TPTP Problem File: SLH0722^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Hales_Jewett/0002_Hales_Jewett/prob_00768_030514__5711428_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1501 ( 412 unt; 224 typ; 0 def)
% Number of atoms : 4229 (1126 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 12851 ( 265 ~; 25 |; 274 &;10178 @)
% ( 0 <=>;2109 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 8 avg)
% Number of types : 24 ( 23 usr)
% Number of type conns : 2210 (2210 >; 0 *; 0 +; 0 <<)
% Number of symbols : 204 ( 201 usr; 26 con; 0-4 aty)
% Number of variables : 4603 ( 446 ^;4016 !; 141 ?;4603 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:39:32.771
%------------------------------------------------------------------------------
% Could-be-implicit typings (23)
thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_nat_nat_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
set_set_nat_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J_Mt__Nat__Onat_J_J,type,
set_nat_set_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_I_Eo_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
set_set_o_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_nat_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
set_nat_nat_nat2: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_set_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
set_set_nat_nat2: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
set_nat_set_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_Eo_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_o_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J,type,
set_nat_nat_o: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J,type,
set_set_nat_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_Eo_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
set_o_set_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_Eo_J_J,type,
set_nat_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_Eo_Mt__Nat__Onat_J_J,type,
set_o_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
set_set_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_Eo_M_Eo_J_J,type,
set_o_o: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_I_Eo_J,type,
set_o: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (201)
thf(sy_c_Complete__Lattices_OInf__class_OInf_001_062_It__Nat__Onat_M_Eo_J,type,
comple6214475593288795910_nat_o: set_nat_o > nat > $o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
comple6608973012141742712at_nat: set_nat_nat > nat > nat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J,type,
comple6797894177231197998et_nat: set_nat_set_nat > nat > set_nat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001_Eo,type,
complete_Inf_Inf_o: set_o > $o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Nat__Onat,type,
complete_Inf_Inf_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_I_062_I_Eo_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
comple6464488905778062255at_nat: set_set_o_nat_nat > set_o_nat_nat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
comple884914421528019101at_nat: set_set_nat_nat_nat > set_nat_nat_nat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
comple439066603627490862at_nat: set_set_nat_nat > set_nat_nat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_I_Eo_J,type,
comple3063163877087187839_set_o: set_set_o > set_o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Nat__Onat_J,type,
comple7806235888213564991et_nat: set_set_nat > set_nat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
comple1065008630642458357et_nat: set_set_set_nat > set_set_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Nat__Onat_M_Eo_J,type,
comple8317665133742190828_nat_o: set_nat_o > nat > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
comple2450677804321093138at_nat: set_nat_nat > nat > nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J,type,
comple2583738152068352712et_nat: set_nat_set_nat > nat > set_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_Eo,type,
complete_Sup_Sup_o: set_o > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
complete_Sup_Sup_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_I_062_I_Eo_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
comple7172370505855214741at_nat: set_set_o_nat_nat > set_o_nat_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
comple8167887107183641911at_nat: set_set_nat_nat_nat > set_nat_nat_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
comple5448282615319421384at_nat: set_set_nat_nat > set_nat_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_I_Eo_J,type,
comple90263536869209701_set_o: set_set_o > set_o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
comple7399068483239264473et_nat: set_set_nat > set_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
comple548664676211718543et_nat: set_set_set_nat > set_set_nat ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Nat__Onat,type,
condit2214826472909112428ve_nat: set_nat > $o ).
thf(sy_c_Finite__Set_OFpow_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
finite_Fpow_nat_nat: set_nat_nat > set_set_nat_nat ).
thf(sy_c_Finite__Set_OFpow_001_Eo,type,
finite_Fpow_o: set_o > set_set_o ).
thf(sy_c_Finite__Set_OFpow_001t__Nat__Onat,type,
finite_Fpow_nat: set_nat > set_set_nat ).
thf(sy_c_Finite__Set_OFpow_001t__Set__Oset_It__Nat__Onat_J,type,
finite_Fpow_set_nat: set_set_nat > set_set_set_nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_FuncSet_OPiE_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
piE_nat_nat_nat_nat: set_nat_nat > ( ( nat > nat ) > set_nat_nat ) > set_nat_nat_nat_nat ).
thf(sy_c_FuncSet_OPiE_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_Eo,type,
piE_nat_nat_o: set_nat_nat > ( ( nat > nat ) > set_o ) > set_nat_nat_o ).
thf(sy_c_FuncSet_OPiE_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
piE_nat_nat_nat: set_nat_nat > ( ( nat > nat ) > set_nat ) > set_nat_nat_nat2 ).
thf(sy_c_FuncSet_OPiE_001_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J_001t__Nat__Onat,type,
piE_nat_set_nat_nat: set_nat_set_nat > ( ( nat > set_nat ) > set_nat ) > set_nat_set_nat_nat ).
thf(sy_c_FuncSet_OPiE_001_Eo_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
piE_o_nat_nat: set_o > ( $o > set_nat_nat ) > set_o_nat_nat ).
thf(sy_c_FuncSet_OPiE_001_Eo_001_Eo,type,
piE_o_o: set_o > ( $o > set_o ) > set_o_o ).
thf(sy_c_FuncSet_OPiE_001_Eo_001t__Nat__Onat,type,
piE_o_nat: set_o > ( $o > set_nat ) > set_o_nat ).
thf(sy_c_FuncSet_OPiE_001_Eo_001t__Set__Oset_It__Nat__Onat_J,type,
piE_o_set_nat: set_o > ( $o > set_set_nat ) > set_o_set_nat ).
thf(sy_c_FuncSet_OPiE_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
piE_nat_nat_nat2: set_nat > ( nat > set_nat_nat ) > set_nat_nat_nat ).
thf(sy_c_FuncSet_OPiE_001t__Nat__Onat_001_Eo,type,
piE_nat_o: set_nat > ( nat > set_o ) > set_nat_o ).
thf(sy_c_FuncSet_OPiE_001t__Nat__Onat_001t__Nat__Onat,type,
piE_nat_nat: set_nat > ( nat > set_nat ) > set_nat_nat ).
thf(sy_c_FuncSet_OPiE_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
piE_nat_set_nat: set_nat > ( nat > set_set_nat ) > set_nat_set_nat ).
thf(sy_c_FuncSet_OPiE_001t__Set__Oset_It__Nat__Onat_J_001_Eo,type,
piE_set_nat_o: set_set_nat > ( set_nat > set_o ) > set_set_nat_o ).
thf(sy_c_FuncSet_OPiE_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
piE_set_nat_nat: set_set_nat > ( set_nat > set_nat ) > set_set_nat_nat2 ).
thf(sy_c_FuncSet_OPi_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
pi_nat_nat_nat_nat: set_nat_nat > ( ( nat > nat ) > set_nat_nat ) > set_nat_nat_nat_nat ).
thf(sy_c_FuncSet_OPi_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_Eo,type,
pi_nat_nat_o: set_nat_nat > ( ( nat > nat ) > set_o ) > set_nat_nat_o ).
thf(sy_c_FuncSet_OPi_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
pi_nat_nat_nat: set_nat_nat > ( ( nat > nat ) > set_nat ) > set_nat_nat_nat2 ).
thf(sy_c_FuncSet_OPi_001_Eo_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
pi_o_nat_nat: set_o > ( $o > set_nat_nat ) > set_o_nat_nat ).
thf(sy_c_FuncSet_OPi_001_Eo_001_Eo,type,
pi_o_o: set_o > ( $o > set_o ) > set_o_o ).
thf(sy_c_FuncSet_OPi_001_Eo_001t__Nat__Onat,type,
pi_o_nat: set_o > ( $o > set_nat ) > set_o_nat ).
thf(sy_c_FuncSet_OPi_001_Eo_001t__Set__Oset_It__Nat__Onat_J,type,
pi_o_set_nat: set_o > ( $o > set_set_nat ) > set_o_set_nat ).
thf(sy_c_FuncSet_OPi_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
pi_nat_nat_nat2: set_nat > ( nat > set_nat_nat ) > set_nat_nat_nat ).
thf(sy_c_FuncSet_OPi_001t__Nat__Onat_001_Eo,type,
pi_nat_o: set_nat > ( nat > set_o ) > set_nat_o ).
thf(sy_c_FuncSet_OPi_001t__Nat__Onat_001t__Nat__Onat,type,
pi_nat_nat: set_nat > ( nat > set_nat ) > set_nat_nat ).
thf(sy_c_FuncSet_OPi_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
pi_nat_set_nat: set_nat > ( nat > set_set_nat ) > set_nat_set_nat ).
thf(sy_c_FuncSet_OPi_001t__Set__Oset_It__Nat__Onat_J_001_Eo,type,
pi_set_nat_o: set_set_nat > ( set_nat > set_o ) > set_set_nat_o ).
thf(sy_c_FuncSet_OPi_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
pi_set_nat_nat: set_set_nat > ( set_nat > set_nat ) > set_set_nat_nat2 ).
thf(sy_c_Hales__Jewett_Ois__subspace,type,
hales_is_subspace: ( ( nat > nat ) > nat > nat ) > nat > nat > nat > $o ).
thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Nat__Onat,type,
lattic8265883725875713057ax_nat: set_nat > nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
bot_bo1568108970253895006_nat_o: ( ( nat > nat ) > nat > nat ) > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
bot_bot_nat_nat_o: ( nat > nat ) > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
bot_bo8210142506433397254_nat_o: ( nat > set_nat ) > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_I_Eo_M_Eo_J,type,
bot_bot_o_o: $o > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
bot_bot_set_nat_o: set_nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_Eo,type,
bot_bot_o: $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
bot_bo3919185967433191911at_nat: set_nat_nat_nat_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
bot_bo945813143650711160at_nat: set_nat_nat_nat2 ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J_Mt__Nat__Onat_J_J,type,
bot_bo4500225246352649262at_nat: set_nat_set_nat_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_I_Eo_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
bot_bo8153995795486405652at_nat: set_o_nat_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_I_Eo_Mt__Nat__Onat_J_J,type,
bot_bot_set_o_nat: set_o_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
bot_bo7445843802507891576at_nat: set_nat_nat_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Nat__Onat_M_Eo_J_J,type,
bot_bot_set_nat_o2: set_nat_o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
bot_bot_set_nat_nat: set_nat_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
bot_bo4007787791999405887et_nat: set_nat_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
bot_bo7208697003875722815at_nat: set_set_nat_nat2 ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_Eo_J,type,
bot_bot_set_o: set_o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_I_062_I_Eo_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
bot_bo6195285094354290676at_nat: set_set_o_nat_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
bot_bo6668270333135560750at_nat: set_set_nat_nat_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
bot_bo7376149671870096959at_nat: set_set_nat_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
bot_bot_set_set_o: set_set_o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bot_bot_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
bot_bo7198184520161983622et_nat: set_set_set_nat ).
thf(sy_c_Orderings_Oord__class_OLeast_001t__Nat__Onat,type,
ord_Least_nat: ( nat > $o ) > nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
ord_le5430825838364970130_nat_o: ( ( ( nat > nat ) > nat > nat ) > $o ) > ( ( ( nat > nat ) > nat > nat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_I_Eo_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
ord_le6871787456926423957_nat_o: ( ( $o > nat > nat ) > $o ) > ( ( $o > nat > nat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
ord_le5384859702510996545_nat_o: ( ( nat > nat > nat ) > $o ) > ( ( nat > nat > nat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
ord_le7366121074344172400_nat_o: ( ( nat > nat ) > $o ) > ( ( nat > nat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
ord_le8865062304692155706_nat_o: ( ( nat > set_nat ) > $o ) > ( ( nat > set_nat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_Eo_J,type,
ord_less_eq_o_o: ( $o > $o ) > ( $o > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6195038898401538645et_nat: ( nat > set_nat ) > ( nat > set_nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
ord_le3964352015994296041_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_Eo,type,
ord_less_eq_o: $o > $o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le5260717879541182899at_nat: set_nat_nat_nat_nat > set_nat_nat_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_Eo_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le8808915593745164104at_nat: set_o_nat_nat > set_o_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_Eo_M_Eo_J_J,type,
ord_less_eq_set_o_o: set_o_o > set_o_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_Eo_Mt__Nat__Onat_J_J,type,
ord_le4981610546006782297_o_nat: set_o_nat > set_o_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_Eo_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
ord_le5010423853744249615et_nat: set_o_set_nat > set_o_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le3211623285424100676at_nat: set_nat_nat_nat > set_nat_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_M_Eo_J_J,type,
ord_le6029213668185085951_nat_o: set_nat_o > set_nat_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le9059583361652607317at_nat: set_nat_nat > set_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
ord_le1585852046946910987et_nat: set_nat_set_nat > set_nat_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J,type,
ord_le1935226719417650633_nat_o: set_set_nat_o > set_set_nat_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
ord_le4786761258823227915at_nat: set_set_nat_nat2 > set_set_nat_nat2 > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
ord_less_eq_set_o: set_o > set_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_I_062_I_Eo_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
ord_le1662808457768676136at_nat: set_set_o_nat_nat > set_set_o_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
ord_le8468300607614202362at_nat: set_set_nat_nat_nat > set_set_nat_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le4954213926817602059at_nat: set_set_nat_nat > set_set_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
ord_le4374716579403074808_set_o: set_set_o > set_set_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
ord_le9131159989063066194et_nat: set_set_set_nat > set_set_set_nat > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
order_Greatest_nat: ( nat > $o ) > nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_I_062_I_Eo_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
order_3307316977169705729at_nat: ( set_o_nat_nat > $o ) > set_o_nat_nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
order_8011581075050426827at_nat: ( set_nat_nat_nat > $o ) > set_nat_nat_nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
order_8228081171942161500at_nat: ( set_nat_nat > $o ) > set_nat_nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_I_Eo_J,type,
order_Greatest_set_o: ( set_o > $o ) > set_o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__Nat__Onat_J,type,
order_5724808138429204845et_nat: ( set_nat > $o ) > set_nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
order_1279421399067128355et_nat: ( set_set_nat > $o ) > set_set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Set_OCollect_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
collec3567154360959927026at_nat: ( ( ( nat > nat ) > nat > nat ) > $o ) > set_nat_nat_nat_nat ).
thf(sy_c_Set_OCollect_001_062_I_Eo_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
collect_o_nat_nat: ( ( $o > nat > nat ) > $o ) > set_o_nat_nat ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
collect_nat_nat_nat: ( ( nat > nat > nat ) > $o ) > set_nat_nat_nat ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
collect_nat_nat: ( ( nat > nat ) > $o ) > set_nat_nat ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J,type,
collect_nat_set_nat: ( ( nat > set_nat ) > $o ) > set_nat_set_nat ).
thf(sy_c_Set_OCollect_001_Eo,type,
collect_o: ( $o > $o ) > set_o ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_3205354838064109189at_nat: ( ( nat > nat ) > nat > nat ) > set_nat_nat > set_nat_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_Eo,type,
image_nat_nat_o: ( ( nat > nat ) > $o ) > set_nat_nat > set_o ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
image_nat_nat_nat: ( ( nat > nat ) > nat ) > set_nat_nat > set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Set__Oset_I_Eo_J,type,
image_nat_nat_set_o: ( ( nat > nat ) > set_o ) > set_nat_nat > set_set_o ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7432509271690132940et_nat: ( ( nat > nat ) > set_nat ) > set_nat_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J_001_Eo,type,
image_nat_set_nat_o: ( ( nat > set_nat ) > $o ) > set_nat_set_nat > set_o ).
thf(sy_c_Set_Oimage_001_Eo_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_o_nat_nat: ( $o > nat > nat ) > set_o > set_nat_nat ).
thf(sy_c_Set_Oimage_001_Eo_001_Eo,type,
image_o_o: ( $o > $o ) > set_o > set_o ).
thf(sy_c_Set_Oimage_001_Eo_001t__Nat__Onat,type,
image_o_nat: ( $o > nat ) > set_o > set_nat ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
image_o_set_nat_nat: ( $o > set_nat_nat ) > set_o > set_set_nat_nat ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_I_Eo_J,type,
image_o_set_o: ( $o > set_o ) > set_o > set_set_o ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_It__Nat__Onat_J,type,
image_o_set_nat: ( $o > set_nat ) > set_o > set_set_nat ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_o_set_set_nat: ( $o > set_set_nat ) > set_o > set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_062_I_Eo_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
image_nat_o_nat_nat: ( nat > $o > nat > nat ) > set_nat > set_o_nat_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
image_6919068903512877573at_nat: ( nat > nat > nat > nat ) > set_nat > set_nat_nat_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_062_It__Nat__Onat_M_Eo_J,type,
image_nat_nat_o2: ( nat > nat > $o ) > set_nat > set_nat_o ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_nat_nat_nat2: ( nat > nat > nat ) > set_nat > set_nat_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J,type,
image_4436799006340492620et_nat: ( nat > nat > set_nat ) > set_nat > set_nat_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_Eo,type,
image_nat_o: ( nat > $o ) > set_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_I_062_I_Eo_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
image_6307125673115463441at_nat: ( nat > set_o_nat_nat ) > set_nat > set_set_o_nat_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
image_6130888460295934395at_nat: ( nat > set_nat_nat_nat ) > set_nat > set_set_nat_nat_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
image_7301343469591561292at_nat: ( nat > set_nat_nat ) > set_nat > set_set_nat_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_I_Eo_J,type,
image_nat_set_o: ( nat > set_o ) > set_nat > set_set_o ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_2194112158459175443et_nat: ( nat > set_set_nat ) > set_nat > set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
image_3832368097948589297at_nat: ( set_nat_nat > set_nat_nat ) > set_set_nat_nat > set_set_nat_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_Eo_J_001t__Set__Oset_I_Eo_J,type,
image_set_o_set_o: ( set_o > set_o ) > set_set_o > set_set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_Eo_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_set_o_set_nat: ( set_o > set_nat ) > set_set_o > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001_Eo,type,
image_set_nat_o: ( set_nat > $o ) > set_set_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
image_set_nat_nat: ( set_nat > nat ) > set_set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_I_Eo_J,type,
image_set_nat_set_o: ( set_nat > set_o ) > set_set_nat > set_set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_6725021117256019401et_nat: ( set_nat > set_set_nat ) > set_set_nat > set_set_set_nat ).
thf(sy_c_Set_Ois__empty_001t__Nat__Onat,type,
is_empty_nat: set_nat > $o ).
thf(sy_c_Set_Othe__elem_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
the_elem_nat_nat: set_nat_nat > nat > nat ).
thf(sy_c_Set_Othe__elem_001_Eo,type,
the_elem_o: set_o > $o ).
thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
thf(sy_c_Set_Othe__elem_001t__Set__Oset_It__Nat__Onat_J,type,
the_elem_set_nat: set_set_nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001_Eo,type,
set_or6416164934427428222Than_o: $o > set_o ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
set_or1210151606488870762an_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001_Eo,type,
set_ord_lessThan_o: $o > set_o ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member952132173341509300at_nat: ( ( nat > nat ) > nat > nat ) > set_nat_nat_nat_nat > $o ).
thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
member_nat_nat_o: ( ( nat > nat ) > $o ) > set_nat_nat_o > $o ).
thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
member_nat_nat_nat: ( ( nat > nat ) > nat ) > set_nat_nat_nat2 > $o ).
thf(sy_c_member_001_062_I_Eo_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member_o_nat_nat: ( $o > nat > nat ) > set_o_nat_nat > $o ).
thf(sy_c_member_001_062_I_Eo_M_Eo_J,type,
member_o_o: ( $o > $o ) > set_o_o > $o ).
thf(sy_c_member_001_062_I_Eo_Mt__Nat__Onat_J,type,
member_o_nat: ( $o > nat ) > set_o_nat > $o ).
thf(sy_c_member_001_062_I_Eo_Mt__Set__Oset_It__Nat__Onat_J_J,type,
member_o_set_nat: ( $o > set_nat ) > set_o_set_nat > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member_nat_nat_nat2: ( nat > nat > nat ) > set_nat_nat_nat > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_M_Eo_J,type,
member_nat_o: ( nat > $o ) > set_nat_o > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
member_nat_nat: ( nat > nat ) > set_nat_nat > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J,type,
member_nat_set_nat: ( nat > set_nat ) > set_nat_set_nat > $o ).
thf(sy_c_member_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
member_set_nat_o: ( set_nat > $o ) > set_set_nat_o > $o ).
thf(sy_c_member_001_062_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J,type,
member_set_nat_nat: ( set_nat > nat ) > set_set_nat_nat2 > $o ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_I_062_I_Eo_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
member_set_o_nat_nat: set_o_nat_nat > set_set_o_nat_nat > $o ).
thf(sy_c_member_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
member8194441297229544571at_nat: set_nat_nat_nat > set_set_nat_nat_nat > $o ).
thf(sy_c_member_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member_set_nat_nat2: set_nat_nat > set_set_nat_nat > $o ).
thf(sy_c_member_001t__Set__Oset_I_Eo_J,type,
member_set_o: set_o > set_set_o > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
member_set_set_nat: set_set_nat > set_set_set_nat > $o ).
thf(sy_v_S,type,
s: ( nat > nat ) > nat > nat ).
thf(sy_v_k,type,
k: nat ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_t,type,
t: nat ).
% Relevant facts (1276)
thf(fact_0_assms,axiom,
hales_is_subspace @ s @ k @ n @ t ).
% assms
thf(fact_1_fun__ex,axiom,
! [A: $o,A2: set_o,B: $o,B2: set_o] :
( ( member_o @ A @ A2 )
=> ( ( member_o @ B @ B2 )
=> ? [X: $o > $o] :
( ( member_o_o @ X
@ ( piE_o_o @ A2
@ ^ [I: $o] : B2 ) )
& ( ( X @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_2_fun__ex,axiom,
! [A: $o,A2: set_o,B: nat,B2: set_nat] :
( ( member_o @ A @ A2 )
=> ( ( member_nat @ B @ B2 )
=> ? [X: $o > nat] :
( ( member_o_nat @ X
@ ( piE_o_nat @ A2
@ ^ [I: $o] : B2 ) )
& ( ( X @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_3_fun__ex,axiom,
! [A: nat,A2: set_nat,B: $o,B2: set_o] :
( ( member_nat @ A @ A2 )
=> ( ( member_o @ B @ B2 )
=> ? [X: nat > $o] :
( ( member_nat_o @ X
@ ( piE_nat_o @ A2
@ ^ [I: nat] : B2 ) )
& ( ( X @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_4_fun__ex,axiom,
! [A: nat,A2: set_nat,B: nat,B2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( member_nat @ B @ B2 )
=> ? [X: nat > nat] :
( ( member_nat_nat @ X
@ ( piE_nat_nat @ A2
@ ^ [I: nat] : B2 ) )
& ( ( X @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_5_fun__ex,axiom,
! [A: $o,A2: set_o,B: set_nat,B2: set_set_nat] :
( ( member_o @ A @ A2 )
=> ( ( member_set_nat @ B @ B2 )
=> ? [X: $o > set_nat] :
( ( member_o_set_nat @ X
@ ( piE_o_set_nat @ A2
@ ^ [I: $o] : B2 ) )
& ( ( X @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_6_fun__ex,axiom,
! [A: set_nat,A2: set_set_nat,B: $o,B2: set_o] :
( ( member_set_nat @ A @ A2 )
=> ( ( member_o @ B @ B2 )
=> ? [X: set_nat > $o] :
( ( member_set_nat_o @ X
@ ( piE_set_nat_o @ A2
@ ^ [I: set_nat] : B2 ) )
& ( ( X @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_7_fun__ex,axiom,
! [A: set_nat,A2: set_set_nat,B: nat,B2: set_nat] :
( ( member_set_nat @ A @ A2 )
=> ( ( member_nat @ B @ B2 )
=> ? [X: set_nat > nat] :
( ( member_set_nat_nat @ X
@ ( piE_set_nat_nat @ A2
@ ^ [I: set_nat] : B2 ) )
& ( ( X @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_8_fun__ex,axiom,
! [A: nat,A2: set_nat,B: set_nat,B2: set_set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( member_set_nat @ B @ B2 )
=> ? [X: nat > set_nat] :
( ( member_nat_set_nat @ X
@ ( piE_nat_set_nat @ A2
@ ^ [I: nat] : B2 ) )
& ( ( X @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_9_fun__ex,axiom,
! [A: nat > nat,A2: set_nat_nat,B: $o,B2: set_o] :
( ( member_nat_nat @ A @ A2 )
=> ( ( member_o @ B @ B2 )
=> ? [X: ( nat > nat ) > $o] :
( ( member_nat_nat_o @ X
@ ( piE_nat_nat_o @ A2
@ ^ [I: nat > nat] : B2 ) )
& ( ( X @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_10_fun__ex,axiom,
! [A: nat > nat,A2: set_nat_nat,B: nat,B2: set_nat] :
( ( member_nat_nat @ A @ A2 )
=> ( ( member_nat @ B @ B2 )
=> ? [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X
@ ( piE_nat_nat_nat @ A2
@ ^ [I: nat > nat] : B2 ) )
& ( ( X @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_11_PiE__uniqueness,axiom,
! [F: $o > nat > nat,A2: set_o,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( image_o_nat_nat @ F @ A2 ) @ B2 )
=> ? [X: $o > nat > nat] :
( ( member_o_nat_nat @ X
@ ( piE_o_nat_nat @ A2
@ ^ [I: $o] : B2 ) )
& ! [Xa: $o] :
( ( member_o @ Xa @ A2 )
=> ( ( X @ Xa )
= ( F @ Xa ) ) )
& ! [Y: $o > nat > nat] :
( ( ( member_o_nat_nat @ Y
@ ( piE_o_nat_nat @ A2
@ ^ [I: $o] : B2 ) )
& ! [Xa2: $o] :
( ( member_o @ Xa2 @ A2 )
=> ( ( Y @ Xa2 )
= ( F @ Xa2 ) ) ) )
=> ( Y = X ) ) ) ) ).
% PiE_uniqueness
thf(fact_12_PiE__uniqueness,axiom,
! [F: nat > nat > nat,A2: set_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ A2 ) @ B2 )
=> ? [X: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X
@ ( piE_nat_nat_nat2 @ A2
@ ^ [I: nat] : B2 ) )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( X @ Xa )
= ( F @ Xa ) ) )
& ! [Y: nat > nat > nat] :
( ( ( member_nat_nat_nat2 @ Y
@ ( piE_nat_nat_nat2 @ A2
@ ^ [I: nat] : B2 ) )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( Y @ Xa2 )
= ( F @ Xa2 ) ) ) )
=> ( Y = X ) ) ) ) ).
% PiE_uniqueness
thf(fact_13_PiE__uniqueness,axiom,
! [F: $o > nat,A2: set_o,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_o_nat @ F @ A2 ) @ B2 )
=> ? [X: $o > nat] :
( ( member_o_nat @ X
@ ( piE_o_nat @ A2
@ ^ [I: $o] : B2 ) )
& ! [Xa: $o] :
( ( member_o @ Xa @ A2 )
=> ( ( X @ Xa )
= ( F @ Xa ) ) )
& ! [Y: $o > nat] :
( ( ( member_o_nat @ Y
@ ( piE_o_nat @ A2
@ ^ [I: $o] : B2 ) )
& ! [Xa2: $o] :
( ( member_o @ Xa2 @ A2 )
=> ( ( Y @ Xa2 )
= ( F @ Xa2 ) ) ) )
=> ( Y = X ) ) ) ) ).
% PiE_uniqueness
thf(fact_14_PiE__uniqueness,axiom,
! [F: $o > $o,A2: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ ( image_o_o @ F @ A2 ) @ B2 )
=> ? [X: $o > $o] :
( ( member_o_o @ X
@ ( piE_o_o @ A2
@ ^ [I: $o] : B2 ) )
& ! [Xa: $o] :
( ( member_o @ Xa @ A2 )
=> ( ( X @ Xa )
= ( F @ Xa ) ) )
& ! [Y: $o > $o] :
( ( ( member_o_o @ Y
@ ( piE_o_o @ A2
@ ^ [I: $o] : B2 ) )
& ! [Xa2: $o] :
( ( member_o @ Xa2 @ A2 )
=> ( ( Y @ Xa2 )
= ( F @ Xa2 ) ) ) )
=> ( Y = X ) ) ) ) ).
% PiE_uniqueness
thf(fact_15_PiE__uniqueness,axiom,
! [F: nat > $o,A2: set_nat,B2: set_o] :
( ( ord_less_eq_set_o @ ( image_nat_o @ F @ A2 ) @ B2 )
=> ? [X: nat > $o] :
( ( member_nat_o @ X
@ ( piE_nat_o @ A2
@ ^ [I: nat] : B2 ) )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( X @ Xa )
= ( F @ Xa ) ) )
& ! [Y: nat > $o] :
( ( ( member_nat_o @ Y
@ ( piE_nat_o @ A2
@ ^ [I: nat] : B2 ) )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( Y @ Xa2 )
= ( F @ Xa2 ) ) ) )
=> ( Y = X ) ) ) ) ).
% PiE_uniqueness
thf(fact_16_PiE__uniqueness,axiom,
! [F: nat > set_nat,A2: set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A2 ) @ B2 )
=> ? [X: nat > set_nat] :
( ( member_nat_set_nat @ X
@ ( piE_nat_set_nat @ A2
@ ^ [I: nat] : B2 ) )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( X @ Xa )
= ( F @ Xa ) ) )
& ! [Y: nat > set_nat] :
( ( ( member_nat_set_nat @ Y
@ ( piE_nat_set_nat @ A2
@ ^ [I: nat] : B2 ) )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( Y @ Xa2 )
= ( F @ Xa2 ) ) ) )
=> ( Y = X ) ) ) ) ).
% PiE_uniqueness
thf(fact_17_PiE__uniqueness,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B2 )
=> ? [X: nat > nat] :
( ( member_nat_nat @ X
@ ( piE_nat_nat @ A2
@ ^ [I: nat] : B2 ) )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( X @ Xa )
= ( F @ Xa ) ) )
& ! [Y: nat > nat] :
( ( ( member_nat_nat @ Y
@ ( piE_nat_nat @ A2
@ ^ [I: nat] : B2 ) )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( Y @ Xa2 )
= ( F @ Xa2 ) ) ) )
=> ( Y = X ) ) ) ) ).
% PiE_uniqueness
thf(fact_18_PiE__uniqueness,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F @ A2 ) @ B2 )
=> ? [X: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X
@ ( piE_nat_nat_nat_nat @ A2
@ ^ [I: nat > nat] : B2 ) )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A2 )
=> ( ( X @ Xa )
= ( F @ Xa ) ) )
& ! [Y: ( nat > nat ) > nat > nat] :
( ( ( member952132173341509300at_nat @ Y
@ ( piE_nat_nat_nat_nat @ A2
@ ^ [I: nat > nat] : B2 ) )
& ! [Xa2: nat > nat] :
( ( member_nat_nat @ Xa2 @ A2 )
=> ( ( Y @ Xa2 )
= ( F @ Xa2 ) ) ) )
=> ( Y = X ) ) ) ) ).
% PiE_uniqueness
thf(fact_19_lessThan__subset__iff,axiom,
! [X2: $o,Y2: $o] :
( ( ord_less_eq_set_o @ ( set_ord_lessThan_o @ X2 ) @ ( set_ord_lessThan_o @ Y2 ) )
= ( ord_less_eq_o @ X2 @ Y2 ) ) ).
% lessThan_subset_iff
thf(fact_20_lessThan__subset__iff,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X2 ) @ ( set_ord_lessThan_nat @ Y2 ) )
= ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% lessThan_subset_iff
thf(fact_21_PiE__mono,axiom,
! [A2: set_nat,B2: nat > set_nat,C: nat > set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ X ) @ ( C @ X ) ) )
=> ( ord_le9059583361652607317at_nat @ ( piE_nat_nat @ A2 @ B2 ) @ ( piE_nat_nat @ A2 @ C ) ) ) ).
% PiE_mono
thf(fact_22_PiE__mono,axiom,
! [A2: set_o,B2: $o > set_nat_nat,C: $o > set_nat_nat] :
( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B2 @ X ) @ ( C @ X ) ) )
=> ( ord_le8808915593745164104at_nat @ ( piE_o_nat_nat @ A2 @ B2 ) @ ( piE_o_nat_nat @ A2 @ C ) ) ) ).
% PiE_mono
thf(fact_23_PiE__mono,axiom,
! [A2: set_nat,B2: nat > set_nat_nat,C: nat > set_nat_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B2 @ X ) @ ( C @ X ) ) )
=> ( ord_le3211623285424100676at_nat @ ( piE_nat_nat_nat2 @ A2 @ B2 ) @ ( piE_nat_nat_nat2 @ A2 @ C ) ) ) ).
% PiE_mono
thf(fact_24_PiE__mono,axiom,
! [A2: set_o,B2: $o > set_nat,C: $o > set_nat] :
( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ X ) @ ( C @ X ) ) )
=> ( ord_le4981610546006782297_o_nat @ ( piE_o_nat @ A2 @ B2 ) @ ( piE_o_nat @ A2 @ C ) ) ) ).
% PiE_mono
thf(fact_25_PiE__mono,axiom,
! [A2: set_o,B2: $o > set_o,C: $o > set_o] :
( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ord_less_eq_set_o @ ( B2 @ X ) @ ( C @ X ) ) )
=> ( ord_less_eq_set_o_o @ ( piE_o_o @ A2 @ B2 ) @ ( piE_o_o @ A2 @ C ) ) ) ).
% PiE_mono
thf(fact_26_PiE__mono,axiom,
! [A2: set_nat,B2: nat > set_o,C: nat > set_o] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_o @ ( B2 @ X ) @ ( C @ X ) ) )
=> ( ord_le6029213668185085951_nat_o @ ( piE_nat_o @ A2 @ B2 ) @ ( piE_nat_o @ A2 @ C ) ) ) ).
% PiE_mono
thf(fact_27_PiE__mono,axiom,
! [A2: set_set_nat,B2: set_nat > set_nat,C: set_nat > set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ X ) @ ( C @ X ) ) )
=> ( ord_le4786761258823227915at_nat @ ( piE_set_nat_nat @ A2 @ B2 ) @ ( piE_set_nat_nat @ A2 @ C ) ) ) ).
% PiE_mono
thf(fact_28_PiE__mono,axiom,
! [A2: set_o,B2: $o > set_set_nat,C: $o > set_set_nat] :
( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( B2 @ X ) @ ( C @ X ) ) )
=> ( ord_le5010423853744249615et_nat @ ( piE_o_set_nat @ A2 @ B2 ) @ ( piE_o_set_nat @ A2 @ C ) ) ) ).
% PiE_mono
thf(fact_29_PiE__mono,axiom,
! [A2: set_nat,B2: nat > set_set_nat,C: nat > set_set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( B2 @ X ) @ ( C @ X ) ) )
=> ( ord_le1585852046946910987et_nat @ ( piE_nat_set_nat @ A2 @ B2 ) @ ( piE_nat_set_nat @ A2 @ C ) ) ) ).
% PiE_mono
thf(fact_30_PiE__mono,axiom,
! [A2: set_set_nat,B2: set_nat > set_o,C: set_nat > set_o] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( ord_less_eq_set_o @ ( B2 @ X ) @ ( C @ X ) ) )
=> ( ord_le1935226719417650633_nat_o @ ( piE_set_nat_o @ A2 @ B2 ) @ ( piE_set_nat_o @ A2 @ C ) ) ) ).
% PiE_mono
thf(fact_31_image__ident,axiom,
! [Y3: set_nat] :
( ( image_nat_nat
@ ^ [X3: nat] : X3
@ Y3 )
= Y3 ) ).
% image_ident
thf(fact_32_image__ident,axiom,
! [Y3: set_o] :
( ( image_o_o
@ ^ [X3: $o] : X3
@ Y3 )
= Y3 ) ).
% image_ident
thf(fact_33_image__ident,axiom,
! [Y3: set_nat_nat] :
( ( image_3205354838064109189at_nat
@ ^ [X3: nat > nat] : X3
@ Y3 )
= Y3 ) ).
% image_ident
thf(fact_34_lessThan__eq__iff,axiom,
! [X2: nat,Y2: nat] :
( ( ( set_ord_lessThan_nat @ X2 )
= ( set_ord_lessThan_nat @ Y2 ) )
= ( X2 = Y2 ) ) ).
% lessThan_eq_iff
thf(fact_35_subsetI,axiom,
! [A2: set_nat_nat_nat_nat,B2: set_nat_nat_nat_nat] :
( ! [X: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X @ A2 )
=> ( member952132173341509300at_nat @ X @ B2 ) )
=> ( ord_le5260717879541182899at_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_36_subsetI,axiom,
! [A2: set_nat_set_nat,B2: set_nat_set_nat] :
( ! [X: nat > set_nat] :
( ( member_nat_set_nat @ X @ A2 )
=> ( member_nat_set_nat @ X @ B2 ) )
=> ( ord_le1585852046946910987et_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_37_subsetI,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( member_set_nat @ X @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_38_subsetI,axiom,
! [A2: set_nat_nat_nat,B2: set_nat_nat_nat] :
( ! [X: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X @ A2 )
=> ( member_nat_nat_nat2 @ X @ B2 ) )
=> ( ord_le3211623285424100676at_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_39_subsetI,axiom,
! [A2: set_o_nat_nat,B2: set_o_nat_nat] :
( ! [X: $o > nat > nat] :
( ( member_o_nat_nat @ X @ A2 )
=> ( member_o_nat_nat @ X @ B2 ) )
=> ( ord_le8808915593745164104at_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_40_subsetI,axiom,
! [A2: set_o,B2: set_o] :
( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( member_o @ X @ B2 ) )
=> ( ord_less_eq_set_o @ A2 @ B2 ) ) ).
% subsetI
thf(fact_41_subsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B2 ) )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_42_subsetI,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ! [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( member_nat_nat @ X @ B2 ) )
=> ( ord_le9059583361652607317at_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_43_subset__antisym,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_44_subset__antisym,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_45_subset__antisym,axiom,
! [A2: set_nat_nat_nat,B2: set_nat_nat_nat] :
( ( ord_le3211623285424100676at_nat @ A2 @ B2 )
=> ( ( ord_le3211623285424100676at_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_46_subset__antisym,axiom,
! [A2: set_o_nat_nat,B2: set_o_nat_nat] :
( ( ord_le8808915593745164104at_nat @ A2 @ B2 )
=> ( ( ord_le8808915593745164104at_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_47_subset__antisym,axiom,
! [A2: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ( ord_less_eq_set_o @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_48_subset__antisym,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_49_image__eqI,axiom,
! [B: $o,F: $o > $o,X2: $o,A2: set_o] :
( ( B
= ( F @ X2 ) )
=> ( ( member_o @ X2 @ A2 )
=> ( member_o @ B @ ( image_o_o @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_50_image__eqI,axiom,
! [B: nat,F: $o > nat,X2: $o,A2: set_o] :
( ( B
= ( F @ X2 ) )
=> ( ( member_o @ X2 @ A2 )
=> ( member_nat @ B @ ( image_o_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_51_image__eqI,axiom,
! [B: $o,F: nat > $o,X2: nat,A2: set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_o @ B @ ( image_nat_o @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_52_image__eqI,axiom,
! [B: nat,F: nat > nat,X2: nat,A2: set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_53_image__eqI,axiom,
! [B: set_nat,F: nat > set_nat,X2: nat,A2: set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_set_nat @ B @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_54_image__eqI,axiom,
! [B: nat > nat,F: ( nat > nat ) > nat > nat,X2: nat > nat,A2: set_nat_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat_nat @ X2 @ A2 )
=> ( member_nat_nat @ B @ ( image_3205354838064109189at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_55_image__eqI,axiom,
! [B: set_nat,F: $o > set_nat,X2: $o,A2: set_o] :
( ( B
= ( F @ X2 ) )
=> ( ( member_o @ X2 @ A2 )
=> ( member_set_nat @ B @ ( image_o_set_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_56_image__eqI,axiom,
! [B: $o,F: set_nat > $o,X2: set_nat,A2: set_set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_set_nat @ X2 @ A2 )
=> ( member_o @ B @ ( image_set_nat_o @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_57_image__eqI,axiom,
! [B: nat,F: set_nat > nat,X2: set_nat,A2: set_set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_set_nat @ X2 @ A2 )
=> ( member_nat @ B @ ( image_set_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_58_image__eqI,axiom,
! [B: nat > nat,F: $o > nat > nat,X2: $o,A2: set_o] :
( ( B
= ( F @ X2 ) )
=> ( ( member_o @ X2 @ A2 )
=> ( member_nat_nat @ B @ ( image_o_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_59_order__refl,axiom,
! [X2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_60_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_61_order__refl,axiom,
! [X2: set_nat] : ( ord_less_eq_set_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_62_order__refl,axiom,
! [X2: set_set_nat] : ( ord_le6893508408891458716et_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_63_order__refl,axiom,
! [X2: set_nat_nat_nat] : ( ord_le3211623285424100676at_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_64_order__refl,axiom,
! [X2: set_o_nat_nat] : ( ord_le8808915593745164104at_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_65_order__refl,axiom,
! [X2: set_o] : ( ord_less_eq_set_o @ X2 @ X2 ) ).
% order_refl
thf(fact_66_dual__order_Orefl,axiom,
! [A: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A @ A ) ).
% dual_order.refl
thf(fact_67_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_68_dual__order_Orefl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% dual_order.refl
thf(fact_69_dual__order_Orefl,axiom,
! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ A @ A ) ).
% dual_order.refl
thf(fact_70_dual__order_Orefl,axiom,
! [A: set_nat_nat_nat] : ( ord_le3211623285424100676at_nat @ A @ A ) ).
% dual_order.refl
thf(fact_71_dual__order_Orefl,axiom,
! [A: set_o_nat_nat] : ( ord_le8808915593745164104at_nat @ A @ A ) ).
% dual_order.refl
thf(fact_72_dual__order_Orefl,axiom,
! [A: set_o] : ( ord_less_eq_set_o @ A @ A ) ).
% dual_order.refl
thf(fact_73_image__Collect__subsetI,axiom,
! [P: $o > $o,F: $o > nat,B2: set_nat] :
( ! [X: $o] :
( ( P @ X )
=> ( member_nat @ ( F @ X ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_o_nat @ F @ ( collect_o @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_74_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > nat,B2: set_nat] :
( ! [X: nat] :
( ( P @ X )
=> ( member_nat @ ( F @ X ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( collect_nat @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_75_image__Collect__subsetI,axiom,
! [P: $o > $o,F: $o > $o,B2: set_o] :
( ! [X: $o] :
( ( P @ X )
=> ( member_o @ ( F @ X ) @ B2 ) )
=> ( ord_less_eq_set_o @ ( image_o_o @ F @ ( collect_o @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_76_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > $o,B2: set_o] :
( ! [X: nat] :
( ( P @ X )
=> ( member_o @ ( F @ X ) @ B2 ) )
=> ( ord_less_eq_set_o @ ( image_nat_o @ F @ ( collect_nat @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_77_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > set_nat,B2: set_set_nat] :
( ! [X: nat] :
( ( P @ X )
=> ( member_set_nat @ ( F @ X ) @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ ( collect_nat @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_78_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > nat > nat,B2: set_nat_nat] :
( ! [X: nat] :
( ( P @ X )
=> ( member_nat_nat @ ( F @ X ) @ B2 ) )
=> ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ ( collect_nat @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_79_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > nat > set_nat,B2: set_nat_set_nat] :
( ! [X: nat] :
( ( P @ X )
=> ( member_nat_set_nat @ ( F @ X ) @ B2 ) )
=> ( ord_le1585852046946910987et_nat @ ( image_4436799006340492620et_nat @ F @ ( collect_nat @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_80_image__Collect__subsetI,axiom,
! [P: ( nat > nat ) > $o,F: ( nat > nat ) > nat > nat,B2: set_nat_nat] :
( ! [X: nat > nat] :
( ( P @ X )
=> ( member_nat_nat @ ( F @ X ) @ B2 ) )
=> ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F @ ( collect_nat_nat @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_81_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > nat > nat > nat,B2: set_nat_nat_nat] :
( ! [X: nat] :
( ( P @ X )
=> ( member_nat_nat_nat2 @ ( F @ X ) @ B2 ) )
=> ( ord_le3211623285424100676at_nat @ ( image_6919068903512877573at_nat @ F @ ( collect_nat @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_82_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > $o > nat > nat,B2: set_o_nat_nat] :
( ! [X: nat] :
( ( P @ X )
=> ( member_o_nat_nat @ ( F @ X ) @ B2 ) )
=> ( ord_le8808915593745164104at_nat @ ( image_nat_o_nat_nat @ F @ ( collect_nat @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_83_image__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_84_image__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > $o] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_o @ ( image_nat_o @ F @ A2 ) @ ( image_nat_o @ F @ B2 ) ) ) ).
% image_mono
thf(fact_85_image__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > nat] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ ( image_o_nat @ F @ A2 ) @ ( image_o_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_86_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_87_image__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A2 ) @ ( image_nat_set_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_88_image__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat,F: set_nat > nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ ( image_set_nat_nat @ F @ A2 ) @ ( image_set_nat_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_89_image__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat,F: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ord_less_eq_set_o @ ( image_set_nat_o @ F @ A2 ) @ ( image_set_nat_o @ F @ B2 ) ) ) ).
% image_mono
thf(fact_90_image__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > set_nat] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( image_o_set_nat @ F @ A2 ) @ ( image_o_set_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_91_image__mono,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,F: ( nat > nat ) > nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ ( image_nat_nat_nat @ F @ A2 ) @ ( image_nat_nat_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_92_image__mono,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,F: ( nat > nat ) > $o] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ord_less_eq_set_o @ ( image_nat_nat_o @ F @ A2 ) @ ( image_nat_nat_o @ F @ B2 ) ) ) ).
% image_mono
thf(fact_93_image__subsetI,axiom,
! [A2: set_o,F: $o > nat,B2: set_nat] :
( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_o_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_94_image__subsetI,axiom,
! [A2: set_nat,F: nat > nat,B2: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_95_image__subsetI,axiom,
! [A2: set_o,F: $o > $o,B2: set_o] :
( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( member_o @ ( F @ X ) @ B2 ) )
=> ( ord_less_eq_set_o @ ( image_o_o @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_96_image__subsetI,axiom,
! [A2: set_nat,F: nat > $o,B2: set_o] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_o @ ( F @ X ) @ B2 ) )
=> ( ord_less_eq_set_o @ ( image_nat_o @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_97_image__subsetI,axiom,
! [A2: set_set_nat,F: set_nat > nat,B2: set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_set_nat_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_98_image__subsetI,axiom,
! [A2: set_o,F: $o > set_nat,B2: set_set_nat] :
( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( member_set_nat @ ( F @ X ) @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ ( image_o_set_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_99_image__subsetI,axiom,
! [A2: set_nat,F: nat > set_nat,B2: set_set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_set_nat @ ( F @ X ) @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_100_image__subsetI,axiom,
! [A2: set_set_nat,F: set_nat > $o,B2: set_o] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( member_o @ ( F @ X ) @ B2 ) )
=> ( ord_less_eq_set_o @ ( image_set_nat_o @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_101_image__subsetI,axiom,
! [A2: set_o,F: $o > nat > nat,B2: set_nat_nat] :
( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( member_nat_nat @ ( F @ X ) @ B2 ) )
=> ( ord_le9059583361652607317at_nat @ ( image_o_nat_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_102_image__subsetI,axiom,
! [A2: set_nat,F: nat > nat > nat,B2: set_nat_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat_nat @ ( F @ X ) @ B2 ) )
=> ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_103_less__eq__set__def,axiom,
( ord_le5260717879541182899at_nat
= ( ^ [A3: set_nat_nat_nat_nat,B3: set_nat_nat_nat_nat] :
( ord_le5430825838364970130_nat_o
@ ^ [X3: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X3 @ A3 )
@ ^ [X3: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X3 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_104_less__eq__set__def,axiom,
( ord_le1585852046946910987et_nat
= ( ^ [A3: set_nat_set_nat,B3: set_nat_set_nat] :
( ord_le8865062304692155706_nat_o
@ ^ [X3: nat > set_nat] : ( member_nat_set_nat @ X3 @ A3 )
@ ^ [X3: nat > set_nat] : ( member_nat_set_nat @ X3 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_105_less__eq__set__def,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( ord_le7366121074344172400_nat_o
@ ^ [X3: nat > nat] : ( member_nat_nat @ X3 @ A3 )
@ ^ [X3: nat > nat] : ( member_nat_nat @ X3 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_106_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ord_less_eq_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A3 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_107_less__eq__set__def,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ord_le3964352015994296041_nat_o
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A3 )
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_108_less__eq__set__def,axiom,
( ord_le3211623285424100676at_nat
= ( ^ [A3: set_nat_nat_nat,B3: set_nat_nat_nat] :
( ord_le5384859702510996545_nat_o
@ ^ [X3: nat > nat > nat] : ( member_nat_nat_nat2 @ X3 @ A3 )
@ ^ [X3: nat > nat > nat] : ( member_nat_nat_nat2 @ X3 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_109_less__eq__set__def,axiom,
( ord_le8808915593745164104at_nat
= ( ^ [A3: set_o_nat_nat,B3: set_o_nat_nat] :
( ord_le6871787456926423957_nat_o
@ ^ [X3: $o > nat > nat] : ( member_o_nat_nat @ X3 @ A3 )
@ ^ [X3: $o > nat > nat] : ( member_o_nat_nat @ X3 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_110_less__eq__set__def,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B3: set_o] :
( ord_less_eq_o_o
@ ^ [X3: $o] : ( member_o @ X3 @ A3 )
@ ^ [X3: $o] : ( member_o @ X3 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_111_order__antisym__conv,axiom,
! [Y2: set_nat_nat,X2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y2 @ X2 )
=> ( ( ord_le9059583361652607317at_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_112_order__antisym__conv,axiom,
! [Y2: nat,X2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_113_order__antisym__conv,axiom,
! [Y2: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_114_order__antisym__conv,axiom,
! [Y2: set_set_nat,X2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y2 @ X2 )
=> ( ( ord_le6893508408891458716et_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_115_order__antisym__conv,axiom,
! [Y2: set_nat_nat_nat,X2: set_nat_nat_nat] :
( ( ord_le3211623285424100676at_nat @ Y2 @ X2 )
=> ( ( ord_le3211623285424100676at_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_116_order__antisym__conv,axiom,
! [Y2: set_o_nat_nat,X2: set_o_nat_nat] :
( ( ord_le8808915593745164104at_nat @ Y2 @ X2 )
=> ( ( ord_le8808915593745164104at_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_117_order__antisym__conv,axiom,
! [Y2: set_o,X2: set_o] :
( ( ord_less_eq_set_o @ Y2 @ X2 )
=> ( ( ord_less_eq_set_o @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_118_linorder__le__cases,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linorder_le_cases
thf(fact_119_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_120_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_nat,C2: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_121_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_o,C2: set_o] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_set_o @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_o @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_122_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C2: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_123_ord__le__eq__subst,axiom,
! [A: set_o,B: set_o,F: set_o > nat,C2: nat] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X: set_o,Y4: set_o] :
( ( ord_less_eq_set_o @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_124_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_set_nat,C2: set_set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_125_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_126_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_o,C2: set_o] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_set_o @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_o @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_127_ord__le__eq__subst,axiom,
! [A: set_set_nat,B: set_set_nat,F: set_set_nat > nat,C2: nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X: set_set_nat,Y4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_128_ord__le__eq__subst,axiom,
! [A: set_o,B: set_o,F: set_o > set_nat,C2: set_nat] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X: set_o,Y4: set_o] :
( ( ord_less_eq_set_o @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_129_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_130_ord__eq__le__subst,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_131_ord__eq__le__subst,axiom,
! [A: set_o,F: nat > set_o,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_set_o @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_o @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_132_ord__eq__le__subst,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C2: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_133_ord__eq__le__subst,axiom,
! [A: nat,F: set_o > nat,B: set_o,C2: set_o] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_o @ B @ C2 )
=> ( ! [X: set_o,Y4: set_o] :
( ( ord_less_eq_set_o @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_134_ord__eq__le__subst,axiom,
! [A: set_set_nat,F: nat > set_set_nat,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le6893508408891458716et_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_135_ord__eq__le__subst,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C2: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_136_ord__eq__le__subst,axiom,
! [A: set_o,F: set_nat > set_o,B: set_nat,C2: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_set_o @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_o @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_137_ord__eq__le__subst,axiom,
! [A: nat,F: set_set_nat > nat,B: set_set_nat,C2: set_set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le6893508408891458716et_nat @ B @ C2 )
=> ( ! [X: set_set_nat,Y4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_138_ord__eq__le__subst,axiom,
! [A: set_nat,F: set_o > set_nat,B: set_o,C2: set_o] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_o @ B @ C2 )
=> ( ! [X: set_o,Y4: set_o] :
( ( ord_less_eq_set_o @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_139_linorder__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linorder_linear
thf(fact_140_order__eq__refl,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat] :
( ( X2 = Y2 )
=> ( ord_le9059583361652607317at_nat @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_141_order__eq__refl,axiom,
! [X2: nat,Y2: nat] :
( ( X2 = Y2 )
=> ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_142_order__eq__refl,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( X2 = Y2 )
=> ( ord_less_eq_set_nat @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_143_order__eq__refl,axiom,
! [X2: set_set_nat,Y2: set_set_nat] :
( ( X2 = Y2 )
=> ( ord_le6893508408891458716et_nat @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_144_order__eq__refl,axiom,
! [X2: set_nat_nat_nat,Y2: set_nat_nat_nat] :
( ( X2 = Y2 )
=> ( ord_le3211623285424100676at_nat @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_145_order__eq__refl,axiom,
! [X2: set_o_nat_nat,Y2: set_o_nat_nat] :
( ( X2 = Y2 )
=> ( ord_le8808915593745164104at_nat @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_146_order__eq__refl,axiom,
! [X2: set_o,Y2: set_o] :
( ( X2 = Y2 )
=> ( ord_less_eq_set_o @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_147_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 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_148_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C2: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_149_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_o,C2: set_o] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_set_o @ ( F @ B ) @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_set_o @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_o @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_150_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C2: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_151_order__subst2,axiom,
! [A: set_o,B: set_o,F: set_o > nat,C2: nat] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X: set_o,Y4: set_o] :
( ( ord_less_eq_set_o @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_152_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_set_nat,C2: set_set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ ( F @ B ) @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_153_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C2 )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_154_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_o,C2: set_o] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_o @ ( F @ B ) @ C2 )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_set_o @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_o @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_155_order__subst2,axiom,
! [A: set_set_nat,B: set_set_nat,F: set_set_nat > nat,C2: nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X: set_set_nat,Y4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_156_order__subst2,axiom,
! [A: set_o,B: set_o,F: set_o > set_nat,C2: set_nat] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C2 )
=> ( ! [X: set_o,Y4: set_o] :
( ( ord_less_eq_set_o @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_157_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 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_158_order__subst1,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_159_order__subst1,axiom,
! [A: nat,F: set_o > nat,B: set_o,C2: set_o] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_o @ B @ C2 )
=> ( ! [X: set_o,Y4: set_o] :
( ( ord_less_eq_set_o @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_160_order__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C2: nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_161_order__subst1,axiom,
! [A: set_o,F: nat > set_o,B: nat,C2: nat] :
( ( ord_less_eq_set_o @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_set_o @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_o @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_162_order__subst1,axiom,
! [A: nat,F: set_set_nat > nat,B: set_set_nat,C2: set_set_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le6893508408891458716et_nat @ B @ C2 )
=> ( ! [X: set_set_nat,Y4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_163_order__subst1,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_164_order__subst1,axiom,
! [A: set_nat,F: set_o > set_nat,B: set_o,C2: set_o] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_o @ B @ C2 )
=> ( ! [X: set_o,Y4: set_o] :
( ( ord_less_eq_set_o @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_165_order__subst1,axiom,
! [A: set_set_nat,F: nat > set_set_nat,B: nat,C2: nat] :
( ( ord_le6893508408891458716et_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le6893508408891458716et_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_166_order__subst1,axiom,
! [A: set_o,F: set_nat > set_o,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_o @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_set_o @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_o @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_167_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_nat_nat,Z: set_nat_nat] : ( Y5 = Z ) )
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A4 @ B4 )
& ( ord_le9059583361652607317at_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_168_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_169_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_nat,Z: set_nat] : ( Y5 = Z ) )
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_170_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_set_nat,Z: set_set_nat] : ( Y5 = Z ) )
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A4 @ B4 )
& ( ord_le6893508408891458716et_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_171_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_nat_nat_nat,Z: set_nat_nat_nat] : ( Y5 = Z ) )
= ( ^ [A4: set_nat_nat_nat,B4: set_nat_nat_nat] :
( ( ord_le3211623285424100676at_nat @ A4 @ B4 )
& ( ord_le3211623285424100676at_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_172_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_o_nat_nat,Z: set_o_nat_nat] : ( Y5 = Z ) )
= ( ^ [A4: set_o_nat_nat,B4: set_o_nat_nat] :
( ( ord_le8808915593745164104at_nat @ A4 @ B4 )
& ( ord_le8808915593745164104at_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_173_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_o,Z: set_o] : ( Y5 = Z ) )
= ( ^ [A4: set_o,B4: set_o] :
( ( ord_less_eq_set_o @ A4 @ B4 )
& ( ord_less_eq_set_o @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_174_antisym,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_175_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_176_antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_177_antisym,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_178_antisym,axiom,
! [A: set_nat_nat_nat,B: set_nat_nat_nat] :
( ( ord_le3211623285424100676at_nat @ A @ B )
=> ( ( ord_le3211623285424100676at_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_179_antisym,axiom,
! [A: set_o_nat_nat,B: set_o_nat_nat] :
( ( ord_le8808915593745164104at_nat @ A @ B )
=> ( ( ord_le8808915593745164104at_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_180_antisym,axiom,
! [A: set_o,B: set_o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( ord_less_eq_set_o @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_181_dual__order_Otrans,axiom,
! [B: set_nat_nat,A: set_nat_nat,C2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B @ A )
=> ( ( ord_le9059583361652607317at_nat @ C2 @ B )
=> ( ord_le9059583361652607317at_nat @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_182_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_183_dual__order_Otrans,axiom,
! [B: set_nat,A: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C2 @ B )
=> ( ord_less_eq_set_nat @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_184_dual__order_Otrans,axiom,
! [B: set_set_nat,A: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( ( ord_le6893508408891458716et_nat @ C2 @ B )
=> ( ord_le6893508408891458716et_nat @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_185_dual__order_Otrans,axiom,
! [B: set_nat_nat_nat,A: set_nat_nat_nat,C2: set_nat_nat_nat] :
( ( ord_le3211623285424100676at_nat @ B @ A )
=> ( ( ord_le3211623285424100676at_nat @ C2 @ B )
=> ( ord_le3211623285424100676at_nat @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_186_dual__order_Otrans,axiom,
! [B: set_o_nat_nat,A: set_o_nat_nat,C2: set_o_nat_nat] :
( ( ord_le8808915593745164104at_nat @ B @ A )
=> ( ( ord_le8808915593745164104at_nat @ C2 @ B )
=> ( ord_le8808915593745164104at_nat @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_187_dual__order_Otrans,axiom,
! [B: set_o,A: set_o,C2: set_o] :
( ( ord_less_eq_set_o @ B @ A )
=> ( ( ord_less_eq_set_o @ C2 @ B )
=> ( ord_less_eq_set_o @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_188_dual__order_Oantisym,axiom,
! [B: set_nat_nat,A: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B @ A )
=> ( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_189_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_190_dual__order_Oantisym,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_191_dual__order_Oantisym,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_192_dual__order_Oantisym,axiom,
! [B: set_nat_nat_nat,A: set_nat_nat_nat] :
( ( ord_le3211623285424100676at_nat @ B @ A )
=> ( ( ord_le3211623285424100676at_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_193_dual__order_Oantisym,axiom,
! [B: set_o_nat_nat,A: set_o_nat_nat] :
( ( ord_le8808915593745164104at_nat @ B @ A )
=> ( ( ord_le8808915593745164104at_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_194_dual__order_Oantisym,axiom,
! [B: set_o,A: set_o] :
( ( ord_less_eq_set_o @ B @ A )
=> ( ( ord_less_eq_set_o @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_195_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_nat_nat,Z: set_nat_nat] : ( Y5 = Z ) )
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B4 @ A4 )
& ( ord_le9059583361652607317at_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_196_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_197_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_nat,Z: set_nat] : ( Y5 = Z ) )
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A4 )
& ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_198_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_set_nat,Z: set_set_nat] : ( Y5 = Z ) )
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B4 @ A4 )
& ( ord_le6893508408891458716et_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_199_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_nat_nat_nat,Z: set_nat_nat_nat] : ( Y5 = Z ) )
= ( ^ [A4: set_nat_nat_nat,B4: set_nat_nat_nat] :
( ( ord_le3211623285424100676at_nat @ B4 @ A4 )
& ( ord_le3211623285424100676at_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_200_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_o_nat_nat,Z: set_o_nat_nat] : ( Y5 = Z ) )
= ( ^ [A4: set_o_nat_nat,B4: set_o_nat_nat] :
( ( ord_le8808915593745164104at_nat @ B4 @ A4 )
& ( ord_le8808915593745164104at_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_201_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_o,Z: set_o] : ( Y5 = Z ) )
= ( ^ [A4: set_o,B4: set_o] :
( ( ord_less_eq_set_o @ B4 @ A4 )
& ( ord_less_eq_set_o @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_202_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: nat,B5: nat] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_203_order__trans,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat,Z2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y2 )
=> ( ( ord_le9059583361652607317at_nat @ Y2 @ Z2 )
=> ( ord_le9059583361652607317at_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_204_order__trans,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ( ord_less_eq_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_205_order__trans,axiom,
! [X2: set_nat,Y2: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_set_nat @ Y2 @ Z2 )
=> ( ord_less_eq_set_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_206_order__trans,axiom,
! [X2: set_set_nat,Y2: set_set_nat,Z2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y2 )
=> ( ( ord_le6893508408891458716et_nat @ Y2 @ Z2 )
=> ( ord_le6893508408891458716et_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_207_order__trans,axiom,
! [X2: set_nat_nat_nat,Y2: set_nat_nat_nat,Z2: set_nat_nat_nat] :
( ( ord_le3211623285424100676at_nat @ X2 @ Y2 )
=> ( ( ord_le3211623285424100676at_nat @ Y2 @ Z2 )
=> ( ord_le3211623285424100676at_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_208_order__trans,axiom,
! [X2: set_o_nat_nat,Y2: set_o_nat_nat,Z2: set_o_nat_nat] :
( ( ord_le8808915593745164104at_nat @ X2 @ Y2 )
=> ( ( ord_le8808915593745164104at_nat @ Y2 @ Z2 )
=> ( ord_le8808915593745164104at_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_209_order__trans,axiom,
! [X2: set_o,Y2: set_o,Z2: set_o] :
( ( ord_less_eq_set_o @ X2 @ Y2 )
=> ( ( ord_less_eq_set_o @ Y2 @ Z2 )
=> ( ord_less_eq_set_o @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_210_order_Otrans,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ B @ C2 )
=> ( ord_le9059583361652607317at_nat @ A @ C2 ) ) ) ).
% order.trans
thf(fact_211_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_212_order_Otrans,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% order.trans
thf(fact_213_order_Otrans,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ C2 )
=> ( ord_le6893508408891458716et_nat @ A @ C2 ) ) ) ).
% order.trans
thf(fact_214_order_Otrans,axiom,
! [A: set_nat_nat_nat,B: set_nat_nat_nat,C2: set_nat_nat_nat] :
( ( ord_le3211623285424100676at_nat @ A @ B )
=> ( ( ord_le3211623285424100676at_nat @ B @ C2 )
=> ( ord_le3211623285424100676at_nat @ A @ C2 ) ) ) ).
% order.trans
thf(fact_215_order_Otrans,axiom,
! [A: set_o_nat_nat,B: set_o_nat_nat,C2: set_o_nat_nat] :
( ( ord_le8808915593745164104at_nat @ A @ B )
=> ( ( ord_le8808915593745164104at_nat @ B @ C2 )
=> ( ord_le8808915593745164104at_nat @ A @ C2 ) ) ) ).
% order.trans
thf(fact_216_order_Otrans,axiom,
! [A: set_o,B: set_o,C2: set_o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( ord_less_eq_set_o @ B @ C2 )
=> ( ord_less_eq_set_o @ A @ C2 ) ) ) ).
% order.trans
thf(fact_217_order__antisym,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y2 )
=> ( ( ord_le9059583361652607317at_nat @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_218_order__antisym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_219_order__antisym,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_set_nat @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_220_order__antisym,axiom,
! [X2: set_set_nat,Y2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y2 )
=> ( ( ord_le6893508408891458716et_nat @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_221_order__antisym,axiom,
! [X2: set_nat_nat_nat,Y2: set_nat_nat_nat] :
( ( ord_le3211623285424100676at_nat @ X2 @ Y2 )
=> ( ( ord_le3211623285424100676at_nat @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_222_order__antisym,axiom,
! [X2: set_o_nat_nat,Y2: set_o_nat_nat] :
( ( ord_le8808915593745164104at_nat @ X2 @ Y2 )
=> ( ( ord_le8808915593745164104at_nat @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_223_order__antisym,axiom,
! [X2: set_o,Y2: set_o] :
( ( ord_less_eq_set_o @ X2 @ Y2 )
=> ( ( ord_less_eq_set_o @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_224_ord__le__eq__trans,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( B = C2 )
=> ( ord_le9059583361652607317at_nat @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_225_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_226_ord__le__eq__trans,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_227_ord__le__eq__trans,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( B = C2 )
=> ( ord_le6893508408891458716et_nat @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_228_ord__le__eq__trans,axiom,
! [A: set_nat_nat_nat,B: set_nat_nat_nat,C2: set_nat_nat_nat] :
( ( ord_le3211623285424100676at_nat @ A @ B )
=> ( ( B = C2 )
=> ( ord_le3211623285424100676at_nat @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_229_ord__le__eq__trans,axiom,
! [A: set_o_nat_nat,B: set_o_nat_nat,C2: set_o_nat_nat] :
( ( ord_le8808915593745164104at_nat @ A @ B )
=> ( ( B = C2 )
=> ( ord_le8808915593745164104at_nat @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_230_ord__le__eq__trans,axiom,
! [A: set_o,B: set_o,C2: set_o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_set_o @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_231_ord__eq__le__trans,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( A = B )
=> ( ( ord_le9059583361652607317at_nat @ B @ C2 )
=> ( ord_le9059583361652607317at_nat @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_232_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_233_ord__eq__le__trans,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( A = B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_234_ord__eq__le__trans,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( A = B )
=> ( ( ord_le6893508408891458716et_nat @ B @ C2 )
=> ( ord_le6893508408891458716et_nat @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_235_ord__eq__le__trans,axiom,
! [A: set_nat_nat_nat,B: set_nat_nat_nat,C2: set_nat_nat_nat] :
( ( A = B )
=> ( ( ord_le3211623285424100676at_nat @ B @ C2 )
=> ( ord_le3211623285424100676at_nat @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_236_ord__eq__le__trans,axiom,
! [A: set_o_nat_nat,B: set_o_nat_nat,C2: set_o_nat_nat] :
( ( A = B )
=> ( ( ord_le8808915593745164104at_nat @ B @ C2 )
=> ( ord_le8808915593745164104at_nat @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_237_ord__eq__le__trans,axiom,
! [A: set_o,B: set_o,C2: set_o] :
( ( A = B )
=> ( ( ord_less_eq_set_o @ B @ C2 )
=> ( ord_less_eq_set_o @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_238_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_nat_nat,Z: set_nat_nat] : ( Y5 = Z ) )
= ( ^ [X3: set_nat_nat,Y6: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y6 )
& ( ord_le9059583361652607317at_nat @ Y6 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_239_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
= ( ^ [X3: nat,Y6: nat] :
( ( ord_less_eq_nat @ X3 @ Y6 )
& ( ord_less_eq_nat @ Y6 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_240_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_nat,Z: set_nat] : ( Y5 = Z ) )
= ( ^ [X3: set_nat,Y6: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y6 )
& ( ord_less_eq_set_nat @ Y6 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_241_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_set_nat,Z: set_set_nat] : ( Y5 = Z ) )
= ( ^ [X3: set_set_nat,Y6: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y6 )
& ( ord_le6893508408891458716et_nat @ Y6 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_242_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_nat_nat_nat,Z: set_nat_nat_nat] : ( Y5 = Z ) )
= ( ^ [X3: set_nat_nat_nat,Y6: set_nat_nat_nat] :
( ( ord_le3211623285424100676at_nat @ X3 @ Y6 )
& ( ord_le3211623285424100676at_nat @ Y6 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_243_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_o_nat_nat,Z: set_o_nat_nat] : ( Y5 = Z ) )
= ( ^ [X3: set_o_nat_nat,Y6: set_o_nat_nat] :
( ( ord_le8808915593745164104at_nat @ X3 @ Y6 )
& ( ord_le8808915593745164104at_nat @ Y6 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_244_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_o,Z: set_o] : ( Y5 = Z ) )
= ( ^ [X3: set_o,Y6: set_o] :
( ( ord_less_eq_set_o @ X3 @ Y6 )
& ( ord_less_eq_set_o @ Y6 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_245_le__cases3,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_246_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_247_rev__image__eqI,axiom,
! [X2: $o,A2: set_o,B: $o,F: $o > $o] :
( ( member_o @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_o @ B @ ( image_o_o @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_248_rev__image__eqI,axiom,
! [X2: $o,A2: set_o,B: nat,F: $o > nat] :
( ( member_o @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_nat @ B @ ( image_o_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_249_rev__image__eqI,axiom,
! [X2: nat,A2: set_nat,B: $o,F: nat > $o] :
( ( member_nat @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_o @ B @ ( image_nat_o @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_250_rev__image__eqI,axiom,
! [X2: nat,A2: set_nat,B: nat,F: nat > nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_251_rev__image__eqI,axiom,
! [X2: $o,A2: set_o,B: set_nat,F: $o > set_nat] :
( ( member_o @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_set_nat @ B @ ( image_o_set_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_252_rev__image__eqI,axiom,
! [X2: nat,A2: set_nat,B: set_nat,F: nat > set_nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_set_nat @ B @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_253_rev__image__eqI,axiom,
! [X2: set_nat,A2: set_set_nat,B: $o,F: set_nat > $o] :
( ( member_set_nat @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_o @ B @ ( image_set_nat_o @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_254_rev__image__eqI,axiom,
! [X2: set_nat,A2: set_set_nat,B: nat,F: set_nat > nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_nat @ B @ ( image_set_nat_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_255_rev__image__eqI,axiom,
! [X2: $o,A2: set_o,B: nat > nat,F: $o > nat > nat] :
( ( member_o @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_nat_nat @ B @ ( image_o_nat_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_256_rev__image__eqI,axiom,
! [X2: nat,A2: set_nat,B: nat > nat,F: nat > nat > nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_nat_nat @ B @ ( image_nat_nat_nat2 @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_257_ball__imageD,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,P: ( nat > nat ) > $o] :
( ! [X: nat > nat] :
( ( member_nat_nat @ X @ ( image_3205354838064109189at_nat @ F @ A2 ) )
=> ( P @ X ) )
=> ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A2 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_258_ball__imageD,axiom,
! [F: nat > set_nat,A2: set_nat,P: set_nat > $o] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ ( image_nat_set_nat @ F @ A2 ) )
=> ( P @ X ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_259_ball__imageD,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ! [X: nat] :
( ( member_nat @ X @ ( image_nat_nat @ F @ A2 ) )
=> ( P @ X ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_260_ball__imageD,axiom,
! [F: nat > $o,A2: set_nat,P: $o > $o] :
( ! [X: $o] :
( ( member_o @ X @ ( image_nat_o @ F @ A2 ) )
=> ( P @ X ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_261_ball__imageD,axiom,
! [F: $o > nat,A2: set_o,P: nat > $o] :
( ! [X: nat] :
( ( member_nat @ X @ ( image_o_nat @ F @ A2 ) )
=> ( P @ X ) )
=> ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_262_ball__imageD,axiom,
! [F: $o > $o,A2: set_o,P: $o > $o] :
( ! [X: $o] :
( ( member_o @ X @ ( image_o_o @ F @ A2 ) )
=> ( P @ X ) )
=> ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_263_image__cong,axiom,
! [M: set_o,N: set_o,F: $o > nat,G: $o > nat] :
( ( M = N )
=> ( ! [X: $o] :
( ( member_o @ X @ N )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_o_nat @ F @ M )
= ( image_o_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_264_image__cong,axiom,
! [M: set_o,N: set_o,F: $o > $o,G: $o > $o] :
( ( M = N )
=> ( ! [X: $o] :
( ( member_o @ X @ N )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_o_o @ F @ M )
= ( image_o_o @ G @ N ) ) ) ) ).
% image_cong
thf(fact_265_image__cong,axiom,
! [M: set_nat,N: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ( M = N )
=> ( ! [X: nat] :
( ( member_nat @ X @ N )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_nat_set_nat @ F @ M )
= ( image_nat_set_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_266_image__cong,axiom,
! [M: set_nat,N: set_nat,F: nat > nat,G: nat > nat] :
( ( M = N )
=> ( ! [X: nat] :
( ( member_nat @ X @ N )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_nat_nat @ F @ M )
= ( image_nat_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_267_image__cong,axiom,
! [M: set_nat,N: set_nat,F: nat > $o,G: nat > $o] :
( ( M = N )
=> ( ! [X: nat] :
( ( member_nat @ X @ N )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_nat_o @ F @ M )
= ( image_nat_o @ G @ N ) ) ) ) ).
% image_cong
thf(fact_268_image__cong,axiom,
! [M: set_nat_nat,N: set_nat_nat,F: ( nat > nat ) > nat > nat,G: ( nat > nat ) > nat > nat] :
( ( M = N )
=> ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ N )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_3205354838064109189at_nat @ F @ M )
= ( image_3205354838064109189at_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_269_mem__Collect__eq,axiom,
! [A: $o,P: $o > $o] :
( ( member_o @ A @ ( collect_o @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_270_mem__Collect__eq,axiom,
! [A: ( nat > nat ) > nat > nat,P: ( ( nat > nat ) > nat > nat ) > $o] :
( ( member952132173341509300at_nat @ A @ ( collec3567154360959927026at_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_271_mem__Collect__eq,axiom,
! [A: nat > nat,P: ( nat > nat ) > $o] :
( ( member_nat_nat @ A @ ( collect_nat_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_272_mem__Collect__eq,axiom,
! [A: nat > set_nat,P: ( nat > set_nat ) > $o] :
( ( member_nat_set_nat @ A @ ( collect_nat_set_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_273_mem__Collect__eq,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( member_set_nat @ A @ ( collect_set_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_274_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_275_Collect__mem__eq,axiom,
! [A2: set_o] :
( ( collect_o
@ ^ [X3: $o] : ( member_o @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_276_Collect__mem__eq,axiom,
! [A2: set_nat_nat_nat_nat] :
( ( collec3567154360959927026at_nat
@ ^ [X3: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_277_Collect__mem__eq,axiom,
! [A2: set_nat_nat] :
( ( collect_nat_nat
@ ^ [X3: nat > nat] : ( member_nat_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_278_Collect__mem__eq,axiom,
! [A2: set_nat_set_nat] :
( ( collect_nat_set_nat
@ ^ [X3: nat > set_nat] : ( member_nat_set_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_279_Collect__mem__eq,axiom,
! [A2: set_set_nat] :
( ( collect_set_nat
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_280_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_281_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X: nat] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_282_bex__imageD,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,P: ( nat > nat ) > $o] :
( ? [X4: nat > nat] :
( ( member_nat_nat @ X4 @ ( image_3205354838064109189at_nat @ F @ A2 ) )
& ( P @ X4 ) )
=> ? [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_283_bex__imageD,axiom,
! [F: nat > set_nat,A2: set_nat,P: set_nat > $o] :
( ? [X4: set_nat] :
( ( member_set_nat @ X4 @ ( image_nat_set_nat @ F @ A2 ) )
& ( P @ X4 ) )
=> ? [X: nat] :
( ( member_nat @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_284_bex__imageD,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ X4 ) )
=> ? [X: nat] :
( ( member_nat @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_285_bex__imageD,axiom,
! [F: nat > $o,A2: set_nat,P: $o > $o] :
( ? [X4: $o] :
( ( member_o @ X4 @ ( image_nat_o @ F @ A2 ) )
& ( P @ X4 ) )
=> ? [X: nat] :
( ( member_nat @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_286_bex__imageD,axiom,
! [F: $o > nat,A2: set_o,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( image_o_nat @ F @ A2 ) )
& ( P @ X4 ) )
=> ? [X: $o] :
( ( member_o @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_287_bex__imageD,axiom,
! [F: $o > $o,A2: set_o,P: $o > $o] :
( ? [X4: $o] :
( ( member_o @ X4 @ ( image_o_o @ F @ A2 ) )
& ( P @ X4 ) )
=> ? [X: $o] :
( ( member_o @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_288_image__iff,axiom,
! [Z2: $o,F: nat > $o,A2: set_nat] :
( ( member_o @ Z2 @ ( image_nat_o @ F @ A2 ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( Z2
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_289_image__iff,axiom,
! [Z2: $o,F: $o > $o,A2: set_o] :
( ( member_o @ Z2 @ ( image_o_o @ F @ A2 ) )
= ( ? [X3: $o] :
( ( member_o @ X3 @ A2 )
& ( Z2
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_290_image__iff,axiom,
! [Z2: nat,F: nat > nat,A2: set_nat] :
( ( member_nat @ Z2 @ ( image_nat_nat @ F @ A2 ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( Z2
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_291_image__iff,axiom,
! [Z2: nat,F: $o > nat,A2: set_o] :
( ( member_nat @ Z2 @ ( image_o_nat @ F @ A2 ) )
= ( ? [X3: $o] :
( ( member_o @ X3 @ A2 )
& ( Z2
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_292_image__iff,axiom,
! [Z2: nat > nat,F: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( member_nat_nat @ Z2 @ ( image_3205354838064109189at_nat @ F @ A2 ) )
= ( ? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
& ( Z2
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_293_image__iff,axiom,
! [Z2: set_nat,F: nat > set_nat,A2: set_nat] :
( ( member_set_nat @ Z2 @ ( image_nat_set_nat @ F @ A2 ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( Z2
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_294_imageI,axiom,
! [X2: $o,A2: set_o,F: $o > $o] :
( ( member_o @ X2 @ A2 )
=> ( member_o @ ( F @ X2 ) @ ( image_o_o @ F @ A2 ) ) ) ).
% imageI
thf(fact_295_imageI,axiom,
! [X2: $o,A2: set_o,F: $o > nat] :
( ( member_o @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ ( image_o_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_296_imageI,axiom,
! [X2: nat,A2: set_nat,F: nat > $o] :
( ( member_nat @ X2 @ A2 )
=> ( member_o @ ( F @ X2 ) @ ( image_nat_o @ F @ A2 ) ) ) ).
% imageI
thf(fact_297_imageI,axiom,
! [X2: nat,A2: set_nat,F: nat > nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ ( image_nat_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_298_imageI,axiom,
! [X2: $o,A2: set_o,F: $o > set_nat] :
( ( member_o @ X2 @ A2 )
=> ( member_set_nat @ ( F @ X2 ) @ ( image_o_set_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_299_imageI,axiom,
! [X2: nat,A2: set_nat,F: nat > set_nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_set_nat @ ( F @ X2 ) @ ( image_nat_set_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_300_imageI,axiom,
! [X2: set_nat,A2: set_set_nat,F: set_nat > $o] :
( ( member_set_nat @ X2 @ A2 )
=> ( member_o @ ( F @ X2 ) @ ( image_set_nat_o @ F @ A2 ) ) ) ).
% imageI
thf(fact_301_imageI,axiom,
! [X2: set_nat,A2: set_set_nat,F: set_nat > nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ ( image_set_nat_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_302_imageI,axiom,
! [X2: $o,A2: set_o,F: $o > nat > nat] :
( ( member_o @ X2 @ A2 )
=> ( member_nat_nat @ ( F @ X2 ) @ ( image_o_nat_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_303_imageI,axiom,
! [X2: nat,A2: set_nat,F: nat > nat > nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat_nat @ ( F @ X2 ) @ ( image_nat_nat_nat2 @ F @ A2 ) ) ) ).
% imageI
thf(fact_304_Collect__mono__iff,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ( ord_le9059583361652607317at_nat @ ( collect_nat_nat @ P ) @ ( collect_nat_nat @ Q ) )
= ( ! [X3: nat > nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_305_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_306_Collect__mono__iff,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) )
= ( ! [X3: set_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_307_Collect__mono__iff,axiom,
! [P: ( nat > nat > nat ) > $o,Q: ( nat > nat > nat ) > $o] :
( ( ord_le3211623285424100676at_nat @ ( collect_nat_nat_nat @ P ) @ ( collect_nat_nat_nat @ Q ) )
= ( ! [X3: nat > nat > nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_308_Collect__mono__iff,axiom,
! [P: ( $o > nat > nat ) > $o,Q: ( $o > nat > nat ) > $o] :
( ( ord_le8808915593745164104at_nat @ ( collect_o_nat_nat @ P ) @ ( collect_o_nat_nat @ Q ) )
= ( ! [X3: $o > nat > nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_309_Collect__mono__iff,axiom,
! [P: $o > $o,Q: $o > $o] :
( ( ord_less_eq_set_o @ ( collect_o @ P ) @ ( collect_o @ Q ) )
= ( ! [X3: $o] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_310_set__eq__subset,axiom,
( ( ^ [Y5: set_nat_nat,Z: set_nat_nat] : ( Y5 = Z ) )
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ B3 )
& ( ord_le9059583361652607317at_nat @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_311_set__eq__subset,axiom,
( ( ^ [Y5: set_nat,Z: set_nat] : ( Y5 = Z ) )
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_312_set__eq__subset,axiom,
( ( ^ [Y5: set_set_nat,Z: set_set_nat] : ( Y5 = Z ) )
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A3 @ B3 )
& ( ord_le6893508408891458716et_nat @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_313_set__eq__subset,axiom,
( ( ^ [Y5: set_nat_nat_nat,Z: set_nat_nat_nat] : ( Y5 = Z ) )
= ( ^ [A3: set_nat_nat_nat,B3: set_nat_nat_nat] :
( ( ord_le3211623285424100676at_nat @ A3 @ B3 )
& ( ord_le3211623285424100676at_nat @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_314_set__eq__subset,axiom,
( ( ^ [Y5: set_o_nat_nat,Z: set_o_nat_nat] : ( Y5 = Z ) )
= ( ^ [A3: set_o_nat_nat,B3: set_o_nat_nat] :
( ( ord_le8808915593745164104at_nat @ A3 @ B3 )
& ( ord_le8808915593745164104at_nat @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_315_set__eq__subset,axiom,
( ( ^ [Y5: set_o,Z: set_o] : ( Y5 = Z ) )
= ( ^ [A3: set_o,B3: set_o] :
( ( ord_less_eq_set_o @ A3 @ B3 )
& ( ord_less_eq_set_o @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_316_subset__trans,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ C )
=> ( ord_le9059583361652607317at_nat @ A2 @ C ) ) ) ).
% subset_trans
thf(fact_317_subset__trans,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% subset_trans
thf(fact_318_subset__trans,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ C )
=> ( ord_le6893508408891458716et_nat @ A2 @ C ) ) ) ).
% subset_trans
thf(fact_319_subset__trans,axiom,
! [A2: set_nat_nat_nat,B2: set_nat_nat_nat,C: set_nat_nat_nat] :
( ( ord_le3211623285424100676at_nat @ A2 @ B2 )
=> ( ( ord_le3211623285424100676at_nat @ B2 @ C )
=> ( ord_le3211623285424100676at_nat @ A2 @ C ) ) ) ).
% subset_trans
thf(fact_320_subset__trans,axiom,
! [A2: set_o_nat_nat,B2: set_o_nat_nat,C: set_o_nat_nat] :
( ( ord_le8808915593745164104at_nat @ A2 @ B2 )
=> ( ( ord_le8808915593745164104at_nat @ B2 @ C )
=> ( ord_le8808915593745164104at_nat @ A2 @ C ) ) ) ).
% subset_trans
thf(fact_321_subset__trans,axiom,
! [A2: set_o,B2: set_o,C: set_o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ( ord_less_eq_set_o @ B2 @ C )
=> ( ord_less_eq_set_o @ A2 @ C ) ) ) ).
% subset_trans
thf(fact_322_Collect__mono,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ! [X: nat > nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le9059583361652607317at_nat @ ( collect_nat_nat @ P ) @ ( collect_nat_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_323_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X: nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_324_Collect__mono,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ! [X: set_nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_325_Collect__mono,axiom,
! [P: ( nat > nat > nat ) > $o,Q: ( nat > nat > nat ) > $o] :
( ! [X: nat > nat > nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le3211623285424100676at_nat @ ( collect_nat_nat_nat @ P ) @ ( collect_nat_nat_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_326_Collect__mono,axiom,
! [P: ( $o > nat > nat ) > $o,Q: ( $o > nat > nat ) > $o] :
( ! [X: $o > nat > nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le8808915593745164104at_nat @ ( collect_o_nat_nat @ P ) @ ( collect_o_nat_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_327_Collect__mono,axiom,
! [P: $o > $o,Q: $o > $o] :
( ! [X: $o] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_o @ ( collect_o @ P ) @ ( collect_o @ Q ) ) ) ).
% Collect_mono
thf(fact_328_subset__refl,axiom,
! [A2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_329_subset__refl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_330_subset__refl,axiom,
! [A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_331_subset__refl,axiom,
! [A2: set_nat_nat_nat] : ( ord_le3211623285424100676at_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_332_subset__refl,axiom,
! [A2: set_o_nat_nat] : ( ord_le8808915593745164104at_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_333_subset__refl,axiom,
! [A2: set_o] : ( ord_less_eq_set_o @ A2 @ A2 ) ).
% subset_refl
thf(fact_334_subset__iff,axiom,
( ord_le5260717879541182899at_nat
= ( ^ [A3: set_nat_nat_nat_nat,B3: set_nat_nat_nat_nat] :
! [T: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ T @ A3 )
=> ( member952132173341509300at_nat @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_335_subset__iff,axiom,
( ord_le1585852046946910987et_nat
= ( ^ [A3: set_nat_set_nat,B3: set_nat_set_nat] :
! [T: nat > set_nat] :
( ( member_nat_set_nat @ T @ A3 )
=> ( member_nat_set_nat @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_336_subset__iff,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
! [T: nat > nat] :
( ( member_nat_nat @ T @ A3 )
=> ( member_nat_nat @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_337_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A3 )
=> ( member_nat @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_338_subset__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
! [T: set_nat] :
( ( member_set_nat @ T @ A3 )
=> ( member_set_nat @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_339_subset__iff,axiom,
( ord_le3211623285424100676at_nat
= ( ^ [A3: set_nat_nat_nat,B3: set_nat_nat_nat] :
! [T: nat > nat > nat] :
( ( member_nat_nat_nat2 @ T @ A3 )
=> ( member_nat_nat_nat2 @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_340_subset__iff,axiom,
( ord_le8808915593745164104at_nat
= ( ^ [A3: set_o_nat_nat,B3: set_o_nat_nat] :
! [T: $o > nat > nat] :
( ( member_o_nat_nat @ T @ A3 )
=> ( member_o_nat_nat @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_341_subset__iff,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B3: set_o] :
! [T: $o] :
( ( member_o @ T @ A3 )
=> ( member_o @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_342_equalityD2,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( A2 = B2 )
=> ( ord_le9059583361652607317at_nat @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_343_equalityD2,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 = B2 )
=> ( ord_less_eq_set_nat @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_344_equalityD2,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( A2 = B2 )
=> ( ord_le6893508408891458716et_nat @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_345_equalityD2,axiom,
! [A2: set_nat_nat_nat,B2: set_nat_nat_nat] :
( ( A2 = B2 )
=> ( ord_le3211623285424100676at_nat @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_346_equalityD2,axiom,
! [A2: set_o_nat_nat,B2: set_o_nat_nat] :
( ( A2 = B2 )
=> ( ord_le8808915593745164104at_nat @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_347_equalityD2,axiom,
! [A2: set_o,B2: set_o] :
( ( A2 = B2 )
=> ( ord_less_eq_set_o @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_348_equalityD1,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( A2 = B2 )
=> ( ord_le9059583361652607317at_nat @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_349_equalityD1,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 = B2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_350_equalityD1,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( A2 = B2 )
=> ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_351_equalityD1,axiom,
! [A2: set_nat_nat_nat,B2: set_nat_nat_nat] :
( ( A2 = B2 )
=> ( ord_le3211623285424100676at_nat @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_352_equalityD1,axiom,
! [A2: set_o_nat_nat,B2: set_o_nat_nat] :
( ( A2 = B2 )
=> ( ord_le8808915593745164104at_nat @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_353_equalityD1,axiom,
! [A2: set_o,B2: set_o] :
( ( A2 = B2 )
=> ( ord_less_eq_set_o @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_354_subset__eq,axiom,
( ord_le5260717879541182899at_nat
= ( ^ [A3: set_nat_nat_nat_nat,B3: set_nat_nat_nat_nat] :
! [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3 @ A3 )
=> ( member952132173341509300at_nat @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_355_subset__eq,axiom,
( ord_le1585852046946910987et_nat
= ( ^ [A3: set_nat_set_nat,B3: set_nat_set_nat] :
! [X3: nat > set_nat] :
( ( member_nat_set_nat @ X3 @ A3 )
=> ( member_nat_set_nat @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_356_subset__eq,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A3 )
=> ( member_nat_nat @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_357_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( member_nat @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_358_subset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
! [X3: set_nat] :
( ( member_set_nat @ X3 @ A3 )
=> ( member_set_nat @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_359_subset__eq,axiom,
( ord_le3211623285424100676at_nat
= ( ^ [A3: set_nat_nat_nat,B3: set_nat_nat_nat] :
! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A3 )
=> ( member_nat_nat_nat2 @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_360_subset__eq,axiom,
( ord_le8808915593745164104at_nat
= ( ^ [A3: set_o_nat_nat,B3: set_o_nat_nat] :
! [X3: $o > nat > nat] :
( ( member_o_nat_nat @ X3 @ A3 )
=> ( member_o_nat_nat @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_361_subset__eq,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B3: set_o] :
! [X3: $o] :
( ( member_o @ X3 @ A3 )
=> ( member_o @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_362_equalityE,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( A2 = B2 )
=> ~ ( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ~ ( ord_le9059583361652607317at_nat @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_363_equalityE,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ~ ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_364_equalityE,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( A2 = B2 )
=> ~ ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ~ ( ord_le6893508408891458716et_nat @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_365_equalityE,axiom,
! [A2: set_nat_nat_nat,B2: set_nat_nat_nat] :
( ( A2 = B2 )
=> ~ ( ( ord_le3211623285424100676at_nat @ A2 @ B2 )
=> ~ ( ord_le3211623285424100676at_nat @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_366_equalityE,axiom,
! [A2: set_o_nat_nat,B2: set_o_nat_nat] :
( ( A2 = B2 )
=> ~ ( ( ord_le8808915593745164104at_nat @ A2 @ B2 )
=> ~ ( ord_le8808915593745164104at_nat @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_367_equalityE,axiom,
! [A2: set_o,B2: set_o] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq_set_o @ A2 @ B2 )
=> ~ ( ord_less_eq_set_o @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_368_subsetD,axiom,
! [A2: set_nat_nat_nat_nat,B2: set_nat_nat_nat_nat,C2: ( nat > nat ) > nat > nat] :
( ( ord_le5260717879541182899at_nat @ A2 @ B2 )
=> ( ( member952132173341509300at_nat @ C2 @ A2 )
=> ( member952132173341509300at_nat @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_369_subsetD,axiom,
! [A2: set_nat_set_nat,B2: set_nat_set_nat,C2: nat > set_nat] :
( ( ord_le1585852046946910987et_nat @ A2 @ B2 )
=> ( ( member_nat_set_nat @ C2 @ A2 )
=> ( member_nat_set_nat @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_370_subsetD,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C2: nat > nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( member_nat_nat @ C2 @ A2 )
=> ( member_nat_nat @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_371_subsetD,axiom,
! [A2: set_nat,B2: set_nat,C2: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C2 @ A2 )
=> ( member_nat @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_372_subsetD,axiom,
! [A2: set_set_nat,B2: set_set_nat,C2: set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( member_set_nat @ C2 @ A2 )
=> ( member_set_nat @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_373_subsetD,axiom,
! [A2: set_nat_nat_nat,B2: set_nat_nat_nat,C2: nat > nat > nat] :
( ( ord_le3211623285424100676at_nat @ A2 @ B2 )
=> ( ( member_nat_nat_nat2 @ C2 @ A2 )
=> ( member_nat_nat_nat2 @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_374_subsetD,axiom,
! [A2: set_o_nat_nat,B2: set_o_nat_nat,C2: $o > nat > nat] :
( ( ord_le8808915593745164104at_nat @ A2 @ B2 )
=> ( ( member_o_nat_nat @ C2 @ A2 )
=> ( member_o_nat_nat @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_375_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_376_in__mono,axiom,
! [A2: set_nat_nat_nat_nat,B2: set_nat_nat_nat_nat,X2: ( nat > nat ) > nat > nat] :
( ( ord_le5260717879541182899at_nat @ A2 @ B2 )
=> ( ( member952132173341509300at_nat @ X2 @ A2 )
=> ( member952132173341509300at_nat @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_377_in__mono,axiom,
! [A2: set_nat_set_nat,B2: set_nat_set_nat,X2: nat > set_nat] :
( ( ord_le1585852046946910987et_nat @ A2 @ B2 )
=> ( ( member_nat_set_nat @ X2 @ A2 )
=> ( member_nat_set_nat @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_378_in__mono,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,X2: nat > nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( member_nat_nat @ X2 @ A2 )
=> ( member_nat_nat @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_379_in__mono,axiom,
! [A2: set_nat,B2: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_nat @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_380_in__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat,X2: set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( member_set_nat @ X2 @ A2 )
=> ( member_set_nat @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_381_in__mono,axiom,
! [A2: set_nat_nat_nat,B2: set_nat_nat_nat,X2: nat > nat > nat] :
( ( ord_le3211623285424100676at_nat @ A2 @ B2 )
=> ( ( member_nat_nat_nat2 @ X2 @ A2 )
=> ( member_nat_nat_nat2 @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_382_in__mono,axiom,
! [A2: set_o_nat_nat,B2: set_o_nat_nat,X2: $o > nat > nat] :
( ( ord_le8808915593745164104at_nat @ A2 @ B2 )
=> ( ( member_o_nat_nat @ X2 @ A2 )
=> ( member_o_nat_nat @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_383_in__mono,axiom,
! [A2: set_o,B2: set_o,X2: $o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ( member_o @ X2 @ A2 )
=> ( member_o @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_384_PiE__cong,axiom,
! [I2: set_nat,A2: nat > set_nat,B2: nat > set_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ I2 )
=> ( ( A2 @ I3 )
= ( B2 @ I3 ) ) )
=> ( ( piE_nat_nat @ I2 @ A2 )
= ( piE_nat_nat @ I2 @ B2 ) ) ) ).
% PiE_cong
thf(fact_385_PiE__cong,axiom,
! [I2: set_o,A2: $o > set_nat_nat,B2: $o > set_nat_nat] :
( ! [I3: $o] :
( ( member_o @ I3 @ I2 )
=> ( ( A2 @ I3 )
= ( B2 @ I3 ) ) )
=> ( ( piE_o_nat_nat @ I2 @ A2 )
= ( piE_o_nat_nat @ I2 @ B2 ) ) ) ).
% PiE_cong
thf(fact_386_PiE__cong,axiom,
! [I2: set_nat,A2: nat > set_o,B2: nat > set_o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ I2 )
=> ( ( A2 @ I3 )
= ( B2 @ I3 ) ) )
=> ( ( piE_nat_o @ I2 @ A2 )
= ( piE_nat_o @ I2 @ B2 ) ) ) ).
% PiE_cong
thf(fact_387_PiE__cong,axiom,
! [I2: set_nat,A2: nat > set_set_nat,B2: nat > set_set_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ I2 )
=> ( ( A2 @ I3 )
= ( B2 @ I3 ) ) )
=> ( ( piE_nat_set_nat @ I2 @ A2 )
= ( piE_nat_set_nat @ I2 @ B2 ) ) ) ).
% PiE_cong
thf(fact_388_PiE__cong,axiom,
! [I2: set_nat,A2: nat > set_nat_nat,B2: nat > set_nat_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ I2 )
=> ( ( A2 @ I3 )
= ( B2 @ I3 ) ) )
=> ( ( piE_nat_nat_nat2 @ I2 @ A2 )
= ( piE_nat_nat_nat2 @ I2 @ B2 ) ) ) ).
% PiE_cong
thf(fact_389_PiE__cong,axiom,
! [I2: set_nat_nat,A2: ( nat > nat ) > set_nat_nat,B2: ( nat > nat ) > set_nat_nat] :
( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ I2 )
=> ( ( A2 @ I3 )
= ( B2 @ I3 ) ) )
=> ( ( piE_nat_nat_nat_nat @ I2 @ A2 )
= ( piE_nat_nat_nat_nat @ I2 @ B2 ) ) ) ).
% PiE_cong
thf(fact_390_PiE__mem,axiom,
! [F: $o > $o,S: set_o,T2: $o > set_o,X2: $o] :
( ( member_o_o @ F @ ( piE_o_o @ S @ T2 ) )
=> ( ( member_o @ X2 @ S )
=> ( member_o @ ( F @ X2 ) @ ( T2 @ X2 ) ) ) ) ).
% PiE_mem
thf(fact_391_PiE__mem,axiom,
! [F: $o > nat,S: set_o,T2: $o > set_nat,X2: $o] :
( ( member_o_nat @ F @ ( piE_o_nat @ S @ T2 ) )
=> ( ( member_o @ X2 @ S )
=> ( member_nat @ ( F @ X2 ) @ ( T2 @ X2 ) ) ) ) ).
% PiE_mem
thf(fact_392_PiE__mem,axiom,
! [F: nat > nat,S: set_nat,T2: nat > set_nat,X2: nat] :
( ( member_nat_nat @ F @ ( piE_nat_nat @ S @ T2 ) )
=> ( ( member_nat @ X2 @ S )
=> ( member_nat @ ( F @ X2 ) @ ( T2 @ X2 ) ) ) ) ).
% PiE_mem
thf(fact_393_PiE__mem,axiom,
! [F: nat > $o,S: set_nat,T2: nat > set_o,X2: nat] :
( ( member_nat_o @ F @ ( piE_nat_o @ S @ T2 ) )
=> ( ( member_nat @ X2 @ S )
=> ( member_o @ ( F @ X2 ) @ ( T2 @ X2 ) ) ) ) ).
% PiE_mem
thf(fact_394_PiE__mem,axiom,
! [F: $o > set_nat,S: set_o,T2: $o > set_set_nat,X2: $o] :
( ( member_o_set_nat @ F @ ( piE_o_set_nat @ S @ T2 ) )
=> ( ( member_o @ X2 @ S )
=> ( member_set_nat @ ( F @ X2 ) @ ( T2 @ X2 ) ) ) ) ).
% PiE_mem
thf(fact_395_PiE__mem,axiom,
! [F: set_nat > $o,S: set_set_nat,T2: set_nat > set_o,X2: set_nat] :
( ( member_set_nat_o @ F @ ( piE_set_nat_o @ S @ T2 ) )
=> ( ( member_set_nat @ X2 @ S )
=> ( member_o @ ( F @ X2 ) @ ( T2 @ X2 ) ) ) ) ).
% PiE_mem
thf(fact_396_PiE__mem,axiom,
! [F: set_nat > nat,S: set_set_nat,T2: set_nat > set_nat,X2: set_nat] :
( ( member_set_nat_nat @ F @ ( piE_set_nat_nat @ S @ T2 ) )
=> ( ( member_set_nat @ X2 @ S )
=> ( member_nat @ ( F @ X2 ) @ ( T2 @ X2 ) ) ) ) ).
% PiE_mem
thf(fact_397_PiE__mem,axiom,
! [F: nat > set_nat,S: set_nat,T2: nat > set_set_nat,X2: nat] :
( ( member_nat_set_nat @ F @ ( piE_nat_set_nat @ S @ T2 ) )
=> ( ( member_nat @ X2 @ S )
=> ( member_set_nat @ ( F @ X2 ) @ ( T2 @ X2 ) ) ) ) ).
% PiE_mem
thf(fact_398_PiE__mem,axiom,
! [F: ( nat > nat ) > $o,S: set_nat_nat,T2: ( nat > nat ) > set_o,X2: nat > nat] :
( ( member_nat_nat_o @ F @ ( piE_nat_nat_o @ S @ T2 ) )
=> ( ( member_nat_nat @ X2 @ S )
=> ( member_o @ ( F @ X2 ) @ ( T2 @ X2 ) ) ) ) ).
% PiE_mem
thf(fact_399_PiE__mem,axiom,
! [F: ( nat > nat ) > nat,S: set_nat_nat,T2: ( nat > nat ) > set_nat,X2: nat > nat] :
( ( member_nat_nat_nat @ F @ ( piE_nat_nat_nat @ S @ T2 ) )
=> ( ( member_nat_nat @ X2 @ S )
=> ( member_nat @ ( F @ X2 ) @ ( T2 @ X2 ) ) ) ) ).
% PiE_mem
thf(fact_400_PiE__ext,axiom,
! [X2: nat > nat,K: set_nat,S2: nat > set_nat,Y2: nat > nat] :
( ( member_nat_nat @ X2 @ ( piE_nat_nat @ K @ S2 ) )
=> ( ( member_nat_nat @ Y2 @ ( piE_nat_nat @ K @ S2 ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ K )
=> ( ( X2 @ I3 )
= ( Y2 @ I3 ) ) )
=> ( X2 = Y2 ) ) ) ) ).
% PiE_ext
thf(fact_401_PiE__ext,axiom,
! [X2: $o > nat > nat,K: set_o,S2: $o > set_nat_nat,Y2: $o > nat > nat] :
( ( member_o_nat_nat @ X2 @ ( piE_o_nat_nat @ K @ S2 ) )
=> ( ( member_o_nat_nat @ Y2 @ ( piE_o_nat_nat @ K @ S2 ) )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ K )
=> ( ( X2 @ I3 )
= ( Y2 @ I3 ) ) )
=> ( X2 = Y2 ) ) ) ) ).
% PiE_ext
thf(fact_402_PiE__ext,axiom,
! [X2: nat > $o,K: set_nat,S2: nat > set_o,Y2: nat > $o] :
( ( member_nat_o @ X2 @ ( piE_nat_o @ K @ S2 ) )
=> ( ( member_nat_o @ Y2 @ ( piE_nat_o @ K @ S2 ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ K )
=> ( ( X2 @ I3 )
= ( Y2 @ I3 ) ) )
=> ( X2 = Y2 ) ) ) ) ).
% PiE_ext
thf(fact_403_PiE__ext,axiom,
! [X2: nat > set_nat,K: set_nat,S2: nat > set_set_nat,Y2: nat > set_nat] :
( ( member_nat_set_nat @ X2 @ ( piE_nat_set_nat @ K @ S2 ) )
=> ( ( member_nat_set_nat @ Y2 @ ( piE_nat_set_nat @ K @ S2 ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ K )
=> ( ( X2 @ I3 )
= ( Y2 @ I3 ) ) )
=> ( X2 = Y2 ) ) ) ) ).
% PiE_ext
thf(fact_404_PiE__ext,axiom,
! [X2: nat > nat > nat,K: set_nat,S2: nat > set_nat_nat,Y2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X2 @ ( piE_nat_nat_nat2 @ K @ S2 ) )
=> ( ( member_nat_nat_nat2 @ Y2 @ ( piE_nat_nat_nat2 @ K @ S2 ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ K )
=> ( ( X2 @ I3 )
= ( Y2 @ I3 ) ) )
=> ( X2 = Y2 ) ) ) ) ).
% PiE_ext
thf(fact_405_PiE__ext,axiom,
! [X2: ( nat > nat ) > nat > nat,K: set_nat_nat,S2: ( nat > nat ) > set_nat_nat,Y2: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X2 @ ( piE_nat_nat_nat_nat @ K @ S2 ) )
=> ( ( member952132173341509300at_nat @ Y2 @ ( piE_nat_nat_nat_nat @ K @ S2 ) )
=> ( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ K )
=> ( ( X2 @ I3 )
= ( Y2 @ I3 ) ) )
=> ( X2 = Y2 ) ) ) ) ).
% PiE_ext
thf(fact_406_Compr__image__eq,axiom,
! [F: $o > $o,A2: set_o,P: $o > $o] :
( ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ ( image_o_o @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_o_o @ F
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_407_Compr__image__eq,axiom,
! [F: nat > $o,A2: set_nat,P: $o > $o] :
( ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ ( image_nat_o @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_nat_o @ F
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_408_Compr__image__eq,axiom,
! [F: $o > nat,A2: set_o,P: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ ( image_o_nat @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_o_nat @ F
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_409_Compr__image__eq,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_410_Compr__image__eq,axiom,
! [F: set_nat > $o,A2: set_set_nat,P: $o > $o] :
( ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ ( image_set_nat_o @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_set_nat_o @ F
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_411_Compr__image__eq,axiom,
! [F: $o > set_nat,A2: set_o,P: set_nat > $o] :
( ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ ( image_o_set_nat @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_o_set_nat @ F
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_412_Compr__image__eq,axiom,
! [F: nat > set_nat,A2: set_nat,P: set_nat > $o] :
( ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ ( image_nat_set_nat @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_nat_set_nat @ F
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_413_Compr__image__eq,axiom,
! [F: set_nat > nat,A2: set_set_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ ( image_set_nat_nat @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_set_nat_nat @ F
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_414_Compr__image__eq,axiom,
! [F: ( nat > nat ) > $o,A2: set_nat_nat,P: $o > $o] :
( ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ ( image_nat_nat_o @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_nat_nat_o @ F
@ ( collect_nat_nat
@ ^ [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_415_Compr__image__eq,axiom,
! [F: $o > nat > nat,A2: set_o,P: ( nat > nat ) > $o] :
( ( collect_nat_nat
@ ^ [X3: nat > nat] :
( ( member_nat_nat @ X3 @ ( image_o_nat_nat @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_o_nat_nat @ F
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_416_image__image,axiom,
! [F: nat > nat,G: nat > nat,A2: set_nat] :
( ( image_nat_nat @ F @ ( image_nat_nat @ G @ A2 ) )
= ( image_nat_nat
@ ^ [X3: nat] : ( F @ ( G @ X3 ) )
@ A2 ) ) ).
% image_image
thf(fact_417_image__image,axiom,
! [F: nat > nat,G: $o > nat,A2: set_o] :
( ( image_nat_nat @ F @ ( image_o_nat @ G @ A2 ) )
= ( image_o_nat
@ ^ [X3: $o] : ( F @ ( G @ X3 ) )
@ A2 ) ) ).
% image_image
thf(fact_418_image__image,axiom,
! [F: nat > $o,G: nat > nat,A2: set_nat] :
( ( image_nat_o @ F @ ( image_nat_nat @ G @ A2 ) )
= ( image_nat_o
@ ^ [X3: nat] : ( F @ ( G @ X3 ) )
@ A2 ) ) ).
% image_image
thf(fact_419_image__image,axiom,
! [F: nat > $o,G: $o > nat,A2: set_o] :
( ( image_nat_o @ F @ ( image_o_nat @ G @ A2 ) )
= ( image_o_o
@ ^ [X3: $o] : ( F @ ( G @ X3 ) )
@ A2 ) ) ).
% image_image
thf(fact_420_image__image,axiom,
! [F: $o > nat,G: nat > $o,A2: set_nat] :
( ( image_o_nat @ F @ ( image_nat_o @ G @ A2 ) )
= ( image_nat_nat
@ ^ [X3: nat] : ( F @ ( G @ X3 ) )
@ A2 ) ) ).
% image_image
thf(fact_421_image__image,axiom,
! [F: $o > nat,G: $o > $o,A2: set_o] :
( ( image_o_nat @ F @ ( image_o_o @ G @ A2 ) )
= ( image_o_nat
@ ^ [X3: $o] : ( F @ ( G @ X3 ) )
@ A2 ) ) ).
% image_image
thf(fact_422_image__image,axiom,
! [F: $o > $o,G: nat > $o,A2: set_nat] :
( ( image_o_o @ F @ ( image_nat_o @ G @ A2 ) )
= ( image_nat_o
@ ^ [X3: nat] : ( F @ ( G @ X3 ) )
@ A2 ) ) ).
% image_image
thf(fact_423_image__image,axiom,
! [F: $o > $o,G: $o > $o,A2: set_o] :
( ( image_o_o @ F @ ( image_o_o @ G @ A2 ) )
= ( image_o_o
@ ^ [X3: $o] : ( F @ ( G @ X3 ) )
@ A2 ) ) ).
% image_image
thf(fact_424_image__image,axiom,
! [F: set_nat > nat,G: nat > set_nat,A2: set_nat] :
( ( image_set_nat_nat @ F @ ( image_nat_set_nat @ G @ A2 ) )
= ( image_nat_nat
@ ^ [X3: nat] : ( F @ ( G @ X3 ) )
@ A2 ) ) ).
% image_image
thf(fact_425_image__image,axiom,
! [F: set_nat > $o,G: nat > set_nat,A2: set_nat] :
( ( image_set_nat_o @ F @ ( image_nat_set_nat @ G @ A2 ) )
= ( image_nat_o
@ ^ [X3: nat] : ( F @ ( G @ X3 ) )
@ A2 ) ) ).
% image_image
thf(fact_426_imageE,axiom,
! [B: $o,F: $o > $o,A2: set_o] :
( ( member_o @ B @ ( image_o_o @ F @ A2 ) )
=> ~ ! [X: $o] :
( ( B
= ( F @ X ) )
=> ~ ( member_o @ X @ A2 ) ) ) ).
% imageE
thf(fact_427_imageE,axiom,
! [B: $o,F: nat > $o,A2: set_nat] :
( ( member_o @ B @ ( image_nat_o @ F @ A2 ) )
=> ~ ! [X: nat] :
( ( B
= ( F @ X ) )
=> ~ ( member_nat @ X @ A2 ) ) ) ).
% imageE
thf(fact_428_imageE,axiom,
! [B: nat,F: $o > nat,A2: set_o] :
( ( member_nat @ B @ ( image_o_nat @ F @ A2 ) )
=> ~ ! [X: $o] :
( ( B
= ( F @ X ) )
=> ~ ( member_o @ X @ A2 ) ) ) ).
% imageE
thf(fact_429_imageE,axiom,
! [B: nat,F: nat > nat,A2: set_nat] :
( ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [X: nat] :
( ( B
= ( F @ X ) )
=> ~ ( member_nat @ X @ A2 ) ) ) ).
% imageE
thf(fact_430_imageE,axiom,
! [B: $o,F: set_nat > $o,A2: set_set_nat] :
( ( member_o @ B @ ( image_set_nat_o @ F @ A2 ) )
=> ~ ! [X: set_nat] :
( ( B
= ( F @ X ) )
=> ~ ( member_set_nat @ X @ A2 ) ) ) ).
% imageE
thf(fact_431_imageE,axiom,
! [B: nat,F: set_nat > nat,A2: set_set_nat] :
( ( member_nat @ B @ ( image_set_nat_nat @ F @ A2 ) )
=> ~ ! [X: set_nat] :
( ( B
= ( F @ X ) )
=> ~ ( member_set_nat @ X @ A2 ) ) ) ).
% imageE
thf(fact_432_imageE,axiom,
! [B: set_nat,F: $o > set_nat,A2: set_o] :
( ( member_set_nat @ B @ ( image_o_set_nat @ F @ A2 ) )
=> ~ ! [X: $o] :
( ( B
= ( F @ X ) )
=> ~ ( member_o @ X @ A2 ) ) ) ).
% imageE
thf(fact_433_imageE,axiom,
! [B: set_nat,F: nat > set_nat,A2: set_nat] :
( ( member_set_nat @ B @ ( image_nat_set_nat @ F @ A2 ) )
=> ~ ! [X: nat] :
( ( B
= ( F @ X ) )
=> ~ ( member_nat @ X @ A2 ) ) ) ).
% imageE
thf(fact_434_imageE,axiom,
! [B: $o,F: ( nat > nat ) > $o,A2: set_nat_nat] :
( ( member_o @ B @ ( image_nat_nat_o @ F @ A2 ) )
=> ~ ! [X: nat > nat] :
( ( B
= ( F @ X ) )
=> ~ ( member_nat_nat @ X @ A2 ) ) ) ).
% imageE
thf(fact_435_imageE,axiom,
! [B: nat,F: ( nat > nat ) > nat,A2: set_nat_nat] :
( ( member_nat @ B @ ( image_nat_nat_nat @ F @ A2 ) )
=> ~ ! [X: nat > nat] :
( ( B
= ( F @ X ) )
=> ~ ( member_nat_nat @ X @ A2 ) ) ) ).
% imageE
thf(fact_436_Collect__restrict,axiom,
! [X5: set_nat_nat_nat_nat,P: ( ( nat > nat ) > nat > nat ) > $o] :
( ord_le5260717879541182899at_nat
@ ( collec3567154360959927026at_nat
@ ^ [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3 @ X5 )
& ( P @ X3 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_437_Collect__restrict,axiom,
! [X5: set_nat_set_nat,P: ( nat > set_nat ) > $o] :
( ord_le1585852046946910987et_nat
@ ( collect_nat_set_nat
@ ^ [X3: nat > set_nat] :
( ( member_nat_set_nat @ X3 @ X5 )
& ( P @ X3 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_438_Collect__restrict,axiom,
! [X5: set_nat_nat,P: ( nat > nat ) > $o] :
( ord_le9059583361652607317at_nat
@ ( collect_nat_nat
@ ^ [X3: nat > nat] :
( ( member_nat_nat @ X3 @ X5 )
& ( P @ X3 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_439_Collect__restrict,axiom,
! [X5: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ X5 )
& ( P @ X3 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_440_Collect__restrict,axiom,
! [X5: set_set_nat,P: set_nat > $o] :
( ord_le6893508408891458716et_nat
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ X5 )
& ( P @ X3 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_441_Collect__restrict,axiom,
! [X5: set_nat_nat_nat,P: ( nat > nat > nat ) > $o] :
( ord_le3211623285424100676at_nat
@ ( collect_nat_nat_nat
@ ^ [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ X5 )
& ( P @ X3 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_442_Collect__restrict,axiom,
! [X5: set_o_nat_nat,P: ( $o > nat > nat ) > $o] :
( ord_le8808915593745164104at_nat
@ ( collect_o_nat_nat
@ ^ [X3: $o > nat > nat] :
( ( member_o_nat_nat @ X3 @ X5 )
& ( P @ X3 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_443_Collect__restrict,axiom,
! [X5: set_o,P: $o > $o] :
( ord_less_eq_set_o
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ X5 )
& ( P @ X3 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_444_prop__restrict,axiom,
! [X2: ( nat > nat ) > nat > nat,Z3: set_nat_nat_nat_nat,X5: set_nat_nat_nat_nat,P: ( ( nat > nat ) > nat > nat ) > $o] :
( ( member952132173341509300at_nat @ X2 @ Z3 )
=> ( ( ord_le5260717879541182899at_nat @ Z3
@ ( collec3567154360959927026at_nat
@ ^ [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3 @ X5 )
& ( P @ X3 ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_445_prop__restrict,axiom,
! [X2: nat > set_nat,Z3: set_nat_set_nat,X5: set_nat_set_nat,P: ( nat > set_nat ) > $o] :
( ( member_nat_set_nat @ X2 @ Z3 )
=> ( ( ord_le1585852046946910987et_nat @ Z3
@ ( collect_nat_set_nat
@ ^ [X3: nat > set_nat] :
( ( member_nat_set_nat @ X3 @ X5 )
& ( P @ X3 ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_446_prop__restrict,axiom,
! [X2: nat > nat,Z3: set_nat_nat,X5: set_nat_nat,P: ( nat > nat ) > $o] :
( ( member_nat_nat @ X2 @ Z3 )
=> ( ( ord_le9059583361652607317at_nat @ Z3
@ ( collect_nat_nat
@ ^ [X3: nat > nat] :
( ( member_nat_nat @ X3 @ X5 )
& ( P @ X3 ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_447_prop__restrict,axiom,
! [X2: nat,Z3: set_nat,X5: set_nat,P: nat > $o] :
( ( member_nat @ X2 @ Z3 )
=> ( ( ord_less_eq_set_nat @ Z3
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ X5 )
& ( P @ X3 ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_448_prop__restrict,axiom,
! [X2: set_nat,Z3: set_set_nat,X5: set_set_nat,P: set_nat > $o] :
( ( member_set_nat @ X2 @ Z3 )
=> ( ( ord_le6893508408891458716et_nat @ Z3
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ X5 )
& ( P @ X3 ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_449_prop__restrict,axiom,
! [X2: nat > nat > nat,Z3: set_nat_nat_nat,X5: set_nat_nat_nat,P: ( nat > nat > nat ) > $o] :
( ( member_nat_nat_nat2 @ X2 @ Z3 )
=> ( ( ord_le3211623285424100676at_nat @ Z3
@ ( collect_nat_nat_nat
@ ^ [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ X5 )
& ( P @ X3 ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_450_prop__restrict,axiom,
! [X2: $o > nat > nat,Z3: set_o_nat_nat,X5: set_o_nat_nat,P: ( $o > nat > nat ) > $o] :
( ( member_o_nat_nat @ X2 @ Z3 )
=> ( ( ord_le8808915593745164104at_nat @ Z3
@ ( collect_o_nat_nat
@ ^ [X3: $o > nat > nat] :
( ( member_o_nat_nat @ X3 @ X5 )
& ( P @ X3 ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_451_prop__restrict,axiom,
! [X2: $o,Z3: set_o,X5: set_o,P: $o > $o] :
( ( member_o @ X2 @ Z3 )
=> ( ( ord_less_eq_set_o @ Z3
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ X5 )
& ( P @ X3 ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_452_Collect__subset,axiom,
! [A2: set_nat_nat_nat_nat,P: ( ( nat > nat ) > nat > nat ) > $o] :
( ord_le5260717879541182899at_nat
@ ( collec3567154360959927026at_nat
@ ^ [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_453_Collect__subset,axiom,
! [A2: set_nat_set_nat,P: ( nat > set_nat ) > $o] :
( ord_le1585852046946910987et_nat
@ ( collect_nat_set_nat
@ ^ [X3: nat > set_nat] :
( ( member_nat_set_nat @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_454_Collect__subset,axiom,
! [A2: set_nat_nat,P: ( nat > nat ) > $o] :
( ord_le9059583361652607317at_nat
@ ( collect_nat_nat
@ ^ [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_455_Collect__subset,axiom,
! [A2: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_456_Collect__subset,axiom,
! [A2: set_set_nat,P: set_nat > $o] :
( ord_le6893508408891458716et_nat
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_457_Collect__subset,axiom,
! [A2: set_nat_nat_nat,P: ( nat > nat > nat ) > $o] :
( ord_le3211623285424100676at_nat
@ ( collect_nat_nat_nat
@ ^ [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_458_Collect__subset,axiom,
! [A2: set_o_nat_nat,P: ( $o > nat > nat ) > $o] :
( ord_le8808915593745164104at_nat
@ ( collect_o_nat_nat
@ ^ [X3: $o > nat > nat] :
( ( member_o_nat_nat @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_459_Collect__subset,axiom,
! [A2: set_o,P: $o > $o] :
( ord_less_eq_set_o
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_460_subset__image__iff,axiom,
! [B2: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B2
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_461_subset__image__iff,axiom,
! [B2: set_nat,F: $o > nat,A2: set_o] :
( ( ord_less_eq_set_nat @ B2 @ ( image_o_nat @ F @ A2 ) )
= ( ? [AA: set_o] :
( ( ord_less_eq_set_o @ AA @ A2 )
& ( B2
= ( image_o_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_462_subset__image__iff,axiom,
! [B2: set_o,F: nat > $o,A2: set_nat] :
( ( ord_less_eq_set_o @ B2 @ ( image_nat_o @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B2
= ( image_nat_o @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_463_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_464_subset__image__iff,axiom,
! [B2: set_nat,F: set_nat > nat,A2: set_set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_set_nat_nat @ F @ A2 ) )
= ( ? [AA: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ AA @ A2 )
& ( B2
= ( image_set_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_465_subset__image__iff,axiom,
! [B2: set_set_nat,F: nat > set_nat,A2: set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_nat_set_nat @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B2
= ( image_nat_set_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_466_subset__image__iff,axiom,
! [B2: set_set_nat,F: $o > set_nat,A2: set_o] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_o_set_nat @ F @ A2 ) )
= ( ? [AA: set_o] :
( ( ord_less_eq_set_o @ AA @ A2 )
& ( B2
= ( image_o_set_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_467_subset__image__iff,axiom,
! [B2: set_o,F: set_nat > $o,A2: set_set_nat] :
( ( ord_less_eq_set_o @ B2 @ ( image_set_nat_o @ F @ A2 ) )
= ( ? [AA: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ AA @ A2 )
& ( B2
= ( image_set_nat_o @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_468_subset__image__iff,axiom,
! [B2: set_nat_nat,F: nat > nat > nat,A2: set_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ ( image_nat_nat_nat2 @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B2
= ( image_nat_nat_nat2 @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_469_subset__image__iff,axiom,
! [B2: set_nat_nat,F: $o > nat > nat,A2: set_o] :
( ( ord_le9059583361652607317at_nat @ B2 @ ( image_o_nat_nat @ F @ A2 ) )
= ( ? [AA: set_o] :
( ( ord_less_eq_set_o @ AA @ A2 )
& ( B2
= ( image_o_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_470_image__subset__iff,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F @ A2 ) @ B2 )
= ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( member_nat_nat @ ( F @ X3 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_471_image__subset__iff,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B2 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_472_image__subset__iff,axiom,
! [F: $o > nat,A2: set_o,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_o_nat @ F @ A2 ) @ B2 )
= ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_473_image__subset__iff,axiom,
! [F: nat > set_nat,A2: set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A2 ) @ B2 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_set_nat @ ( F @ X3 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_474_image__subset__iff,axiom,
! [F: nat > $o,A2: set_nat,B2: set_o] :
( ( ord_less_eq_set_o @ ( image_nat_o @ F @ A2 ) @ B2 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_o @ ( F @ X3 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_475_image__subset__iff,axiom,
! [F: $o > $o,A2: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ ( image_o_o @ F @ A2 ) @ B2 )
= ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( member_o @ ( F @ X3 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_476_subset__imageE,axiom,
! [B2: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A2 )
=> ( B2
!= ( image_nat_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_477_subset__imageE,axiom,
! [B2: set_nat,F: $o > nat,A2: set_o] :
( ( ord_less_eq_set_nat @ B2 @ ( image_o_nat @ F @ A2 ) )
=> ~ ! [C3: set_o] :
( ( ord_less_eq_set_o @ C3 @ A2 )
=> ( B2
!= ( image_o_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_478_subset__imageE,axiom,
! [B2: set_o,F: nat > $o,A2: set_nat] :
( ( ord_less_eq_set_o @ B2 @ ( image_nat_o @ F @ A2 ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A2 )
=> ( B2
!= ( image_nat_o @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_479_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_480_subset__imageE,axiom,
! [B2: set_nat,F: set_nat > nat,A2: set_set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_set_nat_nat @ F @ A2 ) )
=> ~ ! [C3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C3 @ A2 )
=> ( B2
!= ( image_set_nat_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_481_subset__imageE,axiom,
! [B2: set_set_nat,F: nat > set_nat,A2: set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_nat_set_nat @ F @ A2 ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A2 )
=> ( B2
!= ( image_nat_set_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_482_subset__imageE,axiom,
! [B2: set_set_nat,F: $o > set_nat,A2: set_o] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_o_set_nat @ F @ A2 ) )
=> ~ ! [C3: set_o] :
( ( ord_less_eq_set_o @ C3 @ A2 )
=> ( B2
!= ( image_o_set_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_483_subset__imageE,axiom,
! [B2: set_o,F: set_nat > $o,A2: set_set_nat] :
( ( ord_less_eq_set_o @ B2 @ ( image_set_nat_o @ F @ A2 ) )
=> ~ ! [C3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C3 @ A2 )
=> ( B2
!= ( image_set_nat_o @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_484_subset__imageE,axiom,
! [B2: set_nat_nat,F: nat > nat > nat,A2: set_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ ( image_nat_nat_nat2 @ F @ A2 ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A2 )
=> ( B2
!= ( image_nat_nat_nat2 @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_485_subset__imageE,axiom,
! [B2: set_nat_nat,F: $o > nat > nat,A2: set_o] :
( ( ord_le9059583361652607317at_nat @ B2 @ ( image_o_nat_nat @ F @ A2 ) )
=> ~ ! [C3: set_o] :
( ( ord_less_eq_set_o @ C3 @ A2 )
=> ( B2
!= ( image_o_nat_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_486_all__subset__image,axiom,
! [F: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( P @ ( image_nat_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_487_all__subset__image,axiom,
! [F: $o > nat,A2: set_o,P: set_nat > $o] :
( ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ ( image_o_nat @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_o] :
( ( ord_less_eq_set_o @ B3 @ A2 )
=> ( P @ ( image_o_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_488_all__subset__image,axiom,
! [F: nat > $o,A2: set_nat,P: set_o > $o] :
( ( ! [B3: set_o] :
( ( ord_less_eq_set_o @ B3 @ ( image_nat_o @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( P @ ( image_nat_o @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_489_all__subset__image,axiom,
! [F: $o > $o,A2: set_o,P: set_o > $o] :
( ( ! [B3: set_o] :
( ( ord_less_eq_set_o @ B3 @ ( image_o_o @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_o] :
( ( ord_less_eq_set_o @ B3 @ A2 )
=> ( P @ ( image_o_o @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_490_all__subset__image,axiom,
! [F: set_nat > nat,A2: set_set_nat,P: set_nat > $o] :
( ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ ( image_set_nat_nat @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B3 @ A2 )
=> ( P @ ( image_set_nat_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_491_all__subset__image,axiom,
! [F: nat > set_nat,A2: set_nat,P: set_set_nat > $o] :
( ( ! [B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B3 @ ( image_nat_set_nat @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( P @ ( image_nat_set_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_492_all__subset__image,axiom,
! [F: $o > set_nat,A2: set_o,P: set_set_nat > $o] :
( ( ! [B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B3 @ ( image_o_set_nat @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_o] :
( ( ord_less_eq_set_o @ B3 @ A2 )
=> ( P @ ( image_o_set_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_493_all__subset__image,axiom,
! [F: set_nat > $o,A2: set_set_nat,P: set_o > $o] :
( ( ! [B3: set_o] :
( ( ord_less_eq_set_o @ B3 @ ( image_set_nat_o @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B3 @ A2 )
=> ( P @ ( image_set_nat_o @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_494_all__subset__image,axiom,
! [F: nat > nat > nat,A2: set_nat,P: set_nat_nat > $o] :
( ( ! [B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B3 @ ( image_nat_nat_nat2 @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( P @ ( image_nat_nat_nat2 @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_495_all__subset__image,axiom,
! [F: $o > nat > nat,A2: set_o,P: set_nat_nat > $o] :
( ( ! [B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B3 @ ( image_o_nat_nat @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_o] :
( ( ord_less_eq_set_o @ B3 @ A2 )
=> ( P @ ( image_o_nat_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_496_pred__subset__eq,axiom,
! [R: set_nat_nat_nat_nat,S: set_nat_nat_nat_nat] :
( ( ord_le5430825838364970130_nat_o
@ ^ [X3: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X3 @ R )
@ ^ [X3: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X3 @ S ) )
= ( ord_le5260717879541182899at_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_497_pred__subset__eq,axiom,
! [R: set_nat_set_nat,S: set_nat_set_nat] :
( ( ord_le8865062304692155706_nat_o
@ ^ [X3: nat > set_nat] : ( member_nat_set_nat @ X3 @ R )
@ ^ [X3: nat > set_nat] : ( member_nat_set_nat @ X3 @ S ) )
= ( ord_le1585852046946910987et_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_498_pred__subset__eq,axiom,
! [R: set_nat_nat,S: set_nat_nat] :
( ( ord_le7366121074344172400_nat_o
@ ^ [X3: nat > nat] : ( member_nat_nat @ X3 @ R )
@ ^ [X3: nat > nat] : ( member_nat_nat @ X3 @ S ) )
= ( ord_le9059583361652607317at_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_499_pred__subset__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ R )
@ ^ [X3: nat] : ( member_nat @ X3 @ S ) )
= ( ord_less_eq_set_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_500_pred__subset__eq,axiom,
! [R: set_set_nat,S: set_set_nat] :
( ( ord_le3964352015994296041_nat_o
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ R )
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ S ) )
= ( ord_le6893508408891458716et_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_501_pred__subset__eq,axiom,
! [R: set_nat_nat_nat,S: set_nat_nat_nat] :
( ( ord_le5384859702510996545_nat_o
@ ^ [X3: nat > nat > nat] : ( member_nat_nat_nat2 @ X3 @ R )
@ ^ [X3: nat > nat > nat] : ( member_nat_nat_nat2 @ X3 @ S ) )
= ( ord_le3211623285424100676at_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_502_pred__subset__eq,axiom,
! [R: set_o_nat_nat,S: set_o_nat_nat] :
( ( ord_le6871787456926423957_nat_o
@ ^ [X3: $o > nat > nat] : ( member_o_nat_nat @ X3 @ R )
@ ^ [X3: $o > nat > nat] : ( member_o_nat_nat @ X3 @ S ) )
= ( ord_le8808915593745164104at_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_503_pred__subset__eq,axiom,
! [R: set_o,S: set_o] :
( ( ord_less_eq_o_o
@ ^ [X3: $o] : ( member_o @ X3 @ R )
@ ^ [X3: $o] : ( member_o @ X3 @ S ) )
= ( ord_less_eq_set_o @ R @ S ) ) ).
% pred_subset_eq
thf(fact_504_subset__Collect__iff,axiom,
! [B2: set_nat_nat_nat_nat,A2: set_nat_nat_nat_nat,P: ( ( nat > nat ) > nat > nat ) > $o] :
( ( ord_le5260717879541182899at_nat @ B2 @ A2 )
=> ( ( ord_le5260717879541182899at_nat @ B2
@ ( collec3567154360959927026at_nat
@ ^ [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3 @ A2 )
& ( P @ X3 ) ) ) )
= ( ! [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3 @ B2 )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_505_subset__Collect__iff,axiom,
! [B2: set_nat_set_nat,A2: set_nat_set_nat,P: ( nat > set_nat ) > $o] :
( ( ord_le1585852046946910987et_nat @ B2 @ A2 )
=> ( ( ord_le1585852046946910987et_nat @ B2
@ ( collect_nat_set_nat
@ ^ [X3: nat > set_nat] :
( ( member_nat_set_nat @ X3 @ A2 )
& ( P @ X3 ) ) ) )
= ( ! [X3: nat > set_nat] :
( ( member_nat_set_nat @ X3 @ B2 )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_506_subset__Collect__iff,axiom,
! [B2: set_nat_nat,A2: set_nat_nat,P: ( nat > nat ) > $o] :
( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ B2
@ ( collect_nat_nat
@ ^ [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
& ( P @ X3 ) ) ) )
= ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ B2 )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_507_subset__Collect__iff,axiom,
! [B2: set_nat,A2: set_nat,P: nat > $o] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ord_less_eq_set_nat @ B2
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ X3 ) ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_508_subset__Collect__iff,axiom,
! [B2: set_set_nat,A2: set_set_nat,P: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ B2
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
& ( P @ X3 ) ) ) )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ B2 )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_509_subset__Collect__iff,axiom,
! [B2: set_nat_nat_nat,A2: set_nat_nat_nat,P: ( nat > nat > nat ) > $o] :
( ( ord_le3211623285424100676at_nat @ B2 @ A2 )
=> ( ( ord_le3211623285424100676at_nat @ B2
@ ( collect_nat_nat_nat
@ ^ [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A2 )
& ( P @ X3 ) ) ) )
= ( ! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ B2 )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_510_subset__Collect__iff,axiom,
! [B2: set_o_nat_nat,A2: set_o_nat_nat,P: ( $o > nat > nat ) > $o] :
( ( ord_le8808915593745164104at_nat @ B2 @ A2 )
=> ( ( ord_le8808915593745164104at_nat @ B2
@ ( collect_o_nat_nat
@ ^ [X3: $o > nat > nat] :
( ( member_o_nat_nat @ X3 @ A2 )
& ( P @ X3 ) ) ) )
= ( ! [X3: $o > nat > nat] :
( ( member_o_nat_nat @ X3 @ B2 )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_511_subset__Collect__iff,axiom,
! [B2: set_o,A2: set_o,P: $o > $o] :
( ( ord_less_eq_set_o @ B2 @ A2 )
=> ( ( ord_less_eq_set_o @ B2
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ A2 )
& ( P @ X3 ) ) ) )
= ( ! [X3: $o] :
( ( member_o @ X3 @ B2 )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_512_subset__CollectI,axiom,
! [B2: set_nat_nat_nat_nat,A2: set_nat_nat_nat_nat,Q: ( ( nat > nat ) > nat > nat ) > $o,P: ( ( nat > nat ) > nat > nat ) > $o] :
( ( ord_le5260717879541182899at_nat @ B2 @ A2 )
=> ( ! [X: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X @ B2 )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_le5260717879541182899at_nat
@ ( collec3567154360959927026at_nat
@ ^ [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3 @ B2 )
& ( Q @ X3 ) ) )
@ ( collec3567154360959927026at_nat
@ ^ [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3 @ A2 )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_513_subset__CollectI,axiom,
! [B2: set_nat_set_nat,A2: set_nat_set_nat,Q: ( nat > set_nat ) > $o,P: ( nat > set_nat ) > $o] :
( ( ord_le1585852046946910987et_nat @ B2 @ A2 )
=> ( ! [X: nat > set_nat] :
( ( member_nat_set_nat @ X @ B2 )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_le1585852046946910987et_nat
@ ( collect_nat_set_nat
@ ^ [X3: nat > set_nat] :
( ( member_nat_set_nat @ X3 @ B2 )
& ( Q @ X3 ) ) )
@ ( collect_nat_set_nat
@ ^ [X3: nat > set_nat] :
( ( member_nat_set_nat @ X3 @ A2 )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_514_subset__CollectI,axiom,
! [B2: set_nat_nat,A2: set_nat_nat,Q: ( nat > nat ) > $o,P: ( nat > nat ) > $o] :
( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
=> ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ B2 )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_le9059583361652607317at_nat
@ ( collect_nat_nat
@ ^ [X3: nat > nat] :
( ( member_nat_nat @ X3 @ B2 )
& ( Q @ X3 ) ) )
@ ( collect_nat_nat
@ ^ [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_515_subset__CollectI,axiom,
! [B2: set_nat,A2: set_nat,Q: nat > $o,P: nat > $o] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B2 )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ B2 )
& ( Q @ X3 ) ) )
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_516_subset__CollectI,axiom,
! [B2: set_set_nat,A2: set_set_nat,Q: set_nat > $o,P: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ B2 )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_le6893508408891458716et_nat
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ B2 )
& ( Q @ X3 ) ) )
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_517_subset__CollectI,axiom,
! [B2: set_nat_nat_nat,A2: set_nat_nat_nat,Q: ( nat > nat > nat ) > $o,P: ( nat > nat > nat ) > $o] :
( ( ord_le3211623285424100676at_nat @ B2 @ A2 )
=> ( ! [X: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X @ B2 )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_le3211623285424100676at_nat
@ ( collect_nat_nat_nat
@ ^ [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ B2 )
& ( Q @ X3 ) ) )
@ ( collect_nat_nat_nat
@ ^ [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A2 )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_518_subset__CollectI,axiom,
! [B2: set_o_nat_nat,A2: set_o_nat_nat,Q: ( $o > nat > nat ) > $o,P: ( $o > nat > nat ) > $o] :
( ( ord_le8808915593745164104at_nat @ B2 @ A2 )
=> ( ! [X: $o > nat > nat] :
( ( member_o_nat_nat @ X @ B2 )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_le8808915593745164104at_nat
@ ( collect_o_nat_nat
@ ^ [X3: $o > nat > nat] :
( ( member_o_nat_nat @ X3 @ B2 )
& ( Q @ X3 ) ) )
@ ( collect_o_nat_nat
@ ^ [X3: $o > nat > nat] :
( ( member_o_nat_nat @ X3 @ A2 )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_519_subset__CollectI,axiom,
! [B2: set_o,A2: set_o,Q: $o > $o,P: $o > $o] :
( ( ord_less_eq_set_o @ B2 @ A2 )
=> ( ! [X: $o] :
( ( member_o @ X @ B2 )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_less_eq_set_o
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ B2 )
& ( Q @ X3 ) ) )
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ A2 )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_520_conj__subset__def,axiom,
! [A2: set_nat_nat,P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ( ord_le9059583361652607317at_nat @ A2
@ ( collect_nat_nat
@ ^ [X3: nat > nat] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) )
= ( ( ord_le9059583361652607317at_nat @ A2 @ ( collect_nat_nat @ P ) )
& ( ord_le9059583361652607317at_nat @ A2 @ ( collect_nat_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_521_conj__subset__def,axiom,
! [A2: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A2
@ ( collect_nat
@ ^ [X3: nat] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) )
= ( ( ord_less_eq_set_nat @ A2 @ ( collect_nat @ P ) )
& ( ord_less_eq_set_nat @ A2 @ ( collect_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_522_conj__subset__def,axiom,
! [A2: set_set_nat,P: set_nat > $o,Q: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ A2
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) )
= ( ( ord_le6893508408891458716et_nat @ A2 @ ( collect_set_nat @ P ) )
& ( ord_le6893508408891458716et_nat @ A2 @ ( collect_set_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_523_conj__subset__def,axiom,
! [A2: set_nat_nat_nat,P: ( nat > nat > nat ) > $o,Q: ( nat > nat > nat ) > $o] :
( ( ord_le3211623285424100676at_nat @ A2
@ ( collect_nat_nat_nat
@ ^ [X3: nat > nat > nat] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) )
= ( ( ord_le3211623285424100676at_nat @ A2 @ ( collect_nat_nat_nat @ P ) )
& ( ord_le3211623285424100676at_nat @ A2 @ ( collect_nat_nat_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_524_conj__subset__def,axiom,
! [A2: set_o_nat_nat,P: ( $o > nat > nat ) > $o,Q: ( $o > nat > nat ) > $o] :
( ( ord_le8808915593745164104at_nat @ A2
@ ( collect_o_nat_nat
@ ^ [X3: $o > nat > nat] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) )
= ( ( ord_le8808915593745164104at_nat @ A2 @ ( collect_o_nat_nat @ P ) )
& ( ord_le8808915593745164104at_nat @ A2 @ ( collect_o_nat_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_525_conj__subset__def,axiom,
! [A2: set_o,P: $o > $o,Q: $o > $o] :
( ( ord_less_eq_set_o @ A2
@ ( collect_o
@ ^ [X3: $o] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) )
= ( ( ord_less_eq_set_o @ A2 @ ( collect_o @ P ) )
& ( ord_less_eq_set_o @ A2 @ ( collect_o @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_526_Sup_OSUP__identity__eq,axiom,
! [Sup: set_nat_nat > nat > nat,A2: set_nat_nat] :
( ( Sup
@ ( image_3205354838064109189at_nat
@ ^ [X3: nat > nat] : X3
@ A2 ) )
= ( Sup @ A2 ) ) ).
% Sup.SUP_identity_eq
thf(fact_527_Sup_OSUP__identity__eq,axiom,
! [Sup: set_nat > nat,A2: set_nat] :
( ( Sup
@ ( image_nat_nat
@ ^ [X3: nat] : X3
@ A2 ) )
= ( Sup @ A2 ) ) ).
% Sup.SUP_identity_eq
thf(fact_528_Sup_OSUP__identity__eq,axiom,
! [Sup: set_o > $o,A2: set_o] :
( ( Sup
@ ( image_o_o
@ ^ [X3: $o] : X3
@ A2 ) )
= ( Sup @ A2 ) ) ).
% Sup.SUP_identity_eq
thf(fact_529_Inf_OINF__identity__eq,axiom,
! [Inf: set_nat_nat > nat > nat,A2: set_nat_nat] :
( ( Inf
@ ( image_3205354838064109189at_nat
@ ^ [X3: nat > nat] : X3
@ A2 ) )
= ( Inf @ A2 ) ) ).
% Inf.INF_identity_eq
thf(fact_530_Inf_OINF__identity__eq,axiom,
! [Inf: set_nat > nat,A2: set_nat] :
( ( Inf
@ ( image_nat_nat
@ ^ [X3: nat] : X3
@ A2 ) )
= ( Inf @ A2 ) ) ).
% Inf.INF_identity_eq
thf(fact_531_Inf_OINF__identity__eq,axiom,
! [Inf: set_o > $o,A2: set_o] :
( ( Inf
@ ( image_o_o
@ ^ [X3: $o] : X3
@ A2 ) )
= ( Inf @ A2 ) ) ).
% Inf.INF_identity_eq
thf(fact_532_subset__PiE,axiom,
! [I2: set_nat_nat,S: ( nat > nat ) > set_nat_nat,T2: ( nat > nat ) > set_nat_nat] :
( ( ord_le5260717879541182899at_nat @ ( piE_nat_nat_nat_nat @ I2 @ S ) @ ( piE_nat_nat_nat_nat @ I2 @ T2 ) )
= ( ( ( piE_nat_nat_nat_nat @ I2 @ S )
= bot_bo3919185967433191911at_nat )
| ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ I2 )
=> ( ord_le9059583361652607317at_nat @ ( S @ X3 ) @ ( T2 @ X3 ) ) ) ) ) ).
% subset_PiE
thf(fact_533_subset__PiE,axiom,
! [I2: set_nat,S: nat > set_set_nat,T2: nat > set_set_nat] :
( ( ord_le1585852046946910987et_nat @ ( piE_nat_set_nat @ I2 @ S ) @ ( piE_nat_set_nat @ I2 @ T2 ) )
= ( ( ( piE_nat_set_nat @ I2 @ S )
= bot_bo4007787791999405887et_nat )
| ! [X3: nat] :
( ( member_nat @ X3 @ I2 )
=> ( ord_le6893508408891458716et_nat @ ( S @ X3 ) @ ( T2 @ X3 ) ) ) ) ) ).
% subset_PiE
thf(fact_534_subset__PiE,axiom,
! [I2: set_nat,S: nat > set_o,T2: nat > set_o] :
( ( ord_le6029213668185085951_nat_o @ ( piE_nat_o @ I2 @ S ) @ ( piE_nat_o @ I2 @ T2 ) )
= ( ( ( piE_nat_o @ I2 @ S )
= bot_bot_set_nat_o2 )
| ! [X3: nat] :
( ( member_nat @ X3 @ I2 )
=> ( ord_less_eq_set_o @ ( S @ X3 ) @ ( T2 @ X3 ) ) ) ) ) ).
% subset_PiE
thf(fact_535_subset__PiE,axiom,
! [I2: set_nat,S: nat > set_nat,T2: nat > set_nat] :
( ( ord_le9059583361652607317at_nat @ ( piE_nat_nat @ I2 @ S ) @ ( piE_nat_nat @ I2 @ T2 ) )
= ( ( ( piE_nat_nat @ I2 @ S )
= bot_bot_set_nat_nat )
| ! [X3: nat] :
( ( member_nat @ X3 @ I2 )
=> ( ord_less_eq_set_nat @ ( S @ X3 ) @ ( T2 @ X3 ) ) ) ) ) ).
% subset_PiE
thf(fact_536_subset__PiE,axiom,
! [I2: set_nat,S: nat > set_nat_nat,T2: nat > set_nat_nat] :
( ( ord_le3211623285424100676at_nat @ ( piE_nat_nat_nat2 @ I2 @ S ) @ ( piE_nat_nat_nat2 @ I2 @ T2 ) )
= ( ( ( piE_nat_nat_nat2 @ I2 @ S )
= bot_bo7445843802507891576at_nat )
| ! [X3: nat] :
( ( member_nat @ X3 @ I2 )
=> ( ord_le9059583361652607317at_nat @ ( S @ X3 ) @ ( T2 @ X3 ) ) ) ) ) ).
% subset_PiE
thf(fact_537_subset__PiE,axiom,
! [I2: set_o,S: $o > set_nat_nat,T2: $o > set_nat_nat] :
( ( ord_le8808915593745164104at_nat @ ( piE_o_nat_nat @ I2 @ S ) @ ( piE_o_nat_nat @ I2 @ T2 ) )
= ( ( ( piE_o_nat_nat @ I2 @ S )
= bot_bo8153995795486405652at_nat )
| ! [X3: $o] :
( ( member_o @ X3 @ I2 )
=> ( ord_le9059583361652607317at_nat @ ( S @ X3 ) @ ( T2 @ X3 ) ) ) ) ) ).
% subset_PiE
thf(fact_538_Greatest__equality,axiom,
! [P: set_nat_nat > $o,X2: set_nat_nat] :
( ( P @ X2 )
=> ( ! [Y4: set_nat_nat] :
( ( P @ Y4 )
=> ( ord_le9059583361652607317at_nat @ Y4 @ X2 ) )
=> ( ( order_8228081171942161500at_nat @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_539_Greatest__equality,axiom,
! [P: nat > $o,X2: nat] :
( ( P @ X2 )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X2 ) )
=> ( ( order_Greatest_nat @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_540_Greatest__equality,axiom,
! [P: set_nat > $o,X2: set_nat] :
( ( P @ X2 )
=> ( ! [Y4: set_nat] :
( ( P @ Y4 )
=> ( ord_less_eq_set_nat @ Y4 @ X2 ) )
=> ( ( order_5724808138429204845et_nat @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_541_Greatest__equality,axiom,
! [P: set_set_nat > $o,X2: set_set_nat] :
( ( P @ X2 )
=> ( ! [Y4: set_set_nat] :
( ( P @ Y4 )
=> ( ord_le6893508408891458716et_nat @ Y4 @ X2 ) )
=> ( ( order_1279421399067128355et_nat @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_542_Greatest__equality,axiom,
! [P: set_nat_nat_nat > $o,X2: set_nat_nat_nat] :
( ( P @ X2 )
=> ( ! [Y4: set_nat_nat_nat] :
( ( P @ Y4 )
=> ( ord_le3211623285424100676at_nat @ Y4 @ X2 ) )
=> ( ( order_8011581075050426827at_nat @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_543_Greatest__equality,axiom,
! [P: set_o_nat_nat > $o,X2: set_o_nat_nat] :
( ( P @ X2 )
=> ( ! [Y4: set_o_nat_nat] :
( ( P @ Y4 )
=> ( ord_le8808915593745164104at_nat @ Y4 @ X2 ) )
=> ( ( order_3307316977169705729at_nat @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_544_Greatest__equality,axiom,
! [P: set_o > $o,X2: set_o] :
( ( P @ X2 )
=> ( ! [Y4: set_o] :
( ( P @ Y4 )
=> ( ord_less_eq_set_o @ Y4 @ X2 ) )
=> ( ( order_Greatest_set_o @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_545_GreatestI2__order,axiom,
! [P: set_nat_nat > $o,X2: set_nat_nat,Q: set_nat_nat > $o] :
( ( P @ X2 )
=> ( ! [Y4: set_nat_nat] :
( ( P @ Y4 )
=> ( ord_le9059583361652607317at_nat @ Y4 @ X2 ) )
=> ( ! [X: set_nat_nat] :
( ( P @ X )
=> ( ! [Y: set_nat_nat] :
( ( P @ Y )
=> ( ord_le9059583361652607317at_nat @ Y @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_8228081171942161500at_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_546_GreatestI2__order,axiom,
! [P: nat > $o,X2: nat,Q: nat > $o] :
( ( P @ X2 )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X2 ) )
=> ( ! [X: nat] :
( ( P @ X )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_Greatest_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_547_GreatestI2__order,axiom,
! [P: set_nat > $o,X2: set_nat,Q: set_nat > $o] :
( ( P @ X2 )
=> ( ! [Y4: set_nat] :
( ( P @ Y4 )
=> ( ord_less_eq_set_nat @ Y4 @ X2 ) )
=> ( ! [X: set_nat] :
( ( P @ X )
=> ( ! [Y: set_nat] :
( ( P @ Y )
=> ( ord_less_eq_set_nat @ Y @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_5724808138429204845et_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_548_GreatestI2__order,axiom,
! [P: set_set_nat > $o,X2: set_set_nat,Q: set_set_nat > $o] :
( ( P @ X2 )
=> ( ! [Y4: set_set_nat] :
( ( P @ Y4 )
=> ( ord_le6893508408891458716et_nat @ Y4 @ X2 ) )
=> ( ! [X: set_set_nat] :
( ( P @ X )
=> ( ! [Y: set_set_nat] :
( ( P @ Y )
=> ( ord_le6893508408891458716et_nat @ Y @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_1279421399067128355et_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_549_GreatestI2__order,axiom,
! [P: set_nat_nat_nat > $o,X2: set_nat_nat_nat,Q: set_nat_nat_nat > $o] :
( ( P @ X2 )
=> ( ! [Y4: set_nat_nat_nat] :
( ( P @ Y4 )
=> ( ord_le3211623285424100676at_nat @ Y4 @ X2 ) )
=> ( ! [X: set_nat_nat_nat] :
( ( P @ X )
=> ( ! [Y: set_nat_nat_nat] :
( ( P @ Y )
=> ( ord_le3211623285424100676at_nat @ Y @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_8011581075050426827at_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_550_GreatestI2__order,axiom,
! [P: set_o_nat_nat > $o,X2: set_o_nat_nat,Q: set_o_nat_nat > $o] :
( ( P @ X2 )
=> ( ! [Y4: set_o_nat_nat] :
( ( P @ Y4 )
=> ( ord_le8808915593745164104at_nat @ Y4 @ X2 ) )
=> ( ! [X: set_o_nat_nat] :
( ( P @ X )
=> ( ! [Y: set_o_nat_nat] :
( ( P @ Y )
=> ( ord_le8808915593745164104at_nat @ Y @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_3307316977169705729at_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_551_GreatestI2__order,axiom,
! [P: set_o > $o,X2: set_o,Q: set_o > $o] :
( ( P @ X2 )
=> ( ! [Y4: set_o] :
( ( P @ Y4 )
=> ( ord_less_eq_set_o @ Y4 @ X2 ) )
=> ( ! [X: set_o] :
( ( P @ X )
=> ( ! [Y: set_o] :
( ( P @ Y )
=> ( ord_less_eq_set_o @ Y @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_Greatest_set_o @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_552_Sup_OSUP__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > nat,D: $o > nat,Sup: set_nat > nat] :
( ( A2 = B2 )
=> ( ! [X: $o] :
( ( member_o @ X @ B2 )
=> ( ( C @ X )
= ( D @ X ) ) )
=> ( ( Sup @ ( image_o_nat @ C @ A2 ) )
= ( Sup @ ( image_o_nat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_553_Sup_OSUP__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > $o,D: $o > $o,Sup: set_o > $o] :
( ( A2 = B2 )
=> ( ! [X: $o] :
( ( member_o @ X @ B2 )
=> ( ( C @ X )
= ( D @ X ) ) )
=> ( ( Sup @ ( image_o_o @ C @ A2 ) )
= ( Sup @ ( image_o_o @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_554_Sup_OSUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C: nat > set_nat,D: nat > set_nat,Sup: set_set_nat > set_nat] :
( ( A2 = B2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B2 )
=> ( ( C @ X )
= ( D @ X ) ) )
=> ( ( Sup @ ( image_nat_set_nat @ C @ A2 ) )
= ( Sup @ ( image_nat_set_nat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_555_Sup_OSUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C: nat > nat,D: nat > nat,Sup: set_nat > nat] :
( ( A2 = B2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B2 )
=> ( ( C @ X )
= ( D @ X ) ) )
=> ( ( Sup @ ( image_nat_nat @ C @ A2 ) )
= ( Sup @ ( image_nat_nat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_556_Sup_OSUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C: nat > $o,D: nat > $o,Sup: set_o > $o] :
( ( A2 = B2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B2 )
=> ( ( C @ X )
= ( D @ X ) ) )
=> ( ( Sup @ ( image_nat_o @ C @ A2 ) )
= ( Sup @ ( image_nat_o @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_557_Sup_OSUP__cong,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C: ( nat > nat ) > nat > nat,D: ( nat > nat ) > nat > nat,Sup: set_nat_nat > nat > nat] :
( ( A2 = B2 )
=> ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ B2 )
=> ( ( C @ X )
= ( D @ X ) ) )
=> ( ( Sup @ ( image_3205354838064109189at_nat @ C @ A2 ) )
= ( Sup @ ( image_3205354838064109189at_nat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_558_Inf_OINF__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > nat,D: $o > nat,Inf: set_nat > nat] :
( ( A2 = B2 )
=> ( ! [X: $o] :
( ( member_o @ X @ B2 )
=> ( ( C @ X )
= ( D @ X ) ) )
=> ( ( Inf @ ( image_o_nat @ C @ A2 ) )
= ( Inf @ ( image_o_nat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_559_Inf_OINF__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > $o,D: $o > $o,Inf: set_o > $o] :
( ( A2 = B2 )
=> ( ! [X: $o] :
( ( member_o @ X @ B2 )
=> ( ( C @ X )
= ( D @ X ) ) )
=> ( ( Inf @ ( image_o_o @ C @ A2 ) )
= ( Inf @ ( image_o_o @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_560_Inf_OINF__cong,axiom,
! [A2: set_nat,B2: set_nat,C: nat > set_nat,D: nat > set_nat,Inf: set_set_nat > set_nat] :
( ( A2 = B2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B2 )
=> ( ( C @ X )
= ( D @ X ) ) )
=> ( ( Inf @ ( image_nat_set_nat @ C @ A2 ) )
= ( Inf @ ( image_nat_set_nat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_561_Inf_OINF__cong,axiom,
! [A2: set_nat,B2: set_nat,C: nat > nat,D: nat > nat,Inf: set_nat > nat] :
( ( A2 = B2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B2 )
=> ( ( C @ X )
= ( D @ X ) ) )
=> ( ( Inf @ ( image_nat_nat @ C @ A2 ) )
= ( Inf @ ( image_nat_nat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_562_Inf_OINF__cong,axiom,
! [A2: set_nat,B2: set_nat,C: nat > $o,D: nat > $o,Inf: set_o > $o] :
( ( A2 = B2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B2 )
=> ( ( C @ X )
= ( D @ X ) ) )
=> ( ( Inf @ ( image_nat_o @ C @ A2 ) )
= ( Inf @ ( image_nat_o @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_563_Inf_OINF__cong,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C: ( nat > nat ) > nat > nat,D: ( nat > nat ) > nat > nat,Inf: set_nat_nat > nat > nat] :
( ( A2 = B2 )
=> ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ B2 )
=> ( ( C @ X )
= ( D @ X ) ) )
=> ( ( Inf @ ( image_3205354838064109189at_nat @ C @ A2 ) )
= ( Inf @ ( image_3205354838064109189at_nat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_564_empty__iff,axiom,
! [C2: $o] :
~ ( member_o @ C2 @ bot_bot_set_o ) ).
% empty_iff
thf(fact_565_empty__iff,axiom,
! [C2: ( nat > nat ) > nat > nat] :
~ ( member952132173341509300at_nat @ C2 @ bot_bo3919185967433191911at_nat ) ).
% empty_iff
thf(fact_566_empty__iff,axiom,
! [C2: nat > nat] :
~ ( member_nat_nat @ C2 @ bot_bot_set_nat_nat ) ).
% empty_iff
thf(fact_567_empty__iff,axiom,
! [C2: nat > set_nat] :
~ ( member_nat_set_nat @ C2 @ bot_bo4007787791999405887et_nat ) ).
% empty_iff
thf(fact_568_empty__iff,axiom,
! [C2: set_nat] :
~ ( member_set_nat @ C2 @ bot_bot_set_set_nat ) ).
% empty_iff
thf(fact_569_empty__iff,axiom,
! [C2: nat] :
~ ( member_nat @ C2 @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_570_all__not__in__conv,axiom,
! [A2: set_o] :
( ( ! [X3: $o] :
~ ( member_o @ X3 @ A2 ) )
= ( A2 = bot_bot_set_o ) ) ).
% all_not_in_conv
thf(fact_571_all__not__in__conv,axiom,
! [A2: set_nat_nat_nat_nat] :
( ( ! [X3: ( nat > nat ) > nat > nat] :
~ ( member952132173341509300at_nat @ X3 @ A2 ) )
= ( A2 = bot_bo3919185967433191911at_nat ) ) ).
% all_not_in_conv
thf(fact_572_all__not__in__conv,axiom,
! [A2: set_nat_nat] :
( ( ! [X3: nat > nat] :
~ ( member_nat_nat @ X3 @ A2 ) )
= ( A2 = bot_bot_set_nat_nat ) ) ).
% all_not_in_conv
thf(fact_573_all__not__in__conv,axiom,
! [A2: set_nat_set_nat] :
( ( ! [X3: nat > set_nat] :
~ ( member_nat_set_nat @ X3 @ A2 ) )
= ( A2 = bot_bo4007787791999405887et_nat ) ) ).
% all_not_in_conv
thf(fact_574_all__not__in__conv,axiom,
! [A2: set_set_nat] :
( ( ! [X3: set_nat] :
~ ( member_set_nat @ X3 @ A2 ) )
= ( A2 = bot_bot_set_set_nat ) ) ).
% all_not_in_conv
thf(fact_575_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X3: nat] :
~ ( member_nat @ X3 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_576_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_577_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_578_image__empty,axiom,
! [F: ( nat > nat ) > nat > nat] :
( ( image_3205354838064109189at_nat @ F @ bot_bot_set_nat_nat )
= bot_bot_set_nat_nat ) ).
% image_empty
thf(fact_579_image__empty,axiom,
! [F: $o > $o] :
( ( image_o_o @ F @ bot_bot_set_o )
= bot_bot_set_o ) ).
% image_empty
thf(fact_580_image__empty,axiom,
! [F: $o > nat] :
( ( image_o_nat @ F @ bot_bot_set_o )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_581_image__empty,axiom,
! [F: nat > set_nat] :
( ( image_nat_set_nat @ F @ bot_bot_set_nat )
= bot_bot_set_set_nat ) ).
% image_empty
thf(fact_582_image__empty,axiom,
! [F: nat > $o] :
( ( image_nat_o @ F @ bot_bot_set_nat )
= bot_bot_set_o ) ).
% image_empty
thf(fact_583_image__empty,axiom,
! [F: nat > nat] :
( ( image_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_584_empty__is__image,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( bot_bot_set_nat_nat
= ( image_3205354838064109189at_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_nat_nat ) ) ).
% empty_is_image
thf(fact_585_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_586_empty__is__image,axiom,
! [F: nat > set_nat,A2: set_nat] :
( ( bot_bot_set_set_nat
= ( image_nat_set_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_587_empty__is__image,axiom,
! [F: nat > $o,A2: set_nat] :
( ( bot_bot_set_o
= ( image_nat_o @ F @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_588_empty__is__image,axiom,
! [F: $o > nat,A2: set_o] :
( ( bot_bot_set_nat
= ( image_o_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_o ) ) ).
% empty_is_image
thf(fact_589_empty__is__image,axiom,
! [F: nat > nat,A2: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_590_image__is__empty,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( ( image_3205354838064109189at_nat @ F @ A2 )
= bot_bot_set_nat_nat )
= ( A2 = bot_bot_set_nat_nat ) ) ).
% image_is_empty
thf(fact_591_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_592_image__is__empty,axiom,
! [F: nat > set_nat,A2: set_nat] :
( ( ( image_nat_set_nat @ F @ A2 )
= bot_bot_set_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_593_image__is__empty,axiom,
! [F: nat > $o,A2: set_nat] :
( ( ( image_nat_o @ F @ A2 )
= bot_bot_set_o )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_594_image__is__empty,axiom,
! [F: $o > nat,A2: set_o] :
( ( ( image_o_nat @ F @ A2 )
= bot_bot_set_nat )
= ( A2 = bot_bot_set_o ) ) ).
% image_is_empty
thf(fact_595_image__is__empty,axiom,
! [F: nat > nat,A2: set_nat] :
( ( ( image_nat_nat @ F @ A2 )
= bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_596_subset__empty,axiom,
! [A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ bot_bot_set_nat_nat )
= ( A2 = bot_bot_set_nat_nat ) ) ).
% subset_empty
thf(fact_597_subset__empty,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_598_subset__empty,axiom,
! [A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ bot_bot_set_set_nat )
= ( A2 = bot_bot_set_set_nat ) ) ).
% subset_empty
thf(fact_599_subset__empty,axiom,
! [A2: set_nat_nat_nat] :
( ( ord_le3211623285424100676at_nat @ A2 @ bot_bo7445843802507891576at_nat )
= ( A2 = bot_bo7445843802507891576at_nat ) ) ).
% subset_empty
thf(fact_600_subset__empty,axiom,
! [A2: set_o_nat_nat] :
( ( ord_le8808915593745164104at_nat @ A2 @ bot_bo8153995795486405652at_nat )
= ( A2 = bot_bo8153995795486405652at_nat ) ) ).
% subset_empty
thf(fact_601_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_602_empty__subsetI,axiom,
! [A2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ bot_bot_set_nat_nat @ A2 ) ).
% empty_subsetI
thf(fact_603_empty__subsetI,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_604_empty__subsetI,axiom,
! [A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ bot_bot_set_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_605_empty__subsetI,axiom,
! [A2: set_nat_nat_nat] : ( ord_le3211623285424100676at_nat @ bot_bo7445843802507891576at_nat @ A2 ) ).
% empty_subsetI
thf(fact_606_empty__subsetI,axiom,
! [A2: set_o_nat_nat] : ( ord_le8808915593745164104at_nat @ bot_bo8153995795486405652at_nat @ A2 ) ).
% empty_subsetI
thf(fact_607_empty__subsetI,axiom,
! [A2: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A2 ) ).
% empty_subsetI
thf(fact_608_PiE__empty__range,axiom,
! [I4: nat,I2: set_nat,F2: nat > set_o] :
( ( member_nat @ I4 @ I2 )
=> ( ( ( F2 @ I4 )
= bot_bot_set_o )
=> ( ( piE_nat_o @ I2 @ F2 )
= bot_bot_set_nat_o2 ) ) ) ).
% PiE_empty_range
thf(fact_609_PiE__empty__range,axiom,
! [I4: $o,I2: set_o,F2: $o > set_nat] :
( ( member_o @ I4 @ I2 )
=> ( ( ( F2 @ I4 )
= bot_bot_set_nat )
=> ( ( piE_o_nat @ I2 @ F2 )
= bot_bot_set_o_nat ) ) ) ).
% PiE_empty_range
thf(fact_610_PiE__empty__range,axiom,
! [I4: nat,I2: set_nat,F2: nat > set_nat] :
( ( member_nat @ I4 @ I2 )
=> ( ( ( F2 @ I4 )
= bot_bot_set_nat )
=> ( ( piE_nat_nat @ I2 @ F2 )
= bot_bot_set_nat_nat ) ) ) ).
% PiE_empty_range
thf(fact_611_PiE__empty__range,axiom,
! [I4: nat,I2: set_nat,F2: nat > set_set_nat] :
( ( member_nat @ I4 @ I2 )
=> ( ( ( F2 @ I4 )
= bot_bot_set_set_nat )
=> ( ( piE_nat_set_nat @ I2 @ F2 )
= bot_bo4007787791999405887et_nat ) ) ) ).
% PiE_empty_range
thf(fact_612_PiE__empty__range,axiom,
! [I4: set_nat,I2: set_set_nat,F2: set_nat > set_nat] :
( ( member_set_nat @ I4 @ I2 )
=> ( ( ( F2 @ I4 )
= bot_bot_set_nat )
=> ( ( piE_set_nat_nat @ I2 @ F2 )
= bot_bo7208697003875722815at_nat ) ) ) ).
% PiE_empty_range
thf(fact_613_PiE__empty__range,axiom,
! [I4: $o,I2: set_o,F2: $o > set_nat_nat] :
( ( member_o @ I4 @ I2 )
=> ( ( ( F2 @ I4 )
= bot_bot_set_nat_nat )
=> ( ( piE_o_nat_nat @ I2 @ F2 )
= bot_bo8153995795486405652at_nat ) ) ) ).
% PiE_empty_range
thf(fact_614_PiE__empty__range,axiom,
! [I4: nat,I2: set_nat,F2: nat > set_nat_nat] :
( ( member_nat @ I4 @ I2 )
=> ( ( ( F2 @ I4 )
= bot_bot_set_nat_nat )
=> ( ( piE_nat_nat_nat2 @ I2 @ F2 )
= bot_bo7445843802507891576at_nat ) ) ) ).
% PiE_empty_range
thf(fact_615_PiE__empty__range,axiom,
! [I4: nat > nat,I2: set_nat_nat,F2: ( nat > nat ) > set_nat] :
( ( member_nat_nat @ I4 @ I2 )
=> ( ( ( F2 @ I4 )
= bot_bot_set_nat )
=> ( ( piE_nat_nat_nat @ I2 @ F2 )
= bot_bo945813143650711160at_nat ) ) ) ).
% PiE_empty_range
thf(fact_616_PiE__empty__range,axiom,
! [I4: nat > set_nat,I2: set_nat_set_nat,F2: ( nat > set_nat ) > set_nat] :
( ( member_nat_set_nat @ I4 @ I2 )
=> ( ( ( F2 @ I4 )
= bot_bot_set_nat )
=> ( ( piE_nat_set_nat_nat @ I2 @ F2 )
= bot_bo4500225246352649262at_nat ) ) ) ).
% PiE_empty_range
thf(fact_617_PiE__empty__range,axiom,
! [I4: nat > nat,I2: set_nat_nat,F2: ( nat > nat ) > set_nat_nat] :
( ( member_nat_nat @ I4 @ I2 )
=> ( ( ( F2 @ I4 )
= bot_bot_set_nat_nat )
=> ( ( piE_nat_nat_nat_nat @ I2 @ F2 )
= bot_bo3919185967433191911at_nat ) ) ) ).
% PiE_empty_range
thf(fact_618_emptyE,axiom,
! [A: $o] :
~ ( member_o @ A @ bot_bot_set_o ) ).
% emptyE
thf(fact_619_emptyE,axiom,
! [A: ( nat > nat ) > nat > nat] :
~ ( member952132173341509300at_nat @ A @ bot_bo3919185967433191911at_nat ) ).
% emptyE
thf(fact_620_emptyE,axiom,
! [A: nat > nat] :
~ ( member_nat_nat @ A @ bot_bot_set_nat_nat ) ).
% emptyE
thf(fact_621_emptyE,axiom,
! [A: nat > set_nat] :
~ ( member_nat_set_nat @ A @ bot_bo4007787791999405887et_nat ) ).
% emptyE
thf(fact_622_emptyE,axiom,
! [A: set_nat] :
~ ( member_set_nat @ A @ bot_bot_set_set_nat ) ).
% emptyE
thf(fact_623_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_624_equals0D,axiom,
! [A2: set_o,A: $o] :
( ( A2 = bot_bot_set_o )
=> ~ ( member_o @ A @ A2 ) ) ).
% equals0D
thf(fact_625_equals0D,axiom,
! [A2: set_nat_nat_nat_nat,A: ( nat > nat ) > nat > nat] :
( ( A2 = bot_bo3919185967433191911at_nat )
=> ~ ( member952132173341509300at_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_626_equals0D,axiom,
! [A2: set_nat_nat,A: nat > nat] :
( ( A2 = bot_bot_set_nat_nat )
=> ~ ( member_nat_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_627_equals0D,axiom,
! [A2: set_nat_set_nat,A: nat > set_nat] :
( ( A2 = bot_bo4007787791999405887et_nat )
=> ~ ( member_nat_set_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_628_equals0D,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( A2 = bot_bot_set_set_nat )
=> ~ ( member_set_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_629_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_630_equals0I,axiom,
! [A2: set_o] :
( ! [Y4: $o] :
~ ( member_o @ Y4 @ A2 )
=> ( A2 = bot_bot_set_o ) ) ).
% equals0I
thf(fact_631_equals0I,axiom,
! [A2: set_nat_nat_nat_nat] :
( ! [Y4: ( nat > nat ) > nat > nat] :
~ ( member952132173341509300at_nat @ Y4 @ A2 )
=> ( A2 = bot_bo3919185967433191911at_nat ) ) ).
% equals0I
thf(fact_632_equals0I,axiom,
! [A2: set_nat_nat] :
( ! [Y4: nat > nat] :
~ ( member_nat_nat @ Y4 @ A2 )
=> ( A2 = bot_bot_set_nat_nat ) ) ).
% equals0I
thf(fact_633_equals0I,axiom,
! [A2: set_nat_set_nat] :
( ! [Y4: nat > set_nat] :
~ ( member_nat_set_nat @ Y4 @ A2 )
=> ( A2 = bot_bo4007787791999405887et_nat ) ) ).
% equals0I
thf(fact_634_equals0I,axiom,
! [A2: set_set_nat] :
( ! [Y4: set_nat] :
~ ( member_set_nat @ Y4 @ A2 )
=> ( A2 = bot_bot_set_set_nat ) ) ).
% equals0I
thf(fact_635_equals0I,axiom,
! [A2: set_nat] :
( ! [Y4: nat] :
~ ( member_nat @ Y4 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_636_ex__in__conv,axiom,
! [A2: set_o] :
( ( ? [X3: $o] : ( member_o @ X3 @ A2 ) )
= ( A2 != bot_bot_set_o ) ) ).
% ex_in_conv
thf(fact_637_ex__in__conv,axiom,
! [A2: set_nat_nat_nat_nat] :
( ( ? [X3: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X3 @ A2 ) )
= ( A2 != bot_bo3919185967433191911at_nat ) ) ).
% ex_in_conv
thf(fact_638_ex__in__conv,axiom,
! [A2: set_nat_nat] :
( ( ? [X3: nat > nat] : ( member_nat_nat @ X3 @ A2 ) )
= ( A2 != bot_bot_set_nat_nat ) ) ).
% ex_in_conv
thf(fact_639_ex__in__conv,axiom,
! [A2: set_nat_set_nat] :
( ( ? [X3: nat > set_nat] : ( member_nat_set_nat @ X3 @ A2 ) )
= ( A2 != bot_bo4007787791999405887et_nat ) ) ).
% ex_in_conv
thf(fact_640_ex__in__conv,axiom,
! [A2: set_set_nat] :
( ( ? [X3: set_nat] : ( member_set_nat @ X3 @ A2 ) )
= ( A2 != bot_bot_set_set_nat ) ) ).
% ex_in_conv
thf(fact_641_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X3: nat] : ( member_nat @ X3 @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_642_Iio__eq__empty__iff,axiom,
! [N2: nat] :
( ( ( set_ord_lessThan_nat @ N2 )
= bot_bot_set_nat )
= ( N2 = bot_bot_nat ) ) ).
% Iio_eq_empty_iff
thf(fact_643_PiE__eq__empty__iff,axiom,
! [I2: set_o,F2: $o > set_nat_nat] :
( ( ( piE_o_nat_nat @ I2 @ F2 )
= bot_bo8153995795486405652at_nat )
= ( ? [X3: $o] :
( ( member_o @ X3 @ I2 )
& ( ( F2 @ X3 )
= bot_bot_set_nat_nat ) ) ) ) ).
% PiE_eq_empty_iff
thf(fact_644_PiE__eq__empty__iff,axiom,
! [I2: set_nat,F2: nat > set_o] :
( ( ( piE_nat_o @ I2 @ F2 )
= bot_bot_set_nat_o2 )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ I2 )
& ( ( F2 @ X3 )
= bot_bot_set_o ) ) ) ) ).
% PiE_eq_empty_iff
thf(fact_645_PiE__eq__empty__iff,axiom,
! [I2: set_nat,F2: nat > set_set_nat] :
( ( ( piE_nat_set_nat @ I2 @ F2 )
= bot_bo4007787791999405887et_nat )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ I2 )
& ( ( F2 @ X3 )
= bot_bot_set_set_nat ) ) ) ) ).
% PiE_eq_empty_iff
thf(fact_646_PiE__eq__empty__iff,axiom,
! [I2: set_nat,F2: nat > set_nat_nat] :
( ( ( piE_nat_nat_nat2 @ I2 @ F2 )
= bot_bo7445843802507891576at_nat )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ I2 )
& ( ( F2 @ X3 )
= bot_bot_set_nat_nat ) ) ) ) ).
% PiE_eq_empty_iff
thf(fact_647_PiE__eq__empty__iff,axiom,
! [I2: set_nat_nat,F2: ( nat > nat ) > set_nat_nat] :
( ( ( piE_nat_nat_nat_nat @ I2 @ F2 )
= bot_bo3919185967433191911at_nat )
= ( ? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ I2 )
& ( ( F2 @ X3 )
= bot_bot_set_nat_nat ) ) ) ) ).
% PiE_eq_empty_iff
thf(fact_648_PiE__eq__empty__iff,axiom,
! [I2: set_nat,F2: nat > set_nat] :
( ( ( piE_nat_nat @ I2 @ F2 )
= bot_bot_set_nat_nat )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ I2 )
& ( ( F2 @ X3 )
= bot_bot_set_nat ) ) ) ) ).
% PiE_eq_empty_iff
thf(fact_649_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X3: nat] : $false ) ) ).
% empty_def
thf(fact_650_bot_Oextremum,axiom,
! [A: set_nat_nat] : ( ord_le9059583361652607317at_nat @ bot_bot_set_nat_nat @ A ) ).
% bot.extremum
thf(fact_651_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_652_bot_Oextremum,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% bot.extremum
thf(fact_653_bot_Oextremum,axiom,
! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ bot_bot_set_set_nat @ A ) ).
% bot.extremum
thf(fact_654_bot_Oextremum,axiom,
! [A: set_nat_nat_nat] : ( ord_le3211623285424100676at_nat @ bot_bo7445843802507891576at_nat @ A ) ).
% bot.extremum
thf(fact_655_bot_Oextremum,axiom,
! [A: set_o_nat_nat] : ( ord_le8808915593745164104at_nat @ bot_bo8153995795486405652at_nat @ A ) ).
% bot.extremum
thf(fact_656_bot_Oextremum,axiom,
! [A: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A ) ).
% bot.extremum
thf(fact_657_bot_Oextremum__unique,axiom,
! [A: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ bot_bot_set_nat_nat )
= ( A = bot_bot_set_nat_nat ) ) ).
% bot.extremum_unique
thf(fact_658_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_659_bot_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_660_bot_Oextremum__unique,axiom,
! [A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ bot_bot_set_set_nat )
= ( A = bot_bot_set_set_nat ) ) ).
% bot.extremum_unique
thf(fact_661_bot_Oextremum__unique,axiom,
! [A: set_nat_nat_nat] :
( ( ord_le3211623285424100676at_nat @ A @ bot_bo7445843802507891576at_nat )
= ( A = bot_bo7445843802507891576at_nat ) ) ).
% bot.extremum_unique
thf(fact_662_bot_Oextremum__unique,axiom,
! [A: set_o_nat_nat] :
( ( ord_le8808915593745164104at_nat @ A @ bot_bo8153995795486405652at_nat )
= ( A = bot_bo8153995795486405652at_nat ) ) ).
% bot.extremum_unique
thf(fact_663_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_664_bot_Oextremum__uniqueI,axiom,
! [A: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ bot_bot_set_nat_nat )
=> ( A = bot_bot_set_nat_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_665_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_666_bot_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
=> ( A = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_667_bot_Oextremum__uniqueI,axiom,
! [A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ bot_bot_set_set_nat )
=> ( A = bot_bot_set_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_668_bot_Oextremum__uniqueI,axiom,
! [A: set_nat_nat_nat] :
( ( ord_le3211623285424100676at_nat @ A @ bot_bo7445843802507891576at_nat )
=> ( A = bot_bo7445843802507891576at_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_669_bot_Oextremum__uniqueI,axiom,
! [A: set_o_nat_nat] :
( ( ord_le8808915593745164104at_nat @ A @ bot_bo8153995795486405652at_nat )
=> ( A = bot_bo8153995795486405652at_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_670_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_671_subset__emptyI,axiom,
! [A2: set_nat_nat_nat_nat] :
( ! [X: ( nat > nat ) > nat > nat] :
~ ( member952132173341509300at_nat @ X @ A2 )
=> ( ord_le5260717879541182899at_nat @ A2 @ bot_bo3919185967433191911at_nat ) ) ).
% subset_emptyI
thf(fact_672_subset__emptyI,axiom,
! [A2: set_nat_set_nat] :
( ! [X: nat > set_nat] :
~ ( member_nat_set_nat @ X @ A2 )
=> ( ord_le1585852046946910987et_nat @ A2 @ bot_bo4007787791999405887et_nat ) ) ).
% subset_emptyI
thf(fact_673_subset__emptyI,axiom,
! [A2: set_nat_nat] :
( ! [X: nat > nat] :
~ ( member_nat_nat @ X @ A2 )
=> ( ord_le9059583361652607317at_nat @ A2 @ bot_bot_set_nat_nat ) ) ).
% subset_emptyI
thf(fact_674_subset__emptyI,axiom,
! [A2: set_nat] :
( ! [X: nat] :
~ ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_675_subset__emptyI,axiom,
! [A2: set_set_nat] :
( ! [X: set_nat] :
~ ( member_set_nat @ X @ A2 )
=> ( ord_le6893508408891458716et_nat @ A2 @ bot_bot_set_set_nat ) ) ).
% subset_emptyI
thf(fact_676_subset__emptyI,axiom,
! [A2: set_nat_nat_nat] :
( ! [X: nat > nat > nat] :
~ ( member_nat_nat_nat2 @ X @ A2 )
=> ( ord_le3211623285424100676at_nat @ A2 @ bot_bo7445843802507891576at_nat ) ) ).
% subset_emptyI
thf(fact_677_subset__emptyI,axiom,
! [A2: set_o_nat_nat] :
( ! [X: $o > nat > nat] :
~ ( member_o_nat_nat @ X @ A2 )
=> ( ord_le8808915593745164104at_nat @ A2 @ bot_bo8153995795486405652at_nat ) ) ).
% subset_emptyI
thf(fact_678_subset__emptyI,axiom,
! [A2: set_o] :
( ! [X: $o] :
~ ( member_o @ X @ A2 )
=> ( ord_less_eq_set_o @ A2 @ bot_bot_set_o ) ) ).
% subset_emptyI
thf(fact_679_PiE__eq,axiom,
! [I2: set_nat,S: nat > set_nat,T2: nat > set_nat] :
( ( ( piE_nat_nat @ I2 @ S )
= ( piE_nat_nat @ I2 @ T2 ) )
= ( ( ( ( piE_nat_nat @ I2 @ S )
= bot_bot_set_nat_nat )
& ( ( piE_nat_nat @ I2 @ T2 )
= bot_bot_set_nat_nat ) )
| ! [X3: nat] :
( ( member_nat @ X3 @ I2 )
=> ( ( S @ X3 )
= ( T2 @ X3 ) ) ) ) ) ).
% PiE_eq
thf(fact_680_PiE__eq,axiom,
! [I2: set_o,S: $o > set_nat_nat,T2: $o > set_nat_nat] :
( ( ( piE_o_nat_nat @ I2 @ S )
= ( piE_o_nat_nat @ I2 @ T2 ) )
= ( ( ( ( piE_o_nat_nat @ I2 @ S )
= bot_bo8153995795486405652at_nat )
& ( ( piE_o_nat_nat @ I2 @ T2 )
= bot_bo8153995795486405652at_nat ) )
| ! [X3: $o] :
( ( member_o @ X3 @ I2 )
=> ( ( S @ X3 )
= ( T2 @ X3 ) ) ) ) ) ).
% PiE_eq
thf(fact_681_PiE__eq,axiom,
! [I2: set_nat,S: nat > set_o,T2: nat > set_o] :
( ( ( piE_nat_o @ I2 @ S )
= ( piE_nat_o @ I2 @ T2 ) )
= ( ( ( ( piE_nat_o @ I2 @ S )
= bot_bot_set_nat_o2 )
& ( ( piE_nat_o @ I2 @ T2 )
= bot_bot_set_nat_o2 ) )
| ! [X3: nat] :
( ( member_nat @ X3 @ I2 )
=> ( ( S @ X3 )
= ( T2 @ X3 ) ) ) ) ) ).
% PiE_eq
thf(fact_682_PiE__eq,axiom,
! [I2: set_nat,S: nat > set_set_nat,T2: nat > set_set_nat] :
( ( ( piE_nat_set_nat @ I2 @ S )
= ( piE_nat_set_nat @ I2 @ T2 ) )
= ( ( ( ( piE_nat_set_nat @ I2 @ S )
= bot_bo4007787791999405887et_nat )
& ( ( piE_nat_set_nat @ I2 @ T2 )
= bot_bo4007787791999405887et_nat ) )
| ! [X3: nat] :
( ( member_nat @ X3 @ I2 )
=> ( ( S @ X3 )
= ( T2 @ X3 ) ) ) ) ) ).
% PiE_eq
thf(fact_683_PiE__eq,axiom,
! [I2: set_nat,S: nat > set_nat_nat,T2: nat > set_nat_nat] :
( ( ( piE_nat_nat_nat2 @ I2 @ S )
= ( piE_nat_nat_nat2 @ I2 @ T2 ) )
= ( ( ( ( piE_nat_nat_nat2 @ I2 @ S )
= bot_bo7445843802507891576at_nat )
& ( ( piE_nat_nat_nat2 @ I2 @ T2 )
= bot_bo7445843802507891576at_nat ) )
| ! [X3: nat] :
( ( member_nat @ X3 @ I2 )
=> ( ( S @ X3 )
= ( T2 @ X3 ) ) ) ) ) ).
% PiE_eq
thf(fact_684_PiE__eq,axiom,
! [I2: set_nat_nat,S: ( nat > nat ) > set_nat_nat,T2: ( nat > nat ) > set_nat_nat] :
( ( ( piE_nat_nat_nat_nat @ I2 @ S )
= ( piE_nat_nat_nat_nat @ I2 @ T2 ) )
= ( ( ( ( piE_nat_nat_nat_nat @ I2 @ S )
= bot_bo3919185967433191911at_nat )
& ( ( piE_nat_nat_nat_nat @ I2 @ T2 )
= bot_bo3919185967433191911at_nat ) )
| ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ I2 )
=> ( ( S @ X3 )
= ( T2 @ X3 ) ) ) ) ) ).
% PiE_eq
thf(fact_685_all__PiE__elements,axiom,
! [I2: set_nat,S: nat > set_nat,P: nat > nat > $o] :
( ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ ( piE_nat_nat @ I2 @ S ) )
=> ! [Y6: nat] :
( ( member_nat @ Y6 @ I2 )
=> ( P @ Y6 @ ( X3 @ Y6 ) ) ) ) )
= ( ( ( piE_nat_nat @ I2 @ S )
= bot_bot_set_nat_nat )
| ! [X3: nat] :
( ( member_nat @ X3 @ I2 )
=> ! [Y6: nat] :
( ( member_nat @ Y6 @ ( S @ X3 ) )
=> ( P @ X3 @ Y6 ) ) ) ) ) ).
% all_PiE_elements
thf(fact_686_all__PiE__elements,axiom,
! [I2: set_o,S: $o > set_nat_nat,P: $o > ( nat > nat ) > $o] :
( ( ! [X3: $o > nat > nat] :
( ( member_o_nat_nat @ X3 @ ( piE_o_nat_nat @ I2 @ S ) )
=> ! [Y6: $o] :
( ( member_o @ Y6 @ I2 )
=> ( P @ Y6 @ ( X3 @ Y6 ) ) ) ) )
= ( ( ( piE_o_nat_nat @ I2 @ S )
= bot_bo8153995795486405652at_nat )
| ! [X3: $o] :
( ( member_o @ X3 @ I2 )
=> ! [Y6: nat > nat] :
( ( member_nat_nat @ Y6 @ ( S @ X3 ) )
=> ( P @ X3 @ Y6 ) ) ) ) ) ).
% all_PiE_elements
thf(fact_687_all__PiE__elements,axiom,
! [I2: set_nat,S: nat > set_o,P: nat > $o > $o] :
( ( ! [X3: nat > $o] :
( ( member_nat_o @ X3 @ ( piE_nat_o @ I2 @ S ) )
=> ! [Y6: nat] :
( ( member_nat @ Y6 @ I2 )
=> ( P @ Y6 @ ( X3 @ Y6 ) ) ) ) )
= ( ( ( piE_nat_o @ I2 @ S )
= bot_bot_set_nat_o2 )
| ! [X3: nat] :
( ( member_nat @ X3 @ I2 )
=> ! [Y6: $o] :
( ( member_o @ Y6 @ ( S @ X3 ) )
=> ( P @ X3 @ Y6 ) ) ) ) ) ).
% all_PiE_elements
thf(fact_688_all__PiE__elements,axiom,
! [I2: set_nat,S: nat > set_set_nat,P: nat > set_nat > $o] :
( ( ! [X3: nat > set_nat] :
( ( member_nat_set_nat @ X3 @ ( piE_nat_set_nat @ I2 @ S ) )
=> ! [Y6: nat] :
( ( member_nat @ Y6 @ I2 )
=> ( P @ Y6 @ ( X3 @ Y6 ) ) ) ) )
= ( ( ( piE_nat_set_nat @ I2 @ S )
= bot_bo4007787791999405887et_nat )
| ! [X3: nat] :
( ( member_nat @ X3 @ I2 )
=> ! [Y6: set_nat] :
( ( member_set_nat @ Y6 @ ( S @ X3 ) )
=> ( P @ X3 @ Y6 ) ) ) ) ) ).
% all_PiE_elements
thf(fact_689_all__PiE__elements,axiom,
! [I2: set_nat,S: nat > set_nat_nat,P: nat > ( nat > nat ) > $o] :
( ( ! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ ( piE_nat_nat_nat2 @ I2 @ S ) )
=> ! [Y6: nat] :
( ( member_nat @ Y6 @ I2 )
=> ( P @ Y6 @ ( X3 @ Y6 ) ) ) ) )
= ( ( ( piE_nat_nat_nat2 @ I2 @ S )
= bot_bo7445843802507891576at_nat )
| ! [X3: nat] :
( ( member_nat @ X3 @ I2 )
=> ! [Y6: nat > nat] :
( ( member_nat_nat @ Y6 @ ( S @ X3 ) )
=> ( P @ X3 @ Y6 ) ) ) ) ) ).
% all_PiE_elements
thf(fact_690_all__PiE__elements,axiom,
! [I2: set_nat_nat,S: ( nat > nat ) > set_nat_nat,P: ( nat > nat ) > ( nat > nat ) > $o] :
( ( ! [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3 @ ( piE_nat_nat_nat_nat @ I2 @ S ) )
=> ! [Y6: nat > nat] :
( ( member_nat_nat @ Y6 @ I2 )
=> ( P @ Y6 @ ( X3 @ Y6 ) ) ) ) )
= ( ( ( piE_nat_nat_nat_nat @ I2 @ S )
= bot_bo3919185967433191911at_nat )
| ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ I2 )
=> ! [Y6: nat > nat] :
( ( member_nat_nat @ Y6 @ ( S @ X3 ) )
=> ( P @ X3 @ Y6 ) ) ) ) ) ).
% all_PiE_elements
thf(fact_691_PiE__eq__iff,axiom,
! [I2: set_o,F2: $o > set_nat_nat,F3: $o > set_nat_nat] :
( ( ( piE_o_nat_nat @ I2 @ F2 )
= ( piE_o_nat_nat @ I2 @ F3 ) )
= ( ! [X3: $o] :
( ( member_o @ X3 @ I2 )
=> ( ( F2 @ X3 )
= ( F3 @ X3 ) ) )
| ( ? [X3: $o] :
( ( member_o @ X3 @ I2 )
& ( ( F2 @ X3 )
= bot_bot_set_nat_nat ) )
& ? [X3: $o] :
( ( member_o @ X3 @ I2 )
& ( ( F3 @ X3 )
= bot_bot_set_nat_nat ) ) ) ) ) ).
% PiE_eq_iff
thf(fact_692_PiE__eq__iff,axiom,
! [I2: set_nat,F2: nat > set_o,F3: nat > set_o] :
( ( ( piE_nat_o @ I2 @ F2 )
= ( piE_nat_o @ I2 @ F3 ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ I2 )
=> ( ( F2 @ X3 )
= ( F3 @ X3 ) ) )
| ( ? [X3: nat] :
( ( member_nat @ X3 @ I2 )
& ( ( F2 @ X3 )
= bot_bot_set_o ) )
& ? [X3: nat] :
( ( member_nat @ X3 @ I2 )
& ( ( F3 @ X3 )
= bot_bot_set_o ) ) ) ) ) ).
% PiE_eq_iff
thf(fact_693_PiE__eq__iff,axiom,
! [I2: set_nat,F2: nat > set_set_nat,F3: nat > set_set_nat] :
( ( ( piE_nat_set_nat @ I2 @ F2 )
= ( piE_nat_set_nat @ I2 @ F3 ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ I2 )
=> ( ( F2 @ X3 )
= ( F3 @ X3 ) ) )
| ( ? [X3: nat] :
( ( member_nat @ X3 @ I2 )
& ( ( F2 @ X3 )
= bot_bot_set_set_nat ) )
& ? [X3: nat] :
( ( member_nat @ X3 @ I2 )
& ( ( F3 @ X3 )
= bot_bot_set_set_nat ) ) ) ) ) ).
% PiE_eq_iff
thf(fact_694_PiE__eq__iff,axiom,
! [I2: set_nat,F2: nat > set_nat_nat,F3: nat > set_nat_nat] :
( ( ( piE_nat_nat_nat2 @ I2 @ F2 )
= ( piE_nat_nat_nat2 @ I2 @ F3 ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ I2 )
=> ( ( F2 @ X3 )
= ( F3 @ X3 ) ) )
| ( ? [X3: nat] :
( ( member_nat @ X3 @ I2 )
& ( ( F2 @ X3 )
= bot_bot_set_nat_nat ) )
& ? [X3: nat] :
( ( member_nat @ X3 @ I2 )
& ( ( F3 @ X3 )
= bot_bot_set_nat_nat ) ) ) ) ) ).
% PiE_eq_iff
thf(fact_695_PiE__eq__iff,axiom,
! [I2: set_nat_nat,F2: ( nat > nat ) > set_nat_nat,F3: ( nat > nat ) > set_nat_nat] :
( ( ( piE_nat_nat_nat_nat @ I2 @ F2 )
= ( piE_nat_nat_nat_nat @ I2 @ F3 ) )
= ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ I2 )
=> ( ( F2 @ X3 )
= ( F3 @ X3 ) ) )
| ( ? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ I2 )
& ( ( F2 @ X3 )
= bot_bot_set_nat_nat ) )
& ? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ I2 )
& ( ( F3 @ X3 )
= bot_bot_set_nat_nat ) ) ) ) ) ).
% PiE_eq_iff
thf(fact_696_PiE__eq__iff,axiom,
! [I2: set_nat,F2: nat > set_nat,F3: nat > set_nat] :
( ( ( piE_nat_nat @ I2 @ F2 )
= ( piE_nat_nat @ I2 @ F3 ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ I2 )
=> ( ( F2 @ X3 )
= ( F3 @ X3 ) ) )
| ( ? [X3: nat] :
( ( member_nat @ X3 @ I2 )
& ( ( F2 @ X3 )
= bot_bot_set_nat ) )
& ? [X3: nat] :
( ( member_nat @ X3 @ I2 )
& ( ( F3 @ X3 )
= bot_bot_set_nat ) ) ) ) ) ).
% PiE_eq_iff
thf(fact_697_PiE__eq__iff__not__empty,axiom,
! [I2: set_nat,F2: nat > set_o,F3: nat > set_o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ I2 )
=> ( ( F2 @ I3 )
!= bot_bot_set_o ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I2 )
=> ( ( F3 @ I3 )
!= bot_bot_set_o ) )
=> ( ( ( piE_nat_o @ I2 @ F2 )
= ( piE_nat_o @ I2 @ F3 ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ I2 )
=> ( ( F2 @ X3 )
= ( F3 @ X3 ) ) ) ) ) ) ) ).
% PiE_eq_iff_not_empty
thf(fact_698_PiE__eq__iff__not__empty,axiom,
! [I2: set_o,F2: $o > set_nat,F3: $o > set_nat] :
( ! [I3: $o] :
( ( member_o @ I3 @ I2 )
=> ( ( F2 @ I3 )
!= bot_bot_set_nat ) )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I2 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat ) )
=> ( ( ( piE_o_nat @ I2 @ F2 )
= ( piE_o_nat @ I2 @ F3 ) )
= ( ! [X3: $o] :
( ( member_o @ X3 @ I2 )
=> ( ( F2 @ X3 )
= ( F3 @ X3 ) ) ) ) ) ) ) ).
% PiE_eq_iff_not_empty
thf(fact_699_PiE__eq__iff__not__empty,axiom,
! [I2: set_nat,F2: nat > set_nat,F3: nat > set_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ I2 )
=> ( ( F2 @ I3 )
!= bot_bot_set_nat ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I2 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat ) )
=> ( ( ( piE_nat_nat @ I2 @ F2 )
= ( piE_nat_nat @ I2 @ F3 ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ I2 )
=> ( ( F2 @ X3 )
= ( F3 @ X3 ) ) ) ) ) ) ) ).
% PiE_eq_iff_not_empty
thf(fact_700_PiE__eq__iff__not__empty,axiom,
! [I2: set_nat,F2: nat > set_set_nat,F3: nat > set_set_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ I2 )
=> ( ( F2 @ I3 )
!= bot_bot_set_set_nat ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I2 )
=> ( ( F3 @ I3 )
!= bot_bot_set_set_nat ) )
=> ( ( ( piE_nat_set_nat @ I2 @ F2 )
= ( piE_nat_set_nat @ I2 @ F3 ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ I2 )
=> ( ( F2 @ X3 )
= ( F3 @ X3 ) ) ) ) ) ) ) ).
% PiE_eq_iff_not_empty
thf(fact_701_PiE__eq__iff__not__empty,axiom,
! [I2: set_set_nat,F2: set_nat > set_nat,F3: set_nat > set_nat] :
( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ I2 )
=> ( ( F2 @ I3 )
!= bot_bot_set_nat ) )
=> ( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ I2 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat ) )
=> ( ( ( piE_set_nat_nat @ I2 @ F2 )
= ( piE_set_nat_nat @ I2 @ F3 ) )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ I2 )
=> ( ( F2 @ X3 )
= ( F3 @ X3 ) ) ) ) ) ) ) ).
% PiE_eq_iff_not_empty
thf(fact_702_PiE__eq__iff__not__empty,axiom,
! [I2: set_o,F2: $o > set_nat_nat,F3: $o > set_nat_nat] :
( ! [I3: $o] :
( ( member_o @ I3 @ I2 )
=> ( ( F2 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I2 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ( ( piE_o_nat_nat @ I2 @ F2 )
= ( piE_o_nat_nat @ I2 @ F3 ) )
= ( ! [X3: $o] :
( ( member_o @ X3 @ I2 )
=> ( ( F2 @ X3 )
= ( F3 @ X3 ) ) ) ) ) ) ) ).
% PiE_eq_iff_not_empty
thf(fact_703_PiE__eq__iff__not__empty,axiom,
! [I2: set_nat,F2: nat > set_nat_nat,F3: nat > set_nat_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ I2 )
=> ( ( F2 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I2 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ( ( piE_nat_nat_nat2 @ I2 @ F2 )
= ( piE_nat_nat_nat2 @ I2 @ F3 ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ I2 )
=> ( ( F2 @ X3 )
= ( F3 @ X3 ) ) ) ) ) ) ) ).
% PiE_eq_iff_not_empty
thf(fact_704_PiE__eq__iff__not__empty,axiom,
! [I2: set_nat_nat,F2: ( nat > nat ) > set_nat,F3: ( nat > nat ) > set_nat] :
( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ I2 )
=> ( ( F2 @ I3 )
!= bot_bot_set_nat ) )
=> ( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ I2 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat ) )
=> ( ( ( piE_nat_nat_nat @ I2 @ F2 )
= ( piE_nat_nat_nat @ I2 @ F3 ) )
= ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ I2 )
=> ( ( F2 @ X3 )
= ( F3 @ X3 ) ) ) ) ) ) ) ).
% PiE_eq_iff_not_empty
thf(fact_705_PiE__eq__iff__not__empty,axiom,
! [I2: set_nat_set_nat,F2: ( nat > set_nat ) > set_nat,F3: ( nat > set_nat ) > set_nat] :
( ! [I3: nat > set_nat] :
( ( member_nat_set_nat @ I3 @ I2 )
=> ( ( F2 @ I3 )
!= bot_bot_set_nat ) )
=> ( ! [I3: nat > set_nat] :
( ( member_nat_set_nat @ I3 @ I2 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat ) )
=> ( ( ( piE_nat_set_nat_nat @ I2 @ F2 )
= ( piE_nat_set_nat_nat @ I2 @ F3 ) )
= ( ! [X3: nat > set_nat] :
( ( member_nat_set_nat @ X3 @ I2 )
=> ( ( F2 @ X3 )
= ( F3 @ X3 ) ) ) ) ) ) ) ).
% PiE_eq_iff_not_empty
thf(fact_706_PiE__eq__iff__not__empty,axiom,
! [I2: set_nat_nat,F2: ( nat > nat ) > set_nat_nat,F3: ( nat > nat ) > set_nat_nat] :
( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ I2 )
=> ( ( F2 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ I2 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ( ( piE_nat_nat_nat_nat @ I2 @ F2 )
= ( piE_nat_nat_nat_nat @ I2 @ F3 ) )
= ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ I2 )
=> ( ( F2 @ X3 )
= ( F3 @ X3 ) ) ) ) ) ) ) ).
% PiE_eq_iff_not_empty
thf(fact_707_PiE__eq__subset,axiom,
! [I2: set_o,F2: $o > set_nat,F3: $o > set_nat,I4: $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ I2 )
=> ( ( F2 @ I3 )
!= bot_bot_set_nat ) )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I2 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat ) )
=> ( ( ( piE_o_nat @ I2 @ F2 )
= ( piE_o_nat @ I2 @ F3 ) )
=> ( ( member_o @ I4 @ I2 )
=> ( ord_less_eq_set_nat @ ( F2 @ I4 ) @ ( F3 @ I4 ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_708_PiE__eq__subset,axiom,
! [I2: set_nat,F2: nat > set_nat,F3: nat > set_nat,I4: nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ I2 )
=> ( ( F2 @ I3 )
!= bot_bot_set_nat ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I2 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat ) )
=> ( ( ( piE_nat_nat @ I2 @ F2 )
= ( piE_nat_nat @ I2 @ F3 ) )
=> ( ( member_nat @ I4 @ I2 )
=> ( ord_less_eq_set_nat @ ( F2 @ I4 ) @ ( F3 @ I4 ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_709_PiE__eq__subset,axiom,
! [I2: set_o,F2: $o > set_o,F3: $o > set_o,I4: $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ I2 )
=> ( ( F2 @ I3 )
!= bot_bot_set_o ) )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I2 )
=> ( ( F3 @ I3 )
!= bot_bot_set_o ) )
=> ( ( ( piE_o_o @ I2 @ F2 )
= ( piE_o_o @ I2 @ F3 ) )
=> ( ( member_o @ I4 @ I2 )
=> ( ord_less_eq_set_o @ ( F2 @ I4 ) @ ( F3 @ I4 ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_710_PiE__eq__subset,axiom,
! [I2: set_nat,F2: nat > set_o,F3: nat > set_o,I4: nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ I2 )
=> ( ( F2 @ I3 )
!= bot_bot_set_o ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I2 )
=> ( ( F3 @ I3 )
!= bot_bot_set_o ) )
=> ( ( ( piE_nat_o @ I2 @ F2 )
= ( piE_nat_o @ I2 @ F3 ) )
=> ( ( member_nat @ I4 @ I2 )
=> ( ord_less_eq_set_o @ ( F2 @ I4 ) @ ( F3 @ I4 ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_711_PiE__eq__subset,axiom,
! [I2: set_set_nat,F2: set_nat > set_nat,F3: set_nat > set_nat,I4: set_nat] :
( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ I2 )
=> ( ( F2 @ I3 )
!= bot_bot_set_nat ) )
=> ( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ I2 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat ) )
=> ( ( ( piE_set_nat_nat @ I2 @ F2 )
= ( piE_set_nat_nat @ I2 @ F3 ) )
=> ( ( member_set_nat @ I4 @ I2 )
=> ( ord_less_eq_set_nat @ ( F2 @ I4 ) @ ( F3 @ I4 ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_712_PiE__eq__subset,axiom,
! [I2: set_o,F2: $o > set_set_nat,F3: $o > set_set_nat,I4: $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ I2 )
=> ( ( F2 @ I3 )
!= bot_bot_set_set_nat ) )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I2 )
=> ( ( F3 @ I3 )
!= bot_bot_set_set_nat ) )
=> ( ( ( piE_o_set_nat @ I2 @ F2 )
= ( piE_o_set_nat @ I2 @ F3 ) )
=> ( ( member_o @ I4 @ I2 )
=> ( ord_le6893508408891458716et_nat @ ( F2 @ I4 ) @ ( F3 @ I4 ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_713_PiE__eq__subset,axiom,
! [I2: set_nat,F2: nat > set_set_nat,F3: nat > set_set_nat,I4: nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ I2 )
=> ( ( F2 @ I3 )
!= bot_bot_set_set_nat ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I2 )
=> ( ( F3 @ I3 )
!= bot_bot_set_set_nat ) )
=> ( ( ( piE_nat_set_nat @ I2 @ F2 )
= ( piE_nat_set_nat @ I2 @ F3 ) )
=> ( ( member_nat @ I4 @ I2 )
=> ( ord_le6893508408891458716et_nat @ ( F2 @ I4 ) @ ( F3 @ I4 ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_714_PiE__eq__subset,axiom,
! [I2: set_set_nat,F2: set_nat > set_o,F3: set_nat > set_o,I4: set_nat] :
( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ I2 )
=> ( ( F2 @ I3 )
!= bot_bot_set_o ) )
=> ( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ I2 )
=> ( ( F3 @ I3 )
!= bot_bot_set_o ) )
=> ( ( ( piE_set_nat_o @ I2 @ F2 )
= ( piE_set_nat_o @ I2 @ F3 ) )
=> ( ( member_set_nat @ I4 @ I2 )
=> ( ord_less_eq_set_o @ ( F2 @ I4 ) @ ( F3 @ I4 ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_715_PiE__eq__subset,axiom,
! [I2: set_o,F2: $o > set_nat_nat,F3: $o > set_nat_nat,I4: $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ I2 )
=> ( ( F2 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I2 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ( ( piE_o_nat_nat @ I2 @ F2 )
= ( piE_o_nat_nat @ I2 @ F3 ) )
=> ( ( member_o @ I4 @ I2 )
=> ( ord_le9059583361652607317at_nat @ ( F2 @ I4 ) @ ( F3 @ I4 ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_716_PiE__eq__subset,axiom,
! [I2: set_nat,F2: nat > set_nat_nat,F3: nat > set_nat_nat,I4: nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ I2 )
=> ( ( F2 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I2 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ( ( piE_nat_nat_nat2 @ I2 @ F2 )
= ( piE_nat_nat_nat2 @ I2 @ F3 ) )
=> ( ( member_nat @ I4 @ I2 )
=> ( ord_le9059583361652607317at_nat @ ( F2 @ I4 ) @ ( F3 @ I4 ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_717_the__elem__image__unique,axiom,
! [A2: set_o,F: $o > nat,X2: $o] :
( ( A2 != bot_bot_set_o )
=> ( ! [Y4: $o] :
( ( member_o @ Y4 @ A2 )
=> ( ( F @ Y4 )
= ( F @ X2 ) ) )
=> ( ( the_elem_nat @ ( image_o_nat @ F @ A2 ) )
= ( F @ X2 ) ) ) ) ).
% the_elem_image_unique
thf(fact_718_the__elem__image__unique,axiom,
! [A2: set_o,F: $o > $o,X2: $o] :
( ( A2 != bot_bot_set_o )
=> ( ! [Y4: $o] :
( ( member_o @ Y4 @ A2 )
=> ( ( F @ Y4 )
= ( F @ X2 ) ) )
=> ( ( the_elem_o @ ( image_o_o @ F @ A2 ) )
= ( F @ X2 ) ) ) ) ).
% the_elem_image_unique
thf(fact_719_the__elem__image__unique,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > nat > nat,X2: nat > nat] :
( ( A2 != bot_bot_set_nat_nat )
=> ( ! [Y4: nat > nat] :
( ( member_nat_nat @ Y4 @ A2 )
=> ( ( F @ Y4 )
= ( F @ X2 ) ) )
=> ( ( the_elem_nat_nat @ ( image_3205354838064109189at_nat @ F @ A2 ) )
= ( F @ X2 ) ) ) ) ).
% the_elem_image_unique
thf(fact_720_the__elem__image__unique,axiom,
! [A2: set_nat,F: nat > set_nat,X2: nat] :
( ( A2 != bot_bot_set_nat )
=> ( ! [Y4: nat] :
( ( member_nat @ Y4 @ A2 )
=> ( ( F @ Y4 )
= ( F @ X2 ) ) )
=> ( ( the_elem_set_nat @ ( image_nat_set_nat @ F @ A2 ) )
= ( F @ X2 ) ) ) ) ).
% the_elem_image_unique
thf(fact_721_the__elem__image__unique,axiom,
! [A2: set_nat,F: nat > nat,X2: nat] :
( ( A2 != bot_bot_set_nat )
=> ( ! [Y4: nat] :
( ( member_nat @ Y4 @ A2 )
=> ( ( F @ Y4 )
= ( F @ X2 ) ) )
=> ( ( the_elem_nat @ ( image_nat_nat @ F @ A2 ) )
= ( F @ X2 ) ) ) ) ).
% the_elem_image_unique
thf(fact_722_the__elem__image__unique,axiom,
! [A2: set_nat,F: nat > $o,X2: nat] :
( ( A2 != bot_bot_set_nat )
=> ( ! [Y4: nat] :
( ( member_nat @ Y4 @ A2 )
=> ( ( F @ Y4 )
= ( F @ X2 ) ) )
=> ( ( the_elem_o @ ( image_nat_o @ F @ A2 ) )
= ( F @ X2 ) ) ) ) ).
% the_elem_image_unique
thf(fact_723_Set_Ois__empty__def,axiom,
( is_empty_nat
= ( ^ [A3: set_nat] : ( A3 = bot_bot_set_nat ) ) ) ).
% Set.is_empty_def
thf(fact_724_image__Fpow__mono,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F @ A2 ) @ B2 )
=> ( ord_le4954213926817602059at_nat @ ( image_3832368097948589297at_nat @ ( image_3205354838064109189at_nat @ F ) @ ( finite_Fpow_nat_nat @ A2 ) ) @ ( finite_Fpow_nat_nat @ B2 ) ) ) ).
% image_Fpow_mono
thf(fact_725_image__Fpow__mono,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ ( finite_Fpow_nat @ A2 ) ) @ ( finite_Fpow_nat @ B2 ) ) ) ).
% image_Fpow_mono
thf(fact_726_image__Fpow__mono,axiom,
! [F: $o > nat,A2: set_o,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_o_nat @ F @ A2 ) @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( image_set_o_set_nat @ ( image_o_nat @ F ) @ ( finite_Fpow_o @ A2 ) ) @ ( finite_Fpow_nat @ B2 ) ) ) ).
% image_Fpow_mono
thf(fact_727_image__Fpow__mono,axiom,
! [F: nat > set_nat,A2: set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A2 ) @ B2 )
=> ( ord_le9131159989063066194et_nat @ ( image_6725021117256019401et_nat @ ( image_nat_set_nat @ F ) @ ( finite_Fpow_nat @ A2 ) ) @ ( finite_Fpow_set_nat @ B2 ) ) ) ).
% image_Fpow_mono
thf(fact_728_image__Fpow__mono,axiom,
! [F: nat > $o,A2: set_nat,B2: set_o] :
( ( ord_less_eq_set_o @ ( image_nat_o @ F @ A2 ) @ B2 )
=> ( ord_le4374716579403074808_set_o @ ( image_set_nat_set_o @ ( image_nat_o @ F ) @ ( finite_Fpow_nat @ A2 ) ) @ ( finite_Fpow_o @ B2 ) ) ) ).
% image_Fpow_mono
thf(fact_729_image__Fpow__mono,axiom,
! [F: $o > $o,A2: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ ( image_o_o @ F @ A2 ) @ B2 )
=> ( ord_le4374716579403074808_set_o @ ( image_set_o_set_o @ ( image_o_o @ F ) @ ( finite_Fpow_o @ A2 ) ) @ ( finite_Fpow_o @ B2 ) ) ) ).
% image_Fpow_mono
thf(fact_730_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_731_SUP__empty,axiom,
! [F: nat > $o] :
( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ bot_bot_set_nat ) )
= bot_bot_o ) ).
% SUP_empty
thf(fact_732_SUP__empty,axiom,
! [F: nat > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ bot_bot_set_nat ) )
= bot_bot_set_nat ) ).
% SUP_empty
thf(fact_733_SUP__constant,axiom,
! [C2: $o,A2: set_o] :
( ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [Y6: $o] : C2
@ A2 ) )
= ( ( ( A2 = bot_bot_set_o )
=> bot_bot_o )
& ( ( A2 != bot_bot_set_o )
=> C2 ) ) ) ).
% SUP_constant
thf(fact_734_SUP__constant,axiom,
! [C2: $o,A2: set_nat] :
( ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [Y6: nat] : C2
@ A2 ) )
= ( ( ( A2 = bot_bot_set_nat )
=> bot_bot_o )
& ( ( A2 != bot_bot_set_nat )
=> C2 ) ) ) ).
% SUP_constant
thf(fact_735_SUP__constant,axiom,
! [A2: set_nat,C2: set_nat] :
( ( ( A2 = bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y6: nat] : C2
@ A2 ) )
= bot_bot_set_nat ) )
& ( ( A2 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y6: nat] : C2
@ A2 ) )
= C2 ) ) ) ).
% SUP_constant
thf(fact_736_SUP__subset__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > $o,G: nat > $o] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_less_eq_o @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A2 ) ) @ ( complete_Sup_Sup_o @ ( image_nat_o @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_737_SUP__subset__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > $o,G: $o > $o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ord_less_eq_o @ ( F @ X ) @ ( G @ X ) ) )
=> ( 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_738_SUP__subset__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > set_o,G: nat > set_o] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_o @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ F @ A2 ) ) @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_739_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 )
=> ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ord_less_eq_set_o @ ( F @ X ) @ ( G @ X ) ) )
=> ( 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_740_SUP__subset__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat,F: set_nat > $o,G: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( ord_less_eq_o @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_set_nat_o @ F @ A2 ) ) @ ( complete_Sup_Sup_o @ ( image_set_nat_o @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_741_SUP__subset__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_742_SUP__subset__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > set_nat,G: $o > set_nat] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_743_SUP__subset__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > set_set_nat,G: nat > set_set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ F @ A2 ) ) @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_744_SUP__subset__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat,F: set_nat > set_o,G: set_nat > set_o] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( ord_less_eq_set_o @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_set_nat_set_o @ F @ A2 ) ) @ ( comple90263536869209701_set_o @ ( image_set_nat_set_o @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_745_SUP__subset__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > set_set_nat,G: $o > set_set_nat] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ ( image_o_set_set_nat @ F @ A2 ) ) @ ( comple548664676211718543et_nat @ ( image_o_set_set_nat @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_746_INF__superset__mono,axiom,
! [B2: set_nat,A2: set_nat,F: nat > $o,G: nat > $o] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B2 )
=> ( ord_less_eq_o @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_nat_o @ F @ A2 ) ) @ ( complete_Inf_Inf_o @ ( image_nat_o @ G @ B2 ) ) ) ) ) ).
% INF_superset_mono
thf(fact_747_INF__superset__mono,axiom,
! [B2: set_o,A2: set_o,F: $o > $o,G: $o > $o] :
( ( ord_less_eq_set_o @ B2 @ A2 )
=> ( ! [X: $o] :
( ( member_o @ X @ B2 )
=> ( ord_less_eq_o @ ( F @ X ) @ ( G @ X ) ) )
=> ( 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_748_INF__superset__mono,axiom,
! [B2: set_nat,A2: set_nat,F: nat > set_o,G: nat > set_o] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B2 )
=> ( ord_less_eq_set_o @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_o @ ( comple3063163877087187839_set_o @ ( image_nat_set_o @ F @ A2 ) ) @ ( comple3063163877087187839_set_o @ ( image_nat_set_o @ G @ B2 ) ) ) ) ) ).
% INF_superset_mono
thf(fact_749_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 )
=> ( ! [X: $o] :
( ( member_o @ X @ B2 )
=> ( ord_less_eq_set_o @ ( F @ X ) @ ( G @ X ) ) )
=> ( 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_750_INF__superset__mono,axiom,
! [B2: set_set_nat,A2: set_set_nat,F: set_nat > $o,G: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ B2 )
=> ( ord_less_eq_o @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_set_nat_o @ F @ A2 ) ) @ ( complete_Inf_Inf_o @ ( image_set_nat_o @ G @ B2 ) ) ) ) ) ).
% INF_superset_mono
thf(fact_751_INF__superset__mono,axiom,
! [B2: set_nat,A2: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ G @ B2 ) ) ) ) ) ).
% INF_superset_mono
thf(fact_752_INF__superset__mono,axiom,
! [B2: set_o,A2: set_o,F: $o > set_nat,G: $o > set_nat] :
( ( ord_less_eq_set_o @ B2 @ A2 )
=> ( ! [X: $o] :
( ( member_o @ X @ B2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ A2 ) ) @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ G @ B2 ) ) ) ) ) ).
% INF_superset_mono
thf(fact_753_INF__superset__mono,axiom,
! [B2: set_nat,A2: set_nat,F: nat > set_set_nat,G: nat > set_set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_le6893508408891458716et_nat @ ( comple1065008630642458357et_nat @ ( image_2194112158459175443et_nat @ F @ A2 ) ) @ ( comple1065008630642458357et_nat @ ( image_2194112158459175443et_nat @ G @ B2 ) ) ) ) ) ).
% INF_superset_mono
thf(fact_754_INF__superset__mono,axiom,
! [B2: set_set_nat,A2: set_set_nat,F: set_nat > set_o,G: set_nat > set_o] :
( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ B2 )
=> ( ord_less_eq_set_o @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_o @ ( comple3063163877087187839_set_o @ ( image_set_nat_set_o @ F @ A2 ) ) @ ( comple3063163877087187839_set_o @ ( image_set_nat_set_o @ G @ B2 ) ) ) ) ) ).
% INF_superset_mono
thf(fact_755_INF__superset__mono,axiom,
! [B2: set_o,A2: set_o,F: $o > set_set_nat,G: $o > set_set_nat] :
( ( ord_less_eq_set_o @ B2 @ A2 )
=> ( ! [X: $o] :
( ( member_o @ X @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_le6893508408891458716et_nat @ ( comple1065008630642458357et_nat @ ( image_o_set_set_nat @ F @ A2 ) ) @ ( comple1065008630642458357et_nat @ ( image_o_set_set_nat @ G @ B2 ) ) ) ) ) ).
% INF_superset_mono
thf(fact_756_greaterThan__subset__iff,axiom,
! [X2: $o,Y2: $o] :
( ( ord_less_eq_set_o @ ( set_or6416164934427428222Than_o @ X2 ) @ ( set_or6416164934427428222Than_o @ Y2 ) )
= ( ord_less_eq_o @ Y2 @ X2 ) ) ).
% greaterThan_subset_iff
thf(fact_757_greaterThan__subset__iff,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_set_nat @ ( set_or1210151606488870762an_nat @ X2 ) @ ( set_or1210151606488870762an_nat @ Y2 ) )
= ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% greaterThan_subset_iff
thf(fact_758_UN__ball__bex__simps_I4_J,axiom,
! [B2: nat > set_nat,A2: set_nat,P: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat @ X3 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
& ( P @ X3 ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ? [Y6: nat] :
( ( member_nat @ Y6 @ ( B2 @ X3 ) )
& ( P @ Y6 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_759_UN__ball__bex__simps_I2_J,axiom,
! [B2: nat > set_nat,A2: set_nat,P: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat @ X3 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
=> ( P @ X3 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ! [Y6: nat] :
( ( member_nat @ Y6 @ ( B2 @ X3 ) )
=> ( P @ Y6 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_760_bex__UN,axiom,
! [B2: nat > set_nat,A2: set_nat,P: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat @ X3 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
& ( P @ X3 ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ? [Y6: nat] :
( ( member_nat @ Y6 @ ( B2 @ X3 ) )
& ( P @ Y6 ) ) ) ) ) ).
% bex_UN
thf(fact_761_ball__UN,axiom,
! [B2: nat > set_nat,A2: set_nat,P: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat @ X3 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
=> ( P @ X3 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ! [Y6: nat] :
( ( member_nat @ Y6 @ ( B2 @ X3 ) )
=> ( P @ Y6 ) ) ) ) ) ).
% ball_UN
thf(fact_762_Pi__I,axiom,
! [A2: set_o,F: $o > $o,B2: $o > set_o] :
( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( member_o @ ( F @ X ) @ ( B2 @ X ) ) )
=> ( member_o_o @ F @ ( pi_o_o @ A2 @ B2 ) ) ) ).
% Pi_I
thf(fact_763_Pi__I,axiom,
! [A2: set_o,F: $o > nat,B2: $o > set_nat] :
( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( B2 @ X ) ) )
=> ( member_o_nat @ F @ ( pi_o_nat @ A2 @ B2 ) ) ) ).
% Pi_I
thf(fact_764_Pi__I,axiom,
! [A2: set_nat,F: nat > $o,B2: nat > set_o] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_o @ ( F @ X ) @ ( B2 @ X ) ) )
=> ( member_nat_o @ F @ ( pi_nat_o @ A2 @ B2 ) ) ) ).
% Pi_I
thf(fact_765_Pi__I,axiom,
! [A2: set_nat,F: nat > nat,B2: nat > set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( B2 @ X ) ) )
=> ( member_nat_nat @ F @ ( pi_nat_nat @ A2 @ B2 ) ) ) ).
% Pi_I
thf(fact_766_Pi__I,axiom,
! [A2: set_o,F: $o > set_nat,B2: $o > set_set_nat] :
( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( member_set_nat @ ( F @ X ) @ ( B2 @ X ) ) )
=> ( member_o_set_nat @ F @ ( pi_o_set_nat @ A2 @ B2 ) ) ) ).
% Pi_I
thf(fact_767_Pi__I,axiom,
! [A2: set_nat,F: nat > set_nat,B2: nat > set_set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_set_nat @ ( F @ X ) @ ( B2 @ X ) ) )
=> ( member_nat_set_nat @ F @ ( pi_nat_set_nat @ A2 @ B2 ) ) ) ).
% Pi_I
thf(fact_768_Pi__I,axiom,
! [A2: set_set_nat,F: set_nat > $o,B2: set_nat > set_o] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( member_o @ ( F @ X ) @ ( B2 @ X ) ) )
=> ( member_set_nat_o @ F @ ( pi_set_nat_o @ A2 @ B2 ) ) ) ).
% Pi_I
thf(fact_769_Pi__I,axiom,
! [A2: set_set_nat,F: set_nat > nat,B2: set_nat > set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( B2 @ X ) ) )
=> ( member_set_nat_nat @ F @ ( pi_set_nat_nat @ A2 @ B2 ) ) ) ).
% Pi_I
thf(fact_770_Pi__I,axiom,
! [A2: set_o,F: $o > nat > nat,B2: $o > set_nat_nat] :
( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( member_nat_nat @ ( F @ X ) @ ( B2 @ X ) ) )
=> ( member_o_nat_nat @ F @ ( pi_o_nat_nat @ A2 @ B2 ) ) ) ).
% Pi_I
thf(fact_771_Pi__I,axiom,
! [A2: set_nat,F: nat > nat > nat,B2: nat > set_nat_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat_nat @ ( F @ X ) @ ( B2 @ X ) ) )
=> ( member_nat_nat_nat2 @ F @ ( pi_nat_nat_nat2 @ A2 @ B2 ) ) ) ).
% Pi_I
thf(fact_772_greaterThan__eq__iff,axiom,
! [X2: nat,Y2: nat] :
( ( ( set_or1210151606488870762an_nat @ X2 )
= ( set_or1210151606488870762an_nat @ Y2 ) )
= ( X2 = Y2 ) ) ).
% greaterThan_eq_iff
thf(fact_773_INT__iff,axiom,
! [B: nat,B2: nat > set_nat,A2: set_nat] :
( ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ B @ ( B2 @ X3 ) ) ) ) ) ).
% INT_iff
thf(fact_774_INT__I,axiom,
! [A2: set_o,B: $o,B2: $o > set_o] :
( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( member_o @ B @ ( B2 @ X ) ) )
=> ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ B2 @ A2 ) ) ) ) ).
% INT_I
thf(fact_775_INT__I,axiom,
! [A2: set_nat,B: $o,B2: nat > set_o] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_o @ B @ ( B2 @ X ) ) )
=> ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_nat_set_o @ B2 @ A2 ) ) ) ) ).
% INT_I
thf(fact_776_INT__I,axiom,
! [A2: set_o,B: nat,B2: $o > set_nat] :
( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( member_nat @ B @ ( B2 @ X ) ) )
=> ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ B2 @ A2 ) ) ) ) ).
% INT_I
thf(fact_777_INT__I,axiom,
! [A2: set_nat,B: nat,B2: nat > set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ B @ ( B2 @ X ) ) )
=> ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) ) ) ).
% INT_I
thf(fact_778_INT__I,axiom,
! [A2: set_o,B: set_nat,B2: $o > set_set_nat] :
( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( member_set_nat @ B @ ( B2 @ X ) ) )
=> ( member_set_nat @ B @ ( comple1065008630642458357et_nat @ ( image_o_set_set_nat @ B2 @ A2 ) ) ) ) ).
% INT_I
thf(fact_779_INT__I,axiom,
! [A2: set_nat,B: set_nat,B2: nat > set_set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_set_nat @ B @ ( B2 @ X ) ) )
=> ( member_set_nat @ B @ ( comple1065008630642458357et_nat @ ( image_2194112158459175443et_nat @ B2 @ A2 ) ) ) ) ).
% INT_I
thf(fact_780_INT__I,axiom,
! [A2: set_set_nat,B: $o,B2: set_nat > set_o] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( member_o @ B @ ( B2 @ X ) ) )
=> ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_set_nat_set_o @ B2 @ A2 ) ) ) ) ).
% INT_I
thf(fact_781_INT__I,axiom,
! [A2: set_set_nat,B: nat,B2: set_nat > set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( member_nat @ B @ ( B2 @ X ) ) )
=> ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_7916887816326733075et_nat @ B2 @ A2 ) ) ) ) ).
% INT_I
thf(fact_782_INT__I,axiom,
! [A2: set_o,B: nat > nat,B2: $o > set_nat_nat] :
( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( member_nat_nat @ B @ ( B2 @ X ) ) )
=> ( member_nat_nat @ B @ ( comple439066603627490862at_nat @ ( image_o_set_nat_nat @ B2 @ A2 ) ) ) ) ).
% INT_I
thf(fact_783_INT__I,axiom,
! [A2: set_nat,B: nat > nat,B2: nat > set_nat_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat_nat @ B @ ( B2 @ X ) ) )
=> ( member_nat_nat @ B @ ( comple439066603627490862at_nat @ ( image_7301343469591561292at_nat @ B2 @ A2 ) ) ) ) ).
% INT_I
thf(fact_784_UN__iff,axiom,
! [B: nat,B2: nat > set_nat,A2: set_nat] :
( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( member_nat @ B @ ( B2 @ X3 ) ) ) ) ) ).
% UN_iff
thf(fact_785_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_786_UN__I,axiom,
! [A: nat,A2: set_nat,B: $o,B2: nat > set_o] :
( ( member_nat @ A @ A2 )
=> ( ( member_o @ B @ ( B2 @ A ) )
=> ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_787_UN__I,axiom,
! [A: $o,A2: set_o,B: nat,B2: $o > set_nat] :
( ( member_o @ A @ A2 )
=> ( ( member_nat @ B @ ( B2 @ A ) )
=> ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_788_UN__I,axiom,
! [A: nat,A2: set_nat,B: nat,B2: nat > set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( member_nat @ B @ ( B2 @ A ) )
=> ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_789_UN__I,axiom,
! [A: $o,A2: set_o,B: set_nat,B2: $o > set_set_nat] :
( ( member_o @ A @ A2 )
=> ( ( member_set_nat @ B @ ( B2 @ A ) )
=> ( member_set_nat @ B @ ( comple548664676211718543et_nat @ ( image_o_set_set_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_790_UN__I,axiom,
! [A: nat,A2: set_nat,B: set_nat,B2: nat > set_set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( member_set_nat @ B @ ( B2 @ A ) )
=> ( member_set_nat @ B @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_791_UN__I,axiom,
! [A: set_nat,A2: set_set_nat,B: $o,B2: set_nat > set_o] :
( ( member_set_nat @ A @ A2 )
=> ( ( member_o @ B @ ( B2 @ A ) )
=> ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_set_nat_set_o @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_792_UN__I,axiom,
! [A: set_nat,A2: set_set_nat,B: nat,B2: set_nat > set_nat] :
( ( member_set_nat @ A @ A2 )
=> ( ( member_nat @ B @ ( B2 @ A ) )
=> ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_793_UN__I,axiom,
! [A: $o,A2: set_o,B: nat > nat,B2: $o > set_nat_nat] :
( ( member_o @ A @ A2 )
=> ( ( member_nat_nat @ B @ ( B2 @ A ) )
=> ( member_nat_nat @ B @ ( comple5448282615319421384at_nat @ ( image_o_set_nat_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_794_UN__I,axiom,
! [A: nat,A2: set_nat,B: nat > nat,B2: nat > set_nat_nat] :
( ( member_nat @ A @ A2 )
=> ( ( member_nat_nat @ B @ ( B2 @ A ) )
=> ( member_nat_nat @ B @ ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_795_Sup__bot__conv_I2_J,axiom,
! [A2: set_o] :
( ( bot_bot_o
= ( complete_Sup_Sup_o @ A2 ) )
= ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( X3 = bot_bot_o ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_796_Sup__bot__conv_I2_J,axiom,
! [A2: set_set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ A2 ) )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( X3 = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_797_Sup__bot__conv_I1_J,axiom,
! [A2: set_o] :
( ( ( complete_Sup_Sup_o @ A2 )
= bot_bot_o )
= ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( X3 = bot_bot_o ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_798_Sup__bot__conv_I1_J,axiom,
! [A2: set_set_nat] :
( ( ( comple7399068483239264473et_nat @ A2 )
= bot_bot_set_nat )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( X3 = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_799_SUP__identity__eq,axiom,
! [A2: set_nat_nat] :
( ( comple2450677804321093138at_nat
@ ( image_3205354838064109189at_nat
@ ^ [X3: nat > nat] : X3
@ A2 ) )
= ( comple2450677804321093138at_nat @ A2 ) ) ).
% SUP_identity_eq
thf(fact_800_SUP__identity__eq,axiom,
! [A2: set_nat] :
( ( complete_Sup_Sup_nat
@ ( image_nat_nat
@ ^ [X3: nat] : X3
@ A2 ) )
= ( complete_Sup_Sup_nat @ A2 ) ) ).
% SUP_identity_eq
thf(fact_801_SUP__identity__eq,axiom,
! [A2: set_o] :
( ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [X3: $o] : X3
@ A2 ) )
= ( complete_Sup_Sup_o @ A2 ) ) ).
% SUP_identity_eq
thf(fact_802_SUP__identity__eq,axiom,
! [A2: set_set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [X3: set_nat] : X3
@ A2 ) )
= ( comple7399068483239264473et_nat @ A2 ) ) ).
% SUP_identity_eq
thf(fact_803_INF__identity__eq,axiom,
! [A2: set_nat_nat] :
( ( comple6608973012141742712at_nat
@ ( image_3205354838064109189at_nat
@ ^ [X3: nat > nat] : X3
@ A2 ) )
= ( comple6608973012141742712at_nat @ A2 ) ) ).
% INF_identity_eq
thf(fact_804_INF__identity__eq,axiom,
! [A2: set_o] :
( ( complete_Inf_Inf_o
@ ( image_o_o
@ ^ [X3: $o] : X3
@ A2 ) )
= ( complete_Inf_Inf_o @ A2 ) ) ).
% INF_identity_eq
thf(fact_805_INF__identity__eq,axiom,
! [A2: set_set_nat] :
( ( comple7806235888213564991et_nat
@ ( image_7916887816326733075et_nat
@ ^ [X3: set_nat] : X3
@ A2 ) )
= ( comple7806235888213564991et_nat @ A2 ) ) ).
% INF_identity_eq
thf(fact_806_INF__identity__eq,axiom,
! [A2: set_nat] :
( ( complete_Inf_Inf_nat
@ ( image_nat_nat
@ ^ [X3: nat] : X3
@ A2 ) )
= ( complete_Inf_Inf_nat @ A2 ) ) ).
% INF_identity_eq
thf(fact_807_UN__constant,axiom,
! [A2: set_nat,C2: set_nat] :
( ( ( A2 = bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y6: nat] : C2
@ A2 ) )
= bot_bot_set_nat ) )
& ( ( A2 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y6: nat] : C2
@ A2 ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_808_Sup__empty,axiom,
( ( complete_Sup_Sup_o @ bot_bot_set_o )
= bot_bot_o ) ).
% Sup_empty
thf(fact_809_Sup__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% Sup_empty
thf(fact_810_SUP__bot,axiom,
! [A2: set_nat] :
( ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [X3: nat] : bot_bot_o
@ A2 ) )
= bot_bot_o ) ).
% SUP_bot
thf(fact_811_SUP__bot,axiom,
! [A2: set_o] :
( ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [X3: $o] : bot_bot_o
@ A2 ) )
= bot_bot_o ) ).
% SUP_bot
thf(fact_812_SUP__bot,axiom,
! [A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X3: nat] : bot_bot_set_nat
@ A2 ) )
= bot_bot_set_nat ) ).
% SUP_bot
thf(fact_813_SUP__bot__conv_I1_J,axiom,
! [B2: nat > $o,A2: set_nat] :
( ( ( complete_Sup_Sup_o @ ( image_nat_o @ B2 @ A2 ) )
= bot_bot_o )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( B2 @ X3 )
= bot_bot_o ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_814_SUP__bot__conv_I1_J,axiom,
! [B2: $o > $o,A2: set_o] :
( ( ( complete_Sup_Sup_o @ ( image_o_o @ B2 @ A2 ) )
= bot_bot_o )
= ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ( B2 @ X3 )
= bot_bot_o ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_815_SUP__bot__conv_I1_J,axiom,
! [B2: nat > set_nat,A2: set_nat] :
( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) )
= bot_bot_set_nat )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( B2 @ X3 )
= bot_bot_set_nat ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_816_SUP__bot__conv_I2_J,axiom,
! [B2: nat > $o,A2: set_nat] :
( ( bot_bot_o
= ( complete_Sup_Sup_o @ ( image_nat_o @ B2 @ A2 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( B2 @ X3 )
= bot_bot_o ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_817_SUP__bot__conv_I2_J,axiom,
! [B2: $o > $o,A2: set_o] :
( ( bot_bot_o
= ( complete_Sup_Sup_o @ ( image_o_o @ B2 @ A2 ) ) )
= ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ( B2 @ X3 )
= bot_bot_o ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_818_SUP__bot__conv_I2_J,axiom,
! [B2: nat > set_nat,A2: set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( B2 @ X3 )
= bot_bot_set_nat ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_819_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_820_SUP__const,axiom,
! [A2: set_nat,F: $o] :
( ( A2 != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [I: nat] : F
@ A2 ) )
= F ) ) ).
% SUP_const
thf(fact_821_SUP__const,axiom,
! [A2: set_nat,F: set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I: nat] : F
@ A2 ) )
= F ) ) ).
% SUP_const
thf(fact_822_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_823_INF__const,axiom,
! [A2: set_nat,F: $o] :
( ( A2 != bot_bot_set_nat )
=> ( ( complete_Inf_Inf_o
@ ( image_nat_o
@ ^ [I: nat] : F
@ A2 ) )
= F ) ) ).
% INF_const
thf(fact_824_INF__const,axiom,
! [A2: set_nat,F: set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [I: nat] : F
@ A2 ) )
= F ) ) ).
% INF_const
thf(fact_825_SUP__apply,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,X2: nat] :
( ( comple2450677804321093138at_nat @ ( image_3205354838064109189at_nat @ F @ A2 ) @ X2 )
= ( complete_Sup_Sup_nat
@ ( image_nat_nat_nat
@ ^ [Y6: nat > nat] : ( F @ Y6 @ X2 )
@ A2 ) ) ) ).
% SUP_apply
thf(fact_826_INF__apply,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,X2: nat] :
( ( comple6608973012141742712at_nat @ ( image_3205354838064109189at_nat @ F @ A2 ) @ X2 )
= ( complete_Inf_Inf_nat
@ ( image_nat_nat_nat
@ ^ [Y6: nat > nat] : ( F @ Y6 @ X2 )
@ A2 ) ) ) ).
% INF_apply
thf(fact_827_INT__extend__simps_I9_J,axiom,
! [C: nat > set_nat,B2: nat > set_nat,A2: set_nat] :
( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [X3: nat] : ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ C @ ( B2 @ X3 ) ) )
@ A2 ) )
= ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ C @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) ) ) ) ).
% INT_extend_simps(9)
thf(fact_828_INT__extend__simps_I8_J,axiom,
! [B2: nat > set_nat,A2: set_set_nat] :
( ( comple7806235888213564991et_nat
@ ( image_7916887816326733075et_nat
@ ^ [Y6: set_nat] : ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B2 @ Y6 ) )
@ A2 ) )
= ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B2 @ ( comple7399068483239264473et_nat @ A2 ) ) ) ) ).
% INT_extend_simps(8)
thf(fact_829_SUP__UN__eq,axiom,
! [R2: nat > set_nat,S: set_nat] :
( ( comple8317665133742190828_nat_o
@ ( image_nat_nat_o2
@ ^ [I: nat,X3: nat] : ( member_nat @ X3 @ ( R2 @ I ) )
@ S ) )
= ( ^ [X3: nat] : ( member_nat @ X3 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ R2 @ S ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_830_INF__INT__eq,axiom,
! [R2: nat > set_nat,S: set_nat] :
( ( comple6214475593288795910_nat_o
@ ( image_nat_nat_o2
@ ^ [I: nat,X3: nat] : ( member_nat @ X3 @ ( R2 @ I ) )
@ S ) )
= ( ^ [X3: nat] : ( member_nat @ X3 @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ R2 @ S ) ) ) ) ) ).
% INF_INT_eq
thf(fact_831_Inter__subset,axiom,
! [A2: set_set_nat_nat,B2: set_nat_nat] :
( ! [X6: set_nat_nat] :
( ( member_set_nat_nat2 @ X6 @ A2 )
=> ( ord_le9059583361652607317at_nat @ X6 @ B2 ) )
=> ( ( A2 != bot_bo7376149671870096959at_nat )
=> ( ord_le9059583361652607317at_nat @ ( comple439066603627490862at_nat @ A2 ) @ B2 ) ) ) ).
% Inter_subset
thf(fact_832_Inter__subset,axiom,
! [A2: set_set_set_nat,B2: set_set_nat] :
( ! [X6: set_set_nat] :
( ( member_set_set_nat @ X6 @ A2 )
=> ( ord_le6893508408891458716et_nat @ X6 @ B2 ) )
=> ( ( A2 != bot_bo7198184520161983622et_nat )
=> ( ord_le6893508408891458716et_nat @ ( comple1065008630642458357et_nat @ A2 ) @ B2 ) ) ) ).
% Inter_subset
thf(fact_833_Inter__subset,axiom,
! [A2: set_set_nat_nat_nat,B2: set_nat_nat_nat] :
( ! [X6: set_nat_nat_nat] :
( ( member8194441297229544571at_nat @ X6 @ A2 )
=> ( ord_le3211623285424100676at_nat @ X6 @ B2 ) )
=> ( ( A2 != bot_bo6668270333135560750at_nat )
=> ( ord_le3211623285424100676at_nat @ ( comple884914421528019101at_nat @ A2 ) @ B2 ) ) ) ).
% Inter_subset
thf(fact_834_Inter__subset,axiom,
! [A2: set_set_o_nat_nat,B2: set_o_nat_nat] :
( ! [X6: set_o_nat_nat] :
( ( member_set_o_nat_nat @ X6 @ A2 )
=> ( ord_le8808915593745164104at_nat @ X6 @ B2 ) )
=> ( ( A2 != bot_bo6195285094354290676at_nat )
=> ( ord_le8808915593745164104at_nat @ ( comple6464488905778062255at_nat @ A2 ) @ B2 ) ) ) ).
% Inter_subset
thf(fact_835_Inter__subset,axiom,
! [A2: set_set_o,B2: set_o] :
( ! [X6: set_o] :
( ( member_set_o @ X6 @ A2 )
=> ( ord_less_eq_set_o @ X6 @ B2 ) )
=> ( ( A2 != bot_bot_set_set_o )
=> ( ord_less_eq_set_o @ ( comple3063163877087187839_set_o @ A2 ) @ B2 ) ) ) ).
% Inter_subset
thf(fact_836_Inter__subset,axiom,
! [A2: set_set_nat,B2: set_nat] :
( ! [X6: set_nat] :
( ( member_set_nat @ X6 @ A2 )
=> ( ord_less_eq_set_nat @ X6 @ B2 ) )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A2 ) @ B2 ) ) ) ).
% Inter_subset
thf(fact_837_Pi__cong,axiom,
! [A2: set_nat,F: nat > nat,G: nat > nat,B2: nat > set_nat] :
( ! [W: nat] :
( ( member_nat @ W @ A2 )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( member_nat_nat @ F @ ( pi_nat_nat @ A2 @ B2 ) )
= ( member_nat_nat @ G @ ( pi_nat_nat @ A2 @ B2 ) ) ) ) ).
% Pi_cong
thf(fact_838_Pi__cong,axiom,
! [A2: set_nat,F: nat > set_nat,G: nat > set_nat,B2: nat > set_set_nat] :
( ! [W: nat] :
( ( member_nat @ W @ A2 )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( member_nat_set_nat @ F @ ( pi_nat_set_nat @ A2 @ B2 ) )
= ( member_nat_set_nat @ G @ ( pi_nat_set_nat @ A2 @ B2 ) ) ) ) ).
% Pi_cong
thf(fact_839_Pi__cong,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > nat > nat,G: ( nat > nat ) > nat > nat,B2: ( nat > nat ) > set_nat_nat] :
( ! [W: nat > nat] :
( ( member_nat_nat @ W @ A2 )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( member952132173341509300at_nat @ F @ ( pi_nat_nat_nat_nat @ A2 @ B2 ) )
= ( member952132173341509300at_nat @ G @ ( pi_nat_nat_nat_nat @ A2 @ B2 ) ) ) ) ).
% Pi_cong
thf(fact_840_Pi__mem,axiom,
! [F: $o > $o,A2: set_o,B2: $o > set_o,X2: $o] :
( ( member_o_o @ F @ ( pi_o_o @ A2 @ B2 ) )
=> ( ( member_o @ X2 @ A2 )
=> ( member_o @ ( F @ X2 ) @ ( B2 @ X2 ) ) ) ) ).
% Pi_mem
thf(fact_841_Pi__mem,axiom,
! [F: $o > nat,A2: set_o,B2: $o > set_nat,X2: $o] :
( ( member_o_nat @ F @ ( pi_o_nat @ A2 @ B2 ) )
=> ( ( member_o @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ ( B2 @ X2 ) ) ) ) ).
% Pi_mem
thf(fact_842_Pi__mem,axiom,
! [F: nat > $o,A2: set_nat,B2: nat > set_o,X2: nat] :
( ( member_nat_o @ F @ ( pi_nat_o @ A2 @ B2 ) )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_o @ ( F @ X2 ) @ ( B2 @ X2 ) ) ) ) ).
% Pi_mem
thf(fact_843_Pi__mem,axiom,
! [F: nat > nat,A2: set_nat,B2: nat > set_nat,X2: nat] :
( ( member_nat_nat @ F @ ( pi_nat_nat @ A2 @ B2 ) )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ ( B2 @ X2 ) ) ) ) ).
% Pi_mem
thf(fact_844_Pi__mem,axiom,
! [F: $o > set_nat,A2: set_o,B2: $o > set_set_nat,X2: $o] :
( ( member_o_set_nat @ F @ ( pi_o_set_nat @ A2 @ B2 ) )
=> ( ( member_o @ X2 @ A2 )
=> ( member_set_nat @ ( F @ X2 ) @ ( B2 @ X2 ) ) ) ) ).
% Pi_mem
thf(fact_845_Pi__mem,axiom,
! [F: set_nat > $o,A2: set_set_nat,B2: set_nat > set_o,X2: set_nat] :
( ( member_set_nat_o @ F @ ( pi_set_nat_o @ A2 @ B2 ) )
=> ( ( member_set_nat @ X2 @ A2 )
=> ( member_o @ ( F @ X2 ) @ ( B2 @ X2 ) ) ) ) ).
% Pi_mem
thf(fact_846_Pi__mem,axiom,
! [F: set_nat > nat,A2: set_set_nat,B2: set_nat > set_nat,X2: set_nat] :
( ( member_set_nat_nat @ F @ ( pi_set_nat_nat @ A2 @ B2 ) )
=> ( ( member_set_nat @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ ( B2 @ X2 ) ) ) ) ).
% Pi_mem
thf(fact_847_Pi__mem,axiom,
! [F: nat > set_nat,A2: set_nat,B2: nat > set_set_nat,X2: nat] :
( ( member_nat_set_nat @ F @ ( pi_nat_set_nat @ A2 @ B2 ) )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_set_nat @ ( F @ X2 ) @ ( B2 @ X2 ) ) ) ) ).
% Pi_mem
thf(fact_848_Pi__mem,axiom,
! [F: $o > nat > nat,A2: set_o,B2: $o > set_nat_nat,X2: $o] :
( ( member_o_nat_nat @ F @ ( pi_o_nat_nat @ A2 @ B2 ) )
=> ( ( member_o @ X2 @ A2 )
=> ( member_nat_nat @ ( F @ X2 ) @ ( B2 @ X2 ) ) ) ) ).
% Pi_mem
thf(fact_849_Pi__mem,axiom,
! [F: nat > nat > nat,A2: set_nat,B2: nat > set_nat_nat,X2: nat] :
( ( member_nat_nat_nat2 @ F @ ( pi_nat_nat_nat2 @ A2 @ B2 ) )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_nat_nat @ ( F @ X2 ) @ ( B2 @ X2 ) ) ) ) ).
% Pi_mem
thf(fact_850_Pi__iff,axiom,
! [F: ( nat > nat ) > nat > nat,I2: set_nat_nat,X5: ( nat > nat ) > set_nat_nat] :
( ( member952132173341509300at_nat @ F @ ( pi_nat_nat_nat_nat @ I2 @ X5 ) )
= ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ I2 )
=> ( member_nat_nat @ ( F @ X3 ) @ ( X5 @ X3 ) ) ) ) ) ).
% Pi_iff
thf(fact_851_Pi__iff,axiom,
! [F: nat > nat,I2: set_nat,X5: nat > set_nat] :
( ( member_nat_nat @ F @ ( pi_nat_nat @ I2 @ X5 ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ I2 )
=> ( member_nat @ ( F @ X3 ) @ ( X5 @ X3 ) ) ) ) ) ).
% Pi_iff
thf(fact_852_Pi__iff,axiom,
! [F: nat > set_nat,I2: set_nat,X5: nat > set_set_nat] :
( ( member_nat_set_nat @ F @ ( pi_nat_set_nat @ I2 @ X5 ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ I2 )
=> ( member_set_nat @ ( F @ X3 ) @ ( X5 @ X3 ) ) ) ) ) ).
% Pi_iff
thf(fact_853_Pi__I_H,axiom,
! [A2: set_o,F: $o > $o,B2: $o > set_o] :
( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( member_o @ ( F @ X ) @ ( B2 @ X ) ) )
=> ( member_o_o @ F @ ( pi_o_o @ A2 @ B2 ) ) ) ).
% Pi_I'
thf(fact_854_Pi__I_H,axiom,
! [A2: set_o,F: $o > nat,B2: $o > set_nat] :
( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( B2 @ X ) ) )
=> ( member_o_nat @ F @ ( pi_o_nat @ A2 @ B2 ) ) ) ).
% Pi_I'
thf(fact_855_Pi__I_H,axiom,
! [A2: set_nat,F: nat > $o,B2: nat > set_o] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_o @ ( F @ X ) @ ( B2 @ X ) ) )
=> ( member_nat_o @ F @ ( pi_nat_o @ A2 @ B2 ) ) ) ).
% Pi_I'
thf(fact_856_Pi__I_H,axiom,
! [A2: set_nat,F: nat > nat,B2: nat > set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( B2 @ X ) ) )
=> ( member_nat_nat @ F @ ( pi_nat_nat @ A2 @ B2 ) ) ) ).
% Pi_I'
thf(fact_857_Pi__I_H,axiom,
! [A2: set_o,F: $o > set_nat,B2: $o > set_set_nat] :
( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( member_set_nat @ ( F @ X ) @ ( B2 @ X ) ) )
=> ( member_o_set_nat @ F @ ( pi_o_set_nat @ A2 @ B2 ) ) ) ).
% Pi_I'
thf(fact_858_Pi__I_H,axiom,
! [A2: set_nat,F: nat > set_nat,B2: nat > set_set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_set_nat @ ( F @ X ) @ ( B2 @ X ) ) )
=> ( member_nat_set_nat @ F @ ( pi_nat_set_nat @ A2 @ B2 ) ) ) ).
% Pi_I'
thf(fact_859_Pi__I_H,axiom,
! [A2: set_set_nat,F: set_nat > $o,B2: set_nat > set_o] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( member_o @ ( F @ X ) @ ( B2 @ X ) ) )
=> ( member_set_nat_o @ F @ ( pi_set_nat_o @ A2 @ B2 ) ) ) ).
% Pi_I'
thf(fact_860_Pi__I_H,axiom,
! [A2: set_set_nat,F: set_nat > nat,B2: set_nat > set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( B2 @ X ) ) )
=> ( member_set_nat_nat @ F @ ( pi_set_nat_nat @ A2 @ B2 ) ) ) ).
% Pi_I'
thf(fact_861_Pi__I_H,axiom,
! [A2: set_o,F: $o > nat > nat,B2: $o > set_nat_nat] :
( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( member_nat_nat @ ( F @ X ) @ ( B2 @ X ) ) )
=> ( member_o_nat_nat @ F @ ( pi_o_nat_nat @ A2 @ B2 ) ) ) ).
% Pi_I'
thf(fact_862_Pi__I_H,axiom,
! [A2: set_nat,F: nat > nat > nat,B2: nat > set_nat_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat_nat @ ( F @ X ) @ ( B2 @ X ) ) )
=> ( member_nat_nat_nat2 @ F @ ( pi_nat_nat_nat2 @ A2 @ B2 ) ) ) ).
% Pi_I'
thf(fact_863_PiE,axiom,
! [F: $o > $o,A2: set_o,B2: $o > set_o,X2: $o] :
( ( member_o_o @ F @ ( pi_o_o @ A2 @ B2 ) )
=> ( ~ ( member_o @ ( F @ X2 ) @ ( B2 @ X2 ) )
=> ~ ( member_o @ X2 @ A2 ) ) ) ).
% PiE
thf(fact_864_PiE,axiom,
! [F: nat > $o,A2: set_nat,B2: nat > set_o,X2: nat] :
( ( member_nat_o @ F @ ( pi_nat_o @ A2 @ B2 ) )
=> ( ~ ( member_o @ ( F @ X2 ) @ ( B2 @ X2 ) )
=> ~ ( member_nat @ X2 @ A2 ) ) ) ).
% PiE
thf(fact_865_PiE,axiom,
! [F: $o > nat,A2: set_o,B2: $o > set_nat,X2: $o] :
( ( member_o_nat @ F @ ( pi_o_nat @ A2 @ B2 ) )
=> ( ~ ( member_nat @ ( F @ X2 ) @ ( B2 @ X2 ) )
=> ~ ( member_o @ X2 @ A2 ) ) ) ).
% PiE
thf(fact_866_PiE,axiom,
! [F: nat > nat,A2: set_nat,B2: nat > set_nat,X2: nat] :
( ( member_nat_nat @ F @ ( pi_nat_nat @ A2 @ B2 ) )
=> ( ~ ( member_nat @ ( F @ X2 ) @ ( B2 @ X2 ) )
=> ~ ( member_nat @ X2 @ A2 ) ) ) ).
% PiE
thf(fact_867_PiE,axiom,
! [F: set_nat > $o,A2: set_set_nat,B2: set_nat > set_o,X2: set_nat] :
( ( member_set_nat_o @ F @ ( pi_set_nat_o @ A2 @ B2 ) )
=> ( ~ ( member_o @ ( F @ X2 ) @ ( B2 @ X2 ) )
=> ~ ( member_set_nat @ X2 @ A2 ) ) ) ).
% PiE
thf(fact_868_PiE,axiom,
! [F: set_nat > nat,A2: set_set_nat,B2: set_nat > set_nat,X2: set_nat] :
( ( member_set_nat_nat @ F @ ( pi_set_nat_nat @ A2 @ B2 ) )
=> ( ~ ( member_nat @ ( F @ X2 ) @ ( B2 @ X2 ) )
=> ~ ( member_set_nat @ X2 @ A2 ) ) ) ).
% PiE
thf(fact_869_PiE,axiom,
! [F: $o > set_nat,A2: set_o,B2: $o > set_set_nat,X2: $o] :
( ( member_o_set_nat @ F @ ( pi_o_set_nat @ A2 @ B2 ) )
=> ( ~ ( member_set_nat @ ( F @ X2 ) @ ( B2 @ X2 ) )
=> ~ ( member_o @ X2 @ A2 ) ) ) ).
% PiE
thf(fact_870_PiE,axiom,
! [F: nat > set_nat,A2: set_nat,B2: nat > set_set_nat,X2: nat] :
( ( member_nat_set_nat @ F @ ( pi_nat_set_nat @ A2 @ B2 ) )
=> ( ~ ( member_set_nat @ ( F @ X2 ) @ ( B2 @ X2 ) )
=> ~ ( member_nat @ X2 @ A2 ) ) ) ).
% PiE
thf(fact_871_PiE,axiom,
! [F: ( nat > nat ) > $o,A2: set_nat_nat,B2: ( nat > nat ) > set_o,X2: nat > nat] :
( ( member_nat_nat_o @ F @ ( pi_nat_nat_o @ A2 @ B2 ) )
=> ( ~ ( member_o @ ( F @ X2 ) @ ( B2 @ X2 ) )
=> ~ ( member_nat_nat @ X2 @ A2 ) ) ) ).
% PiE
thf(fact_872_PiE,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat,B2: ( nat > nat ) > set_nat,X2: nat > nat] :
( ( member_nat_nat_nat @ F @ ( pi_nat_nat_nat @ A2 @ B2 ) )
=> ( ~ ( member_nat @ ( F @ X2 ) @ ( B2 @ X2 ) )
=> ~ ( member_nat_nat @ X2 @ A2 ) ) ) ).
% PiE
thf(fact_873_funcsetI,axiom,
! [A2: set_o,F: $o > $o,B2: set_o] :
( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( member_o @ ( F @ X ) @ B2 ) )
=> ( member_o_o @ F
@ ( pi_o_o @ A2
@ ^ [Uu: $o] : B2 ) ) ) ).
% funcsetI
thf(fact_874_funcsetI,axiom,
! [A2: set_o,F: $o > nat,B2: set_nat] :
( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ B2 ) )
=> ( member_o_nat @ F
@ ( pi_o_nat @ A2
@ ^ [Uu: $o] : B2 ) ) ) ).
% funcsetI
thf(fact_875_funcsetI,axiom,
! [A2: set_nat,F: nat > $o,B2: set_o] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_o @ ( F @ X ) @ B2 ) )
=> ( member_nat_o @ F
@ ( pi_nat_o @ A2
@ ^ [Uu: nat] : B2 ) ) ) ).
% funcsetI
thf(fact_876_funcsetI,axiom,
! [A2: set_nat,F: nat > nat,B2: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ B2 ) )
=> ( member_nat_nat @ F
@ ( pi_nat_nat @ A2
@ ^ [Uu: nat] : B2 ) ) ) ).
% funcsetI
thf(fact_877_funcsetI,axiom,
! [A2: set_o,F: $o > set_nat,B2: set_set_nat] :
( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( member_set_nat @ ( F @ X ) @ B2 ) )
=> ( member_o_set_nat @ F
@ ( pi_o_set_nat @ A2
@ ^ [Uu: $o] : B2 ) ) ) ).
% funcsetI
thf(fact_878_funcsetI,axiom,
! [A2: set_nat,F: nat > set_nat,B2: set_set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_set_nat @ ( F @ X ) @ B2 ) )
=> ( member_nat_set_nat @ F
@ ( pi_nat_set_nat @ A2
@ ^ [Uu: nat] : B2 ) ) ) ).
% funcsetI
thf(fact_879_funcsetI,axiom,
! [A2: set_set_nat,F: set_nat > $o,B2: set_o] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( member_o @ ( F @ X ) @ B2 ) )
=> ( member_set_nat_o @ F
@ ( pi_set_nat_o @ A2
@ ^ [Uu: set_nat] : B2 ) ) ) ).
% funcsetI
thf(fact_880_funcsetI,axiom,
! [A2: set_set_nat,F: set_nat > nat,B2: set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ B2 ) )
=> ( member_set_nat_nat @ F
@ ( pi_set_nat_nat @ A2
@ ^ [Uu: set_nat] : B2 ) ) ) ).
% funcsetI
thf(fact_881_funcsetI,axiom,
! [A2: set_o,F: $o > nat > nat,B2: set_nat_nat] :
( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( member_nat_nat @ ( F @ X ) @ B2 ) )
=> ( member_o_nat_nat @ F
@ ( pi_o_nat_nat @ A2
@ ^ [Uu: $o] : B2 ) ) ) ).
% funcsetI
thf(fact_882_funcsetI,axiom,
! [A2: set_nat,F: nat > nat > nat,B2: set_nat_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat_nat @ ( F @ X ) @ B2 ) )
=> ( member_nat_nat_nat2 @ F
@ ( pi_nat_nat_nat2 @ A2
@ ^ [Uu: nat] : B2 ) ) ) ).
% funcsetI
thf(fact_883_funcset__id,axiom,
! [A2: set_nat_nat] :
( member952132173341509300at_nat
@ ^ [X3: nat > nat] : X3
@ ( pi_nat_nat_nat_nat @ A2
@ ^ [Uu: nat > nat] : A2 ) ) ).
% funcset_id
thf(fact_884_funcset__id,axiom,
! [A2: set_nat] :
( member_nat_nat
@ ^ [X3: nat] : X3
@ ( pi_nat_nat @ A2
@ ^ [Uu: nat] : A2 ) ) ).
% funcset_id
thf(fact_885_funcset__mem,axiom,
! [F: $o > $o,A2: set_o,B2: set_o,X2: $o] :
( ( member_o_o @ F
@ ( pi_o_o @ A2
@ ^ [Uu: $o] : B2 ) )
=> ( ( member_o @ X2 @ A2 )
=> ( member_o @ ( F @ X2 ) @ B2 ) ) ) ).
% funcset_mem
thf(fact_886_funcset__mem,axiom,
! [F: $o > nat,A2: set_o,B2: set_nat,X2: $o] :
( ( member_o_nat @ F
@ ( pi_o_nat @ A2
@ ^ [Uu: $o] : B2 ) )
=> ( ( member_o @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ B2 ) ) ) ).
% funcset_mem
thf(fact_887_funcset__mem,axiom,
! [F: nat > $o,A2: set_nat,B2: set_o,X2: nat] :
( ( member_nat_o @ F
@ ( pi_nat_o @ A2
@ ^ [Uu: nat] : B2 ) )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_o @ ( F @ X2 ) @ B2 ) ) ) ).
% funcset_mem
thf(fact_888_funcset__mem,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat,X2: nat] :
( ( member_nat_nat @ F
@ ( pi_nat_nat @ A2
@ ^ [Uu: nat] : B2 ) )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ B2 ) ) ) ).
% funcset_mem
thf(fact_889_funcset__mem,axiom,
! [F: $o > set_nat,A2: set_o,B2: set_set_nat,X2: $o] :
( ( member_o_set_nat @ F
@ ( pi_o_set_nat @ A2
@ ^ [Uu: $o] : B2 ) )
=> ( ( member_o @ X2 @ A2 )
=> ( member_set_nat @ ( F @ X2 ) @ B2 ) ) ) ).
% funcset_mem
thf(fact_890_funcset__mem,axiom,
! [F: set_nat > $o,A2: set_set_nat,B2: set_o,X2: set_nat] :
( ( member_set_nat_o @ F
@ ( pi_set_nat_o @ A2
@ ^ [Uu: set_nat] : B2 ) )
=> ( ( member_set_nat @ X2 @ A2 )
=> ( member_o @ ( F @ X2 ) @ B2 ) ) ) ).
% funcset_mem
thf(fact_891_funcset__mem,axiom,
! [F: set_nat > nat,A2: set_set_nat,B2: set_nat,X2: set_nat] :
( ( member_set_nat_nat @ F
@ ( pi_set_nat_nat @ A2
@ ^ [Uu: set_nat] : B2 ) )
=> ( ( member_set_nat @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ B2 ) ) ) ).
% funcset_mem
thf(fact_892_funcset__mem,axiom,
! [F: nat > set_nat,A2: set_nat,B2: set_set_nat,X2: nat] :
( ( member_nat_set_nat @ F
@ ( pi_nat_set_nat @ A2
@ ^ [Uu: nat] : B2 ) )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_set_nat @ ( F @ X2 ) @ B2 ) ) ) ).
% funcset_mem
thf(fact_893_funcset__mem,axiom,
! [F: $o > nat > nat,A2: set_o,B2: set_nat_nat,X2: $o] :
( ( member_o_nat_nat @ F
@ ( pi_o_nat_nat @ A2
@ ^ [Uu: $o] : B2 ) )
=> ( ( member_o @ X2 @ A2 )
=> ( member_nat_nat @ ( F @ X2 ) @ B2 ) ) ) ).
% funcset_mem
thf(fact_894_funcset__mem,axiom,
! [F: nat > nat > nat,A2: set_nat,B2: set_nat_nat,X2: nat] :
( ( member_nat_nat_nat2 @ F
@ ( pi_nat_nat_nat2 @ A2
@ ^ [Uu: nat] : B2 ) )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_nat_nat @ ( F @ X2 ) @ B2 ) ) ) ).
% funcset_mem
thf(fact_895_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_896_INT__E,axiom,
! [B: $o,B2: nat > set_o,A2: set_nat,A: nat] :
( ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_nat_set_o @ B2 @ A2 ) ) )
=> ( ~ ( member_o @ B @ ( B2 @ A ) )
=> ~ ( member_nat @ A @ A2 ) ) ) ).
% INT_E
thf(fact_897_INT__E,axiom,
! [B: nat,B2: $o > set_nat,A2: set_o,A: $o] :
( ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ B2 @ A2 ) ) )
=> ( ~ ( member_nat @ B @ ( B2 @ A ) )
=> ~ ( member_o @ A @ A2 ) ) ) ).
% INT_E
thf(fact_898_INT__E,axiom,
! [B: nat,B2: nat > set_nat,A2: set_nat,A: nat] :
( ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
=> ( ~ ( member_nat @ B @ ( B2 @ A ) )
=> ~ ( member_nat @ A @ A2 ) ) ) ).
% INT_E
thf(fact_899_INT__E,axiom,
! [B: $o,B2: set_nat > set_o,A2: set_set_nat,A: set_nat] :
( ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_set_nat_set_o @ B2 @ A2 ) ) )
=> ( ~ ( member_o @ B @ ( B2 @ A ) )
=> ~ ( member_set_nat @ A @ A2 ) ) ) ).
% INT_E
thf(fact_900_INT__E,axiom,
! [B: set_nat,B2: $o > set_set_nat,A2: set_o,A: $o] :
( ( member_set_nat @ B @ ( comple1065008630642458357et_nat @ ( image_o_set_set_nat @ B2 @ A2 ) ) )
=> ( ~ ( member_set_nat @ B @ ( B2 @ A ) )
=> ~ ( member_o @ A @ A2 ) ) ) ).
% INT_E
thf(fact_901_INT__E,axiom,
! [B: set_nat,B2: nat > set_set_nat,A2: set_nat,A: nat] :
( ( member_set_nat @ B @ ( comple1065008630642458357et_nat @ ( image_2194112158459175443et_nat @ B2 @ A2 ) ) )
=> ( ~ ( member_set_nat @ B @ ( B2 @ A ) )
=> ~ ( member_nat @ A @ A2 ) ) ) ).
% INT_E
thf(fact_902_INT__E,axiom,
! [B: nat,B2: set_nat > set_nat,A2: set_set_nat,A: set_nat] :
( ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_7916887816326733075et_nat @ B2 @ A2 ) ) )
=> ( ~ ( member_nat @ B @ ( B2 @ A ) )
=> ~ ( member_set_nat @ A @ A2 ) ) ) ).
% INT_E
thf(fact_903_INT__E,axiom,
! [B: $o,B2: ( nat > nat ) > set_o,A2: set_nat_nat,A: nat > nat] :
( ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_nat_nat_set_o @ B2 @ A2 ) ) )
=> ( ~ ( member_o @ B @ ( B2 @ A ) )
=> ~ ( member_nat_nat @ A @ A2 ) ) ) ).
% INT_E
thf(fact_904_INT__E,axiom,
! [B: nat > nat,B2: $o > set_nat_nat,A2: set_o,A: $o] :
( ( member_nat_nat @ B @ ( comple439066603627490862at_nat @ ( image_o_set_nat_nat @ B2 @ A2 ) ) )
=> ( ~ ( member_nat_nat @ B @ ( B2 @ A ) )
=> ~ ( member_o @ A @ A2 ) ) ) ).
% INT_E
thf(fact_905_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_906_INT__D,axiom,
! [B: $o,B2: nat > set_o,A2: set_nat,A: nat] :
( ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_nat_set_o @ B2 @ A2 ) ) )
=> ( ( member_nat @ A @ A2 )
=> ( member_o @ B @ ( B2 @ A ) ) ) ) ).
% INT_D
thf(fact_907_INT__D,axiom,
! [B: nat,B2: $o > set_nat,A2: set_o,A: $o] :
( ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ B2 @ A2 ) ) )
=> ( ( member_o @ A @ A2 )
=> ( member_nat @ B @ ( B2 @ A ) ) ) ) ).
% INT_D
thf(fact_908_INT__D,axiom,
! [B: nat,B2: nat > set_nat,A2: set_nat,A: nat] :
( ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
=> ( ( member_nat @ A @ A2 )
=> ( member_nat @ B @ ( B2 @ A ) ) ) ) ).
% INT_D
thf(fact_909_INT__D,axiom,
! [B: $o,B2: set_nat > set_o,A2: set_set_nat,A: set_nat] :
( ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_set_nat_set_o @ B2 @ A2 ) ) )
=> ( ( member_set_nat @ A @ A2 )
=> ( member_o @ B @ ( B2 @ A ) ) ) ) ).
% INT_D
thf(fact_910_INT__D,axiom,
! [B: set_nat,B2: $o > set_set_nat,A2: set_o,A: $o] :
( ( member_set_nat @ B @ ( comple1065008630642458357et_nat @ ( image_o_set_set_nat @ B2 @ A2 ) ) )
=> ( ( member_o @ A @ A2 )
=> ( member_set_nat @ B @ ( B2 @ A ) ) ) ) ).
% INT_D
thf(fact_911_INT__D,axiom,
! [B: set_nat,B2: nat > set_set_nat,A2: set_nat,A: nat] :
( ( member_set_nat @ B @ ( comple1065008630642458357et_nat @ ( image_2194112158459175443et_nat @ B2 @ A2 ) ) )
=> ( ( member_nat @ A @ A2 )
=> ( member_set_nat @ B @ ( B2 @ A ) ) ) ) ).
% INT_D
thf(fact_912_INT__D,axiom,
! [B: nat,B2: set_nat > set_nat,A2: set_set_nat,A: set_nat] :
( ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_7916887816326733075et_nat @ B2 @ A2 ) ) )
=> ( ( member_set_nat @ A @ A2 )
=> ( member_nat @ B @ ( B2 @ A ) ) ) ) ).
% INT_D
thf(fact_913_INT__D,axiom,
! [B: $o,B2: ( nat > nat ) > set_o,A2: set_nat_nat,A: nat > nat] :
( ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_nat_nat_set_o @ B2 @ A2 ) ) )
=> ( ( member_nat_nat @ A @ A2 )
=> ( member_o @ B @ ( B2 @ A ) ) ) ) ).
% INT_D
thf(fact_914_INT__D,axiom,
! [B: nat > nat,B2: $o > set_nat_nat,A2: set_o,A: $o] :
( ( member_nat_nat @ B @ ( comple439066603627490862at_nat @ ( image_o_set_nat_nat @ B2 @ A2 ) ) )
=> ( ( member_o @ A @ A2 )
=> ( member_nat_nat @ B @ ( B2 @ A ) ) ) ) ).
% INT_D
thf(fact_915_UN__UN__flatten,axiom,
! [C: nat > set_nat,B2: nat > set_nat,A2: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y6: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C @ ( B2 @ Y6 ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_916_UN__E,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_E
thf(fact_917_UN__E,axiom,
! [B: $o,B2: nat > set_o,A2: set_nat] :
( ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B2 @ A2 ) ) )
=> ~ ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ~ ( member_o @ B @ ( B2 @ X ) ) ) ) ).
% UN_E
thf(fact_918_UN__E,axiom,
! [B: nat,B2: $o > set_nat,A2: set_o] :
( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ A2 ) ) )
=> ~ ! [X: $o] :
( ( member_o @ X @ A2 )
=> ~ ( member_nat @ B @ ( B2 @ X ) ) ) ) ).
% UN_E
thf(fact_919_UN__E,axiom,
! [B: nat,B2: nat > set_nat,A2: set_nat] :
( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
=> ~ ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ~ ( member_nat @ B @ ( B2 @ X ) ) ) ) ).
% UN_E
thf(fact_920_UN__E,axiom,
! [B: $o,B2: set_nat > set_o,A2: set_set_nat] :
( ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_set_nat_set_o @ B2 @ A2 ) ) )
=> ~ ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ~ ( member_o @ B @ ( B2 @ X ) ) ) ) ).
% UN_E
thf(fact_921_UN__E,axiom,
! [B: set_nat,B2: $o > set_set_nat,A2: set_o] :
( ( member_set_nat @ B @ ( comple548664676211718543et_nat @ ( image_o_set_set_nat @ B2 @ A2 ) ) )
=> ~ ! [X: $o] :
( ( member_o @ X @ A2 )
=> ~ ( member_set_nat @ B @ ( B2 @ X ) ) ) ) ).
% UN_E
thf(fact_922_UN__E,axiom,
! [B: set_nat,B2: nat > set_set_nat,A2: set_nat] :
( ( member_set_nat @ B @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ B2 @ A2 ) ) )
=> ~ ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ~ ( member_set_nat @ B @ ( B2 @ X ) ) ) ) ).
% UN_E
thf(fact_923_UN__E,axiom,
! [B: nat,B2: set_nat > set_nat,A2: set_set_nat] :
( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B2 @ A2 ) ) )
=> ~ ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ~ ( member_nat @ B @ ( B2 @ X ) ) ) ) ).
% UN_E
thf(fact_924_UN__E,axiom,
! [B: $o,B2: ( nat > nat ) > set_o,A2: set_nat_nat] :
( ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_nat_nat_set_o @ B2 @ A2 ) ) )
=> ~ ! [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ~ ( member_o @ B @ ( B2 @ X ) ) ) ) ).
% UN_E
thf(fact_925_UN__E,axiom,
! [B: nat > nat,B2: $o > set_nat_nat,A2: set_o] :
( ( member_nat_nat @ B @ ( comple5448282615319421384at_nat @ ( image_o_set_nat_nat @ B2 @ A2 ) ) )
=> ~ ! [X: $o] :
( ( member_o @ X @ A2 )
=> ~ ( member_nat_nat @ B @ ( B2 @ X ) ) ) ) ).
% UN_E
thf(fact_926_UN__extend__simps_I8_J,axiom,
! [B2: nat > set_nat,A2: set_set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [Y6: set_nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ Y6 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ ( comple7399068483239264473et_nat @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_927_UN__extend__simps_I9_J,axiom,
! [C: nat > set_nat,B2: nat > set_nat,A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X3: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C @ ( B2 @ X3 ) ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_928_SUP__UNION,axiom,
! [F: $o > $o,G: nat > set_o,A2: set_nat] :
( ( complete_Sup_Sup_o @ ( image_o_o @ F @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ G @ A2 ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [Y6: nat] : ( complete_Sup_Sup_o @ ( image_o_o @ F @ ( G @ Y6 ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_929_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
@ ^ [Y6: $o] : ( complete_Sup_Sup_o @ ( image_o_o @ F @ ( G @ Y6 ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_930_SUP__UNION,axiom,
! [F: nat > $o,G: $o > set_nat,A2: set_o] :
( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ G @ A2 ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [Y6: $o] : ( complete_Sup_Sup_o @ ( image_nat_o @ F @ ( G @ Y6 ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_931_SUP__UNION,axiom,
! [F: nat > $o,G: nat > set_nat,A2: set_nat] :
( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ A2 ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [Y6: nat] : ( complete_Sup_Sup_o @ ( image_nat_o @ F @ ( G @ Y6 ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_932_SUP__UNION,axiom,
! [F: nat > set_nat,G: nat > set_nat,A2: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ A2 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y6: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ ( G @ Y6 ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_933_Inf__le__Sup,axiom,
! [A2: set_set_nat_nat] :
( ( A2 != bot_bo7376149671870096959at_nat )
=> ( ord_le9059583361652607317at_nat @ ( comple439066603627490862at_nat @ A2 ) @ ( comple5448282615319421384at_nat @ A2 ) ) ) ).
% Inf_le_Sup
thf(fact_934_Inf__le__Sup,axiom,
! [A2: set_set_set_nat] :
( ( A2 != bot_bo7198184520161983622et_nat )
=> ( ord_le6893508408891458716et_nat @ ( comple1065008630642458357et_nat @ A2 ) @ ( comple548664676211718543et_nat @ A2 ) ) ) ).
% Inf_le_Sup
thf(fact_935_Inf__le__Sup,axiom,
! [A2: set_set_nat_nat_nat] :
( ( A2 != bot_bo6668270333135560750at_nat )
=> ( ord_le3211623285424100676at_nat @ ( comple884914421528019101at_nat @ A2 ) @ ( comple8167887107183641911at_nat @ A2 ) ) ) ).
% Inf_le_Sup
thf(fact_936_Inf__le__Sup,axiom,
! [A2: set_set_o_nat_nat] :
( ( A2 != bot_bo6195285094354290676at_nat )
=> ( ord_le8808915593745164104at_nat @ ( comple6464488905778062255at_nat @ A2 ) @ ( comple7172370505855214741at_nat @ A2 ) ) ) ).
% Inf_le_Sup
thf(fact_937_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_938_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_939_Inf__le__Sup,axiom,
! [A2: set_set_nat] :
( ( A2 != bot_bot_set_set_nat )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ A2 ) ) ) ).
% Inf_le_Sup
thf(fact_940_bot__empty__eq,axiom,
( bot_bot_o_o
= ( ^ [X3: $o] : ( member_o @ X3 @ bot_bot_set_o ) ) ) ).
% bot_empty_eq
thf(fact_941_bot__empty__eq,axiom,
( bot_bo1568108970253895006_nat_o
= ( ^ [X3: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X3 @ bot_bo3919185967433191911at_nat ) ) ) ).
% bot_empty_eq
thf(fact_942_bot__empty__eq,axiom,
( bot_bot_nat_nat_o
= ( ^ [X3: nat > nat] : ( member_nat_nat @ X3 @ bot_bot_set_nat_nat ) ) ) ).
% bot_empty_eq
thf(fact_943_bot__empty__eq,axiom,
( bot_bo8210142506433397254_nat_o
= ( ^ [X3: nat > set_nat] : ( member_nat_set_nat @ X3 @ bot_bo4007787791999405887et_nat ) ) ) ).
% bot_empty_eq
thf(fact_944_bot__empty__eq,axiom,
( bot_bot_set_nat_o
= ( ^ [X3: set_nat] : ( member_set_nat @ X3 @ bot_bot_set_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_945_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X3: nat] : ( member_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_946_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_947_Inf__greatest,axiom,
! [A2: set_nat_set_nat,Z2: nat > set_nat] :
( ! [X: nat > set_nat] :
( ( member_nat_set_nat @ X @ A2 )
=> ( ord_le6195038898401538645et_nat @ Z2 @ X ) )
=> ( ord_le6195038898401538645et_nat @ Z2 @ ( comple6797894177231197998et_nat @ A2 ) ) ) ).
% Inf_greatest
thf(fact_948_Inf__greatest,axiom,
! [A2: set_set_nat_nat,Z2: set_nat_nat] :
( ! [X: set_nat_nat] :
( ( member_set_nat_nat2 @ X @ A2 )
=> ( ord_le9059583361652607317at_nat @ Z2 @ X ) )
=> ( ord_le9059583361652607317at_nat @ Z2 @ ( comple439066603627490862at_nat @ A2 ) ) ) ).
% Inf_greatest
thf(fact_949_Inf__greatest,axiom,
! [A2: set_set_set_nat,Z2: set_set_nat] :
( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ A2 )
=> ( ord_le6893508408891458716et_nat @ Z2 @ X ) )
=> ( ord_le6893508408891458716et_nat @ Z2 @ ( comple1065008630642458357et_nat @ A2 ) ) ) ).
% Inf_greatest
thf(fact_950_Inf__greatest,axiom,
! [A2: set_set_nat_nat_nat,Z2: set_nat_nat_nat] :
( ! [X: set_nat_nat_nat] :
( ( member8194441297229544571at_nat @ X @ A2 )
=> ( ord_le3211623285424100676at_nat @ Z2 @ X ) )
=> ( ord_le3211623285424100676at_nat @ Z2 @ ( comple884914421528019101at_nat @ A2 ) ) ) ).
% Inf_greatest
thf(fact_951_Inf__greatest,axiom,
! [A2: set_set_o_nat_nat,Z2: set_o_nat_nat] :
( ! [X: set_o_nat_nat] :
( ( member_set_o_nat_nat @ X @ A2 )
=> ( ord_le8808915593745164104at_nat @ Z2 @ X ) )
=> ( ord_le8808915593745164104at_nat @ Z2 @ ( comple6464488905778062255at_nat @ A2 ) ) ) ).
% Inf_greatest
thf(fact_952_Inf__greatest,axiom,
! [A2: set_set_o,Z2: set_o] :
( ! [X: set_o] :
( ( member_set_o @ X @ A2 )
=> ( ord_less_eq_set_o @ Z2 @ X ) )
=> ( ord_less_eq_set_o @ Z2 @ ( comple3063163877087187839_set_o @ A2 ) ) ) ).
% Inf_greatest
thf(fact_953_Inf__greatest,axiom,
! [A2: set_o,Z2: $o] :
( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ord_less_eq_o @ Z2 @ X ) )
=> ( ord_less_eq_o @ Z2 @ ( complete_Inf_Inf_o @ A2 ) ) ) ).
% Inf_greatest
thf(fact_954_Inf__greatest,axiom,
! [A2: set_set_nat,Z2: set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ Z2 @ X ) )
=> ( ord_less_eq_set_nat @ Z2 @ ( comple7806235888213564991et_nat @ A2 ) ) ) ).
% Inf_greatest
thf(fact_955_le__Inf__iff,axiom,
! [B: set_nat_nat,A2: set_set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B @ ( comple439066603627490862at_nat @ A2 ) )
= ( ! [X3: set_nat_nat] :
( ( member_set_nat_nat2 @ X3 @ A2 )
=> ( ord_le9059583361652607317at_nat @ B @ X3 ) ) ) ) ).
% le_Inf_iff
thf(fact_956_le__Inf__iff,axiom,
! [B: set_set_nat,A2: set_set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ ( comple1065008630642458357et_nat @ A2 ) )
= ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A2 )
=> ( ord_le6893508408891458716et_nat @ B @ X3 ) ) ) ) ).
% le_Inf_iff
thf(fact_957_le__Inf__iff,axiom,
! [B: set_nat_nat_nat,A2: set_set_nat_nat_nat] :
( ( ord_le3211623285424100676at_nat @ B @ ( comple884914421528019101at_nat @ A2 ) )
= ( ! [X3: set_nat_nat_nat] :
( ( member8194441297229544571at_nat @ X3 @ A2 )
=> ( ord_le3211623285424100676at_nat @ B @ X3 ) ) ) ) ).
% le_Inf_iff
thf(fact_958_le__Inf__iff,axiom,
! [B: set_o_nat_nat,A2: set_set_o_nat_nat] :
( ( ord_le8808915593745164104at_nat @ B @ ( comple6464488905778062255at_nat @ A2 ) )
= ( ! [X3: set_o_nat_nat] :
( ( member_set_o_nat_nat @ X3 @ A2 )
=> ( ord_le8808915593745164104at_nat @ B @ X3 ) ) ) ) ).
% le_Inf_iff
thf(fact_959_le__Inf__iff,axiom,
! [B: set_o,A2: set_set_o] :
( ( ord_less_eq_set_o @ B @ ( comple3063163877087187839_set_o @ A2 ) )
= ( ! [X3: set_o] :
( ( member_set_o @ X3 @ A2 )
=> ( ord_less_eq_set_o @ B @ X3 ) ) ) ) ).
% le_Inf_iff
thf(fact_960_le__Inf__iff,axiom,
! [B: $o,A2: set_o] :
( ( ord_less_eq_o @ B @ ( complete_Inf_Inf_o @ A2 ) )
= ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_less_eq_o @ B @ X3 ) ) ) ) ).
% le_Inf_iff
thf(fact_961_le__Inf__iff,axiom,
! [B: set_nat,A2: set_set_nat] :
( ( ord_less_eq_set_nat @ B @ ( comple7806235888213564991et_nat @ A2 ) )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ B @ X3 ) ) ) ) ).
% le_Inf_iff
thf(fact_962_Inf__lower2,axiom,
! [U: nat > set_nat,A2: set_nat_set_nat,V: nat > set_nat] :
( ( member_nat_set_nat @ U @ A2 )
=> ( ( ord_le6195038898401538645et_nat @ U @ V )
=> ( ord_le6195038898401538645et_nat @ ( comple6797894177231197998et_nat @ A2 ) @ V ) ) ) ).
% Inf_lower2
thf(fact_963_Inf__lower2,axiom,
! [U: set_nat_nat,A2: set_set_nat_nat,V: set_nat_nat] :
( ( member_set_nat_nat2 @ U @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ U @ V )
=> ( ord_le9059583361652607317at_nat @ ( comple439066603627490862at_nat @ A2 ) @ V ) ) ) ).
% Inf_lower2
thf(fact_964_Inf__lower2,axiom,
! [U: set_set_nat,A2: set_set_set_nat,V: set_set_nat] :
( ( member_set_set_nat @ U @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ U @ V )
=> ( ord_le6893508408891458716et_nat @ ( comple1065008630642458357et_nat @ A2 ) @ V ) ) ) ).
% Inf_lower2
thf(fact_965_Inf__lower2,axiom,
! [U: set_nat_nat_nat,A2: set_set_nat_nat_nat,V: set_nat_nat_nat] :
( ( member8194441297229544571at_nat @ U @ A2 )
=> ( ( ord_le3211623285424100676at_nat @ U @ V )
=> ( ord_le3211623285424100676at_nat @ ( comple884914421528019101at_nat @ A2 ) @ V ) ) ) ).
% Inf_lower2
thf(fact_966_Inf__lower2,axiom,
! [U: set_o_nat_nat,A2: set_set_o_nat_nat,V: set_o_nat_nat] :
( ( member_set_o_nat_nat @ U @ A2 )
=> ( ( ord_le8808915593745164104at_nat @ U @ V )
=> ( ord_le8808915593745164104at_nat @ ( comple6464488905778062255at_nat @ A2 ) @ V ) ) ) ).
% Inf_lower2
thf(fact_967_Inf__lower2,axiom,
! [U: set_o,A2: set_set_o,V: set_o] :
( ( member_set_o @ U @ A2 )
=> ( ( ord_less_eq_set_o @ U @ V )
=> ( ord_less_eq_set_o @ ( comple3063163877087187839_set_o @ A2 ) @ V ) ) ) ).
% Inf_lower2
thf(fact_968_Inf__lower2,axiom,
! [U: $o,A2: set_o,V: $o] :
( ( member_o @ U @ A2 )
=> ( ( ord_less_eq_o @ U @ V )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A2 ) @ V ) ) ) ).
% Inf_lower2
thf(fact_969_Inf__lower2,axiom,
! [U: set_nat,A2: set_set_nat,V: set_nat] :
( ( member_set_nat @ U @ A2 )
=> ( ( ord_less_eq_set_nat @ U @ V )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A2 ) @ V ) ) ) ).
% Inf_lower2
thf(fact_970_Inf__lower,axiom,
! [X2: nat > set_nat,A2: set_nat_set_nat] :
( ( member_nat_set_nat @ X2 @ A2 )
=> ( ord_le6195038898401538645et_nat @ ( comple6797894177231197998et_nat @ A2 ) @ X2 ) ) ).
% Inf_lower
thf(fact_971_Inf__lower,axiom,
! [X2: set_nat_nat,A2: set_set_nat_nat] :
( ( member_set_nat_nat2 @ X2 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( comple439066603627490862at_nat @ A2 ) @ X2 ) ) ).
% Inf_lower
thf(fact_972_Inf__lower,axiom,
! [X2: set_set_nat,A2: set_set_set_nat] :
( ( member_set_set_nat @ X2 @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( comple1065008630642458357et_nat @ A2 ) @ X2 ) ) ).
% Inf_lower
thf(fact_973_Inf__lower,axiom,
! [X2: set_nat_nat_nat,A2: set_set_nat_nat_nat] :
( ( member8194441297229544571at_nat @ X2 @ A2 )
=> ( ord_le3211623285424100676at_nat @ ( comple884914421528019101at_nat @ A2 ) @ X2 ) ) ).
% Inf_lower
thf(fact_974_Inf__lower,axiom,
! [X2: set_o_nat_nat,A2: set_set_o_nat_nat] :
( ( member_set_o_nat_nat @ X2 @ A2 )
=> ( ord_le8808915593745164104at_nat @ ( comple6464488905778062255at_nat @ A2 ) @ X2 ) ) ).
% Inf_lower
thf(fact_975_Inf__lower,axiom,
! [X2: set_o,A2: set_set_o] :
( ( member_set_o @ X2 @ A2 )
=> ( ord_less_eq_set_o @ ( comple3063163877087187839_set_o @ A2 ) @ X2 ) ) ).
% Inf_lower
thf(fact_976_Inf__lower,axiom,
! [X2: $o,A2: set_o] :
( ( member_o @ X2 @ A2 )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A2 ) @ X2 ) ) ).
% Inf_lower
thf(fact_977_Inf__lower,axiom,
! [X2: set_nat,A2: set_set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A2 ) @ X2 ) ) ).
% Inf_lower
thf(fact_978_Inf__mono,axiom,
! [B2: set_nat_set_nat,A2: set_nat_set_nat] :
( ! [B5: nat > set_nat] :
( ( member_nat_set_nat @ B5 @ B2 )
=> ? [X4: nat > set_nat] :
( ( member_nat_set_nat @ X4 @ A2 )
& ( ord_le6195038898401538645et_nat @ X4 @ B5 ) ) )
=> ( ord_le6195038898401538645et_nat @ ( comple6797894177231197998et_nat @ A2 ) @ ( comple6797894177231197998et_nat @ B2 ) ) ) ).
% Inf_mono
thf(fact_979_Inf__mono,axiom,
! [B2: set_set_nat_nat,A2: set_set_nat_nat] :
( ! [B5: set_nat_nat] :
( ( member_set_nat_nat2 @ B5 @ B2 )
=> ? [X4: set_nat_nat] :
( ( member_set_nat_nat2 @ X4 @ A2 )
& ( ord_le9059583361652607317at_nat @ X4 @ B5 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( comple439066603627490862at_nat @ A2 ) @ ( comple439066603627490862at_nat @ B2 ) ) ) ).
% Inf_mono
thf(fact_980_Inf__mono,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ! [B5: set_set_nat] :
( ( member_set_set_nat @ B5 @ B2 )
=> ? [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ A2 )
& ( ord_le6893508408891458716et_nat @ X4 @ B5 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( comple1065008630642458357et_nat @ A2 ) @ ( comple1065008630642458357et_nat @ B2 ) ) ) ).
% Inf_mono
thf(fact_981_Inf__mono,axiom,
! [B2: set_set_nat_nat_nat,A2: set_set_nat_nat_nat] :
( ! [B5: set_nat_nat_nat] :
( ( member8194441297229544571at_nat @ B5 @ B2 )
=> ? [X4: set_nat_nat_nat] :
( ( member8194441297229544571at_nat @ X4 @ A2 )
& ( ord_le3211623285424100676at_nat @ X4 @ B5 ) ) )
=> ( ord_le3211623285424100676at_nat @ ( comple884914421528019101at_nat @ A2 ) @ ( comple884914421528019101at_nat @ B2 ) ) ) ).
% Inf_mono
thf(fact_982_Inf__mono,axiom,
! [B2: set_set_o_nat_nat,A2: set_set_o_nat_nat] :
( ! [B5: set_o_nat_nat] :
( ( member_set_o_nat_nat @ B5 @ B2 )
=> ? [X4: set_o_nat_nat] :
( ( member_set_o_nat_nat @ X4 @ A2 )
& ( ord_le8808915593745164104at_nat @ X4 @ B5 ) ) )
=> ( ord_le8808915593745164104at_nat @ ( comple6464488905778062255at_nat @ A2 ) @ ( comple6464488905778062255at_nat @ B2 ) ) ) ).
% Inf_mono
thf(fact_983_Inf__mono,axiom,
! [B2: set_set_o,A2: set_set_o] :
( ! [B5: set_o] :
( ( member_set_o @ B5 @ B2 )
=> ? [X4: set_o] :
( ( member_set_o @ X4 @ A2 )
& ( ord_less_eq_set_o @ X4 @ B5 ) ) )
=> ( ord_less_eq_set_o @ ( comple3063163877087187839_set_o @ A2 ) @ ( comple3063163877087187839_set_o @ B2 ) ) ) ).
% Inf_mono
thf(fact_984_Inf__mono,axiom,
! [B2: set_o,A2: set_o] :
( ! [B5: $o] :
( ( member_o @ B5 @ B2 )
=> ? [X4: $o] :
( ( member_o @ X4 @ A2 )
& ( ord_less_eq_o @ X4 @ B5 ) ) )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A2 ) @ ( complete_Inf_Inf_o @ B2 ) ) ) ).
% Inf_mono
thf(fact_985_Inf__mono,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ! [B5: set_nat] :
( ( member_set_nat @ B5 @ B2 )
=> ? [X4: set_nat] :
( ( member_set_nat @ X4 @ A2 )
& ( ord_less_eq_set_nat @ X4 @ B5 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A2 ) @ ( comple7806235888213564991et_nat @ B2 ) ) ) ).
% Inf_mono
thf(fact_986_Inf__eqI,axiom,
! [A2: set_nat_set_nat,X2: nat > set_nat] :
( ! [I3: nat > set_nat] :
( ( member_nat_set_nat @ I3 @ A2 )
=> ( ord_le6195038898401538645et_nat @ X2 @ I3 ) )
=> ( ! [Y4: nat > set_nat] :
( ! [I5: nat > set_nat] :
( ( member_nat_set_nat @ I5 @ A2 )
=> ( ord_le6195038898401538645et_nat @ Y4 @ I5 ) )
=> ( ord_le6195038898401538645et_nat @ Y4 @ X2 ) )
=> ( ( comple6797894177231197998et_nat @ A2 )
= X2 ) ) ) ).
% Inf_eqI
thf(fact_987_Inf__eqI,axiom,
! [A2: set_set_nat_nat,X2: set_nat_nat] :
( ! [I3: set_nat_nat] :
( ( member_set_nat_nat2 @ I3 @ A2 )
=> ( ord_le9059583361652607317at_nat @ X2 @ I3 ) )
=> ( ! [Y4: set_nat_nat] :
( ! [I5: set_nat_nat] :
( ( member_set_nat_nat2 @ I5 @ A2 )
=> ( ord_le9059583361652607317at_nat @ Y4 @ I5 ) )
=> ( ord_le9059583361652607317at_nat @ Y4 @ X2 ) )
=> ( ( comple439066603627490862at_nat @ A2 )
= X2 ) ) ) ).
% Inf_eqI
thf(fact_988_Inf__eqI,axiom,
! [A2: set_set_set_nat,X2: set_set_nat] :
( ! [I3: set_set_nat] :
( ( member_set_set_nat @ I3 @ A2 )
=> ( ord_le6893508408891458716et_nat @ X2 @ I3 ) )
=> ( ! [Y4: set_set_nat] :
( ! [I5: set_set_nat] :
( ( member_set_set_nat @ I5 @ A2 )
=> ( ord_le6893508408891458716et_nat @ Y4 @ I5 ) )
=> ( ord_le6893508408891458716et_nat @ Y4 @ X2 ) )
=> ( ( comple1065008630642458357et_nat @ A2 )
= X2 ) ) ) ).
% Inf_eqI
thf(fact_989_Inf__eqI,axiom,
! [A2: set_set_nat_nat_nat,X2: set_nat_nat_nat] :
( ! [I3: set_nat_nat_nat] :
( ( member8194441297229544571at_nat @ I3 @ A2 )
=> ( ord_le3211623285424100676at_nat @ X2 @ I3 ) )
=> ( ! [Y4: set_nat_nat_nat] :
( ! [I5: set_nat_nat_nat] :
( ( member8194441297229544571at_nat @ I5 @ A2 )
=> ( ord_le3211623285424100676at_nat @ Y4 @ I5 ) )
=> ( ord_le3211623285424100676at_nat @ Y4 @ X2 ) )
=> ( ( comple884914421528019101at_nat @ A2 )
= X2 ) ) ) ).
% Inf_eqI
thf(fact_990_Inf__eqI,axiom,
! [A2: set_set_o_nat_nat,X2: set_o_nat_nat] :
( ! [I3: set_o_nat_nat] :
( ( member_set_o_nat_nat @ I3 @ A2 )
=> ( ord_le8808915593745164104at_nat @ X2 @ I3 ) )
=> ( ! [Y4: set_o_nat_nat] :
( ! [I5: set_o_nat_nat] :
( ( member_set_o_nat_nat @ I5 @ A2 )
=> ( ord_le8808915593745164104at_nat @ Y4 @ I5 ) )
=> ( ord_le8808915593745164104at_nat @ Y4 @ X2 ) )
=> ( ( comple6464488905778062255at_nat @ A2 )
= X2 ) ) ) ).
% Inf_eqI
thf(fact_991_Inf__eqI,axiom,
! [A2: set_set_o,X2: set_o] :
( ! [I3: set_o] :
( ( member_set_o @ I3 @ A2 )
=> ( ord_less_eq_set_o @ X2 @ I3 ) )
=> ( ! [Y4: set_o] :
( ! [I5: set_o] :
( ( member_set_o @ I5 @ A2 )
=> ( ord_less_eq_set_o @ Y4 @ I5 ) )
=> ( ord_less_eq_set_o @ Y4 @ X2 ) )
=> ( ( comple3063163877087187839_set_o @ A2 )
= X2 ) ) ) ).
% Inf_eqI
thf(fact_992_Inf__eqI,axiom,
! [A2: set_o,X2: $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A2 )
=> ( ord_less_eq_o @ X2 @ I3 ) )
=> ( ! [Y4: $o] :
( ! [I5: $o] :
( ( member_o @ I5 @ A2 )
=> ( ord_less_eq_o @ Y4 @ I5 ) )
=> ( ord_less_eq_o @ Y4 @ X2 ) )
=> ( ( complete_Inf_Inf_o @ A2 )
= X2 ) ) ) ).
% Inf_eqI
thf(fact_993_Inf__eqI,axiom,
! [A2: set_set_nat,X2: set_nat] :
( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ A2 )
=> ( ord_less_eq_set_nat @ X2 @ I3 ) )
=> ( ! [Y4: set_nat] :
( ! [I5: set_nat] :
( ( member_set_nat @ I5 @ A2 )
=> ( ord_less_eq_set_nat @ Y4 @ I5 ) )
=> ( ord_less_eq_set_nat @ Y4 @ X2 ) )
=> ( ( comple7806235888213564991et_nat @ A2 )
= X2 ) ) ) ).
% Inf_eqI
thf(fact_994_Sup__upper2,axiom,
! [U: nat > set_nat,A2: set_nat_set_nat,V: nat > set_nat] :
( ( member_nat_set_nat @ U @ A2 )
=> ( ( ord_le6195038898401538645et_nat @ V @ U )
=> ( ord_le6195038898401538645et_nat @ V @ ( comple2583738152068352712et_nat @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_995_Sup__upper2,axiom,
! [U: set_nat_nat,A2: set_set_nat_nat,V: set_nat_nat] :
( ( member_set_nat_nat2 @ U @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ V @ U )
=> ( ord_le9059583361652607317at_nat @ V @ ( comple5448282615319421384at_nat @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_996_Sup__upper2,axiom,
! [U: set_set_nat,A2: set_set_set_nat,V: set_set_nat] :
( ( member_set_set_nat @ U @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ V @ U )
=> ( ord_le6893508408891458716et_nat @ V @ ( comple548664676211718543et_nat @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_997_Sup__upper2,axiom,
! [U: set_nat_nat_nat,A2: set_set_nat_nat_nat,V: set_nat_nat_nat] :
( ( member8194441297229544571at_nat @ U @ A2 )
=> ( ( ord_le3211623285424100676at_nat @ V @ U )
=> ( ord_le3211623285424100676at_nat @ V @ ( comple8167887107183641911at_nat @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_998_Sup__upper2,axiom,
! [U: set_o_nat_nat,A2: set_set_o_nat_nat,V: set_o_nat_nat] :
( ( member_set_o_nat_nat @ U @ A2 )
=> ( ( ord_le8808915593745164104at_nat @ V @ U )
=> ( ord_le8808915593745164104at_nat @ V @ ( comple7172370505855214741at_nat @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_999_Sup__upper2,axiom,
! [U: set_o,A2: set_set_o,V: set_o] :
( ( member_set_o @ U @ A2 )
=> ( ( ord_less_eq_set_o @ V @ U )
=> ( ord_less_eq_set_o @ V @ ( comple90263536869209701_set_o @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_1000_Sup__upper2,axiom,
! [U: $o,A2: set_o,V: $o] :
( ( member_o @ U @ A2 )
=> ( ( ord_less_eq_o @ V @ U )
=> ( ord_less_eq_o @ V @ ( complete_Sup_Sup_o @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_1001_Sup__upper2,axiom,
! [U: set_nat,A2: set_set_nat,V: set_nat] :
( ( member_set_nat @ U @ A2 )
=> ( ( ord_less_eq_set_nat @ V @ U )
=> ( ord_less_eq_set_nat @ V @ ( comple7399068483239264473et_nat @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_1002_Sup__le__iff,axiom,
! [A2: set_set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ A2 ) @ B )
= ( ! [X3: set_nat_nat] :
( ( member_set_nat_nat2 @ X3 @ A2 )
=> ( ord_le9059583361652607317at_nat @ X3 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_1003_Sup__le__iff,axiom,
! [A2: set_set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ A2 ) @ B )
= ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A2 )
=> ( ord_le6893508408891458716et_nat @ X3 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_1004_Sup__le__iff,axiom,
! [A2: set_set_nat_nat_nat,B: set_nat_nat_nat] :
( ( ord_le3211623285424100676at_nat @ ( comple8167887107183641911at_nat @ A2 ) @ B )
= ( ! [X3: set_nat_nat_nat] :
( ( member8194441297229544571at_nat @ X3 @ A2 )
=> ( ord_le3211623285424100676at_nat @ X3 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_1005_Sup__le__iff,axiom,
! [A2: set_set_o_nat_nat,B: set_o_nat_nat] :
( ( ord_le8808915593745164104at_nat @ ( comple7172370505855214741at_nat @ A2 ) @ B )
= ( ! [X3: set_o_nat_nat] :
( ( member_set_o_nat_nat @ X3 @ A2 )
=> ( ord_le8808915593745164104at_nat @ X3 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_1006_Sup__le__iff,axiom,
! [A2: set_set_o,B: set_o] :
( ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ A2 ) @ B )
= ( ! [X3: set_o] :
( ( member_set_o @ X3 @ A2 )
=> ( ord_less_eq_set_o @ X3 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_1007_Sup__le__iff,axiom,
! [A2: set_o,B: $o] :
( ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A2 ) @ B )
= ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_less_eq_o @ X3 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_1008_Sup__le__iff,axiom,
! [A2: set_set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ B )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ X3 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_1009_Sup__upper,axiom,
! [X2: nat > set_nat,A2: set_nat_set_nat] :
( ( member_nat_set_nat @ X2 @ A2 )
=> ( ord_le6195038898401538645et_nat @ X2 @ ( comple2583738152068352712et_nat @ A2 ) ) ) ).
% Sup_upper
thf(fact_1010_Sup__upper,axiom,
! [X2: set_nat_nat,A2: set_set_nat_nat] :
( ( member_set_nat_nat2 @ X2 @ A2 )
=> ( ord_le9059583361652607317at_nat @ X2 @ ( comple5448282615319421384at_nat @ A2 ) ) ) ).
% Sup_upper
thf(fact_1011_Sup__upper,axiom,
! [X2: set_set_nat,A2: set_set_set_nat] :
( ( member_set_set_nat @ X2 @ A2 )
=> ( ord_le6893508408891458716et_nat @ X2 @ ( comple548664676211718543et_nat @ A2 ) ) ) ).
% Sup_upper
thf(fact_1012_Sup__upper,axiom,
! [X2: set_nat_nat_nat,A2: set_set_nat_nat_nat] :
( ( member8194441297229544571at_nat @ X2 @ A2 )
=> ( ord_le3211623285424100676at_nat @ X2 @ ( comple8167887107183641911at_nat @ A2 ) ) ) ).
% Sup_upper
thf(fact_1013_Sup__upper,axiom,
! [X2: set_o_nat_nat,A2: set_set_o_nat_nat] :
( ( member_set_o_nat_nat @ X2 @ A2 )
=> ( ord_le8808915593745164104at_nat @ X2 @ ( comple7172370505855214741at_nat @ A2 ) ) ) ).
% Sup_upper
thf(fact_1014_Sup__upper,axiom,
! [X2: set_o,A2: set_set_o] :
( ( member_set_o @ X2 @ A2 )
=> ( ord_less_eq_set_o @ X2 @ ( comple90263536869209701_set_o @ A2 ) ) ) ).
% Sup_upper
thf(fact_1015_Sup__upper,axiom,
! [X2: $o,A2: set_o] :
( ( member_o @ X2 @ A2 )
=> ( ord_less_eq_o @ X2 @ ( complete_Sup_Sup_o @ A2 ) ) ) ).
% Sup_upper
thf(fact_1016_Sup__upper,axiom,
! [X2: set_nat,A2: set_set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ X2 @ ( comple7399068483239264473et_nat @ A2 ) ) ) ).
% Sup_upper
thf(fact_1017_Sup__least,axiom,
! [A2: set_nat_set_nat,Z2: nat > set_nat] :
( ! [X: nat > set_nat] :
( ( member_nat_set_nat @ X @ A2 )
=> ( ord_le6195038898401538645et_nat @ X @ Z2 ) )
=> ( ord_le6195038898401538645et_nat @ ( comple2583738152068352712et_nat @ A2 ) @ Z2 ) ) ).
% Sup_least
thf(fact_1018_Sup__least,axiom,
! [A2: set_set_nat_nat,Z2: set_nat_nat] :
( ! [X: set_nat_nat] :
( ( member_set_nat_nat2 @ X @ A2 )
=> ( ord_le9059583361652607317at_nat @ X @ Z2 ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ A2 ) @ Z2 ) ) ).
% Sup_least
thf(fact_1019_Sup__least,axiom,
! [A2: set_set_set_nat,Z2: set_set_nat] :
( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ A2 )
=> ( ord_le6893508408891458716et_nat @ X @ Z2 ) )
=> ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ A2 ) @ Z2 ) ) ).
% Sup_least
thf(fact_1020_Sup__least,axiom,
! [A2: set_set_nat_nat_nat,Z2: set_nat_nat_nat] :
( ! [X: set_nat_nat_nat] :
( ( member8194441297229544571at_nat @ X @ A2 )
=> ( ord_le3211623285424100676at_nat @ X @ Z2 ) )
=> ( ord_le3211623285424100676at_nat @ ( comple8167887107183641911at_nat @ A2 ) @ Z2 ) ) ).
% Sup_least
thf(fact_1021_Sup__least,axiom,
! [A2: set_set_o_nat_nat,Z2: set_o_nat_nat] :
( ! [X: set_o_nat_nat] :
( ( member_set_o_nat_nat @ X @ A2 )
=> ( ord_le8808915593745164104at_nat @ X @ Z2 ) )
=> ( ord_le8808915593745164104at_nat @ ( comple7172370505855214741at_nat @ A2 ) @ Z2 ) ) ).
% Sup_least
thf(fact_1022_Sup__least,axiom,
! [A2: set_set_o,Z2: set_o] :
( ! [X: set_o] :
( ( member_set_o @ X @ A2 )
=> ( ord_less_eq_set_o @ X @ Z2 ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ A2 ) @ Z2 ) ) ).
% Sup_least
thf(fact_1023_Sup__least,axiom,
! [A2: set_o,Z2: $o] :
( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ord_less_eq_o @ X @ Z2 ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A2 ) @ Z2 ) ) ).
% Sup_least
thf(fact_1024_Sup__least,axiom,
! [A2: set_set_nat,Z2: set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ X @ Z2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ Z2 ) ) ).
% Sup_least
thf(fact_1025_Sup__mono,axiom,
! [A2: set_nat_set_nat,B2: set_nat_set_nat] :
( ! [A5: nat > set_nat] :
( ( member_nat_set_nat @ A5 @ A2 )
=> ? [X4: nat > set_nat] :
( ( member_nat_set_nat @ X4 @ B2 )
& ( ord_le6195038898401538645et_nat @ A5 @ X4 ) ) )
=> ( ord_le6195038898401538645et_nat @ ( comple2583738152068352712et_nat @ A2 ) @ ( comple2583738152068352712et_nat @ B2 ) ) ) ).
% Sup_mono
thf(fact_1026_Sup__mono,axiom,
! [A2: set_set_nat_nat,B2: set_set_nat_nat] :
( ! [A5: set_nat_nat] :
( ( member_set_nat_nat2 @ A5 @ A2 )
=> ? [X4: set_nat_nat] :
( ( member_set_nat_nat2 @ X4 @ B2 )
& ( ord_le9059583361652607317at_nat @ A5 @ X4 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ A2 ) @ ( comple5448282615319421384at_nat @ B2 ) ) ) ).
% Sup_mono
thf(fact_1027_Sup__mono,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ! [A5: set_set_nat] :
( ( member_set_set_nat @ A5 @ A2 )
=> ? [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ B2 )
& ( ord_le6893508408891458716et_nat @ A5 @ X4 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ A2 ) @ ( comple548664676211718543et_nat @ B2 ) ) ) ).
% Sup_mono
thf(fact_1028_Sup__mono,axiom,
! [A2: set_set_nat_nat_nat,B2: set_set_nat_nat_nat] :
( ! [A5: set_nat_nat_nat] :
( ( member8194441297229544571at_nat @ A5 @ A2 )
=> ? [X4: set_nat_nat_nat] :
( ( member8194441297229544571at_nat @ X4 @ B2 )
& ( ord_le3211623285424100676at_nat @ A5 @ X4 ) ) )
=> ( ord_le3211623285424100676at_nat @ ( comple8167887107183641911at_nat @ A2 ) @ ( comple8167887107183641911at_nat @ B2 ) ) ) ).
% Sup_mono
thf(fact_1029_Sup__mono,axiom,
! [A2: set_set_o_nat_nat,B2: set_set_o_nat_nat] :
( ! [A5: set_o_nat_nat] :
( ( member_set_o_nat_nat @ A5 @ A2 )
=> ? [X4: set_o_nat_nat] :
( ( member_set_o_nat_nat @ X4 @ B2 )
& ( ord_le8808915593745164104at_nat @ A5 @ X4 ) ) )
=> ( ord_le8808915593745164104at_nat @ ( comple7172370505855214741at_nat @ A2 ) @ ( comple7172370505855214741at_nat @ B2 ) ) ) ).
% Sup_mono
thf(fact_1030_Sup__mono,axiom,
! [A2: set_set_o,B2: set_set_o] :
( ! [A5: set_o] :
( ( member_set_o @ A5 @ A2 )
=> ? [X4: set_o] :
( ( member_set_o @ X4 @ B2 )
& ( ord_less_eq_set_o @ A5 @ X4 ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ A2 ) @ ( comple90263536869209701_set_o @ B2 ) ) ) ).
% Sup_mono
thf(fact_1031_Sup__mono,axiom,
! [A2: set_o,B2: set_o] :
( ! [A5: $o] :
( ( member_o @ A5 @ A2 )
=> ? [X4: $o] :
( ( member_o @ X4 @ B2 )
& ( ord_less_eq_o @ A5 @ X4 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A2 ) @ ( complete_Sup_Sup_o @ B2 ) ) ) ).
% Sup_mono
thf(fact_1032_Sup__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ! [A5: set_nat] :
( ( member_set_nat @ A5 @ A2 )
=> ? [X4: set_nat] :
( ( member_set_nat @ X4 @ B2 )
& ( ord_less_eq_set_nat @ A5 @ X4 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ).
% Sup_mono
thf(fact_1033_Sup__eqI,axiom,
! [A2: set_nat_set_nat,X2: nat > set_nat] :
( ! [Y4: nat > set_nat] :
( ( member_nat_set_nat @ Y4 @ A2 )
=> ( ord_le6195038898401538645et_nat @ Y4 @ X2 ) )
=> ( ! [Y4: nat > set_nat] :
( ! [Z4: nat > set_nat] :
( ( member_nat_set_nat @ Z4 @ A2 )
=> ( ord_le6195038898401538645et_nat @ Z4 @ Y4 ) )
=> ( ord_le6195038898401538645et_nat @ X2 @ Y4 ) )
=> ( ( comple2583738152068352712et_nat @ A2 )
= X2 ) ) ) ).
% Sup_eqI
thf(fact_1034_Sup__eqI,axiom,
! [A2: set_set_nat_nat,X2: set_nat_nat] :
( ! [Y4: set_nat_nat] :
( ( member_set_nat_nat2 @ Y4 @ A2 )
=> ( ord_le9059583361652607317at_nat @ Y4 @ X2 ) )
=> ( ! [Y4: set_nat_nat] :
( ! [Z4: set_nat_nat] :
( ( member_set_nat_nat2 @ Z4 @ A2 )
=> ( ord_le9059583361652607317at_nat @ Z4 @ Y4 ) )
=> ( ord_le9059583361652607317at_nat @ X2 @ Y4 ) )
=> ( ( comple5448282615319421384at_nat @ A2 )
= X2 ) ) ) ).
% Sup_eqI
thf(fact_1035_Sup__eqI,axiom,
! [A2: set_set_set_nat,X2: set_set_nat] :
( ! [Y4: set_set_nat] :
( ( member_set_set_nat @ Y4 @ A2 )
=> ( ord_le6893508408891458716et_nat @ Y4 @ X2 ) )
=> ( ! [Y4: set_set_nat] :
( ! [Z4: set_set_nat] :
( ( member_set_set_nat @ Z4 @ A2 )
=> ( ord_le6893508408891458716et_nat @ Z4 @ Y4 ) )
=> ( ord_le6893508408891458716et_nat @ X2 @ Y4 ) )
=> ( ( comple548664676211718543et_nat @ A2 )
= X2 ) ) ) ).
% Sup_eqI
thf(fact_1036_Sup__eqI,axiom,
! [A2: set_set_nat_nat_nat,X2: set_nat_nat_nat] :
( ! [Y4: set_nat_nat_nat] :
( ( member8194441297229544571at_nat @ Y4 @ A2 )
=> ( ord_le3211623285424100676at_nat @ Y4 @ X2 ) )
=> ( ! [Y4: set_nat_nat_nat] :
( ! [Z4: set_nat_nat_nat] :
( ( member8194441297229544571at_nat @ Z4 @ A2 )
=> ( ord_le3211623285424100676at_nat @ Z4 @ Y4 ) )
=> ( ord_le3211623285424100676at_nat @ X2 @ Y4 ) )
=> ( ( comple8167887107183641911at_nat @ A2 )
= X2 ) ) ) ).
% Sup_eqI
thf(fact_1037_Sup__eqI,axiom,
! [A2: set_set_o_nat_nat,X2: set_o_nat_nat] :
( ! [Y4: set_o_nat_nat] :
( ( member_set_o_nat_nat @ Y4 @ A2 )
=> ( ord_le8808915593745164104at_nat @ Y4 @ X2 ) )
=> ( ! [Y4: set_o_nat_nat] :
( ! [Z4: set_o_nat_nat] :
( ( member_set_o_nat_nat @ Z4 @ A2 )
=> ( ord_le8808915593745164104at_nat @ Z4 @ Y4 ) )
=> ( ord_le8808915593745164104at_nat @ X2 @ Y4 ) )
=> ( ( comple7172370505855214741at_nat @ A2 )
= X2 ) ) ) ).
% Sup_eqI
thf(fact_1038_Sup__eqI,axiom,
! [A2: set_set_o,X2: set_o] :
( ! [Y4: set_o] :
( ( member_set_o @ Y4 @ A2 )
=> ( ord_less_eq_set_o @ Y4 @ X2 ) )
=> ( ! [Y4: set_o] :
( ! [Z4: set_o] :
( ( member_set_o @ Z4 @ A2 )
=> ( ord_less_eq_set_o @ Z4 @ Y4 ) )
=> ( ord_less_eq_set_o @ X2 @ Y4 ) )
=> ( ( comple90263536869209701_set_o @ A2 )
= X2 ) ) ) ).
% Sup_eqI
thf(fact_1039_Sup__eqI,axiom,
! [A2: set_o,X2: $o] :
( ! [Y4: $o] :
( ( member_o @ Y4 @ A2 )
=> ( ord_less_eq_o @ Y4 @ X2 ) )
=> ( ! [Y4: $o] :
( ! [Z4: $o] :
( ( member_o @ Z4 @ A2 )
=> ( ord_less_eq_o @ Z4 @ Y4 ) )
=> ( ord_less_eq_o @ X2 @ Y4 ) )
=> ( ( complete_Sup_Sup_o @ A2 )
= X2 ) ) ) ).
% Sup_eqI
thf(fact_1040_Sup__eqI,axiom,
! [A2: set_set_nat,X2: set_nat] :
( ! [Y4: set_nat] :
( ( member_set_nat @ Y4 @ A2 )
=> ( ord_less_eq_set_nat @ Y4 @ X2 ) )
=> ( ! [Y4: set_nat] :
( ! [Z4: set_nat] :
( ( member_set_nat @ Z4 @ A2 )
=> ( ord_less_eq_set_nat @ Z4 @ Y4 ) )
=> ( ord_less_eq_set_nat @ X2 @ Y4 ) )
=> ( ( comple7399068483239264473et_nat @ A2 )
= X2 ) ) ) ).
% Sup_eqI
thf(fact_1041_INF__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > $o,D: $o > $o] :
( ( A2 = B2 )
=> ( ! [X: $o] :
( ( member_o @ X @ B2 )
=> ( ( C @ X )
= ( D @ X ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ C @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_o_o @ D @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_1042_INF__cong,axiom,
! [A2: set_nat,B2: set_nat,C: nat > $o,D: nat > $o] :
( ( A2 = B2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B2 )
=> ( ( C @ X )
= ( D @ X ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_nat_o @ C @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_nat_o @ D @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_1043_INF__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > nat,D: $o > nat] :
( ( A2 = B2 )
=> ( ! [X: $o] :
( ( member_o @ X @ B2 )
=> ( ( C @ X )
= ( D @ X ) ) )
=> ( ( complete_Inf_Inf_nat @ ( image_o_nat @ C @ A2 ) )
= ( complete_Inf_Inf_nat @ ( image_o_nat @ D @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_1044_INF__cong,axiom,
! [A2: set_nat,B2: set_nat,C: nat > nat,D: nat > nat] :
( ( A2 = B2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B2 )
=> ( ( C @ X )
= ( D @ X ) ) )
=> ( ( complete_Inf_Inf_nat @ ( image_nat_nat @ C @ A2 ) )
= ( complete_Inf_Inf_nat @ ( image_nat_nat @ D @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_1045_INF__cong,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_nat > $o,D: set_nat > $o] :
( ( A2 = B2 )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ B2 )
=> ( ( C @ X )
= ( D @ X ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_set_nat_o @ C @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_set_nat_o @ D @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_1046_INF__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > set_nat,D: $o > set_nat] :
( ( A2 = B2 )
=> ( ! [X: $o] :
( ( member_o @ X @ B2 )
=> ( ( C @ X )
= ( D @ X ) ) )
=> ( ( comple7806235888213564991et_nat @ ( image_o_set_nat @ C @ A2 ) )
= ( comple7806235888213564991et_nat @ ( image_o_set_nat @ D @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_1047_INF__cong,axiom,
! [A2: set_nat,B2: set_nat,C: nat > set_nat,D: nat > set_nat] :
( ( A2 = B2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B2 )
=> ( ( C @ X )
= ( D @ X ) ) )
=> ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ C @ A2 ) )
= ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ D @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_1048_INF__cong,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_nat > nat,D: set_nat > nat] :
( ( A2 = B2 )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ B2 )
=> ( ( C @ X )
= ( D @ X ) ) )
=> ( ( complete_Inf_Inf_nat @ ( image_set_nat_nat @ C @ A2 ) )
= ( complete_Inf_Inf_nat @ ( image_set_nat_nat @ D @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_1049_INF__cong,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C: ( nat > nat ) > $o,D: ( nat > nat ) > $o] :
( ( A2 = B2 )
=> ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ B2 )
=> ( ( C @ X )
= ( D @ X ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_nat_nat_o @ C @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_nat_nat_o @ D @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_1050_INF__cong,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_nat > set_nat,D: set_nat > set_nat] :
( ( A2 = B2 )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ B2 )
=> ( ( C @ X )
= ( D @ X ) ) )
=> ( ( comple7806235888213564991et_nat @ ( image_7916887816326733075et_nat @ C @ A2 ) )
= ( comple7806235888213564991et_nat @ ( image_7916887816326733075et_nat @ D @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_1051_SUP__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > nat,D: $o > nat] :
( ( A2 = B2 )
=> ( ! [X: $o] :
( ( member_o @ X @ B2 )
=> ( ( C @ X )
= ( D @ X ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_o_nat @ C @ A2 ) )
= ( complete_Sup_Sup_nat @ ( image_o_nat @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_1052_SUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C: nat > nat,D: nat > nat] :
( ( A2 = B2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B2 )
=> ( ( C @ X )
= ( D @ X ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat @ C @ A2 ) )
= ( complete_Sup_Sup_nat @ ( image_nat_nat @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_1053_SUP__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > $o,D: $o > $o] :
( ( A2 = B2 )
=> ( ! [X: $o] :
( ( member_o @ X @ B2 )
=> ( ( C @ X )
= ( D @ X ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ C @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_1054_SUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C: nat > $o,D: nat > $o] :
( ( A2 = B2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B2 )
=> ( ( C @ X )
= ( D @ X ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ C @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_1055_SUP__cong,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_nat > $o,D: set_nat > $o] :
( ( A2 = B2 )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ B2 )
=> ( ( C @ X )
= ( D @ X ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_set_nat_o @ C @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_set_nat_o @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_1056_SUP__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > set_nat,D: $o > set_nat] :
( ( A2 = B2 )
=> ( ! [X: $o] :
( ( member_o @ X @ B2 )
=> ( ( C @ X )
= ( D @ X ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ C @ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_o_set_nat @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_1057_SUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C: nat > set_nat,D: nat > set_nat] :
( ( A2 = B2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B2 )
=> ( ( C @ X )
= ( D @ X ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C @ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_1058_SUP__cong,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C: ( nat > nat ) > $o,D: ( nat > nat ) > $o] :
( ( A2 = B2 )
=> ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ B2 )
=> ( ( C @ X )
= ( D @ X ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_nat_o @ C @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_nat_nat_o @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_1059_SUP__cong,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_nat > set_nat,D: set_nat > set_nat] :
( ( A2 = B2 )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ B2 )
=> ( ( C @ X )
= ( D @ X ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ C @ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_1060_SUP__cong,axiom,
! [A2: set_nat_set_nat,B2: set_nat_set_nat,C: ( nat > set_nat ) > $o,D: ( nat > set_nat ) > $o] :
( ( A2 = B2 )
=> ( ! [X: nat > set_nat] :
( ( member_nat_set_nat @ X @ B2 )
=> ( ( C @ X )
= ( D @ X ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_set_nat_o @ C @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_nat_set_nat_o @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_1061_Inter__anti__mono,axiom,
! [B2: set_set_nat_nat,A2: set_set_nat_nat] :
( ( ord_le4954213926817602059at_nat @ B2 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( comple439066603627490862at_nat @ A2 ) @ ( comple439066603627490862at_nat @ B2 ) ) ) ).
% Inter_anti_mono
thf(fact_1062_Inter__anti__mono,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( comple1065008630642458357et_nat @ A2 ) @ ( comple1065008630642458357et_nat @ B2 ) ) ) ).
% Inter_anti_mono
thf(fact_1063_Inter__anti__mono,axiom,
! [B2: set_set_nat_nat_nat,A2: set_set_nat_nat_nat] :
( ( ord_le8468300607614202362at_nat @ B2 @ A2 )
=> ( ord_le3211623285424100676at_nat @ ( comple884914421528019101at_nat @ A2 ) @ ( comple884914421528019101at_nat @ B2 ) ) ) ).
% Inter_anti_mono
thf(fact_1064_Inter__anti__mono,axiom,
! [B2: set_set_o_nat_nat,A2: set_set_o_nat_nat] :
( ( ord_le1662808457768676136at_nat @ B2 @ A2 )
=> ( ord_le8808915593745164104at_nat @ ( comple6464488905778062255at_nat @ A2 ) @ ( comple6464488905778062255at_nat @ B2 ) ) ) ).
% Inter_anti_mono
thf(fact_1065_Inter__anti__mono,axiom,
! [B2: set_set_o,A2: set_set_o] :
( ( ord_le4374716579403074808_set_o @ B2 @ A2 )
=> ( ord_less_eq_set_o @ ( comple3063163877087187839_set_o @ A2 ) @ ( comple3063163877087187839_set_o @ B2 ) ) ) ).
% Inter_anti_mono
thf(fact_1066_Inter__anti__mono,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A2 ) @ ( comple7806235888213564991et_nat @ B2 ) ) ) ).
% Inter_anti_mono
thf(fact_1067_Inter__greatest,axiom,
! [A2: set_set_nat_nat,C: set_nat_nat] :
( ! [X6: set_nat_nat] :
( ( member_set_nat_nat2 @ X6 @ A2 )
=> ( ord_le9059583361652607317at_nat @ C @ X6 ) )
=> ( ord_le9059583361652607317at_nat @ C @ ( comple439066603627490862at_nat @ A2 ) ) ) ).
% Inter_greatest
thf(fact_1068_Inter__greatest,axiom,
! [A2: set_set_set_nat,C: set_set_nat] :
( ! [X6: set_set_nat] :
( ( member_set_set_nat @ X6 @ A2 )
=> ( ord_le6893508408891458716et_nat @ C @ X6 ) )
=> ( ord_le6893508408891458716et_nat @ C @ ( comple1065008630642458357et_nat @ A2 ) ) ) ).
% Inter_greatest
thf(fact_1069_Inter__greatest,axiom,
! [A2: set_set_nat_nat_nat,C: set_nat_nat_nat] :
( ! [X6: set_nat_nat_nat] :
( ( member8194441297229544571at_nat @ X6 @ A2 )
=> ( ord_le3211623285424100676at_nat @ C @ X6 ) )
=> ( ord_le3211623285424100676at_nat @ C @ ( comple884914421528019101at_nat @ A2 ) ) ) ).
% Inter_greatest
thf(fact_1070_Inter__greatest,axiom,
! [A2: set_set_o_nat_nat,C: set_o_nat_nat] :
( ! [X6: set_o_nat_nat] :
( ( member_set_o_nat_nat @ X6 @ A2 )
=> ( ord_le8808915593745164104at_nat @ C @ X6 ) )
=> ( ord_le8808915593745164104at_nat @ C @ ( comple6464488905778062255at_nat @ A2 ) ) ) ).
% Inter_greatest
thf(fact_1071_Inter__greatest,axiom,
! [A2: set_set_o,C: set_o] :
( ! [X6: set_o] :
( ( member_set_o @ X6 @ A2 )
=> ( ord_less_eq_set_o @ C @ X6 ) )
=> ( ord_less_eq_set_o @ C @ ( comple3063163877087187839_set_o @ A2 ) ) ) ).
% Inter_greatest
thf(fact_1072_Inter__greatest,axiom,
! [A2: set_set_nat,C: set_nat] :
( ! [X6: set_nat] :
( ( member_set_nat @ X6 @ A2 )
=> ( ord_less_eq_set_nat @ C @ X6 ) )
=> ( ord_less_eq_set_nat @ C @ ( comple7806235888213564991et_nat @ A2 ) ) ) ).
% Inter_greatest
thf(fact_1073_Inter__lower,axiom,
! [B2: set_nat_nat,A2: set_set_nat_nat] :
( ( member_set_nat_nat2 @ B2 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( comple439066603627490862at_nat @ A2 ) @ B2 ) ) ).
% Inter_lower
thf(fact_1074_Inter__lower,axiom,
! [B2: set_set_nat,A2: set_set_set_nat] :
( ( member_set_set_nat @ B2 @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( comple1065008630642458357et_nat @ A2 ) @ B2 ) ) ).
% Inter_lower
thf(fact_1075_Inter__lower,axiom,
! [B2: set_nat_nat_nat,A2: set_set_nat_nat_nat] :
( ( member8194441297229544571at_nat @ B2 @ A2 )
=> ( ord_le3211623285424100676at_nat @ ( comple884914421528019101at_nat @ A2 ) @ B2 ) ) ).
% Inter_lower
thf(fact_1076_Inter__lower,axiom,
! [B2: set_o_nat_nat,A2: set_set_o_nat_nat] :
( ( member_set_o_nat_nat @ B2 @ A2 )
=> ( ord_le8808915593745164104at_nat @ ( comple6464488905778062255at_nat @ A2 ) @ B2 ) ) ).
% Inter_lower
thf(fact_1077_Inter__lower,axiom,
! [B2: set_o,A2: set_set_o] :
( ( member_set_o @ B2 @ A2 )
=> ( ord_less_eq_set_o @ ( comple3063163877087187839_set_o @ A2 ) @ B2 ) ) ).
% Inter_lower
thf(fact_1078_Inter__lower,axiom,
! [B2: set_nat,A2: set_set_nat] :
( ( member_set_nat @ B2 @ A2 )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A2 ) @ B2 ) ) ).
% Inter_lower
thf(fact_1079_Union__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% Union_empty
thf(fact_1080_Union__empty__conv,axiom,
! [A2: set_set_nat] :
( ( ( comple7399068483239264473et_nat @ A2 )
= bot_bot_set_nat )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( X3 = bot_bot_set_nat ) ) ) ) ).
% Union_empty_conv
thf(fact_1081_empty__Union__conv,axiom,
! [A2: set_set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ A2 ) )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( X3 = bot_bot_set_nat ) ) ) ) ).
% empty_Union_conv
thf(fact_1082_Union__subsetI,axiom,
! [A2: set_set_nat_nat,B2: set_set_nat_nat] :
( ! [X: set_nat_nat] :
( ( member_set_nat_nat2 @ X @ A2 )
=> ? [Y: set_nat_nat] :
( ( member_set_nat_nat2 @ Y @ B2 )
& ( ord_le9059583361652607317at_nat @ X @ Y ) ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ A2 ) @ ( comple5448282615319421384at_nat @ B2 ) ) ) ).
% Union_subsetI
thf(fact_1083_Union__subsetI,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ A2 )
=> ? [Y: set_set_nat] :
( ( member_set_set_nat @ Y @ B2 )
& ( ord_le6893508408891458716et_nat @ X @ Y ) ) )
=> ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ A2 ) @ ( comple548664676211718543et_nat @ B2 ) ) ) ).
% Union_subsetI
thf(fact_1084_Union__subsetI,axiom,
! [A2: set_set_nat_nat_nat,B2: set_set_nat_nat_nat] :
( ! [X: set_nat_nat_nat] :
( ( member8194441297229544571at_nat @ X @ A2 )
=> ? [Y: set_nat_nat_nat] :
( ( member8194441297229544571at_nat @ Y @ B2 )
& ( ord_le3211623285424100676at_nat @ X @ Y ) ) )
=> ( ord_le3211623285424100676at_nat @ ( comple8167887107183641911at_nat @ A2 ) @ ( comple8167887107183641911at_nat @ B2 ) ) ) ).
% Union_subsetI
thf(fact_1085_Union__subsetI,axiom,
! [A2: set_set_o_nat_nat,B2: set_set_o_nat_nat] :
( ! [X: set_o_nat_nat] :
( ( member_set_o_nat_nat @ X @ A2 )
=> ? [Y: set_o_nat_nat] :
( ( member_set_o_nat_nat @ Y @ B2 )
& ( ord_le8808915593745164104at_nat @ X @ Y ) ) )
=> ( ord_le8808915593745164104at_nat @ ( comple7172370505855214741at_nat @ A2 ) @ ( comple7172370505855214741at_nat @ B2 ) ) ) ).
% Union_subsetI
thf(fact_1086_Union__subsetI,axiom,
! [A2: set_set_o,B2: set_set_o] :
( ! [X: set_o] :
( ( member_set_o @ X @ A2 )
=> ? [Y: set_o] :
( ( member_set_o @ Y @ B2 )
& ( ord_less_eq_set_o @ X @ Y ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ A2 ) @ ( comple90263536869209701_set_o @ B2 ) ) ) ).
% Union_subsetI
thf(fact_1087_Union__subsetI,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ? [Y: set_nat] :
( ( member_set_nat @ Y @ B2 )
& ( ord_less_eq_set_nat @ X @ Y ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ).
% Union_subsetI
thf(fact_1088_Union__upper,axiom,
! [B2: set_nat_nat,A2: set_set_nat_nat] :
( ( member_set_nat_nat2 @ B2 @ A2 )
=> ( ord_le9059583361652607317at_nat @ B2 @ ( comple5448282615319421384at_nat @ A2 ) ) ) ).
% Union_upper
thf(fact_1089_Union__upper,axiom,
! [B2: set_set_nat,A2: set_set_set_nat] :
( ( member_set_set_nat @ B2 @ A2 )
=> ( ord_le6893508408891458716et_nat @ B2 @ ( comple548664676211718543et_nat @ A2 ) ) ) ).
% Union_upper
thf(fact_1090_Union__upper,axiom,
! [B2: set_nat_nat_nat,A2: set_set_nat_nat_nat] :
( ( member8194441297229544571at_nat @ B2 @ A2 )
=> ( ord_le3211623285424100676at_nat @ B2 @ ( comple8167887107183641911at_nat @ A2 ) ) ) ).
% Union_upper
thf(fact_1091_Union__upper,axiom,
! [B2: set_o_nat_nat,A2: set_set_o_nat_nat] :
( ( member_set_o_nat_nat @ B2 @ A2 )
=> ( ord_le8808915593745164104at_nat @ B2 @ ( comple7172370505855214741at_nat @ A2 ) ) ) ).
% Union_upper
thf(fact_1092_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_1093_Union__upper,axiom,
! [B2: set_nat,A2: set_set_nat] :
( ( member_set_nat @ B2 @ A2 )
=> ( ord_less_eq_set_nat @ B2 @ ( comple7399068483239264473et_nat @ A2 ) ) ) ).
% Union_upper
thf(fact_1094_Union__least,axiom,
! [A2: set_set_nat_nat,C: set_nat_nat] :
( ! [X6: set_nat_nat] :
( ( member_set_nat_nat2 @ X6 @ A2 )
=> ( ord_le9059583361652607317at_nat @ X6 @ C ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ A2 ) @ C ) ) ).
% Union_least
thf(fact_1095_Union__least,axiom,
! [A2: set_set_set_nat,C: set_set_nat] :
( ! [X6: set_set_nat] :
( ( member_set_set_nat @ X6 @ A2 )
=> ( ord_le6893508408891458716et_nat @ X6 @ C ) )
=> ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ A2 ) @ C ) ) ).
% Union_least
thf(fact_1096_Union__least,axiom,
! [A2: set_set_nat_nat_nat,C: set_nat_nat_nat] :
( ! [X6: set_nat_nat_nat] :
( ( member8194441297229544571at_nat @ X6 @ A2 )
=> ( ord_le3211623285424100676at_nat @ X6 @ C ) )
=> ( ord_le3211623285424100676at_nat @ ( comple8167887107183641911at_nat @ A2 ) @ C ) ) ).
% Union_least
thf(fact_1097_Union__least,axiom,
! [A2: set_set_o_nat_nat,C: set_o_nat_nat] :
( ! [X6: set_o_nat_nat] :
( ( member_set_o_nat_nat @ X6 @ A2 )
=> ( ord_le8808915593745164104at_nat @ X6 @ C ) )
=> ( ord_le8808915593745164104at_nat @ ( comple7172370505855214741at_nat @ A2 ) @ C ) ) ).
% Union_least
thf(fact_1098_Union__least,axiom,
! [A2: set_set_o,C: set_o] :
( ! [X6: set_o] :
( ( member_set_o @ X6 @ A2 )
=> ( ord_less_eq_set_o @ X6 @ C ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ A2 ) @ C ) ) ).
% Union_least
thf(fact_1099_Union__least,axiom,
! [A2: set_set_nat,C: set_nat] :
( ! [X6: set_nat] :
( ( member_set_nat @ X6 @ A2 )
=> ( ord_less_eq_set_nat @ X6 @ C ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ C ) ) ).
% Union_least
thf(fact_1100_Union__mono,axiom,
! [A2: set_set_nat_nat,B2: set_set_nat_nat] :
( ( ord_le4954213926817602059at_nat @ A2 @ B2 )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ A2 ) @ ( comple5448282615319421384at_nat @ B2 ) ) ) ).
% Union_mono
thf(fact_1101_Union__mono,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ A2 ) @ ( comple548664676211718543et_nat @ B2 ) ) ) ).
% Union_mono
thf(fact_1102_Union__mono,axiom,
! [A2: set_set_nat_nat_nat,B2: set_set_nat_nat_nat] :
( ( ord_le8468300607614202362at_nat @ A2 @ B2 )
=> ( ord_le3211623285424100676at_nat @ ( comple8167887107183641911at_nat @ A2 ) @ ( comple8167887107183641911at_nat @ B2 ) ) ) ).
% Union_mono
thf(fact_1103_Union__mono,axiom,
! [A2: set_set_o_nat_nat,B2: set_set_o_nat_nat] :
( ( ord_le1662808457768676136at_nat @ A2 @ B2 )
=> ( ord_le8808915593745164104at_nat @ ( comple7172370505855214741at_nat @ A2 ) @ ( comple7172370505855214741at_nat @ B2 ) ) ) ).
% Union_mono
thf(fact_1104_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_1105_Union__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ).
% Union_mono
thf(fact_1106_INF__le__SUP,axiom,
! [A2: set_nat,F: nat > set_nat_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ord_le9059583361652607317at_nat @ ( comple439066603627490862at_nat @ ( image_7301343469591561292at_nat @ F @ A2 ) ) @ ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ F @ A2 ) ) ) ) ).
% INF_le_SUP
thf(fact_1107_INF__le__SUP,axiom,
! [A2: set_nat,F: nat > set_set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ord_le6893508408891458716et_nat @ ( comple1065008630642458357et_nat @ ( image_2194112158459175443et_nat @ F @ A2 ) ) @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ F @ A2 ) ) ) ) ).
% INF_le_SUP
thf(fact_1108_INF__le__SUP,axiom,
! [A2: set_nat,F: nat > set_nat_nat_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ord_le3211623285424100676at_nat @ ( comple884914421528019101at_nat @ ( image_6130888460295934395at_nat @ F @ A2 ) ) @ ( comple8167887107183641911at_nat @ ( image_6130888460295934395at_nat @ F @ A2 ) ) ) ) ).
% INF_le_SUP
thf(fact_1109_INF__le__SUP,axiom,
! [A2: set_nat,F: nat > set_o_nat_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ord_le8808915593745164104at_nat @ ( comple6464488905778062255at_nat @ ( image_6307125673115463441at_nat @ F @ A2 ) ) @ ( comple7172370505855214741at_nat @ ( image_6307125673115463441at_nat @ F @ A2 ) ) ) ) ).
% INF_le_SUP
thf(fact_1110_INF__le__SUP,axiom,
! [A2: set_nat,F: nat > set_o] :
( ( A2 != bot_bot_set_nat )
=> ( ord_less_eq_set_o @ ( comple3063163877087187839_set_o @ ( image_nat_set_o @ F @ A2 ) ) @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ F @ A2 ) ) ) ) ).
% INF_le_SUP
thf(fact_1111_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_1112_INF__le__SUP,axiom,
! [A2: set_nat,F: nat > $o] :
( ( A2 != bot_bot_set_nat )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_nat_o @ F @ A2 ) ) @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A2 ) ) ) ) ).
% INF_le_SUP
thf(fact_1113_INF__le__SUP,axiom,
! [A2: set_nat,F: nat > set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ).
% INF_le_SUP
thf(fact_1114_greaterThan__non__empty,axiom,
! [X2: nat] :
( ( set_or1210151606488870762an_nat @ X2 )
!= bot_bot_set_nat ) ).
% greaterThan_non_empty
thf(fact_1115_INF__commute,axiom,
! [F: nat > nat > $o,B2: set_nat,A2: set_nat] :
( ( complete_Inf_Inf_o
@ ( image_nat_o
@ ^ [I: nat] : ( complete_Inf_Inf_o @ ( image_nat_o @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( complete_Inf_Inf_o
@ ( image_nat_o
@ ^ [J: nat] :
( complete_Inf_Inf_o
@ ( image_nat_o
@ ^ [I: nat] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% INF_commute
thf(fact_1116_INF__commute,axiom,
! [F: nat > $o > $o,B2: set_o,A2: set_nat] :
( ( complete_Inf_Inf_o
@ ( image_nat_o
@ ^ [I: nat] : ( 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_nat_o
@ ^ [I: nat] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% INF_commute
thf(fact_1117_INF__commute,axiom,
! [F: $o > nat > $o,B2: set_nat,A2: set_o] :
( ( complete_Inf_Inf_o
@ ( image_o_o
@ ^ [I: $o] : ( complete_Inf_Inf_o @ ( image_nat_o @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( complete_Inf_Inf_o
@ ( image_nat_o
@ ^ [J: nat] :
( complete_Inf_Inf_o
@ ( image_o_o
@ ^ [I: $o] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% INF_commute
thf(fact_1118_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_1119_INF__commute,axiom,
! [F: nat > nat > set_nat,B2: set_nat,A2: set_nat] :
( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [I: nat] : ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [J: nat] :
( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [I: nat] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% INF_commute
thf(fact_1120_SUP__commute,axiom,
! [F: nat > nat > $o,B2: set_nat,A2: set_nat] :
( ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [I: nat] : ( complete_Sup_Sup_o @ ( image_nat_o @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [J: nat] :
( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [I: nat] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_1121_SUP__commute,axiom,
! [F: nat > $o > $o,B2: set_o,A2: set_nat] :
( ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [I: nat] : ( 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_nat_o
@ ^ [I: nat] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_1122_SUP__commute,axiom,
! [F: $o > nat > $o,B2: set_nat,A2: set_o] :
( ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [I: $o] : ( complete_Sup_Sup_o @ ( image_nat_o @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [J: nat] :
( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [I: $o] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_1123_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_1124_SUP__commute,axiom,
! [F: nat > nat > set_nat,B2: set_nat,A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [J: nat] :
( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I: nat] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_1125_INT__extend__simps_I10_J,axiom,
! [B2: ( nat > nat ) > set_nat,F: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( comple7806235888213564991et_nat
@ ( image_7432509271690132940et_nat
@ ^ [A4: nat > nat] : ( B2 @ ( F @ A4 ) )
@ A2 ) )
= ( comple7806235888213564991et_nat @ ( image_7432509271690132940et_nat @ B2 @ ( image_3205354838064109189at_nat @ F @ A2 ) ) ) ) ).
% INT_extend_simps(10)
thf(fact_1126_INT__extend__simps_I10_J,axiom,
! [B2: $o > set_nat,F: $o > $o,A2: set_o] :
( ( comple7806235888213564991et_nat
@ ( image_o_set_nat
@ ^ [A4: $o] : ( B2 @ ( F @ A4 ) )
@ A2 ) )
= ( comple7806235888213564991et_nat @ ( image_o_set_nat @ B2 @ ( image_o_o @ F @ A2 ) ) ) ) ).
% INT_extend_simps(10)
thf(fact_1127_INT__extend__simps_I10_J,axiom,
! [B2: nat > set_nat,F: $o > nat,A2: set_o] :
( ( comple7806235888213564991et_nat
@ ( image_o_set_nat
@ ^ [A4: $o] : ( B2 @ ( F @ A4 ) )
@ A2 ) )
= ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B2 @ ( image_o_nat @ F @ A2 ) ) ) ) ).
% INT_extend_simps(10)
thf(fact_1128_INT__extend__simps_I10_J,axiom,
! [B2: set_nat > set_nat,F: nat > set_nat,A2: set_nat] :
( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [A4: nat] : ( B2 @ ( F @ A4 ) )
@ A2 ) )
= ( comple7806235888213564991et_nat @ ( image_7916887816326733075et_nat @ B2 @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ).
% INT_extend_simps(10)
thf(fact_1129_INT__extend__simps_I10_J,axiom,
! [B2: $o > set_nat,F: nat > $o,A2: set_nat] :
( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [A4: nat] : ( B2 @ ( F @ A4 ) )
@ A2 ) )
= ( comple7806235888213564991et_nat @ ( image_o_set_nat @ B2 @ ( image_nat_o @ F @ A2 ) ) ) ) ).
% INT_extend_simps(10)
thf(fact_1130_INT__extend__simps_I10_J,axiom,
! [B2: nat > set_nat,F: nat > nat,A2: set_nat] :
( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [A4: nat] : ( B2 @ ( F @ A4 ) )
@ A2 ) )
= ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% INT_extend_simps(10)
thf(fact_1131_INT__subset__iff,axiom,
! [B2: set_nat,A2: nat > set_nat,I2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ A2 @ I2 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ I2 )
=> ( ord_less_eq_set_nat @ B2 @ ( A2 @ X3 ) ) ) ) ) ).
% INT_subset_iff
thf(fact_1132_INT__anti__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > set_o,G: nat > set_o] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_o @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_o @ ( comple3063163877087187839_set_o @ ( image_nat_set_o @ F @ B2 ) ) @ ( comple3063163877087187839_set_o @ ( image_nat_set_o @ G @ A2 ) ) ) ) ) ).
% INT_anti_mono
thf(fact_1133_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 )
=> ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ord_less_eq_set_o @ ( F @ X ) @ ( G @ X ) ) )
=> ( 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_1134_INT__anti__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ B2 ) ) @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ G @ A2 ) ) ) ) ) ).
% INT_anti_mono
thf(fact_1135_INT__anti__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > set_nat,G: $o > set_nat] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ B2 ) ) @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ G @ A2 ) ) ) ) ) ).
% INT_anti_mono
thf(fact_1136_INT__anti__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > set_set_nat,G: nat > set_set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_le6893508408891458716et_nat @ ( comple1065008630642458357et_nat @ ( image_2194112158459175443et_nat @ F @ B2 ) ) @ ( comple1065008630642458357et_nat @ ( image_2194112158459175443et_nat @ G @ A2 ) ) ) ) ) ).
% INT_anti_mono
thf(fact_1137_INT__anti__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat,F: set_nat > set_o,G: set_nat > set_o] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( ord_less_eq_set_o @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_o @ ( comple3063163877087187839_set_o @ ( image_set_nat_set_o @ F @ B2 ) ) @ ( comple3063163877087187839_set_o @ ( image_set_nat_set_o @ G @ A2 ) ) ) ) ) ).
% INT_anti_mono
thf(fact_1138_INT__anti__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > set_set_nat,G: $o > set_set_nat] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_le6893508408891458716et_nat @ ( comple1065008630642458357et_nat @ ( image_o_set_set_nat @ F @ B2 ) ) @ ( comple1065008630642458357et_nat @ ( image_o_set_set_nat @ G @ A2 ) ) ) ) ) ).
% INT_anti_mono
thf(fact_1139_INT__anti__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat,F: set_nat > set_nat,G: set_nat > set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_7916887816326733075et_nat @ F @ B2 ) ) @ ( comple7806235888213564991et_nat @ ( image_7916887816326733075et_nat @ G @ A2 ) ) ) ) ) ).
% INT_anti_mono
thf(fact_1140_INT__anti__mono,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,F: ( nat > nat ) > set_o,G: ( nat > nat ) > set_o] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( ord_less_eq_set_o @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_o @ ( comple3063163877087187839_set_o @ ( image_nat_nat_set_o @ F @ B2 ) ) @ ( comple3063163877087187839_set_o @ ( image_nat_nat_set_o @ G @ A2 ) ) ) ) ) ).
% INT_anti_mono
thf(fact_1141_INT__anti__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > set_nat_nat,G: nat > set_nat_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_le9059583361652607317at_nat @ ( comple439066603627490862at_nat @ ( image_7301343469591561292at_nat @ F @ B2 ) ) @ ( comple439066603627490862at_nat @ ( image_7301343469591561292at_nat @ G @ A2 ) ) ) ) ) ).
% INT_anti_mono
thf(fact_1142_INT__greatest,axiom,
! [A2: set_o,C: set_o,B2: $o > set_o] :
( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ord_less_eq_set_o @ C @ ( B2 @ X ) ) )
=> ( ord_less_eq_set_o @ C @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ B2 @ A2 ) ) ) ) ).
% INT_greatest
thf(fact_1143_INT__greatest,axiom,
! [A2: set_nat,C: set_o,B2: nat > set_o] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_o @ C @ ( B2 @ X ) ) )
=> ( ord_less_eq_set_o @ C @ ( comple3063163877087187839_set_o @ ( image_nat_set_o @ B2 @ A2 ) ) ) ) ).
% INT_greatest
thf(fact_1144_INT__greatest,axiom,
! [A2: set_o,C: set_nat,B2: $o > set_nat] :
( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ord_less_eq_set_nat @ C @ ( B2 @ X ) ) )
=> ( ord_less_eq_set_nat @ C @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ B2 @ A2 ) ) ) ) ).
% INT_greatest
thf(fact_1145_INT__greatest,axiom,
! [A2: set_nat,C: set_nat,B2: nat > set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ C @ ( B2 @ X ) ) )
=> ( ord_less_eq_set_nat @ C @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) ) ) ).
% INT_greatest
thf(fact_1146_INT__greatest,axiom,
! [A2: set_o,C: set_set_nat,B2: $o > set_set_nat] :
( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ord_le6893508408891458716et_nat @ C @ ( B2 @ X ) ) )
=> ( ord_le6893508408891458716et_nat @ C @ ( comple1065008630642458357et_nat @ ( image_o_set_set_nat @ B2 @ A2 ) ) ) ) ).
% INT_greatest
thf(fact_1147_INT__greatest,axiom,
! [A2: set_nat,C: set_set_nat,B2: nat > set_set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_le6893508408891458716et_nat @ C @ ( B2 @ X ) ) )
=> ( ord_le6893508408891458716et_nat @ C @ ( comple1065008630642458357et_nat @ ( image_2194112158459175443et_nat @ B2 @ A2 ) ) ) ) ).
% INT_greatest
thf(fact_1148_INT__greatest,axiom,
! [A2: set_set_nat,C: set_o,B2: set_nat > set_o] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( ord_less_eq_set_o @ C @ ( B2 @ X ) ) )
=> ( ord_less_eq_set_o @ C @ ( comple3063163877087187839_set_o @ ( image_set_nat_set_o @ B2 @ A2 ) ) ) ) ).
% INT_greatest
thf(fact_1149_INT__greatest,axiom,
! [A2: set_set_nat,C: set_nat,B2: set_nat > set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ C @ ( B2 @ X ) ) )
=> ( ord_less_eq_set_nat @ C @ ( comple7806235888213564991et_nat @ ( image_7916887816326733075et_nat @ B2 @ A2 ) ) ) ) ).
% INT_greatest
thf(fact_1150_INT__greatest,axiom,
! [A2: set_o,C: set_nat_nat,B2: $o > set_nat_nat] :
( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ord_le9059583361652607317at_nat @ C @ ( B2 @ X ) ) )
=> ( ord_le9059583361652607317at_nat @ C @ ( comple439066603627490862at_nat @ ( image_o_set_nat_nat @ B2 @ A2 ) ) ) ) ).
% INT_greatest
thf(fact_1151_INT__greatest,axiom,
! [A2: set_nat,C: set_nat_nat,B2: nat > set_nat_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_le9059583361652607317at_nat @ C @ ( B2 @ X ) ) )
=> ( ord_le9059583361652607317at_nat @ C @ ( comple439066603627490862at_nat @ ( image_7301343469591561292at_nat @ B2 @ A2 ) ) ) ) ).
% INT_greatest
thf(fact_1152_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_1153_INT__lower,axiom,
! [A: nat,A2: set_nat,B2: nat > set_o] :
( ( member_nat @ A @ A2 )
=> ( ord_less_eq_set_o @ ( comple3063163877087187839_set_o @ ( image_nat_set_o @ B2 @ A2 ) ) @ ( B2 @ A ) ) ) ).
% INT_lower
thf(fact_1154_INT__lower,axiom,
! [A: $o,A2: set_o,B2: $o > set_nat] :
( ( member_o @ A @ A2 )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ B2 @ A2 ) ) @ ( B2 @ A ) ) ) ).
% INT_lower
thf(fact_1155_INT__lower,axiom,
! [A: nat,A2: set_nat,B2: nat > set_nat] :
( ( member_nat @ A @ A2 )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) @ ( B2 @ A ) ) ) ).
% INT_lower
thf(fact_1156_INT__lower,axiom,
! [A: $o,A2: set_o,B2: $o > set_set_nat] :
( ( member_o @ A @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( comple1065008630642458357et_nat @ ( image_o_set_set_nat @ B2 @ A2 ) ) @ ( B2 @ A ) ) ) ).
% INT_lower
thf(fact_1157_INT__lower,axiom,
! [A: nat,A2: set_nat,B2: nat > set_set_nat] :
( ( member_nat @ A @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( comple1065008630642458357et_nat @ ( image_2194112158459175443et_nat @ B2 @ A2 ) ) @ ( B2 @ A ) ) ) ).
% INT_lower
thf(fact_1158_INT__lower,axiom,
! [A: set_nat,A2: set_set_nat,B2: set_nat > set_o] :
( ( member_set_nat @ A @ A2 )
=> ( ord_less_eq_set_o @ ( comple3063163877087187839_set_o @ ( image_set_nat_set_o @ B2 @ A2 ) ) @ ( B2 @ A ) ) ) ).
% INT_lower
thf(fact_1159_INT__lower,axiom,
! [A: set_nat,A2: set_set_nat,B2: set_nat > set_nat] :
( ( member_set_nat @ A @ A2 )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_7916887816326733075et_nat @ B2 @ A2 ) ) @ ( B2 @ A ) ) ) ).
% INT_lower
thf(fact_1160_INT__lower,axiom,
! [A: $o,A2: set_o,B2: $o > set_nat_nat] :
( ( member_o @ A @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( comple439066603627490862at_nat @ ( image_o_set_nat_nat @ B2 @ A2 ) ) @ ( B2 @ A ) ) ) ).
% INT_lower
thf(fact_1161_INT__lower,axiom,
! [A: nat,A2: set_nat,B2: nat > set_nat_nat] :
( ( member_nat @ A @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( comple439066603627490862at_nat @ ( image_7301343469591561292at_nat @ B2 @ A2 ) ) @ ( B2 @ A ) ) ) ).
% INT_lower
thf(fact_1162_image__Union,axiom,
! [F: ( nat > nat ) > nat > nat,S: set_set_nat_nat] :
( ( image_3205354838064109189at_nat @ F @ ( comple5448282615319421384at_nat @ S ) )
= ( comple5448282615319421384at_nat @ ( image_3832368097948589297at_nat @ ( image_3205354838064109189at_nat @ F ) @ S ) ) ) ).
% image_Union
thf(fact_1163_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_1164_image__Union,axiom,
! [F: $o > nat,S: set_set_o] :
( ( image_o_nat @ F @ ( comple90263536869209701_set_o @ S ) )
= ( comple7399068483239264473et_nat @ ( image_set_o_set_nat @ ( image_o_nat @ F ) @ S ) ) ) ).
% image_Union
thf(fact_1165_image__Union,axiom,
! [F: nat > set_nat,S: set_set_nat] :
( ( image_nat_set_nat @ F @ ( comple7399068483239264473et_nat @ S ) )
= ( comple548664676211718543et_nat @ ( image_6725021117256019401et_nat @ ( image_nat_set_nat @ F ) @ S ) ) ) ).
% image_Union
thf(fact_1166_image__Union,axiom,
! [F: nat > $o,S: set_set_nat] :
( ( image_nat_o @ F @ ( comple7399068483239264473et_nat @ S ) )
= ( comple90263536869209701_set_o @ ( image_set_nat_set_o @ ( image_nat_o @ F ) @ S ) ) ) ).
% image_Union
thf(fact_1167_image__Union,axiom,
! [F: nat > nat,S: set_set_nat] :
( ( image_nat_nat @ F @ ( comple7399068483239264473et_nat @ S ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ S ) ) ) ).
% image_Union
thf(fact_1168_image__UN,axiom,
! [F: $o > nat,B2: nat > set_o,A2: set_nat] :
( ( image_o_nat @ F @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B2 @ A2 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X3: nat] : ( image_o_nat @ F @ ( B2 @ X3 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_1169_image__UN,axiom,
! [F: nat > set_nat,B2: nat > set_nat,A2: set_nat] :
( ( image_nat_set_nat @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
= ( comple548664676211718543et_nat
@ ( image_2194112158459175443et_nat
@ ^ [X3: nat] : ( image_nat_set_nat @ F @ ( B2 @ X3 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_1170_image__UN,axiom,
! [F: nat > $o,B2: nat > set_nat,A2: set_nat] :
( ( image_nat_o @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
= ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [X3: nat] : ( image_nat_o @ F @ ( B2 @ X3 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_1171_image__UN,axiom,
! [F: nat > nat,B2: nat > set_nat,A2: set_nat] :
( ( image_nat_nat @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X3: nat] : ( image_nat_nat @ F @ ( B2 @ X3 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_1172_UN__extend__simps_I10_J,axiom,
! [B2: ( nat > nat ) > set_nat,F: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( comple7399068483239264473et_nat
@ ( image_7432509271690132940et_nat
@ ^ [A4: nat > nat] : ( B2 @ ( F @ A4 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ B2 @ ( image_3205354838064109189at_nat @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_1173_UN__extend__simps_I10_J,axiom,
! [B2: $o > set_nat,F: $o > $o,A2: set_o] :
( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [A4: $o] : ( B2 @ ( F @ A4 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ ( image_o_o @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_1174_UN__extend__simps_I10_J,axiom,
! [B2: nat > set_nat,F: $o > nat,A2: set_o] :
( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [A4: $o] : ( B2 @ ( F @ A4 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ ( image_o_nat @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_1175_UN__extend__simps_I10_J,axiom,
! [B2: set_nat > set_nat,F: nat > set_nat,A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [A4: nat] : ( B2 @ ( F @ A4 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B2 @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_1176_UN__extend__simps_I10_J,axiom,
! [B2: $o > set_nat,F: nat > $o,A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [A4: nat] : ( B2 @ ( F @ A4 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ ( image_nat_o @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_1177_UN__extend__simps_I10_J,axiom,
! [B2: nat > set_nat,F: nat > nat,A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [A4: nat] : ( B2 @ ( F @ A4 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_1178_UN__empty2,axiom,
! [A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X3: nat] : bot_bot_set_nat
@ A2 ) )
= bot_bot_set_nat ) ).
% UN_empty2
thf(fact_1179_UN__empty,axiom,
! [B2: nat > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ bot_bot_set_nat ) )
= bot_bot_set_nat ) ).
% UN_empty
thf(fact_1180_UNION__empty__conv_I1_J,axiom,
! [B2: nat > set_nat,A2: set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( B2 @ X3 )
= bot_bot_set_nat ) ) ) ) ).
% UNION_empty_conv(1)
thf(fact_1181_UNION__empty__conv_I2_J,axiom,
! [B2: nat > set_nat,A2: set_nat] :
( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) )
= bot_bot_set_nat )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( B2 @ X3 )
= bot_bot_set_nat ) ) ) ) ).
% UNION_empty_conv(2)
thf(fact_1182_UN__subset__iff,axiom,
! [A2: nat > set_nat,I2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ I2 ) ) @ B2 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ I2 )
=> ( ord_less_eq_set_nat @ ( A2 @ X3 ) @ B2 ) ) ) ) ).
% UN_subset_iff
thf(fact_1183_UN__upper,axiom,
! [A: $o,A2: set_o,B2: $o > set_o] :
( ( member_o @ A @ A2 )
=> ( ord_less_eq_set_o @ ( B2 @ A ) @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_1184_UN__upper,axiom,
! [A: nat,A2: set_nat,B2: nat > set_o] :
( ( member_nat @ A @ A2 )
=> ( ord_less_eq_set_o @ ( B2 @ A ) @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_1185_UN__upper,axiom,
! [A: $o,A2: set_o,B2: $o > set_nat] :
( ( member_o @ A @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ A ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_1186_UN__upper,axiom,
! [A: nat,A2: set_nat,B2: nat > set_nat] :
( ( member_nat @ A @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ A ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_1187_UN__upper,axiom,
! [A: $o,A2: set_o,B2: $o > set_set_nat] :
( ( member_o @ A @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( B2 @ A ) @ ( comple548664676211718543et_nat @ ( image_o_set_set_nat @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_1188_UN__upper,axiom,
! [A: nat,A2: set_nat,B2: nat > set_set_nat] :
( ( member_nat @ A @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( B2 @ A ) @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_1189_UN__upper,axiom,
! [A: set_nat,A2: set_set_nat,B2: set_nat > set_o] :
( ( member_set_nat @ A @ A2 )
=> ( ord_less_eq_set_o @ ( B2 @ A ) @ ( comple90263536869209701_set_o @ ( image_set_nat_set_o @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_1190_UN__upper,axiom,
! [A: set_nat,A2: set_set_nat,B2: set_nat > set_nat] :
( ( member_set_nat @ A @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ A ) @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_1191_UN__upper,axiom,
! [A: $o,A2: set_o,B2: $o > set_nat_nat] :
( ( member_o @ A @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B2 @ A ) @ ( comple5448282615319421384at_nat @ ( image_o_set_nat_nat @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_1192_UN__upper,axiom,
! [A: nat,A2: set_nat,B2: nat > set_nat_nat] :
( ( member_nat @ A @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B2 @ A ) @ ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_1193_UN__least,axiom,
! [A2: set_o,B2: $o > set_o,C: set_o] :
( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ord_less_eq_set_o @ ( B2 @ X ) @ C ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) @ C ) ) ).
% UN_least
thf(fact_1194_UN__least,axiom,
! [A2: set_nat,B2: nat > set_o,C: set_o] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_o @ ( B2 @ X ) @ C ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B2 @ A2 ) ) @ C ) ) ).
% UN_least
thf(fact_1195_UN__least,axiom,
! [A2: set_o,B2: $o > set_nat,C: set_nat] :
( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ X ) @ C ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ A2 ) ) @ C ) ) ).
% UN_least
thf(fact_1196_UN__least,axiom,
! [A2: set_nat,B2: nat > set_nat,C: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ X ) @ C ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) @ C ) ) ).
% UN_least
thf(fact_1197_UN__least,axiom,
! [A2: set_o,B2: $o > set_set_nat,C: set_set_nat] :
( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( B2 @ X ) @ C ) )
=> ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ ( image_o_set_set_nat @ B2 @ A2 ) ) @ C ) ) ).
% UN_least
thf(fact_1198_UN__least,axiom,
! [A2: set_nat,B2: nat > set_set_nat,C: set_set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( B2 @ X ) @ C ) )
=> ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ B2 @ A2 ) ) @ C ) ) ).
% UN_least
thf(fact_1199_UN__least,axiom,
! [A2: set_set_nat,B2: set_nat > set_o,C: set_o] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( ord_less_eq_set_o @ ( B2 @ X ) @ C ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_set_nat_set_o @ B2 @ A2 ) ) @ C ) ) ).
% UN_least
thf(fact_1200_UN__least,axiom,
! [A2: set_set_nat,B2: set_nat > set_nat,C: set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ X ) @ C ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B2 @ A2 ) ) @ C ) ) ).
% UN_least
thf(fact_1201_UN__least,axiom,
! [A2: set_o,B2: $o > set_nat_nat,C: set_nat_nat] :
( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B2 @ X ) @ C ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ ( image_o_set_nat_nat @ B2 @ A2 ) ) @ C ) ) ).
% UN_least
thf(fact_1202_UN__least,axiom,
! [A2: set_nat,B2: nat > set_nat_nat,C: set_nat_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B2 @ X ) @ C ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ B2 @ A2 ) ) @ C ) ) ).
% UN_least
thf(fact_1203_UN__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > set_o,G: nat > set_o] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_o @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ F @ A2 ) ) @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_1204_UN__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > set_o,G: $o > set_o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ord_less_eq_set_o @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ A2 ) ) @ ( comple90263536869209701_set_o @ ( image_o_set_o @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_1205_UN__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_1206_UN__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > set_nat,G: $o > set_nat] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_1207_UN__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > set_set_nat,G: nat > set_set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ F @ A2 ) ) @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_1208_UN__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat,F: set_nat > set_o,G: set_nat > set_o] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( ord_less_eq_set_o @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_set_nat_set_o @ F @ A2 ) ) @ ( comple90263536869209701_set_o @ ( image_set_nat_set_o @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_1209_UN__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > set_set_nat,G: $o > set_set_nat] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ ( image_o_set_set_nat @ F @ A2 ) ) @ ( comple548664676211718543et_nat @ ( image_o_set_set_nat @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_1210_UN__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat,F: set_nat > set_nat,G: set_nat > set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_1211_UN__mono,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,F: ( nat > nat ) > set_o,G: ( nat > nat ) > set_o] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( ord_less_eq_set_o @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_nat_nat_set_o @ F @ A2 ) ) @ ( comple90263536869209701_set_o @ ( image_nat_nat_set_o @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_1212_UN__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > set_nat_nat,G: nat > set_nat_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ F @ A2 ) ) @ ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_1213_INF__eq,axiom,
! [A2: set_o,B2: set_o,G: $o > $o,F: $o > $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A2 )
=> ? [X4: $o] :
( ( member_o @ X4 @ B2 )
& ( ord_less_eq_o @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B2 )
=> ? [X4: $o] :
( ( member_o @ X4 @ A2 )
& ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ J2 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_o_o @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_1214_INF__eq,axiom,
! [A2: set_o,B2: set_nat,G: nat > $o,F: $o > $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B2 )
& ( ord_less_eq_o @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B2 )
=> ? [X4: $o] :
( ( member_o @ X4 @ A2 )
& ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ J2 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_nat_o @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_1215_INF__eq,axiom,
! [A2: set_nat,B2: set_o,G: $o > $o,F: nat > $o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ? [X4: $o] :
( ( member_o @ X4 @ B2 )
& ( ord_less_eq_o @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ J2 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_nat_o @ F @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_o_o @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_1216_INF__eq,axiom,
! [A2: set_nat,B2: set_nat,G: nat > $o,F: nat > $o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B2 )
& ( ord_less_eq_o @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ J2 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_nat_o @ F @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_nat_o @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_1217_INF__eq,axiom,
! [A2: set_o,B2: set_o,G: $o > set_o,F: $o > set_o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A2 )
=> ? [X4: $o] :
( ( member_o @ X4 @ B2 )
& ( ord_less_eq_set_o @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B2 )
=> ? [X4: $o] :
( ( member_o @ X4 @ A2 )
& ( ord_less_eq_set_o @ ( F @ X4 ) @ ( G @ J2 ) ) ) )
=> ( ( comple3063163877087187839_set_o @ ( image_o_set_o @ F @ A2 ) )
= ( comple3063163877087187839_set_o @ ( image_o_set_o @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_1218_INF__eq,axiom,
! [A2: set_o,B2: set_nat,G: nat > set_o,F: $o > set_o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B2 )
& ( ord_less_eq_set_o @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B2 )
=> ? [X4: $o] :
( ( member_o @ X4 @ A2 )
& ( ord_less_eq_set_o @ ( F @ X4 ) @ ( G @ J2 ) ) ) )
=> ( ( comple3063163877087187839_set_o @ ( image_o_set_o @ F @ A2 ) )
= ( comple3063163877087187839_set_o @ ( image_nat_set_o @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_1219_INF__eq,axiom,
! [A2: set_nat,B2: set_o,G: $o > set_o,F: nat > set_o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ? [X4: $o] :
( ( member_o @ X4 @ B2 )
& ( ord_less_eq_set_o @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_eq_set_o @ ( F @ X4 ) @ ( G @ J2 ) ) ) )
=> ( ( comple3063163877087187839_set_o @ ( image_nat_set_o @ F @ A2 ) )
= ( comple3063163877087187839_set_o @ ( image_o_set_o @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_1220_INF__eq,axiom,
! [A2: set_nat,B2: set_nat,G: nat > set_o,F: nat > set_o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B2 )
& ( ord_less_eq_set_o @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_eq_set_o @ ( F @ X4 ) @ ( G @ J2 ) ) ) )
=> ( ( comple3063163877087187839_set_o @ ( image_nat_set_o @ F @ A2 ) )
= ( comple3063163877087187839_set_o @ ( image_nat_set_o @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_1221_INF__eq,axiom,
! [A2: set_o,B2: set_set_nat,G: set_nat > $o,F: $o > $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A2 )
=> ? [X4: set_nat] :
( ( member_set_nat @ X4 @ B2 )
& ( ord_less_eq_o @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J2: set_nat] :
( ( member_set_nat @ J2 @ B2 )
=> ? [X4: $o] :
( ( member_o @ X4 @ A2 )
& ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ J2 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_set_nat_o @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_1222_INF__eq,axiom,
! [A2: set_nat,B2: set_set_nat,G: set_nat > $o,F: nat > $o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ? [X4: set_nat] :
( ( member_set_nat @ X4 @ B2 )
& ( ord_less_eq_o @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J2: set_nat] :
( ( member_set_nat @ J2 @ B2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ J2 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_nat_o @ F @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_set_nat_o @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_1223_Inf__less__eq,axiom,
! [A2: set_nat_set_nat,U: nat > set_nat] :
( ! [V2: nat > set_nat] :
( ( member_nat_set_nat @ V2 @ A2 )
=> ( ord_le6195038898401538645et_nat @ V2 @ U ) )
=> ( ( A2 != bot_bo4007787791999405887et_nat )
=> ( ord_le6195038898401538645et_nat @ ( comple6797894177231197998et_nat @ A2 ) @ U ) ) ) ).
% Inf_less_eq
thf(fact_1224_Inf__less__eq,axiom,
! [A2: set_set_nat_nat,U: set_nat_nat] :
( ! [V2: set_nat_nat] :
( ( member_set_nat_nat2 @ V2 @ A2 )
=> ( ord_le9059583361652607317at_nat @ V2 @ U ) )
=> ( ( A2 != bot_bo7376149671870096959at_nat )
=> ( ord_le9059583361652607317at_nat @ ( comple439066603627490862at_nat @ A2 ) @ U ) ) ) ).
% Inf_less_eq
thf(fact_1225_Inf__less__eq,axiom,
! [A2: set_set_set_nat,U: set_set_nat] :
( ! [V2: set_set_nat] :
( ( member_set_set_nat @ V2 @ A2 )
=> ( ord_le6893508408891458716et_nat @ V2 @ U ) )
=> ( ( A2 != bot_bo7198184520161983622et_nat )
=> ( ord_le6893508408891458716et_nat @ ( comple1065008630642458357et_nat @ A2 ) @ U ) ) ) ).
% Inf_less_eq
thf(fact_1226_Inf__less__eq,axiom,
! [A2: set_set_nat_nat_nat,U: set_nat_nat_nat] :
( ! [V2: set_nat_nat_nat] :
( ( member8194441297229544571at_nat @ V2 @ A2 )
=> ( ord_le3211623285424100676at_nat @ V2 @ U ) )
=> ( ( A2 != bot_bo6668270333135560750at_nat )
=> ( ord_le3211623285424100676at_nat @ ( comple884914421528019101at_nat @ A2 ) @ U ) ) ) ).
% Inf_less_eq
thf(fact_1227_Inf__less__eq,axiom,
! [A2: set_set_o_nat_nat,U: set_o_nat_nat] :
( ! [V2: set_o_nat_nat] :
( ( member_set_o_nat_nat @ V2 @ A2 )
=> ( ord_le8808915593745164104at_nat @ V2 @ U ) )
=> ( ( A2 != bot_bo6195285094354290676at_nat )
=> ( ord_le8808915593745164104at_nat @ ( comple6464488905778062255at_nat @ A2 ) @ U ) ) ) ).
% Inf_less_eq
thf(fact_1228_Inf__less__eq,axiom,
! [A2: set_set_o,U: set_o] :
( ! [V2: set_o] :
( ( member_set_o @ V2 @ A2 )
=> ( ord_less_eq_set_o @ V2 @ U ) )
=> ( ( A2 != bot_bot_set_set_o )
=> ( ord_less_eq_set_o @ ( comple3063163877087187839_set_o @ A2 ) @ U ) ) ) ).
% Inf_less_eq
thf(fact_1229_Inf__less__eq,axiom,
! [A2: set_o,U: $o] :
( ! [V2: $o] :
( ( member_o @ V2 @ A2 )
=> ( ord_less_eq_o @ V2 @ U ) )
=> ( ( A2 != bot_bot_set_o )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A2 ) @ U ) ) ) ).
% Inf_less_eq
thf(fact_1230_Inf__less__eq,axiom,
! [A2: set_set_nat,U: set_nat] :
( ! [V2: set_nat] :
( ( member_set_nat @ V2 @ A2 )
=> ( ord_less_eq_set_nat @ V2 @ U ) )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A2 ) @ U ) ) ) ).
% Inf_less_eq
thf(fact_1231_SUP__eq,axiom,
! [A2: set_o,B2: set_o,F: $o > $o,G: $o > $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A2 )
=> ? [X4: $o] :
( ( member_o @ X4 @ B2 )
& ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B2 )
=> ? [X4: $o] :
( ( member_o @ X4 @ A2 )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X4 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_1232_SUP__eq,axiom,
! [A2: set_o,B2: set_nat,F: $o > $o,G: nat > $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B2 )
& ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B2 )
=> ? [X4: $o] :
( ( member_o @ X4 @ A2 )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X4 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_1233_SUP__eq,axiom,
! [A2: set_nat,B2: set_o,F: nat > $o,G: $o > $o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ? [X4: $o] :
( ( member_o @ X4 @ B2 )
& ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X4 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_1234_SUP__eq,axiom,
! [A2: set_nat,B2: set_nat,F: nat > $o,G: nat > $o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B2 )
& ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X4 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_1235_SUP__eq,axiom,
! [A2: set_o,B2: set_o,F: $o > set_o,G: $o > set_o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A2 )
=> ? [X4: $o] :
( ( member_o @ X4 @ B2 )
& ( ord_less_eq_set_o @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B2 )
=> ? [X4: $o] :
( ( member_o @ X4 @ A2 )
& ( ord_less_eq_set_o @ ( G @ J2 ) @ ( F @ X4 ) ) ) )
=> ( ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ A2 ) )
= ( comple90263536869209701_set_o @ ( image_o_set_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_1236_SUP__eq,axiom,
! [A2: set_o,B2: set_nat,F: $o > set_o,G: nat > set_o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B2 )
& ( ord_less_eq_set_o @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B2 )
=> ? [X4: $o] :
( ( member_o @ X4 @ A2 )
& ( ord_less_eq_set_o @ ( G @ J2 ) @ ( F @ X4 ) ) ) )
=> ( ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ A2 ) )
= ( comple90263536869209701_set_o @ ( image_nat_set_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_1237_SUP__eq,axiom,
! [A2: set_nat,B2: set_o,F: nat > set_o,G: $o > set_o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ? [X4: $o] :
( ( member_o @ X4 @ B2 )
& ( ord_less_eq_set_o @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_eq_set_o @ ( G @ J2 ) @ ( F @ X4 ) ) ) )
=> ( ( comple90263536869209701_set_o @ ( image_nat_set_o @ F @ A2 ) )
= ( comple90263536869209701_set_o @ ( image_o_set_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_1238_SUP__eq,axiom,
! [A2: set_nat,B2: set_nat,F: nat > set_o,G: nat > set_o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B2 )
& ( ord_less_eq_set_o @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_eq_set_o @ ( G @ J2 ) @ ( F @ X4 ) ) ) )
=> ( ( comple90263536869209701_set_o @ ( image_nat_set_o @ F @ A2 ) )
= ( comple90263536869209701_set_o @ ( image_nat_set_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_1239_SUP__eq,axiom,
! [A2: set_o,B2: set_set_nat,F: $o > $o,G: set_nat > $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A2 )
=> ? [X4: set_nat] :
( ( member_set_nat @ X4 @ B2 )
& ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J2: set_nat] :
( ( member_set_nat @ J2 @ B2 )
=> ? [X4: $o] :
( ( member_o @ X4 @ A2 )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X4 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_set_nat_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_1240_SUP__eq,axiom,
! [A2: set_nat,B2: set_set_nat,F: nat > $o,G: set_nat > $o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ? [X4: set_nat] :
( ( member_set_nat @ X4 @ B2 )
& ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J2: set_nat] :
( ( member_set_nat @ J2 @ B2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X4 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_set_nat_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_1241_less__eq__Sup,axiom,
! [A2: set_nat_set_nat,U: nat > set_nat] :
( ! [V2: nat > set_nat] :
( ( member_nat_set_nat @ V2 @ A2 )
=> ( ord_le6195038898401538645et_nat @ U @ V2 ) )
=> ( ( A2 != bot_bo4007787791999405887et_nat )
=> ( ord_le6195038898401538645et_nat @ U @ ( comple2583738152068352712et_nat @ A2 ) ) ) ) ).
% less_eq_Sup
thf(fact_1242_less__eq__Sup,axiom,
! [A2: set_set_nat_nat,U: set_nat_nat] :
( ! [V2: set_nat_nat] :
( ( member_set_nat_nat2 @ V2 @ A2 )
=> ( ord_le9059583361652607317at_nat @ U @ V2 ) )
=> ( ( A2 != bot_bo7376149671870096959at_nat )
=> ( ord_le9059583361652607317at_nat @ U @ ( comple5448282615319421384at_nat @ A2 ) ) ) ) ).
% less_eq_Sup
thf(fact_1243_less__eq__Sup,axiom,
! [A2: set_set_set_nat,U: set_set_nat] :
( ! [V2: set_set_nat] :
( ( member_set_set_nat @ V2 @ A2 )
=> ( ord_le6893508408891458716et_nat @ U @ V2 ) )
=> ( ( A2 != bot_bo7198184520161983622et_nat )
=> ( ord_le6893508408891458716et_nat @ U @ ( comple548664676211718543et_nat @ A2 ) ) ) ) ).
% less_eq_Sup
thf(fact_1244_less__eq__Sup,axiom,
! [A2: set_set_nat_nat_nat,U: set_nat_nat_nat] :
( ! [V2: set_nat_nat_nat] :
( ( member8194441297229544571at_nat @ V2 @ A2 )
=> ( ord_le3211623285424100676at_nat @ U @ V2 ) )
=> ( ( A2 != bot_bo6668270333135560750at_nat )
=> ( ord_le3211623285424100676at_nat @ U @ ( comple8167887107183641911at_nat @ A2 ) ) ) ) ).
% less_eq_Sup
thf(fact_1245_less__eq__Sup,axiom,
! [A2: set_set_o_nat_nat,U: set_o_nat_nat] :
( ! [V2: set_o_nat_nat] :
( ( member_set_o_nat_nat @ V2 @ A2 )
=> ( ord_le8808915593745164104at_nat @ U @ V2 ) )
=> ( ( A2 != bot_bo6195285094354290676at_nat )
=> ( ord_le8808915593745164104at_nat @ U @ ( comple7172370505855214741at_nat @ A2 ) ) ) ) ).
% less_eq_Sup
thf(fact_1246_less__eq__Sup,axiom,
! [A2: set_set_o,U: set_o] :
( ! [V2: set_o] :
( ( member_set_o @ V2 @ A2 )
=> ( ord_less_eq_set_o @ U @ V2 ) )
=> ( ( A2 != bot_bot_set_set_o )
=> ( ord_less_eq_set_o @ U @ ( comple90263536869209701_set_o @ A2 ) ) ) ) ).
% less_eq_Sup
thf(fact_1247_less__eq__Sup,axiom,
! [A2: set_o,U: $o] :
( ! [V2: $o] :
( ( member_o @ V2 @ A2 )
=> ( ord_less_eq_o @ U @ V2 ) )
=> ( ( A2 != bot_bot_set_o )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ A2 ) ) ) ) ).
% less_eq_Sup
thf(fact_1248_less__eq__Sup,axiom,
! [A2: set_set_nat,U: set_nat] :
( ! [V2: set_nat] :
( ( member_set_nat @ V2 @ A2 )
=> ( ord_less_eq_set_nat @ U @ V2 ) )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ A2 ) ) ) ) ).
% less_eq_Sup
thf(fact_1249_Inf__superset__mono,axiom,
! [B2: set_set_nat_nat,A2: set_set_nat_nat] :
( ( ord_le4954213926817602059at_nat @ B2 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( comple439066603627490862at_nat @ A2 ) @ ( comple439066603627490862at_nat @ B2 ) ) ) ).
% Inf_superset_mono
thf(fact_1250_Inf__superset__mono,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( comple1065008630642458357et_nat @ A2 ) @ ( comple1065008630642458357et_nat @ B2 ) ) ) ).
% Inf_superset_mono
thf(fact_1251_Inf__superset__mono,axiom,
! [B2: set_set_nat_nat_nat,A2: set_set_nat_nat_nat] :
( ( ord_le8468300607614202362at_nat @ B2 @ A2 )
=> ( ord_le3211623285424100676at_nat @ ( comple884914421528019101at_nat @ A2 ) @ ( comple884914421528019101at_nat @ B2 ) ) ) ).
% Inf_superset_mono
thf(fact_1252_Inf__superset__mono,axiom,
! [B2: set_set_o_nat_nat,A2: set_set_o_nat_nat] :
( ( ord_le1662808457768676136at_nat @ B2 @ A2 )
=> ( ord_le8808915593745164104at_nat @ ( comple6464488905778062255at_nat @ A2 ) @ ( comple6464488905778062255at_nat @ B2 ) ) ) ).
% Inf_superset_mono
thf(fact_1253_Inf__superset__mono,axiom,
! [B2: set_set_o,A2: set_set_o] :
( ( ord_le4374716579403074808_set_o @ B2 @ A2 )
=> ( ord_less_eq_set_o @ ( comple3063163877087187839_set_o @ A2 ) @ ( comple3063163877087187839_set_o @ B2 ) ) ) ).
% Inf_superset_mono
thf(fact_1254_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_1255_Inf__superset__mono,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A2 ) @ ( comple7806235888213564991et_nat @ B2 ) ) ) ).
% Inf_superset_mono
thf(fact_1256_Sup__bool__def,axiom,
( complete_Sup_Sup_o
= ( member_o @ $true ) ) ).
% Sup_bool_def
thf(fact_1257_Inf__bool__def,axiom,
( complete_Inf_Inf_o
= ( ^ [A3: set_o] :
~ ( member_o @ $false @ A3 ) ) ) ).
% Inf_bool_def
thf(fact_1258_UN__lessThan__UNIV,axiom,
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_lessThan_nat @ top_top_set_nat ) )
= top_top_set_nat ) ).
% UN_lessThan_UNIV
thf(fact_1259_INT__greaterThan__UNIV,axiom,
( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ set_or1210151606488870762an_nat @ top_top_set_nat ) )
= bot_bot_set_nat ) ).
% INT_greaterThan_UNIV
thf(fact_1260_Inf__nat__def1,axiom,
! [K2: set_nat] :
( ( K2 != bot_bot_set_nat )
=> ( member_nat @ ( complete_Inf_Inf_nat @ K2 ) @ K2 ) ) ).
% Inf_nat_def1
thf(fact_1261_finite__lessThan,axiom,
! [K: nat] : ( finite_finite_nat @ ( set_ord_lessThan_nat @ K ) ) ).
% finite_lessThan
thf(fact_1262_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N3: nat] : ( ord_less_eq_nat @ N3 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_1263_infinite__UNIV__nat,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_nat
thf(fact_1264_finite__less__ub,axiom,
! [F: nat > nat,U: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ N4 @ ( F @ N4 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_1265_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M2: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N5 )
=> ( ord_less_eq_nat @ X3 @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1266_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N3: nat] : ( ord_less_nat @ N3 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_1267_bounded__nat__set__is__finite,axiom,
! [N: set_nat,N2: nat] :
( ! [X: nat] :
( ( member_nat @ X @ N )
=> ( ord_less_nat @ X @ N2 ) )
=> ( finite_finite_nat @ N ) ) ).
% bounded_nat_set_is_finite
thf(fact_1268_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M2: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N5 )
=> ( ord_less_nat @ X3 @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_1269_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I4: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( P @ K3 )
& ( ord_less_nat @ K3 @ I4 ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_1270_bdd__above__nat,axiom,
condit2214826472909112428ve_nat = finite_finite_nat ).
% bdd_above_nat
thf(fact_1271_card__lessThan,axiom,
! [U: nat] :
( ( finite_card_nat @ ( set_ord_lessThan_nat @ U ) )
= U ) ).
% card_lessThan
thf(fact_1272_card__Collect__less__nat,axiom,
! [N2: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_nat @ I @ N2 ) ) )
= N2 ) ).
% card_Collect_less_nat
thf(fact_1273_card__Collect__le__nat,axiom,
! [N2: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_eq_nat @ I @ N2 ) ) )
= ( suc @ N2 ) ) ).
% card_Collect_le_nat
thf(fact_1274_card__le__Suc__Max,axiom,
! [S: set_nat] :
( ( finite_finite_nat @ S )
=> ( ord_less_eq_nat @ ( finite_card_nat @ S ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S ) ) ) ) ).
% card_le_Suc_Max
thf(fact_1275_Inf__nat__def,axiom,
( complete_Inf_Inf_nat
= ( ^ [X7: set_nat] :
( ord_Least_nat
@ ^ [N3: nat] : ( member_nat @ N3 @ X7 ) ) ) ) ).
% Inf_nat_def
% Conjectures (1)
thf(conj_0,conjecture,
( ord_le9059583361652607317at_nat
@ ( image_3205354838064109189at_nat @ s
@ ( piE_nat_nat @ ( set_ord_lessThan_nat @ k )
@ ^ [I: nat] : ( set_ord_lessThan_nat @ t ) ) )
@ ( piE_nat_nat @ ( set_ord_lessThan_nat @ n )
@ ^ [I: nat] : ( set_ord_lessThan_nat @ t ) ) ) ).
%------------------------------------------------------------------------------