set_fm @ ( image_fm_set_fm @ B2 @ A2 ) ) )
=> ( ( member_fm @ A @ A2 )
=> ( member_fm @ B @ ( B2 @ A ) ) ) ) ).
% INT_D
thf(fact_1175_INT__D,axiom,
! [B: fm,B2: produc340336539035504054m_rule > set_fm,A2: set_Pr1822751329126368876m_rule,A: produc340336539035504054m_rule] :
( ( member_fm @ B @ ( comple1024113957266099930set_fm @ ( image_377115208147135190set_fm @ B2 @ A2 ) ) )
=> ( ( member7231649785386036813m_rule @ A @ A2 )
=> ( member_fm @ B @ ( B2 @ A ) ) ) ) ).
% INT_D
thf(fact_1176_INT__D,axiom,
! [B: fm,B2: $o > set_fm,A2: set_o,A: $o] :
( ( member_fm @ B @ ( comple1024113957266099930set_fm @ ( image_o_set_fm @ B2 @ A2 ) ) )
=> ( ( member_o @ A @ A2 )
=> ( member_fm @ B @ ( B2 @ A ) ) ) ) ).
% INT_D
thf(fact_1177_INT__D,axiom,
! [B: produc340336539035504054m_rule,B2: fm > set_Pr1822751329126368876m_rule,A2: set_fm,A: fm] :
( ( member7231649785386036813m_rule @ B @ ( comple1222484001442006309m_rule @ ( image_7929583154892365210m_rule @ B2 @ A2 ) ) )
=> ( ( member_fm @ A @ A2 )
=> ( member7231649785386036813m_rule @ B @ ( B2 @ A ) ) ) ) ).
% INT_D
thf(fact_1178_INT__D,axiom,
! [B: produc340336539035504054m_rule,B2: produc340336539035504054m_rule > set_Pr1822751329126368876m_rule,A2: set_Pr1822751329126368876m_rule,A: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ B @ ( comple1222484001442006309m_rule @ ( image_2683796897779739177m_rule @ B2 @ A2 ) ) )
=> ( ( member7231649785386036813m_rule @ A @ A2 )
=> ( member7231649785386036813m_rule @ B @ ( B2 @ A ) ) ) ) ).
% INT_D
thf(fact_1179_INT__D,axiom,
! [B: produc340336539035504054m_rule,B2: $o > set_Pr1822751329126368876m_rule,A2: set_o,A: $o] :
( ( member7231649785386036813m_rule @ B @ ( comple1222484001442006309m_rule @ ( image_6392605097815222207m_rule @ B2 @ A2 ) ) )
=> ( ( member_o @ A @ A2 )
=> ( member7231649785386036813m_rule @ B @ ( B2 @ A ) ) ) ) ).
% INT_D
thf(fact_1180_INT__D,axiom,
! [B: $o,B2: fm > set_o,A2: set_fm,A: fm] :
( ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_fm_set_o @ B2 @ A2 ) ) )
=> ( ( member_fm @ A @ A2 )
=> ( member_o @ B @ ( B2 @ A ) ) ) ) ).
% INT_D
thf(fact_1181_INT__D,axiom,
! [B: $o,B2: produc340336539035504054m_rule > set_o,A2: set_Pr1822751329126368876m_rule,A: produc340336539035504054m_rule] :
( ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_2025595867862115707_set_o @ B2 @ A2 ) ) )
=> ( ( member7231649785386036813m_rule @ A @ A2 )
=> ( member_o @ B @ ( B2 @ A ) ) ) ) ).
% INT_D
thf(fact_1182_INT__D,axiom,
! [B: $o,B2: $o > set_o,A2: set_o,A: $o] :
( ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
=> ( ( member_o @ A @ A2 )
=> ( member_o @ B @ ( B2 @ A ) ) ) ) ).
% INT_D
thf(fact_1183_times__eq__iff,axiom,
! [A2: set_o,B2: set_o,C: set_o,D: set_o] :
( ( ( product_Sigma_o_o @ A2
@ ^ [Uu: $o] : B2 )
= ( product_Sigma_o_o @ C
@ ^ [Uu: $o] : D ) )
= ( ( ( A2 = C )
& ( B2 = D ) )
| ( ( ( A2 = bot_bot_set_o )
| ( B2 = bot_bot_set_o ) )
& ( ( C = bot_bot_set_o )
| ( D = bot_bot_set_o ) ) ) ) ) ).
% times_eq_iff
thf(fact_1184_range__subsetD,axiom,
! [F: nat > rat,B2: set_rat,I2: nat] :
( ( ord_less_eq_set_rat @ ( image_nat_rat @ F @ top_top_set_nat ) @ B2 )
=> ( member_rat @ ( F @ I2 ) @ B2 ) ) ).
% range_subsetD
thf(fact_1185_range__subsetD,axiom,
! [F: nat > fm,B2: set_fm,I2: nat] :
( ( ord_less_eq_set_fm @ ( image_nat_fm @ F @ top_top_set_nat ) @ B2 )
=> ( member_fm @ ( F @ I2 ) @ B2 ) ) ).
% range_subsetD
thf(fact_1186_range__subsetD,axiom,
! [F: nat > $o,B2: set_o,I2: nat] :
( ( ord_less_eq_set_o @ ( image_nat_o @ F @ top_top_set_nat ) @ B2 )
=> ( member_o @ ( F @ I2 ) @ B2 ) ) ).
% range_subsetD
thf(fact_1187_range__subsetD,axiom,
! [F: rat > fm,B2: set_fm,I2: rat] :
( ( ord_less_eq_set_fm @ ( image_rat_fm @ F @ top_top_set_rat ) @ B2 )
=> ( member_fm @ ( F @ I2 ) @ B2 ) ) ).
% range_subsetD
thf(fact_1188_range__subsetD,axiom,
! [F: rat > $o,B2: set_o,I2: rat] :
( ( ord_less_eq_set_o @ ( image_rat_o @ F @ top_top_set_rat ) @ B2 )
=> ( member_o @ ( F @ I2 ) @ B2 ) ) ).
% range_subsetD
thf(fact_1189_range__subsetD,axiom,
! [F: $o > fm,B2: set_fm,I2: $o] :
( ( ord_less_eq_set_fm @ ( image_o_fm @ F @ top_top_set_o ) @ B2 )
=> ( member_fm @ ( F @ I2 ) @ B2 ) ) ).
% range_subsetD
thf(fact_1190_range__subsetD,axiom,
! [F: $o > $o,B2: set_o,I2: $o] :
( ( ord_less_eq_set_o @ ( image_o_o @ F @ top_top_set_o ) @ B2 )
=> ( member_o @ ( F @ I2 ) @ B2 ) ) ).
% range_subsetD
thf(fact_1191_range__subsetD,axiom,
! [F: $o > set_fm,B2: set_set_fm,I2: $o] :
( ( ord_le5844446314808584147set_fm @ ( image_o_set_fm @ F @ top_top_set_o ) @ B2 )
=> ( member_set_fm @ ( F @ I2 ) @ B2 ) ) ).
% range_subsetD
thf(fact_1192_range__subsetD,axiom,
! [F: $o > set_o,B2: set_set_o,I2: $o] :
( ( ord_le4374716579403074808_set_o @ ( image_o_set_o @ F @ top_top_set_o ) @ B2 )
=> ( member_set_o @ ( F @ I2 ) @ B2 ) ) ).
% range_subsetD
thf(fact_1193_range__subsetD,axiom,
! [F: set_fm > set_fm,B2: set_set_fm,I2: set_fm] :
( ( ord_le5844446314808584147set_fm @ ( image_set_fm_set_fm @ F @ top_top_set_set_fm ) @ B2 )
=> ( member_set_fm @ ( F @ I2 ) @ B2 ) ) ).
% range_subsetD
thf(fact_1194_SUP__eq,axiom,
! [A2: set_fm,B2: set_fm,F: fm > $o,G: fm > $o] :
( ! [I4: fm] :
( ( member_fm @ I4 @ A2 )
=> ? [X6: fm] :
( ( member_fm @ X6 @ B2 )
& ( ord_less_eq_o @ ( F @ I4 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: fm] :
( ( member_fm @ J2 @ B2 )
=> ? [X6: fm] :
( ( member_fm @ X6 @ A2 )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_fm_o @ F @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_fm_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_1195_SUP__eq,axiom,
! [A2: set_fm,B2: set_o,F: fm > $o,G: $o > $o] :
( ! [I4: fm] :
( ( member_fm @ I4 @ A2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ B2 )
& ( ord_less_eq_o @ ( F @ I4 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B2 )
=> ? [X6: fm] :
( ( member_fm @ X6 @ A2 )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_fm_o @ F @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_1196_SUP__eq,axiom,
! [A2: set_o,B2: set_fm,F: $o > $o,G: fm > $o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ? [X6: fm] :
( ( member_fm @ X6 @ B2 )
& ( ord_less_eq_o @ ( F @ I4 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: fm] :
( ( member_fm @ J2 @ B2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A2 )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_fm_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_1197_SUP__eq,axiom,
! [A2: set_o,B2: set_o,F: $o > $o,G: $o > $o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ B2 )
& ( ord_less_eq_o @ ( F @ I4 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A2 )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_1198_SUP__eq,axiom,
! [A2: set_fm,B2: set_fm,F: fm > set_fm,G: fm > set_fm] :
( ! [I4: fm] :
( ( member_fm @ I4 @ A2 )
=> ? [X6: fm] :
( ( member_fm @ X6 @ B2 )
& ( ord_less_eq_set_fm @ ( F @ I4 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: fm] :
( ( member_fm @ J2 @ B2 )
=> ? [X6: fm] :
( ( member_fm @ X6 @ A2 )
& ( ord_less_eq_set_fm @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
=> ( ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ F @ A2 ) )
= ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_1199_SUP__eq,axiom,
! [A2: set_fm,B2: set_o,F: fm > set_fm,G: $o > set_fm] :
( ! [I4: fm] :
( ( member_fm @ I4 @ A2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ B2 )
& ( ord_less_eq_set_fm @ ( F @ I4 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B2 )
=> ? [X6: fm] :
( ( member_fm @ X6 @ A2 )
& ( ord_less_eq_set_fm @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
=> ( ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ F @ A2 ) )
= ( comple2134933779557159616set_fm @ ( image_o_set_fm @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_1200_SUP__eq,axiom,
! [A2: set_o,B2: set_fm,F: $o > set_fm,G: fm > set_fm] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ? [X6: fm] :
( ( member_fm @ X6 @ B2 )
& ( ord_less_eq_set_fm @ ( F @ I4 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: fm] :
( ( member_fm @ J2 @ B2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A2 )
& ( ord_less_eq_set_fm @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
=> ( ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ A2 ) )
= ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_1201_SUP__eq,axiom,
! [A2: set_o,B2: set_o,F: $o > set_fm,G: $o > set_fm] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ B2 )
& ( ord_less_eq_set_fm @ ( F @ I4 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A2 )
& ( ord_less_eq_set_fm @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
=> ( ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ A2 ) )
= ( comple2134933779557159616set_fm @ ( image_o_set_fm @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_1202_SUP__eq,axiom,
! [A2: set_fm,B2: set_fm,F: fm > set_o,G: fm > set_o] :
( ! [I4: fm] :
( ( member_fm @ I4 @ A2 )
=> ? [X6: fm] :
( ( member_fm @ X6 @ B2 )
& ( ord_less_eq_set_o @ ( F @ I4 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: fm] :
( ( member_fm @ J2 @ B2 )
=> ? [X6: fm] :
( ( member_fm @ X6 @ A2 )
& ( ord_less_eq_set_o @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
=> ( ( comple90263536869209701_set_o @ ( image_fm_set_o @ F @ A2 ) )
= ( comple90263536869209701_set_o @ ( image_fm_set_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_1203_SUP__eq,axiom,
! [A2: set_fm,B2: set_o,F: fm > set_o,G: $o > set_o] :
( ! [I4: fm] :
( ( member_fm @ I4 @ A2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ B2 )
& ( ord_less_eq_set_o @ ( F @ I4 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B2 )
=> ? [X6: fm] :
( ( member_fm @ X6 @ A2 )
& ( ord_less_eq_set_o @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
=> ( ( comple90263536869209701_set_o @ ( image_fm_set_o @ F @ A2 ) )
= ( comple90263536869209701_set_o @ ( image_o_set_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_1204_SUP__eq__const,axiom,
! [I3: set_fm,F: fm > $o,X5: $o] :
( ( I3 != bot_bot_set_fm )
=> ( ! [I4: fm] :
( ( member_fm @ I4 @ I3 )
=> ( ( F @ I4 )
= X5 ) )
=> ( ( complete_Sup_Sup_o @ ( image_fm_o @ F @ I3 ) )
= X5 ) ) ) ).
% SUP_eq_const
thf(fact_1205_SUP__eq__const,axiom,
! [I3: set_o,F: $o > $o,X5: $o] :
( ( I3 != bot_bot_set_o )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ I3 )
=> ( ( F @ I4 )
= X5 ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ I3 ) )
= X5 ) ) ) ).
% SUP_eq_const
thf(fact_1206_SUP__eq__const,axiom,
! [I3: set_fm,F: fm > set_fm,X5: set_fm] :
( ( I3 != bot_bot_set_fm )
=> ( ! [I4: fm] :
( ( member_fm @ I4 @ I3 )
=> ( ( F @ I4 )
= X5 ) )
=> ( ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ F @ I3 ) )
= X5 ) ) ) ).
% SUP_eq_const
thf(fact_1207_SUP__eq__const,axiom,
! [I3: set_o,F: $o > set_fm,X5: set_fm] :
( ( I3 != bot_bot_set_o )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ I3 )
=> ( ( F @ I4 )
= X5 ) )
=> ( ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ I3 ) )
= X5 ) ) ) ).
% SUP_eq_const
thf(fact_1208_SUP__eq__const,axiom,
! [I3: set_fm,F: fm > set_o,X5: set_o] :
( ( I3 != bot_bot_set_fm )
=> ( ! [I4: fm] :
( ( member_fm @ I4 @ I3 )
=> ( ( F @ I4 )
= X5 ) )
=> ( ( comple90263536869209701_set_o @ ( image_fm_set_o @ F @ I3 ) )
= X5 ) ) ) ).
% SUP_eq_const
thf(fact_1209_SUP__eq__const,axiom,
! [I3: set_o,F: $o > set_o,X5: set_o] :
( ( I3 != bot_bot_set_o )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ I3 )
=> ( ( F @ I4 )
= X5 ) )
=> ( ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ I3 ) )
= X5 ) ) ) ).
% SUP_eq_const
thf(fact_1210_SUP__eq__const,axiom,
! [I3: set_set_fm,F: set_fm > set_fm,X5: set_fm] :
( ( I3 != bot_bot_set_set_fm )
=> ( ! [I4: set_fm] :
( ( member_set_fm @ I4 @ I3 )
=> ( ( F @ I4 )
= X5 ) )
=> ( ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ F @ I3 ) )
= X5 ) ) ) ).
% SUP_eq_const
thf(fact_1211_SUP__eq__const,axiom,
! [I3: set_Pr1822751329126368876m_rule,F: produc340336539035504054m_rule > $o,X5: $o] :
( ( I3 != bot_bo3426410157013319936m_rule )
=> ( ! [I4: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ I4 @ I3 )
=> ( ( F @ I4 )
= X5 ) )
=> ( ( complete_Sup_Sup_o @ ( image_8969478372357302939rule_o @ F @ I3 ) )
= X5 ) ) ) ).
% SUP_eq_const
thf(fact_1212_SUP__eq__const,axiom,
! [I3: set_Pr1822751329126368876m_rule,F: produc340336539035504054m_rule > set_fm,X5: set_fm] :
( ( I3 != bot_bo3426410157013319936m_rule )
=> ( ! [I4: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ I4 @ I3 )
=> ( ( F @ I4 )
= X5 ) )
=> ( ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ F @ I3 ) )
= X5 ) ) ) ).
% SUP_eq_const
thf(fact_1213_SUP__eq__const,axiom,
! [I3: set_Pr1822751329126368876m_rule,F: produc340336539035504054m_rule > set_o,X5: set_o] :
( ( I3 != bot_bo3426410157013319936m_rule )
=> ( ! [I4: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ I4 @ I3 )
=> ( ( F @ I4 )
= X5 ) )
=> ( ( comple90263536869209701_set_o @ ( image_2025595867862115707_set_o @ F @ I3 ) )
= X5 ) ) ) ).
% SUP_eq_const
thf(fact_1214_less__INF__D,axiom,
! [Y5: set_fm,F: set_fm > set_fm,A2: set_set_fm,I2: set_fm] :
( ( ord_less_set_fm @ Y5 @ ( comple1024113957266099930set_fm @ ( image_set_fm_set_fm @ F @ A2 ) ) )
=> ( ( member_set_fm @ I2 @ A2 )
=> ( ord_less_set_fm @ Y5 @ ( F @ I2 ) ) ) ) ).
% less_INF_D
thf(fact_1215_less__INF__D,axiom,
! [Y5: set_fm,F: produc340336539035504054m_rule > set_fm,A2: set_Pr1822751329126368876m_rule,I2: produc340336539035504054m_rule] :
( ( ord_less_set_fm @ Y5 @ ( comple1024113957266099930set_fm @ ( image_377115208147135190set_fm @ F @ A2 ) ) )
=> ( ( member7231649785386036813m_rule @ I2 @ A2 )
=> ( ord_less_set_fm @ Y5 @ ( F @ I2 ) ) ) ) ).
% less_INF_D
thf(fact_1216_less__INF__D,axiom,
! [Y5: set_Pr1822751329126368876m_rule,F: $o > set_Pr1822751329126368876m_rule,A2: set_o,I2: $o] :
( ( ord_le4093866177137961152m_rule @ Y5 @ ( comple1222484001442006309m_rule @ ( image_6392605097815222207m_rule @ F @ A2 ) ) )
=> ( ( member_o @ I2 @ A2 )
=> ( ord_le4093866177137961152m_rule @ Y5 @ ( F @ I2 ) ) ) ) ).
% less_INF_D
thf(fact_1217_less__INF__D,axiom,
! [Y5: set_fm,F: $o > set_fm,A2: set_o,I2: $o] :
( ( ord_less_set_fm @ Y5 @ ( comple1024113957266099930set_fm @ ( image_o_set_fm @ F @ A2 ) ) )
=> ( ( member_o @ I2 @ A2 )
=> ( ord_less_set_fm @ Y5 @ ( F @ I2 ) ) ) ) ).
% less_INF_D
thf(fact_1218_less__INF__D,axiom,
! [Y5: set_o,F: $o > set_o,A2: set_o,I2: $o] :
( ( ord_less_set_o @ Y5 @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ F @ A2 ) ) )
=> ( ( member_o @ I2 @ A2 )
=> ( ord_less_set_o @ Y5 @ ( F @ I2 ) ) ) ) ).
% less_INF_D
thf(fact_1219_less__INF__D,axiom,
! [Y5: $o,F: fm > $o,A2: set_fm,I2: fm] :
( ( ord_less_o @ Y5 @ ( complete_Inf_Inf_o @ ( image_fm_o @ F @ A2 ) ) )
=> ( ( member_fm @ I2 @ A2 )
=> ( ord_less_o @ Y5 @ ( F @ I2 ) ) ) ) ).
% less_INF_D
thf(fact_1220_less__INF__D,axiom,
! [Y5: $o,F: produc340336539035504054m_rule > $o,A2: set_Pr1822751329126368876m_rule,I2: produc340336539035504054m_rule] :
( ( ord_less_o @ Y5 @ ( complete_Inf_Inf_o @ ( image_8969478372357302939rule_o @ F @ A2 ) ) )
=> ( ( member7231649785386036813m_rule @ I2 @ A2 )
=> ( ord_less_o @ Y5 @ ( F @ I2 ) ) ) ) ).
% less_INF_D
thf(fact_1221_less__INF__D,axiom,
! [Y5: $o,F: $o > $o,A2: set_o,I2: $o] :
( ( ord_less_o @ Y5 @ ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) ) )
=> ( ( member_o @ I2 @ A2 )
=> ( ord_less_o @ Y5 @ ( F @ I2 ) ) ) ) ).
% less_INF_D
thf(fact_1222_INTER__UNIV__conv_I1_J,axiom,
! [B2: produc340336539035504054m_rule > set_fm,A2: set_Pr1822751329126368876m_rule] :
( ( top_top_set_fm
= ( comple1024113957266099930set_fm @ ( image_377115208147135190set_fm @ B2 @ A2 ) ) )
= ( ! [X: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X @ A2 )
=> ( ( B2 @ X )
= top_top_set_fm ) ) ) ) ).
% INTER_UNIV_conv(1)
thf(fact_1223_INTER__UNIV__conv_I1_J,axiom,
! [B2: $o > set_Pr1822751329126368876m_rule,A2: set_o] :
( ( top_to8442024021002298652m_rule
= ( comple1222484001442006309m_rule @ ( image_6392605097815222207m_rule @ B2 @ A2 ) ) )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ( B2 @ X )
= top_to8442024021002298652m_rule ) ) ) ) ).
% INTER_UNIV_conv(1)
thf(fact_1224_INTER__UNIV__conv_I1_J,axiom,
! [B2: $o > set_fm,A2: set_o] :
( ( top_top_set_fm
= ( comple1024113957266099930set_fm @ ( image_o_set_fm @ B2 @ A2 ) ) )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ( B2 @ X )
= top_top_set_fm ) ) ) ) ).
% INTER_UNIV_conv(1)
thf(fact_1225_INTER__UNIV__conv_I1_J,axiom,
! [B2: set_fm > set_fm,A2: set_set_fm] :
( ( top_top_set_fm
= ( comple1024113957266099930set_fm @ ( image_set_fm_set_fm @ B2 @ A2 ) ) )
= ( ! [X: set_fm] :
( ( member_set_fm @ X @ A2 )
=> ( ( B2 @ X )
= top_top_set_fm ) ) ) ) ).
% INTER_UNIV_conv(1)
thf(fact_1226_INTER__UNIV__conv_I1_J,axiom,
! [B2: $o > set_o,A2: set_o] :
( ( top_top_set_o
= ( comple3063163877087187839_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ( B2 @ X )
= top_top_set_o ) ) ) ) ).
% INTER_UNIV_conv(1)
thf(fact_1227_INTER__UNIV__conv_I2_J,axiom,
! [B2: produc340336539035504054m_rule > set_fm,A2: set_Pr1822751329126368876m_rule] :
( ( ( comple1024113957266099930set_fm @ ( image_377115208147135190set_fm @ B2 @ A2 ) )
= top_top_set_fm )
= ( ! [X: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X @ A2 )
=> ( ( B2 @ X )
= top_top_set_fm ) ) ) ) ).
% INTER_UNIV_conv(2)
thf(fact_1228_INTER__UNIV__conv_I2_J,axiom,
! [B2: $o > set_Pr1822751329126368876m_rule,A2: set_o] :
( ( ( comple1222484001442006309m_rule @ ( image_6392605097815222207m_rule @ B2 @ A2 ) )
= top_to8442024021002298652m_rule )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ( B2 @ X )
= top_to8442024021002298652m_rule ) ) ) ) ).
% INTER_UNIV_conv(2)
thf(fact_1229_INTER__UNIV__conv_I2_J,axiom,
! [B2: $o > set_fm,A2: set_o] :
( ( ( comple1024113957266099930set_fm @ ( image_o_set_fm @ B2 @ A2 ) )
= top_top_set_fm )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ( B2 @ X )
= top_top_set_fm ) ) ) ) ).
% INTER_UNIV_conv(2)
thf(fact_1230_INTER__UNIV__conv_I2_J,axiom,
! [B2: set_fm > set_fm,A2: set_set_fm] :
( ( ( comple1024113957266099930set_fm @ ( image_set_fm_set_fm @ B2 @ A2 ) )
= top_top_set_fm )
= ( ! [X: set_fm] :
( ( member_set_fm @ X @ A2 )
=> ( ( B2 @ X )
= top_top_set_fm ) ) ) ) ).
% INTER_UNIV_conv(2)
thf(fact_1231_INTER__UNIV__conv_I2_J,axiom,
! [B2: $o > set_o,A2: set_o] :
( ( ( comple3063163877087187839_set_o @ ( image_o_set_o @ B2 @ A2 ) )
= top_top_set_o )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ( B2 @ X )
= top_top_set_o ) ) ) ) ).
% INTER_UNIV_conv(2)
thf(fact_1232_INT__extend__simps_I10_J,axiom,
! [B2: $o > set_fm,F: $o > $o,A2: set_o] :
( ( comple1024113957266099930set_fm
@ ( image_o_set_fm
@ ^ [A3: $o] : ( B2 @ ( F @ A3 ) )
@ A2 ) )
= ( comple1024113957266099930set_fm @ ( image_o_set_fm @ B2 @ ( image_o_o @ F @ A2 ) ) ) ) ).
% INT_extend_simps(10)
thf(fact_1233_INT__extend__simps_I10_J,axiom,
! [B2: $o > set_o,F: $o > $o,A2: set_o] :
( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [A3: $o] : ( B2 @ ( F @ A3 ) )
@ A2 ) )
= ( comple3063163877087187839_set_o @ ( image_o_set_o @ B2 @ ( image_o_o @ F @ A2 ) ) ) ) ).
% INT_extend_simps(10)
thf(fact_1234_INT__extend__simps_I10_J,axiom,
! [B2: set_o > set_fm,F: $o > set_o,A2: set_o] :
( ( comple1024113957266099930set_fm
@ ( image_o_set_fm
@ ^ [A3: $o] : ( B2 @ ( F @ A3 ) )
@ A2 ) )
= ( comple1024113957266099930set_fm @ ( image_set_o_set_fm @ B2 @ ( image_o_set_o @ F @ A2 ) ) ) ) ).
% INT_extend_simps(10)
thf(fact_1235_INT__extend__simps_I10_J,axiom,
! [B2: set_fm > set_fm,F: $o > set_fm,A2: set_o] :
( ( comple1024113957266099930set_fm
@ ( image_o_set_fm
@ ^ [A3: $o] : ( B2 @ ( F @ A3 ) )
@ A2 ) )
= ( comple1024113957266099930set_fm @ ( image_set_fm_set_fm @ B2 @ ( image_o_set_fm @ F @ A2 ) ) ) ) ).
% INT_extend_simps(10)
thf(fact_1236_INT__extend__simps_I10_J,axiom,
! [B2: set_fm > set_o,F: $o > set_fm,A2: set_o] :
( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [A3: $o] : ( B2 @ ( F @ A3 ) )
@ A2 ) )
= ( comple3063163877087187839_set_o @ ( image_set_fm_set_o @ B2 @ ( image_o_set_fm @ F @ A2 ) ) ) ) ).
% INT_extend_simps(10)
thf(fact_1237_INT__extend__simps_I10_J,axiom,
! [B2: set_o > set_o,F: $o > set_o,A2: set_o] :
( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [A3: $o] : ( B2 @ ( F @ A3 ) )
@ A2 ) )
= ( comple3063163877087187839_set_o @ ( image_set_o_set_o @ B2 @ ( image_o_set_o @ F @ A2 ) ) ) ) ).
% INT_extend_simps(10)
thf(fact_1238_INT__extend__simps_I10_J,axiom,
! [B2: $o > set_fm,F: set_fm > $o,A2: set_set_fm] :
( ( comple1024113957266099930set_fm
@ ( image_set_fm_set_fm
@ ^ [A3: set_fm] : ( B2 @ ( F @ A3 ) )
@ A2 ) )
= ( comple1024113957266099930set_fm @ ( image_o_set_fm @ B2 @ ( image_set_fm_o @ F @ A2 ) ) ) ) ).
% INT_extend_simps(10)
thf(fact_1239_INT__extend__simps_I10_J,axiom,
! [B2: set_fm > set_fm,F: set_fm > set_fm,A2: set_set_fm] :
( ( comple1024113957266099930set_fm
@ ( image_set_fm_set_fm
@ ^ [A3: set_fm] : ( B2 @ ( F @ A3 ) )
@ A2 ) )
= ( comple1024113957266099930set_fm @ ( image_set_fm_set_fm @ B2 @ ( image_set_fm_set_fm @ F @ A2 ) ) ) ) ).
% INT_extend_simps(10)
thf(fact_1240_INT__extend__simps_I10_J,axiom,
! [B2: produc340336539035504054m_rule > set_fm,F: nat > produc340336539035504054m_rule,A2: set_nat] :
( ( comple1024113957266099930set_fm
@ ( image_nat_set_fm
@ ^ [A3: nat] : ( B2 @ ( F @ A3 ) )
@ A2 ) )
= ( comple1024113957266099930set_fm @ ( image_377115208147135190set_fm @ B2 @ ( image_8361736131336225677m_rule @ F @ A2 ) ) ) ) ).
% INT_extend_simps(10)
thf(fact_1241_INT__extend__simps_I10_J,axiom,
! [B2: $o > set_fm,F: produc340336539035504054m_rule > $o,A2: set_Pr1822751329126368876m_rule] :
( ( comple1024113957266099930set_fm
@ ( image_377115208147135190set_fm
@ ^ [A3: produc340336539035504054m_rule] : ( B2 @ ( F @ A3 ) )
@ A2 ) )
= ( comple1024113957266099930set_fm @ ( image_o_set_fm @ B2 @ ( image_8969478372357302939rule_o @ F @ A2 ) ) ) ) ).
% INT_extend_simps(10)
thf(fact_1242_INT__extend__simps_I9_J,axiom,
! [C: produc340336539035504054m_rule > set_fm,B2: produc340336539035504054m_rule > set_Pr1822751329126368876m_rule,A2: set_Pr1822751329126368876m_rule] :
( ( comple1024113957266099930set_fm
@ ( image_377115208147135190set_fm
@ ^ [X: produc340336539035504054m_rule] : ( comple1024113957266099930set_fm @ ( image_377115208147135190set_fm @ C @ ( B2 @ X ) ) )
@ A2 ) )
= ( comple1024113957266099930set_fm @ ( image_377115208147135190set_fm @ C @ ( comple5648871819318716351m_rule @ ( image_2683796897779739177m_rule @ B2 @ A2 ) ) ) ) ) ).
% INT_extend_simps(9)
thf(fact_1243_INT__extend__simps_I9_J,axiom,
! [C: produc340336539035504054m_rule > set_Pr1822751329126368876m_rule,B2: $o > set_Pr1822751329126368876m_rule,A2: set_o] :
( ( comple1222484001442006309m_rule
@ ( image_6392605097815222207m_rule
@ ^ [X: $o] : ( comple1222484001442006309m_rule @ ( image_2683796897779739177m_rule @ C @ ( B2 @ X ) ) )
@ A2 ) )
= ( comple1222484001442006309m_rule @ ( image_2683796897779739177m_rule @ C @ ( comple5648871819318716351m_rule @ ( image_6392605097815222207m_rule @ B2 @ A2 ) ) ) ) ) ).
% INT_extend_simps(9)
thf(fact_1244_INT__extend__simps_I9_J,axiom,
! [C: produc340336539035504054m_rule > set_fm,B2: $o > set_Pr1822751329126368876m_rule,A2: set_o] :
( ( comple1024113957266099930set_fm
@ ( image_o_set_fm
@ ^ [X: $o] : ( comple1024113957266099930set_fm @ ( image_377115208147135190set_fm @ C @ ( B2 @ X ) ) )
@ A2 ) )
= ( comple1024113957266099930set_fm @ ( image_377115208147135190set_fm @ C @ ( comple5648871819318716351m_rule @ ( image_6392605097815222207m_rule @ B2 @ A2 ) ) ) ) ) ).
% INT_extend_simps(9)
thf(fact_1245_INT__extend__simps_I9_J,axiom,
! [C: produc340336539035504054m_rule > set_o,B2: $o > set_Pr1822751329126368876m_rule,A2: set_o] :
( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( comple3063163877087187839_set_o @ ( image_2025595867862115707_set_o @ C @ ( B2 @ X ) ) )
@ A2 ) )
= ( comple3063163877087187839_set_o @ ( image_2025595867862115707_set_o @ C @ ( comple5648871819318716351m_rule @ ( image_6392605097815222207m_rule @ B2 @ A2 ) ) ) ) ) ).
% INT_extend_simps(9)
thf(fact_1246_INT__extend__simps_I9_J,axiom,
! [C: produc340336539035504054m_rule > set_fm,B2: set_fm > set_Pr1822751329126368876m_rule,A2: set_set_fm] :
( ( comple1024113957266099930set_fm
@ ( image_set_fm_set_fm
@ ^ [X: set_fm] : ( comple1024113957266099930set_fm @ ( image_377115208147135190set_fm @ C @ ( B2 @ X ) ) )
@ A2 ) )
= ( comple1024113957266099930set_fm @ ( image_377115208147135190set_fm @ C @ ( comple5648871819318716351m_rule @ ( image_2204476524374002298m_rule @ B2 @ A2 ) ) ) ) ) ).
% INT_extend_simps(9)
thf(fact_1247_Inf__bool__def,axiom,
( complete_Inf_Inf_o
= ( ^ [A4: set_o] :
~ ( member_o @ $false @ A4 ) ) ) ).
% Inf_bool_def
thf(fact_1248_Rats__eq__range__nat__to__rat__surj,axiom,
( field_6020823756834552118ts_rat
= ( image_nat_rat @ nat_to_rat_surj @ top_top_set_nat ) ) ).
% Rats_eq_range_nat_to_rat_surj
thf(fact_1249_UNIV__bool,axiom,
( top_top_set_o
= ( insert_o @ $false @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).
% UNIV_bool
thf(fact_1250_nat__descend__induct,axiom,
! [N2: nat,P: nat > $o,M2: nat] :
( ! [K: nat] :
( ( ord_less_nat @ N2 @ K )
=> ( P @ K ) )
=> ( ! [K: nat] :
( ( ord_less_eq_nat @ K @ N2 )
=> ( ! [I5: nat] :
( ( ord_less_nat @ K @ I5 )
=> ( P @ I5 ) )
=> ( P @ K ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_1251_diff__diff__cancel,axiom,
! [I2: nat,N2: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
=> ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_1252_diff__le__mono2,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_1253_le__diff__iff_H,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A ) @ ( minus_minus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1254_diff__le__self,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ).
% diff_le_self
thf(fact_1255_diff__le__mono,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).
% diff_le_mono
thf(fact_1256_Nat_Odiff__diff__eq,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N2 @ K2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1257_le__diff__iff,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N2 @ K2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_1258_eq__diff__iff,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( ( minus_minus_nat @ M2 @ K2 )
= ( minus_minus_nat @ N2 @ K2 ) )
= ( M2 = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_1259_less__diff__iff,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N2 @ K2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ) ) ).
% less_diff_iff
thf(fact_1260_diff__less__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C2 ) @ ( minus_minus_nat @ B @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_1261_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I2: nat,J3: nat] :
( ! [I4: nat,J2: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ord_less_nat @ ( F @ I4 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J3 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J3 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1262_le__neq__implies__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( M2 != N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_1263_less__or__eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_1264_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
| ( M3 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1265_less__imp__le__nat,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_imp_le_nat
thf(fact_1266_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
& ( M3 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_1267_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_1268_le__trans,axiom,
! [I2: nat,J3: nat,K2: nat] :
( ( ord_less_eq_nat @ I2 @ J3 )
=> ( ( ord_less_eq_nat @ J3 @ K2 )
=> ( ord_less_eq_nat @ I2 @ K2 ) ) ) ).
% le_trans
thf(fact_1269_eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( M2 = N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% eq_imp_le
thf(fact_1270_le__antisym,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M2 )
=> ( M2 = N2 ) ) ) ).
% le_antisym
thf(fact_1271_nat__le__linear,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
| ( ord_less_eq_nat @ N2 @ M2 ) ) ).
% nat_le_linear
thf(fact_1272_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K2: nat,B: nat] :
( ( P @ K2 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
% Conjectures (2)
thf(conj_0,hypothesis,
( member_fm @ p
@ ( comple2134933779557159616set_fm
@ ( image_377115208147135190set_fm
@ ^ [Ss: produc340336539035504054m_rule] : ( set_fm2 @ ( produc3242453064496107283ist_fm @ ( produc993771546023551940m_rule @ Ss ) ) )
@ ( sset_P4484857331586881186m_rule @ steps ) ) ) ) ).
thf(conj_1,conjecture,
? [N4: nat] : ( member_fm @ p @ ( set_fm2 @ ( produc3242453064496107283ist_fm @ ( produc993771546023551940m_rule @ ( snth_P6679518042731451922m_rule @ steps @ N4 ) ) ) ) ) ).
%------------------------------------------------------------------------------