TPTP Problem File: SLH0562^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Equivalence_Relation_Enumeration/0007_Equivalence_Relation_Enumeration/prob_00100_004138__11774310_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1522 ( 650 unt; 246 typ; 0 def)
% Number of atoms : 3522 (1070 equ; 0 cnn)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 9704 ( 166 ~; 26 |; 183 &;8149 @)
% ( 0 <=>;1180 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 34 ( 33 usr)
% Number of type conns : 1319 (1319 >; 0 *; 0 +; 0 <<)
% Number of symbols : 215 ( 213 usr; 33 con; 0-4 aty)
% Number of variables : 3352 ( 499 ^;2821 !; 32 ?;3352 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:13:20.673
%------------------------------------------------------------------------------
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collect_set_a: ( set_a > $o ) > set_set_a ).
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thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_Itf__a_J,type,
image_a_set_a: ( a > set_a ) > set_a > set_set_a ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
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thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__a_J,type,
insert_set_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
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thf(sy_c_Wellfounded_Olex__prod_001tf__a_001tf__a,type,
lex_prod_a_a: set_Product_prod_a_a > set_Product_prod_a_a > set_Pr8600417178894128327od_a_a ).
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thf(sy_c_Zorn_OChains_001tf__a,type,
chains_a: set_Product_prod_a_a > set_set_a ).
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thf(sy_c_Zorn_Ochain__subset_001tf__a,type,
chain_subset_a: set_set_a > $o ).
thf(sy_c_Zorn_Ochains_001tf__a,type,
chains_a2: set_set_a > set_set_set_a ).
thf(sy_c_Zorn_Opred__on_Ochain_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
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thf(sy_c_Zorn_Opred__on_Ochain_001t__Set__Oset_Itf__a_J,type,
pred_chain_set_a: set_set_a > ( set_a > set_a > $o ) > set_set_a > $o ).
thf(sy_c_Zorn_Opred__on_Ochain_001tf__a,type,
pred_chain_a: set_a > ( a > a > $o ) > set_a > $o ).
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thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
member_set_set_a: set_set_a > set_set_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_A,type,
a2: set_a ).
thf(sy_v_B,type,
b: set_a ).
thf(sy_v_p,type,
p: set_Product_prod_a_a ).
% Relevant facts (1275)
thf(fact_0_calculation,axiom,
( refl_on_a @ b
@ ( inf_in8905007599844390133od_a_a @ p
@ ( product_Sigma_a_a @ b
@ ^ [Uu: a] : b ) ) ) ).
% calculation
thf(fact_1_UNIV__Times__UNIV,axiom,
( ( product_Sigma_a_a @ top_top_set_a
@ ^ [Uu: a] : top_top_set_a )
= top_to8063371432257647191od_a_a ) ).
% UNIV_Times_UNIV
thf(fact_2_UNIV__Times__UNIV,axiom,
( ( produc6342321021181284593od_a_a @ top_top_set_a
@ ^ [Uu: a] : top_to8063371432257647191od_a_a )
= top_to4273090908018391168od_a_a ) ).
% UNIV_Times_UNIV
thf(fact_3_UNIV__Times__UNIV,axiom,
( ( produc2379640491490746847_a_a_a @ top_to8063371432257647191od_a_a
@ ^ [Uu: product_prod_a_a] : top_top_set_a )
= top_to7619527732258454254_a_a_a ) ).
% UNIV_Times_UNIV
thf(fact_4_UNIV__Times__UNIV,axiom,
( ( produc5899993699339346696od_a_a @ top_to8063371432257647191od_a_a
@ ^ [Uu: product_prod_a_a] : top_to8063371432257647191od_a_a )
= top_to7953187161931797015od_a_a ) ).
% UNIV_Times_UNIV
thf(fact_5_UNIV__Times__UNIV,axiom,
( ( produc7797748338049884712_set_a @ top_top_set_a
@ ^ [Uu: a] : top_top_set_set_a )
= top_to1705781063600484279_set_a ) ).
% UNIV_Times_UNIV
thf(fact_6_UNIV__Times__UNIV,axiom,
( ( produc8014260396509227880et_a_a @ top_top_set_set_a
@ ^ [Uu: set_a] : top_top_set_a )
= top_to6952078089992083703et_a_a ) ).
% UNIV_Times_UNIV
thf(fact_7_UNIV__Times__UNIV,axiom,
( ( produc6033315442965015752_set_a @ top_top_set_set_a
@ ^ [Uu: set_a] : top_top_set_set_a )
= top_to466452440834987607_set_a ) ).
% UNIV_Times_UNIV
thf(fact_8_UNIV__Times__UNIV,axiom,
( ( produc5663638008720543953od_a_a @ top_top_set_a
@ ^ [Uu: a] : top_to1047947862415971895od_a_a )
= top_to3028490459629974624od_a_a ) ).
% UNIV_Times_UNIV
thf(fact_9_UNIV__Times__UNIV,axiom,
( ( produc1070844763932510015_set_a @ top_to8063371432257647191od_a_a
@ ^ [Uu: product_prod_a_a] : top_top_set_set_a )
= top_to6960155520164011086_set_a ) ).
% UNIV_Times_UNIV
thf(fact_10_UNIV__Times__UNIV,axiom,
( ( produc5649525414689246719_a_a_a @ top_to1047947862415971895od_a_a
@ ^ [Uu: set_Product_prod_a_a] : top_top_set_a )
= top_to6203144423882474254_a_a_a ) ).
% UNIV_Times_UNIV
thf(fact_11_Int__UNIV,axiom,
! [A: set_se5735800977113168103od_a_a,B: set_se5735800977113168103od_a_a] :
( ( ( inf_in3339382566020358357od_a_a @ A @ B )
= top_to1047947862415971895od_a_a )
= ( ( A = top_to1047947862415971895od_a_a )
& ( B = top_to1047947862415971895od_a_a ) ) ) ).
% Int_UNIV
thf(fact_12_Int__UNIV,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ A @ B )
= top_top_set_set_a )
= ( ( A = top_top_set_set_a )
& ( B = top_top_set_set_a ) ) ) ).
% Int_UNIV
thf(fact_13_Int__UNIV,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ A @ B )
= top_to8063371432257647191od_a_a )
= ( ( A = top_to8063371432257647191od_a_a )
& ( B = top_to8063371432257647191od_a_a ) ) ) ).
% Int_UNIV
thf(fact_14_Int__UNIV,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= top_top_set_a )
= ( ( A = top_top_set_a )
& ( B = top_top_set_a ) ) ) ).
% Int_UNIV
thf(fact_15_inf__top__left,axiom,
! [X: set_se5735800977113168103od_a_a] :
( ( inf_in3339382566020358357od_a_a @ top_to1047947862415971895od_a_a @ X )
= X ) ).
% inf_top_left
thf(fact_16_inf__top__left,axiom,
! [X: set_set_a] :
( ( inf_inf_set_set_a @ top_top_set_set_a @ X )
= X ) ).
% inf_top_left
thf(fact_17_inf__top__left,axiom,
! [X: product_prod_a_a > $o] :
( ( inf_in2559554923042384936_a_a_o @ top_to8687885267596698950_a_a_o @ X )
= X ) ).
% inf_top_left
thf(fact_18_inf__top__left,axiom,
! [X: a > $o] :
( ( inf_inf_a_o @ top_top_a_o @ X )
= X ) ).
% inf_top_left
thf(fact_19_inf__top__left,axiom,
! [X: a > a > $o] :
( ( inf_inf_a_a_o @ top_top_a_a_o @ X )
= X ) ).
% inf_top_left
thf(fact_20_inf__top__left,axiom,
! [X: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ top_to8063371432257647191od_a_a @ X )
= X ) ).
% inf_top_left
thf(fact_21_inf__top__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ top_top_set_a @ X )
= X ) ).
% inf_top_left
thf(fact_22_inf__top__right,axiom,
! [X: set_se5735800977113168103od_a_a] :
( ( inf_in3339382566020358357od_a_a @ X @ top_to1047947862415971895od_a_a )
= X ) ).
% inf_top_right
thf(fact_23_inf__top__right,axiom,
! [X: set_set_a] :
( ( inf_inf_set_set_a @ X @ top_top_set_set_a )
= X ) ).
% inf_top_right
thf(fact_24_inf__top__right,axiom,
! [X: product_prod_a_a > $o] :
( ( inf_in2559554923042384936_a_a_o @ X @ top_to8687885267596698950_a_a_o )
= X ) ).
% inf_top_right
thf(fact_25_inf__top__right,axiom,
! [X: a > $o] :
( ( inf_inf_a_o @ X @ top_top_a_o )
= X ) ).
% inf_top_right
thf(fact_26_inf__top__right,axiom,
! [X: a > a > $o] :
( ( inf_inf_a_a_o @ X @ top_top_a_a_o )
= X ) ).
% inf_top_right
thf(fact_27_inf__top__right,axiom,
! [X: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ X @ top_to8063371432257647191od_a_a )
= X ) ).
% inf_top_right
thf(fact_28_inf__top__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ top_top_set_a )
= X ) ).
% inf_top_right
thf(fact_29_inf__eq__top__iff,axiom,
! [X: set_se5735800977113168103od_a_a,Y: set_se5735800977113168103od_a_a] :
( ( ( inf_in3339382566020358357od_a_a @ X @ Y )
= top_to1047947862415971895od_a_a )
= ( ( X = top_to1047947862415971895od_a_a )
& ( Y = top_to1047947862415971895od_a_a ) ) ) ).
% inf_eq_top_iff
thf(fact_30_inf__eq__top__iff,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( ( inf_inf_set_set_a @ X @ Y )
= top_top_set_set_a )
= ( ( X = top_top_set_set_a )
& ( Y = top_top_set_set_a ) ) ) ).
% inf_eq_top_iff
thf(fact_31_inf__eq__top__iff,axiom,
! [X: product_prod_a_a > $o,Y: product_prod_a_a > $o] :
( ( ( inf_in2559554923042384936_a_a_o @ X @ Y )
= top_to8687885267596698950_a_a_o )
= ( ( X = top_to8687885267596698950_a_a_o )
& ( Y = top_to8687885267596698950_a_a_o ) ) ) ).
% inf_eq_top_iff
thf(fact_32_inf__eq__top__iff,axiom,
! [X: a > $o,Y: a > $o] :
( ( ( inf_inf_a_o @ X @ Y )
= top_top_a_o )
= ( ( X = top_top_a_o )
& ( Y = top_top_a_o ) ) ) ).
% inf_eq_top_iff
thf(fact_33_inf__eq__top__iff,axiom,
! [X: a > a > $o,Y: a > a > $o] :
( ( ( inf_inf_a_a_o @ X @ Y )
= top_top_a_a_o )
= ( ( X = top_top_a_a_o )
& ( Y = top_top_a_a_o ) ) ) ).
% inf_eq_top_iff
thf(fact_34_inf__eq__top__iff,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ X @ Y )
= top_to8063371432257647191od_a_a )
= ( ( X = top_to8063371432257647191od_a_a )
& ( Y = top_to8063371432257647191od_a_a ) ) ) ).
% inf_eq_top_iff
thf(fact_35_inf__eq__top__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ( inf_inf_set_a @ X @ Y )
= top_top_set_a )
= ( ( X = top_top_set_a )
& ( Y = top_top_set_a ) ) ) ).
% inf_eq_top_iff
thf(fact_36_top__eq__inf__iff,axiom,
! [X: set_se5735800977113168103od_a_a,Y: set_se5735800977113168103od_a_a] :
( ( top_to1047947862415971895od_a_a
= ( inf_in3339382566020358357od_a_a @ X @ Y ) )
= ( ( X = top_to1047947862415971895od_a_a )
& ( Y = top_to1047947862415971895od_a_a ) ) ) ).
% top_eq_inf_iff
thf(fact_37_top__eq__inf__iff,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( top_top_set_set_a
= ( inf_inf_set_set_a @ X @ Y ) )
= ( ( X = top_top_set_set_a )
& ( Y = top_top_set_set_a ) ) ) ).
% top_eq_inf_iff
thf(fact_38_top__eq__inf__iff,axiom,
! [X: product_prod_a_a > $o,Y: product_prod_a_a > $o] :
( ( top_to8687885267596698950_a_a_o
= ( inf_in2559554923042384936_a_a_o @ X @ Y ) )
= ( ( X = top_to8687885267596698950_a_a_o )
& ( Y = top_to8687885267596698950_a_a_o ) ) ) ).
% top_eq_inf_iff
thf(fact_39_top__eq__inf__iff,axiom,
! [X: a > $o,Y: a > $o] :
( ( top_top_a_o
= ( inf_inf_a_o @ X @ Y ) )
= ( ( X = top_top_a_o )
& ( Y = top_top_a_o ) ) ) ).
% top_eq_inf_iff
thf(fact_40_top__eq__inf__iff,axiom,
! [X: a > a > $o,Y: a > a > $o] :
( ( top_top_a_a_o
= ( inf_inf_a_a_o @ X @ Y ) )
= ( ( X = top_top_a_a_o )
& ( Y = top_top_a_a_o ) ) ) ).
% top_eq_inf_iff
thf(fact_41_top__eq__inf__iff,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( top_to8063371432257647191od_a_a
= ( inf_in8905007599844390133od_a_a @ X @ Y ) )
= ( ( X = top_to8063371432257647191od_a_a )
& ( Y = top_to8063371432257647191od_a_a ) ) ) ).
% top_eq_inf_iff
thf(fact_42_top__eq__inf__iff,axiom,
! [X: set_a,Y: set_a] :
( ( top_top_set_a
= ( inf_inf_set_a @ X @ Y ) )
= ( ( X = top_top_set_a )
& ( Y = top_top_set_a ) ) ) ).
% top_eq_inf_iff
thf(fact_43_inf__top_Oeq__neutr__iff,axiom,
! [A2: set_se5735800977113168103od_a_a,B2: set_se5735800977113168103od_a_a] :
( ( ( inf_in3339382566020358357od_a_a @ A2 @ B2 )
= top_to1047947862415971895od_a_a )
= ( ( A2 = top_to1047947862415971895od_a_a )
& ( B2 = top_to1047947862415971895od_a_a ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_44_inf__top_Oeq__neutr__iff,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ( inf_inf_set_set_a @ A2 @ B2 )
= top_top_set_set_a )
= ( ( A2 = top_top_set_set_a )
& ( B2 = top_top_set_set_a ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_45_inf__top_Oeq__neutr__iff,axiom,
! [A2: product_prod_a_a > $o,B2: product_prod_a_a > $o] :
( ( ( inf_in2559554923042384936_a_a_o @ A2 @ B2 )
= top_to8687885267596698950_a_a_o )
= ( ( A2 = top_to8687885267596698950_a_a_o )
& ( B2 = top_to8687885267596698950_a_a_o ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_46_inf__top_Oeq__neutr__iff,axiom,
! [A2: a > $o,B2: a > $o] :
( ( ( inf_inf_a_o @ A2 @ B2 )
= top_top_a_o )
= ( ( A2 = top_top_a_o )
& ( B2 = top_top_a_o ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_47_inf__top_Oeq__neutr__iff,axiom,
! [A2: a > a > $o,B2: a > a > $o] :
( ( ( inf_inf_a_a_o @ A2 @ B2 )
= top_top_a_a_o )
= ( ( A2 = top_top_a_a_o )
& ( B2 = top_top_a_a_o ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_48_inf__top_Oeq__neutr__iff,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ A2 @ B2 )
= top_to8063371432257647191od_a_a )
= ( ( A2 = top_to8063371432257647191od_a_a )
& ( B2 = top_to8063371432257647191od_a_a ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_49_inf__top_Oeq__neutr__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ A2 @ B2 )
= top_top_set_a )
= ( ( A2 = top_top_set_a )
& ( B2 = top_top_set_a ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_50_inf__top_Oleft__neutral,axiom,
! [A2: set_se5735800977113168103od_a_a] :
( ( inf_in3339382566020358357od_a_a @ top_to1047947862415971895od_a_a @ A2 )
= A2 ) ).
% inf_top.left_neutral
thf(fact_51_inf__top_Oleft__neutral,axiom,
! [A2: set_set_a] :
( ( inf_inf_set_set_a @ top_top_set_set_a @ A2 )
= A2 ) ).
% inf_top.left_neutral
thf(fact_52_inf__top_Oleft__neutral,axiom,
! [A2: product_prod_a_a > $o] :
( ( inf_in2559554923042384936_a_a_o @ top_to8687885267596698950_a_a_o @ A2 )
= A2 ) ).
% inf_top.left_neutral
thf(fact_53_inf__top_Oleft__neutral,axiom,
! [A2: a > $o] :
( ( inf_inf_a_o @ top_top_a_o @ A2 )
= A2 ) ).
% inf_top.left_neutral
thf(fact_54_inf__top_Oleft__neutral,axiom,
! [A2: a > a > $o] :
( ( inf_inf_a_a_o @ top_top_a_a_o @ A2 )
= A2 ) ).
% inf_top.left_neutral
thf(fact_55_inf__top_Oleft__neutral,axiom,
! [A2: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ top_to8063371432257647191od_a_a @ A2 )
= A2 ) ).
% inf_top.left_neutral
thf(fact_56_inf__top_Oleft__neutral,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ top_top_set_a @ A2 )
= A2 ) ).
% inf_top.left_neutral
thf(fact_57_inf__top_Oneutr__eq__iff,axiom,
! [A2: set_se5735800977113168103od_a_a,B2: set_se5735800977113168103od_a_a] :
( ( top_to1047947862415971895od_a_a
= ( inf_in3339382566020358357od_a_a @ A2 @ B2 ) )
= ( ( A2 = top_to1047947862415971895od_a_a )
& ( B2 = top_to1047947862415971895od_a_a ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_58_inf__top_Oneutr__eq__iff,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( top_top_set_set_a
= ( inf_inf_set_set_a @ A2 @ B2 ) )
= ( ( A2 = top_top_set_set_a )
& ( B2 = top_top_set_set_a ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_59_inf__top_Oneutr__eq__iff,axiom,
! [A2: product_prod_a_a > $o,B2: product_prod_a_a > $o] :
( ( top_to8687885267596698950_a_a_o
= ( inf_in2559554923042384936_a_a_o @ A2 @ B2 ) )
= ( ( A2 = top_to8687885267596698950_a_a_o )
& ( B2 = top_to8687885267596698950_a_a_o ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_60_inf__top_Oneutr__eq__iff,axiom,
! [A2: a > $o,B2: a > $o] :
( ( top_top_a_o
= ( inf_inf_a_o @ A2 @ B2 ) )
= ( ( A2 = top_top_a_o )
& ( B2 = top_top_a_o ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_61_inf__top_Oneutr__eq__iff,axiom,
! [A2: a > a > $o,B2: a > a > $o] :
( ( top_top_a_a_o
= ( inf_inf_a_a_o @ A2 @ B2 ) )
= ( ( A2 = top_top_a_a_o )
& ( B2 = top_top_a_a_o ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_62_inf__top_Oneutr__eq__iff,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( top_to8063371432257647191od_a_a
= ( inf_in8905007599844390133od_a_a @ A2 @ B2 ) )
= ( ( A2 = top_to8063371432257647191od_a_a )
& ( B2 = top_to8063371432257647191od_a_a ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_63_inf__top_Oneutr__eq__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( top_top_set_a
= ( inf_inf_set_a @ A2 @ B2 ) )
= ( ( A2 = top_top_set_a )
& ( B2 = top_top_set_a ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_64_inf__top_Oright__neutral,axiom,
! [A2: set_se5735800977113168103od_a_a] :
( ( inf_in3339382566020358357od_a_a @ A2 @ top_to1047947862415971895od_a_a )
= A2 ) ).
% inf_top.right_neutral
thf(fact_65_inf__top_Oright__neutral,axiom,
! [A2: set_set_a] :
( ( inf_inf_set_set_a @ A2 @ top_top_set_set_a )
= A2 ) ).
% inf_top.right_neutral
thf(fact_66_inf__top_Oright__neutral,axiom,
! [A2: product_prod_a_a > $o] :
( ( inf_in2559554923042384936_a_a_o @ A2 @ top_to8687885267596698950_a_a_o )
= A2 ) ).
% inf_top.right_neutral
thf(fact_67_inf__top_Oright__neutral,axiom,
! [A2: a > $o] :
( ( inf_inf_a_o @ A2 @ top_top_a_o )
= A2 ) ).
% inf_top.right_neutral
thf(fact_68_inf__top_Oright__neutral,axiom,
! [A2: a > a > $o] :
( ( inf_inf_a_a_o @ A2 @ top_top_a_a_o )
= A2 ) ).
% inf_top.right_neutral
thf(fact_69_inf__top_Oright__neutral,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ top_top_set_a )
= A2 ) ).
% inf_top.right_neutral
thf(fact_70_inf__top_Oright__neutral,axiom,
! [A2: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ A2 @ top_to8063371432257647191od_a_a )
= A2 ) ).
% inf_top.right_neutral
thf(fact_71_assms_I1_J,axiom,
ord_less_eq_set_a @ b @ a2 ).
% assms(1)
thf(fact_72_UNIV__I,axiom,
! [X: set_set_a] : ( member_set_set_a @ X @ top_to4027821306633060462_set_a ) ).
% UNIV_I
thf(fact_73_UNIV__I,axiom,
! [X: set_Product_prod_a_a] : ( member1816616512716248880od_a_a @ X @ top_to1047947862415971895od_a_a ) ).
% UNIV_I
thf(fact_74_UNIV__I,axiom,
! [X: set_a] : ( member_set_a @ X @ top_top_set_set_a ) ).
% UNIV_I
thf(fact_75_UNIV__I,axiom,
! [X: a] : ( member_a @ X @ top_top_set_a ) ).
% UNIV_I
thf(fact_76_UNIV__I,axiom,
! [X: product_prod_a_a] : ( member1426531477525435216od_a_a @ X @ top_to8063371432257647191od_a_a ) ).
% UNIV_I
thf(fact_77_subsetI,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A )
=> ( member1426531477525435216od_a_a @ X2 @ B ) )
=> ( ord_le746702958409616551od_a_a @ A @ B ) ) ).
% subsetI
thf(fact_78_subsetI,axiom,
! [A: set_set_set_a,B: set_set_set_a] :
( ! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ A )
=> ( member_set_set_a @ X2 @ B ) )
=> ( ord_le5722252365846178494_set_a @ A @ B ) ) ).
% subsetI
thf(fact_79_subsetI,axiom,
! [A: set_se5735800977113168103od_a_a,B: set_se5735800977113168103od_a_a] :
( ! [X2: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X2 @ A )
=> ( member1816616512716248880od_a_a @ X2 @ B ) )
=> ( ord_le1995061765932249223od_a_a @ A @ B ) ) ).
% subsetI
thf(fact_80_subsetI,axiom,
! [A: set_set_a,B: set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( member_set_a @ X2 @ B ) )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% subsetI
thf(fact_81_subsetI,axiom,
! [A: set_a,B: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_a @ X2 @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% subsetI
thf(fact_82_subset__antisym,axiom,
! [A: set_se5735800977113168103od_a_a,B: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ A @ B )
=> ( ( ord_le1995061765932249223od_a_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_83_subset__antisym,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_84_subset__antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_85_inf__apply,axiom,
( inf_in2559554923042384936_a_a_o
= ( ^ [F: product_prod_a_a > $o,G: product_prod_a_a > $o,X3: product_prod_a_a] : ( inf_inf_o @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ).
% inf_apply
thf(fact_86_inf__apply,axiom,
( inf_inf_a_o
= ( ^ [F: a > $o,G: a > $o,X3: a] : ( inf_inf_o @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ).
% inf_apply
thf(fact_87_inf__apply,axiom,
( inf_inf_a_a_o
= ( ^ [F: a > a > $o,G: a > a > $o,X3: a] : ( inf_inf_a_o @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ).
% inf_apply
thf(fact_88_inf__right__idem,axiom,
! [X: set_se5735800977113168103od_a_a,Y: set_se5735800977113168103od_a_a] :
( ( inf_in3339382566020358357od_a_a @ ( inf_in3339382566020358357od_a_a @ X @ Y ) @ Y )
= ( inf_in3339382566020358357od_a_a @ X @ Y ) ) ).
% inf_right_idem
thf(fact_89_inf__right__idem,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ X @ Y ) @ Y )
= ( inf_inf_set_set_a @ X @ Y ) ) ).
% inf_right_idem
thf(fact_90_inf__right__idem,axiom,
! [X: product_prod_a_a > $o,Y: product_prod_a_a > $o] :
( ( inf_in2559554923042384936_a_a_o @ ( inf_in2559554923042384936_a_a_o @ X @ Y ) @ Y )
= ( inf_in2559554923042384936_a_a_o @ X @ Y ) ) ).
% inf_right_idem
thf(fact_91_inf__right__idem,axiom,
! [X: a > $o,Y: a > $o] :
( ( inf_inf_a_o @ ( inf_inf_a_o @ X @ Y ) @ Y )
= ( inf_inf_a_o @ X @ Y ) ) ).
% inf_right_idem
thf(fact_92_inf__right__idem,axiom,
! [X: a > a > $o,Y: a > a > $o] :
( ( inf_inf_a_a_o @ ( inf_inf_a_a_o @ X @ Y ) @ Y )
= ( inf_inf_a_a_o @ X @ Y ) ) ).
% inf_right_idem
thf(fact_93_inf__right__idem,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ ( inf_in8905007599844390133od_a_a @ X @ Y ) @ Y )
= ( inf_in8905007599844390133od_a_a @ X @ Y ) ) ).
% inf_right_idem
thf(fact_94_inf__right__idem,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y )
= ( inf_inf_set_a @ X @ Y ) ) ).
% inf_right_idem
thf(fact_95_inf_Oright__idem,axiom,
! [A2: set_se5735800977113168103od_a_a,B2: set_se5735800977113168103od_a_a] :
( ( inf_in3339382566020358357od_a_a @ ( inf_in3339382566020358357od_a_a @ A2 @ B2 ) @ B2 )
= ( inf_in3339382566020358357od_a_a @ A2 @ B2 ) ) ).
% inf.right_idem
thf(fact_96_inf_Oright__idem,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ A2 @ B2 ) @ B2 )
= ( inf_inf_set_set_a @ A2 @ B2 ) ) ).
% inf.right_idem
thf(fact_97_inf_Oright__idem,axiom,
! [A2: product_prod_a_a > $o,B2: product_prod_a_a > $o] :
( ( inf_in2559554923042384936_a_a_o @ ( inf_in2559554923042384936_a_a_o @ A2 @ B2 ) @ B2 )
= ( inf_in2559554923042384936_a_a_o @ A2 @ B2 ) ) ).
% inf.right_idem
thf(fact_98_inf_Oright__idem,axiom,
! [A2: a > $o,B2: a > $o] :
( ( inf_inf_a_o @ ( inf_inf_a_o @ A2 @ B2 ) @ B2 )
= ( inf_inf_a_o @ A2 @ B2 ) ) ).
% inf.right_idem
thf(fact_99_inf_Oright__idem,axiom,
! [A2: a > a > $o,B2: a > a > $o] :
( ( inf_inf_a_a_o @ ( inf_inf_a_a_o @ A2 @ B2 ) @ B2 )
= ( inf_inf_a_a_o @ A2 @ B2 ) ) ).
% inf.right_idem
thf(fact_100_inf_Oright__idem,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ ( inf_in8905007599844390133od_a_a @ A2 @ B2 ) @ B2 )
= ( inf_in8905007599844390133od_a_a @ A2 @ B2 ) ) ).
% inf.right_idem
thf(fact_101_inf_Oright__idem,axiom,
! [A2: set_a,B2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ B2 )
= ( inf_inf_set_a @ A2 @ B2 ) ) ).
% inf.right_idem
thf(fact_102_inf__left__idem,axiom,
! [X: set_se5735800977113168103od_a_a,Y: set_se5735800977113168103od_a_a] :
( ( inf_in3339382566020358357od_a_a @ X @ ( inf_in3339382566020358357od_a_a @ X @ Y ) )
= ( inf_in3339382566020358357od_a_a @ X @ Y ) ) ).
% inf_left_idem
thf(fact_103_inf__left__idem,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( inf_inf_set_set_a @ X @ ( inf_inf_set_set_a @ X @ Y ) )
= ( inf_inf_set_set_a @ X @ Y ) ) ).
% inf_left_idem
thf(fact_104_inf__left__idem,axiom,
! [X: product_prod_a_a > $o,Y: product_prod_a_a > $o] :
( ( inf_in2559554923042384936_a_a_o @ X @ ( inf_in2559554923042384936_a_a_o @ X @ Y ) )
= ( inf_in2559554923042384936_a_a_o @ X @ Y ) ) ).
% inf_left_idem
thf(fact_105_inf__left__idem,axiom,
! [X: a > $o,Y: a > $o] :
( ( inf_inf_a_o @ X @ ( inf_inf_a_o @ X @ Y ) )
= ( inf_inf_a_o @ X @ Y ) ) ).
% inf_left_idem
thf(fact_106_inf__left__idem,axiom,
! [X: a > a > $o,Y: a > a > $o] :
( ( inf_inf_a_a_o @ X @ ( inf_inf_a_a_o @ X @ Y ) )
= ( inf_inf_a_a_o @ X @ Y ) ) ).
% inf_left_idem
thf(fact_107_inf__left__idem,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ X @ ( inf_in8905007599844390133od_a_a @ X @ Y ) )
= ( inf_in8905007599844390133od_a_a @ X @ Y ) ) ).
% inf_left_idem
thf(fact_108_inf__left__idem,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
= ( inf_inf_set_a @ X @ Y ) ) ).
% inf_left_idem
thf(fact_109_inf_Oleft__idem,axiom,
! [A2: set_se5735800977113168103od_a_a,B2: set_se5735800977113168103od_a_a] :
( ( inf_in3339382566020358357od_a_a @ A2 @ ( inf_in3339382566020358357od_a_a @ A2 @ B2 ) )
= ( inf_in3339382566020358357od_a_a @ A2 @ B2 ) ) ).
% inf.left_idem
thf(fact_110_inf_Oleft__idem,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( inf_inf_set_set_a @ A2 @ ( inf_inf_set_set_a @ A2 @ B2 ) )
= ( inf_inf_set_set_a @ A2 @ B2 ) ) ).
% inf.left_idem
thf(fact_111_inf_Oleft__idem,axiom,
! [A2: product_prod_a_a > $o,B2: product_prod_a_a > $o] :
( ( inf_in2559554923042384936_a_a_o @ A2 @ ( inf_in2559554923042384936_a_a_o @ A2 @ B2 ) )
= ( inf_in2559554923042384936_a_a_o @ A2 @ B2 ) ) ).
% inf.left_idem
thf(fact_112_inf_Oleft__idem,axiom,
! [A2: a > $o,B2: a > $o] :
( ( inf_inf_a_o @ A2 @ ( inf_inf_a_o @ A2 @ B2 ) )
= ( inf_inf_a_o @ A2 @ B2 ) ) ).
% inf.left_idem
thf(fact_113_inf_Oleft__idem,axiom,
! [A2: a > a > $o,B2: a > a > $o] :
( ( inf_inf_a_a_o @ A2 @ ( inf_inf_a_a_o @ A2 @ B2 ) )
= ( inf_inf_a_a_o @ A2 @ B2 ) ) ).
% inf.left_idem
thf(fact_114_inf_Oleft__idem,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ A2 @ ( inf_in8905007599844390133od_a_a @ A2 @ B2 ) )
= ( inf_in8905007599844390133od_a_a @ A2 @ B2 ) ) ).
% inf.left_idem
thf(fact_115_inf_Oleft__idem,axiom,
! [A2: set_a,B2: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ A2 @ B2 ) )
= ( inf_inf_set_a @ A2 @ B2 ) ) ).
% inf.left_idem
thf(fact_116_inf__idem,axiom,
! [X: set_se5735800977113168103od_a_a] :
( ( inf_in3339382566020358357od_a_a @ X @ X )
= X ) ).
% inf_idem
thf(fact_117_inf__idem,axiom,
! [X: set_set_a] :
( ( inf_inf_set_set_a @ X @ X )
= X ) ).
% inf_idem
thf(fact_118_inf__idem,axiom,
! [X: product_prod_a_a > $o] :
( ( inf_in2559554923042384936_a_a_o @ X @ X )
= X ) ).
% inf_idem
thf(fact_119_inf__idem,axiom,
! [X: a > $o] :
( ( inf_inf_a_o @ X @ X )
= X ) ).
% inf_idem
thf(fact_120_inf__idem,axiom,
! [X: a > a > $o] :
( ( inf_inf_a_a_o @ X @ X )
= X ) ).
% inf_idem
thf(fact_121_inf__idem,axiom,
! [X: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ X @ X )
= X ) ).
% inf_idem
thf(fact_122_inf__idem,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ X )
= X ) ).
% inf_idem
thf(fact_123_inf_Oidem,axiom,
! [A2: set_se5735800977113168103od_a_a] :
( ( inf_in3339382566020358357od_a_a @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_124_inf_Oidem,axiom,
! [A2: set_set_a] :
( ( inf_inf_set_set_a @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_125_inf_Oidem,axiom,
! [A2: product_prod_a_a > $o] :
( ( inf_in2559554923042384936_a_a_o @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_126_inf_Oidem,axiom,
! [A2: a > $o] :
( ( inf_inf_a_o @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_127_inf_Oidem,axiom,
! [A2: a > a > $o] :
( ( inf_inf_a_a_o @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_128_inf_Oidem,axiom,
! [A2: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_129_inf_Oidem,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_130_Int__iff,axiom,
! [C: set_set_a,A: set_set_set_a,B: set_set_set_a] :
( ( member_set_set_a @ C @ ( inf_in1205276777018777868_set_a @ A @ B ) )
= ( ( member_set_set_a @ C @ A )
& ( member_set_set_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_131_Int__iff,axiom,
! [C: set_Product_prod_a_a,A: set_se5735800977113168103od_a_a,B: set_se5735800977113168103od_a_a] :
( ( member1816616512716248880od_a_a @ C @ ( inf_in3339382566020358357od_a_a @ A @ B ) )
= ( ( member1816616512716248880od_a_a @ C @ A )
& ( member1816616512716248880od_a_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_132_Int__iff,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) )
= ( ( member_set_a @ C @ A )
& ( member_set_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_133_Int__iff,axiom,
! [C: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A @ B ) )
= ( ( member1426531477525435216od_a_a @ C @ A )
& ( member1426531477525435216od_a_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_134_Int__iff,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
= ( ( member_a @ C @ A )
& ( member_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_135_IntI,axiom,
! [C: set_set_a,A: set_set_set_a,B: set_set_set_a] :
( ( member_set_set_a @ C @ A )
=> ( ( member_set_set_a @ C @ B )
=> ( member_set_set_a @ C @ ( inf_in1205276777018777868_set_a @ A @ B ) ) ) ) ).
% IntI
thf(fact_136_IntI,axiom,
! [C: set_Product_prod_a_a,A: set_se5735800977113168103od_a_a,B: set_se5735800977113168103od_a_a] :
( ( member1816616512716248880od_a_a @ C @ A )
=> ( ( member1816616512716248880od_a_a @ C @ B )
=> ( member1816616512716248880od_a_a @ C @ ( inf_in3339382566020358357od_a_a @ A @ B ) ) ) ) ).
% IntI
thf(fact_137_IntI,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ A )
=> ( ( member_set_a @ C @ B )
=> ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% IntI
thf(fact_138_IntI,axiom,
! [C: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ A )
=> ( ( member1426531477525435216od_a_a @ C @ B )
=> ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A @ B ) ) ) ) ).
% IntI
thf(fact_139_IntI,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ A )
=> ( ( member_a @ C @ B )
=> ( member_a @ C @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% IntI
thf(fact_140_inf_Obounded__iff,axiom,
! [A2: product_prod_a_a > $o,B2: product_prod_a_a > $o,C: product_prod_a_a > $o] :
( ( ord_le1591150415168442102_a_a_o @ A2 @ ( inf_in2559554923042384936_a_a_o @ B2 @ C ) )
= ( ( ord_le1591150415168442102_a_a_o @ A2 @ B2 )
& ( ord_le1591150415168442102_a_a_o @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_141_inf_Obounded__iff,axiom,
! [A2: set_se5735800977113168103od_a_a,B2: set_se5735800977113168103od_a_a,C: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ A2 @ ( inf_in3339382566020358357od_a_a @ B2 @ C ) )
= ( ( ord_le1995061765932249223od_a_a @ A2 @ B2 )
& ( ord_le1995061765932249223od_a_a @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_142_inf_Obounded__iff,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( inf_inf_set_set_a @ B2 @ C ) )
= ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
& ( ord_le3724670747650509150_set_a @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_143_inf_Obounded__iff,axiom,
! [A2: product_prod_a_a > product_prod_a_a > $o,B2: product_prod_a_a > product_prod_a_a > $o,C: product_prod_a_a > product_prod_a_a > $o] :
( ( ord_le540031911545281998_a_a_o @ A2 @ ( inf_in6214470387022126620_a_a_o @ B2 @ C ) )
= ( ( ord_le540031911545281998_a_a_o @ A2 @ B2 )
& ( ord_le540031911545281998_a_a_o @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_144_inf_Obounded__iff,axiom,
! [A2: a > $o,B2: a > $o,C: a > $o] :
( ( ord_less_eq_a_o @ A2 @ ( inf_inf_a_o @ B2 @ C ) )
= ( ( ord_less_eq_a_o @ A2 @ B2 )
& ( ord_less_eq_a_o @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_145_inf_Obounded__iff,axiom,
! [A2: a > a > $o,B2: a > a > $o,C: a > a > $o] :
( ( ord_less_eq_a_a_o @ A2 @ ( inf_inf_a_a_o @ B2 @ C ) )
= ( ( ord_less_eq_a_a_o @ A2 @ B2 )
& ( ord_less_eq_a_a_o @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_146_inf_Obounded__iff,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ ( inf_in8905007599844390133od_a_a @ B2 @ C ) )
= ( ( ord_le746702958409616551od_a_a @ A2 @ B2 )
& ( ord_le746702958409616551od_a_a @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_147_inf_Obounded__iff,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) )
= ( ( ord_less_eq_set_a @ A2 @ B2 )
& ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_148_le__inf__iff,axiom,
! [X: product_prod_a_a > $o,Y: product_prod_a_a > $o,Z: product_prod_a_a > $o] :
( ( ord_le1591150415168442102_a_a_o @ X @ ( inf_in2559554923042384936_a_a_o @ Y @ Z ) )
= ( ( ord_le1591150415168442102_a_a_o @ X @ Y )
& ( ord_le1591150415168442102_a_a_o @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_149_le__inf__iff,axiom,
! [X: set_se5735800977113168103od_a_a,Y: set_se5735800977113168103od_a_a,Z: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ X @ ( inf_in3339382566020358357od_a_a @ Y @ Z ) )
= ( ( ord_le1995061765932249223od_a_a @ X @ Y )
& ( ord_le1995061765932249223od_a_a @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_150_le__inf__iff,axiom,
! [X: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ ( inf_inf_set_set_a @ Y @ Z ) )
= ( ( ord_le3724670747650509150_set_a @ X @ Y )
& ( ord_le3724670747650509150_set_a @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_151_le__inf__iff,axiom,
! [X: product_prod_a_a > product_prod_a_a > $o,Y: product_prod_a_a > product_prod_a_a > $o,Z: product_prod_a_a > product_prod_a_a > $o] :
( ( ord_le540031911545281998_a_a_o @ X @ ( inf_in6214470387022126620_a_a_o @ Y @ Z ) )
= ( ( ord_le540031911545281998_a_a_o @ X @ Y )
& ( ord_le540031911545281998_a_a_o @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_152_le__inf__iff,axiom,
! [X: a > $o,Y: a > $o,Z: a > $o] :
( ( ord_less_eq_a_o @ X @ ( inf_inf_a_o @ Y @ Z ) )
= ( ( ord_less_eq_a_o @ X @ Y )
& ( ord_less_eq_a_o @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_153_le__inf__iff,axiom,
! [X: a > a > $o,Y: a > a > $o,Z: a > a > $o] :
( ( ord_less_eq_a_a_o @ X @ ( inf_inf_a_a_o @ Y @ Z ) )
= ( ( ord_less_eq_a_a_o @ X @ Y )
& ( ord_less_eq_a_a_o @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_154_le__inf__iff,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X @ ( inf_in8905007599844390133od_a_a @ Y @ Z ) )
= ( ( ord_le746702958409616551od_a_a @ X @ Y )
& ( ord_le746702958409616551od_a_a @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_155_le__inf__iff,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( ( ord_less_eq_set_a @ X @ Y )
& ( ord_less_eq_set_a @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_156_Int__subset__iff,axiom,
! [C2: set_se5735800977113168103od_a_a,A: set_se5735800977113168103od_a_a,B: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ C2 @ ( inf_in3339382566020358357od_a_a @ A @ B ) )
= ( ( ord_le1995061765932249223od_a_a @ C2 @ A )
& ( ord_le1995061765932249223od_a_a @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_157_Int__subset__iff,axiom,
! [C2: set_set_a,A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C2 @ ( inf_inf_set_set_a @ A @ B ) )
= ( ( ord_le3724670747650509150_set_a @ C2 @ A )
& ( ord_le3724670747650509150_set_a @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_158_Int__subset__iff,axiom,
! [C2: set_Product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ C2 @ ( inf_in8905007599844390133od_a_a @ A @ B ) )
= ( ( ord_le746702958409616551od_a_a @ C2 @ A )
& ( ord_le746702958409616551od_a_a @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_159_Int__subset__iff,axiom,
! [C2: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A @ B ) )
= ( ( ord_less_eq_set_a @ C2 @ A )
& ( ord_less_eq_set_a @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_160_assms_I2_J,axiom,
equiv_equiv_a @ a2 @ p ).
% assms(2)
thf(fact_161_in__mono,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,X: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( ( member1426531477525435216od_a_a @ X @ A )
=> ( member1426531477525435216od_a_a @ X @ B ) ) ) ).
% in_mono
thf(fact_162_in__mono,axiom,
! [A: set_set_set_a,B: set_set_set_a,X: set_set_a] :
( ( ord_le5722252365846178494_set_a @ A @ B )
=> ( ( member_set_set_a @ X @ A )
=> ( member_set_set_a @ X @ B ) ) ) ).
% in_mono
thf(fact_163_in__mono,axiom,
! [A: set_se5735800977113168103od_a_a,B: set_se5735800977113168103od_a_a,X: set_Product_prod_a_a] :
( ( ord_le1995061765932249223od_a_a @ A @ B )
=> ( ( member1816616512716248880od_a_a @ X @ A )
=> ( member1816616512716248880od_a_a @ X @ B ) ) ) ).
% in_mono
thf(fact_164_in__mono,axiom,
! [A: set_set_a,B: set_set_a,X: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( member_set_a @ X @ A )
=> ( member_set_a @ X @ B ) ) ) ).
% in_mono
thf(fact_165_in__mono,axiom,
! [A: set_a,B: set_a,X: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ X @ A )
=> ( member_a @ X @ B ) ) ) ).
% in_mono
thf(fact_166_subsetD,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( ( member1426531477525435216od_a_a @ C @ A )
=> ( member1426531477525435216od_a_a @ C @ B ) ) ) ).
% subsetD
thf(fact_167_subsetD,axiom,
! [A: set_set_set_a,B: set_set_set_a,C: set_set_a] :
( ( ord_le5722252365846178494_set_a @ A @ B )
=> ( ( member_set_set_a @ C @ A )
=> ( member_set_set_a @ C @ B ) ) ) ).
% subsetD
thf(fact_168_subsetD,axiom,
! [A: set_se5735800977113168103od_a_a,B: set_se5735800977113168103od_a_a,C: set_Product_prod_a_a] :
( ( ord_le1995061765932249223od_a_a @ A @ B )
=> ( ( member1816616512716248880od_a_a @ C @ A )
=> ( member1816616512716248880od_a_a @ C @ B ) ) ) ).
% subsetD
thf(fact_169_subsetD,axiom,
! [A: set_set_a,B: set_set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( member_set_a @ C @ A )
=> ( member_set_a @ C @ B ) ) ) ).
% subsetD
thf(fact_170_subsetD,axiom,
! [A: set_a,B: set_a,C: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% subsetD
thf(fact_171_equalityE,axiom,
! [A: set_se5735800977113168103od_a_a,B: set_se5735800977113168103od_a_a] :
( ( A = B )
=> ~ ( ( ord_le1995061765932249223od_a_a @ A @ B )
=> ~ ( ord_le1995061765932249223od_a_a @ B @ A ) ) ) ).
% equalityE
thf(fact_172_equalityE,axiom,
! [A: set_set_a,B: set_set_a] :
( ( A = B )
=> ~ ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ~ ( ord_le3724670747650509150_set_a @ B @ A ) ) ) ).
% equalityE
thf(fact_173_equalityE,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ~ ( ord_less_eq_set_a @ B @ A ) ) ) ).
% equalityE
thf(fact_174_subset__eq,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
! [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A3 )
=> ( member1426531477525435216od_a_a @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_175_subset__eq,axiom,
( ord_le5722252365846178494_set_a
= ( ^ [A3: set_set_set_a,B3: set_set_set_a] :
! [X3: set_set_a] :
( ( member_set_set_a @ X3 @ A3 )
=> ( member_set_set_a @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_176_subset__eq,axiom,
( ord_le1995061765932249223od_a_a
= ( ^ [A3: set_se5735800977113168103od_a_a,B3: set_se5735800977113168103od_a_a] :
! [X3: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X3 @ A3 )
=> ( member1816616512716248880od_a_a @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_177_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ A3 )
=> ( member_set_a @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_178_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ( member_a @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_179_equalityD1,axiom,
! [A: set_se5735800977113168103od_a_a,B: set_se5735800977113168103od_a_a] :
( ( A = B )
=> ( ord_le1995061765932249223od_a_a @ A @ B ) ) ).
% equalityD1
thf(fact_180_equalityD1,axiom,
! [A: set_set_a,B: set_set_a] :
( ( A = B )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% equalityD1
thf(fact_181_equalityD1,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% equalityD1
thf(fact_182_equalityD2,axiom,
! [A: set_se5735800977113168103od_a_a,B: set_se5735800977113168103od_a_a] :
( ( A = B )
=> ( ord_le1995061765932249223od_a_a @ B @ A ) ) ).
% equalityD2
thf(fact_183_equalityD2,axiom,
! [A: set_set_a,B: set_set_a] :
( ( A = B )
=> ( ord_le3724670747650509150_set_a @ B @ A ) ) ).
% equalityD2
thf(fact_184_equalityD2,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% equalityD2
thf(fact_185_subset__iff,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
! [T: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ T @ A3 )
=> ( member1426531477525435216od_a_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_186_subset__iff,axiom,
( ord_le5722252365846178494_set_a
= ( ^ [A3: set_set_set_a,B3: set_set_set_a] :
! [T: set_set_a] :
( ( member_set_set_a @ T @ A3 )
=> ( member_set_set_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_187_subset__iff,axiom,
( ord_le1995061765932249223od_a_a
= ( ^ [A3: set_se5735800977113168103od_a_a,B3: set_se5735800977113168103od_a_a] :
! [T: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ T @ A3 )
=> ( member1816616512716248880od_a_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_188_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
! [T: set_a] :
( ( member_set_a @ T @ A3 )
=> ( member_set_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_189_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
! [T: a] :
( ( member_a @ T @ A3 )
=> ( member_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_190_subset__refl,axiom,
! [A: set_se5735800977113168103od_a_a] : ( ord_le1995061765932249223od_a_a @ A @ A ) ).
% subset_refl
thf(fact_191_subset__refl,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ A ) ).
% subset_refl
thf(fact_192_subset__refl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% subset_refl
thf(fact_193_Collect__mono,axiom,
! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ! [X2: product_prod_a_a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le746702958409616551od_a_a @ ( collec3336397797384452498od_a_a @ P ) @ ( collec3336397797384452498od_a_a @ Q ) ) ) ).
% Collect_mono
thf(fact_194_Collect__mono,axiom,
! [P: set_Product_prod_a_a > $o,Q: set_Product_prod_a_a > $o] :
( ! [X2: set_Product_prod_a_a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le1995061765932249223od_a_a @ ( collec1673347964119250290od_a_a @ P ) @ ( collec1673347964119250290od_a_a @ Q ) ) ) ).
% Collect_mono
thf(fact_195_Collect__mono,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X2: set_a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).
% Collect_mono
thf(fact_196_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_197_subset__trans,axiom,
! [A: set_se5735800977113168103od_a_a,B: set_se5735800977113168103od_a_a,C2: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ A @ B )
=> ( ( ord_le1995061765932249223od_a_a @ B @ C2 )
=> ( ord_le1995061765932249223od_a_a @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_198_subset__trans,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C2 )
=> ( ord_le3724670747650509150_set_a @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_199_subset__trans,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_200_set__eq__subset,axiom,
( ( ^ [Y2: set_se5735800977113168103od_a_a,Z2: set_se5735800977113168103od_a_a] : ( Y2 = Z2 ) )
= ( ^ [A3: set_se5735800977113168103od_a_a,B3: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ A3 @ B3 )
& ( ord_le1995061765932249223od_a_a @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_201_set__eq__subset,axiom,
( ( ^ [Y2: set_set_a,Z2: set_set_a] : ( Y2 = Z2 ) )
= ( ^ [A3: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B3 )
& ( ord_le3724670747650509150_set_a @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_202_set__eq__subset,axiom,
( ( ^ [Y2: set_a,Z2: set_a] : ( Y2 = Z2 ) )
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_203_Collect__subset,axiom,
! [A: set_set_set_a,P: set_set_a > $o] :
( ord_le5722252365846178494_set_a
@ ( collect_set_set_a
@ ^ [X3: set_set_a] :
( ( member_set_set_a @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_204_Collect__subset,axiom,
! [A: set_Product_prod_a_a,P: product_prod_a_a > $o] :
( ord_le746702958409616551od_a_a
@ ( collec3336397797384452498od_a_a
@ ^ [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_205_Collect__subset,axiom,
! [A: set_se5735800977113168103od_a_a,P: set_Product_prod_a_a > $o] :
( ord_le1995061765932249223od_a_a
@ ( collec1673347964119250290od_a_a
@ ^ [X3: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_206_Collect__subset,axiom,
! [A: set_set_a,P: set_a > $o] :
( ord_le3724670747650509150_set_a
@ ( collect_set_a
@ ^ [X3: set_a] :
( ( member_set_a @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_207_Collect__subset,axiom,
! [A: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_208_Collect__mono__iff,axiom,
! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ( ord_le746702958409616551od_a_a @ ( collec3336397797384452498od_a_a @ P ) @ ( collec3336397797384452498od_a_a @ Q ) )
= ( ! [X3: product_prod_a_a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_209_Collect__mono__iff,axiom,
! [P: set_Product_prod_a_a > $o,Q: set_Product_prod_a_a > $o] :
( ( ord_le1995061765932249223od_a_a @ ( collec1673347964119250290od_a_a @ P ) @ ( collec1673347964119250290od_a_a @ Q ) )
= ( ! [X3: set_Product_prod_a_a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_210_Collect__mono__iff,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) )
= ( ! [X3: set_a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_211_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_212_Sigma__mono,axiom,
! [A: set_a,C2: set_a,B: a > set_a,D: a > set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_less_eq_set_a @ ( B @ X2 ) @ ( D @ X2 ) ) )
=> ( ord_le746702958409616551od_a_a @ ( product_Sigma_a_a @ A @ B ) @ ( product_Sigma_a_a @ C2 @ D ) ) ) ) ).
% Sigma_mono
thf(fact_213_Sigma__mono,axiom,
! [A: set_a,C2: set_a,B: a > set_set_a,D: a > set_set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_le3724670747650509150_set_a @ ( B @ X2 ) @ ( D @ X2 ) ) )
=> ( ord_le2652894967116761607_set_a @ ( produc7797748338049884712_set_a @ A @ B ) @ ( produc7797748338049884712_set_a @ C2 @ D ) ) ) ) ).
% Sigma_mono
thf(fact_214_Sigma__mono,axiom,
! [A: set_set_a,C2: set_set_a,B: set_a > set_a,D: set_a > set_a] :
( ( ord_le3724670747650509150_set_a @ A @ C2 )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( ord_less_eq_set_a @ ( B @ X2 ) @ ( D @ X2 ) ) )
=> ( ord_le7899191993508361031et_a_a @ ( produc8014260396509227880et_a_a @ A @ B ) @ ( produc8014260396509227880et_a_a @ C2 @ D ) ) ) ) ).
% Sigma_mono
thf(fact_215_Sigma__mono,axiom,
! [A: set_set_set_a,C2: set_set_set_a,B: set_set_a > set_a,D: set_set_a > set_a] :
( ( ord_le5722252365846178494_set_a @ A @ C2 )
=> ( ! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ A )
=> ( ord_less_eq_set_a @ ( B @ X2 ) @ ( D @ X2 ) ) )
=> ( ord_le2152028902748708327et_a_a @ ( produc3336215955681524232et_a_a @ A @ B ) @ ( produc3336215955681524232et_a_a @ C2 @ D ) ) ) ) ).
% Sigma_mono
thf(fact_216_Sigma__mono,axiom,
! [A: set_Product_prod_a_a,C2: set_Product_prod_a_a,B: product_prod_a_a > set_a,D: product_prod_a_a > set_a] :
( ( ord_le746702958409616551od_a_a @ A @ C2 )
=> ( ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A )
=> ( ord_less_eq_set_a @ ( B @ X2 ) @ ( D @ X2 ) ) )
=> ( ord_le3461320655694136638_a_a_a @ ( produc2379640491490746847_a_a_a @ A @ B ) @ ( produc2379640491490746847_a_a_a @ C2 @ D ) ) ) ) ).
% Sigma_mono
thf(fact_217_Sigma__mono,axiom,
! [A: set_a,C2: set_a,B: a > set_Product_prod_a_a,D: a > set_Product_prod_a_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_le746702958409616551od_a_a @ ( B @ X2 ) @ ( D @ X2 ) ) )
=> ( ord_le114883831454073552od_a_a @ ( produc6342321021181284593od_a_a @ A @ B ) @ ( produc6342321021181284593od_a_a @ C2 @ D ) ) ) ) ).
% Sigma_mono
thf(fact_218_Sigma__mono,axiom,
! [A: set_set_a,C2: set_set_a,B: set_a > set_set_a,D: set_a > set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ C2 )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( ord_le3724670747650509150_set_a @ ( B @ X2 ) @ ( D @ X2 ) ) )
=> ( ord_le8376522849517564071_set_a @ ( produc6033315442965015752_set_a @ A @ B ) @ ( produc6033315442965015752_set_a @ C2 @ D ) ) ) ) ).
% Sigma_mono
thf(fact_219_Sigma__mono,axiom,
! [A: set_Product_prod_a_a,C2: set_Product_prod_a_a,B: product_prod_a_a > set_set_a,D: product_prod_a_a > set_set_a] :
( ( ord_le746702958409616551od_a_a @ A @ C2 )
=> ( ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A )
=> ( ord_le3724670747650509150_set_a @ ( B @ X2 ) @ ( D @ X2 ) ) )
=> ( ord_le7000946053975811742_set_a @ ( produc1070844763932510015_set_a @ A @ B ) @ ( produc1070844763932510015_set_a @ C2 @ D ) ) ) ) ).
% Sigma_mono
thf(fact_220_Sigma__mono,axiom,
! [A: set_set_set_a,C2: set_set_set_a,B: set_set_a > set_set_a,D: set_set_a > set_set_a] :
( ( ord_le5722252365846178494_set_a @ A @ C2 )
=> ( ! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ A )
=> ( ord_le3724670747650509150_set_a @ ( B @ X2 ) @ ( D @ X2 ) ) )
=> ( ord_le7603424477761657671_set_a @ ( produc2571717186284272488_set_a @ A @ B ) @ ( produc2571717186284272488_set_a @ C2 @ D ) ) ) ) ).
% Sigma_mono
thf(fact_221_Sigma__mono,axiom,
! [A: set_a,C2: set_a,B: a > set_se5735800977113168103od_a_a,D: a > set_se5735800977113168103od_a_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_le1995061765932249223od_a_a @ ( B @ X2 ) @ ( D @ X2 ) ) )
=> ( ord_le3069280993441775280od_a_a @ ( produc5663638008720543953od_a_a @ A @ B ) @ ( produc5663638008720543953od_a_a @ C2 @ D ) ) ) ) ).
% Sigma_mono
thf(fact_222_Times__subset__cancel2,axiom,
! [X: a,C2: set_a,A: set_a,B: set_a] :
( ( member_a @ X @ C2 )
=> ( ( ord_le746702958409616551od_a_a
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : C2 )
@ ( product_Sigma_a_a @ B
@ ^ [Uu: a] : C2 ) )
= ( ord_less_eq_set_a @ A @ B ) ) ) ).
% Times_subset_cancel2
thf(fact_223_Times__subset__cancel2,axiom,
! [X: set_a,C2: set_set_a,A: set_a,B: set_a] :
( ( member_set_a @ X @ C2 )
=> ( ( ord_le2652894967116761607_set_a
@ ( produc7797748338049884712_set_a @ A
@ ^ [Uu: a] : C2 )
@ ( produc7797748338049884712_set_a @ B
@ ^ [Uu: a] : C2 ) )
= ( ord_less_eq_set_a @ A @ B ) ) ) ).
% Times_subset_cancel2
thf(fact_224_Times__subset__cancel2,axiom,
! [X: a,C2: set_a,A: set_set_a,B: set_set_a] :
( ( member_a @ X @ C2 )
=> ( ( ord_le7899191993508361031et_a_a
@ ( produc8014260396509227880et_a_a @ A
@ ^ [Uu: set_a] : C2 )
@ ( produc8014260396509227880et_a_a @ B
@ ^ [Uu: set_a] : C2 ) )
= ( ord_le3724670747650509150_set_a @ A @ B ) ) ) ).
% Times_subset_cancel2
thf(fact_225_Times__subset__cancel2,axiom,
! [X: a,C2: set_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member_a @ X @ C2 )
=> ( ( ord_le3461320655694136638_a_a_a
@ ( produc2379640491490746847_a_a_a @ A
@ ^ [Uu: product_prod_a_a] : C2 )
@ ( produc2379640491490746847_a_a_a @ B
@ ^ [Uu: product_prod_a_a] : C2 ) )
= ( ord_le746702958409616551od_a_a @ A @ B ) ) ) ).
% Times_subset_cancel2
thf(fact_226_Times__subset__cancel2,axiom,
! [X: set_set_a,C2: set_set_set_a,A: set_a,B: set_a] :
( ( member_set_set_a @ X @ C2 )
=> ( ( ord_le5210530899523273575_set_a
@ ( produc9054356137160646536_set_a @ A
@ ^ [Uu: a] : C2 )
@ ( produc9054356137160646536_set_a @ B
@ ^ [Uu: a] : C2 ) )
= ( ord_less_eq_set_a @ A @ B ) ) ) ).
% Times_subset_cancel2
thf(fact_227_Times__subset__cancel2,axiom,
! [X: product_prod_a_a,C2: set_Product_prod_a_a,A: set_a,B: set_a] :
( ( member1426531477525435216od_a_a @ X @ C2 )
=> ( ( ord_le114883831454073552od_a_a
@ ( produc6342321021181284593od_a_a @ A
@ ^ [Uu: a] : C2 )
@ ( produc6342321021181284593od_a_a @ B
@ ^ [Uu: a] : C2 ) )
= ( ord_less_eq_set_a @ A @ B ) ) ) ).
% Times_subset_cancel2
thf(fact_228_Times__subset__cancel2,axiom,
! [X: set_a,C2: set_set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ X @ C2 )
=> ( ( ord_le8376522849517564071_set_a
@ ( produc6033315442965015752_set_a @ A
@ ^ [Uu: set_a] : C2 )
@ ( produc6033315442965015752_set_a @ B
@ ^ [Uu: set_a] : C2 ) )
= ( ord_le3724670747650509150_set_a @ A @ B ) ) ) ).
% Times_subset_cancel2
thf(fact_229_Times__subset__cancel2,axiom,
! [X: set_Product_prod_a_a,C2: set_se5735800977113168103od_a_a,A: set_a,B: set_a] :
( ( member1816616512716248880od_a_a @ X @ C2 )
=> ( ( ord_le3069280993441775280od_a_a
@ ( produc5663638008720543953od_a_a @ A
@ ^ [Uu: a] : C2 )
@ ( produc5663638008720543953od_a_a @ B
@ ^ [Uu: a] : C2 ) )
= ( ord_less_eq_set_a @ A @ B ) ) ) ).
% Times_subset_cancel2
thf(fact_230_Times__subset__cancel2,axiom,
! [X: a,C2: set_a,A: set_se5735800977113168103od_a_a,B: set_se5735800977113168103od_a_a] :
( ( member_a @ X @ C2 )
=> ( ( ord_le6243934957694274910_a_a_a
@ ( produc5649525414689246719_a_a_a @ A
@ ^ [Uu: set_Product_prod_a_a] : C2 )
@ ( produc5649525414689246719_a_a_a @ B
@ ^ [Uu: set_Product_prod_a_a] : C2 ) )
= ( ord_le1995061765932249223od_a_a @ A @ B ) ) ) ).
% Times_subset_cancel2
thf(fact_231_Times__subset__cancel2,axiom,
! [X: product_prod_a_a,C2: set_Product_prod_a_a,A: set_set_a,B: set_set_a] :
( ( member1426531477525435216od_a_a @ X @ C2 )
=> ( ( ord_le1380148221811214704od_a_a
@ ( produc8919425190596410257od_a_a @ A
@ ^ [Uu: set_a] : C2 )
@ ( produc8919425190596410257od_a_a @ B
@ ^ [Uu: set_a] : C2 ) )
= ( ord_le3724670747650509150_set_a @ A @ B ) ) ) ).
% Times_subset_cancel2
thf(fact_232_top__set__def,axiom,
( top_to1047947862415971895od_a_a
= ( collec1673347964119250290od_a_a @ top_to8405625822285162854_a_a_o ) ) ).
% top_set_def
thf(fact_233_top__set__def,axiom,
( top_top_set_set_a
= ( collect_set_a @ top_top_set_a_o ) ) ).
% top_set_def
thf(fact_234_top__set__def,axiom,
( top_top_set_a
= ( collect_a @ top_top_a_o ) ) ).
% top_set_def
thf(fact_235_top__set__def,axiom,
( top_to8063371432257647191od_a_a
= ( collec3336397797384452498od_a_a @ top_to8687885267596698950_a_a_o ) ) ).
% top_set_def
thf(fact_236_subset__UNIV,axiom,
! [A: set_se5735800977113168103od_a_a] : ( ord_le1995061765932249223od_a_a @ A @ top_to1047947862415971895od_a_a ) ).
% subset_UNIV
thf(fact_237_subset__UNIV,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ top_top_set_set_a ) ).
% subset_UNIV
thf(fact_238_subset__UNIV,axiom,
! [A: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ A @ top_to8063371432257647191od_a_a ) ).
% subset_UNIV
thf(fact_239_subset__UNIV,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ top_top_set_a ) ).
% subset_UNIV
thf(fact_240_mem__Collect__eq,axiom,
! [A2: set_set_a,P: set_set_a > $o] :
( ( member_set_set_a @ A2 @ ( collect_set_set_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_241_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_242_mem__Collect__eq,axiom,
! [A2: product_prod_a_a,P: product_prod_a_a > $o] :
( ( member1426531477525435216od_a_a @ A2 @ ( collec3336397797384452498od_a_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_243_mem__Collect__eq,axiom,
! [A2: set_Product_prod_a_a,P: set_Product_prod_a_a > $o] :
( ( member1816616512716248880od_a_a @ A2 @ ( collec1673347964119250290od_a_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_244_mem__Collect__eq,axiom,
! [A2: set_a,P: set_a > $o] :
( ( member_set_a @ A2 @ ( collect_set_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_245_Collect__mem__eq,axiom,
! [A: set_set_set_a] :
( ( collect_set_set_a
@ ^ [X3: set_set_a] : ( member_set_set_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_246_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_247_Collect__mem__eq,axiom,
! [A: set_Product_prod_a_a] :
( ( collec3336397797384452498od_a_a
@ ^ [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_248_Collect__mem__eq,axiom,
! [A: set_se5735800977113168103od_a_a] :
( ( collec1673347964119250290od_a_a
@ ^ [X3: set_Product_prod_a_a] : ( member1816616512716248880od_a_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_249_Collect__mem__eq,axiom,
! [A: set_set_a] :
( ( collect_set_a
@ ^ [X3: set_a] : ( member_set_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_250_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_251_Collect__cong,axiom,
! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ! [X2: product_prod_a_a] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collec3336397797384452498od_a_a @ P )
= ( collec3336397797384452498od_a_a @ Q ) ) ) ).
% Collect_cong
thf(fact_252_Collect__cong,axiom,
! [P: set_Product_prod_a_a > $o,Q: set_Product_prod_a_a > $o] :
( ! [X2: set_Product_prod_a_a] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collec1673347964119250290od_a_a @ P )
= ( collec1673347964119250290od_a_a @ Q ) ) ) ).
% Collect_cong
thf(fact_253_Collect__cong,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X2: set_a] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_set_a @ P )
= ( collect_set_a @ Q ) ) ) ).
% Collect_cong
thf(fact_254_inf_OcoboundedI2,axiom,
! [B2: product_prod_a_a > $o,C: product_prod_a_a > $o,A2: product_prod_a_a > $o] :
( ( ord_le1591150415168442102_a_a_o @ B2 @ C )
=> ( ord_le1591150415168442102_a_a_o @ ( inf_in2559554923042384936_a_a_o @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_255_inf_OcoboundedI2,axiom,
! [B2: set_se5735800977113168103od_a_a,C: set_se5735800977113168103od_a_a,A2: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ B2 @ C )
=> ( ord_le1995061765932249223od_a_a @ ( inf_in3339382566020358357od_a_a @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_256_inf_OcoboundedI2,axiom,
! [B2: set_set_a,C: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_257_inf_OcoboundedI2,axiom,
! [B2: product_prod_a_a > product_prod_a_a > $o,C: product_prod_a_a > product_prod_a_a > $o,A2: product_prod_a_a > product_prod_a_a > $o] :
( ( ord_le540031911545281998_a_a_o @ B2 @ C )
=> ( ord_le540031911545281998_a_a_o @ ( inf_in6214470387022126620_a_a_o @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_258_inf_OcoboundedI2,axiom,
! [B2: a > $o,C: a > $o,A2: a > $o] :
( ( ord_less_eq_a_o @ B2 @ C )
=> ( ord_less_eq_a_o @ ( inf_inf_a_o @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_259_inf_OcoboundedI2,axiom,
! [B2: a > a > $o,C: a > a > $o,A2: a > a > $o] :
( ( ord_less_eq_a_a_o @ B2 @ C )
=> ( ord_less_eq_a_a_o @ ( inf_inf_a_a_o @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_260_inf_OcoboundedI2,axiom,
! [B2: set_Product_prod_a_a,C: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B2 @ C )
=> ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_261_inf_OcoboundedI2,axiom,
! [B2: set_a,C: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_262_inf_OcoboundedI1,axiom,
! [A2: product_prod_a_a > $o,C: product_prod_a_a > $o,B2: product_prod_a_a > $o] :
( ( ord_le1591150415168442102_a_a_o @ A2 @ C )
=> ( ord_le1591150415168442102_a_a_o @ ( inf_in2559554923042384936_a_a_o @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_263_inf_OcoboundedI1,axiom,
! [A2: set_se5735800977113168103od_a_a,C: set_se5735800977113168103od_a_a,B2: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ A2 @ C )
=> ( ord_le1995061765932249223od_a_a @ ( inf_in3339382566020358357od_a_a @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_264_inf_OcoboundedI1,axiom,
! [A2: set_set_a,C: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ C )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_265_inf_OcoboundedI1,axiom,
! [A2: product_prod_a_a > product_prod_a_a > $o,C: product_prod_a_a > product_prod_a_a > $o,B2: product_prod_a_a > product_prod_a_a > $o] :
( ( ord_le540031911545281998_a_a_o @ A2 @ C )
=> ( ord_le540031911545281998_a_a_o @ ( inf_in6214470387022126620_a_a_o @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_266_inf_OcoboundedI1,axiom,
! [A2: a > $o,C: a > $o,B2: a > $o] :
( ( ord_less_eq_a_o @ A2 @ C )
=> ( ord_less_eq_a_o @ ( inf_inf_a_o @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_267_inf_OcoboundedI1,axiom,
! [A2: a > a > $o,C: a > a > $o,B2: a > a > $o] :
( ( ord_less_eq_a_a_o @ A2 @ C )
=> ( ord_less_eq_a_a_o @ ( inf_inf_a_a_o @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_268_inf_OcoboundedI1,axiom,
! [A2: set_Product_prod_a_a,C: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ C )
=> ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_269_inf_OcoboundedI1,axiom,
! [A2: set_a,C: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_270_inf_Oabsorb__iff2,axiom,
( ord_le1591150415168442102_a_a_o
= ( ^ [B4: product_prod_a_a > $o,A4: product_prod_a_a > $o] :
( ( inf_in2559554923042384936_a_a_o @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_271_inf_Oabsorb__iff2,axiom,
( ord_le1995061765932249223od_a_a
= ( ^ [B4: set_se5735800977113168103od_a_a,A4: set_se5735800977113168103od_a_a] :
( ( inf_in3339382566020358357od_a_a @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_272_inf_Oabsorb__iff2,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [B4: set_set_a,A4: set_set_a] :
( ( inf_inf_set_set_a @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_273_inf_Oabsorb__iff2,axiom,
( ord_le540031911545281998_a_a_o
= ( ^ [B4: product_prod_a_a > product_prod_a_a > $o,A4: product_prod_a_a > product_prod_a_a > $o] :
( ( inf_in6214470387022126620_a_a_o @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_274_inf_Oabsorb__iff2,axiom,
( ord_less_eq_a_o
= ( ^ [B4: a > $o,A4: a > $o] :
( ( inf_inf_a_o @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_275_inf_Oabsorb__iff2,axiom,
( ord_less_eq_a_a_o
= ( ^ [B4: a > a > $o,A4: a > a > $o] :
( ( inf_inf_a_a_o @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_276_inf_Oabsorb__iff2,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [B4: set_Product_prod_a_a,A4: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_277_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A4: set_a] :
( ( inf_inf_set_a @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_278_inf_Oabsorb__iff1,axiom,
( ord_le1591150415168442102_a_a_o
= ( ^ [A4: product_prod_a_a > $o,B4: product_prod_a_a > $o] :
( ( inf_in2559554923042384936_a_a_o @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_279_inf_Oabsorb__iff1,axiom,
( ord_le1995061765932249223od_a_a
= ( ^ [A4: set_se5735800977113168103od_a_a,B4: set_se5735800977113168103od_a_a] :
( ( inf_in3339382566020358357od_a_a @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_280_inf_Oabsorb__iff1,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] :
( ( inf_inf_set_set_a @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_281_inf_Oabsorb__iff1,axiom,
( ord_le540031911545281998_a_a_o
= ( ^ [A4: product_prod_a_a > product_prod_a_a > $o,B4: product_prod_a_a > product_prod_a_a > $o] :
( ( inf_in6214470387022126620_a_a_o @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_282_inf_Oabsorb__iff1,axiom,
( ord_less_eq_a_o
= ( ^ [A4: a > $o,B4: a > $o] :
( ( inf_inf_a_o @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_283_inf_Oabsorb__iff1,axiom,
( ord_less_eq_a_a_o
= ( ^ [A4: a > a > $o,B4: a > a > $o] :
( ( inf_inf_a_a_o @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_284_inf_Oabsorb__iff1,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A4: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_285_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( inf_inf_set_a @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_286_inf_Ocobounded2,axiom,
! [A2: product_prod_a_a > $o,B2: product_prod_a_a > $o] : ( ord_le1591150415168442102_a_a_o @ ( inf_in2559554923042384936_a_a_o @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_287_inf_Ocobounded2,axiom,
! [A2: set_se5735800977113168103od_a_a,B2: set_se5735800977113168103od_a_a] : ( ord_le1995061765932249223od_a_a @ ( inf_in3339382566020358357od_a_a @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_288_inf_Ocobounded2,axiom,
! [A2: set_set_a,B2: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_289_inf_Ocobounded2,axiom,
! [A2: product_prod_a_a > product_prod_a_a > $o,B2: product_prod_a_a > product_prod_a_a > $o] : ( ord_le540031911545281998_a_a_o @ ( inf_in6214470387022126620_a_a_o @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_290_inf_Ocobounded2,axiom,
! [A2: a > $o,B2: a > $o] : ( ord_less_eq_a_o @ ( inf_inf_a_o @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_291_inf_Ocobounded2,axiom,
! [A2: a > a > $o,B2: a > a > $o] : ( ord_less_eq_a_a_o @ ( inf_inf_a_a_o @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_292_inf_Ocobounded2,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_293_inf_Ocobounded2,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_294_inf_Ocobounded1,axiom,
! [A2: product_prod_a_a > $o,B2: product_prod_a_a > $o] : ( ord_le1591150415168442102_a_a_o @ ( inf_in2559554923042384936_a_a_o @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_295_inf_Ocobounded1,axiom,
! [A2: set_se5735800977113168103od_a_a,B2: set_se5735800977113168103od_a_a] : ( ord_le1995061765932249223od_a_a @ ( inf_in3339382566020358357od_a_a @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_296_inf_Ocobounded1,axiom,
! [A2: set_set_a,B2: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_297_inf_Ocobounded1,axiom,
! [A2: product_prod_a_a > product_prod_a_a > $o,B2: product_prod_a_a > product_prod_a_a > $o] : ( ord_le540031911545281998_a_a_o @ ( inf_in6214470387022126620_a_a_o @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_298_inf_Ocobounded1,axiom,
! [A2: a > $o,B2: a > $o] : ( ord_less_eq_a_o @ ( inf_inf_a_o @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_299_inf_Ocobounded1,axiom,
! [A2: a > a > $o,B2: a > a > $o] : ( ord_less_eq_a_a_o @ ( inf_inf_a_a_o @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_300_inf_Ocobounded1,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_301_inf_Ocobounded1,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_302_inf_Oorder__iff,axiom,
( ord_le1591150415168442102_a_a_o
= ( ^ [A4: product_prod_a_a > $o,B4: product_prod_a_a > $o] :
( A4
= ( inf_in2559554923042384936_a_a_o @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_303_inf_Oorder__iff,axiom,
( ord_le1995061765932249223od_a_a
= ( ^ [A4: set_se5735800977113168103od_a_a,B4: set_se5735800977113168103od_a_a] :
( A4
= ( inf_in3339382566020358357od_a_a @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_304_inf_Oorder__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] :
( A4
= ( inf_inf_set_set_a @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_305_inf_Oorder__iff,axiom,
( ord_le540031911545281998_a_a_o
= ( ^ [A4: product_prod_a_a > product_prod_a_a > $o,B4: product_prod_a_a > product_prod_a_a > $o] :
( A4
= ( inf_in6214470387022126620_a_a_o @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_306_inf_Oorder__iff,axiom,
( ord_less_eq_a_o
= ( ^ [A4: a > $o,B4: a > $o] :
( A4
= ( inf_inf_a_o @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_307_inf_Oorder__iff,axiom,
( ord_less_eq_a_a_o
= ( ^ [A4: a > a > $o,B4: a > a > $o] :
( A4
= ( inf_inf_a_a_o @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_308_inf_Oorder__iff,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A4: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( A4
= ( inf_in8905007599844390133od_a_a @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_309_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( A4
= ( inf_inf_set_a @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_310_inf__greatest,axiom,
! [X: product_prod_a_a > $o,Y: product_prod_a_a > $o,Z: product_prod_a_a > $o] :
( ( ord_le1591150415168442102_a_a_o @ X @ Y )
=> ( ( ord_le1591150415168442102_a_a_o @ X @ Z )
=> ( ord_le1591150415168442102_a_a_o @ X @ ( inf_in2559554923042384936_a_a_o @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_311_inf__greatest,axiom,
! [X: set_se5735800977113168103od_a_a,Y: set_se5735800977113168103od_a_a,Z: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ X @ Y )
=> ( ( ord_le1995061765932249223od_a_a @ X @ Z )
=> ( ord_le1995061765932249223od_a_a @ X @ ( inf_in3339382566020358357od_a_a @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_312_inf__greatest,axiom,
! [X: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y )
=> ( ( ord_le3724670747650509150_set_a @ X @ Z )
=> ( ord_le3724670747650509150_set_a @ X @ ( inf_inf_set_set_a @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_313_inf__greatest,axiom,
! [X: product_prod_a_a > product_prod_a_a > $o,Y: product_prod_a_a > product_prod_a_a > $o,Z: product_prod_a_a > product_prod_a_a > $o] :
( ( ord_le540031911545281998_a_a_o @ X @ Y )
=> ( ( ord_le540031911545281998_a_a_o @ X @ Z )
=> ( ord_le540031911545281998_a_a_o @ X @ ( inf_in6214470387022126620_a_a_o @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_314_inf__greatest,axiom,
! [X: a > $o,Y: a > $o,Z: a > $o] :
( ( ord_less_eq_a_o @ X @ Y )
=> ( ( ord_less_eq_a_o @ X @ Z )
=> ( ord_less_eq_a_o @ X @ ( inf_inf_a_o @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_315_inf__greatest,axiom,
! [X: a > a > $o,Y: a > a > $o,Z: a > a > $o] :
( ( ord_less_eq_a_a_o @ X @ Y )
=> ( ( ord_less_eq_a_a_o @ X @ Z )
=> ( ord_less_eq_a_a_o @ X @ ( inf_inf_a_a_o @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_316_inf__greatest,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X @ Y )
=> ( ( ord_le746702958409616551od_a_a @ X @ Z )
=> ( ord_le746702958409616551od_a_a @ X @ ( inf_in8905007599844390133od_a_a @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_317_inf__greatest,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ X @ Z )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_318_inf_OboundedI,axiom,
! [A2: product_prod_a_a > $o,B2: product_prod_a_a > $o,C: product_prod_a_a > $o] :
( ( ord_le1591150415168442102_a_a_o @ A2 @ B2 )
=> ( ( ord_le1591150415168442102_a_a_o @ A2 @ C )
=> ( ord_le1591150415168442102_a_a_o @ A2 @ ( inf_in2559554923042384936_a_a_o @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_319_inf_OboundedI,axiom,
! [A2: set_se5735800977113168103od_a_a,B2: set_se5735800977113168103od_a_a,C: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ A2 @ B2 )
=> ( ( ord_le1995061765932249223od_a_a @ A2 @ C )
=> ( ord_le1995061765932249223od_a_a @ A2 @ ( inf_in3339382566020358357od_a_a @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_320_inf_OboundedI,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ C )
=> ( ord_le3724670747650509150_set_a @ A2 @ ( inf_inf_set_set_a @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_321_inf_OboundedI,axiom,
! [A2: product_prod_a_a > product_prod_a_a > $o,B2: product_prod_a_a > product_prod_a_a > $o,C: product_prod_a_a > product_prod_a_a > $o] :
( ( ord_le540031911545281998_a_a_o @ A2 @ B2 )
=> ( ( ord_le540031911545281998_a_a_o @ A2 @ C )
=> ( ord_le540031911545281998_a_a_o @ A2 @ ( inf_in6214470387022126620_a_a_o @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_322_inf_OboundedI,axiom,
! [A2: a > $o,B2: a > $o,C: a > $o] :
( ( ord_less_eq_a_o @ A2 @ B2 )
=> ( ( ord_less_eq_a_o @ A2 @ C )
=> ( ord_less_eq_a_o @ A2 @ ( inf_inf_a_o @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_323_inf_OboundedI,axiom,
! [A2: a > a > $o,B2: a > a > $o,C: a > a > $o] :
( ( ord_less_eq_a_a_o @ A2 @ B2 )
=> ( ( ord_less_eq_a_a_o @ A2 @ C )
=> ( ord_less_eq_a_a_o @ A2 @ ( inf_inf_a_a_o @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_324_inf_OboundedI,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B2 )
=> ( ( ord_le746702958409616551od_a_a @ A2 @ C )
=> ( ord_le746702958409616551od_a_a @ A2 @ ( inf_in8905007599844390133od_a_a @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_325_inf_OboundedI,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ C )
=> ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_326_inf_OboundedE,axiom,
! [A2: product_prod_a_a > $o,B2: product_prod_a_a > $o,C: product_prod_a_a > $o] :
( ( ord_le1591150415168442102_a_a_o @ A2 @ ( inf_in2559554923042384936_a_a_o @ B2 @ C ) )
=> ~ ( ( ord_le1591150415168442102_a_a_o @ A2 @ B2 )
=> ~ ( ord_le1591150415168442102_a_a_o @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_327_inf_OboundedE,axiom,
! [A2: set_se5735800977113168103od_a_a,B2: set_se5735800977113168103od_a_a,C: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ A2 @ ( inf_in3339382566020358357od_a_a @ B2 @ C ) )
=> ~ ( ( ord_le1995061765932249223od_a_a @ A2 @ B2 )
=> ~ ( ord_le1995061765932249223od_a_a @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_328_inf_OboundedE,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( inf_inf_set_set_a @ B2 @ C ) )
=> ~ ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ~ ( ord_le3724670747650509150_set_a @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_329_inf_OboundedE,axiom,
! [A2: product_prod_a_a > product_prod_a_a > $o,B2: product_prod_a_a > product_prod_a_a > $o,C: product_prod_a_a > product_prod_a_a > $o] :
( ( ord_le540031911545281998_a_a_o @ A2 @ ( inf_in6214470387022126620_a_a_o @ B2 @ C ) )
=> ~ ( ( ord_le540031911545281998_a_a_o @ A2 @ B2 )
=> ~ ( ord_le540031911545281998_a_a_o @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_330_inf_OboundedE,axiom,
! [A2: a > $o,B2: a > $o,C: a > $o] :
( ( ord_less_eq_a_o @ A2 @ ( inf_inf_a_o @ B2 @ C ) )
=> ~ ( ( ord_less_eq_a_o @ A2 @ B2 )
=> ~ ( ord_less_eq_a_o @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_331_inf_OboundedE,axiom,
! [A2: a > a > $o,B2: a > a > $o,C: a > a > $o] :
( ( ord_less_eq_a_a_o @ A2 @ ( inf_inf_a_a_o @ B2 @ C ) )
=> ~ ( ( ord_less_eq_a_a_o @ A2 @ B2 )
=> ~ ( ord_less_eq_a_a_o @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_332_inf_OboundedE,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ ( inf_in8905007599844390133od_a_a @ B2 @ C ) )
=> ~ ( ( ord_le746702958409616551od_a_a @ A2 @ B2 )
=> ~ ( ord_le746702958409616551od_a_a @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_333_inf_OboundedE,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ~ ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_334_inf__absorb2,axiom,
! [Y: product_prod_a_a > $o,X: product_prod_a_a > $o] :
( ( ord_le1591150415168442102_a_a_o @ Y @ X )
=> ( ( inf_in2559554923042384936_a_a_o @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_335_inf__absorb2,axiom,
! [Y: set_se5735800977113168103od_a_a,X: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ Y @ X )
=> ( ( inf_in3339382566020358357od_a_a @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_336_inf__absorb2,axiom,
! [Y: set_set_a,X: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y @ X )
=> ( ( inf_inf_set_set_a @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_337_inf__absorb2,axiom,
! [Y: product_prod_a_a > product_prod_a_a > $o,X: product_prod_a_a > product_prod_a_a > $o] :
( ( ord_le540031911545281998_a_a_o @ Y @ X )
=> ( ( inf_in6214470387022126620_a_a_o @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_338_inf__absorb2,axiom,
! [Y: a > $o,X: a > $o] :
( ( ord_less_eq_a_o @ Y @ X )
=> ( ( inf_inf_a_o @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_339_inf__absorb2,axiom,
! [Y: a > a > $o,X: a > a > $o] :
( ( ord_less_eq_a_a_o @ Y @ X )
=> ( ( inf_inf_a_a_o @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_340_inf__absorb2,axiom,
! [Y: set_Product_prod_a_a,X: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ Y @ X )
=> ( ( inf_in8905007599844390133od_a_a @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_341_inf__absorb2,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( inf_inf_set_a @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_342_inf__absorb1,axiom,
! [X: product_prod_a_a > $o,Y: product_prod_a_a > $o] :
( ( ord_le1591150415168442102_a_a_o @ X @ Y )
=> ( ( inf_in2559554923042384936_a_a_o @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_343_inf__absorb1,axiom,
! [X: set_se5735800977113168103od_a_a,Y: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ X @ Y )
=> ( ( inf_in3339382566020358357od_a_a @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_344_inf__absorb1,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y )
=> ( ( inf_inf_set_set_a @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_345_inf__absorb1,axiom,
! [X: product_prod_a_a > product_prod_a_a > $o,Y: product_prod_a_a > product_prod_a_a > $o] :
( ( ord_le540031911545281998_a_a_o @ X @ Y )
=> ( ( inf_in6214470387022126620_a_a_o @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_346_inf__absorb1,axiom,
! [X: a > $o,Y: a > $o] :
( ( ord_less_eq_a_o @ X @ Y )
=> ( ( inf_inf_a_o @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_347_inf__absorb1,axiom,
! [X: a > a > $o,Y: a > a > $o] :
( ( ord_less_eq_a_a_o @ X @ Y )
=> ( ( inf_inf_a_a_o @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_348_inf__absorb1,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X @ Y )
=> ( ( inf_in8905007599844390133od_a_a @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_349_inf__absorb1,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( inf_inf_set_a @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_350_inf_Oabsorb2,axiom,
! [B2: product_prod_a_a > $o,A2: product_prod_a_a > $o] :
( ( ord_le1591150415168442102_a_a_o @ B2 @ A2 )
=> ( ( inf_in2559554923042384936_a_a_o @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_351_inf_Oabsorb2,axiom,
! [B2: set_se5735800977113168103od_a_a,A2: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ B2 @ A2 )
=> ( ( inf_in3339382566020358357od_a_a @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_352_inf_Oabsorb2,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_353_inf_Oabsorb2,axiom,
! [B2: product_prod_a_a > product_prod_a_a > $o,A2: product_prod_a_a > product_prod_a_a > $o] :
( ( ord_le540031911545281998_a_a_o @ B2 @ A2 )
=> ( ( inf_in6214470387022126620_a_a_o @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_354_inf_Oabsorb2,axiom,
! [B2: a > $o,A2: a > $o] :
( ( ord_less_eq_a_o @ B2 @ A2 )
=> ( ( inf_inf_a_o @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_355_inf_Oabsorb2,axiom,
! [B2: a > a > $o,A2: a > a > $o] :
( ( ord_less_eq_a_a_o @ B2 @ A2 )
=> ( ( inf_inf_a_a_o @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_356_inf_Oabsorb2,axiom,
! [B2: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B2 @ A2 )
=> ( ( inf_in8905007599844390133od_a_a @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_357_inf_Oabsorb2,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_358_inf_Oabsorb1,axiom,
! [A2: product_prod_a_a > $o,B2: product_prod_a_a > $o] :
( ( ord_le1591150415168442102_a_a_o @ A2 @ B2 )
=> ( ( inf_in2559554923042384936_a_a_o @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_359_inf_Oabsorb1,axiom,
! [A2: set_se5735800977113168103od_a_a,B2: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ A2 @ B2 )
=> ( ( inf_in3339382566020358357od_a_a @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_360_inf_Oabsorb1,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( inf_inf_set_set_a @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_361_inf_Oabsorb1,axiom,
! [A2: product_prod_a_a > product_prod_a_a > $o,B2: product_prod_a_a > product_prod_a_a > $o] :
( ( ord_le540031911545281998_a_a_o @ A2 @ B2 )
=> ( ( inf_in6214470387022126620_a_a_o @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_362_inf_Oabsorb1,axiom,
! [A2: a > $o,B2: a > $o] :
( ( ord_less_eq_a_o @ A2 @ B2 )
=> ( ( inf_inf_a_o @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_363_inf_Oabsorb1,axiom,
! [A2: a > a > $o,B2: a > a > $o] :
( ( ord_less_eq_a_a_o @ A2 @ B2 )
=> ( ( inf_inf_a_a_o @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_364_inf_Oabsorb1,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B2 )
=> ( ( inf_in8905007599844390133od_a_a @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_365_inf_Oabsorb1,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_366_le__iff__inf,axiom,
( ord_le1591150415168442102_a_a_o
= ( ^ [X3: product_prod_a_a > $o,Y3: product_prod_a_a > $o] :
( ( inf_in2559554923042384936_a_a_o @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_367_le__iff__inf,axiom,
( ord_le1995061765932249223od_a_a
= ( ^ [X3: set_se5735800977113168103od_a_a,Y3: set_se5735800977113168103od_a_a] :
( ( inf_in3339382566020358357od_a_a @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_368_le__iff__inf,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [X3: set_set_a,Y3: set_set_a] :
( ( inf_inf_set_set_a @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_369_le__iff__inf,axiom,
( ord_le540031911545281998_a_a_o
= ( ^ [X3: product_prod_a_a > product_prod_a_a > $o,Y3: product_prod_a_a > product_prod_a_a > $o] :
( ( inf_in6214470387022126620_a_a_o @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_370_le__iff__inf,axiom,
( ord_less_eq_a_o
= ( ^ [X3: a > $o,Y3: a > $o] :
( ( inf_inf_a_o @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_371_le__iff__inf,axiom,
( ord_less_eq_a_a_o
= ( ^ [X3: a > a > $o,Y3: a > a > $o] :
( ( inf_inf_a_a_o @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_372_le__iff__inf,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [X3: set_Product_prod_a_a,Y3: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_373_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y3: set_a] :
( ( inf_inf_set_a @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_374_inf__unique,axiom,
! [F2: ( product_prod_a_a > $o ) > ( product_prod_a_a > $o ) > product_prod_a_a > $o,X: product_prod_a_a > $o,Y: product_prod_a_a > $o] :
( ! [X2: product_prod_a_a > $o,Y4: product_prod_a_a > $o] : ( ord_le1591150415168442102_a_a_o @ ( F2 @ X2 @ Y4 ) @ X2 )
=> ( ! [X2: product_prod_a_a > $o,Y4: product_prod_a_a > $o] : ( ord_le1591150415168442102_a_a_o @ ( F2 @ X2 @ Y4 ) @ Y4 )
=> ( ! [X2: product_prod_a_a > $o,Y4: product_prod_a_a > $o,Z3: product_prod_a_a > $o] :
( ( ord_le1591150415168442102_a_a_o @ X2 @ Y4 )
=> ( ( ord_le1591150415168442102_a_a_o @ X2 @ Z3 )
=> ( ord_le1591150415168442102_a_a_o @ X2 @ ( F2 @ Y4 @ Z3 ) ) ) )
=> ( ( inf_in2559554923042384936_a_a_o @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_375_inf__unique,axiom,
! [F2: set_se5735800977113168103od_a_a > set_se5735800977113168103od_a_a > set_se5735800977113168103od_a_a,X: set_se5735800977113168103od_a_a,Y: set_se5735800977113168103od_a_a] :
( ! [X2: set_se5735800977113168103od_a_a,Y4: set_se5735800977113168103od_a_a] : ( ord_le1995061765932249223od_a_a @ ( F2 @ X2 @ Y4 ) @ X2 )
=> ( ! [X2: set_se5735800977113168103od_a_a,Y4: set_se5735800977113168103od_a_a] : ( ord_le1995061765932249223od_a_a @ ( F2 @ X2 @ Y4 ) @ Y4 )
=> ( ! [X2: set_se5735800977113168103od_a_a,Y4: set_se5735800977113168103od_a_a,Z3: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ X2 @ Y4 )
=> ( ( ord_le1995061765932249223od_a_a @ X2 @ Z3 )
=> ( ord_le1995061765932249223od_a_a @ X2 @ ( F2 @ Y4 @ Z3 ) ) ) )
=> ( ( inf_in3339382566020358357od_a_a @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_376_inf__unique,axiom,
! [F2: set_set_a > set_set_a > set_set_a,X: set_set_a,Y: set_set_a] :
( ! [X2: set_set_a,Y4: set_set_a] : ( ord_le3724670747650509150_set_a @ ( F2 @ X2 @ Y4 ) @ X2 )
=> ( ! [X2: set_set_a,Y4: set_set_a] : ( ord_le3724670747650509150_set_a @ ( F2 @ X2 @ Y4 ) @ Y4 )
=> ( ! [X2: set_set_a,Y4: set_set_a,Z3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y4 )
=> ( ( ord_le3724670747650509150_set_a @ X2 @ Z3 )
=> ( ord_le3724670747650509150_set_a @ X2 @ ( F2 @ Y4 @ Z3 ) ) ) )
=> ( ( inf_inf_set_set_a @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_377_inf__unique,axiom,
! [F2: ( product_prod_a_a > product_prod_a_a > $o ) > ( product_prod_a_a > product_prod_a_a > $o ) > product_prod_a_a > product_prod_a_a > $o,X: product_prod_a_a > product_prod_a_a > $o,Y: product_prod_a_a > product_prod_a_a > $o] :
( ! [X2: product_prod_a_a > product_prod_a_a > $o,Y4: product_prod_a_a > product_prod_a_a > $o] : ( ord_le540031911545281998_a_a_o @ ( F2 @ X2 @ Y4 ) @ X2 )
=> ( ! [X2: product_prod_a_a > product_prod_a_a > $o,Y4: product_prod_a_a > product_prod_a_a > $o] : ( ord_le540031911545281998_a_a_o @ ( F2 @ X2 @ Y4 ) @ Y4 )
=> ( ! [X2: product_prod_a_a > product_prod_a_a > $o,Y4: product_prod_a_a > product_prod_a_a > $o,Z3: product_prod_a_a > product_prod_a_a > $o] :
( ( ord_le540031911545281998_a_a_o @ X2 @ Y4 )
=> ( ( ord_le540031911545281998_a_a_o @ X2 @ Z3 )
=> ( ord_le540031911545281998_a_a_o @ X2 @ ( F2 @ Y4 @ Z3 ) ) ) )
=> ( ( inf_in6214470387022126620_a_a_o @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_378_inf__unique,axiom,
! [F2: ( a > $o ) > ( a > $o ) > a > $o,X: a > $o,Y: a > $o] :
( ! [X2: a > $o,Y4: a > $o] : ( ord_less_eq_a_o @ ( F2 @ X2 @ Y4 ) @ X2 )
=> ( ! [X2: a > $o,Y4: a > $o] : ( ord_less_eq_a_o @ ( F2 @ X2 @ Y4 ) @ Y4 )
=> ( ! [X2: a > $o,Y4: a > $o,Z3: a > $o] :
( ( ord_less_eq_a_o @ X2 @ Y4 )
=> ( ( ord_less_eq_a_o @ X2 @ Z3 )
=> ( ord_less_eq_a_o @ X2 @ ( F2 @ Y4 @ Z3 ) ) ) )
=> ( ( inf_inf_a_o @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_379_inf__unique,axiom,
! [F2: ( a > a > $o ) > ( a > a > $o ) > a > a > $o,X: a > a > $o,Y: a > a > $o] :
( ! [X2: a > a > $o,Y4: a > a > $o] : ( ord_less_eq_a_a_o @ ( F2 @ X2 @ Y4 ) @ X2 )
=> ( ! [X2: a > a > $o,Y4: a > a > $o] : ( ord_less_eq_a_a_o @ ( F2 @ X2 @ Y4 ) @ Y4 )
=> ( ! [X2: a > a > $o,Y4: a > a > $o,Z3: a > a > $o] :
( ( ord_less_eq_a_a_o @ X2 @ Y4 )
=> ( ( ord_less_eq_a_a_o @ X2 @ Z3 )
=> ( ord_less_eq_a_a_o @ X2 @ ( F2 @ Y4 @ Z3 ) ) ) )
=> ( ( inf_inf_a_a_o @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_380_inf__unique,axiom,
! [F2: set_Product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a,X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ! [X2: set_Product_prod_a_a,Y4: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( F2 @ X2 @ Y4 ) @ X2 )
=> ( ! [X2: set_Product_prod_a_a,Y4: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( F2 @ X2 @ Y4 ) @ Y4 )
=> ( ! [X2: set_Product_prod_a_a,Y4: set_Product_prod_a_a,Z3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X2 @ Y4 )
=> ( ( ord_le746702958409616551od_a_a @ X2 @ Z3 )
=> ( ord_le746702958409616551od_a_a @ X2 @ ( F2 @ Y4 @ Z3 ) ) ) )
=> ( ( inf_in8905007599844390133od_a_a @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_381_inf__unique,axiom,
! [F2: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X2: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( F2 @ X2 @ Y4 ) @ X2 )
=> ( ! [X2: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( F2 @ X2 @ Y4 ) @ Y4 )
=> ( ! [X2: set_a,Y4: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ( ord_less_eq_set_a @ X2 @ Z3 )
=> ( ord_less_eq_set_a @ X2 @ ( F2 @ Y4 @ Z3 ) ) ) )
=> ( ( inf_inf_set_a @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_382_inf_OorderI,axiom,
! [A2: product_prod_a_a > $o,B2: product_prod_a_a > $o] :
( ( A2
= ( inf_in2559554923042384936_a_a_o @ A2 @ B2 ) )
=> ( ord_le1591150415168442102_a_a_o @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_383_inf_OorderI,axiom,
! [A2: set_se5735800977113168103od_a_a,B2: set_se5735800977113168103od_a_a] :
( ( A2
= ( inf_in3339382566020358357od_a_a @ A2 @ B2 ) )
=> ( ord_le1995061765932249223od_a_a @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_384_inf_OorderI,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( A2
= ( inf_inf_set_set_a @ A2 @ B2 ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_385_inf_OorderI,axiom,
! [A2: product_prod_a_a > product_prod_a_a > $o,B2: product_prod_a_a > product_prod_a_a > $o] :
( ( A2
= ( inf_in6214470387022126620_a_a_o @ A2 @ B2 ) )
=> ( ord_le540031911545281998_a_a_o @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_386_inf_OorderI,axiom,
! [A2: a > $o,B2: a > $o] :
( ( A2
= ( inf_inf_a_o @ A2 @ B2 ) )
=> ( ord_less_eq_a_o @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_387_inf_OorderI,axiom,
! [A2: a > a > $o,B2: a > a > $o] :
( ( A2
= ( inf_inf_a_a_o @ A2 @ B2 ) )
=> ( ord_less_eq_a_a_o @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_388_inf_OorderI,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( A2
= ( inf_in8905007599844390133od_a_a @ A2 @ B2 ) )
=> ( ord_le746702958409616551od_a_a @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_389_inf_OorderI,axiom,
! [A2: set_a,B2: set_a] :
( ( A2
= ( inf_inf_set_a @ A2 @ B2 ) )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_390_inf_OorderE,axiom,
! [A2: product_prod_a_a > $o,B2: product_prod_a_a > $o] :
( ( ord_le1591150415168442102_a_a_o @ A2 @ B2 )
=> ( A2
= ( inf_in2559554923042384936_a_a_o @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_391_inf_OorderE,axiom,
! [A2: set_se5735800977113168103od_a_a,B2: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ A2 @ B2 )
=> ( A2
= ( inf_in3339382566020358357od_a_a @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_392_inf_OorderE,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( A2
= ( inf_inf_set_set_a @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_393_inf_OorderE,axiom,
! [A2: product_prod_a_a > product_prod_a_a > $o,B2: product_prod_a_a > product_prod_a_a > $o] :
( ( ord_le540031911545281998_a_a_o @ A2 @ B2 )
=> ( A2
= ( inf_in6214470387022126620_a_a_o @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_394_inf_OorderE,axiom,
! [A2: a > $o,B2: a > $o] :
( ( ord_less_eq_a_o @ A2 @ B2 )
=> ( A2
= ( inf_inf_a_o @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_395_inf_OorderE,axiom,
! [A2: a > a > $o,B2: a > a > $o] :
( ( ord_less_eq_a_a_o @ A2 @ B2 )
=> ( A2
= ( inf_inf_a_a_o @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_396_inf_OorderE,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B2 )
=> ( A2
= ( inf_in8905007599844390133od_a_a @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_397_inf_OorderE,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( A2
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_398_le__infI2,axiom,
! [B2: product_prod_a_a > $o,X: product_prod_a_a > $o,A2: product_prod_a_a > $o] :
( ( ord_le1591150415168442102_a_a_o @ B2 @ X )
=> ( ord_le1591150415168442102_a_a_o @ ( inf_in2559554923042384936_a_a_o @ A2 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_399_le__infI2,axiom,
! [B2: set_se5735800977113168103od_a_a,X: set_se5735800977113168103od_a_a,A2: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ B2 @ X )
=> ( ord_le1995061765932249223od_a_a @ ( inf_in3339382566020358357od_a_a @ A2 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_400_le__infI2,axiom,
! [B2: set_set_a,X: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ X )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_401_le__infI2,axiom,
! [B2: product_prod_a_a > product_prod_a_a > $o,X: product_prod_a_a > product_prod_a_a > $o,A2: product_prod_a_a > product_prod_a_a > $o] :
( ( ord_le540031911545281998_a_a_o @ B2 @ X )
=> ( ord_le540031911545281998_a_a_o @ ( inf_in6214470387022126620_a_a_o @ A2 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_402_le__infI2,axiom,
! [B2: a > $o,X: a > $o,A2: a > $o] :
( ( ord_less_eq_a_o @ B2 @ X )
=> ( ord_less_eq_a_o @ ( inf_inf_a_o @ A2 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_403_le__infI2,axiom,
! [B2: a > a > $o,X: a > a > $o,A2: a > a > $o] :
( ( ord_less_eq_a_a_o @ B2 @ X )
=> ( ord_less_eq_a_a_o @ ( inf_inf_a_a_o @ A2 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_404_le__infI2,axiom,
! [B2: set_Product_prod_a_a,X: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B2 @ X )
=> ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A2 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_405_le__infI2,axiom,
! [B2: set_a,X: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_406_le__infI1,axiom,
! [A2: product_prod_a_a > $o,X: product_prod_a_a > $o,B2: product_prod_a_a > $o] :
( ( ord_le1591150415168442102_a_a_o @ A2 @ X )
=> ( ord_le1591150415168442102_a_a_o @ ( inf_in2559554923042384936_a_a_o @ A2 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_407_le__infI1,axiom,
! [A2: set_se5735800977113168103od_a_a,X: set_se5735800977113168103od_a_a,B2: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ A2 @ X )
=> ( ord_le1995061765932249223od_a_a @ ( inf_in3339382566020358357od_a_a @ A2 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_408_le__infI1,axiom,
! [A2: set_set_a,X: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ X )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_409_le__infI1,axiom,
! [A2: product_prod_a_a > product_prod_a_a > $o,X: product_prod_a_a > product_prod_a_a > $o,B2: product_prod_a_a > product_prod_a_a > $o] :
( ( ord_le540031911545281998_a_a_o @ A2 @ X )
=> ( ord_le540031911545281998_a_a_o @ ( inf_in6214470387022126620_a_a_o @ A2 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_410_le__infI1,axiom,
! [A2: a > $o,X: a > $o,B2: a > $o] :
( ( ord_less_eq_a_o @ A2 @ X )
=> ( ord_less_eq_a_o @ ( inf_inf_a_o @ A2 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_411_le__infI1,axiom,
! [A2: a > a > $o,X: a > a > $o,B2: a > a > $o] :
( ( ord_less_eq_a_a_o @ A2 @ X )
=> ( ord_less_eq_a_a_o @ ( inf_inf_a_a_o @ A2 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_412_le__infI1,axiom,
! [A2: set_Product_prod_a_a,X: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ X )
=> ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A2 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_413_le__infI1,axiom,
! [A2: set_a,X: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_414_inf__mono,axiom,
! [A2: product_prod_a_a > $o,C: product_prod_a_a > $o,B2: product_prod_a_a > $o,D2: product_prod_a_a > $o] :
( ( ord_le1591150415168442102_a_a_o @ A2 @ C )
=> ( ( ord_le1591150415168442102_a_a_o @ B2 @ D2 )
=> ( ord_le1591150415168442102_a_a_o @ ( inf_in2559554923042384936_a_a_o @ A2 @ B2 ) @ ( inf_in2559554923042384936_a_a_o @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_415_inf__mono,axiom,
! [A2: set_se5735800977113168103od_a_a,C: set_se5735800977113168103od_a_a,B2: set_se5735800977113168103od_a_a,D2: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ A2 @ C )
=> ( ( ord_le1995061765932249223od_a_a @ B2 @ D2 )
=> ( ord_le1995061765932249223od_a_a @ ( inf_in3339382566020358357od_a_a @ A2 @ B2 ) @ ( inf_in3339382566020358357od_a_a @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_416_inf__mono,axiom,
! [A2: set_set_a,C: set_set_a,B2: set_set_a,D2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ C )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ D2 )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B2 ) @ ( inf_inf_set_set_a @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_417_inf__mono,axiom,
! [A2: product_prod_a_a > product_prod_a_a > $o,C: product_prod_a_a > product_prod_a_a > $o,B2: product_prod_a_a > product_prod_a_a > $o,D2: product_prod_a_a > product_prod_a_a > $o] :
( ( ord_le540031911545281998_a_a_o @ A2 @ C )
=> ( ( ord_le540031911545281998_a_a_o @ B2 @ D2 )
=> ( ord_le540031911545281998_a_a_o @ ( inf_in6214470387022126620_a_a_o @ A2 @ B2 ) @ ( inf_in6214470387022126620_a_a_o @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_418_inf__mono,axiom,
! [A2: a > $o,C: a > $o,B2: a > $o,D2: a > $o] :
( ( ord_less_eq_a_o @ A2 @ C )
=> ( ( ord_less_eq_a_o @ B2 @ D2 )
=> ( ord_less_eq_a_o @ ( inf_inf_a_o @ A2 @ B2 ) @ ( inf_inf_a_o @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_419_inf__mono,axiom,
! [A2: a > a > $o,C: a > a > $o,B2: a > a > $o,D2: a > a > $o] :
( ( ord_less_eq_a_a_o @ A2 @ C )
=> ( ( ord_less_eq_a_a_o @ B2 @ D2 )
=> ( ord_less_eq_a_a_o @ ( inf_inf_a_a_o @ A2 @ B2 ) @ ( inf_inf_a_a_o @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_420_inf__mono,axiom,
! [A2: set_Product_prod_a_a,C: set_Product_prod_a_a,B2: set_Product_prod_a_a,D2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ C )
=> ( ( ord_le746702958409616551od_a_a @ B2 @ D2 )
=> ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A2 @ B2 ) @ ( inf_in8905007599844390133od_a_a @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_421_inf__mono,axiom,
! [A2: set_a,C: set_a,B2: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B2 @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ ( inf_inf_set_a @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_422_le__infI,axiom,
! [X: product_prod_a_a > $o,A2: product_prod_a_a > $o,B2: product_prod_a_a > $o] :
( ( ord_le1591150415168442102_a_a_o @ X @ A2 )
=> ( ( ord_le1591150415168442102_a_a_o @ X @ B2 )
=> ( ord_le1591150415168442102_a_a_o @ X @ ( inf_in2559554923042384936_a_a_o @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_423_le__infI,axiom,
! [X: set_se5735800977113168103od_a_a,A2: set_se5735800977113168103od_a_a,B2: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ X @ A2 )
=> ( ( ord_le1995061765932249223od_a_a @ X @ B2 )
=> ( ord_le1995061765932249223od_a_a @ X @ ( inf_in3339382566020358357od_a_a @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_424_le__infI,axiom,
! [X: set_set_a,A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ X @ B2 )
=> ( ord_le3724670747650509150_set_a @ X @ ( inf_inf_set_set_a @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_425_le__infI,axiom,
! [X: product_prod_a_a > product_prod_a_a > $o,A2: product_prod_a_a > product_prod_a_a > $o,B2: product_prod_a_a > product_prod_a_a > $o] :
( ( ord_le540031911545281998_a_a_o @ X @ A2 )
=> ( ( ord_le540031911545281998_a_a_o @ X @ B2 )
=> ( ord_le540031911545281998_a_a_o @ X @ ( inf_in6214470387022126620_a_a_o @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_426_le__infI,axiom,
! [X: a > $o,A2: a > $o,B2: a > $o] :
( ( ord_less_eq_a_o @ X @ A2 )
=> ( ( ord_less_eq_a_o @ X @ B2 )
=> ( ord_less_eq_a_o @ X @ ( inf_inf_a_o @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_427_le__infI,axiom,
! [X: a > a > $o,A2: a > a > $o,B2: a > a > $o] :
( ( ord_less_eq_a_a_o @ X @ A2 )
=> ( ( ord_less_eq_a_a_o @ X @ B2 )
=> ( ord_less_eq_a_a_o @ X @ ( inf_inf_a_a_o @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_428_le__infI,axiom,
! [X: set_Product_prod_a_a,A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X @ A2 )
=> ( ( ord_le746702958409616551od_a_a @ X @ B2 )
=> ( ord_le746702958409616551od_a_a @ X @ ( inf_in8905007599844390133od_a_a @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_429_le__infI,axiom,
! [X: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ X @ B2 )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_430_le__infE,axiom,
! [X: product_prod_a_a > $o,A2: product_prod_a_a > $o,B2: product_prod_a_a > $o] :
( ( ord_le1591150415168442102_a_a_o @ X @ ( inf_in2559554923042384936_a_a_o @ A2 @ B2 ) )
=> ~ ( ( ord_le1591150415168442102_a_a_o @ X @ A2 )
=> ~ ( ord_le1591150415168442102_a_a_o @ X @ B2 ) ) ) ).
% le_infE
thf(fact_431_le__infE,axiom,
! [X: set_se5735800977113168103od_a_a,A2: set_se5735800977113168103od_a_a,B2: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ X @ ( inf_in3339382566020358357od_a_a @ A2 @ B2 ) )
=> ~ ( ( ord_le1995061765932249223od_a_a @ X @ A2 )
=> ~ ( ord_le1995061765932249223od_a_a @ X @ B2 ) ) ) ).
% le_infE
thf(fact_432_le__infE,axiom,
! [X: set_set_a,A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ ( inf_inf_set_set_a @ A2 @ B2 ) )
=> ~ ( ( ord_le3724670747650509150_set_a @ X @ A2 )
=> ~ ( ord_le3724670747650509150_set_a @ X @ B2 ) ) ) ).
% le_infE
thf(fact_433_le__infE,axiom,
! [X: product_prod_a_a > product_prod_a_a > $o,A2: product_prod_a_a > product_prod_a_a > $o,B2: product_prod_a_a > product_prod_a_a > $o] :
( ( ord_le540031911545281998_a_a_o @ X @ ( inf_in6214470387022126620_a_a_o @ A2 @ B2 ) )
=> ~ ( ( ord_le540031911545281998_a_a_o @ X @ A2 )
=> ~ ( ord_le540031911545281998_a_a_o @ X @ B2 ) ) ) ).
% le_infE
thf(fact_434_le__infE,axiom,
! [X: a > $o,A2: a > $o,B2: a > $o] :
( ( ord_less_eq_a_o @ X @ ( inf_inf_a_o @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_a_o @ X @ A2 )
=> ~ ( ord_less_eq_a_o @ X @ B2 ) ) ) ).
% le_infE
thf(fact_435_le__infE,axiom,
! [X: a > a > $o,A2: a > a > $o,B2: a > a > $o] :
( ( ord_less_eq_a_a_o @ X @ ( inf_inf_a_a_o @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_a_a_o @ X @ A2 )
=> ~ ( ord_less_eq_a_a_o @ X @ B2 ) ) ) ).
% le_infE
thf(fact_436_le__infE,axiom,
! [X: set_Product_prod_a_a,A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X @ ( inf_in8905007599844390133od_a_a @ A2 @ B2 ) )
=> ~ ( ( ord_le746702958409616551od_a_a @ X @ A2 )
=> ~ ( ord_le746702958409616551od_a_a @ X @ B2 ) ) ) ).
% le_infE
thf(fact_437_le__infE,axiom,
! [X: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_set_a @ X @ A2 )
=> ~ ( ord_less_eq_set_a @ X @ B2 ) ) ) ).
% le_infE
thf(fact_438_inf__le2,axiom,
! [X: product_prod_a_a > $o,Y: product_prod_a_a > $o] : ( ord_le1591150415168442102_a_a_o @ ( inf_in2559554923042384936_a_a_o @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_439_inf__le2,axiom,
! [X: set_se5735800977113168103od_a_a,Y: set_se5735800977113168103od_a_a] : ( ord_le1995061765932249223od_a_a @ ( inf_in3339382566020358357od_a_a @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_440_inf__le2,axiom,
! [X: set_set_a,Y: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_441_inf__le2,axiom,
! [X: product_prod_a_a > product_prod_a_a > $o,Y: product_prod_a_a > product_prod_a_a > $o] : ( ord_le540031911545281998_a_a_o @ ( inf_in6214470387022126620_a_a_o @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_442_inf__le2,axiom,
! [X: a > $o,Y: a > $o] : ( ord_less_eq_a_o @ ( inf_inf_a_o @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_443_inf__le2,axiom,
! [X: a > a > $o,Y: a > a > $o] : ( ord_less_eq_a_a_o @ ( inf_inf_a_a_o @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_444_inf__le2,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_445_inf__le2,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_446_inf__le1,axiom,
! [X: product_prod_a_a > $o,Y: product_prod_a_a > $o] : ( ord_le1591150415168442102_a_a_o @ ( inf_in2559554923042384936_a_a_o @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_447_inf__le1,axiom,
! [X: set_se5735800977113168103od_a_a,Y: set_se5735800977113168103od_a_a] : ( ord_le1995061765932249223od_a_a @ ( inf_in3339382566020358357od_a_a @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_448_inf__le1,axiom,
! [X: set_set_a,Y: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_449_inf__le1,axiom,
! [X: product_prod_a_a > product_prod_a_a > $o,Y: product_prod_a_a > product_prod_a_a > $o] : ( ord_le540031911545281998_a_a_o @ ( inf_in6214470387022126620_a_a_o @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_450_inf__le1,axiom,
! [X: a > $o,Y: a > $o] : ( ord_less_eq_a_o @ ( inf_inf_a_o @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_451_inf__le1,axiom,
! [X: a > a > $o,Y: a > a > $o] : ( ord_less_eq_a_a_o @ ( inf_inf_a_a_o @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_452_inf__le1,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_453_inf__le1,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_454_inf__sup__ord_I1_J,axiom,
! [X: product_prod_a_a > $o,Y: product_prod_a_a > $o] : ( ord_le1591150415168442102_a_a_o @ ( inf_in2559554923042384936_a_a_o @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_455_inf__sup__ord_I1_J,axiom,
! [X: set_se5735800977113168103od_a_a,Y: set_se5735800977113168103od_a_a] : ( ord_le1995061765932249223od_a_a @ ( inf_in3339382566020358357od_a_a @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_456_inf__sup__ord_I1_J,axiom,
! [X: set_set_a,Y: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_457_inf__sup__ord_I1_J,axiom,
! [X: product_prod_a_a > product_prod_a_a > $o,Y: product_prod_a_a > product_prod_a_a > $o] : ( ord_le540031911545281998_a_a_o @ ( inf_in6214470387022126620_a_a_o @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_458_inf__sup__ord_I1_J,axiom,
! [X: a > $o,Y: a > $o] : ( ord_less_eq_a_o @ ( inf_inf_a_o @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_459_inf__sup__ord_I1_J,axiom,
! [X: a > a > $o,Y: a > a > $o] : ( ord_less_eq_a_a_o @ ( inf_inf_a_a_o @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_460_inf__sup__ord_I1_J,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_461_inf__sup__ord_I1_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_462_inf__sup__ord_I2_J,axiom,
! [X: product_prod_a_a > $o,Y: product_prod_a_a > $o] : ( ord_le1591150415168442102_a_a_o @ ( inf_in2559554923042384936_a_a_o @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_463_inf__sup__ord_I2_J,axiom,
! [X: set_se5735800977113168103od_a_a,Y: set_se5735800977113168103od_a_a] : ( ord_le1995061765932249223od_a_a @ ( inf_in3339382566020358357od_a_a @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_464_inf__sup__ord_I2_J,axiom,
! [X: set_set_a,Y: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_465_inf__sup__ord_I2_J,axiom,
! [X: product_prod_a_a > product_prod_a_a > $o,Y: product_prod_a_a > product_prod_a_a > $o] : ( ord_le540031911545281998_a_a_o @ ( inf_in6214470387022126620_a_a_o @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_466_inf__sup__ord_I2_J,axiom,
! [X: a > $o,Y: a > $o] : ( ord_less_eq_a_o @ ( inf_inf_a_o @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_467_inf__sup__ord_I2_J,axiom,
! [X: a > a > $o,Y: a > a > $o] : ( ord_less_eq_a_a_o @ ( inf_inf_a_a_o @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_468_inf__sup__ord_I2_J,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_469_inf__sup__ord_I2_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_470_Int__Collect__mono,axiom,
! [A: set_set_set_a,B: set_set_set_a,P: set_set_a > $o,Q: set_set_a > $o] :
( ( ord_le5722252365846178494_set_a @ A @ B )
=> ( ! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ A )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le5722252365846178494_set_a @ ( inf_in1205276777018777868_set_a @ A @ ( collect_set_set_a @ P ) ) @ ( inf_in1205276777018777868_set_a @ B @ ( collect_set_set_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_471_Int__Collect__mono,axiom,
! [A: set_se5735800977113168103od_a_a,B: set_se5735800977113168103od_a_a,P: set_Product_prod_a_a > $o,Q: set_Product_prod_a_a > $o] :
( ( ord_le1995061765932249223od_a_a @ A @ B )
=> ( ! [X2: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X2 @ A )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le1995061765932249223od_a_a @ ( inf_in3339382566020358357od_a_a @ A @ ( collec1673347964119250290od_a_a @ P ) ) @ ( inf_in3339382566020358357od_a_a @ B @ ( collec1673347964119250290od_a_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_472_Int__Collect__mono,axiom,
! [A: set_set_a,B: set_set_a,P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ ( collect_set_a @ P ) ) @ ( inf_inf_set_set_a @ B @ ( collect_set_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_473_Int__Collect__mono,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A @ ( collec3336397797384452498od_a_a @ P ) ) @ ( inf_in8905007599844390133od_a_a @ B @ ( collec3336397797384452498od_a_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_474_Int__Collect__mono,axiom,
! [A: set_a,B: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_475_Int__greatest,axiom,
! [C2: set_se5735800977113168103od_a_a,A: set_se5735800977113168103od_a_a,B: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ C2 @ A )
=> ( ( ord_le1995061765932249223od_a_a @ C2 @ B )
=> ( ord_le1995061765932249223od_a_a @ C2 @ ( inf_in3339382566020358357od_a_a @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_476_Int__greatest,axiom,
! [C2: set_set_a,A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C2 @ A )
=> ( ( ord_le3724670747650509150_set_a @ C2 @ B )
=> ( ord_le3724670747650509150_set_a @ C2 @ ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_477_Int__greatest,axiom,
! [C2: set_Product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ C2 @ A )
=> ( ( ord_le746702958409616551od_a_a @ C2 @ B )
=> ( ord_le746702958409616551od_a_a @ C2 @ ( inf_in8905007599844390133od_a_a @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_478_Int__greatest,axiom,
! [C2: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ A )
=> ( ( ord_less_eq_set_a @ C2 @ B )
=> ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_479_Int__absorb2,axiom,
! [A: set_se5735800977113168103od_a_a,B: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ A @ B )
=> ( ( inf_in3339382566020358357od_a_a @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_480_Int__absorb2,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( inf_inf_set_set_a @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_481_Int__absorb2,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( ( inf_in8905007599844390133od_a_a @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_482_Int__absorb2,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( inf_inf_set_a @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_483_Int__absorb1,axiom,
! [B: set_se5735800977113168103od_a_a,A: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ B @ A )
=> ( ( inf_in3339382566020358357od_a_a @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_484_Int__absorb1,axiom,
! [B: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ( inf_inf_set_set_a @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_485_Int__absorb1,axiom,
! [B: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B @ A )
=> ( ( inf_in8905007599844390133od_a_a @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_486_Int__absorb1,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( inf_inf_set_a @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_487_Int__lower2,axiom,
! [A: set_se5735800977113168103od_a_a,B: set_se5735800977113168103od_a_a] : ( ord_le1995061765932249223od_a_a @ ( inf_in3339382566020358357od_a_a @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_488_Int__lower2,axiom,
! [A: set_set_a,B: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_489_Int__lower2,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_490_Int__lower2,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_491_Int__lower1,axiom,
! [A: set_se5735800977113168103od_a_a,B: set_se5735800977113168103od_a_a] : ( ord_le1995061765932249223od_a_a @ ( inf_in3339382566020358357od_a_a @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_492_Int__lower1,axiom,
! [A: set_set_a,B: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_493_Int__lower1,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_494_Int__lower1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_495_Int__mono,axiom,
! [A: set_se5735800977113168103od_a_a,C2: set_se5735800977113168103od_a_a,B: set_se5735800977113168103od_a_a,D: set_se5735800977113168103od_a_a] :
( ( ord_le1995061765932249223od_a_a @ A @ C2 )
=> ( ( ord_le1995061765932249223od_a_a @ B @ D )
=> ( ord_le1995061765932249223od_a_a @ ( inf_in3339382566020358357od_a_a @ A @ B ) @ ( inf_in3339382566020358357od_a_a @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_496_Int__mono,axiom,
! [A: set_set_a,C2: set_set_a,B: set_set_a,D: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ C2 )
=> ( ( ord_le3724670747650509150_set_a @ B @ D )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B ) @ ( inf_inf_set_set_a @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_497_Int__mono,axiom,
! [A: set_Product_prod_a_a,C2: set_Product_prod_a_a,B: set_Product_prod_a_a,D: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ C2 )
=> ( ( ord_le746702958409616551od_a_a @ B @ D )
=> ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A @ B ) @ ( inf_in8905007599844390133od_a_a @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_498_Int__mono,axiom,
! [A: set_a,C2: set_a,B: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_499_inf__set__def,axiom,
( inf_in1205276777018777868_set_a
= ( ^ [A3: set_set_set_a,B3: set_set_set_a] :
( collect_set_set_a
@ ( inf_inf_set_set_a_o
@ ^ [X3: set_set_a] : ( member_set_set_a @ X3 @ A3 )
@ ^ [X3: set_set_a] : ( member_set_set_a @ X3 @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_500_inf__set__def,axiom,
( inf_in3339382566020358357od_a_a
= ( ^ [A3: set_se5735800977113168103od_a_a,B3: set_se5735800977113168103od_a_a] :
( collec1673347964119250290od_a_a
@ ( inf_in1700971893745756232_a_a_o
@ ^ [X3: set_Product_prod_a_a] : ( member1816616512716248880od_a_a @ X3 @ A3 )
@ ^ [X3: set_Product_prod_a_a] : ( member1816616512716248880od_a_a @ X3 @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_501_inf__set__def,axiom,
( inf_inf_set_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
( collect_set_a
@ ( inf_inf_set_a_o
@ ^ [X3: set_a] : ( member_set_a @ X3 @ A3 )
@ ^ [X3: set_a] : ( member_set_a @ X3 @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_502_inf__set__def,axiom,
( inf_in8905007599844390133od_a_a
= ( ^ [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( collec3336397797384452498od_a_a
@ ( inf_in2559554923042384936_a_a_o
@ ^ [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ A3 )
@ ^ [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_503_inf__set__def,axiom,
( inf_inf_set_a
= ( ^ [A3: set_a,B3: set_a] :
( collect_a
@ ( inf_inf_a_o
@ ^ [X3: a] : ( member_a @ X3 @ A3 )
@ ^ [X3: a] : ( member_a @ X3 @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_504_UNIV__witness,axiom,
? [X2: set_set_a] : ( member_set_set_a @ X2 @ top_to4027821306633060462_set_a ) ).
% UNIV_witness
thf(fact_505_UNIV__witness,axiom,
? [X2: set_Product_prod_a_a] : ( member1816616512716248880od_a_a @ X2 @ top_to1047947862415971895od_a_a ) ).
% UNIV_witness
thf(fact_506_UNIV__witness,axiom,
? [X2: set_a] : ( member_set_a @ X2 @ top_top_set_set_a ) ).
% UNIV_witness
thf(fact_507_UNIV__witness,axiom,
? [X2: a] : ( member_a @ X2 @ top_top_set_a ) ).
% UNIV_witness
thf(fact_508_UNIV__witness,axiom,
? [X2: product_prod_a_a] : ( member1426531477525435216od_a_a @ X2 @ top_to8063371432257647191od_a_a ) ).
% UNIV_witness
thf(fact_509_UNIV__eq__I,axiom,
! [A: set_set_set_a] :
( ! [X2: set_set_a] : ( member_set_set_a @ X2 @ A )
=> ( top_to4027821306633060462_set_a = A ) ) ).
% UNIV_eq_I
thf(fact_510_UNIV__eq__I,axiom,
! [A: set_se5735800977113168103od_a_a] :
( ! [X2: set_Product_prod_a_a] : ( member1816616512716248880od_a_a @ X2 @ A )
=> ( top_to1047947862415971895od_a_a = A ) ) ).
% UNIV_eq_I
thf(fact_511_UNIV__eq__I,axiom,
! [A: set_set_a] :
( ! [X2: set_a] : ( member_set_a @ X2 @ A )
=> ( top_top_set_set_a = A ) ) ).
% UNIV_eq_I
thf(fact_512_UNIV__eq__I,axiom,
! [A: set_a] :
( ! [X2: a] : ( member_a @ X2 @ A )
=> ( top_top_set_a = A ) ) ).
% UNIV_eq_I
thf(fact_513_UNIV__eq__I,axiom,
! [A: set_Product_prod_a_a] :
( ! [X2: product_prod_a_a] : ( member1426531477525435216od_a_a @ X2 @ A )
=> ( top_to8063371432257647191od_a_a = A ) ) ).
% UNIV_eq_I
thf(fact_514_inf__fun__def,axiom,
( inf_in2559554923042384936_a_a_o
= ( ^ [F: product_prod_a_a > $o,G: product_prod_a_a > $o,X3: product_prod_a_a] : ( inf_inf_o @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ).
% inf_fun_def
thf(fact_515_inf__fun__def,axiom,
( inf_inf_a_o
= ( ^ [F: a > $o,G: a > $o,X3: a] : ( inf_inf_o @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ).
% inf_fun_def
thf(fact_516_inf__fun__def,axiom,
( inf_inf_a_a_o
= ( ^ [F: a > a > $o,G: a > a > $o,X3: a] : ( inf_inf_a_o @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ).
% inf_fun_def
thf(fact_517_inf__left__commute,axiom,
! [X: set_se5735800977113168103od_a_a,Y: set_se5735800977113168103od_a_a,Z: set_se5735800977113168103od_a_a] :
( ( inf_in3339382566020358357od_a_a @ X @ ( inf_in3339382566020358357od_a_a @ Y @ Z ) )
= ( inf_in3339382566020358357od_a_a @ Y @ ( inf_in3339382566020358357od_a_a @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_518_inf__left__commute,axiom,
! [X: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( inf_inf_set_set_a @ X @ ( inf_inf_set_set_a @ Y @ Z ) )
= ( inf_inf_set_set_a @ Y @ ( inf_inf_set_set_a @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_519_inf__left__commute,axiom,
! [X: product_prod_a_a > $o,Y: product_prod_a_a > $o,Z: product_prod_a_a > $o] :
( ( inf_in2559554923042384936_a_a_o @ X @ ( inf_in2559554923042384936_a_a_o @ Y @ Z ) )
= ( inf_in2559554923042384936_a_a_o @ Y @ ( inf_in2559554923042384936_a_a_o @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_520_inf__left__commute,axiom,
! [X: a > $o,Y: a > $o,Z: a > $o] :
( ( inf_inf_a_o @ X @ ( inf_inf_a_o @ Y @ Z ) )
= ( inf_inf_a_o @ Y @ ( inf_inf_a_o @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_521_inf__left__commute,axiom,
! [X: a > a > $o,Y: a > a > $o,Z: a > a > $o] :
( ( inf_inf_a_a_o @ X @ ( inf_inf_a_a_o @ Y @ Z ) )
= ( inf_inf_a_a_o @ Y @ ( inf_inf_a_a_o @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_522_inf__left__commute,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ X @ ( inf_in8905007599844390133od_a_a @ Y @ Z ) )
= ( inf_in8905007599844390133od_a_a @ Y @ ( inf_in8905007599844390133od_a_a @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_523_inf__left__commute,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_524_inf_Oleft__commute,axiom,
! [B2: set_se5735800977113168103od_a_a,A2: set_se5735800977113168103od_a_a,C: set_se5735800977113168103od_a_a] :
( ( inf_in3339382566020358357od_a_a @ B2 @ ( inf_in3339382566020358357od_a_a @ A2 @ C ) )
= ( inf_in3339382566020358357od_a_a @ A2 @ ( inf_in3339382566020358357od_a_a @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_525_inf_Oleft__commute,axiom,
! [B2: set_set_a,A2: set_set_a,C: set_set_a] :
( ( inf_inf_set_set_a @ B2 @ ( inf_inf_set_set_a @ A2 @ C ) )
= ( inf_inf_set_set_a @ A2 @ ( inf_inf_set_set_a @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_526_inf_Oleft__commute,axiom,
! [B2: product_prod_a_a > $o,A2: product_prod_a_a > $o,C: product_prod_a_a > $o] :
( ( inf_in2559554923042384936_a_a_o @ B2 @ ( inf_in2559554923042384936_a_a_o @ A2 @ C ) )
= ( inf_in2559554923042384936_a_a_o @ A2 @ ( inf_in2559554923042384936_a_a_o @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_527_inf_Oleft__commute,axiom,
! [B2: a > $o,A2: a > $o,C: a > $o] :
( ( inf_inf_a_o @ B2 @ ( inf_inf_a_o @ A2 @ C ) )
= ( inf_inf_a_o @ A2 @ ( inf_inf_a_o @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_528_inf_Oleft__commute,axiom,
! [B2: a > a > $o,A2: a > a > $o,C: a > a > $o] :
( ( inf_inf_a_a_o @ B2 @ ( inf_inf_a_a_o @ A2 @ C ) )
= ( inf_inf_a_a_o @ A2 @ ( inf_inf_a_a_o @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_529_inf_Oleft__commute,axiom,
! [B2: set_Product_prod_a_a,A2: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ B2 @ ( inf_in8905007599844390133od_a_a @ A2 @ C ) )
= ( inf_in8905007599844390133od_a_a @ A2 @ ( inf_in8905007599844390133od_a_a @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_530_inf_Oleft__commute,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( inf_inf_set_a @ B2 @ ( inf_inf_set_a @ A2 @ C ) )
= ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_531_inf__commute,axiom,
( inf_in3339382566020358357od_a_a
= ( ^ [X3: set_se5735800977113168103od_a_a,Y3: set_se5735800977113168103od_a_a] : ( inf_in3339382566020358357od_a_a @ Y3 @ X3 ) ) ) ).
% inf_commute
thf(fact_532_inf__commute,axiom,
( inf_inf_set_set_a
= ( ^ [X3: set_set_a,Y3: set_set_a] : ( inf_inf_set_set_a @ Y3 @ X3 ) ) ) ).
% inf_commute
thf(fact_533_inf__commute,axiom,
( inf_in2559554923042384936_a_a_o
= ( ^ [X3: product_prod_a_a > $o,Y3: product_prod_a_a > $o] : ( inf_in2559554923042384936_a_a_o @ Y3 @ X3 ) ) ) ).
% inf_commute
thf(fact_534_inf__commute,axiom,
( inf_inf_a_o
= ( ^ [X3: a > $o,Y3: a > $o] : ( inf_inf_a_o @ Y3 @ X3 ) ) ) ).
% inf_commute
thf(fact_535_inf__commute,axiom,
( inf_inf_a_a_o
= ( ^ [X3: a > a > $o,Y3: a > a > $o] : ( inf_inf_a_a_o @ Y3 @ X3 ) ) ) ).
% inf_commute
thf(fact_536_inf__commute,axiom,
( inf_in8905007599844390133od_a_a
= ( ^ [X3: set_Product_prod_a_a,Y3: set_Product_prod_a_a] : ( inf_in8905007599844390133od_a_a @ Y3 @ X3 ) ) ) ).
% inf_commute
thf(fact_537_inf__commute,axiom,
( inf_inf_set_a
= ( ^ [X3: set_a,Y3: set_a] : ( inf_inf_set_a @ Y3 @ X3 ) ) ) ).
% inf_commute
thf(fact_538_inf_Ocommute,axiom,
( inf_in3339382566020358357od_a_a
= ( ^ [A4: set_se5735800977113168103od_a_a,B4: set_se5735800977113168103od_a_a] : ( inf_in3339382566020358357od_a_a @ B4 @ A4 ) ) ) ).
% inf.commute
thf(fact_539_inf_Ocommute,axiom,
( inf_inf_set_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] : ( inf_inf_set_set_a @ B4 @ A4 ) ) ) ).
% inf.commute
thf(fact_540_inf_Ocommute,axiom,
( inf_in2559554923042384936_a_a_o
= ( ^ [A4: product_prod_a_a > $o,B4: product_prod_a_a > $o] : ( inf_in2559554923042384936_a_a_o @ B4 @ A4 ) ) ) ).
% inf.commute
thf(fact_541_inf_Ocommute,axiom,
( inf_inf_a_o
= ( ^ [A4: a > $o,B4: a > $o] : ( inf_inf_a_o @ B4 @ A4 ) ) ) ).
% inf.commute
thf(fact_542_inf_Ocommute,axiom,
( inf_inf_a_a_o
= ( ^ [A4: a > a > $o,B4: a > a > $o] : ( inf_inf_a_a_o @ B4 @ A4 ) ) ) ).
% inf.commute
thf(fact_543_inf_Ocommute,axiom,
( inf_in8905007599844390133od_a_a
= ( ^ [A4: set_Product_prod_a_a,B4: set_Product_prod_a_a] : ( inf_in8905007599844390133od_a_a @ B4 @ A4 ) ) ) ).
% inf.commute
thf(fact_544_inf_Ocommute,axiom,
( inf_inf_set_a
= ( ^ [A4: set_a,B4: set_a] : ( inf_inf_set_a @ B4 @ A4 ) ) ) ).
% inf.commute
thf(fact_545_inf__assoc,axiom,
! [X: set_se5735800977113168103od_a_a,Y: set_se5735800977113168103od_a_a,Z: set_se5735800977113168103od_a_a] :
( ( inf_in3339382566020358357od_a_a @ ( inf_in3339382566020358357od_a_a @ X @ Y ) @ Z )
= ( inf_in3339382566020358357od_a_a @ X @ ( inf_in3339382566020358357od_a_a @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_546_inf__assoc,axiom,
! [X: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ X @ Y ) @ Z )
= ( inf_inf_set_set_a @ X @ ( inf_inf_set_set_a @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_547_inf__assoc,axiom,
! [X: product_prod_a_a > $o,Y: product_prod_a_a > $o,Z: product_prod_a_a > $o] :
( ( inf_in2559554923042384936_a_a_o @ ( inf_in2559554923042384936_a_a_o @ X @ Y ) @ Z )
= ( inf_in2559554923042384936_a_a_o @ X @ ( inf_in2559554923042384936_a_a_o @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_548_inf__assoc,axiom,
! [X: a > $o,Y: a > $o,Z: a > $o] :
( ( inf_inf_a_o @ ( inf_inf_a_o @ X @ Y ) @ Z )
= ( inf_inf_a_o @ X @ ( inf_inf_a_o @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_549_inf__assoc,axiom,
! [X: a > a > $o,Y: a > a > $o,Z: a > a > $o] :
( ( inf_inf_a_a_o @ ( inf_inf_a_a_o @ X @ Y ) @ Z )
= ( inf_inf_a_a_o @ X @ ( inf_inf_a_a_o @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_550_inf__assoc,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ ( inf_in8905007599844390133od_a_a @ X @ Y ) @ Z )
= ( inf_in8905007599844390133od_a_a @ X @ ( inf_in8905007599844390133od_a_a @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_551_inf__assoc,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_552_inf_Oassoc,axiom,
! [A2: set_se5735800977113168103od_a_a,B2: set_se5735800977113168103od_a_a,C: set_se5735800977113168103od_a_a] :
( ( inf_in3339382566020358357od_a_a @ ( inf_in3339382566020358357od_a_a @ A2 @ B2 ) @ C )
= ( inf_in3339382566020358357od_a_a @ A2 @ ( inf_in3339382566020358357od_a_a @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_553_inf_Oassoc,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ A2 @ B2 ) @ C )
= ( inf_inf_set_set_a @ A2 @ ( inf_inf_set_set_a @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_554_inf_Oassoc,axiom,
! [A2: product_prod_a_a > $o,B2: product_prod_a_a > $o,C: product_prod_a_a > $o] :
( ( inf_in2559554923042384936_a_a_o @ ( inf_in2559554923042384936_a_a_o @ A2 @ B2 ) @ C )
= ( inf_in2559554923042384936_a_a_o @ A2 @ ( inf_in2559554923042384936_a_a_o @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_555_inf_Oassoc,axiom,
! [A2: a > $o,B2: a > $o,C: a > $o] :
( ( inf_inf_a_o @ ( inf_inf_a_o @ A2 @ B2 ) @ C )
= ( inf_inf_a_o @ A2 @ ( inf_inf_a_o @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_556_inf_Oassoc,axiom,
! [A2: a > a > $o,B2: a > a > $o,C: a > a > $o] :
( ( inf_inf_a_a_o @ ( inf_inf_a_a_o @ A2 @ B2 ) @ C )
= ( inf_inf_a_a_o @ A2 @ ( inf_inf_a_a_o @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_557_inf_Oassoc,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ ( inf_in8905007599844390133od_a_a @ A2 @ B2 ) @ C )
= ( inf_in8905007599844390133od_a_a @ A2 @ ( inf_in8905007599844390133od_a_a @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_558_inf_Oassoc,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C )
= ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_559_inf__sup__aci_I1_J,axiom,
( inf_in3339382566020358357od_a_a
= ( ^ [X3: set_se5735800977113168103od_a_a,Y3: set_se5735800977113168103od_a_a] : ( inf_in3339382566020358357od_a_a @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_560_inf__sup__aci_I1_J,axiom,
( inf_inf_set_set_a
= ( ^ [X3: set_set_a,Y3: set_set_a] : ( inf_inf_set_set_a @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_561_inf__sup__aci_I1_J,axiom,
( inf_in2559554923042384936_a_a_o
= ( ^ [X3: product_prod_a_a > $o,Y3: product_prod_a_a > $o] : ( inf_in2559554923042384936_a_a_o @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_562_inf__sup__aci_I1_J,axiom,
( inf_inf_a_o
= ( ^ [X3: a > $o,Y3: a > $o] : ( inf_inf_a_o @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_563_inf__sup__aci_I1_J,axiom,
( inf_inf_a_a_o
= ( ^ [X3: a > a > $o,Y3: a > a > $o] : ( inf_inf_a_a_o @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_564_inf__sup__aci_I1_J,axiom,
( inf_in8905007599844390133od_a_a
= ( ^ [X3: set_Product_prod_a_a,Y3: set_Product_prod_a_a] : ( inf_in8905007599844390133od_a_a @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_565_inf__sup__aci_I1_J,axiom,
( inf_inf_set_a
= ( ^ [X3: set_a,Y3: set_a] : ( inf_inf_set_a @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_566_inf__sup__aci_I2_J,axiom,
! [X: set_se5735800977113168103od_a_a,Y: set_se5735800977113168103od_a_a,Z: set_se5735800977113168103od_a_a] :
( ( inf_in3339382566020358357od_a_a @ ( inf_in3339382566020358357od_a_a @ X @ Y ) @ Z )
= ( inf_in3339382566020358357od_a_a @ X @ ( inf_in3339382566020358357od_a_a @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_567_inf__sup__aci_I2_J,axiom,
! [X: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ X @ Y ) @ Z )
= ( inf_inf_set_set_a @ X @ ( inf_inf_set_set_a @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_568_inf__sup__aci_I2_J,axiom,
! [X: product_prod_a_a > $o,Y: product_prod_a_a > $o,Z: product_prod_a_a > $o] :
( ( inf_in2559554923042384936_a_a_o @ ( inf_in2559554923042384936_a_a_o @ X @ Y ) @ Z )
= ( inf_in2559554923042384936_a_a_o @ X @ ( inf_in2559554923042384936_a_a_o @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_569_inf__sup__aci_I2_J,axiom,
! [X: a > $o,Y: a > $o,Z: a > $o] :
( ( inf_inf_a_o @ ( inf_inf_a_o @ X @ Y ) @ Z )
= ( inf_inf_a_o @ X @ ( inf_inf_a_o @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_570_inf__sup__aci_I2_J,axiom,
! [X: a > a > $o,Y: a > a > $o,Z: a > a > $o] :
( ( inf_inf_a_a_o @ ( inf_inf_a_a_o @ X @ Y ) @ Z )
= ( inf_inf_a_a_o @ X @ ( inf_inf_a_a_o @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_571_inf__sup__aci_I2_J,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ ( inf_in8905007599844390133od_a_a @ X @ Y ) @ Z )
= ( inf_in8905007599844390133od_a_a @ X @ ( inf_in8905007599844390133od_a_a @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_572_inf__sup__aci_I2_J,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_573_inf__sup__aci_I3_J,axiom,
! [X: set_se5735800977113168103od_a_a,Y: set_se5735800977113168103od_a_a,Z: set_se5735800977113168103od_a_a] :
( ( inf_in3339382566020358357od_a_a @ X @ ( inf_in3339382566020358357od_a_a @ Y @ Z ) )
= ( inf_in3339382566020358357od_a_a @ Y @ ( inf_in3339382566020358357od_a_a @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_574_inf__sup__aci_I3_J,axiom,
! [X: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( inf_inf_set_set_a @ X @ ( inf_inf_set_set_a @ Y @ Z ) )
= ( inf_inf_set_set_a @ Y @ ( inf_inf_set_set_a @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_575_inf__sup__aci_I3_J,axiom,
! [X: product_prod_a_a > $o,Y: product_prod_a_a > $o,Z: product_prod_a_a > $o] :
( ( inf_in2559554923042384936_a_a_o @ X @ ( inf_in2559554923042384936_a_a_o @ Y @ Z ) )
= ( inf_in2559554923042384936_a_a_o @ Y @ ( inf_in2559554923042384936_a_a_o @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_576_inf__sup__aci_I3_J,axiom,
! [X: a > $o,Y: a > $o,Z: a > $o] :
( ( inf_inf_a_o @ X @ ( inf_inf_a_o @ Y @ Z ) )
= ( inf_inf_a_o @ Y @ ( inf_inf_a_o @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_577_inf__sup__aci_I3_J,axiom,
! [X: a > a > $o,Y: a > a > $o,Z: a > a > $o] :
( ( inf_inf_a_a_o @ X @ ( inf_inf_a_a_o @ Y @ Z ) )
= ( inf_inf_a_a_o @ Y @ ( inf_inf_a_a_o @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_578_inf__sup__aci_I3_J,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ X @ ( inf_in8905007599844390133od_a_a @ Y @ Z ) )
= ( inf_in8905007599844390133od_a_a @ Y @ ( inf_in8905007599844390133od_a_a @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_579_inf__sup__aci_I3_J,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_580_inf__sup__aci_I4_J,axiom,
! [X: set_se5735800977113168103od_a_a,Y: set_se5735800977113168103od_a_a] :
( ( inf_in3339382566020358357od_a_a @ X @ ( inf_in3339382566020358357od_a_a @ X @ Y ) )
= ( inf_in3339382566020358357od_a_a @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_581_inf__sup__aci_I4_J,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( inf_inf_set_set_a @ X @ ( inf_inf_set_set_a @ X @ Y ) )
= ( inf_inf_set_set_a @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_582_inf__sup__aci_I4_J,axiom,
! [X: product_prod_a_a > $o,Y: product_prod_a_a > $o] :
( ( inf_in2559554923042384936_a_a_o @ X @ ( inf_in2559554923042384936_a_a_o @ X @ Y ) )
= ( inf_in2559554923042384936_a_a_o @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_583_inf__sup__aci_I4_J,axiom,
! [X: a > $o,Y: a > $o] :
( ( inf_inf_a_o @ X @ ( inf_inf_a_o @ X @ Y ) )
= ( inf_inf_a_o @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_584_inf__sup__aci_I4_J,axiom,
! [X: a > a > $o,Y: a > a > $o] :
( ( inf_inf_a_a_o @ X @ ( inf_inf_a_a_o @ X @ Y ) )
= ( inf_inf_a_a_o @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_585_inf__sup__aci_I4_J,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ X @ ( inf_in8905007599844390133od_a_a @ X @ Y ) )
= ( inf_in8905007599844390133od_a_a @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_586_inf__sup__aci_I4_J,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
= ( inf_inf_set_a @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_587_Int__left__commute,axiom,
! [A: set_se5735800977113168103od_a_a,B: set_se5735800977113168103od_a_a,C2: set_se5735800977113168103od_a_a] :
( ( inf_in3339382566020358357od_a_a @ A @ ( inf_in3339382566020358357od_a_a @ B @ C2 ) )
= ( inf_in3339382566020358357od_a_a @ B @ ( inf_in3339382566020358357od_a_a @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_588_Int__left__commute,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( inf_inf_set_set_a @ A @ ( inf_inf_set_set_a @ B @ C2 ) )
= ( inf_inf_set_set_a @ B @ ( inf_inf_set_set_a @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_589_Int__left__commute,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ A @ ( inf_in8905007599844390133od_a_a @ B @ C2 ) )
= ( inf_in8905007599844390133od_a_a @ B @ ( inf_in8905007599844390133od_a_a @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_590_Int__left__commute,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C2 ) )
= ( inf_inf_set_a @ B @ ( inf_inf_set_a @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_591_Int__left__absorb,axiom,
! [A: set_se5735800977113168103od_a_a,B: set_se5735800977113168103od_a_a] :
( ( inf_in3339382566020358357od_a_a @ A @ ( inf_in3339382566020358357od_a_a @ A @ B ) )
= ( inf_in3339382566020358357od_a_a @ A @ B ) ) ).
% Int_left_absorb
thf(fact_592_Int__left__absorb,axiom,
! [A: set_set_a,B: set_set_a] :
( ( inf_inf_set_set_a @ A @ ( inf_inf_set_set_a @ A @ B ) )
= ( inf_inf_set_set_a @ A @ B ) ) ).
% Int_left_absorb
thf(fact_593_Int__left__absorb,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ A @ ( inf_in8905007599844390133od_a_a @ A @ B ) )
= ( inf_in8905007599844390133od_a_a @ A @ B ) ) ).
% Int_left_absorb
thf(fact_594_Int__left__absorb,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ A @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ).
% Int_left_absorb
thf(fact_595_Int__commute,axiom,
( inf_in3339382566020358357od_a_a
= ( ^ [A3: set_se5735800977113168103od_a_a,B3: set_se5735800977113168103od_a_a] : ( inf_in3339382566020358357od_a_a @ B3 @ A3 ) ) ) ).
% Int_commute
thf(fact_596_Int__commute,axiom,
( inf_inf_set_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] : ( inf_inf_set_set_a @ B3 @ A3 ) ) ) ).
% Int_commute
thf(fact_597_Int__commute,axiom,
( inf_in8905007599844390133od_a_a
= ( ^ [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] : ( inf_in8905007599844390133od_a_a @ B3 @ A3 ) ) ) ).
% Int_commute
thf(fact_598_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A3: set_a,B3: set_a] : ( inf_inf_set_a @ B3 @ A3 ) ) ) ).
% Int_commute
thf(fact_599_Int__absorb,axiom,
! [A: set_se5735800977113168103od_a_a] :
( ( inf_in3339382566020358357od_a_a @ A @ A )
= A ) ).
% Int_absorb
thf(fact_600_Int__absorb,axiom,
! [A: set_set_a] :
( ( inf_inf_set_set_a @ A @ A )
= A ) ).
% Int_absorb
thf(fact_601_Int__absorb,axiom,
! [A: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ A @ A )
= A ) ).
% Int_absorb
thf(fact_602_Int__absorb,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ A )
= A ) ).
% Int_absorb
thf(fact_603_Int__assoc,axiom,
! [A: set_se5735800977113168103od_a_a,B: set_se5735800977113168103od_a_a,C2: set_se5735800977113168103od_a_a] :
( ( inf_in3339382566020358357od_a_a @ ( inf_in3339382566020358357od_a_a @ A @ B ) @ C2 )
= ( inf_in3339382566020358357od_a_a @ A @ ( inf_in3339382566020358357od_a_a @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_604_Int__assoc,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ A @ B ) @ C2 )
= ( inf_inf_set_set_a @ A @ ( inf_inf_set_set_a @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_605_Int__assoc,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ ( inf_in8905007599844390133od_a_a @ A @ B ) @ C2 )
= ( inf_in8905007599844390133od_a_a @ A @ ( inf_in8905007599844390133od_a_a @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_606_Int__assoc,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ C2 )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_607_IntD2,axiom,
! [C: set_set_a,A: set_set_set_a,B: set_set_set_a] :
( ( member_set_set_a @ C @ ( inf_in1205276777018777868_set_a @ A @ B ) )
=> ( member_set_set_a @ C @ B ) ) ).
% IntD2
thf(fact_608_IntD2,axiom,
! [C: set_Product_prod_a_a,A: set_se5735800977113168103od_a_a,B: set_se5735800977113168103od_a_a] :
( ( member1816616512716248880od_a_a @ C @ ( inf_in3339382566020358357od_a_a @ A @ B ) )
=> ( member1816616512716248880od_a_a @ C @ B ) ) ).
% IntD2
thf(fact_609_IntD2,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) )
=> ( member_set_a @ C @ B ) ) ).
% IntD2
thf(fact_610_IntD2,axiom,
! [C: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A @ B ) )
=> ( member1426531477525435216od_a_a @ C @ B ) ) ).
% IntD2
thf(fact_611_IntD2,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
=> ( member_a @ C @ B ) ) ).
% IntD2
thf(fact_612_IntD1,axiom,
! [C: set_set_a,A: set_set_set_a,B: set_set_set_a] :
( ( member_set_set_a @ C @ ( inf_in1205276777018777868_set_a @ A @ B ) )
=> ( member_set_set_a @ C @ A ) ) ).
% IntD1
thf(fact_613_IntD1,axiom,
! [C: set_Product_prod_a_a,A: set_se5735800977113168103od_a_a,B: set_se5735800977113168103od_a_a] :
( ( member1816616512716248880od_a_a @ C @ ( inf_in3339382566020358357od_a_a @ A @ B ) )
=> ( member1816616512716248880od_a_a @ C @ A ) ) ).
% IntD1
thf(fact_614_IntD1,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) )
=> ( member_set_a @ C @ A ) ) ).
% IntD1
thf(fact_615_IntD1,axiom,
! [C: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A @ B ) )
=> ( member1426531477525435216od_a_a @ C @ A ) ) ).
% IntD1
thf(fact_616_IntD1,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
=> ( member_a @ C @ A ) ) ).
% IntD1
thf(fact_617_IntE,axiom,
! [C: set_set_a,A: set_set_set_a,B: set_set_set_a] :
( ( member_set_set_a @ C @ ( inf_in1205276777018777868_set_a @ A @ B ) )
=> ~ ( ( member_set_set_a @ C @ A )
=> ~ ( member_set_set_a @ C @ B ) ) ) ).
% IntE
thf(fact_618_IntE,axiom,
! [C: set_Product_prod_a_a,A: set_se5735800977113168103od_a_a,B: set_se5735800977113168103od_a_a] :
( ( member1816616512716248880od_a_a @ C @ ( inf_in3339382566020358357od_a_a @ A @ B ) )
=> ~ ( ( member1816616512716248880od_a_a @ C @ A )
=> ~ ( member1816616512716248880od_a_a @ C @ B ) ) ) ).
% IntE
thf(fact_619_IntE,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) )
=> ~ ( ( member_set_a @ C @ A )
=> ~ ( member_set_a @ C @ B ) ) ) ).
% IntE
thf(fact_620_IntE,axiom,
! [C: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A @ B ) )
=> ~ ( ( member1426531477525435216od_a_a @ C @ A )
=> ~ ( member1426531477525435216od_a_a @ C @ B ) ) ) ).
% IntE
thf(fact_621_IntE,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
=> ~ ( ( member_a @ C @ A )
=> ~ ( member_a @ C @ B ) ) ) ).
% IntE
thf(fact_622_UNIV__def,axiom,
( top_to1047947862415971895od_a_a
= ( collec1673347964119250290od_a_a
@ ^ [X3: set_Product_prod_a_a] : $true ) ) ).
% UNIV_def
thf(fact_623_UNIV__def,axiom,
( top_top_set_set_a
= ( collect_set_a
@ ^ [X3: set_a] : $true ) ) ).
% UNIV_def
thf(fact_624_UNIV__def,axiom,
( top_top_set_a
= ( collect_a
@ ^ [X3: a] : $true ) ) ).
% UNIV_def
thf(fact_625_UNIV__def,axiom,
( top_to8063371432257647191od_a_a
= ( collec3336397797384452498od_a_a
@ ^ [X3: product_prod_a_a] : $true ) ) ).
% UNIV_def
thf(fact_626_Collect__conj__eq,axiom,
! [P: set_Product_prod_a_a > $o,Q: set_Product_prod_a_a > $o] :
( ( collec1673347964119250290od_a_a
@ ^ [X3: set_Product_prod_a_a] :
( ( P @ X3 )
& ( Q @ X3 ) ) )
= ( inf_in3339382566020358357od_a_a @ ( collec1673347964119250290od_a_a @ P ) @ ( collec1673347964119250290od_a_a @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_627_Collect__conj__eq,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( collect_set_a
@ ^ [X3: set_a] :
( ( P @ X3 )
& ( Q @ X3 ) ) )
= ( inf_inf_set_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_628_Collect__conj__eq,axiom,
! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ( collec3336397797384452498od_a_a
@ ^ [X3: product_prod_a_a] :
( ( P @ X3 )
& ( Q @ X3 ) ) )
= ( inf_in8905007599844390133od_a_a @ ( collec3336397797384452498od_a_a @ P ) @ ( collec3336397797384452498od_a_a @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_629_Collect__conj__eq,axiom,
! [P: a > $o,Q: a > $o] :
( ( collect_a
@ ^ [X3: a] :
( ( P @ X3 )
& ( Q @ X3 ) ) )
= ( inf_inf_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_630_Int__Collect,axiom,
! [X: set_set_a,A: set_set_set_a,P: set_set_a > $o] :
( ( member_set_set_a @ X @ ( inf_in1205276777018777868_set_a @ A @ ( collect_set_set_a @ P ) ) )
= ( ( member_set_set_a @ X @ A )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_631_Int__Collect,axiom,
! [X: set_Product_prod_a_a,A: set_se5735800977113168103od_a_a,P: set_Product_prod_a_a > $o] :
( ( member1816616512716248880od_a_a @ X @ ( inf_in3339382566020358357od_a_a @ A @ ( collec1673347964119250290od_a_a @ P ) ) )
= ( ( member1816616512716248880od_a_a @ X @ A )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_632_Int__Collect,axiom,
! [X: set_a,A: set_set_a,P: set_a > $o] :
( ( member_set_a @ X @ ( inf_inf_set_set_a @ A @ ( collect_set_a @ P ) ) )
= ( ( member_set_a @ X @ A )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_633_Int__Collect,axiom,
! [X: product_prod_a_a,A: set_Product_prod_a_a,P: product_prod_a_a > $o] :
( ( member1426531477525435216od_a_a @ X @ ( inf_in8905007599844390133od_a_a @ A @ ( collec3336397797384452498od_a_a @ P ) ) )
= ( ( member1426531477525435216od_a_a @ X @ A )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_634_Int__Collect,axiom,
! [X: a,A: set_a,P: a > $o] :
( ( member_a @ X @ ( inf_inf_set_a @ A @ ( collect_a @ P ) ) )
= ( ( member_a @ X @ A )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_635_Int__def,axiom,
( inf_in1205276777018777868_set_a
= ( ^ [A3: set_set_set_a,B3: set_set_set_a] :
( collect_set_set_a
@ ^ [X3: set_set_a] :
( ( member_set_set_a @ X3 @ A3 )
& ( member_set_set_a @ X3 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_636_Int__def,axiom,
( inf_in3339382566020358357od_a_a
= ( ^ [A3: set_se5735800977113168103od_a_a,B3: set_se5735800977113168103od_a_a] :
( collec1673347964119250290od_a_a
@ ^ [X3: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X3 @ A3 )
& ( member1816616512716248880od_a_a @ X3 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_637_Int__def,axiom,
( inf_inf_set_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
( collect_set_a
@ ^ [X3: set_a] :
( ( member_set_a @ X3 @ A3 )
& ( member_set_a @ X3 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_638_Int__def,axiom,
( inf_in8905007599844390133od_a_a
= ( ^ [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( collec3336397797384452498od_a_a
@ ^ [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A3 )
& ( member1426531477525435216od_a_a @ X3 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_639_Int__def,axiom,
( inf_inf_set_a
= ( ^ [A3: set_a,B3: set_a] :
( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A3 )
& ( member_a @ X3 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_640_Times__eq__cancel2,axiom,
! [X: product_prod_a_a,C2: set_Product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ C2 )
=> ( ( ( produc5899993699339346696od_a_a @ A
@ ^ [Uu: product_prod_a_a] : C2 )
= ( produc5899993699339346696od_a_a @ B
@ ^ [Uu: product_prod_a_a] : C2 ) )
= ( A = B ) ) ) ).
% Times_eq_cancel2
thf(fact_641_Times__eq__cancel2,axiom,
! [X: a,C2: set_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member_a @ X @ C2 )
=> ( ( ( produc2379640491490746847_a_a_a @ A
@ ^ [Uu: product_prod_a_a] : C2 )
= ( produc2379640491490746847_a_a_a @ B
@ ^ [Uu: product_prod_a_a] : C2 ) )
= ( A = B ) ) ) ).
% Times_eq_cancel2
thf(fact_642_Times__eq__cancel2,axiom,
! [X: product_prod_a_a,C2: set_Product_prod_a_a,A: set_a,B: set_a] :
( ( member1426531477525435216od_a_a @ X @ C2 )
=> ( ( ( produc6342321021181284593od_a_a @ A
@ ^ [Uu: a] : C2 )
= ( produc6342321021181284593od_a_a @ B
@ ^ [Uu: a] : C2 ) )
= ( A = B ) ) ) ).
% Times_eq_cancel2
thf(fact_643_Times__eq__cancel2,axiom,
! [X: a,C2: set_a,A: set_a,B: set_a] :
( ( member_a @ X @ C2 )
=> ( ( ( product_Sigma_a_a @ A
@ ^ [Uu: a] : C2 )
= ( product_Sigma_a_a @ B
@ ^ [Uu: a] : C2 ) )
= ( A = B ) ) ) ).
% Times_eq_cancel2
thf(fact_644_Sigma__cong,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C2: product_prod_a_a > set_Product_prod_a_a,D: product_prod_a_a > set_Product_prod_a_a] :
( ( A = B )
=> ( ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( produc5899993699339346696od_a_a @ A @ C2 )
= ( produc5899993699339346696od_a_a @ B @ D ) ) ) ) ).
% Sigma_cong
thf(fact_645_Sigma__cong,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C2: product_prod_a_a > set_a,D: product_prod_a_a > set_a] :
( ( A = B )
=> ( ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( produc2379640491490746847_a_a_a @ A @ C2 )
= ( produc2379640491490746847_a_a_a @ B @ D ) ) ) ) ).
% Sigma_cong
thf(fact_646_Sigma__cong,axiom,
! [A: set_a,B: set_a,C2: a > set_Product_prod_a_a,D: a > set_Product_prod_a_a] :
( ( A = B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( produc6342321021181284593od_a_a @ A @ C2 )
= ( produc6342321021181284593od_a_a @ B @ D ) ) ) ) ).
% Sigma_cong
thf(fact_647_Sigma__cong,axiom,
! [A: set_a,B: set_a,C2: a > set_a,D: a > set_a] :
( ( A = B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( product_Sigma_a_a @ A @ C2 )
= ( product_Sigma_a_a @ B @ D ) ) ) ) ).
% Sigma_cong
thf(fact_648_Int__UNIV__right,axiom,
! [A: set_se5735800977113168103od_a_a] :
( ( inf_in3339382566020358357od_a_a @ A @ top_to1047947862415971895od_a_a )
= A ) ).
% Int_UNIV_right
thf(fact_649_Int__UNIV__right,axiom,
! [A: set_set_a] :
( ( inf_inf_set_set_a @ A @ top_top_set_set_a )
= A ) ).
% Int_UNIV_right
thf(fact_650_Int__UNIV__right,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ top_top_set_a )
= A ) ).
% Int_UNIV_right
thf(fact_651_Int__UNIV__right,axiom,
! [A: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ A @ top_to8063371432257647191od_a_a )
= A ) ).
% Int_UNIV_right
thf(fact_652_Int__UNIV__left,axiom,
! [B: set_set_a] :
( ( inf_inf_set_set_a @ top_top_set_set_a @ B )
= B ) ).
% Int_UNIV_left
thf(fact_653_Int__UNIV__left,axiom,
! [B: set_a] :
( ( inf_inf_set_a @ top_top_set_a @ B )
= B ) ).
% Int_UNIV_left
thf(fact_654_Int__UNIV__left,axiom,
! [B: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ top_to8063371432257647191od_a_a @ B )
= B ) ).
% Int_UNIV_left
thf(fact_655_Sigma__Int__distrib1,axiom,
! [I: set_a,J: set_a,C2: a > set_a] :
( ( product_Sigma_a_a @ ( inf_inf_set_a @ I @ J ) @ C2 )
= ( inf_in8905007599844390133od_a_a @ ( product_Sigma_a_a @ I @ C2 ) @ ( product_Sigma_a_a @ J @ C2 ) ) ) ).
% Sigma_Int_distrib1
thf(fact_656_Times__Int__distrib1,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( product_Sigma_a_a @ ( inf_inf_set_a @ A @ B )
@ ^ [Uu: a] : C2 )
= ( inf_in8905007599844390133od_a_a
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : C2 )
@ ( product_Sigma_a_a @ B
@ ^ [Uu: a] : C2 ) ) ) ).
% Times_Int_distrib1
thf(fact_657_Sigma__Int__distrib2,axiom,
! [I: set_a,A: a > set_a,B: a > set_a] :
( ( product_Sigma_a_a @ I
@ ^ [I2: a] : ( inf_inf_set_a @ ( A @ I2 ) @ ( B @ I2 ) ) )
= ( inf_in8905007599844390133od_a_a @ ( product_Sigma_a_a @ I @ A ) @ ( product_Sigma_a_a @ I @ B ) ) ) ).
% Sigma_Int_distrib2
thf(fact_658_Times__Int__Times,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C2: set_Product_prod_a_a,D: set_Product_prod_a_a] :
( ( inf_in4058781255473215669od_a_a
@ ( produc5899993699339346696od_a_a @ A
@ ^ [Uu: product_prod_a_a] : B )
@ ( produc5899993699339346696od_a_a @ C2
@ ^ [Uu: product_prod_a_a] : D ) )
= ( produc5899993699339346696od_a_a @ ( inf_in8905007599844390133od_a_a @ A @ C2 )
@ ^ [Uu: product_prod_a_a] : ( inf_in8905007599844390133od_a_a @ B @ D ) ) ) ).
% Times_Int_Times
thf(fact_659_Times__Int__Times,axiom,
! [A: set_Product_prod_a_a,B: set_a,C2: set_Product_prod_a_a,D: set_a] :
( ( inf_in690575969517268876_a_a_a
@ ( produc2379640491490746847_a_a_a @ A
@ ^ [Uu: product_prod_a_a] : B )
@ ( produc2379640491490746847_a_a_a @ C2
@ ^ [Uu: product_prod_a_a] : D ) )
= ( produc2379640491490746847_a_a_a @ ( inf_in8905007599844390133od_a_a @ A @ C2 )
@ ^ [Uu: product_prod_a_a] : ( inf_inf_set_a @ B @ D ) ) ) ).
% Times_Int_Times
thf(fact_660_Times__Int__Times,axiom,
! [A: set_a,B: set_Product_prod_a_a,C2: set_a,D: set_Product_prod_a_a] :
( ( inf_in6567511182131981598od_a_a
@ ( produc6342321021181284593od_a_a @ A
@ ^ [Uu: a] : B )
@ ( produc6342321021181284593od_a_a @ C2
@ ^ [Uu: a] : D ) )
= ( produc6342321021181284593od_a_a @ ( inf_inf_set_a @ A @ C2 )
@ ^ [Uu: a] : ( inf_in8905007599844390133od_a_a @ B @ D ) ) ) ).
% Times_Int_Times
thf(fact_661_Times__Int__Times,axiom,
! [A: set_a,B: set_a,C2: set_a,D: set_a] :
( ( inf_in8905007599844390133od_a_a
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : B )
@ ( product_Sigma_a_a @ C2
@ ^ [Uu: a] : D ) )
= ( product_Sigma_a_a @ ( inf_inf_set_a @ A @ C2 )
@ ^ [Uu: a] : ( inf_inf_set_a @ B @ D ) ) ) ).
% Times_Int_Times
thf(fact_662_Restr__subset,axiom,
! [A: set_a,B: set_a,R: set_Product_prod_a_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( inf_in8905007599844390133od_a_a
@ ( inf_in8905007599844390133od_a_a @ R
@ ( product_Sigma_a_a @ B
@ ^ [Uu: a] : B ) )
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : A ) )
= ( inf_in8905007599844390133od_a_a @ R
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : A ) ) ) ) ).
% Restr_subset
thf(fact_663_refl__on__Int,axiom,
! [A: set_Product_prod_a_a,R: set_Pr8600417178894128327od_a_a,B: set_Product_prod_a_a,S: set_Pr8600417178894128327od_a_a] :
( ( refl_o7745108929832855590od_a_a @ A @ R )
=> ( ( refl_o7745108929832855590od_a_a @ B @ S )
=> ( refl_o7745108929832855590od_a_a @ ( inf_in8905007599844390133od_a_a @ A @ B ) @ ( inf_in4058781255473215669od_a_a @ R @ S ) ) ) ) ).
% refl_on_Int
thf(fact_664_refl__on__Int,axiom,
! [A: set_a,R: set_Product_prod_a_a,B: set_a,S: set_Product_prod_a_a] :
( ( refl_on_a @ A @ R )
=> ( ( refl_on_a @ B @ S )
=> ( refl_on_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_in8905007599844390133od_a_a @ R @ S ) ) ) ) ).
% refl_on_Int
thf(fact_665_sym__Int,axiom,
! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( sym_on_a @ top_top_set_a @ R )
=> ( ( sym_on_a @ top_top_set_a @ S )
=> ( sym_on_a @ top_top_set_a @ ( inf_in8905007599844390133od_a_a @ R @ S ) ) ) ) ).
% sym_Int
thf(fact_666_sym__Int,axiom,
! [R: set_Pr8600417178894128327od_a_a,S: set_Pr8600417178894128327od_a_a] :
( ( sym_on5631557199876295240od_a_a @ top_to8063371432257647191od_a_a @ R )
=> ( ( sym_on5631557199876295240od_a_a @ top_to8063371432257647191od_a_a @ S )
=> ( sym_on5631557199876295240od_a_a @ top_to8063371432257647191od_a_a @ ( inf_in4058781255473215669od_a_a @ R @ S ) ) ) ) ).
% sym_Int
thf(fact_667_iso__tuple__UNIV__I,axiom,
! [X: a] : ( member_a @ X @ top_top_set_a ) ).
% iso_tuple_UNIV_I
thf(fact_668_iso__tuple__UNIV__I,axiom,
! [X: product_prod_a_a] : ( member1426531477525435216od_a_a @ X @ top_to8063371432257647191od_a_a ) ).
% iso_tuple_UNIV_I
thf(fact_669_order__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% order_refl
thf(fact_670_dual__order_Orefl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_671_sym__on__subset,axiom,
! [A: set_a,R: set_Product_prod_a_a,B: set_a] :
( ( sym_on_a @ A @ R )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( sym_on_a @ B @ R ) ) ) ).
% sym_on_subset
thf(fact_672_boolean__algebra_Oconj__one__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ top_top_set_a )
= X ) ).
% boolean_algebra.conj_one_right
thf(fact_673_boolean__algebra_Oconj__one__right,axiom,
! [X: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ X @ top_to8063371432257647191od_a_a )
= X ) ).
% boolean_algebra.conj_one_right
thf(fact_674_top_Oextremum__uniqueI,axiom,
! [A2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ top_to8063371432257647191od_a_a @ A2 )
=> ( A2 = top_to8063371432257647191od_a_a ) ) ).
% top.extremum_uniqueI
thf(fact_675_top_Oextremum__uniqueI,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A2 )
=> ( A2 = top_top_set_a ) ) ).
% top.extremum_uniqueI
thf(fact_676_top_Oextremum__unique,axiom,
! [A2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ top_to8063371432257647191od_a_a @ A2 )
= ( A2 = top_to8063371432257647191od_a_a ) ) ).
% top.extremum_unique
thf(fact_677_top_Oextremum__unique,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A2 )
= ( A2 = top_top_set_a ) ) ).
% top.extremum_unique
thf(fact_678_top__greatest,axiom,
! [A2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ A2 @ top_to8063371432257647191od_a_a ) ).
% top_greatest
thf(fact_679_top__greatest,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ top_top_set_a ) ).
% top_greatest
thf(fact_680_equiv__on__unique,axiom,
! [A: set_a,P2: set_Product_prod_a_a,B: set_a] :
( ( equiv_equiv_a @ A @ P2 )
=> ( ( equiv_equiv_a @ B @ P2 )
=> ( A = B ) ) ) ).
% equiv_on_unique
thf(fact_681_pred__subset__eq,axiom,
! [R2: set_a,S2: set_a] :
( ( ord_less_eq_a_o
@ ^ [X3: a] : ( member_a @ X3 @ R2 )
@ ^ [X3: a] : ( member_a @ X3 @ S2 ) )
= ( ord_less_eq_set_a @ R2 @ S2 ) ) ).
% pred_subset_eq
thf(fact_682_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ord_less_eq_a_o
@ ^ [X3: a] : ( member_a @ X3 @ A3 )
@ ^ [X3: a] : ( member_a @ X3 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_683_order__antisym__conv,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_684_ord__le__eq__subst,axiom,
! [A2: set_a,B2: set_a,F2: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ( F2 @ B2 )
= C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F2 @ X2 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F2 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_685_ord__eq__le__subst,axiom,
! [A2: set_a,F2: set_a > set_a,B2: set_a,C: set_a] :
( ( A2
= ( F2 @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F2 @ X2 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_686_order__eq__refl,axiom,
! [X: set_a,Y: set_a] :
( ( X = Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_687_order__subst2,axiom,
! [A2: set_a,B2: set_a,F2: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ ( F2 @ B2 ) @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F2 @ X2 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_688_order__subst1,axiom,
! [A2: set_a,F2: set_a > set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F2 @ X2 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_689_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_a,Z2: set_a] : ( Y2 = Z2 ) )
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_690_antisym,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_691_dual__order_Otrans,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ C @ B2 )
=> ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_692_dual__order_Oantisym,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_693_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_a,Z2: set_a] : ( Y2 = Z2 ) )
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A4 )
& ( ord_less_eq_set_a @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_694_order__trans,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z )
=> ( ord_less_eq_set_a @ X @ Z ) ) ) ).
% order_trans
thf(fact_695_order_Otrans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_696_order__antisym,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_697_ord__le__eq__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_698_ord__eq__le__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( A2 = B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_699_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_a,Z2: set_a] : ( Y2 = Z2 ) )
= ( ^ [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
& ( ord_less_eq_set_a @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_700_top__empty__eq,axiom,
( top_top_a_o
= ( ^ [X3: a] : ( member_a @ X3 @ top_top_set_a ) ) ) ).
% top_empty_eq
thf(fact_701_top__empty__eq,axiom,
( top_to8687885267596698950_a_a_o
= ( ^ [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ top_to8063371432257647191od_a_a ) ) ) ).
% top_empty_eq
thf(fact_702_boolean__algebra__cancel_Oinf1,axiom,
! [A: set_Product_prod_a_a,K: set_Product_prod_a_a,A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( A
= ( inf_in8905007599844390133od_a_a @ K @ A2 ) )
=> ( ( inf_in8905007599844390133od_a_a @ A @ B2 )
= ( inf_in8905007599844390133od_a_a @ K @ ( inf_in8905007599844390133od_a_a @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_703_boolean__algebra__cancel_Oinf1,axiom,
! [A: set_a,K: set_a,A2: set_a,B2: set_a] :
( ( A
= ( inf_inf_set_a @ K @ A2 ) )
=> ( ( inf_inf_set_a @ A @ B2 )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_704_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_Product_prod_a_a,K: set_Product_prod_a_a,B2: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( B
= ( inf_in8905007599844390133od_a_a @ K @ B2 ) )
=> ( ( inf_in8905007599844390133od_a_a @ A2 @ B )
= ( inf_in8905007599844390133od_a_a @ K @ ( inf_in8905007599844390133od_a_a @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_705_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_a,K: set_a,B2: set_a,A2: set_a] :
( ( B
= ( inf_inf_set_a @ K @ B2 ) )
=> ( ( inf_inf_set_a @ A2 @ B )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_706_inf__Int__eq,axiom,
! [R2: set_Product_prod_a_a,S2: set_Product_prod_a_a] :
( ( inf_in2559554923042384936_a_a_o
@ ^ [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ R2 )
@ ^ [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ S2 ) )
= ( ^ [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ ( inf_in8905007599844390133od_a_a @ R2 @ S2 ) ) ) ) ).
% inf_Int_eq
thf(fact_707_inf__Int__eq,axiom,
! [R2: set_a,S2: set_a] :
( ( inf_inf_a_o
@ ^ [X3: a] : ( member_a @ X3 @ R2 )
@ ^ [X3: a] : ( member_a @ X3 @ S2 ) )
= ( ^ [X3: a] : ( member_a @ X3 @ ( inf_inf_set_a @ R2 @ S2 ) ) ) ) ).
% inf_Int_eq
thf(fact_708_equiv__type,axiom,
! [A: set_a,R: set_Product_prod_a_a] :
( ( equiv_equiv_a @ A @ R )
=> ( ord_le746702958409616551od_a_a @ R
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : A ) ) ) ).
% equiv_type
thf(fact_709_Product__Type_Oproduct__def,axiom,
( product_product_a_a
= ( ^ [A3: set_a,B3: set_a] :
( product_Sigma_a_a @ A3
@ ^ [Uu: a] : B3 ) ) ) ).
% Product_Type.product_def
thf(fact_710_member__product,axiom,
! [X: product_prod_a_a,A: set_a,B: set_a] :
( ( member1426531477525435216od_a_a @ X @ ( product_product_a_a @ A @ B ) )
= ( member1426531477525435216od_a_a @ X
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : B ) ) ) ).
% member_product
thf(fact_711_Refl__Field__Restr,axiom,
! [R: set_Pr8600417178894128327od_a_a,A: set_Product_prod_a_a] :
( ( refl_o7745108929832855590od_a_a @ ( field_1126092520709947252od_a_a @ R ) @ R )
=> ( ( field_1126092520709947252od_a_a
@ ( inf_in4058781255473215669od_a_a @ R
@ ( produc5899993699339346696od_a_a @ A
@ ^ [Uu: product_prod_a_a] : A ) ) )
= ( inf_in8905007599844390133od_a_a @ ( field_1126092520709947252od_a_a @ R ) @ A ) ) ) ).
% Refl_Field_Restr
thf(fact_712_Refl__Field__Restr,axiom,
! [R: set_Product_prod_a_a,A: set_a] :
( ( refl_on_a @ ( field_a @ R ) @ R )
=> ( ( field_a
@ ( inf_in8905007599844390133od_a_a @ R
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : A ) ) )
= ( inf_inf_set_a @ ( field_a @ R ) @ A ) ) ) ).
% Refl_Field_Restr
thf(fact_713_Refl__Field__Restr2,axiom,
! [R: set_Product_prod_a_a,A: set_a] :
( ( refl_on_a @ ( field_a @ R ) @ R )
=> ( ( ord_less_eq_set_a @ A @ ( field_a @ R ) )
=> ( ( field_a
@ ( inf_in8905007599844390133od_a_a @ R
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : A ) ) )
= A ) ) ) ).
% Refl_Field_Restr2
thf(fact_714_subset__Collect__iff,axiom,
! [B: set_a,A: set_a,P: a > $o] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ B
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A )
& ( P @ X3 ) ) ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ B )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_715_subset__CollectI,axiom,
! [B: set_a,A: set_a,Q: a > $o,P: a > $o] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( ( Q @ X2 )
=> ( P @ X2 ) ) )
=> ( ord_less_eq_set_a
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ B )
& ( Q @ X3 ) ) )
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_716_Field__square,axiom,
! [X: set_a] :
( ( field_a
@ ( product_Sigma_a_a @ X
@ ^ [Uu: a] : X ) )
= X ) ).
% Field_square
thf(fact_717_mono__Field,axiom,
! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ R @ S )
=> ( ord_less_eq_set_a @ ( field_a @ R ) @ ( field_a @ S ) ) ) ).
% mono_Field
thf(fact_718_Restr__Field,axiom,
! [R: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ R
@ ( product_Sigma_a_a @ ( field_a @ R )
@ ^ [Uu: a] : ( field_a @ R ) ) )
= R ) ).
% Restr_Field
thf(fact_719_Field__Restr__subset,axiom,
! [R: set_Product_prod_a_a,A: set_a] :
( ord_less_eq_set_a
@ ( field_a
@ ( inf_in8905007599844390133od_a_a @ R
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : A ) ) )
@ A ) ).
% Field_Restr_subset
thf(fact_720_Refl__Restr,axiom,
! [R: set_Product_prod_a_a,A: set_a] :
( ( refl_on_a @ ( field_a @ R ) @ R )
=> ( refl_on_a
@ ( field_a
@ ( inf_in8905007599844390133od_a_a @ R
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : A ) ) )
@ ( inf_in8905007599844390133od_a_a @ R
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : A ) ) ) ) ).
% Refl_Restr
thf(fact_721_Preorder__Restr,axiom,
! [R: set_Product_prod_a_a,A: set_a] :
( ( order_preorder_on_a @ ( field_a @ R ) @ R )
=> ( order_preorder_on_a
@ ( field_a
@ ( inf_in8905007599844390133od_a_a @ R
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : A ) ) )
@ ( inf_in8905007599844390133od_a_a @ R
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : A ) ) ) ) ).
% Preorder_Restr
thf(fact_722_Partial__order__Restr,axiom,
! [R: set_Product_prod_a_a,A: set_a] :
( ( order_5272072345360262643r_on_a @ ( field_a @ R ) @ R )
=> ( order_5272072345360262643r_on_a
@ ( field_a
@ ( inf_in8905007599844390133od_a_a @ R
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : A ) ) )
@ ( inf_in8905007599844390133od_a_a @ R
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : A ) ) ) ) ).
% Partial_order_Restr
thf(fact_723_Linear__order__Restr,axiom,
! [R: set_Product_prod_a_a,A: set_a] :
( ( order_8768733634509060147r_on_a @ ( field_a @ R ) @ R )
=> ( order_8768733634509060147r_on_a
@ ( field_a
@ ( inf_in8905007599844390133od_a_a @ R
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : A ) ) )
@ ( inf_in8905007599844390133od_a_a @ R
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : A ) ) ) ) ).
% Linear_order_Restr
thf(fact_724_conj__subset__def,axiom,
! [A: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A
@ ( collect_a
@ ^ [X3: a] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) )
= ( ( ord_less_eq_set_a @ A @ ( collect_a @ P ) )
& ( ord_less_eq_set_a @ A @ ( collect_a @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_725_prop__restrict,axiom,
! [X: a,Z4: set_a,X4: set_a,P: a > $o] :
( ( member_a @ X @ Z4 )
=> ( ( ord_less_eq_set_a @ Z4
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ X4 )
& ( P @ X3 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_726_Collect__restrict,axiom,
! [X4: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ X4 )
& ( P @ X3 ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_727_equiv__def,axiom,
( equiv_equiv_a
= ( ^ [A3: set_a,R3: set_Product_prod_a_a] :
( ( refl_on_a @ A3 @ R3 )
& ( sym_on_a @ top_top_set_a @ R3 )
& ( trans_on_a @ top_top_set_a @ R3 ) ) ) ) ).
% equiv_def
thf(fact_728_equiv__def,axiom,
( equiv_4910910634973128413od_a_a
= ( ^ [A3: set_Product_prod_a_a,R3: set_Pr8600417178894128327od_a_a] :
( ( refl_o7745108929832855590od_a_a @ A3 @ R3 )
& ( sym_on5631557199876295240od_a_a @ top_to8063371432257647191od_a_a @ R3 )
& ( trans_8517530828069161671od_a_a @ top_to8063371432257647191od_a_a @ R3 ) ) ) ) ).
% equiv_def
thf(fact_729_equivI,axiom,
! [A: set_a,R: set_Product_prod_a_a] :
( ( refl_on_a @ A @ R )
=> ( ( sym_on_a @ top_top_set_a @ R )
=> ( ( trans_on_a @ top_top_set_a @ R )
=> ( equiv_equiv_a @ A @ R ) ) ) ) ).
% equivI
thf(fact_730_equivI,axiom,
! [A: set_Product_prod_a_a,R: set_Pr8600417178894128327od_a_a] :
( ( refl_o7745108929832855590od_a_a @ A @ R )
=> ( ( sym_on5631557199876295240od_a_a @ top_to8063371432257647191od_a_a @ R )
=> ( ( trans_8517530828069161671od_a_a @ top_to8063371432257647191od_a_a @ R )
=> ( equiv_4910910634973128413od_a_a @ A @ R ) ) ) ) ).
% equivI
thf(fact_731_trans__on__subset,axiom,
! [A: set_a,R: set_Product_prod_a_a,B: set_a] :
( ( trans_on_a @ A @ R )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( trans_on_a @ B @ R ) ) ) ).
% trans_on_subset
thf(fact_732_trans__Int,axiom,
! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( trans_on_a @ top_top_set_a @ R )
=> ( ( trans_on_a @ top_top_set_a @ S )
=> ( trans_on_a @ top_top_set_a @ ( inf_in8905007599844390133od_a_a @ R @ S ) ) ) ) ).
% trans_Int
thf(fact_733_trans__Int,axiom,
! [R: set_Pr8600417178894128327od_a_a,S: set_Pr8600417178894128327od_a_a] :
( ( trans_8517530828069161671od_a_a @ top_to8063371432257647191od_a_a @ R )
=> ( ( trans_8517530828069161671od_a_a @ top_to8063371432257647191od_a_a @ S )
=> ( trans_8517530828069161671od_a_a @ top_to8063371432257647191od_a_a @ ( inf_in4058781255473215669od_a_a @ R @ S ) ) ) ) ).
% trans_Int
thf(fact_734_trans__Restr,axiom,
! [R: set_Product_prod_a_a,A: set_a] :
( ( trans_on_a @ top_top_set_a @ R )
=> ( trans_on_a @ top_top_set_a
@ ( inf_in8905007599844390133od_a_a @ R
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : A ) ) ) ) ).
% trans_Restr
thf(fact_735_trans__Restr,axiom,
! [R: set_Pr8600417178894128327od_a_a,A: set_Product_prod_a_a] :
( ( trans_8517530828069161671od_a_a @ top_to8063371432257647191od_a_a @ R )
=> ( trans_8517530828069161671od_a_a @ top_to8063371432257647191od_a_a
@ ( inf_in4058781255473215669od_a_a @ R
@ ( produc5899993699339346696od_a_a @ A
@ ^ [Uu: product_prod_a_a] : A ) ) ) ) ).
% trans_Restr
thf(fact_736_equivE,axiom,
! [A: set_a,R: set_Product_prod_a_a] :
( ( equiv_equiv_a @ A @ R )
=> ~ ( ( refl_on_a @ A @ R )
=> ( ( sym_on_a @ top_top_set_a @ R )
=> ~ ( trans_on_a @ top_top_set_a @ R ) ) ) ) ).
% equivE
thf(fact_737_equivE,axiom,
! [A: set_Product_prod_a_a,R: set_Pr8600417178894128327od_a_a] :
( ( equiv_4910910634973128413od_a_a @ A @ R )
=> ~ ( ( refl_o7745108929832855590od_a_a @ A @ R )
=> ( ( sym_on5631557199876295240od_a_a @ top_to8063371432257647191od_a_a @ R )
=> ~ ( trans_8517530828069161671od_a_a @ top_to8063371432257647191od_a_a @ R ) ) ) ) ).
% equivE
thf(fact_738_preorder__on__def,axiom,
( order_preorder_on_a
= ( ^ [A3: set_a,R3: set_Product_prod_a_a] :
( ( refl_on_a @ A3 @ R3 )
& ( trans_on_a @ top_top_set_a @ R3 ) ) ) ) ).
% preorder_on_def
thf(fact_739_preorder__on__def,axiom,
( order_3202267349275844158od_a_a
= ( ^ [A3: set_Product_prod_a_a,R3: set_Pr8600417178894128327od_a_a] :
( ( refl_o7745108929832855590od_a_a @ A3 @ R3 )
& ( trans_8517530828069161671od_a_a @ top_to8063371432257647191od_a_a @ R3 ) ) ) ) ).
% preorder_on_def
thf(fact_740_linear__order__on__Restr,axiom,
! [A: set_Product_prod_a_a,R: set_Pr8600417178894128327od_a_a,X: product_prod_a_a] :
( ( order_435519263512085852od_a_a @ A @ R )
=> ( order_435519263512085852od_a_a @ ( inf_in8905007599844390133od_a_a @ A @ ( order_147278640462067751od_a_a @ R @ X ) )
@ ( inf_in4058781255473215669od_a_a @ R
@ ( produc5899993699339346696od_a_a @ ( order_147278640462067751od_a_a @ R @ X )
@ ^ [Uu: product_prod_a_a] : ( order_147278640462067751od_a_a @ R @ X ) ) ) ) ) ).
% linear_order_on_Restr
thf(fact_741_linear__order__on__Restr,axiom,
! [A: set_a,R: set_Product_prod_a_a,X: a] :
( ( order_8768733634509060147r_on_a @ A @ R )
=> ( order_8768733634509060147r_on_a @ ( inf_inf_set_a @ A @ ( order_above_a @ R @ X ) )
@ ( inf_in8905007599844390133od_a_a @ R
@ ( product_Sigma_a_a @ ( order_above_a @ R @ X )
@ ^ [Uu: a] : ( order_above_a @ R @ X ) ) ) ) ) ).
% linear_order_on_Restr
thf(fact_742_partial__order__onD_I2_J,axiom,
! [A: set_a,R: set_Product_prod_a_a] :
( ( order_5272072345360262643r_on_a @ A @ R )
=> ( trans_on_a @ top_top_set_a @ R ) ) ).
% partial_order_onD(2)
thf(fact_743_partial__order__onD_I2_J,axiom,
! [A: set_Product_prod_a_a,R: set_Pr8600417178894128327od_a_a] :
( ( order_7408868903334687516od_a_a @ A @ R )
=> ( trans_8517530828069161671od_a_a @ top_to8063371432257647191od_a_a @ R ) ) ).
% partial_order_onD(2)
thf(fact_744_partial__order__onD_I1_J,axiom,
! [A: set_a,R: set_Product_prod_a_a] :
( ( order_5272072345360262643r_on_a @ A @ R )
=> ( refl_on_a @ A @ R ) ) ).
% partial_order_onD(1)
thf(fact_745_antisym__Restr,axiom,
! [R: set_Product_prod_a_a,A: set_a] :
( ( antisym_on_a @ top_top_set_a @ R )
=> ( antisym_on_a @ top_top_set_a
@ ( inf_in8905007599844390133od_a_a @ R
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : A ) ) ) ) ).
% antisym_Restr
thf(fact_746_antisym__Restr,axiom,
! [R: set_Pr8600417178894128327od_a_a,A: set_Product_prod_a_a] :
( ( antisy9008168496540133130od_a_a @ top_to8063371432257647191od_a_a @ R )
=> ( antisy9008168496540133130od_a_a @ top_to8063371432257647191od_a_a
@ ( inf_in4058781255473215669od_a_a @ R
@ ( produc5899993699339346696od_a_a @ A
@ ^ [Uu: product_prod_a_a] : A ) ) ) ) ).
% antisym_Restr
thf(fact_747_well__order__on__Restr,axiom,
! [R: set_Product_prod_a_a,A: set_a] :
( ( order_6972113574731384220r_on_a @ ( field_a @ R ) @ R )
=> ( ( ord_less_eq_set_a @ A @ ( field_a @ R ) )
=> ( order_6972113574731384220r_on_a @ A
@ ( inf_in8905007599844390133od_a_a @ R
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : A ) ) ) ) ) ).
% well_order_on_Restr
thf(fact_748_Total__Restr,axiom,
! [R: set_Product_prod_a_a,A: set_a] :
( ( total_on_a @ ( field_a @ R ) @ R )
=> ( total_on_a
@ ( field_a
@ ( inf_in8905007599844390133od_a_a @ R
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : A ) ) )
@ ( inf_in8905007599844390133od_a_a @ R
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : A ) ) ) ) ).
% Total_Restr
thf(fact_749_total__on__imp__Total__Restr,axiom,
! [A: set_a,R: set_Product_prod_a_a] :
( ( total_on_a @ A @ R )
=> ( total_on_a
@ ( field_a
@ ( inf_in8905007599844390133od_a_a @ R
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : A ) ) )
@ ( inf_in8905007599844390133od_a_a @ R
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : A ) ) ) ) ).
% total_on_imp_Total_Restr
thf(fact_750_total__on__subset,axiom,
! [A: set_a,R: set_Product_prod_a_a,B: set_a] :
( ( total_on_a @ A @ R )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( total_on_a @ B @ R ) ) ) ).
% total_on_subset
thf(fact_751_antisym__on__subset,axiom,
! [A: set_a,R: set_Product_prod_a_a,B: set_a] :
( ( antisym_on_a @ A @ R )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( antisym_on_a @ B @ R ) ) ) ).
% antisym_on_subset
thf(fact_752_partial__order__onD_I3_J,axiom,
! [A: set_a,R: set_Product_prod_a_a] :
( ( order_5272072345360262643r_on_a @ A @ R )
=> ( antisym_on_a @ top_top_set_a @ R ) ) ).
% partial_order_onD(3)
thf(fact_753_partial__order__onD_I3_J,axiom,
! [A: set_Product_prod_a_a,R: set_Pr8600417178894128327od_a_a] :
( ( order_7408868903334687516od_a_a @ A @ R )
=> ( antisy9008168496540133130od_a_a @ top_to8063371432257647191od_a_a @ R ) ) ).
% partial_order_onD(3)
thf(fact_754_antisym__subset,axiom,
! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ R @ S )
=> ( ( antisym_on_a @ top_top_set_a @ S )
=> ( antisym_on_a @ top_top_set_a @ R ) ) ) ).
% antisym_subset
thf(fact_755_antisym__subset,axiom,
! [R: set_Pr8600417178894128327od_a_a,S: set_Pr8600417178894128327od_a_a] :
( ( ord_le3469131294019144807od_a_a @ R @ S )
=> ( ( antisy9008168496540133130od_a_a @ top_to8063371432257647191od_a_a @ S )
=> ( antisy9008168496540133130od_a_a @ top_to8063371432257647191od_a_a @ R ) ) ) ).
% antisym_subset
thf(fact_756_partial__order__on__def,axiom,
( order_5272072345360262643r_on_a
= ( ^ [A3: set_a,R3: set_Product_prod_a_a] :
( ( order_preorder_on_a @ A3 @ R3 )
& ( antisym_on_a @ top_top_set_a @ R3 ) ) ) ) ).
% partial_order_on_def
thf(fact_757_partial__order__on__def,axiom,
( order_7408868903334687516od_a_a
= ( ^ [A3: set_Product_prod_a_a,R3: set_Pr8600417178894128327od_a_a] :
( ( order_3202267349275844158od_a_a @ A3 @ R3 )
& ( antisy9008168496540133130od_a_a @ top_to8063371432257647191od_a_a @ R3 ) ) ) ) ).
% partial_order_on_def
thf(fact_758_Well__order__Restr,axiom,
! [R: set_Product_prod_a_a,A: set_a] :
( ( order_6972113574731384220r_on_a @ ( field_a @ R ) @ R )
=> ( order_6972113574731384220r_on_a
@ ( field_a
@ ( inf_in8905007599844390133od_a_a @ R
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : A ) ) )
@ ( inf_in8905007599844390133od_a_a @ R
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : A ) ) ) ) ).
% Well_order_Restr
thf(fact_759_well__ordering,axiom,
? [R4: set_Product_prod_a_a] :
( ( order_6972113574731384220r_on_a @ ( field_a @ R4 ) @ R4 )
& ( ( field_a @ R4 )
= top_top_set_a ) ) ).
% well_ordering
thf(fact_760_well__ordering,axiom,
? [R4: set_Pr8600417178894128327od_a_a] :
( ( order_1514034925359088069od_a_a @ ( field_1126092520709947252od_a_a @ R4 ) @ R4 )
& ( ( field_1126092520709947252od_a_a @ R4 )
= top_to8063371432257647191od_a_a ) ) ).
% well_ordering
thf(fact_761_ofilter__Restr__Int,axiom,
! [R: set_Pr8600417178894128327od_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( order_1514034925359088069od_a_a @ ( field_1126092520709947252od_a_a @ R ) @ R )
=> ( ( order_4948930231512469437od_a_a @ R @ A )
=> ( order_4948930231512469437od_a_a
@ ( inf_in4058781255473215669od_a_a @ R
@ ( produc5899993699339346696od_a_a @ B
@ ^ [Uu: product_prod_a_a] : B ) )
@ ( inf_in8905007599844390133od_a_a @ A @ B ) ) ) ) ).
% ofilter_Restr_Int
thf(fact_762_ofilter__Restr__Int,axiom,
! [R: set_Product_prod_a_a,A: set_a,B: set_a] :
( ( order_6972113574731384220r_on_a @ ( field_a @ R ) @ R )
=> ( ( order_ofilter_a @ R @ A )
=> ( order_ofilter_a
@ ( inf_in8905007599844390133od_a_a @ R
@ ( product_Sigma_a_a @ B
@ ^ [Uu: a] : B ) )
@ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% ofilter_Restr_Int
thf(fact_763_ofilter__Restr__subset,axiom,
! [R: set_Product_prod_a_a,A: set_a,B: set_a] :
( ( order_6972113574731384220r_on_a @ ( field_a @ R ) @ R )
=> ( ( order_ofilter_a @ R @ A )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( order_ofilter_a
@ ( inf_in8905007599844390133od_a_a @ R
@ ( product_Sigma_a_a @ B
@ ^ [Uu: a] : B ) )
@ A ) ) ) ) ).
% ofilter_Restr_subset
thf(fact_764_bsqr__antisym,axiom,
! [R: set_Product_prod_a_a] :
( ( order_6972113574731384220r_on_a @ ( field_a @ R ) @ R )
=> ( antisy9008168496540133130od_a_a @ top_to8063371432257647191od_a_a @ ( bNF_Wellorder_bsqr_a @ R ) ) ) ).
% bsqr_antisym
thf(fact_765_bsqr__Trans,axiom,
! [R: set_Product_prod_a_a] :
( ( order_6972113574731384220r_on_a @ ( field_a @ R ) @ R )
=> ( trans_8517530828069161671od_a_a @ top_to8063371432257647191od_a_a @ ( bNF_Wellorder_bsqr_a @ R ) ) ) ).
% bsqr_Trans
thf(fact_766_Field__Restr__ofilter,axiom,
! [R: set_Product_prod_a_a,A: set_a] :
( ( order_6972113574731384220r_on_a @ ( field_a @ R ) @ R )
=> ( ( order_ofilter_a @ R @ A )
=> ( ( field_a
@ ( inf_in8905007599844390133od_a_a @ R
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : A ) ) )
= A ) ) ) ).
% Field_Restr_ofilter
thf(fact_767_trans__lex__prod,axiom,
! [R_A: set_Product_prod_a_a,R_B: set_Product_prod_a_a] :
( ( trans_on_a @ top_top_set_a @ R_A )
=> ( ( trans_on_a @ top_top_set_a @ R_B )
=> ( trans_8517530828069161671od_a_a @ top_to8063371432257647191od_a_a @ ( lex_prod_a_a @ R_A @ R_B ) ) ) ) ).
% trans_lex_prod
thf(fact_768_trans__lex__prod,axiom,
! [R_A: set_Product_prod_a_a,R_B: set_Pr8600417178894128327od_a_a] :
( ( trans_on_a @ top_top_set_a @ R_A )
=> ( ( trans_8517530828069161671od_a_a @ top_to8063371432257647191od_a_a @ R_B )
=> ( trans_5289099189627683440od_a_a @ top_to4273090908018391168od_a_a @ ( lex_pr1992703945780986968od_a_a @ R_A @ R_B ) ) ) ) ).
% trans_lex_prod
thf(fact_769_trans__lex__prod,axiom,
! [R_A: set_Pr8600417178894128327od_a_a,R_B: set_Product_prod_a_a] :
( ( trans_8517530828069161671od_a_a @ top_to8063371432257647191od_a_a @ R_A )
=> ( ( trans_on_a @ top_top_set_a @ R_B )
=> ( trans_5047893653580667870_a_a_a @ top_to7619527732258454254_a_a_a @ ( lex_pr7253395452945225030_a_a_a @ R_A @ R_B ) ) ) ) ).
% trans_lex_prod
thf(fact_770_trans__lex__prod,axiom,
! [R_A: set_Pr8600417178894128327od_a_a,R_B: set_Pr8600417178894128327od_a_a] :
( ( trans_8517530828069161671od_a_a @ top_to8063371432257647191od_a_a @ R_A )
=> ( ( trans_8517530828069161671od_a_a @ top_to8063371432257647191od_a_a @ R_B )
=> ( trans_2240019041493443719od_a_a @ top_to7953187161931797015od_a_a @ ( lex_pr3295122962250848111od_a_a @ R_A @ R_B ) ) ) ) ).
% trans_lex_prod
thf(fact_771_trans__on__lex__prod,axiom,
! [A: set_a,R_A: set_Product_prod_a_a,B: set_a,R_B: set_Product_prod_a_a] :
( ( trans_on_a @ A @ R_A )
=> ( ( trans_on_a @ B @ R_B )
=> ( trans_8517530828069161671od_a_a
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : B )
@ ( lex_prod_a_a @ R_A @ R_B ) ) ) ) ).
% trans_on_lex_prod
thf(fact_772_total__on__lex__prod,axiom,
! [A: set_a,R_A: set_Product_prod_a_a,B: set_a,R_B: set_Product_prod_a_a] :
( ( total_on_a @ A @ R_A )
=> ( ( total_on_a @ B @ R_B )
=> ( total_3048927264068743339od_a_a
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : B )
@ ( lex_prod_a_a @ R_A @ R_B ) ) ) ) ).
% total_on_lex_prod
thf(fact_773_sym__on__lex__prod,axiom,
! [A: set_a,R_A: set_Product_prod_a_a,B: set_a,R_B: set_Product_prod_a_a] :
( ( sym_on_a @ A @ R_A )
=> ( ( sym_on_a @ B @ R_B )
=> ( sym_on5631557199876295240od_a_a
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : B )
@ ( lex_prod_a_a @ R_A @ R_B ) ) ) ) ).
% sym_on_lex_prod
thf(fact_774_refl__lex__prod,axiom,
! [R_B: set_Product_prod_a_a,R_A: set_Product_prod_a_a] :
( ( refl_on_a @ top_top_set_a @ R_B )
=> ( refl_o7745108929832855590od_a_a @ top_to8063371432257647191od_a_a @ ( lex_prod_a_a @ R_A @ R_B ) ) ) ).
% refl_lex_prod
thf(fact_775_total__lex__prod,axiom,
! [R_A: set_Product_prod_a_a,R_B: set_Product_prod_a_a] :
( ( total_on_a @ top_top_set_a @ R_A )
=> ( ( total_on_a @ top_top_set_a @ R_B )
=> ( total_3048927264068743339od_a_a @ top_to8063371432257647191od_a_a @ ( lex_prod_a_a @ R_A @ R_B ) ) ) ) ).
% total_lex_prod
thf(fact_776_total__lex__prod,axiom,
! [R_A: set_Product_prod_a_a,R_B: set_Pr8600417178894128327od_a_a] :
( ( total_on_a @ top_top_set_a @ R_A )
=> ( ( total_3048927264068743339od_a_a @ top_to8063371432257647191od_a_a @ R_B )
=> ( total_5054530119720166484od_a_a @ top_to4273090908018391168od_a_a @ ( lex_pr1992703945780986968od_a_a @ R_A @ R_B ) ) ) ) ).
% total_lex_prod
thf(fact_777_total__lex__prod,axiom,
! [R_A: set_Pr8600417178894128327od_a_a,R_B: set_Product_prod_a_a] :
( ( total_3048927264068743339od_a_a @ top_to8063371432257647191od_a_a @ R_A )
=> ( ( total_on_a @ top_top_set_a @ R_B )
=> ( total_4813324583673150914_a_a_a @ top_to7619527732258454254_a_a_a @ ( lex_pr7253395452945225030_a_a_a @ R_A @ R_B ) ) ) ) ).
% total_lex_prod
thf(fact_778_total__lex__prod,axiom,
! [R_A: set_Pr8600417178894128327od_a_a,R_B: set_Pr8600417178894128327od_a_a] :
( ( total_3048927264068743339od_a_a @ top_to8063371432257647191od_a_a @ R_A )
=> ( ( total_3048927264068743339od_a_a @ top_to8063371432257647191od_a_a @ R_B )
=> ( total_755287270405358187od_a_a @ top_to7953187161931797015od_a_a @ ( lex_pr3295122962250848111od_a_a @ R_A @ R_B ) ) ) ) ).
% total_lex_prod
thf(fact_779_sym__lex__prod,axiom,
! [R_A: set_Product_prod_a_a,R_B: set_Product_prod_a_a] :
( ( sym_on_a @ top_top_set_a @ R_A )
=> ( ( sym_on_a @ top_top_set_a @ R_B )
=> ( sym_on5631557199876295240od_a_a @ top_to8063371432257647191od_a_a @ ( lex_prod_a_a @ R_A @ R_B ) ) ) ) ).
% sym_lex_prod
thf(fact_780_sym__lex__prod,axiom,
! [R_A: set_Product_prod_a_a,R_B: set_Pr8600417178894128327od_a_a] :
( ( sym_on_a @ top_top_set_a @ R_A )
=> ( ( sym_on5631557199876295240od_a_a @ top_to8063371432257647191od_a_a @ R_B )
=> ( sym_on3335588482587531505od_a_a @ top_to4273090908018391168od_a_a @ ( lex_pr1992703945780986968od_a_a @ R_A @ R_B ) ) ) ) ).
% sym_lex_prod
thf(fact_781_sym__lex__prod,axiom,
! [R_A: set_Pr8600417178894128327od_a_a,R_B: set_Product_prod_a_a] :
( ( sym_on5631557199876295240od_a_a @ top_to8063371432257647191od_a_a @ R_A )
=> ( ( sym_on_a @ top_top_set_a @ R_B )
=> ( sym_on3094382946540515935_a_a_a @ top_to7619527732258454254_a_a_a @ ( lex_pr7253395452945225030_a_a_a @ R_A @ R_B ) ) ) ) ).
% sym_lex_prod
thf(fact_782_sym__lex__prod,axiom,
! [R_A: set_Pr8600417178894128327od_a_a,R_B: set_Pr8600417178894128327od_a_a] :
( ( sym_on5631557199876295240od_a_a @ top_to8063371432257647191od_a_a @ R_A )
=> ( ( sym_on5631557199876295240od_a_a @ top_to8063371432257647191od_a_a @ R_B )
=> ( sym_on6500787941963452936od_a_a @ top_to7953187161931797015od_a_a @ ( lex_pr3295122962250848111od_a_a @ R_A @ R_B ) ) ) ) ).
% sym_lex_prod
thf(fact_783_Field__relation__of,axiom,
! [A: set_a,P: a > a > $o] :
( ( refl_on_a @ A @ ( order_relation_of_a @ P @ A ) )
=> ( ( field_a @ ( order_relation_of_a @ P @ A ) )
= A ) ) ).
% Field_relation_of
thf(fact_784_Field__bsqr,axiom,
! [R: set_Product_prod_a_a] :
( ( field_1126092520709947252od_a_a @ ( bNF_Wellorder_bsqr_a @ R ) )
= ( product_Sigma_a_a @ ( field_a @ R )
@ ^ [Uu: a] : ( field_a @ R ) ) ) ).
% Field_bsqr
thf(fact_785_ofilter__Restr__under,axiom,
! [R: set_Product_prod_a_a,A: set_a,A2: a] :
( ( order_6972113574731384220r_on_a @ ( field_a @ R ) @ R )
=> ( ( order_ofilter_a @ R @ A )
=> ( ( member_a @ A2 @ A )
=> ( ( order_under_a
@ ( inf_in8905007599844390133od_a_a @ R
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : A ) )
@ A2 )
= ( order_under_a @ R @ A2 ) ) ) ) ) ).
% ofilter_Restr_under
thf(fact_786_Chains__relation__of,axiom,
! [C2: set_a,P: a > a > $o,A: set_a] :
( ( member_set_a @ C2 @ ( chains_a @ ( order_relation_of_a @ P @ A ) ) )
=> ( ord_less_eq_set_a @ C2 @ A ) ) ).
% Chains_relation_of
thf(fact_787_ofilter__def,axiom,
( order_ofilter_a
= ( ^ [R3: set_Product_prod_a_a,A3: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( field_a @ R3 ) )
& ! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ( ord_less_eq_set_a @ ( order_under_a @ R3 @ X3 ) @ A3 ) ) ) ) ) ).
% ofilter_def
thf(fact_788_times__subset__iff,axiom,
! [A: set_a,C2: set_a,B: set_a,D: set_a] :
( ( ord_le746702958409616551od_a_a
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : C2 )
@ ( product_Sigma_a_a @ B
@ ^ [Uu: a] : D ) )
= ( ( A = bot_bot_set_a )
| ( C2 = bot_bot_set_a )
| ( ( ord_less_eq_set_a @ A @ B )
& ( ord_less_eq_set_a @ C2 @ D ) ) ) ) ).
% times_subset_iff
thf(fact_789_refl__on__def,axiom,
( refl_on_a
= ( ^ [A3: set_a,R3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ R3
@ ( product_Sigma_a_a @ A3
@ ^ [Uu: a] : A3 ) )
& ! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ X3 ) @ R3 ) ) ) ) ) ).
% refl_on_def
thf(fact_790_refl__onI,axiom,
! [R: set_Product_prod_a_a,A: set_a] :
( ( ord_le746702958409616551od_a_a @ R
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : A ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ X2 ) @ R ) )
=> ( refl_on_a @ A @ R ) ) ) ).
% refl_onI
thf(fact_791_subset__empty,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_792_empty__subsetI,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% empty_subsetI
thf(fact_793_inf__bot__right,axiom,
! [X: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ X @ bot_bo3357376287454694259od_a_a )
= bot_bo3357376287454694259od_a_a ) ).
% inf_bot_right
thf(fact_794_inf__bot__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ bot_bot_set_a )
= bot_bot_set_a ) ).
% inf_bot_right
thf(fact_795_inf__bot__left,axiom,
! [X: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ bot_bo3357376287454694259od_a_a @ X )
= bot_bo3357376287454694259od_a_a ) ).
% inf_bot_left
thf(fact_796_inf__bot__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X )
= bot_bot_set_a ) ).
% inf_bot_left
thf(fact_797_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ X @ bot_bo3357376287454694259od_a_a )
= bot_bo3357376287454694259od_a_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_798_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ bot_bot_set_a )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_799_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ bot_bo3357376287454694259od_a_a @ X )
= bot_bo3357376287454694259od_a_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_800_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_801_mem__Sigma__iff,axiom,
! [A2: a,B2: a,A: set_a,B: a > set_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B2 ) @ ( product_Sigma_a_a @ A @ B ) )
= ( ( member_a @ A2 @ A )
& ( member_a @ B2 @ ( B @ A2 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_802_SigmaI,axiom,
! [A2: a,A: set_a,B2: a,B: a > set_a] :
( ( member_a @ A2 @ A )
=> ( ( member_a @ B2 @ ( B @ A2 ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B2 ) @ ( product_Sigma_a_a @ A @ B ) ) ) ) ).
% SigmaI
thf(fact_803_Sigma__empty1,axiom,
! [B: a > set_a] :
( ( product_Sigma_a_a @ bot_bot_set_a @ B )
= bot_bo3357376287454694259od_a_a ) ).
% Sigma_empty1
thf(fact_804_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collect_a
@ ^ [S3: a] : P )
= top_top_set_a ) )
& ( ~ P
=> ( ( collect_a
@ ^ [S3: a] : P )
= bot_bot_set_a ) ) ) ).
% Collect_const
thf(fact_805_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collec3336397797384452498od_a_a
@ ^ [S3: product_prod_a_a] : P )
= top_to8063371432257647191od_a_a ) )
& ( ~ P
=> ( ( collec3336397797384452498od_a_a
@ ^ [S3: product_prod_a_a] : P )
= bot_bo3357376287454694259od_a_a ) ) ) ).
% Collect_const
thf(fact_806_Sigma__empty2,axiom,
! [A: set_a] :
( ( product_Sigma_a_a @ A
@ ^ [Uu: a] : bot_bot_set_a )
= bot_bo3357376287454694259od_a_a ) ).
% Sigma_empty2
thf(fact_807_Times__empty,axiom,
! [A: set_a,B: set_a] :
( ( ( product_Sigma_a_a @ A
@ ^ [Uu: a] : B )
= bot_bo3357376287454694259od_a_a )
= ( ( A = bot_bot_set_a )
| ( B = bot_bot_set_a ) ) ) ).
% Times_empty
thf(fact_808_refl__on__empty,axiom,
refl_on_a @ bot_bot_set_a @ bot_bo3357376287454694259od_a_a ).
% refl_on_empty
thf(fact_809_Sigma__empty__iff,axiom,
! [I: set_a,X4: a > set_a] :
( ( ( product_Sigma_a_a @ I @ X4 )
= bot_bo3357376287454694259od_a_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ I )
=> ( ( X4 @ X3 )
= bot_bot_set_a ) ) ) ) ).
% Sigma_empty_iff
thf(fact_810_bot_Oextremum,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% bot.extremum
thf(fact_811_bot_Oextremum__unique,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_812_bot_Oextremum__uniqueI,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
=> ( A2 = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_813_SigmaE2,axiom,
! [A2: a,B2: a,A: set_a,B: a > set_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B2 ) @ ( product_Sigma_a_a @ A @ B ) )
=> ~ ( ( member_a @ A2 @ A )
=> ~ ( member_a @ B2 @ ( B @ A2 ) ) ) ) ).
% SigmaE2
thf(fact_814_SigmaD2,axiom,
! [A2: a,B2: a,A: set_a,B: a > set_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B2 ) @ ( product_Sigma_a_a @ A @ B ) )
=> ( member_a @ B2 @ ( B @ A2 ) ) ) ).
% SigmaD2
thf(fact_815_SigmaD1,axiom,
! [A2: a,B2: a,A: set_a,B: a > set_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B2 ) @ ( product_Sigma_a_a @ A @ B ) )
=> ( member_a @ A2 @ A ) ) ).
% SigmaD1
thf(fact_816_SigmaE,axiom,
! [C: product_prod_a_a,A: set_a,B: a > set_a] :
( ( member1426531477525435216od_a_a @ C @ ( product_Sigma_a_a @ A @ B ) )
=> ~ ! [X2: a] :
( ( member_a @ X2 @ A )
=> ! [Y4: a] :
( ( member_a @ Y4 @ ( B @ X2 ) )
=> ( C
!= ( product_Pair_a_a @ X2 @ Y4 ) ) ) ) ) ).
% SigmaE
thf(fact_817_empty__not__UNIV,axiom,
bot_bot_set_a != top_top_set_a ).
% empty_not_UNIV
thf(fact_818_empty__not__UNIV,axiom,
bot_bo3357376287454694259od_a_a != top_to8063371432257647191od_a_a ).
% empty_not_UNIV
thf(fact_819_subset__emptyI,axiom,
! [A: set_a] :
( ! [X2: a] :
~ ( member_a @ X2 @ A )
=> ( ord_less_eq_set_a @ A @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_820_disjoint__iff__not__equal,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ A @ B )
= bot_bo3357376287454694259od_a_a )
= ( ! [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A )
=> ! [Y3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y3 @ B )
=> ( X3 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_821_disjoint__iff__not__equal,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ! [Y3: a] :
( ( member_a @ Y3 @ B )
=> ( X3 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_822_Int__empty__right,axiom,
! [A: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ A @ bot_bo3357376287454694259od_a_a )
= bot_bo3357376287454694259od_a_a ) ).
% Int_empty_right
thf(fact_823_Int__empty__right,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Int_empty_right
thf(fact_824_Int__empty__left,axiom,
! [B: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ bot_bo3357376287454694259od_a_a @ B )
= bot_bo3357376287454694259od_a_a ) ).
% Int_empty_left
thf(fact_825_Int__empty__left,axiom,
! [B: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ B )
= bot_bot_set_a ) ).
% Int_empty_left
thf(fact_826_disjoint__iff,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ A @ B )
= bot_bo3357376287454694259od_a_a )
= ( ! [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A )
=> ~ ( member1426531477525435216od_a_a @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_827_disjoint__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ~ ( member_a @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_828_Int__emptyI,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A )
=> ~ ( member1426531477525435216od_a_a @ X2 @ B ) )
=> ( ( inf_in8905007599844390133od_a_a @ A @ B )
= bot_bo3357376287454694259od_a_a ) ) ).
% Int_emptyI
thf(fact_829_Int__emptyI,axiom,
! [A: set_a,B: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ~ ( member_a @ X2 @ B ) )
=> ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_830_refl__on__domain,axiom,
! [A: set_a,R: set_Product_prod_a_a,A2: a,B2: a] :
( ( refl_on_a @ A @ R )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B2 ) @ R )
=> ( ( member_a @ A2 @ A )
& ( member_a @ B2 @ A ) ) ) ) ).
% refl_on_domain
thf(fact_831_refl__onD,axiom,
! [A: set_a,R: set_Product_prod_a_a,A2: a] :
( ( refl_on_a @ A @ R )
=> ( ( member_a @ A2 @ A )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ A2 ) @ R ) ) ) ).
% refl_onD
thf(fact_832_refl__onD1,axiom,
! [A: set_a,R: set_Product_prod_a_a,X: a,Y: a] :
( ( refl_on_a @ A @ R )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
=> ( member_a @ X @ A ) ) ) ).
% refl_onD1
thf(fact_833_refl__onD2,axiom,
! [A: set_a,R: set_Product_prod_a_a,X: a,Y: a] :
( ( refl_on_a @ A @ R )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
=> ( member_a @ Y @ A ) ) ) ).
% refl_onD2
thf(fact_834_sym__on__def,axiom,
( sym_on_a
= ( ^ [A3: set_a,R3: set_Product_prod_a_a] :
! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ! [Y3: a] :
( ( member_a @ Y3 @ A3 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R3 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ X3 ) @ R3 ) ) ) ) ) ) ).
% sym_on_def
thf(fact_835_sym__onI,axiom,
! [A: set_a,R: set_Product_prod_a_a] :
( ! [X2: a,Y4: a] :
( ( member_a @ X2 @ A )
=> ( ( member_a @ Y4 @ A )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y4 ) @ R )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ X2 ) @ R ) ) ) )
=> ( sym_on_a @ A @ R ) ) ).
% sym_onI
thf(fact_836_sym__onD,axiom,
! [A: set_a,R: set_Product_prod_a_a,X: a,Y: a] :
( ( sym_on_a @ A @ R )
=> ( ( member_a @ X @ A )
=> ( ( member_a @ Y @ A )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ X ) @ R ) ) ) ) ) ).
% sym_onD
thf(fact_837_under__incr,axiom,
! [R: set_Pr8600417178894128327od_a_a,A2: product_prod_a_a,B2: product_prod_a_a] :
( ( trans_8517530828069161671od_a_a @ top_to8063371432257647191od_a_a @ R )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A2 @ B2 ) @ R )
=> ( ord_le746702958409616551od_a_a @ ( order_865818352684159372od_a_a @ R @ A2 ) @ ( order_865818352684159372od_a_a @ R @ B2 ) ) ) ) ).
% under_incr
thf(fact_838_under__incr,axiom,
! [R: set_Product_prod_a_a,A2: a,B2: a] :
( ( trans_on_a @ top_top_set_a @ R )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B2 ) @ R )
=> ( ord_less_eq_set_a @ ( order_under_a @ R @ A2 ) @ ( order_under_a @ R @ B2 ) ) ) ) ).
% under_incr
thf(fact_839_inf__Int__eq2,axiom,
! [R2: set_Product_prod_a_a,S2: set_Product_prod_a_a] :
( ( inf_inf_a_a_o
@ ^ [X3: a,Y3: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R2 )
@ ^ [X3: a,Y3: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ S2 ) )
= ( ^ [X3: a,Y3: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ ( inf_in8905007599844390133od_a_a @ R2 @ S2 ) ) ) ) ).
% inf_Int_eq2
thf(fact_840_times__eq__iff,axiom,
! [A: set_a,B: set_a,C2: set_a,D: set_a] :
( ( ( product_Sigma_a_a @ A
@ ^ [Uu: a] : B )
= ( product_Sigma_a_a @ C2
@ ^ [Uu: a] : D ) )
= ( ( ( A = C2 )
& ( B = D ) )
| ( ( ( A = bot_bot_set_a )
| ( B = bot_bot_set_a ) )
& ( ( C2 = bot_bot_set_a )
| ( D = bot_bot_set_a ) ) ) ) ) ).
% times_eq_iff
thf(fact_841_top__empty__eq2,axiom,
( top_top_a_a_o
= ( ^ [X3: a,Y3: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ top_to8063371432257647191od_a_a ) ) ) ).
% top_empty_eq2
thf(fact_842_Linear__order__Well__order__iff,axiom,
! [R: set_Product_prod_a_a] :
( ( order_8768733634509060147r_on_a @ ( field_a @ R ) @ R )
=> ( ( order_6972113574731384220r_on_a @ ( field_a @ R ) @ R )
= ( ! [A3: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( field_a @ R ) )
=> ( ( A3 != bot_bot_set_a )
=> ? [X3: a] :
( ( member_a @ X3 @ A3 )
& ! [Y3: a] :
( ( member_a @ Y3 @ A3 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_843_transD,axiom,
! [R: set_Product_prod_a_a,X: a,Y: a,Z: a] :
( ( trans_on_a @ top_top_set_a @ R )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ R )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Z ) @ R ) ) ) ) ).
% transD
thf(fact_844_transD,axiom,
! [R: set_Pr8600417178894128327od_a_a,X: product_prod_a_a,Y: product_prod_a_a,Z: product_prod_a_a] :
( ( trans_8517530828069161671od_a_a @ top_to8063371432257647191od_a_a @ R )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Y ) @ R )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y @ Z ) @ R )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Z ) @ R ) ) ) ) ).
% transD
thf(fact_845_transE,axiom,
! [R: set_Product_prod_a_a,X: a,Y: a,Z: a] :
( ( trans_on_a @ top_top_set_a @ R )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ R )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Z ) @ R ) ) ) ) ).
% transE
thf(fact_846_transE,axiom,
! [R: set_Pr8600417178894128327od_a_a,X: product_prod_a_a,Y: product_prod_a_a,Z: product_prod_a_a] :
( ( trans_8517530828069161671od_a_a @ top_to8063371432257647191od_a_a @ R )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Y ) @ R )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y @ Z ) @ R )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Z ) @ R ) ) ) ) ).
% transE
thf(fact_847_transI,axiom,
! [R: set_Product_prod_a_a] :
( ! [X2: a,Y4: a,Z3: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y4 ) @ R )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ Z3 ) @ R )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Z3 ) @ R ) ) )
=> ( trans_on_a @ top_top_set_a @ R ) ) ).
% transI
thf(fact_848_transI,axiom,
! [R: set_Pr8600417178894128327od_a_a] :
( ! [X2: product_prod_a_a,Y4: product_prod_a_a,Z3: product_prod_a_a] :
( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X2 @ Y4 ) @ R )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y4 @ Z3 ) @ R )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X2 @ Z3 ) @ R ) ) )
=> ( trans_8517530828069161671od_a_a @ top_to8063371432257647191od_a_a @ R ) ) ).
% transI
thf(fact_849_reflD,axiom,
! [R: set_Product_prod_a_a,A2: a] :
( ( refl_on_a @ top_top_set_a @ R )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ A2 ) @ R ) ) ).
% reflD
thf(fact_850_reflD,axiom,
! [R: set_Pr8600417178894128327od_a_a,A2: product_prod_a_a] :
( ( refl_o7745108929832855590od_a_a @ top_to8063371432257647191od_a_a @ R )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A2 @ A2 ) @ R ) ) ).
% reflD
thf(fact_851_reflI,axiom,
! [R: set_Product_prod_a_a] :
( ! [X2: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ X2 ) @ R )
=> ( refl_on_a @ top_top_set_a @ R ) ) ).
% reflI
thf(fact_852_reflI,axiom,
! [R: set_Pr8600417178894128327od_a_a] :
( ! [X2: product_prod_a_a] : ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X2 @ X2 ) @ R )
=> ( refl_o7745108929832855590od_a_a @ top_to8063371432257647191od_a_a @ R ) ) ).
% reflI
thf(fact_853_totalI,axiom,
! [R: set_Product_prod_a_a] :
( ! [X2: a,Y4: a] :
( ( X2 != Y4 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y4 ) @ R )
| ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ X2 ) @ R ) ) )
=> ( total_on_a @ top_top_set_a @ R ) ) ).
% totalI
thf(fact_854_totalI,axiom,
! [R: set_Pr8600417178894128327od_a_a] :
( ! [X2: product_prod_a_a,Y4: product_prod_a_a] :
( ( X2 != Y4 )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X2 @ Y4 ) @ R )
| ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y4 @ X2 ) @ R ) ) )
=> ( total_3048927264068743339od_a_a @ top_to8063371432257647191od_a_a @ R ) ) ).
% totalI
thf(fact_855_symI,axiom,
! [R: set_Product_prod_a_a] :
( ! [X2: a,Y4: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y4 ) @ R )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ X2 ) @ R ) )
=> ( sym_on_a @ top_top_set_a @ R ) ) ).
% symI
thf(fact_856_symI,axiom,
! [R: set_Pr8600417178894128327od_a_a] :
( ! [X2: product_prod_a_a,Y4: product_prod_a_a] :
( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X2 @ Y4 ) @ R )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y4 @ X2 ) @ R ) )
=> ( sym_on5631557199876295240od_a_a @ top_to8063371432257647191od_a_a @ R ) ) ).
% symI
thf(fact_857_symE,axiom,
! [R: set_Product_prod_a_a,B2: a,A2: a] :
( ( sym_on_a @ top_top_set_a @ R )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B2 @ A2 ) @ R )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B2 ) @ R ) ) ) ).
% symE
thf(fact_858_symE,axiom,
! [R: set_Pr8600417178894128327od_a_a,B2: product_prod_a_a,A2: product_prod_a_a] :
( ( sym_on5631557199876295240od_a_a @ top_to8063371432257647191od_a_a @ R )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ B2 @ A2 ) @ R )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A2 @ B2 ) @ R ) ) ) ).
% symE
thf(fact_859_symD,axiom,
! [R: set_Product_prod_a_a,X: a,Y: a] :
( ( sym_on_a @ top_top_set_a @ R )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ X ) @ R ) ) ) ).
% symD
thf(fact_860_symD,axiom,
! [R: set_Pr8600417178894128327od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
( ( sym_on5631557199876295240od_a_a @ top_to8063371432257647191od_a_a @ R )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Y ) @ R )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y @ X ) @ R ) ) ) ).
% symD
thf(fact_861_antisymD,axiom,
! [R: set_Product_prod_a_a,X: a,Y: a] :
( ( antisym_on_a @ top_top_set_a @ R )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ X ) @ R )
=> ( X = Y ) ) ) ) ).
% antisymD
thf(fact_862_antisymD,axiom,
! [R: set_Pr8600417178894128327od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
( ( antisy9008168496540133130od_a_a @ top_to8063371432257647191od_a_a @ R )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Y ) @ R )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y @ X ) @ R )
=> ( X = Y ) ) ) ) ).
% antisymD
thf(fact_863_antisymI,axiom,
! [R: set_Product_prod_a_a] :
( ! [X2: a,Y4: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y4 ) @ R )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ X2 ) @ R )
=> ( X2 = Y4 ) ) )
=> ( antisym_on_a @ top_top_set_a @ R ) ) ).
% antisymI
thf(fact_864_antisymI,axiom,
! [R: set_Pr8600417178894128327od_a_a] :
( ! [X2: product_prod_a_a,Y4: product_prod_a_a] :
( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X2 @ Y4 ) @ R )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y4 @ X2 ) @ R )
=> ( X2 = Y4 ) ) )
=> ( antisy9008168496540133130od_a_a @ top_to8063371432257647191od_a_a @ R ) ) ).
% antisymI
thf(fact_865_trans__empty,axiom,
trans_on_a @ top_top_set_a @ bot_bo3357376287454694259od_a_a ).
% trans_empty
thf(fact_866_trans__empty,axiom,
trans_8517530828069161671od_a_a @ top_to8063371432257647191od_a_a @ bot_bo510284599550014259od_a_a ).
% trans_empty
thf(fact_867_antisym__empty,axiom,
antisym_on_a @ top_top_set_a @ bot_bo3357376287454694259od_a_a ).
% antisym_empty
thf(fact_868_antisym__empty,axiom,
antisy9008168496540133130od_a_a @ top_to8063371432257647191od_a_a @ bot_bo510284599550014259od_a_a ).
% antisym_empty
thf(fact_869_under__Field,axiom,
! [R: set_Product_prod_a_a,A2: a] : ( ord_less_eq_set_a @ ( order_under_a @ R @ A2 ) @ ( field_a @ R ) ) ).
% under_Field
thf(fact_870_Refl__under__in,axiom,
! [R: set_Product_prod_a_a,A2: a] :
( ( refl_on_a @ ( field_a @ R ) @ R )
=> ( ( member_a @ A2 @ ( field_a @ R ) )
=> ( member_a @ A2 @ ( order_under_a @ R @ A2 ) ) ) ) ).
% Refl_under_in
thf(fact_871_ofilter__subset__ordLeq,axiom,
! [R: set_Product_prod_a_a,A: set_a,B: set_a] :
( ( order_6972113574731384220r_on_a @ ( field_a @ R ) @ R )
=> ( ( order_ofilter_a @ R @ A )
=> ( ( order_ofilter_a @ R @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
= ( member4903802553882211088od_a_a
@ ( produc3372137660326521687od_a_a
@ ( inf_in8905007599844390133od_a_a @ R
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : A ) )
@ ( inf_in8905007599844390133od_a_a @ R
@ ( product_Sigma_a_a @ B
@ ^ [Uu: a] : B ) ) )
@ bNF_We6465643273610285415eq_a_a ) ) ) ) ) ).
% ofilter_subset_ordLeq
thf(fact_872_underS__incr,axiom,
! [R: set_Pr8600417178894128327od_a_a,A2: product_prod_a_a,B2: product_prod_a_a] :
( ( trans_8517530828069161671od_a_a @ top_to8063371432257647191od_a_a @ R )
=> ( ( antisy9008168496540133130od_a_a @ top_to8063371432257647191od_a_a @ R )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A2 @ B2 ) @ R )
=> ( ord_le746702958409616551od_a_a @ ( order_8408155380505527533od_a_a @ R @ A2 ) @ ( order_8408155380505527533od_a_a @ R @ B2 ) ) ) ) ) ).
% underS_incr
thf(fact_873_underS__incr,axiom,
! [R: set_Product_prod_a_a,A2: a,B2: a] :
( ( trans_on_a @ top_top_set_a @ R )
=> ( ( antisym_on_a @ top_top_set_a @ R )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B2 ) @ R )
=> ( ord_less_eq_set_a @ ( order_underS_a @ R @ A2 ) @ ( order_underS_a @ R @ B2 ) ) ) ) ) ).
% underS_incr
thf(fact_874_Chains__alt__def,axiom,
! [R: set_Product_prod_a_a] :
( ( refl_on_a @ top_top_set_a @ R )
=> ( ( chains_a @ R )
= ( collect_set_a
@ ( pred_chain_a @ top_top_set_a
@ ^ [X3: a,Y3: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R ) ) ) ) ) ).
% Chains_alt_def
thf(fact_875_Chains__alt__def,axiom,
! [R: set_Pr8600417178894128327od_a_a] :
( ( refl_o7745108929832855590od_a_a @ top_to8063371432257647191od_a_a @ R )
=> ( ( chains7467221010881542615od_a_a @ R )
= ( collec1673347964119250290od_a_a
@ ( pred_c8998633432913954770od_a_a @ top_to8063371432257647191od_a_a
@ ^ [X3: product_prod_a_a,Y3: product_prod_a_a] : ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X3 @ Y3 ) @ R ) ) ) ) ) ).
% Chains_alt_def
thf(fact_876_Order__Relation_OunderS__Field,axiom,
! [R: set_Product_prod_a_a,A2: a] : ( ord_less_eq_set_a @ ( order_underS_a @ R @ A2 ) @ ( field_a @ R ) ) ).
% Order_Relation.underS_Field
thf(fact_877_underS__subset__under,axiom,
! [R: set_Product_prod_a_a,A2: a] : ( ord_less_eq_set_a @ ( order_underS_a @ R @ A2 ) @ ( order_under_a @ R @ A2 ) ) ).
% underS_subset_under
thf(fact_878_Chains__subset,axiom,
! [R: set_Product_prod_a_a] :
( ord_le3724670747650509150_set_a @ ( chains_a @ R )
@ ( collect_set_a
@ ( pred_chain_a @ top_top_set_a
@ ^ [X3: a,Y3: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R ) ) ) ) ).
% Chains_subset
thf(fact_879_Chains__subset,axiom,
! [R: set_Pr8600417178894128327od_a_a] :
( ord_le1995061765932249223od_a_a @ ( chains7467221010881542615od_a_a @ R )
@ ( collec1673347964119250290od_a_a
@ ( pred_c8998633432913954770od_a_a @ top_to8063371432257647191od_a_a
@ ^ [X3: product_prod_a_a,Y3: product_prod_a_a] : ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X3 @ Y3 ) @ R ) ) ) ) ).
% Chains_subset
thf(fact_880_underS__incl__iff,axiom,
! [R: set_Product_prod_a_a,A2: a,B2: a] :
( ( order_8768733634509060147r_on_a @ ( field_a @ R ) @ R )
=> ( ( member_a @ A2 @ ( field_a @ R ) )
=> ( ( member_a @ B2 @ ( field_a @ R ) )
=> ( ( ord_less_eq_set_a @ ( order_underS_a @ R @ A2 ) @ ( order_underS_a @ R @ B2 ) )
= ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B2 ) @ R ) ) ) ) ) ).
% underS_incl_iff
thf(fact_881_Chains__subset_H,axiom,
! [R: set_Product_prod_a_a] :
( ( refl_on_a @ top_top_set_a @ R )
=> ( ord_le3724670747650509150_set_a
@ ( collect_set_a
@ ( pred_chain_a @ top_top_set_a
@ ^ [X3: a,Y3: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R ) ) )
@ ( chains_a @ R ) ) ) ).
% Chains_subset'
thf(fact_882_Chains__subset_H,axiom,
! [R: set_Pr8600417178894128327od_a_a] :
( ( refl_o7745108929832855590od_a_a @ top_to8063371432257647191od_a_a @ R )
=> ( ord_le1995061765932249223od_a_a
@ ( collec1673347964119250290od_a_a
@ ( pred_c8998633432913954770od_a_a @ top_to8063371432257647191od_a_a
@ ^ [X3: product_prod_a_a,Y3: product_prod_a_a] : ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X3 @ Y3 ) @ R ) ) )
@ ( chains7467221010881542615od_a_a @ R ) ) ) ).
% Chains_subset'
thf(fact_883_underS__Restr__ordLess,axiom,
! [R: set_Product_prod_a_a,A2: a] :
( ( order_6972113574731384220r_on_a @ ( field_a @ R ) @ R )
=> ( ( ( field_a @ R )
!= bot_bot_set_a )
=> ( member4903802553882211088od_a_a
@ ( produc3372137660326521687od_a_a
@ ( inf_in8905007599844390133od_a_a @ R
@ ( product_Sigma_a_a @ ( order_underS_a @ R @ A2 )
@ ^ [Uu: a] : ( order_underS_a @ R @ A2 ) ) )
@ R )
@ bNF_We3281422121813167946ss_a_a ) ) ) ).
% underS_Restr_ordLess
thf(fact_884_ofilter__subset__ordLess,axiom,
! [R: set_Product_prod_a_a,A: set_a,B: set_a] :
( ( order_6972113574731384220r_on_a @ ( field_a @ R ) @ R )
=> ( ( order_ofilter_a @ R @ A )
=> ( ( order_ofilter_a @ R @ B )
=> ( ( ord_less_set_a @ A @ B )
= ( member4903802553882211088od_a_a
@ ( produc3372137660326521687od_a_a
@ ( inf_in8905007599844390133od_a_a @ R
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : A ) )
@ ( inf_in8905007599844390133od_a_a @ R
@ ( product_Sigma_a_a @ B
@ ^ [Uu: a] : B ) ) )
@ bNF_We3281422121813167946ss_a_a ) ) ) ) ) ).
% ofilter_subset_ordLess
thf(fact_885_ofilter__ordLess,axiom,
! [R: set_Product_prod_a_a,A: set_a] :
( ( order_6972113574731384220r_on_a @ ( field_a @ R ) @ R )
=> ( ( order_ofilter_a @ R @ A )
=> ( ( ord_less_set_a @ A @ ( field_a @ R ) )
= ( member4903802553882211088od_a_a
@ ( produc3372137660326521687od_a_a
@ ( inf_in8905007599844390133od_a_a @ R
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : A ) )
@ R )
@ bNF_We3281422121813167946ss_a_a ) ) ) ) ).
% ofilter_ordLess
thf(fact_886_antisymp__antisym__eq,axiom,
! [R: set_Product_prod_a_a] :
( ( antisymp_on_a @ top_top_set_a
@ ^ [X3: a,Y3: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R ) )
= ( antisym_on_a @ top_top_set_a @ R ) ) ).
% antisymp_antisym_eq
thf(fact_887_antisymp__antisym__eq,axiom,
! [R: set_Pr8600417178894128327od_a_a] :
( ( antisy8373617738294204244od_a_a @ top_to8063371432257647191od_a_a
@ ^ [X3: product_prod_a_a,Y3: product_prod_a_a] : ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X3 @ Y3 ) @ R ) )
= ( antisy9008168496540133130od_a_a @ top_to8063371432257647191od_a_a @ R ) ) ).
% antisymp_antisym_eq
thf(fact_888_psubsetI,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_a @ A @ B ) ) ) ).
% psubsetI
thf(fact_889_order__le__imp__less__or__eq,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_set_a @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_890_order__less__le__subst1,axiom,
! [A2: set_a,F2: set_a > set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F2 @ X2 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_set_a @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_891_order__le__less__subst2,axiom,
! [A2: set_a,B2: set_a,F2: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ ( F2 @ B2 ) @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F2 @ X2 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_set_a @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_892_order__less__le__trans,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z )
=> ( ord_less_set_a @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_893_order__le__less__trans,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_set_a @ Y @ Z )
=> ( ord_less_set_a @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_894_order__neq__le__trans,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 != B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_set_a @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_895_order__le__neq__trans,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_a @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_896_order__less__imp__le,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_897_order__less__le,axiom,
( ord_less_set_a
= ( ^ [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
& ( X3 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_898_order__le__less,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y3: set_a] :
( ( ord_less_set_a @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_899_dual__order_Ostrict__implies__order,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_900_order_Ostrict__implies__order,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_901_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [B4: set_a,A4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A4 )
& ~ ( ord_less_eq_set_a @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_902_dual__order_Ostrict__trans2,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ C @ B2 )
=> ( ord_less_set_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_903_dual__order_Ostrict__trans1,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_set_a @ C @ B2 )
=> ( ord_less_set_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_904_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [B4: set_a,A4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_905_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A4: set_a] :
( ( ord_less_set_a @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_906_order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ~ ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_907_order_Ostrict__trans2,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_908_order_Ostrict__trans1,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_909_order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_910_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_set_a @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_911_less__le__not__le,axiom,
( ord_less_set_a
= ( ^ [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
& ~ ( ord_less_eq_set_a @ Y3 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_912_antisym__conv2,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ~ ( ord_less_set_a @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_913_antisym__conv1,axiom,
! [X: set_a,Y: set_a] :
( ~ ( ord_less_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_914_nless__le,axiom,
! [A2: set_a,B2: set_a] :
( ( ~ ( ord_less_set_a @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_set_a @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_915_leD,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ~ ( ord_less_set_a @ X @ Y ) ) ).
% leD
thf(fact_916_top_Oextremum__strict,axiom,
! [A2: set_a] :
~ ( ord_less_set_a @ top_top_set_a @ A2 ) ).
% top.extremum_strict
thf(fact_917_top_Oextremum__strict,axiom,
! [A2: set_Product_prod_a_a] :
~ ( ord_le6819997720685908915od_a_a @ top_to8063371432257647191od_a_a @ A2 ) ).
% top.extremum_strict
thf(fact_918_top_Onot__eq__extremum,axiom,
! [A2: set_a] :
( ( A2 != top_top_set_a )
= ( ord_less_set_a @ A2 @ top_top_set_a ) ) ).
% top.not_eq_extremum
thf(fact_919_top_Onot__eq__extremum,axiom,
! [A2: set_Product_prod_a_a] :
( ( A2 != top_to8063371432257647191od_a_a )
= ( ord_le6819997720685908915od_a_a @ A2 @ top_to8063371432257647191od_a_a ) ) ).
% top.not_eq_extremum
thf(fact_920_less__infI1,axiom,
! [A2: set_Product_prod_a_a,X: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le6819997720685908915od_a_a @ A2 @ X )
=> ( ord_le6819997720685908915od_a_a @ ( inf_in8905007599844390133od_a_a @ A2 @ B2 ) @ X ) ) ).
% less_infI1
thf(fact_921_less__infI1,axiom,
! [A2: set_a,X: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ X )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ X ) ) ).
% less_infI1
thf(fact_922_less__infI2,axiom,
! [B2: set_Product_prod_a_a,X: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( ord_le6819997720685908915od_a_a @ B2 @ X )
=> ( ord_le6819997720685908915od_a_a @ ( inf_in8905007599844390133od_a_a @ A2 @ B2 ) @ X ) ) ).
% less_infI2
thf(fact_923_less__infI2,axiom,
! [B2: set_a,X: set_a,A2: set_a] :
( ( ord_less_set_a @ B2 @ X )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ X ) ) ).
% less_infI2
thf(fact_924_inf_Oabsorb3,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le6819997720685908915od_a_a @ A2 @ B2 )
=> ( ( inf_in8905007599844390133od_a_a @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb3
thf(fact_925_inf_Oabsorb3,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb3
thf(fact_926_inf_Oabsorb4,axiom,
! [B2: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( ord_le6819997720685908915od_a_a @ B2 @ A2 )
=> ( ( inf_in8905007599844390133od_a_a @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb4
thf(fact_927_inf_Oabsorb4,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb4
thf(fact_928_inf_Ostrict__boundedE,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le6819997720685908915od_a_a @ A2 @ ( inf_in8905007599844390133od_a_a @ B2 @ C ) )
=> ~ ( ( ord_le6819997720685908915od_a_a @ A2 @ B2 )
=> ~ ( ord_le6819997720685908915od_a_a @ A2 @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_929_inf_Ostrict__boundedE,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) )
=> ~ ( ( ord_less_set_a @ A2 @ B2 )
=> ~ ( ord_less_set_a @ A2 @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_930_inf_Ostrict__order__iff,axiom,
( ord_le6819997720685908915od_a_a
= ( ^ [A4: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( A4
= ( inf_in8905007599844390133od_a_a @ A4 @ B4 ) )
& ( A4 != B4 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_931_inf_Ostrict__order__iff,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( A4
= ( inf_inf_set_a @ A4 @ B4 ) )
& ( A4 != B4 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_932_inf_Ostrict__coboundedI1,axiom,
! [A2: set_Product_prod_a_a,C: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le6819997720685908915od_a_a @ A2 @ C )
=> ( ord_le6819997720685908915od_a_a @ ( inf_in8905007599844390133od_a_a @ A2 @ B2 ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_933_inf_Ostrict__coboundedI1,axiom,
! [A2: set_a,C: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ C )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_934_inf_Ostrict__coboundedI2,axiom,
! [B2: set_Product_prod_a_a,C: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( ord_le6819997720685908915od_a_a @ B2 @ C )
=> ( ord_le6819997720685908915od_a_a @ ( inf_in8905007599844390133od_a_a @ A2 @ B2 ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_935_inf_Ostrict__coboundedI2,axiom,
! [B2: set_a,C: set_a,A2: set_a] :
( ( ord_less_set_a @ B2 @ C )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_936_subset__Zorn,axiom,
! [A: set_set_a] :
( ! [C3: set_set_a] :
( ( pred_chain_set_a @ A @ ord_less_set_a @ C3 )
=> ? [X5: set_a] :
( ( member_set_a @ X5 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ C3 )
=> ( ord_less_eq_set_a @ Xa @ X5 ) ) ) )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A )
=> ( ( ord_less_eq_set_a @ X2 @ Xa2 )
=> ( Xa2 = X2 ) ) ) ) ) ).
% subset_Zorn
thf(fact_937_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_set_a @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_938_subset__psubset__trans,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_set_a @ B @ C2 )
=> ( ord_less_set_a @ A @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_939_subset__not__subset__eq,axiom,
( ord_less_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ~ ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_940_psubset__subset__trans,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_set_a @ A @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_941_psubset__imp__subset,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_942_psubset__eq,axiom,
( ord_less_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_943_psubsetE,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ B @ A ) ) ) ).
% psubsetE
thf(fact_944_antisymp__on__le,axiom,
! [A: set_set_a] : ( antisymp_on_set_a @ A @ ord_less_eq_set_a ) ).
% antisymp_on_le
thf(fact_945_antisympD,axiom,
! [R2: a > a > $o,X: a,Y: a] :
( ( antisymp_on_a @ top_top_set_a @ R2 )
=> ( ( R2 @ X @ Y )
=> ( ( R2 @ Y @ X )
=> ( X = Y ) ) ) ) ).
% antisympD
thf(fact_946_antisympD,axiom,
! [R2: product_prod_a_a > product_prod_a_a > $o,X: product_prod_a_a,Y: product_prod_a_a] :
( ( antisy8373617738294204244od_a_a @ top_to8063371432257647191od_a_a @ R2 )
=> ( ( R2 @ X @ Y )
=> ( ( R2 @ Y @ X )
=> ( X = Y ) ) ) ) ).
% antisympD
thf(fact_947_antisympI,axiom,
! [R2: a > a > $o] :
( ! [X2: a,Y4: a] :
( ( R2 @ X2 @ Y4 )
=> ( ( R2 @ Y4 @ X2 )
=> ( X2 = Y4 ) ) )
=> ( antisymp_on_a @ top_top_set_a @ R2 ) ) ).
% antisympI
thf(fact_948_antisympI,axiom,
! [R2: product_prod_a_a > product_prod_a_a > $o] :
( ! [X2: product_prod_a_a,Y4: product_prod_a_a] :
( ( R2 @ X2 @ Y4 )
=> ( ( R2 @ Y4 @ X2 )
=> ( X2 = Y4 ) ) )
=> ( antisy8373617738294204244od_a_a @ top_to8063371432257647191od_a_a @ R2 ) ) ).
% antisympI
thf(fact_949_antisymp__equality,axiom,
( antisymp_on_a @ top_top_set_a
@ ^ [Y2: a,Z2: a] : ( Y2 = Z2 ) ) ).
% antisymp_equality
thf(fact_950_antisymp__equality,axiom,
( antisy8373617738294204244od_a_a @ top_to8063371432257647191od_a_a
@ ^ [Y2: product_prod_a_a,Z2: product_prod_a_a] : ( Y2 = Z2 ) ) ).
% antisymp_equality
thf(fact_951_antisymp__on__subset,axiom,
! [A: set_a,R2: a > a > $o,B: set_a] :
( ( antisymp_on_a @ A @ R2 )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( antisymp_on_a @ B @ R2 ) ) ) ).
% antisymp_on_subset
thf(fact_952_antisymp__on__ge,axiom,
! [A: set_set_a] :
( antisymp_on_set_a @ A
@ ^ [X3: set_a,Y3: set_a] : ( ord_less_eq_set_a @ Y3 @ X3 ) ) ).
% antisymp_on_ge
thf(fact_953_subset__chain__def,axiom,
! [A5: set_set_a,C4: set_set_a] :
( ( pred_chain_set_a @ A5 @ ord_less_set_a @ C4 )
= ( ( ord_le3724670747650509150_set_a @ C4 @ A5 )
& ! [X3: set_a] :
( ( member_set_a @ X3 @ C4 )
=> ! [Y3: set_a] :
( ( member_set_a @ Y3 @ C4 )
=> ( ( ord_less_eq_set_a @ X3 @ Y3 )
| ( ord_less_eq_set_a @ Y3 @ X3 ) ) ) ) ) ) ).
% subset_chain_def
thf(fact_954_antisymp__less__eq,axiom,
! [R: a > a > $o,S: a > a > $o] :
( ( ord_less_eq_a_a_o @ R @ S )
=> ( ( antisymp_on_a @ top_top_set_a @ S )
=> ( antisymp_on_a @ top_top_set_a @ R ) ) ) ).
% antisymp_less_eq
thf(fact_955_antisymp__less__eq,axiom,
! [R: product_prod_a_a > product_prod_a_a > $o,S: product_prod_a_a > product_prod_a_a > $o] :
( ( ord_le540031911545281998_a_a_o @ R @ S )
=> ( ( antisy8373617738294204244od_a_a @ top_to8063371432257647191od_a_a @ S )
=> ( antisy8373617738294204244od_a_a @ top_to8063371432257647191od_a_a @ R ) ) ) ).
% antisymp_less_eq
thf(fact_956_antisym__bot,axiom,
antisymp_on_a @ top_top_set_a @ bot_bot_a_a_o ).
% antisym_bot
thf(fact_957_antisym__bot,axiom,
antisy8373617738294204244od_a_a @ top_to8063371432257647191od_a_a @ bot_bo74666019906784410_a_a_o ).
% antisym_bot
thf(fact_958_bsqr__ofilter,axiom,
! [R: set_Product_prod_a_a,D: set_Product_prod_a_a] :
( ( order_6972113574731384220r_on_a @ ( field_a @ R ) @ R )
=> ( ( order_4948930231512469437od_a_a @ ( bNF_Wellorder_bsqr_a @ R ) @ D )
=> ( ( ord_le6819997720685908915od_a_a @ D
@ ( product_Sigma_a_a @ ( field_a @ R )
@ ^ [Uu: a] : ( field_a @ R ) ) )
=> ( ~ ? [A6: a] :
( ( field_a @ R )
= ( order_under_a @ R @ A6 ) )
=> ? [A7: set_a] :
( ( order_ofilter_a @ R @ A7 )
& ( ord_less_set_a @ A7 @ ( field_a @ R ) )
& ( ord_le746702958409616551od_a_a @ D
@ ( product_Sigma_a_a @ A7
@ ^ [Uu: a] : A7 ) ) ) ) ) ) ) ).
% bsqr_ofilter
thf(fact_959_verit__comp__simplify1_I2_J,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_960_inf__top_Osemilattice__neutr__order__axioms,axiom,
semila3049385046964979381od_a_a @ inf_in8905007599844390133od_a_a @ top_to8063371432257647191od_a_a @ ord_le746702958409616551od_a_a @ ord_le6819997720685908915od_a_a ).
% inf_top.semilattice_neutr_order_axioms
thf(fact_961_inf__top_Osemilattice__neutr__order__axioms,axiom,
semila2496817875450240012_set_a @ inf_inf_set_a @ top_top_set_a @ ord_less_eq_set_a @ ord_less_set_a ).
% inf_top.semilattice_neutr_order_axioms
thf(fact_962_bot_Oordering__top__axioms,axiom,
( ordering_top_set_a
@ ^ [X3: set_a,Y3: set_a] : ( ord_less_eq_set_a @ Y3 @ X3 )
@ ^ [X3: set_a,Y3: set_a] : ( ord_less_set_a @ Y3 @ X3 )
@ bot_bot_set_a ) ).
% bot.ordering_top_axioms
thf(fact_963_Zorn__Lemma2,axiom,
! [A: set_set_a] :
( ! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ ( chains_a2 @ A ) )
=> ? [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A )
& ! [Xb: set_a] :
( ( member_set_a @ Xb @ X2 )
=> ( ord_less_eq_set_a @ Xb @ Xa2 ) ) ) )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A )
=> ( ( ord_less_eq_set_a @ X2 @ Xa2 )
=> ( Xa2 = X2 ) ) ) ) ) ).
% Zorn_Lemma2
thf(fact_964_chainsD,axiom,
! [C: set_set_a,S2: set_set_a,X: set_a,Y: set_a] :
( ( member_set_set_a @ C @ ( chains_a2 @ S2 ) )
=> ( ( member_set_a @ X @ C )
=> ( ( member_set_a @ Y @ C )
=> ( ( ord_less_eq_set_a @ X @ Y )
| ( ord_less_eq_set_a @ Y @ X ) ) ) ) ) ).
% chainsD
thf(fact_965_top_Oordering__top__axioms,axiom,
orderi8249066106010698168od_a_a @ ord_le746702958409616551od_a_a @ ord_le6819997720685908915od_a_a @ top_to8063371432257647191od_a_a ).
% top.ordering_top_axioms
thf(fact_966_top_Oordering__top__axioms,axiom,
ordering_top_set_a @ ord_less_eq_set_a @ ord_less_set_a @ top_top_set_a ).
% top.ordering_top_axioms
thf(fact_967_sup__bot_Osemilattice__neutr__order__axioms,axiom,
( semila2496817875450240012_set_a @ sup_sup_set_a @ bot_bot_set_a
@ ^ [X3: set_a,Y3: set_a] : ( ord_less_eq_set_a @ Y3 @ X3 )
@ ^ [X3: set_a,Y3: set_a] : ( ord_less_set_a @ Y3 @ X3 ) ) ).
% sup_bot.semilattice_neutr_order_axioms
thf(fact_968_subset__Zorn__nonempty,axiom,
! [A5: set_set_a] :
( ( A5 != bot_bot_set_set_a )
=> ( ! [C5: set_set_a] :
( ( C5 != bot_bot_set_set_a )
=> ( ( pred_chain_set_a @ A5 @ ord_less_set_a @ C5 )
=> ( member_set_a @ ( comple2307003609928055243_set_a @ C5 ) @ A5 ) ) )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A5 )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A5 )
=> ( ( ord_less_eq_set_a @ X2 @ Xa2 )
=> ( Xa2 = X2 ) ) ) ) ) ) ).
% subset_Zorn_nonempty
thf(fact_969_Un__subset__iff,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ C2 )
= ( ( ord_less_eq_set_a @ A @ C2 )
& ( ord_less_eq_set_a @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_970_Int__Un__eq_I4_J,axiom,
! [T2: set_Product_prod_a_a,S2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ T2 @ ( inf_in8905007599844390133od_a_a @ S2 @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_971_Int__Un__eq_I4_J,axiom,
! [T2: set_a,S2: set_a] :
( ( sup_sup_set_a @ T2 @ ( inf_inf_set_a @ S2 @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_972_Int__Un__eq_I3_J,axiom,
! [S2: set_Product_prod_a_a,T2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ S2 @ ( inf_in8905007599844390133od_a_a @ S2 @ T2 ) )
= S2 ) ).
% Int_Un_eq(3)
thf(fact_973_Int__Un__eq_I3_J,axiom,
! [S2: set_a,T2: set_a] :
( ( sup_sup_set_a @ S2 @ ( inf_inf_set_a @ S2 @ T2 ) )
= S2 ) ).
% Int_Un_eq(3)
thf(fact_974_Int__Un__eq_I2_J,axiom,
! [S2: set_Product_prod_a_a,T2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( inf_in8905007599844390133od_a_a @ S2 @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_975_Int__Un__eq_I2_J,axiom,
! [S2: set_a,T2: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S2 @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_976_Int__Un__eq_I1_J,axiom,
! [S2: set_Product_prod_a_a,T2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( inf_in8905007599844390133od_a_a @ S2 @ T2 ) @ S2 )
= S2 ) ).
% Int_Un_eq(1)
thf(fact_977_Int__Un__eq_I1_J,axiom,
! [S2: set_a,T2: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S2 @ T2 ) @ S2 )
= S2 ) ).
% Int_Un_eq(1)
thf(fact_978_Un__Int__eq_I4_J,axiom,
! [T2: set_Product_prod_a_a,S2: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ T2 @ ( sup_su3048258781599657691od_a_a @ S2 @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_979_Un__Int__eq_I4_J,axiom,
! [T2: set_a,S2: set_a] :
( ( inf_inf_set_a @ T2 @ ( sup_sup_set_a @ S2 @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_980_Un__Int__eq_I3_J,axiom,
! [S2: set_Product_prod_a_a,T2: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ S2 @ ( sup_su3048258781599657691od_a_a @ S2 @ T2 ) )
= S2 ) ).
% Un_Int_eq(3)
thf(fact_981_Un__Int__eq_I3_J,axiom,
! [S2: set_a,T2: set_a] :
( ( inf_inf_set_a @ S2 @ ( sup_sup_set_a @ S2 @ T2 ) )
= S2 ) ).
% Un_Int_eq(3)
thf(fact_982_Un__Int__eq_I2_J,axiom,
! [S2: set_Product_prod_a_a,T2: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ ( sup_su3048258781599657691od_a_a @ S2 @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_983_Un__Int__eq_I2_J,axiom,
! [S2: set_a,T2: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S2 @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_984_Un__Int__eq_I1_J,axiom,
! [S2: set_Product_prod_a_a,T2: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ ( sup_su3048258781599657691od_a_a @ S2 @ T2 ) @ S2 )
= S2 ) ).
% Un_Int_eq(1)
thf(fact_985_Un__Int__eq_I1_J,axiom,
! [S2: set_a,T2: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S2 @ T2 ) @ S2 )
= S2 ) ).
% Un_Int_eq(1)
thf(fact_986_sup_Obounded__iff,axiom,
! [B2: set_a,C: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A2 )
= ( ( ord_less_eq_set_a @ B2 @ A2 )
& ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_987_le__sup__iff,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z )
= ( ( ord_less_eq_set_a @ X @ Z )
& ( ord_less_eq_set_a @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_988_boolean__algebra_Odisj__one__left,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ top_top_set_a @ X )
= top_top_set_a ) ).
% boolean_algebra.disj_one_left
thf(fact_989_boolean__algebra_Odisj__one__left,axiom,
! [X: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ top_to8063371432257647191od_a_a @ X )
= top_to8063371432257647191od_a_a ) ).
% boolean_algebra.disj_one_left
thf(fact_990_boolean__algebra_Odisj__one__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ top_top_set_a )
= top_top_set_a ) ).
% boolean_algebra.disj_one_right
thf(fact_991_boolean__algebra_Odisj__one__right,axiom,
! [X: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X @ top_to8063371432257647191od_a_a )
= top_to8063371432257647191od_a_a ) ).
% boolean_algebra.disj_one_right
thf(fact_992_sup__top__left,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ top_top_set_a @ X )
= top_top_set_a ) ).
% sup_top_left
thf(fact_993_sup__top__left,axiom,
! [X: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ top_to8063371432257647191od_a_a @ X )
= top_to8063371432257647191od_a_a ) ).
% sup_top_left
thf(fact_994_sup__top__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ top_top_set_a )
= top_top_set_a ) ).
% sup_top_right
thf(fact_995_sup__top__right,axiom,
! [X: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X @ top_to8063371432257647191od_a_a )
= top_to8063371432257647191od_a_a ) ).
% sup_top_right
thf(fact_996_inf__sup__absorb,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ X @ ( sup_su3048258781599657691od_a_a @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_997_inf__sup__absorb,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_998_sup__inf__absorb,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X @ ( inf_in8905007599844390133od_a_a @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_999_sup__inf__absorb,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_1000_cSup__eq__maximum,axiom,
! [Z: set_a,X4: set_set_a] :
( ( member_set_a @ Z @ X4 )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ X4 )
=> ( ord_less_eq_set_a @ X2 @ Z ) )
=> ( ( comple2307003609928055243_set_a @ X4 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_1001_Un__UNIV__left,axiom,
! [B: set_a] :
( ( sup_sup_set_a @ top_top_set_a @ B )
= top_top_set_a ) ).
% Un_UNIV_left
thf(fact_1002_Un__UNIV__left,axiom,
! [B: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ top_to8063371432257647191od_a_a @ B )
= top_to8063371432257647191od_a_a ) ).
% Un_UNIV_left
thf(fact_1003_Un__UNIV__right,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ top_top_set_a )
= top_top_set_a ) ).
% Un_UNIV_right
thf(fact_1004_Un__UNIV__right,axiom,
! [A: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A @ top_to8063371432257647191od_a_a )
= top_to8063371432257647191od_a_a ) ).
% Un_UNIV_right
thf(fact_1005_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( sup_sup_set_a @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_1006_subset__UnE,axiom,
! [C2: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A @ B ) )
=> ~ ! [A8: set_a] :
( ( ord_less_eq_set_a @ A8 @ A )
=> ! [B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ B )
=> ( C2
!= ( sup_sup_set_a @ A8 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_1007_Un__absorb2,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( sup_sup_set_a @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_1008_Un__absorb1,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( sup_sup_set_a @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_1009_Un__upper2,axiom,
! [B: set_a,A: set_a] : ( ord_less_eq_set_a @ B @ ( sup_sup_set_a @ A @ B ) ) ).
% Un_upper2
thf(fact_1010_Un__upper1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ A @ B ) ) ).
% Un_upper1
thf(fact_1011_Un__least,axiom,
! [A: set_a,C2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_1012_Un__mono,axiom,
! [A: set_a,C2: set_a,B: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B @ D )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_1013_Un__Int__crazy,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( sup_su3048258781599657691od_a_a @ ( inf_in8905007599844390133od_a_a @ A @ B ) @ ( inf_in8905007599844390133od_a_a @ B @ C2 ) ) @ ( inf_in8905007599844390133od_a_a @ C2 @ A ) )
= ( inf_in8905007599844390133od_a_a @ ( inf_in8905007599844390133od_a_a @ ( sup_su3048258781599657691od_a_a @ A @ B ) @ ( sup_su3048258781599657691od_a_a @ B @ C2 ) ) @ ( sup_su3048258781599657691od_a_a @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_1014_Un__Int__crazy,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ B @ C2 ) ) @ ( inf_inf_set_a @ C2 @ A ) )
= ( inf_inf_set_a @ ( inf_inf_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ B @ C2 ) ) @ ( sup_sup_set_a @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_1015_Int__Un__distrib,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ A @ ( sup_su3048258781599657691od_a_a @ B @ C2 ) )
= ( sup_su3048258781599657691od_a_a @ ( inf_in8905007599844390133od_a_a @ A @ B ) @ ( inf_in8905007599844390133od_a_a @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_1016_Int__Un__distrib,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( inf_inf_set_a @ A @ ( sup_sup_set_a @ B @ C2 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_1017_Un__Int__distrib,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A @ ( inf_in8905007599844390133od_a_a @ B @ C2 ) )
= ( inf_in8905007599844390133od_a_a @ ( sup_su3048258781599657691od_a_a @ A @ B ) @ ( sup_su3048258781599657691od_a_a @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_1018_Un__Int__distrib,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( sup_sup_set_a @ A @ ( inf_inf_set_a @ B @ C2 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_1019_Int__Un__distrib2,axiom,
! [B: set_Product_prod_a_a,C2: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ ( sup_su3048258781599657691od_a_a @ B @ C2 ) @ A )
= ( sup_su3048258781599657691od_a_a @ ( inf_in8905007599844390133od_a_a @ B @ A ) @ ( inf_in8905007599844390133od_a_a @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_1020_Int__Un__distrib2,axiom,
! [B: set_a,C2: set_a,A: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ B @ C2 ) @ A )
= ( sup_sup_set_a @ ( inf_inf_set_a @ B @ A ) @ ( inf_inf_set_a @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_1021_Un__Int__distrib2,axiom,
! [B: set_Product_prod_a_a,C2: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( inf_in8905007599844390133od_a_a @ B @ C2 ) @ A )
= ( inf_in8905007599844390133od_a_a @ ( sup_su3048258781599657691od_a_a @ B @ A ) @ ( sup_su3048258781599657691od_a_a @ C2 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_1022_Un__Int__distrib2,axiom,
! [B: set_a,C2: set_a,A: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ B @ C2 ) @ A )
= ( inf_inf_set_a @ ( sup_sup_set_a @ B @ A ) @ ( sup_sup_set_a @ C2 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_1023_Sigma__Un__distrib2,axiom,
! [I: set_a,A: a > set_a,B: a > set_a] :
( ( product_Sigma_a_a @ I
@ ^ [I2: a] : ( sup_sup_set_a @ ( A @ I2 ) @ ( B @ I2 ) ) )
= ( sup_su3048258781599657691od_a_a @ ( product_Sigma_a_a @ I @ A ) @ ( product_Sigma_a_a @ I @ B ) ) ) ).
% Sigma_Un_distrib2
thf(fact_1024_Times__Un__distrib1,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( product_Sigma_a_a @ ( sup_sup_set_a @ A @ B )
@ ^ [Uu: a] : C2 )
= ( sup_su3048258781599657691od_a_a
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : C2 )
@ ( product_Sigma_a_a @ B
@ ^ [Uu: a] : C2 ) ) ) ).
% Times_Un_distrib1
thf(fact_1025_Sigma__Un__distrib1,axiom,
! [I: set_a,J: set_a,C2: a > set_a] :
( ( product_Sigma_a_a @ ( sup_sup_set_a @ I @ J ) @ C2 )
= ( sup_su3048258781599657691od_a_a @ ( product_Sigma_a_a @ I @ C2 ) @ ( product_Sigma_a_a @ J @ C2 ) ) ) ).
% Sigma_Un_distrib1
thf(fact_1026_refl__on__Un,axiom,
! [A: set_a,R: set_Product_prod_a_a,B: set_a,S: set_Product_prod_a_a] :
( ( refl_on_a @ A @ R )
=> ( ( refl_on_a @ B @ S )
=> ( refl_on_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_su3048258781599657691od_a_a @ R @ S ) ) ) ) ).
% refl_on_Un
thf(fact_1027_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ X @ ( sup_su3048258781599657691od_a_a @ Y @ Z ) )
= ( sup_su3048258781599657691od_a_a @ ( inf_in8905007599844390133od_a_a @ X @ Y ) @ ( inf_in8905007599844390133od_a_a @ X @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_1028_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_1029_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X @ ( inf_in8905007599844390133od_a_a @ Y @ Z ) )
= ( inf_in8905007599844390133od_a_a @ ( sup_su3048258781599657691od_a_a @ X @ Y ) @ ( sup_su3048258781599657691od_a_a @ X @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_1030_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_1031_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_Product_prod_a_a,Z: set_Product_prod_a_a,X: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ ( sup_su3048258781599657691od_a_a @ Y @ Z ) @ X )
= ( sup_su3048258781599657691od_a_a @ ( inf_in8905007599844390133od_a_a @ Y @ X ) @ ( inf_in8905007599844390133od_a_a @ Z @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_1032_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_a,Z: set_a,X: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ Z ) @ X )
= ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ X ) @ ( inf_inf_set_a @ Z @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_1033_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_Product_prod_a_a,Z: set_Product_prod_a_a,X: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( inf_in8905007599844390133od_a_a @ Y @ Z ) @ X )
= ( inf_in8905007599844390133od_a_a @ ( sup_su3048258781599657691od_a_a @ Y @ X ) @ ( sup_su3048258781599657691od_a_a @ Z @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_1034_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_a,Z: set_a,X: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ Z ) @ X )
= ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ X ) @ ( sup_sup_set_a @ Z @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_1035_distrib__imp1,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] :
( ! [X2: set_Product_prod_a_a,Y4: set_Product_prod_a_a,Z3: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ X2 @ ( sup_su3048258781599657691od_a_a @ Y4 @ Z3 ) )
= ( sup_su3048258781599657691od_a_a @ ( inf_in8905007599844390133od_a_a @ X2 @ Y4 ) @ ( inf_in8905007599844390133od_a_a @ X2 @ Z3 ) ) )
=> ( ( sup_su3048258781599657691od_a_a @ X @ ( inf_in8905007599844390133od_a_a @ Y @ Z ) )
= ( inf_in8905007599844390133od_a_a @ ( sup_su3048258781599657691od_a_a @ X @ Y ) @ ( sup_su3048258781599657691od_a_a @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_1036_distrib__imp1,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ! [X2: set_a,Y4: set_a,Z3: set_a] :
( ( inf_inf_set_a @ X2 @ ( sup_sup_set_a @ Y4 @ Z3 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X2 @ Y4 ) @ ( inf_inf_set_a @ X2 @ Z3 ) ) )
=> ( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_1037_distrib__imp2,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] :
( ! [X2: set_Product_prod_a_a,Y4: set_Product_prod_a_a,Z3: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X2 @ ( inf_in8905007599844390133od_a_a @ Y4 @ Z3 ) )
= ( inf_in8905007599844390133od_a_a @ ( sup_su3048258781599657691od_a_a @ X2 @ Y4 ) @ ( sup_su3048258781599657691od_a_a @ X2 @ Z3 ) ) )
=> ( ( inf_in8905007599844390133od_a_a @ X @ ( sup_su3048258781599657691od_a_a @ Y @ Z ) )
= ( sup_su3048258781599657691od_a_a @ ( inf_in8905007599844390133od_a_a @ X @ Y ) @ ( inf_in8905007599844390133od_a_a @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_1038_distrib__imp2,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ! [X2: set_a,Y4: set_a,Z3: set_a] :
( ( sup_sup_set_a @ X2 @ ( inf_inf_set_a @ Y4 @ Z3 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X2 @ Y4 ) @ ( sup_sup_set_a @ X2 @ Z3 ) ) )
=> ( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_1039_inf__sup__distrib1,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ X @ ( sup_su3048258781599657691od_a_a @ Y @ Z ) )
= ( sup_su3048258781599657691od_a_a @ ( inf_in8905007599844390133od_a_a @ X @ Y ) @ ( inf_in8905007599844390133od_a_a @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_1040_inf__sup__distrib1,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_1041_inf__sup__distrib2,axiom,
! [Y: set_Product_prod_a_a,Z: set_Product_prod_a_a,X: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ ( sup_su3048258781599657691od_a_a @ Y @ Z ) @ X )
= ( sup_su3048258781599657691od_a_a @ ( inf_in8905007599844390133od_a_a @ Y @ X ) @ ( inf_in8905007599844390133od_a_a @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_1042_inf__sup__distrib2,axiom,
! [Y: set_a,Z: set_a,X: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ Z ) @ X )
= ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ X ) @ ( inf_inf_set_a @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_1043_sup__inf__distrib1,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X @ ( inf_in8905007599844390133od_a_a @ Y @ Z ) )
= ( inf_in8905007599844390133od_a_a @ ( sup_su3048258781599657691od_a_a @ X @ Y ) @ ( sup_su3048258781599657691od_a_a @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_1044_sup__inf__distrib1,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_1045_sup__inf__distrib2,axiom,
! [Y: set_Product_prod_a_a,Z: set_Product_prod_a_a,X: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( inf_in8905007599844390133od_a_a @ Y @ Z ) @ X )
= ( inf_in8905007599844390133od_a_a @ ( sup_su3048258781599657691od_a_a @ Y @ X ) @ ( sup_su3048258781599657691od_a_a @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_1046_sup__inf__distrib2,axiom,
! [Y: set_a,Z: set_a,X: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ Z ) @ X )
= ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ X ) @ ( sup_sup_set_a @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_1047_sup_OcoboundedI2,axiom,
! [C: set_a,B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ C @ B2 )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_1048_sup_OcoboundedI1,axiom,
! [C: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C @ A2 )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_1049_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( sup_sup_set_a @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_1050_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A4: set_a] :
( ( sup_sup_set_a @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_1051_sup_Ocobounded2,axiom,
! [B2: set_a,A2: set_a] : ( ord_less_eq_set_a @ B2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_1052_sup_Ocobounded1,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_1053_sup_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A4: set_a] :
( A4
= ( sup_sup_set_a @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_1054_sup_OboundedI,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ C @ A2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_1055_sup_OboundedE,axiom,
! [B2: set_a,C: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ~ ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_1056_sup__absorb2,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( sup_sup_set_a @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_1057_sup__absorb1,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( sup_sup_set_a @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_1058_sup_Oabsorb2,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_1059_sup_Oabsorb1,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_1060_sup__unique,axiom,
! [F2: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X2: set_a,Y4: set_a] : ( ord_less_eq_set_a @ X2 @ ( F2 @ X2 @ Y4 ) )
=> ( ! [X2: set_a,Y4: set_a] : ( ord_less_eq_set_a @ Y4 @ ( F2 @ X2 @ Y4 ) )
=> ( ! [X2: set_a,Y4: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ Y4 @ X2 )
=> ( ( ord_less_eq_set_a @ Z3 @ X2 )
=> ( ord_less_eq_set_a @ ( F2 @ Y4 @ Z3 ) @ X2 ) ) )
=> ( ( sup_sup_set_a @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_1061_sup_OorderI,axiom,
! [A2: set_a,B2: set_a] :
( ( A2
= ( sup_sup_set_a @ A2 @ B2 ) )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_1062_sup_OorderE,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2
= ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_1063_le__iff__sup,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y3: set_a] :
( ( sup_sup_set_a @ X3 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_1064_sup__least,axiom,
! [Y: set_a,X: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ Z @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_1065_sup__mono,axiom,
! [A2: set_a,C: set_a,B2: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B2 @ D2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ ( sup_sup_set_a @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_1066_sup_Omono,axiom,
! [C: set_a,A2: set_a,D2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C @ A2 )
=> ( ( ord_less_eq_set_a @ D2 @ B2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ C @ D2 ) @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_1067_le__supI2,axiom,
! [X: set_a,B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ X @ B2 )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_1068_le__supI1,axiom,
! [X: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X @ A2 )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_1069_sup__ge2,axiom,
! [Y: set_a,X: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X @ Y ) ) ).
% sup_ge2
thf(fact_1070_sup__ge1,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y ) ) ).
% sup_ge1
thf(fact_1071_le__supI,axiom,
! [A2: set_a,X: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ X )
=> ( ( ord_less_eq_set_a @ B2 @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_1072_le__supE,axiom,
! [A2: set_a,B2: set_a,X: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ X )
=> ~ ( ( ord_less_eq_set_a @ A2 @ X )
=> ~ ( ord_less_eq_set_a @ B2 @ X ) ) ) ).
% le_supE
thf(fact_1073_inf__sup__ord_I3_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_1074_inf__sup__ord_I4_J,axiom,
! [Y: set_a,X: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_1075_chain__subset__antisym__Union,axiom,
! [R2: set_se5735800977113168103od_a_a] :
( ( chain_7119315380238734464od_a_a @ R2 )
=> ( ! [X2: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X2 @ R2 )
=> ( antisym_on_a @ top_top_set_a @ X2 ) )
=> ( antisym_on_a @ top_top_set_a @ ( comple8421679170691845492od_a_a @ R2 ) ) ) ) ).
% chain_subset_antisym_Union
thf(fact_1076_chain__subset__antisym__Union,axiom,
! [R2: set_se7936120678642572967od_a_a] :
( ( chain_951454323991718976od_a_a @ R2 )
=> ( ! [X2: set_Pr8600417178894128327od_a_a] :
( ( member92994164477555952od_a_a @ X2 @ R2 )
=> ( antisy9008168496540133130od_a_a @ top_to8063371432257647191od_a_a @ X2 ) )
=> ( antisy9008168496540133130od_a_a @ top_to8063371432257647191od_a_a @ ( comple6965155415733691188od_a_a @ R2 ) ) ) ) ).
% chain_subset_antisym_Union
thf(fact_1077_chain__subset__trans__Union,axiom,
! [R2: set_se5735800977113168103od_a_a] :
( ( chain_7119315380238734464od_a_a @ R2 )
=> ( ! [X2: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X2 @ R2 )
=> ( trans_on_a @ top_top_set_a @ X2 ) )
=> ( trans_on_a @ top_top_set_a @ ( comple8421679170691845492od_a_a @ R2 ) ) ) ) ).
% chain_subset_trans_Union
thf(fact_1078_chain__subset__trans__Union,axiom,
! [R2: set_se7936120678642572967od_a_a] :
( ( chain_951454323991718976od_a_a @ R2 )
=> ( ! [X2: set_Pr8600417178894128327od_a_a] :
( ( member92994164477555952od_a_a @ X2 @ R2 )
=> ( trans_8517530828069161671od_a_a @ top_to8063371432257647191od_a_a @ X2 ) )
=> ( trans_8517530828069161671od_a_a @ top_to8063371432257647191od_a_a @ ( comple6965155415733691188od_a_a @ R2 ) ) ) ) ).
% chain_subset_trans_Union
thf(fact_1079_cSup__least,axiom,
! [X4: set_set_a,Z: set_a] :
( ( X4 != bot_bot_set_set_a )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ X4 )
=> ( ord_less_eq_set_a @ X2 @ Z ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ X4 ) @ Z ) ) ) ).
% cSup_least
thf(fact_1080_cSup__eq__non__empty,axiom,
! [X4: set_set_a,A2: set_a] :
( ( X4 != bot_bot_set_set_a )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ X4 )
=> ( ord_less_eq_set_a @ X2 @ A2 ) )
=> ( ! [Y4: set_a] :
( ! [X5: set_a] :
( ( member_set_a @ X5 @ X4 )
=> ( ord_less_eq_set_a @ X5 @ Y4 ) )
=> ( ord_less_eq_set_a @ A2 @ Y4 ) )
=> ( ( comple2307003609928055243_set_a @ X4 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_1081_distrib__inf__le,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ ( inf_in8905007599844390133od_a_a @ X @ Y ) @ ( inf_in8905007599844390133od_a_a @ X @ Z ) ) @ ( inf_in8905007599844390133od_a_a @ X @ ( sup_su3048258781599657691od_a_a @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_1082_distrib__inf__le,axiom,
! [X: set_a,Y: set_a,Z: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z ) ) @ ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_1083_distrib__sup__le,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ X @ ( inf_in8905007599844390133od_a_a @ Y @ Z ) ) @ ( inf_in8905007599844390133od_a_a @ ( sup_su3048258781599657691od_a_a @ X @ Y ) @ ( sup_su3048258781599657691od_a_a @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_1084_distrib__sup__le,axiom,
! [X: set_a,Y: set_a,Z: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) @ ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_1085_pred__on_OchainI,axiom,
! [C2: set_a,A: set_a,P: a > a > $o] :
( ( ord_less_eq_set_a @ C2 @ A )
=> ( ! [X2: a,Y4: a] :
( ( member_a @ X2 @ C2 )
=> ( ( member_a @ Y4 @ C2 )
=> ( ( sup_sup_a_a_o @ P
@ ^ [Y2: a,Z2: a] : ( Y2 = Z2 )
@ X2
@ Y4 )
| ( sup_sup_a_a_o @ P
@ ^ [Y2: a,Z2: a] : ( Y2 = Z2 )
@ Y4
@ X2 ) ) ) )
=> ( pred_chain_a @ A @ P @ C2 ) ) ) ).
% pred_on.chainI
thf(fact_1086_pred__on_Ochain__def,axiom,
( pred_chain_a
= ( ^ [A3: set_a,P3: a > a > $o,C6: set_a] :
( ( ord_less_eq_set_a @ C6 @ A3 )
& ! [X3: a] :
( ( member_a @ X3 @ C6 )
=> ! [Y3: a] :
( ( member_a @ Y3 @ C6 )
=> ( ( sup_sup_a_a_o @ P3
@ ^ [Y2: a,Z2: a] : ( Y2 = Z2 )
@ X3
@ Y3 )
| ( sup_sup_a_a_o @ P3
@ ^ [Y2: a,Z2: a] : ( Y2 = Z2 )
@ Y3
@ X3 ) ) ) ) ) ) ) ).
% pred_on.chain_def
thf(fact_1087_Un__Int__assoc__eq,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( ( sup_su3048258781599657691od_a_a @ ( inf_in8905007599844390133od_a_a @ A @ B ) @ C2 )
= ( inf_in8905007599844390133od_a_a @ A @ ( sup_su3048258781599657691od_a_a @ B @ C2 ) ) )
= ( ord_le746702958409616551od_a_a @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_1088_Un__Int__assoc__eq,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B ) @ C2 )
= ( inf_inf_set_a @ A @ ( sup_sup_set_a @ B @ C2 ) ) )
= ( ord_less_eq_set_a @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_1089_sym__Un,axiom,
! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( sym_on_a @ top_top_set_a @ R )
=> ( ( sym_on_a @ top_top_set_a @ S )
=> ( sym_on_a @ top_top_set_a @ ( sup_su3048258781599657691od_a_a @ R @ S ) ) ) ) ).
% sym_Un
thf(fact_1090_sym__Un,axiom,
! [R: set_Pr8600417178894128327od_a_a,S: set_Pr8600417178894128327od_a_a] :
( ( sym_on5631557199876295240od_a_a @ top_to8063371432257647191od_a_a @ R )
=> ( ( sym_on5631557199876295240od_a_a @ top_to8063371432257647191od_a_a @ S )
=> ( sym_on5631557199876295240od_a_a @ top_to8063371432257647191od_a_a @ ( sup_su8193676293155882651od_a_a @ R @ S ) ) ) ) ).
% sym_Un
thf(fact_1091_Zorn__Lemma,axiom,
! [A: set_set_a] :
( ! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ ( chains_a2 @ A ) )
=> ( member_set_a @ ( comple2307003609928055243_set_a @ X2 ) @ A ) )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A )
=> ( ( ord_less_eq_set_a @ X2 @ Xa2 )
=> ( Xa2 = X2 ) ) ) ) ) ).
% Zorn_Lemma
thf(fact_1092_boolean__algebra_Ocomplement__unique,axiom,
! [A2: set_a,X: set_a,Y: set_a] :
( ( ( inf_inf_set_a @ A2 @ X )
= bot_bot_set_a )
=> ( ( ( sup_sup_set_a @ A2 @ X )
= top_top_set_a )
=> ( ( ( inf_inf_set_a @ A2 @ Y )
= bot_bot_set_a )
=> ( ( ( sup_sup_set_a @ A2 @ Y )
= top_top_set_a )
=> ( X = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_1093_boolean__algebra_Ocomplement__unique,axiom,
! [A2: set_Product_prod_a_a,X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ A2 @ X )
= bot_bo3357376287454694259od_a_a )
=> ( ( ( sup_su3048258781599657691od_a_a @ A2 @ X )
= top_to8063371432257647191od_a_a )
=> ( ( ( inf_in8905007599844390133od_a_a @ A2 @ Y )
= bot_bo3357376287454694259od_a_a )
=> ( ( ( sup_su3048258781599657691od_a_a @ A2 @ Y )
= top_to8063371432257647191od_a_a )
=> ( X = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_1094_chain__subset__def,axiom,
( chain_subset_a
= ( ^ [C6: set_set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ C6 )
=> ! [Y3: set_a] :
( ( member_set_a @ Y3 @ C6 )
=> ( ( ord_less_eq_set_a @ X3 @ Y3 )
| ( ord_less_eq_set_a @ Y3 @ X3 ) ) ) ) ) ) ).
% chain_subset_def
thf(fact_1095_subset__Zorn_H,axiom,
! [A: set_set_a] :
( ! [C3: set_set_a] :
( ( pred_chain_set_a @ A @ ord_less_set_a @ C3 )
=> ( member_set_a @ ( comple2307003609928055243_set_a @ C3 ) @ A ) )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A )
=> ( ( ord_less_eq_set_a @ X2 @ Xa2 )
=> ( Xa2 = X2 ) ) ) ) ) ).
% subset_Zorn'
thf(fact_1096_Sup__UNIV,axiom,
( ( comple2307003609928055243_set_a @ top_top_set_set_a )
= top_top_set_a ) ).
% Sup_UNIV
thf(fact_1097_Sup__UNIV,axiom,
( ( comple8421679170691845492od_a_a @ top_to1047947862415971895od_a_a )
= top_to8063371432257647191od_a_a ) ).
% Sup_UNIV
thf(fact_1098_Sup__inter__less__eq,axiom,
! [A: set_se5735800977113168103od_a_a,B: set_se5735800977113168103od_a_a] : ( ord_le746702958409616551od_a_a @ ( comple8421679170691845492od_a_a @ ( inf_in3339382566020358357od_a_a @ A @ B ) ) @ ( inf_in8905007599844390133od_a_a @ ( comple8421679170691845492od_a_a @ A ) @ ( comple8421679170691845492od_a_a @ B ) ) ) ).
% Sup_inter_less_eq
thf(fact_1099_Sup__inter__less__eq,axiom,
! [A: set_set_a,B: set_set_a] : ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( inf_inf_set_set_a @ A @ B ) ) @ ( inf_inf_set_a @ ( comple2307003609928055243_set_a @ A ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Sup_inter_less_eq
thf(fact_1100_Union__UNIV,axiom,
( ( comple2307003609928055243_set_a @ top_top_set_set_a )
= top_top_set_a ) ).
% Union_UNIV
thf(fact_1101_Union__UNIV,axiom,
( ( comple8421679170691845492od_a_a @ top_to1047947862415971895od_a_a )
= top_to8063371432257647191od_a_a ) ).
% Union_UNIV
thf(fact_1102_Sup__eqI,axiom,
! [A: set_set_a,X: set_a] :
( ! [Y4: set_a] :
( ( member_set_a @ Y4 @ A )
=> ( ord_less_eq_set_a @ Y4 @ X ) )
=> ( ! [Y4: set_a] :
( ! [Z5: set_a] :
( ( member_set_a @ Z5 @ A )
=> ( ord_less_eq_set_a @ Z5 @ Y4 ) )
=> ( ord_less_eq_set_a @ X @ Y4 ) )
=> ( ( comple2307003609928055243_set_a @ A )
= X ) ) ) ).
% Sup_eqI
thf(fact_1103_Sup__mono,axiom,
! [A: set_set_a,B: set_set_a] :
( ! [A6: set_a] :
( ( member_set_a @ A6 @ A )
=> ? [X5: set_a] :
( ( member_set_a @ X5 @ B )
& ( ord_less_eq_set_a @ A6 @ X5 ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Sup_mono
thf(fact_1104_Sup__least,axiom,
! [A: set_set_a,Z: set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( ord_less_eq_set_a @ X2 @ Z ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A ) @ Z ) ) ).
% Sup_least
thf(fact_1105_Sup__upper,axiom,
! [X: set_a,A: set_set_a] :
( ( member_set_a @ X @ A )
=> ( ord_less_eq_set_a @ X @ ( comple2307003609928055243_set_a @ A ) ) ) ).
% Sup_upper
thf(fact_1106_Sup__le__iff,axiom,
! [A: set_set_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A ) @ B2 )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A )
=> ( ord_less_eq_set_a @ X3 @ B2 ) ) ) ) ).
% Sup_le_iff
thf(fact_1107_Sup__upper2,axiom,
! [U: set_a,A: set_set_a,V: set_a] :
( ( member_set_a @ U @ A )
=> ( ( ord_less_eq_set_a @ V @ U )
=> ( ord_less_eq_set_a @ V @ ( comple2307003609928055243_set_a @ A ) ) ) ) ).
% Sup_upper2
thf(fact_1108_Union__least,axiom,
! [A: set_set_a,C2: set_a] :
( ! [X6: set_a] :
( ( member_set_a @ X6 @ A )
=> ( ord_less_eq_set_a @ X6 @ C2 ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A ) @ C2 ) ) ).
% Union_least
thf(fact_1109_Union__upper,axiom,
! [B: set_a,A: set_set_a] :
( ( member_set_a @ B @ A )
=> ( ord_less_eq_set_a @ B @ ( comple2307003609928055243_set_a @ A ) ) ) ).
% Union_upper
thf(fact_1110_Union__subsetI,axiom,
! [A: set_set_a,B: set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ? [Y5: set_a] :
( ( member_set_a @ Y5 @ B )
& ( ord_less_eq_set_a @ X2 @ Y5 ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Union_subsetI
thf(fact_1111_less__eq__Sup,axiom,
! [A: set_set_a,U: set_a] :
( ! [V2: set_a] :
( ( member_set_a @ V2 @ A )
=> ( ord_less_eq_set_a @ U @ V2 ) )
=> ( ( A != bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ U @ ( comple2307003609928055243_set_a @ A ) ) ) ) ).
% less_eq_Sup
thf(fact_1112_Sup__subset__mono,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Sup_subset_mono
thf(fact_1113_Union__disjoint,axiom,
! [C2: set_se5735800977113168103od_a_a,A: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ ( comple8421679170691845492od_a_a @ C2 ) @ A )
= bot_bo3357376287454694259od_a_a )
= ( ! [X3: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X3 @ C2 )
=> ( ( inf_in8905007599844390133od_a_a @ X3 @ A )
= bot_bo3357376287454694259od_a_a ) ) ) ) ).
% Union_disjoint
thf(fact_1114_Union__disjoint,axiom,
! [C2: set_set_a,A: set_a] :
( ( ( inf_inf_set_a @ ( comple2307003609928055243_set_a @ C2 ) @ A )
= bot_bot_set_a )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ C2 )
=> ( ( inf_inf_set_a @ X3 @ A )
= bot_bot_set_a ) ) ) ) ).
% Union_disjoint
thf(fact_1115_Union__Int__subset,axiom,
! [A: set_se5735800977113168103od_a_a,B: set_se5735800977113168103od_a_a] : ( ord_le746702958409616551od_a_a @ ( comple8421679170691845492od_a_a @ ( inf_in3339382566020358357od_a_a @ A @ B ) ) @ ( inf_in8905007599844390133od_a_a @ ( comple8421679170691845492od_a_a @ A ) @ ( comple8421679170691845492od_a_a @ B ) ) ) ).
% Union_Int_subset
thf(fact_1116_Union__Int__subset,axiom,
! [A: set_set_a,B: set_set_a] : ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( inf_inf_set_set_a @ A @ B ) ) @ ( inf_inf_set_a @ ( comple2307003609928055243_set_a @ A ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Union_Int_subset
thf(fact_1117_Union__mono,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Union_mono
thf(fact_1118_injectivity__union,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C2: set_Product_prod_a_a,D: set_Product_prod_a_a,P: set_Product_prod_a_a > $o,Q: set_Product_prod_a_a > $o] :
( ( ( sup_su3048258781599657691od_a_a @ A @ B )
= ( sup_su3048258781599657691od_a_a @ C2 @ D ) )
=> ( ( P @ A )
=> ( ( P @ C2 )
=> ( ( Q @ B )
=> ( ( Q @ D )
=> ( ! [S4: set_Product_prod_a_a,T3: set_Product_prod_a_a] :
( ( P @ S4 )
=> ( ( Q @ T3 )
=> ( ( inf_in8905007599844390133od_a_a @ S4 @ T3 )
= bot_bo3357376287454694259od_a_a ) ) )
=> ( ( A = C2 )
& ( B = D ) ) ) ) ) ) ) ) ).
% injectivity_union
thf(fact_1119_injectivity__union,axiom,
! [A: set_a,B: set_a,C2: set_a,D: set_a,P: set_a > $o,Q: set_a > $o] :
( ( ( sup_sup_set_a @ A @ B )
= ( sup_sup_set_a @ C2 @ D ) )
=> ( ( P @ A )
=> ( ( P @ C2 )
=> ( ( Q @ B )
=> ( ( Q @ D )
=> ( ! [S4: set_a,T3: set_a] :
( ( P @ S4 )
=> ( ( Q @ T3 )
=> ( ( inf_inf_set_a @ S4 @ T3 )
= bot_bot_set_a ) ) )
=> ( ( A = C2 )
& ( B = D ) ) ) ) ) ) ) ) ).
% injectivity_union
thf(fact_1120_Sup__inf__eq__bot__iff,axiom,
! [B: set_se5735800977113168103od_a_a,A2: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ ( comple8421679170691845492od_a_a @ B ) @ A2 )
= bot_bo3357376287454694259od_a_a )
= ( ! [X3: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X3 @ B )
=> ( ( inf_in8905007599844390133od_a_a @ X3 @ A2 )
= bot_bo3357376287454694259od_a_a ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_1121_Sup__inf__eq__bot__iff,axiom,
! [B: set_set_a,A2: set_a] :
( ( ( inf_inf_set_a @ ( comple2307003609928055243_set_a @ B ) @ A2 )
= bot_bot_set_a )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ B )
=> ( ( inf_inf_set_a @ X3 @ A2 )
= bot_bot_set_a ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_1122_Refl__under__underS,axiom,
! [R: set_Product_prod_a_a,A2: a] :
( ( refl_on_a @ ( field_a @ R ) @ R )
=> ( ( member_a @ A2 @ ( field_a @ R ) )
=> ( ( order_under_a @ R @ A2 )
= ( sup_sup_set_a @ ( order_underS_a @ R @ A2 ) @ ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ) ).
% Refl_under_underS
thf(fact_1123_finite__subset__Union__chain,axiom,
! [A: set_a,B6: set_set_a,A5: set_set_a] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_a @ A @ ( comple2307003609928055243_set_a @ B6 ) )
=> ( ( B6 != bot_bot_set_set_a )
=> ( ( pred_chain_set_a @ A5 @ ord_less_set_a @ B6 )
=> ~ ! [B7: set_a] :
( ( member_set_a @ B7 @ B6 )
=> ~ ( ord_less_eq_set_a @ A @ B7 ) ) ) ) ) ) ).
% finite_subset_Union_chain
thf(fact_1124_cSup__inter__less__eq,axiom,
! [A: set_set_a,B: set_set_a] :
( ( condit3373647341569784514_set_a @ A )
=> ( ( condit3373647341569784514_set_a @ B )
=> ( ( ( inf_inf_set_set_a @ A @ B )
!= bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( inf_inf_set_set_a @ A @ B ) ) @ ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ A ) @ ( comple2307003609928055243_set_a @ B ) ) ) ) ) ) ).
% cSup_inter_less_eq
thf(fact_1125_insert__subset,axiom,
! [X: a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A ) @ B )
= ( ( member_a @ X @ B )
& ( ord_less_eq_set_a @ A @ B ) ) ) ).
% insert_subset
thf(fact_1126_Int__insert__left__if0,axiom,
! [A2: product_prod_a_a,C2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ A2 @ C2 )
=> ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A2 @ B ) @ C2 )
= ( inf_in8905007599844390133od_a_a @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_1127_Int__insert__left__if0,axiom,
! [A2: a,C2: set_a,B: set_a] :
( ~ ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_a @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_1128_Int__insert__left__if1,axiom,
! [A2: product_prod_a_a,C2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ C2 )
=> ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A2 @ B ) @ C2 )
= ( insert4534936382041156343od_a_a @ A2 @ ( inf_in8905007599844390133od_a_a @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1129_Int__insert__left__if1,axiom,
! [A2: a,C2: set_a,B: set_a] :
( ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( insert_a @ A2 @ ( inf_inf_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1130_insert__inter__insert,axiom,
! [A2: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A2 @ A ) @ ( insert4534936382041156343od_a_a @ A2 @ B ) )
= ( insert4534936382041156343od_a_a @ A2 @ ( inf_in8905007599844390133od_a_a @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_1131_insert__inter__insert,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ ( insert_a @ A2 @ B ) )
= ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_1132_Int__insert__right__if0,axiom,
! [A2: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ A2 @ A )
=> ( ( inf_in8905007599844390133od_a_a @ A @ ( insert4534936382041156343od_a_a @ A2 @ B ) )
= ( inf_in8905007599844390133od_a_a @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_1133_Int__insert__right__if0,axiom,
! [A2: a,A: set_a,B: set_a] :
( ~ ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_1134_Int__insert__right__if1,axiom,
! [A2: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ A )
=> ( ( inf_in8905007599844390133od_a_a @ A @ ( insert4534936382041156343od_a_a @ A2 @ B ) )
= ( insert4534936382041156343od_a_a @ A2 @ ( inf_in8905007599844390133od_a_a @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1135_Int__insert__right__if1,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1136_bdd__above_OI,axiom,
! [A: set_set_a,M: set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( ord_less_eq_set_a @ X2 @ M ) )
=> ( condit3373647341569784514_set_a @ A ) ) ).
% bdd_above.I
thf(fact_1137_singleton__insert__inj__eq_H,axiom,
! [A2: a,A: set_a,B2: a] :
( ( ( insert_a @ A2 @ A )
= ( insert_a @ B2 @ bot_bot_set_a ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_a @ A @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_1138_singleton__insert__inj__eq,axiom,
! [B2: a,A2: a,A: set_a] :
( ( ( insert_a @ B2 @ bot_bot_set_a )
= ( insert_a @ A2 @ A ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_a @ A @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_1139_disjoint__insert_I2_J,axiom,
! [A: set_Product_prod_a_a,B2: product_prod_a_a,B: set_Product_prod_a_a] :
( ( bot_bo3357376287454694259od_a_a
= ( inf_in8905007599844390133od_a_a @ A @ ( insert4534936382041156343od_a_a @ B2 @ B ) ) )
= ( ~ ( member1426531477525435216od_a_a @ B2 @ A )
& ( bot_bo3357376287454694259od_a_a
= ( inf_in8905007599844390133od_a_a @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1140_disjoint__insert_I2_J,axiom,
! [A: set_a,B2: a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A @ ( insert_a @ B2 @ B ) ) )
= ( ~ ( member_a @ B2 @ A )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1141_disjoint__insert_I1_J,axiom,
! [B: set_Product_prod_a_a,A2: product_prod_a_a,A: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ B @ ( insert4534936382041156343od_a_a @ A2 @ A ) )
= bot_bo3357376287454694259od_a_a )
= ( ~ ( member1426531477525435216od_a_a @ A2 @ B )
& ( ( inf_in8905007599844390133od_a_a @ B @ A )
= bot_bo3357376287454694259od_a_a ) ) ) ).
% disjoint_insert(1)
thf(fact_1142_disjoint__insert_I1_J,axiom,
! [B: set_a,A2: a,A: set_a] :
( ( ( inf_inf_set_a @ B @ ( insert_a @ A2 @ A ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A2 @ B )
& ( ( inf_inf_set_a @ B @ A )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_1143_insert__disjoint_I2_J,axiom,
! [A2: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( bot_bo3357376287454694259od_a_a
= ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A2 @ A ) @ B ) )
= ( ~ ( member1426531477525435216od_a_a @ A2 @ B )
& ( bot_bo3357376287454694259od_a_a
= ( inf_in8905007599844390133od_a_a @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1144_insert__disjoint_I2_J,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ B ) )
= ( ~ ( member_a @ A2 @ B )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1145_insert__disjoint_I1_J,axiom,
! [A2: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A2 @ A ) @ B )
= bot_bo3357376287454694259od_a_a )
= ( ~ ( member1426531477525435216od_a_a @ A2 @ B )
& ( ( inf_in8905007599844390133od_a_a @ A @ B )
= bot_bo3357376287454694259od_a_a ) ) ) ).
% insert_disjoint(1)
thf(fact_1146_insert__disjoint_I1_J,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ B )
= bot_bot_set_a )
= ( ~ ( member_a @ A2 @ B )
& ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_1147_insert__Times__insert,axiom,
! [A2: a,A: set_a,B2: a,B: set_a] :
( ( product_Sigma_a_a @ ( insert_a @ A2 @ A )
@ ^ [Uu: a] : ( insert_a @ B2 @ B ) )
= ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ A2 @ B2 )
@ ( sup_su3048258781599657691od_a_a
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : ( insert_a @ B2 @ B ) )
@ ( product_Sigma_a_a @ ( insert_a @ A2 @ A )
@ ^ [Uu: a] : B ) ) ) ) ).
% insert_Times_insert
thf(fact_1148_finite__subset__Union,axiom,
! [A: set_a,B6: set_set_a] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_a @ A @ ( comple2307003609928055243_set_a @ B6 ) )
=> ~ ! [F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ( ord_le3724670747650509150_set_a @ F3 @ B6 )
=> ~ ( ord_less_eq_set_a @ A @ ( comple2307003609928055243_set_a @ F3 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_1149_Int__insert__left,axiom,
! [A2: product_prod_a_a,C2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ( member1426531477525435216od_a_a @ A2 @ C2 )
=> ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A2 @ B ) @ C2 )
= ( insert4534936382041156343od_a_a @ A2 @ ( inf_in8905007599844390133od_a_a @ B @ C2 ) ) ) )
& ( ~ ( member1426531477525435216od_a_a @ A2 @ C2 )
=> ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A2 @ B ) @ C2 )
= ( inf_in8905007599844390133od_a_a @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1150_Int__insert__left,axiom,
! [A2: a,C2: set_a,B: set_a] :
( ( ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( insert_a @ A2 @ ( inf_inf_set_a @ B @ C2 ) ) ) )
& ( ~ ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1151_Int__insert__right,axiom,
! [A2: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ( member1426531477525435216od_a_a @ A2 @ A )
=> ( ( inf_in8905007599844390133od_a_a @ A @ ( insert4534936382041156343od_a_a @ A2 @ B ) )
= ( insert4534936382041156343od_a_a @ A2 @ ( inf_in8905007599844390133od_a_a @ A @ B ) ) ) )
& ( ~ ( member1426531477525435216od_a_a @ A2 @ A )
=> ( ( inf_in8905007599844390133od_a_a @ A @ ( insert4534936382041156343od_a_a @ A2 @ B ) )
= ( inf_in8905007599844390133od_a_a @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_1152_Int__insert__right,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) )
& ( ~ ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_1153_subset__singletonD,axiom,
! [A: set_a,X: a] :
( ( ord_less_eq_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) )
=> ( ( A = bot_bot_set_a )
| ( A
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_1154_subset__singleton__iff,axiom,
! [X4: set_a,A2: a] :
( ( ord_less_eq_set_a @ X4 @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( ( X4 = bot_bot_set_a )
| ( X4
= ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_1155_le__cSup__finite,axiom,
! [X4: set_set_a,X: set_a] :
( ( finite_finite_set_a @ X4 )
=> ( ( member_set_a @ X @ X4 )
=> ( ord_less_eq_set_a @ X @ ( comple2307003609928055243_set_a @ X4 ) ) ) ) ).
% le_cSup_finite
thf(fact_1156_insert__mono,axiom,
! [C2: set_a,D: set_a,A2: a] :
( ( ord_less_eq_set_a @ C2 @ D )
=> ( ord_less_eq_set_a @ ( insert_a @ A2 @ C2 ) @ ( insert_a @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_1157_subset__insert,axiom,
! [X: a,A: set_a,B: set_a] :
( ~ ( member_a @ X @ A )
=> ( ( ord_less_eq_set_a @ A @ ( insert_a @ X @ B ) )
= ( ord_less_eq_set_a @ A @ B ) ) ) ).
% subset_insert
thf(fact_1158_subset__insertI,axiom,
! [B: set_a,A2: a] : ( ord_less_eq_set_a @ B @ ( insert_a @ A2 @ B ) ) ).
% subset_insertI
thf(fact_1159_subset__insertI2,axiom,
! [A: set_a,B: set_a,B2: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ A @ ( insert_a @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_1160_insert__subsetI,axiom,
! [X: a,A: set_a,X4: set_a] :
( ( member_a @ X @ A )
=> ( ( ord_less_eq_set_a @ X4 @ A )
=> ( ord_less_eq_set_a @ ( insert_a @ X @ X4 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_1161_bdd__above_Ounfold,axiom,
( condit3373647341569784514_set_a
= ( ^ [A3: set_set_a] :
? [M2: set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ A3 )
=> ( ord_less_eq_set_a @ X3 @ M2 ) ) ) ) ).
% bdd_above.unfold
thf(fact_1162_bdd__above_OE,axiom,
! [A: set_set_a] :
( ( condit3373647341569784514_set_a @ A )
=> ~ ! [M3: set_a] :
~ ! [X5: set_a] :
( ( member_set_a @ X5 @ A )
=> ( ord_less_eq_set_a @ X5 @ M3 ) ) ) ).
% bdd_above.E
thf(fact_1163_cSup__upper2,axiom,
! [X: set_a,X4: set_set_a,Y: set_a] :
( ( member_set_a @ X @ X4 )
=> ( ( ord_less_eq_set_a @ Y @ X )
=> ( ( condit3373647341569784514_set_a @ X4 )
=> ( ord_less_eq_set_a @ Y @ ( comple2307003609928055243_set_a @ X4 ) ) ) ) ) ).
% cSup_upper2
thf(fact_1164_cSup__upper,axiom,
! [X: set_a,X4: set_set_a] :
( ( member_set_a @ X @ X4 )
=> ( ( condit3373647341569784514_set_a @ X4 )
=> ( ord_less_eq_set_a @ X @ ( comple2307003609928055243_set_a @ X4 ) ) ) ) ).
% cSup_upper
thf(fact_1165_refl__on__singleton,axiom,
! [X: a] : ( refl_on_a @ ( insert_a @ X @ bot_bot_set_a ) @ ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ X @ X ) @ bot_bo3357376287454694259od_a_a ) ) ).
% refl_on_singleton
thf(fact_1166_insert__UNIV,axiom,
! [X: a] :
( ( insert_a @ X @ top_top_set_a )
= top_top_set_a ) ).
% insert_UNIV
thf(fact_1167_insert__UNIV,axiom,
! [X: product_prod_a_a] :
( ( insert4534936382041156343od_a_a @ X @ top_to8063371432257647191od_a_a )
= top_to8063371432257647191od_a_a ) ).
% insert_UNIV
thf(fact_1168_cSup__mono,axiom,
! [B: set_set_a,A: set_set_a] :
( ( B != bot_bot_set_set_a )
=> ( ( condit3373647341569784514_set_a @ A )
=> ( ! [B8: set_a] :
( ( member_set_a @ B8 @ B )
=> ? [X5: set_a] :
( ( member_set_a @ X5 @ A )
& ( ord_less_eq_set_a @ B8 @ X5 ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ B ) @ ( comple2307003609928055243_set_a @ A ) ) ) ) ) ).
% cSup_mono
thf(fact_1169_cSup__le__iff,axiom,
! [S2: set_set_a,A2: set_a] :
( ( S2 != bot_bot_set_set_a )
=> ( ( condit3373647341569784514_set_a @ S2 )
=> ( ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ S2 ) @ A2 )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ S2 )
=> ( ord_less_eq_set_a @ X3 @ A2 ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_1170_subset__chain__insert,axiom,
! [A5: set_set_a,B: set_a,B6: set_set_a] :
( ( pred_chain_set_a @ A5 @ ord_less_set_a @ ( insert_set_a @ B @ B6 ) )
= ( ( member_set_a @ B @ A5 )
& ! [X3: set_a] :
( ( member_set_a @ X3 @ B6 )
=> ( ( ord_less_eq_set_a @ X3 @ B )
| ( ord_less_eq_set_a @ B @ X3 ) ) )
& ( pred_chain_set_a @ A5 @ ord_less_set_a @ B6 ) ) ) ).
% subset_chain_insert
thf(fact_1171_antisym__singleton,axiom,
! [X: product_prod_a_a] : ( antisym_on_a @ top_top_set_a @ ( insert4534936382041156343od_a_a @ X @ bot_bo3357376287454694259od_a_a ) ) ).
% antisym_singleton
thf(fact_1172_antisym__singleton,axiom,
! [X: produc3498347346309940967od_a_a] : ( antisy9008168496540133130od_a_a @ top_to8063371432257647191od_a_a @ ( insert8933621553020740791od_a_a @ X @ bot_bo510284599550014259od_a_a ) ) ).
% antisym_singleton
thf(fact_1173_chains__extend,axiom,
! [C: set_set_a,S2: set_set_a,Z: set_a] :
( ( member_set_set_a @ C @ ( chains_a2 @ S2 ) )
=> ( ( member_set_a @ Z @ S2 )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ C )
=> ( ord_less_eq_set_a @ X2 @ Z ) )
=> ( member_set_set_a @ ( sup_sup_set_set_a @ ( insert_set_a @ Z @ bot_bot_set_set_a ) @ C ) @ ( chains_a2 @ S2 ) ) ) ) ) ).
% chains_extend
thf(fact_1174_cSup__subset__mono,axiom,
! [A: set_set_a,B: set_set_a] :
( ( A != bot_bot_set_set_a )
=> ( ( condit3373647341569784514_set_a @ B )
=> ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A ) @ ( comple2307003609928055243_set_a @ B ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_1175_trans__singleton,axiom,
! [A2: a] : ( trans_on_a @ top_top_set_a @ ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ A2 @ A2 ) @ bot_bo3357376287454694259od_a_a ) ) ).
% trans_singleton
thf(fact_1176_trans__singleton,axiom,
! [A2: product_prod_a_a] : ( trans_8517530828069161671od_a_a @ top_to8063371432257647191od_a_a @ ( insert8933621553020740791od_a_a @ ( produc7886510207707329367od_a_a @ A2 @ A2 ) @ bot_bo510284599550014259od_a_a ) ) ).
% trans_singleton
thf(fact_1177_finite__SigmaI,axiom,
! [A: set_a,B: a > set_a] :
( ( finite_finite_a @ A )
=> ( ! [A6: a] :
( ( member_a @ A6 @ A )
=> ( finite_finite_a @ ( B @ A6 ) ) )
=> ( finite6544458595007987280od_a_a @ ( product_Sigma_a_a @ A @ B ) ) ) ) ).
% finite_SigmaI
thf(fact_1178_finite__Collect__subsets,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( finite_finite_set_a
@ ( collect_set_a
@ ^ [B3: set_a] : ( ord_less_eq_set_a @ B3 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_1179_finite__Collect__not,axiom,
! [P: a > $o] :
( ( finite_finite_a @ ( collect_a @ P ) )
=> ( ( finite_finite_a
@ ( collect_a
@ ^ [X3: a] :
~ ( P @ X3 ) ) )
= ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_Collect_not
thf(fact_1180_finite__Collect__not,axiom,
! [P: product_prod_a_a > $o] :
( ( finite6544458595007987280od_a_a @ ( collec3336397797384452498od_a_a @ P ) )
=> ( ( finite6544458595007987280od_a_a
@ ( collec3336397797384452498od_a_a
@ ^ [X3: product_prod_a_a] :
~ ( P @ X3 ) ) )
= ( finite6544458595007987280od_a_a @ top_to8063371432257647191od_a_a ) ) ) ).
% finite_Collect_not
thf(fact_1181_finite__Plus__UNIV__iff,axiom,
( ( finite51705147264084924um_a_a @ top_to8848906000605539851um_a_a )
= ( ( finite_finite_a @ top_top_set_a )
& ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_1182_finite__Plus__UNIV__iff,axiom,
( ( finite1811503200718464613od_a_a @ top_to888355444713840692od_a_a )
= ( ( finite_finite_a @ top_top_set_a )
& ( finite6544458595007987280od_a_a @ top_to8063371432257647191od_a_a ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_1183_finite__Plus__UNIV__iff,axiom,
( ( finite1570297664671449043_a_a_a @ top_to4234792268953903778_a_a_a )
= ( ( finite6544458595007987280od_a_a @ top_to8063371432257647191od_a_a )
& ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_1184_finite__Plus__UNIV__iff,axiom,
( ( finite8318556799873141628od_a_a @ top_to7388749512536171979od_a_a )
= ( ( finite6544458595007987280od_a_a @ top_to8063371432257647191od_a_a )
& ( finite6544458595007987280od_a_a @ top_to8063371432257647191od_a_a ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_1185_finite__Int,axiom,
! [F4: set_Product_prod_a_a,G2: set_Product_prod_a_a] :
( ( ( finite6544458595007987280od_a_a @ F4 )
| ( finite6544458595007987280od_a_a @ G2 ) )
=> ( finite6544458595007987280od_a_a @ ( inf_in8905007599844390133od_a_a @ F4 @ G2 ) ) ) ).
% finite_Int
thf(fact_1186_finite__Int,axiom,
! [F4: set_a,G2: set_a] :
( ( ( finite_finite_a @ F4 )
| ( finite_finite_a @ G2 ) )
=> ( finite_finite_a @ ( inf_inf_set_a @ F4 @ G2 ) ) ) ).
% finite_Int
thf(fact_1187_finite__has__minimal2,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ( ord_less_eq_set_a @ X2 @ A2 )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A )
=> ( ( ord_less_eq_set_a @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1188_finite__has__maximal2,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ( ord_less_eq_set_a @ A2 @ X2 )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A )
=> ( ( ord_less_eq_set_a @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1189_ex__new__if__finite,axiom,
! [A: set_a] :
( ~ ( finite_finite_a @ top_top_set_a )
=> ( ( finite_finite_a @ A )
=> ? [A6: a] :
~ ( member_a @ A6 @ A ) ) ) ).
% ex_new_if_finite
thf(fact_1190_ex__new__if__finite,axiom,
! [A: set_Product_prod_a_a] :
( ~ ( finite6544458595007987280od_a_a @ top_to8063371432257647191od_a_a )
=> ( ( finite6544458595007987280od_a_a @ A )
=> ? [A6: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ A6 @ A ) ) ) ).
% ex_new_if_finite
thf(fact_1191_rev__finite__subset,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( finite_finite_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_1192_infinite__super,axiom,
! [S2: set_a,T2: set_a] :
( ( ord_less_eq_set_a @ S2 @ T2 )
=> ( ~ ( finite_finite_a @ S2 )
=> ~ ( finite_finite_a @ T2 ) ) ) ).
% infinite_super
thf(fact_1193_finite__subset,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ A ) ) ) ).
% finite_subset
thf(fact_1194_finite__cartesian__product,axiom,
! [A: set_a,B: set_a] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B )
=> ( finite6544458595007987280od_a_a
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : B ) ) ) ) ).
% finite_cartesian_product
thf(fact_1195_finite__has__minimal,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A )
=> ( ( ord_less_eq_set_a @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1196_finite__has__maximal,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A )
=> ( ( ord_less_eq_set_a @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1197_insert__partition,axiom,
! [X: set_Product_prod_a_a,F4: set_se5735800977113168103od_a_a] :
( ~ ( member1816616512716248880od_a_a @ X @ F4 )
=> ( ! [X2: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X2 @ ( insert914553114930139863od_a_a @ X @ F4 ) )
=> ! [Xa: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ Xa @ ( insert914553114930139863od_a_a @ X @ F4 ) )
=> ( ( X2 != Xa )
=> ( ( inf_in8905007599844390133od_a_a @ X2 @ Xa )
= bot_bo3357376287454694259od_a_a ) ) ) )
=> ( ( inf_in8905007599844390133od_a_a @ X @ ( comple8421679170691845492od_a_a @ F4 ) )
= bot_bo3357376287454694259od_a_a ) ) ) ).
% insert_partition
thf(fact_1198_insert__partition,axiom,
! [X: set_a,F4: set_set_a] :
( ~ ( member_set_a @ X @ F4 )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ ( insert_set_a @ X @ F4 ) )
=> ! [Xa: set_a] :
( ( member_set_a @ Xa @ ( insert_set_a @ X @ F4 ) )
=> ( ( X2 != Xa )
=> ( ( inf_inf_set_a @ X2 @ Xa )
= bot_bot_set_a ) ) ) )
=> ( ( inf_inf_set_a @ X @ ( comple2307003609928055243_set_a @ F4 ) )
= bot_bot_set_a ) ) ) ).
% insert_partition
thf(fact_1199_finite__prod,axiom,
( ( finite5848958031409366265od_a_a @ top_to4273090908018391168od_a_a )
= ( ( finite_finite_a @ top_top_set_a )
& ( finite6544458595007987280od_a_a @ top_to8063371432257647191od_a_a ) ) ) ).
% finite_prod
thf(fact_1200_finite__prod,axiom,
( ( finite5607752495362350695_a_a_a @ top_to7619527732258454254_a_a_a )
= ( ( finite6544458595007987280od_a_a @ top_to8063371432257647191od_a_a )
& ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_prod
thf(fact_1201_finite__prod,axiom,
( ( finite20598220287489552od_a_a @ top_to7953187161931797015od_a_a )
= ( ( finite6544458595007987280od_a_a @ top_to8063371432257647191od_a_a )
& ( finite6544458595007987280od_a_a @ top_to8063371432257647191od_a_a ) ) ) ).
% finite_prod
thf(fact_1202_finite__prod,axiom,
( ( finite6544458595007987280od_a_a @ top_to8063371432257647191od_a_a )
= ( ( finite_finite_a @ top_top_set_a )
& ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_prod
thf(fact_1203_finite__Prod__UNIV,axiom,
( ( finite_finite_a @ top_top_set_a )
=> ( ( finite_finite_a @ top_top_set_a )
=> ( finite6544458595007987280od_a_a @ top_to8063371432257647191od_a_a ) ) ) ).
% finite_Prod_UNIV
thf(fact_1204_finite__Prod__UNIV,axiom,
( ( finite_finite_a @ top_top_set_a )
=> ( ( finite6544458595007987280od_a_a @ top_to8063371432257647191od_a_a )
=> ( finite5848958031409366265od_a_a @ top_to4273090908018391168od_a_a ) ) ) ).
% finite_Prod_UNIV
thf(fact_1205_finite__Prod__UNIV,axiom,
( ( finite6544458595007987280od_a_a @ top_to8063371432257647191od_a_a )
=> ( ( finite_finite_a @ top_top_set_a )
=> ( finite5607752495362350695_a_a_a @ top_to7619527732258454254_a_a_a ) ) ) ).
% finite_Prod_UNIV
thf(fact_1206_finite__Prod__UNIV,axiom,
( ( finite6544458595007987280od_a_a @ top_to8063371432257647191od_a_a )
=> ( ( finite6544458595007987280od_a_a @ top_to8063371432257647191od_a_a )
=> ( finite20598220287489552od_a_a @ top_to7953187161931797015od_a_a ) ) ) ).
% finite_Prod_UNIV
thf(fact_1207_Finite__Set_Ofinite__set,axiom,
( ( finite_finite_set_a @ top_top_set_set_a )
= ( finite_finite_a @ top_top_set_a ) ) ).
% Finite_Set.finite_set
thf(fact_1208_Finite__Set_Ofinite__set,axiom,
( ( finite8717734299975451184od_a_a @ top_to1047947862415971895od_a_a )
= ( finite6544458595007987280od_a_a @ top_to8063371432257647191od_a_a ) ) ).
% Finite_Set.finite_set
thf(fact_1209_finite__SigmaI2,axiom,
! [A: set_a,B: a > set_a] :
( ( finite_finite_a
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A )
& ( ( B @ X3 )
!= bot_bot_set_a ) ) ) )
=> ( ! [A6: a] :
( ( member_a @ A6 @ A )
=> ( finite_finite_a @ ( B @ A6 ) ) )
=> ( finite6544458595007987280od_a_a @ ( product_Sigma_a_a @ A @ B ) ) ) ) ).
% finite_SigmaI2
thf(fact_1210_finite__cartesian__productD1,axiom,
! [A: set_a,B: set_a] :
( ( finite6544458595007987280od_a_a
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : B ) )
=> ( ( B != bot_bot_set_a )
=> ( finite_finite_a @ A ) ) ) ).
% finite_cartesian_productD1
thf(fact_1211_finite__cartesian__productD2,axiom,
! [A: set_a,B: set_a] :
( ( finite6544458595007987280od_a_a
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : B ) )
=> ( ( A != bot_bot_set_a )
=> ( finite_finite_a @ B ) ) ) ).
% finite_cartesian_productD2
thf(fact_1212_finite__cartesian__product__iff,axiom,
! [A: set_a,B: set_a] :
( ( finite6544458595007987280od_a_a
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : B ) )
= ( ( A = bot_bot_set_a )
| ( B = bot_bot_set_a )
| ( ( finite_finite_a @ A )
& ( finite_finite_a @ B ) ) ) ) ).
% finite_cartesian_product_iff
thf(fact_1213_finite__subset__induct_H,axiom,
! [F4: set_a,A: set_a,P: set_a > $o] :
( ( finite_finite_a @ F4 )
=> ( ( ord_less_eq_set_a @ F4 @ A )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A6: a,F5: set_a] :
( ( finite_finite_a @ F5 )
=> ( ( member_a @ A6 @ A )
=> ( ( ord_less_eq_set_a @ F5 @ A )
=> ( ~ ( member_a @ A6 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert_a @ A6 @ F5 ) ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1214_finite__subset__induct,axiom,
! [F4: set_a,A: set_a,P: set_a > $o] :
( ( finite_finite_a @ F4 )
=> ( ( ord_less_eq_set_a @ F4 @ A )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A6: a,F5: set_a] :
( ( finite_finite_a @ F5 )
=> ( ( member_a @ A6 @ A )
=> ( ~ ( member_a @ A6 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert_a @ A6 @ F5 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1215_finite__option__UNIV,axiom,
( ( finite1674126218327898605tion_a @ top_top_set_option_a )
= ( finite_finite_a @ top_top_set_a ) ) ).
% finite_option_UNIV
thf(fact_1216_finite__option__UNIV,axiom,
( ( finite1824108633307372438od_a_a @ top_to5085949387790111389od_a_a )
= ( finite6544458595007987280od_a_a @ top_to8063371432257647191od_a_a ) ) ).
% finite_option_UNIV
thf(fact_1217_infinite__cartesian__product,axiom,
! [A: set_a,B: set_a] :
( ~ ( finite_finite_a @ A )
=> ( ~ ( finite_finite_a @ B )
=> ~ ( finite6544458595007987280od_a_a
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : B ) ) ) ) ).
% infinite_cartesian_product
thf(fact_1218_proj__iff,axiom,
! [A: set_a,R: set_Product_prod_a_a,X: a,Y: a] :
( ( equiv_equiv_a @ A @ R )
=> ( ( ord_less_eq_set_a @ ( insert_a @ X @ ( insert_a @ Y @ bot_bot_set_a ) ) @ A )
=> ( ( ( equiv_proj_a_a @ R @ X )
= ( equiv_proj_a_a @ R @ Y ) )
= ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R ) ) ) ) ).
% proj_iff
thf(fact_1219_finite__equiv,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( finite8717734299975451184od_a_a @ ( collec1673347964119250290od_a_a @ ( equiv_equiv_a @ A ) ) ) ) ).
% finite_equiv
thf(fact_1220_trans__Id,axiom,
trans_on_a @ top_top_set_a @ id_a ).
% trans_Id
thf(fact_1221_trans__Id,axiom,
trans_8517530828069161671od_a_a @ top_to8063371432257647191od_a_a @ id_Product_prod_a_a ).
% trans_Id
thf(fact_1222_refl__Id,axiom,
refl_on_a @ top_top_set_a @ id_a ).
% refl_Id
thf(fact_1223_refl__Id,axiom,
refl_o7745108929832855590od_a_a @ top_to8063371432257647191od_a_a @ id_Product_prod_a_a ).
% refl_Id
thf(fact_1224_sym__Id,axiom,
sym_on_a @ top_top_set_a @ id_a ).
% sym_Id
thf(fact_1225_sym__Id,axiom,
sym_on5631557199876295240od_a_a @ top_to8063371432257647191od_a_a @ id_Product_prod_a_a ).
% sym_Id
thf(fact_1226_antisym__Id,axiom,
antisym_on_a @ top_top_set_a @ id_a ).
% antisym_Id
thf(fact_1227_antisym__Id,axiom,
antisy9008168496540133130od_a_a @ top_to8063371432257647191od_a_a @ id_Product_prod_a_a ).
% antisym_Id
thf(fact_1228_refl__reflcl,axiom,
! [R: set_Product_prod_a_a] : ( refl_on_a @ top_top_set_a @ ( sup_su3048258781599657691od_a_a @ R @ id_a ) ) ).
% refl_reflcl
thf(fact_1229_refl__reflcl,axiom,
! [R: set_Pr8600417178894128327od_a_a] : ( refl_o7745108929832855590od_a_a @ top_to8063371432257647191od_a_a @ ( sup_su8193676293155882651od_a_a @ R @ id_Product_prod_a_a ) ) ).
% refl_reflcl
thf(fact_1230_Refl__antisym__eq__Image1__Image1__iff,axiom,
! [R: set_Product_prod_a_a,A2: a,B2: a] :
( ( refl_on_a @ ( field_a @ R ) @ R )
=> ( ( antisym_on_a @ top_top_set_a @ R )
=> ( ( member_a @ A2 @ ( field_a @ R ) )
=> ( ( member_a @ B2 @ ( field_a @ R ) )
=> ( ( ( image_a_a @ R @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( image_a_a @ R @ ( insert_a @ B2 @ bot_bot_set_a ) ) )
= ( A2 = B2 ) ) ) ) ) ) ).
% Refl_antisym_eq_Image1_Image1_iff
thf(fact_1231_Refl__antisym__eq__Image1__Image1__iff,axiom,
! [R: set_Pr8600417178894128327od_a_a,A2: product_prod_a_a,B2: product_prod_a_a] :
( ( refl_o7745108929832855590od_a_a @ ( field_1126092520709947252od_a_a @ R ) @ R )
=> ( ( antisy9008168496540133130od_a_a @ top_to8063371432257647191od_a_a @ R )
=> ( ( member1426531477525435216od_a_a @ A2 @ ( field_1126092520709947252od_a_a @ R ) )
=> ( ( member1426531477525435216od_a_a @ B2 @ ( field_1126092520709947252od_a_a @ R ) )
=> ( ( ( image_9076584400576816019od_a_a @ R @ ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) )
= ( image_9076584400576816019od_a_a @ R @ ( insert4534936382041156343od_a_a @ B2 @ bot_bo3357376287454694259od_a_a ) ) )
= ( A2 = B2 ) ) ) ) ) ) ).
% Refl_antisym_eq_Image1_Image1_iff
thf(fact_1232_subset__Image1__Image1__iff,axiom,
! [R: set_Product_prod_a_a,A2: a,B2: a] :
( ( order_preorder_on_a @ ( field_a @ R ) @ R )
=> ( ( member_a @ A2 @ ( field_a @ R ) )
=> ( ( member_a @ B2 @ ( field_a @ R ) )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ R @ ( insert_a @ A2 @ bot_bot_set_a ) ) @ ( image_a_a @ R @ ( insert_a @ B2 @ bot_bot_set_a ) ) )
= ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B2 @ A2 ) @ R ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_1233_Image__mono,axiom,
! [R5: set_Product_prod_a_a,R: set_Product_prod_a_a,A9: set_a,A: set_a] :
( ( ord_le746702958409616551od_a_a @ R5 @ R )
=> ( ( ord_less_eq_set_a @ A9 @ A )
=> ( ord_less_eq_set_a @ ( image_a_a @ R5 @ A9 ) @ ( image_a_a @ R @ A ) ) ) ) ).
% Image_mono
thf(fact_1234_Image__Int__subset,axiom,
! [R2: set_Pr8600417178894128327od_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( image_9076584400576816019od_a_a @ R2 @ ( inf_in8905007599844390133od_a_a @ A @ B ) ) @ ( inf_in8905007599844390133od_a_a @ ( image_9076584400576816019od_a_a @ R2 @ A ) @ ( image_9076584400576816019od_a_a @ R2 @ B ) ) ) ).
% Image_Int_subset
thf(fact_1235_Image__Int__subset,axiom,
! [R2: set_Pr5530083903271594800od_a_a,A: set_a,B: set_a] : ( ord_le746702958409616551od_a_a @ ( image_2799180466780705916od_a_a @ R2 @ ( inf_inf_set_a @ A @ B ) ) @ ( inf_in8905007599844390133od_a_a @ ( image_2799180466780705916od_a_a @ R2 @ A ) @ ( image_2799180466780705916od_a_a @ R2 @ B ) ) ) ).
% Image_Int_subset
thf(fact_1236_Image__Int__subset,axiom,
! [R2: set_Pr8876520727511657886_a_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] : ( ord_less_eq_set_a @ ( image_8059871973944943978_a_a_a @ R2 @ ( inf_in8905007599844390133od_a_a @ A @ B ) ) @ ( inf_inf_set_a @ ( image_8059871973944943978_a_a_a @ R2 @ A ) @ ( image_8059871973944943978_a_a_a @ R2 @ B ) ) ) ).
% Image_Int_subset
thf(fact_1237_Image__Int__subset,axiom,
! [R2: set_Product_prod_a_a,A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( image_a_a @ R2 @ ( inf_inf_set_a @ A @ B ) ) @ ( inf_inf_set_a @ ( image_a_a @ R2 @ A ) @ ( image_a_a @ R2 @ B ) ) ) ).
% Image_Int_subset
thf(fact_1238_equiv__class__self,axiom,
! [A: set_a,R: set_Product_prod_a_a,A2: a] :
( ( equiv_equiv_a @ A @ R )
=> ( ( member_a @ A2 @ A )
=> ( member_a @ A2 @ ( image_a_a @ R @ ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ) ).
% equiv_class_self
thf(fact_1239_Image__subset,axiom,
! [R: set_Product_prod_a_a,A: set_a,B: set_a,C2: set_a] :
( ( ord_le746702958409616551od_a_a @ R
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : B ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ R @ C2 ) @ B ) ) ).
% Image_subset
thf(fact_1240_eq__equiv__class,axiom,
! [R: set_Product_prod_a_a,A2: a,B2: a,A: set_a] :
( ( ( image_a_a @ R @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( image_a_a @ R @ ( insert_a @ B2 @ bot_bot_set_a ) ) )
=> ( ( equiv_equiv_a @ A @ R )
=> ( ( member_a @ B2 @ A )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B2 ) @ R ) ) ) ) ).
% eq_equiv_class
thf(fact_1241_equiv__class__eq,axiom,
! [A: set_a,R: set_Product_prod_a_a,A2: a,B2: a] :
( ( equiv_equiv_a @ A @ R )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B2 ) @ R )
=> ( ( image_a_a @ R @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( image_a_a @ R @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ) ).
% equiv_class_eq
thf(fact_1242_eq__equiv__class__iff,axiom,
! [A: set_a,R: set_Product_prod_a_a,X: a,Y: a] :
( ( equiv_equiv_a @ A @ R )
=> ( ( member_a @ X @ A )
=> ( ( member_a @ Y @ A )
=> ( ( ( image_a_a @ R @ ( insert_a @ X @ bot_bot_set_a ) )
= ( image_a_a @ R @ ( insert_a @ Y @ bot_bot_set_a ) ) )
= ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R ) ) ) ) ) ).
% eq_equiv_class_iff
thf(fact_1243_equiv__class__eq__iff,axiom,
! [A: set_a,R: set_Product_prod_a_a,X: a,Y: a] :
( ( equiv_equiv_a @ A @ R )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
= ( ( ( image_a_a @ R @ ( insert_a @ X @ bot_bot_set_a ) )
= ( image_a_a @ R @ ( insert_a @ Y @ bot_bot_set_a ) ) )
& ( member_a @ X @ A )
& ( member_a @ Y @ A ) ) ) ) ).
% equiv_class_eq_iff
thf(fact_1244_refines__equiv__class__eq2,axiom,
! [R2: set_Product_prod_a_a,S2: set_Product_prod_a_a,A: set_a,A2: a] :
( ( ord_le746702958409616551od_a_a @ R2 @ S2 )
=> ( ( equiv_equiv_a @ A @ R2 )
=> ( ( equiv_equiv_a @ A @ S2 )
=> ( ( image_a_a @ S2 @ ( image_a_a @ R2 @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
= ( image_a_a @ S2 @ ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ) ) ).
% refines_equiv_class_eq2
thf(fact_1245_refines__equiv__class__eq,axiom,
! [R2: set_Product_prod_a_a,S2: set_Product_prod_a_a,A: set_a,A2: a] :
( ( ord_le746702958409616551od_a_a @ R2 @ S2 )
=> ( ( equiv_equiv_a @ A @ R2 )
=> ( ( equiv_equiv_a @ A @ S2 )
=> ( ( image_a_a @ R2 @ ( image_a_a @ S2 @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
= ( image_a_a @ S2 @ ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ) ) ).
% refines_equiv_class_eq
thf(fact_1246_subset__Image__Image__iff,axiom,
! [R: set_Product_prod_a_a,A: set_a,B: set_a] :
( ( order_preorder_on_a @ ( field_a @ R ) @ R )
=> ( ( ord_less_eq_set_a @ A @ ( field_a @ R ) )
=> ( ( ord_less_eq_set_a @ B @ ( field_a @ R ) )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ R @ A ) @ ( image_a_a @ R @ B ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ? [Y3: a] :
( ( member_a @ Y3 @ B )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ X3 ) @ R ) ) ) ) ) ) ) ) ).
% subset_Image_Image_iff
thf(fact_1247_equiv__class__subset,axiom,
! [A: set_a,R: set_Product_prod_a_a,A2: a,B2: a] :
( ( equiv_equiv_a @ A @ R )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B2 ) @ R )
=> ( ord_less_eq_set_a @ ( image_a_a @ R @ ( insert_a @ A2 @ bot_bot_set_a ) ) @ ( image_a_a @ R @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ) ).
% equiv_class_subset
thf(fact_1248_subset__equiv__class,axiom,
! [A: set_a,R: set_Product_prod_a_a,B2: a,A2: a] :
( ( equiv_equiv_a @ A @ R )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ R @ ( insert_a @ B2 @ bot_bot_set_a ) ) @ ( image_a_a @ R @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
=> ( ( member_a @ B2 @ A )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B2 ) @ R ) ) ) ) ).
% subset_equiv_class
thf(fact_1249_equiv__class__nondisjoint,axiom,
! [A: set_Product_prod_a_a,R: set_Pr8600417178894128327od_a_a,X: product_prod_a_a,A2: product_prod_a_a,B2: product_prod_a_a] :
( ( equiv_4910910634973128413od_a_a @ A @ R )
=> ( ( member1426531477525435216od_a_a @ X @ ( inf_in8905007599844390133od_a_a @ ( image_9076584400576816019od_a_a @ R @ ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) ) @ ( image_9076584400576816019od_a_a @ R @ ( insert4534936382041156343od_a_a @ B2 @ bot_bo3357376287454694259od_a_a ) ) ) )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A2 @ B2 ) @ R ) ) ) ).
% equiv_class_nondisjoint
thf(fact_1250_equiv__class__nondisjoint,axiom,
! [A: set_a,R: set_Product_prod_a_a,X: a,A2: a,B2: a] :
( ( equiv_equiv_a @ A @ R )
=> ( ( member_a @ X @ ( inf_inf_set_a @ ( image_a_a @ R @ ( insert_a @ A2 @ bot_bot_set_a ) ) @ ( image_a_a @ R @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B2 ) @ R ) ) ) ).
% equiv_class_nondisjoint
thf(fact_1251_Sigma__def,axiom,
( product_Sigma_a_a
= ( ^ [A3: set_a,B3: a > set_a] :
( comple8421679170691845492od_a_a
@ ( image_4421510592991446670od_a_a
@ ^ [X3: a] :
( comple8421679170691845492od_a_a
@ ( image_4421510592991446670od_a_a
@ ^ [Y3: a] : ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ bot_bo3357376287454694259od_a_a )
@ ( B3 @ X3 ) ) )
@ A3 ) ) ) ) ).
% Sigma_def
thf(fact_1252_Inf__fin_Oinsert,axiom,
! [A: set_se5735800977113168103od_a_a,X: set_Product_prod_a_a] :
( ( finite8717734299975451184od_a_a @ A )
=> ( ( A != bot_bo777872063958040403od_a_a )
=> ( ( lattic7400613087766618148od_a_a @ ( insert914553114930139863od_a_a @ X @ A ) )
= ( inf_in8905007599844390133od_a_a @ X @ ( lattic7400613087766618148od_a_a @ A ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_1253_Inf__fin_Oinsert,axiom,
! [A: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ( lattic8209813465164889211_set_a @ ( insert_set_a @ X @ A ) )
= ( inf_inf_set_a @ X @ ( lattic8209813465164889211_set_a @ A ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_1254_Inf__fin_Oin__idem,axiom,
! [A: set_se5735800977113168103od_a_a,X: set_Product_prod_a_a] :
( ( finite8717734299975451184od_a_a @ A )
=> ( ( member1816616512716248880od_a_a @ X @ A )
=> ( ( inf_in8905007599844390133od_a_a @ X @ ( lattic7400613087766618148od_a_a @ A ) )
= ( lattic7400613087766618148od_a_a @ A ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_1255_Inf__fin_Oin__idem,axiom,
! [A: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ X @ A )
=> ( ( inf_inf_set_a @ X @ ( lattic8209813465164889211_set_a @ A ) )
= ( lattic8209813465164889211_set_a @ A ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_1256_Inf__fin_Ohom__commute,axiom,
! [H: set_Product_prod_a_a > set_Product_prod_a_a,N: set_se5735800977113168103od_a_a] :
( ! [X2: set_Product_prod_a_a,Y4: set_Product_prod_a_a] :
( ( H @ ( inf_in8905007599844390133od_a_a @ X2 @ Y4 ) )
= ( inf_in8905007599844390133od_a_a @ ( H @ X2 ) @ ( H @ Y4 ) ) )
=> ( ( finite8717734299975451184od_a_a @ N )
=> ( ( N != bot_bo777872063958040403od_a_a )
=> ( ( H @ ( lattic7400613087766618148od_a_a @ N ) )
= ( lattic7400613087766618148od_a_a @ ( image_4506799131697958853od_a_a @ H @ N ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_1257_Inf__fin_Ohom__commute,axiom,
! [H: set_a > set_a,N: set_set_a] :
( ! [X2: set_a,Y4: set_a] :
( ( H @ ( inf_inf_set_a @ X2 @ Y4 ) )
= ( inf_inf_set_a @ ( H @ X2 ) @ ( H @ Y4 ) ) )
=> ( ( finite_finite_set_a @ N )
=> ( ( N != bot_bot_set_set_a )
=> ( ( H @ ( lattic8209813465164889211_set_a @ N ) )
= ( lattic8209813465164889211_set_a @ ( image_set_a_set_a @ H @ N ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_1258_all__finite__subset__image,axiom,
! [F2: a > a,A: set_a,P: set_a > $o] :
( ( ! [B3: set_a] :
( ( ( finite_finite_a @ B3 )
& ( ord_less_eq_set_a @ B3 @ ( image_a_a2 @ F2 @ A ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_a] :
( ( ( finite_finite_a @ B3 )
& ( ord_less_eq_set_a @ B3 @ A ) )
=> ( P @ ( image_a_a2 @ F2 @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1259_ex__finite__subset__image,axiom,
! [F2: a > a,A: set_a,P: set_a > $o] :
( ( ? [B3: set_a] :
( ( finite_finite_a @ B3 )
& ( ord_less_eq_set_a @ B3 @ ( image_a_a2 @ F2 @ A ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_a] :
( ( finite_finite_a @ B3 )
& ( ord_less_eq_set_a @ B3 @ A )
& ( P @ ( image_a_a2 @ F2 @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1260_finite__subset__image,axiom,
! [B: set_a,F2: a > a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ B @ ( image_a_a2 @ F2 @ A ) )
=> ? [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A )
& ( finite_finite_a @ C3 )
& ( B
= ( image_a_a2 @ F2 @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1261_Inf__fin_OcoboundedI,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ( ord_less_eq_set_a @ ( lattic8209813465164889211_set_a @ A ) @ A2 ) ) ) ).
% Inf_fin.coboundedI
thf(fact_1262_all__subset__image,axiom,
! [F2: a > a,A: set_a,P: set_a > $o] :
( ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ ( image_a_a2 @ F2 @ A ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A )
=> ( P @ ( image_a_a2 @ F2 @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_1263_Sup__inf,axiom,
! [B: set_se5735800977113168103od_a_a,A2: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ ( comple8421679170691845492od_a_a @ B ) @ A2 )
= ( comple8421679170691845492od_a_a
@ ( image_4506799131697958853od_a_a
@ ^ [B4: set_Product_prod_a_a] : ( inf_in8905007599844390133od_a_a @ B4 @ A2 )
@ B ) ) ) ).
% Sup_inf
thf(fact_1264_Sup__inf,axiom,
! [B: set_set_a,A2: set_a] :
( ( inf_inf_set_a @ ( comple2307003609928055243_set_a @ B ) @ A2 )
= ( comple2307003609928055243_set_a
@ ( image_set_a_set_a
@ ^ [B4: set_a] : ( inf_inf_set_a @ B4 @ A2 )
@ B ) ) ) ).
% Sup_inf
thf(fact_1265_UN__mono,axiom,
! [A: set_a,B: set_a,F2: a > set_a,G3: a > set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_less_eq_set_a @ ( F2 @ X2 ) @ ( G3 @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F2 @ A ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ G3 @ B ) ) ) ) ) ).
% UN_mono
thf(fact_1266_inf__Sup,axiom,
! [A2: set_Product_prod_a_a,B: set_se5735800977113168103od_a_a] :
( ( inf_in8905007599844390133od_a_a @ A2 @ ( comple8421679170691845492od_a_a @ B ) )
= ( comple8421679170691845492od_a_a @ ( image_4506799131697958853od_a_a @ ( inf_in8905007599844390133od_a_a @ A2 ) @ B ) ) ) ).
% inf_Sup
thf(fact_1267_inf__Sup,axiom,
! [A2: set_a,B: set_set_a] :
( ( inf_inf_set_a @ A2 @ ( comple2307003609928055243_set_a @ B ) )
= ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ ( inf_inf_set_a @ A2 ) @ B ) ) ) ).
% inf_Sup
thf(fact_1268_image__Int__subset,axiom,
! [F2: product_prod_a_a > product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( image_4636654165204879301od_a_a @ F2 @ ( inf_in8905007599844390133od_a_a @ A @ B ) ) @ ( inf_in8905007599844390133od_a_a @ ( image_4636654165204879301od_a_a @ F2 @ A ) @ ( image_4636654165204879301od_a_a @ F2 @ B ) ) ) ).
% image_Int_subset
thf(fact_1269_image__Int__subset,axiom,
! [F2: a > product_prod_a_a,A: set_a,B: set_a] : ( ord_le746702958409616551od_a_a @ ( image_7400625782589995694od_a_a @ F2 @ ( inf_inf_set_a @ A @ B ) ) @ ( inf_in8905007599844390133od_a_a @ ( image_7400625782589995694od_a_a @ F2 @ A ) @ ( image_7400625782589995694od_a_a @ F2 @ B ) ) ) ).
% image_Int_subset
thf(fact_1270_image__Int__subset,axiom,
! [F2: product_prod_a_a > a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] : ( ord_less_eq_set_a @ ( image_3437945252899457948_a_a_a @ F2 @ ( inf_in8905007599844390133od_a_a @ A @ B ) ) @ ( inf_inf_set_a @ ( image_3437945252899457948_a_a_a @ F2 @ A ) @ ( image_3437945252899457948_a_a_a @ F2 @ B ) ) ) ).
% image_Int_subset
thf(fact_1271_image__Int__subset,axiom,
! [F2: a > a,A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( image_a_a2 @ F2 @ ( inf_inf_set_a @ A @ B ) ) @ ( inf_inf_set_a @ ( image_a_a2 @ F2 @ A ) @ ( image_a_a2 @ F2 @ B ) ) ) ).
% image_Int_subset
thf(fact_1272_image__mono,axiom,
! [A: set_a,B: set_a,F2: a > a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ ( image_a_a2 @ F2 @ A ) @ ( image_a_a2 @ F2 @ B ) ) ) ).
% image_mono
thf(fact_1273_subset__imageE,axiom,
! [B: set_a,F2: a > a,A: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a2 @ F2 @ A ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A )
=> ( B
!= ( image_a_a2 @ F2 @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_1274_subset__image__iff,axiom,
! [B: set_a,F2: a > a,A: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a2 @ F2 @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B
= ( image_a_a2 @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
% Conjectures (1)
thf(conj_0,conjecture,
( sym_on_a @ top_top_set_a
@ ( inf_in8905007599844390133od_a_a @ p
@ ( product_Sigma_a_a @ b
@ ^ [Uu: a] : b ) ) ) ).
%------------------------------------------------------------------------------