TPTP Problem File: SLH0456^1.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 : Pluennecke_Ruzsa_Inequality/0003_Pluennecke_Ruzsa_Inequality/prob_00109_003560__12038616_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1413 ( 683 unt; 130 typ; 0 def)
% Number of atoms : 3474 (1339 equ; 0 cnn)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 11289 ( 321 ~; 34 |; 259 &;9347 @)
% ( 0 <=>;1328 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 6 avg)
% Number of types : 10 ( 9 usr)
% Number of type conns : 899 ( 899 >; 0 *; 0 +; 0 <<)
% Number of symbols : 124 ( 121 usr; 19 con; 0-5 aty)
% Number of variables : 3556 ( 377 ^;3087 !; 92 ?;3556 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:19:41.412
%------------------------------------------------------------------------------
% Could-be-implicit typings (9)
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% Explicit typings (121)
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chain_subset_a: set_set_a > $o ).
thf(sy_c_Zorn_Ochains_001tf__a,type,
chains_a: set_set_a > set_set_set_a ).
thf(sy_c_member_001_062_Itf__a_M_Eo_J,type,
member_a_o: ( a > $o ) > set_a_o > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
member_set_set_set_a: set_set_set_a > set_set_set_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
member_set_set_a: set_set_a > set_set_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_A,type,
a2: set_a ).
thf(sy_v_B,type,
b: set_a ).
thf(sy_v_G,type,
g: set_a ).
thf(sy_v_addition,type,
addition: a > a > a ).
thf(sy_v_zero,type,
zero: a ).
% Relevant facts (1277)
thf(fact_0_assms_I2_J,axiom,
finite_finite_a @ ( inf_inf_set_a @ b @ g ) ).
% assms(2)
thf(fact_1_assms_I1_J,axiom,
finite_finite_a @ ( inf_inf_set_a @ a2 @ g ) ).
% assms(1)
thf(fact_2_commutative,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( ( addition @ X @ Y )
= ( addition @ Y @ X ) ) ) ) ).
% commutative
thf(fact_3_sumset_Ocases,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( member_a @ A @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) )
=> ~ ! [A3: a,B2: a] :
( ( A
= ( addition @ A3 @ B2 ) )
=> ( ( member_a @ A3 @ A2 )
=> ( ( member_a @ A3 @ g )
=> ( ( member_a @ B2 @ B )
=> ~ ( member_a @ B2 @ g ) ) ) ) ) ) ).
% sumset.cases
thf(fact_4_sumset_Osimps,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( member_a @ A @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) )
= ( ? [A4: a,B3: a] :
( ( A
= ( addition @ A4 @ B3 ) )
& ( member_a @ A4 @ A2 )
& ( member_a @ A4 @ g )
& ( member_a @ B3 @ B )
& ( member_a @ B3 @ g ) ) ) ) ).
% sumset.simps
thf(fact_5_sumset_OsumsetI,axiom,
! [A: a,A2: set_a,B4: a,B: set_a] :
( ( member_a @ A @ A2 )
=> ( ( member_a @ A @ g )
=> ( ( member_a @ B4 @ B )
=> ( ( member_a @ B4 @ g )
=> ( member_a @ ( addition @ A @ B4 ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ) ) ) ) ).
% sumset.sumsetI
thf(fact_6_sumset__assoc,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) @ C )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( pluenn3038260743871226533mset_a @ g @ addition @ B @ C ) ) ) ).
% sumset_assoc
thf(fact_7_sumset__commute,axiom,
! [A2: set_a,B: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B )
= ( pluenn3038260743871226533mset_a @ g @ addition @ B @ A2 ) ) ).
% sumset_commute
thf(fact_8_finite__sumset,axiom,
! [A2: set_a,B: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ) ) ).
% finite_sumset
thf(fact_9_associative,axiom,
! [A: a,B4: a,C2: a] :
( ( member_a @ A @ g )
=> ( ( member_a @ B4 @ g )
=> ( ( member_a @ C2 @ g )
=> ( ( addition @ ( addition @ A @ B4 ) @ C2 )
= ( addition @ A @ ( addition @ B4 @ C2 ) ) ) ) ) ) ).
% associative
thf(fact_10_composition__closed,axiom,
! [A: a,B4: a] :
( ( member_a @ A @ g )
=> ( ( member_a @ B4 @ g )
=> ( member_a @ ( addition @ A @ B4 ) @ g ) ) ) ).
% composition_closed
thf(fact_11_additive__abelian__group_Osumset_Ocong,axiom,
pluenn3038260743871226533mset_a = pluenn3038260743871226533mset_a ).
% additive_abelian_group.sumset.cong
thf(fact_12_sumset__Int__carrier__eq_I2_J,axiom,
! [A2: set_a,B: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( inf_inf_set_a @ A2 @ g ) @ B )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ).
% sumset_Int_carrier_eq(2)
thf(fact_13_sumset__Int__carrier__eq_I1_J,axiom,
! [A2: set_a,B: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( inf_inf_set_a @ B @ g ) )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ).
% sumset_Int_carrier_eq(1)
thf(fact_14_sumset__Int__carrier,axiom,
! [A2: set_a,B: set_a] :
( ( inf_inf_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) @ g )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ).
% sumset_Int_carrier
thf(fact_15_sumsetp_Ocases,axiom,
! [A2: a > $o,B: a > $o,A: a] :
( ( pluenn895083305082786853setp_a @ g @ addition @ A2 @ B @ A )
=> ~ ! [A3: a,B2: a] :
( ( A
= ( addition @ A3 @ B2 ) )
=> ( ( A2 @ A3 )
=> ( ( member_a @ A3 @ g )
=> ( ( B @ B2 )
=> ~ ( member_a @ B2 @ g ) ) ) ) ) ) ).
% sumsetp.cases
thf(fact_16_sumsetp_Osimps,axiom,
! [A2: a > $o,B: a > $o,A: a] :
( ( pluenn895083305082786853setp_a @ g @ addition @ A2 @ B @ A )
= ( ? [A4: a,B3: a] :
( ( A
= ( addition @ A4 @ B3 ) )
& ( A2 @ A4 )
& ( member_a @ A4 @ g )
& ( B @ B3 )
& ( member_a @ B3 @ g ) ) ) ) ).
% sumsetp.simps
thf(fact_17_sumsetp_OsumsetI,axiom,
! [A2: a > $o,A: a,B: a > $o,B4: a] :
( ( A2 @ A )
=> ( ( member_a @ A @ g )
=> ( ( B @ B4 )
=> ( ( member_a @ B4 @ g )
=> ( pluenn895083305082786853setp_a @ g @ addition @ A2 @ B @ ( addition @ A @ B4 ) ) ) ) ) ) ).
% sumsetp.sumsetI
thf(fact_18_sumset__mono,axiom,
! [A5: set_a,A2: set_a,B5: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A5 @ A2 )
=> ( ( ord_less_eq_set_a @ B5 @ B )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A5 @ B5 ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ) ) ).
% sumset_mono
thf(fact_19_sumset__subset__carrier,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) @ g ) ).
% sumset_subset_carrier
thf(fact_20_sumset__subset__Un1,axiom,
! [A2: set_a,A5: set_a,B: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( sup_sup_set_a @ A2 @ A5 ) @ B )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A5 @ B ) ) ) ).
% sumset_subset_Un1
thf(fact_21_sumset__subset__Un2,axiom,
! [A2: set_a,B: set_a,B5: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( sup_sup_set_a @ B @ B5 ) )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B5 ) ) ) ).
% sumset_subset_Un2
thf(fact_22_sumset__subset__Un_I1_J,axiom,
! [A2: set_a,B: set_a,C: set_a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( sup_sup_set_a @ B @ C ) ) ) ).
% sumset_subset_Un(1)
thf(fact_23_sumset__subset__Un_I2_J,axiom,
! [A2: set_a,B: set_a,C: set_a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ ( sup_sup_set_a @ A2 @ C ) @ B ) ) ).
% sumset_subset_Un(2)
thf(fact_24_finite__Int,axiom,
! [F: set_Sum_sum_a_a,G: set_Sum_sum_a_a] :
( ( ( finite51705147264084924um_a_a @ F )
| ( finite51705147264084924um_a_a @ G ) )
=> ( finite51705147264084924um_a_a @ ( inf_in4219250158259668461um_a_a @ F @ G ) ) ) ).
% finite_Int
thf(fact_25_finite__Int,axiom,
! [F: set_Product_prod_a_a,G: set_Product_prod_a_a] :
( ( ( finite6544458595007987280od_a_a @ F )
| ( finite6544458595007987280od_a_a @ G ) )
=> ( finite6544458595007987280od_a_a @ ( inf_in8905007599844390133od_a_a @ F @ G ) ) ) ).
% finite_Int
thf(fact_26_finite__Int,axiom,
! [F: set_set_a,G: set_set_a] :
( ( ( finite_finite_set_a @ F )
| ( finite_finite_set_a @ G ) )
=> ( finite_finite_set_a @ ( inf_inf_set_set_a @ F @ G ) ) ) ).
% finite_Int
thf(fact_27_finite__Int,axiom,
! [F: set_a,G: set_a] :
( ( ( finite_finite_a @ F )
| ( finite_finite_a @ G ) )
=> ( finite_finite_a @ ( inf_inf_set_a @ F @ G ) ) ) ).
% finite_Int
thf(fact_28_finite__Un,axiom,
! [F: set_Sum_sum_a_a,G: set_Sum_sum_a_a] :
( ( finite51705147264084924um_a_a @ ( sup_su5637237897826962567um_a_a @ F @ G ) )
= ( ( finite51705147264084924um_a_a @ F )
& ( finite51705147264084924um_a_a @ G ) ) ) ).
% finite_Un
thf(fact_29_finite__Un,axiom,
! [F: set_Product_prod_a_a,G: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ ( sup_su3048258781599657691od_a_a @ F @ G ) )
= ( ( finite6544458595007987280od_a_a @ F )
& ( finite6544458595007987280od_a_a @ G ) ) ) ).
% finite_Un
thf(fact_30_finite__Un,axiom,
! [F: set_set_a,G: set_set_a] :
( ( finite_finite_set_a @ ( sup_sup_set_set_a @ F @ G ) )
= ( ( finite_finite_set_a @ F )
& ( finite_finite_set_a @ G ) ) ) ).
% finite_Un
thf(fact_31_finite__Un,axiom,
! [F: set_a,G: set_a] :
( ( finite_finite_a @ ( sup_sup_set_a @ F @ G ) )
= ( ( finite_finite_a @ F )
& ( finite_finite_a @ G ) ) ) ).
% finite_Un
thf(fact_32_additive__abelian__group_Osumsetp_Ocong,axiom,
pluenn895083305082786853setp_a = pluenn895083305082786853setp_a ).
% additive_abelian_group.sumsetp.cong
thf(fact_33_infinite__Un,axiom,
! [S: set_Sum_sum_a_a,T: set_Sum_sum_a_a] :
( ( ~ ( finite51705147264084924um_a_a @ ( sup_su5637237897826962567um_a_a @ S @ T ) ) )
= ( ~ ( finite51705147264084924um_a_a @ S )
| ~ ( finite51705147264084924um_a_a @ T ) ) ) ).
% infinite_Un
thf(fact_34_infinite__Un,axiom,
! [S: set_Product_prod_a_a,T: set_Product_prod_a_a] :
( ( ~ ( finite6544458595007987280od_a_a @ ( sup_su3048258781599657691od_a_a @ S @ T ) ) )
= ( ~ ( finite6544458595007987280od_a_a @ S )
| ~ ( finite6544458595007987280od_a_a @ T ) ) ) ).
% infinite_Un
thf(fact_35_infinite__Un,axiom,
! [S: set_set_a,T: set_set_a] :
( ( ~ ( finite_finite_set_a @ ( sup_sup_set_set_a @ S @ T ) ) )
= ( ~ ( finite_finite_set_a @ S )
| ~ ( finite_finite_set_a @ T ) ) ) ).
% infinite_Un
thf(fact_36_infinite__Un,axiom,
! [S: set_a,T: set_a] :
( ( ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T ) ) )
= ( ~ ( finite_finite_a @ S )
| ~ ( finite_finite_a @ T ) ) ) ).
% infinite_Un
thf(fact_37_Un__infinite,axiom,
! [S: set_Sum_sum_a_a,T: set_Sum_sum_a_a] :
( ~ ( finite51705147264084924um_a_a @ S )
=> ~ ( finite51705147264084924um_a_a @ ( sup_su5637237897826962567um_a_a @ S @ T ) ) ) ).
% Un_infinite
thf(fact_38_Un__infinite,axiom,
! [S: set_Product_prod_a_a,T: set_Product_prod_a_a] :
( ~ ( finite6544458595007987280od_a_a @ S )
=> ~ ( finite6544458595007987280od_a_a @ ( sup_su3048258781599657691od_a_a @ S @ T ) ) ) ).
% Un_infinite
thf(fact_39_Un__infinite,axiom,
! [S: set_set_a,T: set_set_a] :
( ~ ( finite_finite_set_a @ S )
=> ~ ( finite_finite_set_a @ ( sup_sup_set_set_a @ S @ T ) ) ) ).
% Un_infinite
thf(fact_40_Un__infinite,axiom,
! [S: set_a,T: set_a] :
( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T ) ) ) ).
% Un_infinite
thf(fact_41_finite__UnI,axiom,
! [F: set_Sum_sum_a_a,G: set_Sum_sum_a_a] :
( ( finite51705147264084924um_a_a @ F )
=> ( ( finite51705147264084924um_a_a @ G )
=> ( finite51705147264084924um_a_a @ ( sup_su5637237897826962567um_a_a @ F @ G ) ) ) ) ).
% finite_UnI
thf(fact_42_finite__UnI,axiom,
! [F: set_Product_prod_a_a,G: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ F )
=> ( ( finite6544458595007987280od_a_a @ G )
=> ( finite6544458595007987280od_a_a @ ( sup_su3048258781599657691od_a_a @ F @ G ) ) ) ) ).
% finite_UnI
thf(fact_43_finite__UnI,axiom,
! [F: set_set_a,G: set_set_a] :
( ( finite_finite_set_a @ F )
=> ( ( finite_finite_set_a @ G )
=> ( finite_finite_set_a @ ( sup_sup_set_set_a @ F @ G ) ) ) ) ).
% finite_UnI
thf(fact_44_finite__UnI,axiom,
! [F: set_a,G: set_a] :
( ( finite_finite_a @ F )
=> ( ( finite_finite_a @ G )
=> ( finite_finite_a @ ( sup_sup_set_a @ F @ G ) ) ) ) ).
% finite_UnI
thf(fact_45_finite__has__minimal2,axiom,
! [A2: set_set_set_a,A: set_set_a] :
( ( finite7209287970140883943_set_a @ A2 )
=> ( ( member_set_set_a @ A @ A2 )
=> ? [X2: set_set_a] :
( ( member_set_set_a @ X2 @ A2 )
& ( ord_le3724670747650509150_set_a @ X2 @ A )
& ! [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_46_finite__has__minimal2,axiom,
! [A2: set_set_set_set_a,A: set_set_set_a] :
( ( finite5318320746233006407_set_a @ A2 )
=> ( ( member_set_set_set_a @ A @ A2 )
=> ? [X2: set_set_set_a] :
( ( member_set_set_set_a @ X2 @ A2 )
& ( ord_le5722252365846178494_set_a @ X2 @ A )
& ! [Xa: set_set_set_a] :
( ( member_set_set_set_a @ Xa @ A2 )
=> ( ( ord_le5722252365846178494_set_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_47_finite__has__minimal2,axiom,
! [A2: set_a_o,A: a > $o] :
( ( finite_finite_a_o @ A2 )
=> ( ( member_a_o @ A @ A2 )
=> ? [X2: a > $o] :
( ( member_a_o @ X2 @ A2 )
& ( ord_less_eq_a_o @ X2 @ A )
& ! [Xa: a > $o] :
( ( member_a_o @ Xa @ A2 )
=> ( ( ord_less_eq_a_o @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_48_finite__has__minimal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ( ord_less_eq_set_a @ X2 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_49_finite__has__maximal2,axiom,
! [A2: set_set_set_a,A: set_set_a] :
( ( finite7209287970140883943_set_a @ A2 )
=> ( ( member_set_set_a @ A @ A2 )
=> ? [X2: set_set_a] :
( ( member_set_set_a @ X2 @ A2 )
& ( ord_le3724670747650509150_set_a @ A @ X2 )
& ! [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_50_finite__has__maximal2,axiom,
! [A2: set_set_set_set_a,A: set_set_set_a] :
( ( finite5318320746233006407_set_a @ A2 )
=> ( ( member_set_set_set_a @ A @ A2 )
=> ? [X2: set_set_set_a] :
( ( member_set_set_set_a @ X2 @ A2 )
& ( ord_le5722252365846178494_set_a @ A @ X2 )
& ! [Xa: set_set_set_a] :
( ( member_set_set_set_a @ Xa @ A2 )
=> ( ( ord_le5722252365846178494_set_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_51_finite__has__maximal2,axiom,
! [A2: set_a_o,A: a > $o] :
( ( finite_finite_a_o @ A2 )
=> ( ( member_a_o @ A @ A2 )
=> ? [X2: a > $o] :
( ( member_a_o @ X2 @ A2 )
& ( ord_less_eq_a_o @ A @ X2 )
& ! [Xa: a > $o] :
( ( member_a_o @ Xa @ A2 )
=> ( ( ord_less_eq_a_o @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_52_finite__has__maximal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ( ord_less_eq_set_a @ A @ X2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_53_rev__finite__subset,axiom,
! [B: set_Sum_sum_a_a,A2: set_Sum_sum_a_a] :
( ( finite51705147264084924um_a_a @ B )
=> ( ( ord_le8948759482780970939um_a_a @ A2 @ B )
=> ( finite51705147264084924um_a_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_54_rev__finite__subset,axiom,
! [B: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ B )
=> ( ( ord_le746702958409616551od_a_a @ A2 @ B )
=> ( finite6544458595007987280od_a_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_55_rev__finite__subset,axiom,
! [B: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( finite_finite_set_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_56_rev__finite__subset,axiom,
! [B: set_set_set_a,A2: set_set_set_a] :
( ( finite7209287970140883943_set_a @ B )
=> ( ( ord_le5722252365846178494_set_a @ A2 @ B )
=> ( finite7209287970140883943_set_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_57_rev__finite__subset,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( finite_finite_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_58_infinite__super,axiom,
! [S: set_Sum_sum_a_a,T: set_Sum_sum_a_a] :
( ( ord_le8948759482780970939um_a_a @ S @ T )
=> ( ~ ( finite51705147264084924um_a_a @ S )
=> ~ ( finite51705147264084924um_a_a @ T ) ) ) ).
% infinite_super
thf(fact_59_infinite__super,axiom,
! [S: set_Product_prod_a_a,T: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ S @ T )
=> ( ~ ( finite6544458595007987280od_a_a @ S )
=> ~ ( finite6544458595007987280od_a_a @ T ) ) ) ).
% infinite_super
thf(fact_60_infinite__super,axiom,
! [S: set_set_a,T: set_set_a] :
( ( ord_le3724670747650509150_set_a @ S @ T )
=> ( ~ ( finite_finite_set_a @ S )
=> ~ ( finite_finite_set_a @ T ) ) ) ).
% infinite_super
thf(fact_61_infinite__super,axiom,
! [S: set_set_set_a,T: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ S @ T )
=> ( ~ ( finite7209287970140883943_set_a @ S )
=> ~ ( finite7209287970140883943_set_a @ T ) ) ) ).
% infinite_super
thf(fact_62_infinite__super,axiom,
! [S: set_a,T: set_a] :
( ( ord_less_eq_set_a @ S @ T )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ T ) ) ) ).
% infinite_super
thf(fact_63_finite__subset,axiom,
! [A2: set_Sum_sum_a_a,B: set_Sum_sum_a_a] :
( ( ord_le8948759482780970939um_a_a @ A2 @ B )
=> ( ( finite51705147264084924um_a_a @ B )
=> ( finite51705147264084924um_a_a @ A2 ) ) ) ).
% finite_subset
thf(fact_64_finite__subset,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B )
=> ( ( finite6544458595007987280od_a_a @ B )
=> ( finite6544458595007987280od_a_a @ A2 ) ) ) ).
% finite_subset
thf(fact_65_finite__subset,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( finite_finite_set_a @ B )
=> ( finite_finite_set_a @ A2 ) ) ) ).
% finite_subset
thf(fact_66_finite__subset,axiom,
! [A2: set_set_set_a,B: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A2 @ B )
=> ( ( finite7209287970140883943_set_a @ B )
=> ( finite7209287970140883943_set_a @ A2 ) ) ) ).
% finite_subset
thf(fact_67_finite__subset,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ A2 ) ) ) ).
% finite_subset
thf(fact_68_Int__Un__eq_I4_J,axiom,
! [T: set_set_a,S: set_set_a] :
( ( sup_sup_set_set_a @ T @ ( inf_inf_set_set_a @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_69_Int__Un__eq_I4_J,axiom,
! [T: set_a,S: set_a] :
( ( sup_sup_set_a @ T @ ( inf_inf_set_a @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_70_Int__Un__eq_I3_J,axiom,
! [S: set_set_a,T: set_set_a] :
( ( sup_sup_set_set_a @ S @ ( inf_inf_set_set_a @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_71_Int__Un__eq_I3_J,axiom,
! [S: set_a,T: set_a] :
( ( sup_sup_set_a @ S @ ( inf_inf_set_a @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_72_Int__Un__eq_I2_J,axiom,
! [S: set_set_a,T: set_set_a] :
( ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_73_Int__Un__eq_I2_J,axiom,
! [S: set_a,T: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_74_Int__Un__eq_I1_J,axiom,
! [S: set_set_a,T: set_set_a] :
( ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_75_Int__Un__eq_I1_J,axiom,
! [S: set_a,T: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_76_Un__Int__eq_I4_J,axiom,
! [T: set_set_a,S: set_set_a] :
( ( inf_inf_set_set_a @ T @ ( sup_sup_set_set_a @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_77_Un__Int__eq_I4_J,axiom,
! [T: set_a,S: set_a] :
( ( inf_inf_set_a @ T @ ( sup_sup_set_a @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_78_mem__Collect__eq,axiom,
! [A: set_set_a,P: set_set_a > $o] :
( ( member_set_set_a @ A @ ( collect_set_set_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_79_mem__Collect__eq,axiom,
! [A: set_a,P: set_a > $o] :
( ( member_set_a @ A @ ( collect_set_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_80_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_81_Collect__mem__eq,axiom,
! [A2: set_set_set_a] :
( ( collect_set_set_a
@ ^ [X3: set_set_a] : ( member_set_set_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_82_Collect__mem__eq,axiom,
! [A2: set_set_a] :
( ( collect_set_a
@ ^ [X3: set_a] : ( member_set_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_83_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_84_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_85_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_86_Un__Int__eq_I3_J,axiom,
! [S: set_set_a,T: set_set_a] :
( ( inf_inf_set_set_a @ S @ ( sup_sup_set_set_a @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_87_Un__Int__eq_I3_J,axiom,
! [S: set_a,T: set_a] :
( ( inf_inf_set_a @ S @ ( sup_sup_set_a @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_88_Un__Int__eq_I2_J,axiom,
! [S: set_set_a,T: set_set_a] :
( ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_89_Un__Int__eq_I2_J,axiom,
! [S: set_a,T: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_90_Un__Int__eq_I1_J,axiom,
! [S: set_set_a,T: set_set_a] :
( ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_91_Un__Int__eq_I1_J,axiom,
! [S: set_a,T: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_92_sumset__empty_H_I1_J,axiom,
! [A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A2 @ g )
= bot_bot_set_a )
=> ( ( pluenn3038260743871226533mset_a @ g @ addition @ B @ A2 )
= bot_bot_set_a ) ) ).
% sumset_empty'(1)
thf(fact_93_sumset__empty_H_I2_J,axiom,
! [A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A2 @ g )
= bot_bot_set_a )
=> ( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B )
= bot_bot_set_a ) ) ).
% sumset_empty'(2)
thf(fact_94_Un__subset__iff,axiom,
! [A2: set_set_a,B: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A2 @ B ) @ C )
= ( ( ord_le3724670747650509150_set_a @ A2 @ C )
& ( ord_le3724670747650509150_set_a @ B @ C ) ) ) ).
% Un_subset_iff
thf(fact_95_Un__subset__iff,axiom,
! [A2: set_set_set_a,B: set_set_set_a,C: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ ( sup_su2076012971530813682_set_a @ A2 @ B ) @ C )
= ( ( ord_le5722252365846178494_set_a @ A2 @ C )
& ( ord_le5722252365846178494_set_a @ B @ C ) ) ) ).
% Un_subset_iff
thf(fact_96_Un__subset__iff,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B ) @ C )
= ( ( ord_less_eq_set_a @ A2 @ C )
& ( ord_less_eq_set_a @ B @ C ) ) ) ).
% Un_subset_iff
thf(fact_97_empty__Collect__eq,axiom,
! [P: set_a > $o] :
( ( bot_bot_set_set_a
= ( collect_set_a @ P ) )
= ( ! [X3: set_a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_98_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_99_Collect__empty__eq,axiom,
! [P: set_a > $o] :
( ( ( collect_set_a @ P )
= bot_bot_set_set_a )
= ( ! [X3: set_a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_100_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_101_all__not__in__conv,axiom,
! [A2: set_set_set_a] :
( ( ! [X3: set_set_a] :
~ ( member_set_set_a @ X3 @ A2 ) )
= ( A2 = bot_bo3380559777022489994_set_a ) ) ).
% all_not_in_conv
thf(fact_102_all__not__in__conv,axiom,
! [A2: set_set_a] :
( ( ! [X3: set_a] :
~ ( member_set_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_103_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X3: a] :
~ ( member_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_104_empty__iff,axiom,
! [C2: set_set_a] :
~ ( member_set_set_a @ C2 @ bot_bo3380559777022489994_set_a ) ).
% empty_iff
thf(fact_105_empty__iff,axiom,
! [C2: set_a] :
~ ( member_set_a @ C2 @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_106_empty__iff,axiom,
! [C2: a] :
~ ( member_a @ C2 @ bot_bot_set_a ) ).
% empty_iff
thf(fact_107_subset__antisym,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_108_subset__antisym,axiom,
! [A2: set_set_set_a,B: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A2 @ B )
=> ( ( ord_le5722252365846178494_set_a @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_109_subset__antisym,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_110_subsetI,axiom,
! [A2: set_set_a,B: set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( member_set_a @ X2 @ B ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ B ) ) ).
% subsetI
thf(fact_111_subsetI,axiom,
! [A2: set_set_set_a,B: set_set_set_a] :
( ! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ A2 )
=> ( member_set_set_a @ X2 @ B ) )
=> ( ord_le5722252365846178494_set_a @ A2 @ B ) ) ).
% subsetI
thf(fact_112_subsetI,axiom,
! [A2: set_a,B: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_a @ X2 @ B ) )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% subsetI
thf(fact_113_Int__iff,axiom,
! [C2: set_set_a,A2: set_set_set_a,B: set_set_set_a] :
( ( member_set_set_a @ C2 @ ( inf_in1205276777018777868_set_a @ A2 @ B ) )
= ( ( member_set_set_a @ C2 @ A2 )
& ( member_set_set_a @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_114_Int__iff,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A2 @ B ) )
= ( ( member_set_a @ C2 @ A2 )
& ( member_set_a @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_115_Int__iff,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) )
= ( ( member_a @ C2 @ A2 )
& ( member_a @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_116_IntI,axiom,
! [C2: set_set_a,A2: set_set_set_a,B: set_set_set_a] :
( ( member_set_set_a @ C2 @ A2 )
=> ( ( member_set_set_a @ C2 @ B )
=> ( member_set_set_a @ C2 @ ( inf_in1205276777018777868_set_a @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_117_IntI,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ A2 )
=> ( ( member_set_a @ C2 @ B )
=> ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_118_IntI,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ A2 )
=> ( ( member_a @ C2 @ B )
=> ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_119_Un__iff,axiom,
! [C2: set_set_a,A2: set_set_set_a,B: set_set_set_a] :
( ( member_set_set_a @ C2 @ ( sup_su2076012971530813682_set_a @ A2 @ B ) )
= ( ( member_set_set_a @ C2 @ A2 )
| ( member_set_set_a @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_120_Un__iff,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( sup_sup_set_set_a @ A2 @ B ) )
= ( ( member_set_a @ C2 @ A2 )
| ( member_set_a @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_121_Un__iff,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( sup_sup_set_a @ A2 @ B ) )
= ( ( member_a @ C2 @ A2 )
| ( member_a @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_122_UnCI,axiom,
! [C2: set_set_a,B: set_set_set_a,A2: set_set_set_a] :
( ( ~ ( member_set_set_a @ C2 @ B )
=> ( member_set_set_a @ C2 @ A2 ) )
=> ( member_set_set_a @ C2 @ ( sup_su2076012971530813682_set_a @ A2 @ B ) ) ) ).
% UnCI
thf(fact_123_UnCI,axiom,
! [C2: set_a,B: set_set_a,A2: set_set_a] :
( ( ~ ( member_set_a @ C2 @ B )
=> ( member_set_a @ C2 @ A2 ) )
=> ( member_set_a @ C2 @ ( sup_sup_set_set_a @ A2 @ B ) ) ) ).
% UnCI
thf(fact_124_UnCI,axiom,
! [C2: a,B: set_a,A2: set_a] :
( ( ~ ( member_a @ C2 @ B )
=> ( member_a @ C2 @ A2 ) )
=> ( member_a @ C2 @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% UnCI
thf(fact_125_empty__subsetI,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A2 ) ).
% empty_subsetI
thf(fact_126_empty__subsetI,axiom,
! [A2: set_set_set_a] : ( ord_le5722252365846178494_set_a @ bot_bo3380559777022489994_set_a @ A2 ) ).
% empty_subsetI
thf(fact_127_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_128_subset__empty,axiom,
! [A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ bot_bot_set_set_a )
= ( A2 = bot_bot_set_set_a ) ) ).
% subset_empty
thf(fact_129_subset__empty,axiom,
! [A2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A2 @ bot_bo3380559777022489994_set_a )
= ( A2 = bot_bo3380559777022489994_set_a ) ) ).
% subset_empty
thf(fact_130_subset__empty,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_131_Int__subset__iff,axiom,
! [C: set_set_a,A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C @ ( inf_inf_set_set_a @ A2 @ B ) )
= ( ( ord_le3724670747650509150_set_a @ C @ A2 )
& ( ord_le3724670747650509150_set_a @ C @ B ) ) ) ).
% Int_subset_iff
thf(fact_132_Int__subset__iff,axiom,
! [C: set_set_set_a,A2: set_set_set_a,B: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ C @ ( inf_in1205276777018777868_set_a @ A2 @ B ) )
= ( ( ord_le5722252365846178494_set_a @ C @ A2 )
& ( ord_le5722252365846178494_set_a @ C @ B ) ) ) ).
% Int_subset_iff
thf(fact_133_Int__subset__iff,axiom,
! [C: set_a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ ( inf_inf_set_a @ A2 @ B ) )
= ( ( ord_less_eq_set_a @ C @ A2 )
& ( ord_less_eq_set_a @ C @ B ) ) ) ).
% Int_subset_iff
thf(fact_134_Un__empty,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ( sup_sup_set_set_a @ A2 @ B )
= bot_bot_set_set_a )
= ( ( A2 = bot_bot_set_set_a )
& ( B = bot_bot_set_set_a ) ) ) ).
% Un_empty
thf(fact_135_Un__empty,axiom,
! [A2: set_a,B: set_a] :
( ( ( sup_sup_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ( A2 = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% Un_empty
thf(fact_136_sumset__empty_I2_J,axiom,
! [A2: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ bot_bot_set_a @ A2 )
= bot_bot_set_a ) ).
% sumset_empty(2)
thf(fact_137_sumset__empty_I1_J,axiom,
! [A2: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% sumset_empty(1)
thf(fact_138_sumset__is__empty__iff,axiom,
! [A2: set_a,B: set_a] :
( ( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B )
= bot_bot_set_a )
= ( ( ( inf_inf_set_a @ A2 @ g )
= bot_bot_set_a )
| ( ( inf_inf_set_a @ B @ g )
= bot_bot_set_a ) ) ) ).
% sumset_is_empty_iff
thf(fact_139_ex__in__conv,axiom,
! [A2: set_set_set_a] :
( ( ? [X3: set_set_a] : ( member_set_set_a @ X3 @ A2 ) )
= ( A2 != bot_bo3380559777022489994_set_a ) ) ).
% ex_in_conv
thf(fact_140_ex__in__conv,axiom,
! [A2: set_set_a] :
( ( ? [X3: set_a] : ( member_set_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_141_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X3: a] : ( member_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_142_equals0I,axiom,
! [A2: set_set_set_a] :
( ! [Y2: set_set_a] :
~ ( member_set_set_a @ Y2 @ A2 )
=> ( A2 = bot_bo3380559777022489994_set_a ) ) ).
% equals0I
thf(fact_143_equals0I,axiom,
! [A2: set_set_a] :
( ! [Y2: set_a] :
~ ( member_set_a @ Y2 @ A2 )
=> ( A2 = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_144_equals0I,axiom,
! [A2: set_a] :
( ! [Y2: a] :
~ ( member_a @ Y2 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_145_equals0D,axiom,
! [A2: set_set_set_a,A: set_set_a] :
( ( A2 = bot_bo3380559777022489994_set_a )
=> ~ ( member_set_set_a @ A @ A2 ) ) ).
% equals0D
thf(fact_146_equals0D,axiom,
! [A2: set_set_a,A: set_a] :
( ( A2 = bot_bot_set_set_a )
=> ~ ( member_set_a @ A @ A2 ) ) ).
% equals0D
thf(fact_147_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_148_emptyE,axiom,
! [A: set_set_a] :
~ ( member_set_set_a @ A @ bot_bo3380559777022489994_set_a ) ).
% emptyE
thf(fact_149_emptyE,axiom,
! [A: set_a] :
~ ( member_set_a @ A @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_150_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_151_disjoint__iff__not__equal,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ A2 @ B )
= bot_bot_set_set_a )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ! [Y3: set_a] :
( ( member_set_a @ Y3 @ B )
=> ( X3 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_152_disjoint__iff__not__equal,axiom,
! [A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ! [Y3: a] :
( ( member_a @ Y3 @ B )
=> ( X3 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_153_Int__empty__right,axiom,
! [A2: set_set_a] :
( ( inf_inf_set_set_a @ A2 @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% Int_empty_right
thf(fact_154_Int__empty__right,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Int_empty_right
thf(fact_155_Int__empty__left,axiom,
! [B: set_set_a] :
( ( inf_inf_set_set_a @ bot_bot_set_set_a @ B )
= bot_bot_set_set_a ) ).
% Int_empty_left
thf(fact_156_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_157_disjoint__iff,axiom,
! [A2: set_set_set_a,B: set_set_set_a] :
( ( ( inf_in1205276777018777868_set_a @ A2 @ B )
= bot_bo3380559777022489994_set_a )
= ( ! [X3: set_set_a] :
( ( member_set_set_a @ X3 @ A2 )
=> ~ ( member_set_set_a @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_158_disjoint__iff,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ A2 @ B )
= bot_bot_set_set_a )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ~ ( member_set_a @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_159_disjoint__iff,axiom,
! [A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ~ ( member_a @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_160_Int__emptyI,axiom,
! [A2: set_set_set_a,B: set_set_set_a] :
( ! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ A2 )
=> ~ ( member_set_set_a @ X2 @ B ) )
=> ( ( inf_in1205276777018777868_set_a @ A2 @ B )
= bot_bo3380559777022489994_set_a ) ) ).
% Int_emptyI
thf(fact_161_Int__emptyI,axiom,
! [A2: set_set_a,B: set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ~ ( member_set_a @ X2 @ B ) )
=> ( ( inf_inf_set_set_a @ A2 @ B )
= bot_bot_set_set_a ) ) ).
% Int_emptyI
thf(fact_162_Int__emptyI,axiom,
! [A2: set_a,B: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ~ ( member_a @ X2 @ B ) )
=> ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_163_Un__empty__right,axiom,
! [A2: set_set_a] :
( ( sup_sup_set_set_a @ A2 @ bot_bot_set_set_a )
= A2 ) ).
% Un_empty_right
thf(fact_164_Un__empty__right,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Un_empty_right
thf(fact_165_Un__empty__left,axiom,
! [B: set_set_a] :
( ( sup_sup_set_set_a @ bot_bot_set_set_a @ B )
= B ) ).
% Un_empty_left
thf(fact_166_Un__empty__left,axiom,
! [B: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ B )
= B ) ).
% Un_empty_left
thf(fact_167_finite_OemptyI,axiom,
finite51705147264084924um_a_a @ bot_bo8673002622815711727um_a_a ).
% finite.emptyI
thf(fact_168_finite_OemptyI,axiom,
finite6544458595007987280od_a_a @ bot_bo3357376287454694259od_a_a ).
% finite.emptyI
thf(fact_169_finite_OemptyI,axiom,
finite_finite_set_a @ bot_bot_set_set_a ).
% finite.emptyI
thf(fact_170_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_171_infinite__imp__nonempty,axiom,
! [S: set_Sum_sum_a_a] :
( ~ ( finite51705147264084924um_a_a @ S )
=> ( S != bot_bo8673002622815711727um_a_a ) ) ).
% infinite_imp_nonempty
thf(fact_172_infinite__imp__nonempty,axiom,
! [S: set_Product_prod_a_a] :
( ~ ( finite6544458595007987280od_a_a @ S )
=> ( S != bot_bo3357376287454694259od_a_a ) ) ).
% infinite_imp_nonempty
thf(fact_173_infinite__imp__nonempty,axiom,
! [S: set_set_a] :
( ~ ( finite_finite_set_a @ S )
=> ( S != bot_bot_set_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_174_infinite__imp__nonempty,axiom,
! [S: set_a] :
( ~ ( finite_finite_a @ S )
=> ( S != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_175_finite__has__minimal,axiom,
! [A2: set_set_set_a] :
( ( finite7209287970140883943_set_a @ A2 )
=> ( ( A2 != bot_bo3380559777022489994_set_a )
=> ? [X2: set_set_a] :
( ( member_set_set_a @ X2 @ A2 )
& ! [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_176_finite__has__minimal,axiom,
! [A2: set_set_set_set_a] :
( ( finite5318320746233006407_set_a @ A2 )
=> ( ( A2 != bot_bo4178452617224790762_set_a )
=> ? [X2: set_set_set_a] :
( ( member_set_set_set_a @ X2 @ A2 )
& ! [Xa: set_set_set_a] :
( ( member_set_set_set_a @ Xa @ A2 )
=> ( ( ord_le5722252365846178494_set_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_177_finite__has__minimal,axiom,
! [A2: set_a_o] :
( ( finite_finite_a_o @ A2 )
=> ( ( A2 != bot_bot_set_a_o2 )
=> ? [X2: a > $o] :
( ( member_a_o @ X2 @ A2 )
& ! [Xa: a > $o] :
( ( member_a_o @ Xa @ A2 )
=> ( ( ord_less_eq_a_o @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_178_finite__has__minimal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_179_finite__has__maximal,axiom,
! [A2: set_set_set_a] :
( ( finite7209287970140883943_set_a @ A2 )
=> ( ( A2 != bot_bo3380559777022489994_set_a )
=> ? [X2: set_set_a] :
( ( member_set_set_a @ X2 @ A2 )
& ! [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_180_finite__has__maximal,axiom,
! [A2: set_set_set_set_a] :
( ( finite5318320746233006407_set_a @ A2 )
=> ( ( A2 != bot_bo4178452617224790762_set_a )
=> ? [X2: set_set_set_a] :
( ( member_set_set_set_a @ X2 @ A2 )
& ! [Xa: set_set_set_a] :
( ( member_set_set_set_a @ Xa @ A2 )
=> ( ( ord_le5722252365846178494_set_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_181_finite__has__maximal,axiom,
! [A2: set_a_o] :
( ( finite_finite_a_o @ A2 )
=> ( ( A2 != bot_bot_set_a_o2 )
=> ? [X2: a > $o] :
( ( member_a_o @ X2 @ A2 )
& ! [Xa: a > $o] :
( ( member_a_o @ Xa @ A2 )
=> ( ( ord_less_eq_a_o @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_182_finite__has__maximal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_183_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_184_Collect__mono__iff,axiom,
! [P: set_set_a > $o,Q: set_set_a > $o] :
( ( ord_le5722252365846178494_set_a @ ( collect_set_set_a @ P ) @ ( collect_set_set_a @ Q ) )
= ( ! [X3: set_set_a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_185_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_186_set__eq__subset,axiom,
( ( ^ [Y4: set_set_a,Z: set_set_a] : ( Y4 = Z ) )
= ( ^ [A6: set_set_a,B6: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A6 @ B6 )
& ( ord_le3724670747650509150_set_a @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_187_set__eq__subset,axiom,
( ( ^ [Y4: set_set_set_a,Z: set_set_set_a] : ( Y4 = Z ) )
= ( ^ [A6: set_set_set_a,B6: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A6 @ B6 )
& ( ord_le5722252365846178494_set_a @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_188_set__eq__subset,axiom,
( ( ^ [Y4: set_a,Z: set_a] : ( Y4 = Z ) )
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A6 @ B6 )
& ( ord_less_eq_set_a @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_189_subset__trans,axiom,
! [A2: set_set_a,B: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C )
=> ( ord_le3724670747650509150_set_a @ A2 @ C ) ) ) ).
% subset_trans
thf(fact_190_subset__trans,axiom,
! [A2: set_set_set_a,B: set_set_set_a,C: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A2 @ B )
=> ( ( ord_le5722252365846178494_set_a @ B @ C )
=> ( ord_le5722252365846178494_set_a @ A2 @ C ) ) ) ).
% subset_trans
thf(fact_191_subset__trans,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% subset_trans
thf(fact_192_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_193_Collect__mono,axiom,
! [P: set_set_a > $o,Q: set_set_a > $o] :
( ! [X2: set_set_a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le5722252365846178494_set_a @ ( collect_set_set_a @ P ) @ ( collect_set_set_a @ Q ) ) ) ).
% Collect_mono
thf(fact_194_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_195_subset__refl,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_196_subset__refl,axiom,
! [A2: set_set_set_a] : ( ord_le5722252365846178494_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_197_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_198_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A6: set_set_a,B6: set_set_a] :
! [T2: set_a] :
( ( member_set_a @ T2 @ A6 )
=> ( member_set_a @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_199_subset__iff,axiom,
( ord_le5722252365846178494_set_a
= ( ^ [A6: set_set_set_a,B6: set_set_set_a] :
! [T2: set_set_a] :
( ( member_set_set_a @ T2 @ A6 )
=> ( member_set_set_a @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_200_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
! [T2: a] :
( ( member_a @ T2 @ A6 )
=> ( member_a @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_201_equalityD2,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( A2 = B )
=> ( ord_le3724670747650509150_set_a @ B @ A2 ) ) ).
% equalityD2
thf(fact_202_equalityD2,axiom,
! [A2: set_set_set_a,B: set_set_set_a] :
( ( A2 = B )
=> ( ord_le5722252365846178494_set_a @ B @ A2 ) ) ).
% equalityD2
thf(fact_203_equalityD2,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ( ord_less_eq_set_a @ B @ A2 ) ) ).
% equalityD2
thf(fact_204_equalityD1,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( A2 = B )
=> ( ord_le3724670747650509150_set_a @ A2 @ B ) ) ).
% equalityD1
thf(fact_205_equalityD1,axiom,
! [A2: set_set_set_a,B: set_set_set_a] :
( ( A2 = B )
=> ( ord_le5722252365846178494_set_a @ A2 @ B ) ) ).
% equalityD1
thf(fact_206_equalityD1,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% equalityD1
thf(fact_207_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A6: set_set_a,B6: set_set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ A6 )
=> ( member_set_a @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_208_subset__eq,axiom,
( ord_le5722252365846178494_set_a
= ( ^ [A6: set_set_set_a,B6: set_set_set_a] :
! [X3: set_set_a] :
( ( member_set_set_a @ X3 @ A6 )
=> ( member_set_set_a @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_209_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A6 )
=> ( member_a @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_210_equalityE,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( A2 = B )
=> ~ ( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ~ ( ord_le3724670747650509150_set_a @ B @ A2 ) ) ) ).
% equalityE
thf(fact_211_equalityE,axiom,
! [A2: set_set_set_a,B: set_set_set_a] :
( ( A2 = B )
=> ~ ( ( ord_le5722252365846178494_set_a @ A2 @ B )
=> ~ ( ord_le5722252365846178494_set_a @ B @ A2 ) ) ) ).
% equalityE
thf(fact_212_equalityE,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B )
=> ~ ( ord_less_eq_set_a @ B @ A2 ) ) ) ).
% equalityE
thf(fact_213_subsetD,axiom,
! [A2: set_set_a,B: set_set_a,C2: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( member_set_a @ C2 @ A2 )
=> ( member_set_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_214_subsetD,axiom,
! [A2: set_set_set_a,B: set_set_set_a,C2: set_set_a] :
( ( ord_le5722252365846178494_set_a @ A2 @ B )
=> ( ( member_set_set_a @ C2 @ A2 )
=> ( member_set_set_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_215_subsetD,axiom,
! [A2: set_a,B: set_a,C2: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_216_in__mono,axiom,
! [A2: set_set_a,B: set_set_a,X: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( member_set_a @ X @ A2 )
=> ( member_set_a @ X @ B ) ) ) ).
% in_mono
thf(fact_217_in__mono,axiom,
! [A2: set_set_set_a,B: set_set_set_a,X: set_set_a] :
( ( ord_le5722252365846178494_set_a @ A2 @ B )
=> ( ( member_set_set_a @ X @ A2 )
=> ( member_set_set_a @ X @ B ) ) ) ).
% in_mono
thf(fact_218_in__mono,axiom,
! [A2: set_a,B: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ X @ B ) ) ) ).
% in_mono
thf(fact_219_Int__left__commute,axiom,
! [A2: set_set_a,B: set_set_a,C: set_set_a] :
( ( inf_inf_set_set_a @ A2 @ ( inf_inf_set_set_a @ B @ C ) )
= ( inf_inf_set_set_a @ B @ ( inf_inf_set_set_a @ A2 @ C ) ) ) ).
% Int_left_commute
thf(fact_220_Int__left__commute,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B @ C ) )
= ( inf_inf_set_a @ B @ ( inf_inf_set_a @ A2 @ C ) ) ) ).
% Int_left_commute
thf(fact_221_Int__left__absorb,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( inf_inf_set_set_a @ A2 @ ( inf_inf_set_set_a @ A2 @ B ) )
= ( inf_inf_set_set_a @ A2 @ B ) ) ).
% Int_left_absorb
thf(fact_222_Int__left__absorb,axiom,
! [A2: set_a,B: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ A2 @ B ) )
= ( inf_inf_set_a @ A2 @ B ) ) ).
% Int_left_absorb
thf(fact_223_Int__commute,axiom,
( inf_inf_set_set_a
= ( ^ [A6: set_set_a,B6: set_set_a] : ( inf_inf_set_set_a @ B6 @ A6 ) ) ) ).
% Int_commute
thf(fact_224_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A6: set_a,B6: set_a] : ( inf_inf_set_a @ B6 @ A6 ) ) ) ).
% Int_commute
thf(fact_225_Int__absorb,axiom,
! [A2: set_set_a] :
( ( inf_inf_set_set_a @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_226_Int__absorb,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_227_Int__assoc,axiom,
! [A2: set_set_a,B: set_set_a,C: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ A2 @ B ) @ C )
= ( inf_inf_set_set_a @ A2 @ ( inf_inf_set_set_a @ B @ C ) ) ) ).
% Int_assoc
thf(fact_228_Int__assoc,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B ) @ C )
= ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B @ C ) ) ) ).
% Int_assoc
thf(fact_229_IntD2,axiom,
! [C2: set_set_a,A2: set_set_set_a,B: set_set_set_a] :
( ( member_set_set_a @ C2 @ ( inf_in1205276777018777868_set_a @ A2 @ B ) )
=> ( member_set_set_a @ C2 @ B ) ) ).
% IntD2
thf(fact_230_IntD2,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A2 @ B ) )
=> ( member_set_a @ C2 @ B ) ) ).
% IntD2
thf(fact_231_IntD2,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) )
=> ( member_a @ C2 @ B ) ) ).
% IntD2
thf(fact_232_IntD1,axiom,
! [C2: set_set_a,A2: set_set_set_a,B: set_set_set_a] :
( ( member_set_set_a @ C2 @ ( inf_in1205276777018777868_set_a @ A2 @ B ) )
=> ( member_set_set_a @ C2 @ A2 ) ) ).
% IntD1
thf(fact_233_IntD1,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A2 @ B ) )
=> ( member_set_a @ C2 @ A2 ) ) ).
% IntD1
thf(fact_234_IntD1,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) )
=> ( member_a @ C2 @ A2 ) ) ).
% IntD1
thf(fact_235_IntE,axiom,
! [C2: set_set_a,A2: set_set_set_a,B: set_set_set_a] :
( ( member_set_set_a @ C2 @ ( inf_in1205276777018777868_set_a @ A2 @ B ) )
=> ~ ( ( member_set_set_a @ C2 @ A2 )
=> ~ ( member_set_set_a @ C2 @ B ) ) ) ).
% IntE
thf(fact_236_IntE,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A2 @ B ) )
=> ~ ( ( member_set_a @ C2 @ A2 )
=> ~ ( member_set_a @ C2 @ B ) ) ) ).
% IntE
thf(fact_237_IntE,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) )
=> ~ ( ( member_a @ C2 @ A2 )
=> ~ ( member_a @ C2 @ B ) ) ) ).
% IntE
thf(fact_238_Un__left__commute,axiom,
! [A2: set_set_a,B: set_set_a,C: set_set_a] :
( ( sup_sup_set_set_a @ A2 @ ( sup_sup_set_set_a @ B @ C ) )
= ( sup_sup_set_set_a @ B @ ( sup_sup_set_set_a @ A2 @ C ) ) ) ).
% Un_left_commute
thf(fact_239_Un__left__commute,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B @ C ) )
= ( sup_sup_set_a @ B @ ( sup_sup_set_a @ A2 @ C ) ) ) ).
% Un_left_commute
thf(fact_240_Un__left__absorb,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( sup_sup_set_set_a @ A2 @ ( sup_sup_set_set_a @ A2 @ B ) )
= ( sup_sup_set_set_a @ A2 @ B ) ) ).
% Un_left_absorb
thf(fact_241_Un__left__absorb,axiom,
! [A2: set_a,B: set_a] :
( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B ) )
= ( sup_sup_set_a @ A2 @ B ) ) ).
% Un_left_absorb
thf(fact_242_Un__commute,axiom,
( sup_sup_set_set_a
= ( ^ [A6: set_set_a,B6: set_set_a] : ( sup_sup_set_set_a @ B6 @ A6 ) ) ) ).
% Un_commute
thf(fact_243_Un__commute,axiom,
( sup_sup_set_a
= ( ^ [A6: set_a,B6: set_a] : ( sup_sup_set_a @ B6 @ A6 ) ) ) ).
% Un_commute
thf(fact_244_Un__absorb,axiom,
! [A2: set_set_a] :
( ( sup_sup_set_set_a @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_245_Un__absorb,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_246_Un__assoc,axiom,
! [A2: set_set_a,B: set_set_a,C: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ A2 @ B ) @ C )
= ( sup_sup_set_set_a @ A2 @ ( sup_sup_set_set_a @ B @ C ) ) ) ).
% Un_assoc
thf(fact_247_Un__assoc,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B ) @ C )
= ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B @ C ) ) ) ).
% Un_assoc
thf(fact_248_ball__Un,axiom,
! [A2: set_set_a,B: set_set_a,P: set_a > $o] :
( ( ! [X3: set_a] :
( ( member_set_a @ X3 @ ( sup_sup_set_set_a @ A2 @ B ) )
=> ( P @ X3 ) ) )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( P @ X3 ) )
& ! [X3: set_a] :
( ( member_set_a @ X3 @ B )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_249_ball__Un,axiom,
! [A2: set_a,B: set_a,P: a > $o] :
( ( ! [X3: a] :
( ( member_a @ X3 @ ( sup_sup_set_a @ A2 @ B ) )
=> ( P @ X3 ) ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( P @ X3 ) )
& ! [X3: a] :
( ( member_a @ X3 @ B )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_250_bex__Un,axiom,
! [A2: set_set_a,B: set_set_a,P: set_a > $o] :
( ( ? [X3: set_a] :
( ( member_set_a @ X3 @ ( sup_sup_set_set_a @ A2 @ B ) )
& ( P @ X3 ) ) )
= ( ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ( P @ X3 ) )
| ? [X3: set_a] :
( ( member_set_a @ X3 @ B )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_251_bex__Un,axiom,
! [A2: set_a,B: set_a,P: a > $o] :
( ( ? [X3: a] :
( ( member_a @ X3 @ ( sup_sup_set_a @ A2 @ B ) )
& ( P @ X3 ) ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A2 )
& ( P @ X3 ) )
| ? [X3: a] :
( ( member_a @ X3 @ B )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_252_UnI2,axiom,
! [C2: set_set_a,B: set_set_set_a,A2: set_set_set_a] :
( ( member_set_set_a @ C2 @ B )
=> ( member_set_set_a @ C2 @ ( sup_su2076012971530813682_set_a @ A2 @ B ) ) ) ).
% UnI2
thf(fact_253_UnI2,axiom,
! [C2: set_a,B: set_set_a,A2: set_set_a] :
( ( member_set_a @ C2 @ B )
=> ( member_set_a @ C2 @ ( sup_sup_set_set_a @ A2 @ B ) ) ) ).
% UnI2
thf(fact_254_UnI2,axiom,
! [C2: a,B: set_a,A2: set_a] :
( ( member_a @ C2 @ B )
=> ( member_a @ C2 @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% UnI2
thf(fact_255_UnI1,axiom,
! [C2: set_set_a,A2: set_set_set_a,B: set_set_set_a] :
( ( member_set_set_a @ C2 @ A2 )
=> ( member_set_set_a @ C2 @ ( sup_su2076012971530813682_set_a @ A2 @ B ) ) ) ).
% UnI1
thf(fact_256_UnI1,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ A2 )
=> ( member_set_a @ C2 @ ( sup_sup_set_set_a @ A2 @ B ) ) ) ).
% UnI1
thf(fact_257_UnI1,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% UnI1
thf(fact_258_UnE,axiom,
! [C2: set_set_a,A2: set_set_set_a,B: set_set_set_a] :
( ( member_set_set_a @ C2 @ ( sup_su2076012971530813682_set_a @ A2 @ B ) )
=> ( ~ ( member_set_set_a @ C2 @ A2 )
=> ( member_set_set_a @ C2 @ B ) ) ) ).
% UnE
thf(fact_259_UnE,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( sup_sup_set_set_a @ A2 @ B ) )
=> ( ~ ( member_set_a @ C2 @ A2 )
=> ( member_set_a @ C2 @ B ) ) ) ).
% UnE
thf(fact_260_UnE,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( sup_sup_set_a @ A2 @ B ) )
=> ( ~ ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ B ) ) ) ).
% UnE
thf(fact_261_Int__Collect__mono,axiom,
! [A2: set_set_a,B: set_set_a,P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ ( collect_set_a @ P ) ) @ ( inf_inf_set_set_a @ B @ ( collect_set_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_262_Int__Collect__mono,axiom,
! [A2: set_set_set_a,B: set_set_set_a,P: set_set_a > $o,Q: set_set_a > $o] :
( ( ord_le5722252365846178494_set_a @ A2 @ B )
=> ( ! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le5722252365846178494_set_a @ ( inf_in1205276777018777868_set_a @ A2 @ ( collect_set_set_a @ P ) ) @ ( inf_in1205276777018777868_set_a @ B @ ( collect_set_set_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_263_Int__Collect__mono,axiom,
! [A2: set_a,B: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_264_Int__greatest,axiom,
! [C: set_set_a,A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ C @ B )
=> ( ord_le3724670747650509150_set_a @ C @ ( inf_inf_set_set_a @ A2 @ B ) ) ) ) ).
% Int_greatest
thf(fact_265_Int__greatest,axiom,
! [C: set_set_set_a,A2: set_set_set_a,B: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ C @ A2 )
=> ( ( ord_le5722252365846178494_set_a @ C @ B )
=> ( ord_le5722252365846178494_set_a @ C @ ( inf_in1205276777018777868_set_a @ A2 @ B ) ) ) ) ).
% Int_greatest
thf(fact_266_Int__greatest,axiom,
! [C: set_a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ A2 )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% Int_greatest
thf(fact_267_Int__absorb2,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( inf_inf_set_set_a @ A2 @ B )
= A2 ) ) ).
% Int_absorb2
thf(fact_268_Int__absorb2,axiom,
! [A2: set_set_set_a,B: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A2 @ B )
=> ( ( inf_in1205276777018777868_set_a @ A2 @ B )
= A2 ) ) ).
% Int_absorb2
thf(fact_269_Int__absorb2,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( inf_inf_set_a @ A2 @ B )
= A2 ) ) ).
% Int_absorb2
thf(fact_270_Int__absorb1,axiom,
! [B: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_271_Int__absorb1,axiom,
! [B: set_set_set_a,A2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ B @ A2 )
=> ( ( inf_in1205276777018777868_set_a @ A2 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_272_Int__absorb1,axiom,
! [B: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_273_Int__lower2,axiom,
! [A2: set_set_a,B: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B ) @ B ) ).
% Int_lower2
thf(fact_274_Int__lower2,axiom,
! [A2: set_set_set_a,B: set_set_set_a] : ( ord_le5722252365846178494_set_a @ ( inf_in1205276777018777868_set_a @ A2 @ B ) @ B ) ).
% Int_lower2
thf(fact_275_Int__lower2,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ B ) ).
% Int_lower2
thf(fact_276_Int__lower1,axiom,
! [A2: set_set_a,B: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B ) @ A2 ) ).
% Int_lower1
thf(fact_277_Int__lower1,axiom,
! [A2: set_set_set_a,B: set_set_set_a] : ( ord_le5722252365846178494_set_a @ ( inf_in1205276777018777868_set_a @ A2 @ B ) @ A2 ) ).
% Int_lower1
thf(fact_278_Int__lower1,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ A2 ) ).
% Int_lower1
thf(fact_279_Int__mono,axiom,
! [A2: set_set_a,C: set_set_a,B: set_set_a,D: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ C )
=> ( ( ord_le3724670747650509150_set_a @ B @ D )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B ) @ ( inf_inf_set_set_a @ C @ D ) ) ) ) ).
% Int_mono
thf(fact_280_Int__mono,axiom,
! [A2: set_set_set_a,C: set_set_set_a,B: set_set_set_a,D: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A2 @ C )
=> ( ( ord_le5722252365846178494_set_a @ B @ D )
=> ( ord_le5722252365846178494_set_a @ ( inf_in1205276777018777868_set_a @ A2 @ B ) @ ( inf_in1205276777018777868_set_a @ C @ D ) ) ) ) ).
% Int_mono
thf(fact_281_Int__mono,axiom,
! [A2: set_a,C: set_a,B: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( inf_inf_set_a @ C @ D ) ) ) ) ).
% Int_mono
thf(fact_282_subset__Un__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A6: set_set_a,B6: set_set_a] :
( ( sup_sup_set_set_a @ A6 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_283_subset__Un__eq,axiom,
( ord_le5722252365846178494_set_a
= ( ^ [A6: set_set_set_a,B6: set_set_set_a] :
( ( sup_su2076012971530813682_set_a @ A6 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_284_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
( ( sup_sup_set_a @ A6 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_285_subset__UnE,axiom,
! [C: set_set_a,A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C @ ( sup_sup_set_set_a @ A2 @ B ) )
=> ~ ! [A7: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A7 @ A2 )
=> ! [B7: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B7 @ B )
=> ( C
!= ( sup_sup_set_set_a @ A7 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_286_subset__UnE,axiom,
! [C: set_set_set_a,A2: set_set_set_a,B: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ C @ ( sup_su2076012971530813682_set_a @ A2 @ B ) )
=> ~ ! [A7: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A7 @ A2 )
=> ! [B7: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ B7 @ B )
=> ( C
!= ( sup_su2076012971530813682_set_a @ A7 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_287_subset__UnE,axiom,
! [C: set_a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A2 @ B ) )
=> ~ ! [A7: set_a] :
( ( ord_less_eq_set_a @ A7 @ A2 )
=> ! [B7: set_a] :
( ( ord_less_eq_set_a @ B7 @ B )
=> ( C
!= ( sup_sup_set_a @ A7 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_288_Un__absorb2,axiom,
! [B: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A2 )
=> ( ( sup_sup_set_set_a @ A2 @ B )
= A2 ) ) ).
% Un_absorb2
thf(fact_289_Un__absorb2,axiom,
! [B: set_set_set_a,A2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ B @ A2 )
=> ( ( sup_su2076012971530813682_set_a @ A2 @ B )
= A2 ) ) ).
% Un_absorb2
thf(fact_290_Un__absorb2,axiom,
! [B: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( sup_sup_set_a @ A2 @ B )
= A2 ) ) ).
% Un_absorb2
thf(fact_291_Un__absorb1,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( sup_sup_set_set_a @ A2 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_292_Un__absorb1,axiom,
! [A2: set_set_set_a,B: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A2 @ B )
=> ( ( sup_su2076012971530813682_set_a @ A2 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_293_Un__absorb1,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( sup_sup_set_a @ A2 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_294_Un__upper2,axiom,
! [B: set_set_a,A2: set_set_a] : ( ord_le3724670747650509150_set_a @ B @ ( sup_sup_set_set_a @ A2 @ B ) ) ).
% Un_upper2
thf(fact_295_Un__upper2,axiom,
! [B: set_set_set_a,A2: set_set_set_a] : ( ord_le5722252365846178494_set_a @ B @ ( sup_su2076012971530813682_set_a @ A2 @ B ) ) ).
% Un_upper2
thf(fact_296_Un__upper2,axiom,
! [B: set_a,A2: set_a] : ( ord_less_eq_set_a @ B @ ( sup_sup_set_a @ A2 @ B ) ) ).
% Un_upper2
thf(fact_297_Un__upper1,axiom,
! [A2: set_set_a,B: set_set_a] : ( ord_le3724670747650509150_set_a @ A2 @ ( sup_sup_set_set_a @ A2 @ B ) ) ).
% Un_upper1
thf(fact_298_Un__upper1,axiom,
! [A2: set_set_set_a,B: set_set_set_a] : ( ord_le5722252365846178494_set_a @ A2 @ ( sup_su2076012971530813682_set_a @ A2 @ B ) ) ).
% Un_upper1
thf(fact_299_Un__upper1,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B ) ) ).
% Un_upper1
thf(fact_300_Un__least,axiom,
! [A2: set_set_a,C: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ C )
=> ( ( ord_le3724670747650509150_set_a @ B @ C )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A2 @ B ) @ C ) ) ) ).
% Un_least
thf(fact_301_Un__least,axiom,
! [A2: set_set_set_a,C: set_set_set_a,B: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A2 @ C )
=> ( ( ord_le5722252365846178494_set_a @ B @ C )
=> ( ord_le5722252365846178494_set_a @ ( sup_su2076012971530813682_set_a @ A2 @ B ) @ C ) ) ) ).
% Un_least
thf(fact_302_Un__least,axiom,
! [A2: set_a,C: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B ) @ C ) ) ) ).
% Un_least
thf(fact_303_Un__mono,axiom,
! [A2: set_set_a,C: set_set_a,B: set_set_a,D: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ C )
=> ( ( ord_le3724670747650509150_set_a @ B @ D )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A2 @ B ) @ ( sup_sup_set_set_a @ C @ D ) ) ) ) ).
% Un_mono
thf(fact_304_Un__mono,axiom,
! [A2: set_set_set_a,C: set_set_set_a,B: set_set_set_a,D: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A2 @ C )
=> ( ( ord_le5722252365846178494_set_a @ B @ D )
=> ( ord_le5722252365846178494_set_a @ ( sup_su2076012971530813682_set_a @ A2 @ B ) @ ( sup_su2076012971530813682_set_a @ C @ D ) ) ) ) ).
% Un_mono
thf(fact_305_Un__mono,axiom,
! [A2: set_a,C: set_a,B: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B @ D )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B ) @ ( sup_sup_set_a @ C @ D ) ) ) ) ).
% Un_mono
thf(fact_306_Un__Int__distrib2,axiom,
! [B: set_set_a,C: set_set_a,A2: set_set_a] :
( ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ B @ C ) @ A2 )
= ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ B @ A2 ) @ ( sup_sup_set_set_a @ C @ A2 ) ) ) ).
% Un_Int_distrib2
thf(fact_307_Un__Int__distrib2,axiom,
! [B: set_a,C: set_a,A2: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ B @ C ) @ A2 )
= ( inf_inf_set_a @ ( sup_sup_set_a @ B @ A2 ) @ ( sup_sup_set_a @ C @ A2 ) ) ) ).
% Un_Int_distrib2
thf(fact_308_Int__Un__distrib2,axiom,
! [B: set_set_a,C: set_set_a,A2: set_set_a] :
( ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ B @ C ) @ A2 )
= ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ B @ A2 ) @ ( inf_inf_set_set_a @ C @ A2 ) ) ) ).
% Int_Un_distrib2
thf(fact_309_Int__Un__distrib2,axiom,
! [B: set_a,C: set_a,A2: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ B @ C ) @ A2 )
= ( sup_sup_set_a @ ( inf_inf_set_a @ B @ A2 ) @ ( inf_inf_set_a @ C @ A2 ) ) ) ).
% Int_Un_distrib2
thf(fact_310_Un__Int__distrib,axiom,
! [A2: set_set_a,B: set_set_a,C: set_set_a] :
( ( sup_sup_set_set_a @ A2 @ ( inf_inf_set_set_a @ B @ C ) )
= ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ A2 @ B ) @ ( sup_sup_set_set_a @ A2 @ C ) ) ) ).
% Un_Int_distrib
thf(fact_311_Un__Int__distrib,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ A2 @ ( inf_inf_set_a @ B @ C ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ A2 @ B ) @ ( sup_sup_set_a @ A2 @ C ) ) ) ).
% Un_Int_distrib
thf(fact_312_Int__Un__distrib,axiom,
! [A2: set_set_a,B: set_set_a,C: set_set_a] :
( ( inf_inf_set_set_a @ A2 @ ( sup_sup_set_set_a @ B @ C ) )
= ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ A2 @ B ) @ ( inf_inf_set_set_a @ A2 @ C ) ) ) ).
% Int_Un_distrib
thf(fact_313_Int__Un__distrib,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ A2 @ ( sup_sup_set_a @ B @ C ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( inf_inf_set_a @ A2 @ C ) ) ) ).
% Int_Un_distrib
thf(fact_314_Un__Int__crazy,axiom,
! [A2: set_set_a,B: set_set_a,C: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ A2 @ B ) @ ( inf_inf_set_set_a @ B @ C ) ) @ ( inf_inf_set_set_a @ C @ A2 ) )
= ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ A2 @ B ) @ ( sup_sup_set_set_a @ B @ C ) ) @ ( sup_sup_set_set_a @ C @ A2 ) ) ) ).
% Un_Int_crazy
thf(fact_315_Un__Int__crazy,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( inf_inf_set_a @ B @ C ) ) @ ( inf_inf_set_a @ C @ A2 ) )
= ( inf_inf_set_a @ ( inf_inf_set_a @ ( sup_sup_set_a @ A2 @ B ) @ ( sup_sup_set_a @ B @ C ) ) @ ( sup_sup_set_a @ C @ A2 ) ) ) ).
% Un_Int_crazy
thf(fact_316_Un__Int__assoc__eq,axiom,
! [A2: set_set_a,B: set_set_a,C: set_set_a] :
( ( ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ A2 @ B ) @ C )
= ( inf_inf_set_set_a @ A2 @ ( sup_sup_set_set_a @ B @ C ) ) )
= ( ord_le3724670747650509150_set_a @ C @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_317_Un__Int__assoc__eq,axiom,
! [A2: set_set_set_a,B: set_set_set_a,C: set_set_set_a] :
( ( ( sup_su2076012971530813682_set_a @ ( inf_in1205276777018777868_set_a @ A2 @ B ) @ C )
= ( inf_in1205276777018777868_set_a @ A2 @ ( sup_su2076012971530813682_set_a @ B @ C ) ) )
= ( ord_le5722252365846178494_set_a @ C @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_318_Un__Int__assoc__eq,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ( sup_sup_set_a @ ( inf_inf_set_a @ A2 @ B ) @ C )
= ( inf_inf_set_a @ A2 @ ( sup_sup_set_a @ B @ C ) ) )
= ( ord_less_eq_set_a @ C @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_319_sup__inf__absorb,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( sup_sup_set_set_a @ X @ ( inf_inf_set_set_a @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_320_sup__inf__absorb,axiom,
! [X: a > $o,Y: a > $o] :
( ( sup_sup_a_o @ X @ ( inf_inf_a_o @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_321_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_322_inf__sup__absorb,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( inf_inf_set_set_a @ X @ ( sup_sup_set_set_a @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_323_inf__sup__absorb,axiom,
! [X: a > $o,Y: a > $o] :
( ( inf_inf_a_o @ X @ ( sup_sup_a_o @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_324_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_325_sup__bot_Oright__neutral,axiom,
! [A: set_set_a] :
( ( sup_sup_set_set_a @ A @ bot_bot_set_set_a )
= A ) ).
% sup_bot.right_neutral
thf(fact_326_sup__bot_Oright__neutral,axiom,
! [A: a > $o] :
( ( sup_sup_a_o @ A @ bot_bot_a_o )
= A ) ).
% sup_bot.right_neutral
thf(fact_327_sup__bot_Oright__neutral,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ bot_bot_set_a )
= A ) ).
% sup_bot.right_neutral
thf(fact_328_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_set_a,B4: set_set_a] :
( ( bot_bot_set_set_a
= ( sup_sup_set_set_a @ A @ B4 ) )
= ( ( A = bot_bot_set_set_a )
& ( B4 = bot_bot_set_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_329_sup__bot_Oneutr__eq__iff,axiom,
! [A: a > $o,B4: a > $o] :
( ( bot_bot_a_o
= ( sup_sup_a_o @ A @ B4 ) )
= ( ( A = bot_bot_a_o )
& ( B4 = bot_bot_a_o ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_330_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_a,B4: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ A @ B4 ) )
= ( ( A = bot_bot_set_a )
& ( B4 = bot_bot_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_331_sup__bot_Oleft__neutral,axiom,
! [A: set_set_a] :
( ( sup_sup_set_set_a @ bot_bot_set_set_a @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_332_sup__bot_Oleft__neutral,axiom,
! [A: a > $o] :
( ( sup_sup_a_o @ bot_bot_a_o @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_333_sup__bot_Oleft__neutral,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_334_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_set_a,B4: set_set_a] :
( ( ( sup_sup_set_set_a @ A @ B4 )
= bot_bot_set_set_a )
= ( ( A = bot_bot_set_set_a )
& ( B4 = bot_bot_set_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_335_sup__bot_Oeq__neutr__iff,axiom,
! [A: a > $o,B4: a > $o] :
( ( ( sup_sup_a_o @ A @ B4 )
= bot_bot_a_o )
= ( ( A = bot_bot_a_o )
& ( B4 = bot_bot_a_o ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_336_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_a,B4: set_a] :
( ( ( sup_sup_set_a @ A @ B4 )
= bot_bot_set_a )
= ( ( A = bot_bot_set_a )
& ( B4 = bot_bot_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_337_sup__eq__bot__iff,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( ( sup_sup_set_set_a @ X @ Y )
= bot_bot_set_set_a )
= ( ( X = bot_bot_set_set_a )
& ( Y = bot_bot_set_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_338_sup__eq__bot__iff,axiom,
! [X: a > $o,Y: a > $o] :
( ( ( sup_sup_a_o @ X @ Y )
= bot_bot_a_o )
= ( ( X = bot_bot_a_o )
& ( Y = bot_bot_a_o ) ) ) ).
% sup_eq_bot_iff
thf(fact_339_sup__eq__bot__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ( sup_sup_set_a @ X @ Y )
= bot_bot_set_a )
= ( ( X = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_340_bot__eq__sup__iff,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( bot_bot_set_set_a
= ( sup_sup_set_set_a @ X @ Y ) )
= ( ( X = bot_bot_set_set_a )
& ( Y = bot_bot_set_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_341_bot__eq__sup__iff,axiom,
! [X: a > $o,Y: a > $o] :
( ( bot_bot_a_o
= ( sup_sup_a_o @ X @ Y ) )
= ( ( X = bot_bot_a_o )
& ( Y = bot_bot_a_o ) ) ) ).
% bot_eq_sup_iff
thf(fact_342_bot__eq__sup__iff,axiom,
! [X: set_a,Y: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ X @ Y ) )
= ( ( X = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_343_sup__bot__right,axiom,
! [X: set_set_a] :
( ( sup_sup_set_set_a @ X @ bot_bot_set_set_a )
= X ) ).
% sup_bot_right
thf(fact_344_sup__bot__right,axiom,
! [X: a > $o] :
( ( sup_sup_a_o @ X @ bot_bot_a_o )
= X ) ).
% sup_bot_right
thf(fact_345_sup__bot__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ bot_bot_set_a )
= X ) ).
% sup_bot_right
thf(fact_346_sup__bot__left,axiom,
! [X: set_set_a] :
( ( sup_sup_set_set_a @ bot_bot_set_set_a @ X )
= X ) ).
% sup_bot_left
thf(fact_347_sup__bot__left,axiom,
! [X: a > $o] :
( ( sup_sup_a_o @ bot_bot_a_o @ X )
= X ) ).
% sup_bot_left
thf(fact_348_sup__bot__left,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ X )
= X ) ).
% sup_bot_left
thf(fact_349_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_set_a] :
( ( inf_inf_set_set_a @ X @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_350_boolean__algebra_Oconj__zero__right,axiom,
! [X: a > $o] :
( ( inf_inf_a_o @ X @ bot_bot_a_o )
= bot_bot_a_o ) ).
% boolean_algebra.conj_zero_right
thf(fact_351_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_352_inf_Oidem,axiom,
! [A: set_set_a] :
( ( inf_inf_set_set_a @ A @ A )
= A ) ).
% inf.idem
thf(fact_353_inf_Oidem,axiom,
! [A: a > $o] :
( ( inf_inf_a_o @ A @ A )
= A ) ).
% inf.idem
thf(fact_354_inf_Oidem,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ A )
= A ) ).
% inf.idem
thf(fact_355_inf__idem,axiom,
! [X: set_set_a] :
( ( inf_inf_set_set_a @ X @ X )
= X ) ).
% inf_idem
thf(fact_356_inf__idem,axiom,
! [X: a > $o] :
( ( inf_inf_a_o @ X @ X )
= X ) ).
% inf_idem
thf(fact_357_inf__idem,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ X )
= X ) ).
% inf_idem
thf(fact_358_inf_Oleft__idem,axiom,
! [A: set_set_a,B4: set_set_a] :
( ( inf_inf_set_set_a @ A @ ( inf_inf_set_set_a @ A @ B4 ) )
= ( inf_inf_set_set_a @ A @ B4 ) ) ).
% inf.left_idem
thf(fact_359_inf_Oleft__idem,axiom,
! [A: a > $o,B4: a > $o] :
( ( inf_inf_a_o @ A @ ( inf_inf_a_o @ A @ B4 ) )
= ( inf_inf_a_o @ A @ B4 ) ) ).
% inf.left_idem
thf(fact_360_inf_Oleft__idem,axiom,
! [A: set_a,B4: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ A @ B4 ) )
= ( inf_inf_set_a @ A @ B4 ) ) ).
% inf.left_idem
thf(fact_361_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_362_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_363_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_364_inf_Oright__idem,axiom,
! [A: set_set_a,B4: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ A @ B4 ) @ B4 )
= ( inf_inf_set_set_a @ A @ B4 ) ) ).
% inf.right_idem
thf(fact_365_inf_Oright__idem,axiom,
! [A: a > $o,B4: a > $o] :
( ( inf_inf_a_o @ ( inf_inf_a_o @ A @ B4 ) @ B4 )
= ( inf_inf_a_o @ A @ B4 ) ) ).
% inf.right_idem
thf(fact_366_inf_Oright__idem,axiom,
! [A: set_a,B4: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B4 ) @ B4 )
= ( inf_inf_set_a @ A @ B4 ) ) ).
% inf.right_idem
thf(fact_367_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_368_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_369_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_370_inf__apply,axiom,
( inf_inf_a_o
= ( ^ [F2: a > $o,G2: a > $o,X3: a] : ( inf_inf_o @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% inf_apply
thf(fact_371_sup_Oidem,axiom,
! [A: set_set_a] :
( ( sup_sup_set_set_a @ A @ A )
= A ) ).
% sup.idem
thf(fact_372_sup_Oidem,axiom,
! [A: a > $o] :
( ( sup_sup_a_o @ A @ A )
= A ) ).
% sup.idem
thf(fact_373_sup_Oidem,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ A )
= A ) ).
% sup.idem
thf(fact_374_sup__idem,axiom,
! [X: set_set_a] :
( ( sup_sup_set_set_a @ X @ X )
= X ) ).
% sup_idem
thf(fact_375_sup__idem,axiom,
! [X: a > $o] :
( ( sup_sup_a_o @ X @ X )
= X ) ).
% sup_idem
thf(fact_376_sup__idem,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ X )
= X ) ).
% sup_idem
thf(fact_377_sup_Oleft__idem,axiom,
! [A: set_set_a,B4: set_set_a] :
( ( sup_sup_set_set_a @ A @ ( sup_sup_set_set_a @ A @ B4 ) )
= ( sup_sup_set_set_a @ A @ B4 ) ) ).
% sup.left_idem
thf(fact_378_sup_Oleft__idem,axiom,
! [A: a > $o,B4: a > $o] :
( ( sup_sup_a_o @ A @ ( sup_sup_a_o @ A @ B4 ) )
= ( sup_sup_a_o @ A @ B4 ) ) ).
% sup.left_idem
thf(fact_379_sup_Oleft__idem,axiom,
! [A: set_a,B4: set_a] :
( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ A @ B4 ) )
= ( sup_sup_set_a @ A @ B4 ) ) ).
% sup.left_idem
thf(fact_380_sup__left__idem,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( sup_sup_set_set_a @ X @ ( sup_sup_set_set_a @ X @ Y ) )
= ( sup_sup_set_set_a @ X @ Y ) ) ).
% sup_left_idem
thf(fact_381_sup__left__idem,axiom,
! [X: a > $o,Y: a > $o] :
( ( sup_sup_a_o @ X @ ( sup_sup_a_o @ X @ Y ) )
= ( sup_sup_a_o @ X @ Y ) ) ).
% sup_left_idem
thf(fact_382_sup__left__idem,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
= ( sup_sup_set_a @ X @ Y ) ) ).
% sup_left_idem
thf(fact_383_sup_Oright__idem,axiom,
! [A: set_set_a,B4: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ A @ B4 ) @ B4 )
= ( sup_sup_set_set_a @ A @ B4 ) ) ).
% sup.right_idem
thf(fact_384_sup_Oright__idem,axiom,
! [A: a > $o,B4: a > $o] :
( ( sup_sup_a_o @ ( sup_sup_a_o @ A @ B4 ) @ B4 )
= ( sup_sup_a_o @ A @ B4 ) ) ).
% sup.right_idem
thf(fact_385_sup_Oright__idem,axiom,
! [A: set_a,B4: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B4 ) @ B4 )
= ( sup_sup_set_a @ A @ B4 ) ) ).
% sup.right_idem
thf(fact_386_sup__apply,axiom,
( sup_sup_a_o
= ( ^ [F2: a > $o,G2: a > $o,X3: a] : ( sup_sup_o @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% sup_apply
thf(fact_387_le__inf__iff,axiom,
! [X: set_set_a,Y: set_set_a,Z2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ ( inf_inf_set_set_a @ Y @ Z2 ) )
= ( ( ord_le3724670747650509150_set_a @ X @ Y )
& ( ord_le3724670747650509150_set_a @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_388_le__inf__iff,axiom,
! [X: set_set_set_a,Y: set_set_set_a,Z2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ X @ ( inf_in1205276777018777868_set_a @ Y @ Z2 ) )
= ( ( ord_le5722252365846178494_set_a @ X @ Y )
& ( ord_le5722252365846178494_set_a @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_389_le__inf__iff,axiom,
! [X: a > $o,Y: a > $o,Z2: a > $o] :
( ( ord_less_eq_a_o @ X @ ( inf_inf_a_o @ Y @ Z2 ) )
= ( ( ord_less_eq_a_o @ X @ Y )
& ( ord_less_eq_a_o @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_390_le__inf__iff,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) )
= ( ( ord_less_eq_set_a @ X @ Y )
& ( ord_less_eq_set_a @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_391_inf_Obounded__iff,axiom,
! [A: set_set_a,B4: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ ( inf_inf_set_set_a @ B4 @ C2 ) )
= ( ( ord_le3724670747650509150_set_a @ A @ B4 )
& ( ord_le3724670747650509150_set_a @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_392_inf_Obounded__iff,axiom,
! [A: set_set_set_a,B4: set_set_set_a,C2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A @ ( inf_in1205276777018777868_set_a @ B4 @ C2 ) )
= ( ( ord_le5722252365846178494_set_a @ A @ B4 )
& ( ord_le5722252365846178494_set_a @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_393_inf_Obounded__iff,axiom,
! [A: a > $o,B4: a > $o,C2: a > $o] :
( ( ord_less_eq_a_o @ A @ ( inf_inf_a_o @ B4 @ C2 ) )
= ( ( ord_less_eq_a_o @ A @ B4 )
& ( ord_less_eq_a_o @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_394_inf_Obounded__iff,axiom,
! [A: set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B4 @ C2 ) )
= ( ( ord_less_eq_set_a @ A @ B4 )
& ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_395_le__sup__iff,axiom,
! [X: set_set_a,Y: set_set_a,Z2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ X @ Y ) @ Z2 )
= ( ( ord_le3724670747650509150_set_a @ X @ Z2 )
& ( ord_le3724670747650509150_set_a @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_396_le__sup__iff,axiom,
! [X: set_set_set_a,Y: set_set_set_a,Z2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ ( sup_su2076012971530813682_set_a @ X @ Y ) @ Z2 )
= ( ( ord_le5722252365846178494_set_a @ X @ Z2 )
& ( ord_le5722252365846178494_set_a @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_397_le__sup__iff,axiom,
! [X: a > $o,Y: a > $o,Z2: a > $o] :
( ( ord_less_eq_a_o @ ( sup_sup_a_o @ X @ Y ) @ Z2 )
= ( ( ord_less_eq_a_o @ X @ Z2 )
& ( ord_less_eq_a_o @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_398_le__sup__iff,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z2 )
= ( ( ord_less_eq_set_a @ X @ Z2 )
& ( ord_less_eq_set_a @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_399_sup_Obounded__iff,axiom,
! [B4: set_set_a,C2: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ B4 @ C2 ) @ A )
= ( ( ord_le3724670747650509150_set_a @ B4 @ A )
& ( ord_le3724670747650509150_set_a @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_400_sup_Obounded__iff,axiom,
! [B4: set_set_set_a,C2: set_set_set_a,A: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ ( sup_su2076012971530813682_set_a @ B4 @ C2 ) @ A )
= ( ( ord_le5722252365846178494_set_a @ B4 @ A )
& ( ord_le5722252365846178494_set_a @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_401_sup_Obounded__iff,axiom,
! [B4: a > $o,C2: a > $o,A: a > $o] :
( ( ord_less_eq_a_o @ ( sup_sup_a_o @ B4 @ C2 ) @ A )
= ( ( ord_less_eq_a_o @ B4 @ A )
& ( ord_less_eq_a_o @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_402_sup_Obounded__iff,axiom,
! [B4: set_a,C2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B4 @ C2 ) @ A )
= ( ( ord_less_eq_set_a @ B4 @ A )
& ( ord_less_eq_set_a @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_403_inf__bot__left,axiom,
! [X: set_set_a] :
( ( inf_inf_set_set_a @ bot_bot_set_set_a @ X )
= bot_bot_set_set_a ) ).
% inf_bot_left
thf(fact_404_inf__bot__left,axiom,
! [X: a > $o] :
( ( inf_inf_a_o @ bot_bot_a_o @ X )
= bot_bot_a_o ) ).
% inf_bot_left
thf(fact_405_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_406_inf__bot__right,axiom,
! [X: set_set_a] :
( ( inf_inf_set_set_a @ X @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% inf_bot_right
thf(fact_407_inf__bot__right,axiom,
! [X: a > $o] :
( ( inf_inf_a_o @ X @ bot_bot_a_o )
= bot_bot_a_o ) ).
% inf_bot_right
thf(fact_408_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_409_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_set_a] :
( ( inf_inf_set_set_a @ bot_bot_set_set_a @ X )
= bot_bot_set_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_410_boolean__algebra_Oconj__zero__left,axiom,
! [X: a > $o] :
( ( inf_inf_a_o @ bot_bot_a_o @ X )
= bot_bot_a_o ) ).
% boolean_algebra.conj_zero_left
thf(fact_411_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_412_bot__set__def,axiom,
( bot_bot_set_set_a
= ( collect_set_a @ bot_bot_set_a_o ) ) ).
% bot_set_def
thf(fact_413_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_414_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_415_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_416_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_417_inf__sup__aci_I3_J,axiom,
! [X: set_set_a,Y: set_set_a,Z2: set_set_a] :
( ( inf_inf_set_set_a @ X @ ( inf_inf_set_set_a @ Y @ Z2 ) )
= ( inf_inf_set_set_a @ Y @ ( inf_inf_set_set_a @ X @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_418_inf__sup__aci_I3_J,axiom,
! [X: a > $o,Y: a > $o,Z2: a > $o] :
( ( inf_inf_a_o @ X @ ( inf_inf_a_o @ Y @ Z2 ) )
= ( inf_inf_a_o @ Y @ ( inf_inf_a_o @ X @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_419_inf__sup__aci_I3_J,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_420_inf__sup__aci_I2_J,axiom,
! [X: set_set_a,Y: set_set_a,Z2: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ X @ Y ) @ Z2 )
= ( inf_inf_set_set_a @ X @ ( inf_inf_set_set_a @ Y @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_421_inf__sup__aci_I2_J,axiom,
! [X: a > $o,Y: a > $o,Z2: a > $o] :
( ( inf_inf_a_o @ ( inf_inf_a_o @ X @ Y ) @ Z2 )
= ( inf_inf_a_o @ X @ ( inf_inf_a_o @ Y @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_422_inf__sup__aci_I2_J,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z2 )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_423_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_424_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_425_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_426_inf_Oassoc,axiom,
! [A: set_set_a,B4: set_set_a,C2: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ A @ B4 ) @ C2 )
= ( inf_inf_set_set_a @ A @ ( inf_inf_set_set_a @ B4 @ C2 ) ) ) ).
% inf.assoc
thf(fact_427_inf_Oassoc,axiom,
! [A: a > $o,B4: a > $o,C2: a > $o] :
( ( inf_inf_a_o @ ( inf_inf_a_o @ A @ B4 ) @ C2 )
= ( inf_inf_a_o @ A @ ( inf_inf_a_o @ B4 @ C2 ) ) ) ).
% inf.assoc
thf(fact_428_inf_Oassoc,axiom,
! [A: set_a,B4: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B4 ) @ C2 )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B4 @ C2 ) ) ) ).
% inf.assoc
thf(fact_429_inf__assoc,axiom,
! [X: set_set_a,Y: set_set_a,Z2: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ X @ Y ) @ Z2 )
= ( inf_inf_set_set_a @ X @ ( inf_inf_set_set_a @ Y @ Z2 ) ) ) ).
% inf_assoc
thf(fact_430_inf__assoc,axiom,
! [X: a > $o,Y: a > $o,Z2: a > $o] :
( ( inf_inf_a_o @ ( inf_inf_a_o @ X @ Y ) @ Z2 )
= ( inf_inf_a_o @ X @ ( inf_inf_a_o @ Y @ Z2 ) ) ) ).
% inf_assoc
thf(fact_431_inf__assoc,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z2 )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) ) ) ).
% inf_assoc
thf(fact_432_inf_Ocommute,axiom,
( inf_inf_set_set_a
= ( ^ [A4: set_set_a,B3: set_set_a] : ( inf_inf_set_set_a @ B3 @ A4 ) ) ) ).
% inf.commute
thf(fact_433_inf_Ocommute,axiom,
( inf_inf_a_o
= ( ^ [A4: a > $o,B3: a > $o] : ( inf_inf_a_o @ B3 @ A4 ) ) ) ).
% inf.commute
thf(fact_434_inf_Ocommute,axiom,
( inf_inf_set_a
= ( ^ [A4: set_a,B3: set_a] : ( inf_inf_set_a @ B3 @ A4 ) ) ) ).
% inf.commute
thf(fact_435_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_436_inf__commute,axiom,
( inf_inf_a_o
= ( ^ [X3: a > $o,Y3: a > $o] : ( inf_inf_a_o @ Y3 @ X3 ) ) ) ).
% inf_commute
thf(fact_437_inf__commute,axiom,
( inf_inf_set_a
= ( ^ [X3: set_a,Y3: set_a] : ( inf_inf_set_a @ Y3 @ X3 ) ) ) ).
% inf_commute
thf(fact_438_boolean__algebra__cancel_Oinf1,axiom,
! [A2: set_set_a,K: set_set_a,A: set_set_a,B4: set_set_a] :
( ( A2
= ( inf_inf_set_set_a @ K @ A ) )
=> ( ( inf_inf_set_set_a @ A2 @ B4 )
= ( inf_inf_set_set_a @ K @ ( inf_inf_set_set_a @ A @ B4 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_439_boolean__algebra__cancel_Oinf1,axiom,
! [A2: a > $o,K: a > $o,A: a > $o,B4: a > $o] :
( ( A2
= ( inf_inf_a_o @ K @ A ) )
=> ( ( inf_inf_a_o @ A2 @ B4 )
= ( inf_inf_a_o @ K @ ( inf_inf_a_o @ A @ B4 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_440_boolean__algebra__cancel_Oinf1,axiom,
! [A2: set_a,K: set_a,A: set_a,B4: set_a] :
( ( A2
= ( inf_inf_set_a @ K @ A ) )
=> ( ( inf_inf_set_a @ A2 @ B4 )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A @ B4 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_441_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_set_a,K: set_set_a,B4: set_set_a,A: set_set_a] :
( ( B
= ( inf_inf_set_set_a @ K @ B4 ) )
=> ( ( inf_inf_set_set_a @ A @ B )
= ( inf_inf_set_set_a @ K @ ( inf_inf_set_set_a @ A @ B4 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_442_boolean__algebra__cancel_Oinf2,axiom,
! [B: a > $o,K: a > $o,B4: a > $o,A: a > $o] :
( ( B
= ( inf_inf_a_o @ K @ B4 ) )
=> ( ( inf_inf_a_o @ A @ B )
= ( inf_inf_a_o @ K @ ( inf_inf_a_o @ A @ B4 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_443_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_a,K: set_a,B4: set_a,A: set_a] :
( ( B
= ( inf_inf_set_a @ K @ B4 ) )
=> ( ( inf_inf_set_a @ A @ B )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A @ B4 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_444_inf_Oleft__commute,axiom,
! [B4: set_set_a,A: set_set_a,C2: set_set_a] :
( ( inf_inf_set_set_a @ B4 @ ( inf_inf_set_set_a @ A @ C2 ) )
= ( inf_inf_set_set_a @ A @ ( inf_inf_set_set_a @ B4 @ C2 ) ) ) ).
% inf.left_commute
thf(fact_445_inf_Oleft__commute,axiom,
! [B4: a > $o,A: a > $o,C2: a > $o] :
( ( inf_inf_a_o @ B4 @ ( inf_inf_a_o @ A @ C2 ) )
= ( inf_inf_a_o @ A @ ( inf_inf_a_o @ B4 @ C2 ) ) ) ).
% inf.left_commute
thf(fact_446_inf_Oleft__commute,axiom,
! [B4: set_a,A: set_a,C2: set_a] :
( ( inf_inf_set_a @ B4 @ ( inf_inf_set_a @ A @ C2 ) )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B4 @ C2 ) ) ) ).
% inf.left_commute
thf(fact_447_inf__left__commute,axiom,
! [X: set_set_a,Y: set_set_a,Z2: set_set_a] :
( ( inf_inf_set_set_a @ X @ ( inf_inf_set_set_a @ Y @ Z2 ) )
= ( inf_inf_set_set_a @ Y @ ( inf_inf_set_set_a @ X @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_448_inf__left__commute,axiom,
! [X: a > $o,Y: a > $o,Z2: a > $o] :
( ( inf_inf_a_o @ X @ ( inf_inf_a_o @ Y @ Z2 ) )
= ( inf_inf_a_o @ Y @ ( inf_inf_a_o @ X @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_449_inf__left__commute,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_450_inf__fun__def,axiom,
( inf_inf_a_o
= ( ^ [F2: a > $o,G2: a > $o,X3: a] : ( inf_inf_o @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% inf_fun_def
thf(fact_451_inf__sup__aci_I8_J,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( sup_sup_set_set_a @ X @ ( sup_sup_set_set_a @ X @ Y ) )
= ( sup_sup_set_set_a @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_452_inf__sup__aci_I8_J,axiom,
! [X: a > $o,Y: a > $o] :
( ( sup_sup_a_o @ X @ ( sup_sup_a_o @ X @ Y ) )
= ( sup_sup_a_o @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_453_inf__sup__aci_I8_J,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
= ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_454_inf__sup__aci_I7_J,axiom,
! [X: set_set_a,Y: set_set_a,Z2: set_set_a] :
( ( sup_sup_set_set_a @ X @ ( sup_sup_set_set_a @ Y @ Z2 ) )
= ( sup_sup_set_set_a @ Y @ ( sup_sup_set_set_a @ X @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_455_inf__sup__aci_I7_J,axiom,
! [X: a > $o,Y: a > $o,Z2: a > $o] :
( ( sup_sup_a_o @ X @ ( sup_sup_a_o @ Y @ Z2 ) )
= ( sup_sup_a_o @ Y @ ( sup_sup_a_o @ X @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_456_inf__sup__aci_I7_J,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z2 ) )
= ( sup_sup_set_a @ Y @ ( sup_sup_set_a @ X @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_457_inf__sup__aci_I6_J,axiom,
! [X: set_set_a,Y: set_set_a,Z2: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ X @ Y ) @ Z2 )
= ( sup_sup_set_set_a @ X @ ( sup_sup_set_set_a @ Y @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_458_inf__sup__aci_I6_J,axiom,
! [X: a > $o,Y: a > $o,Z2: a > $o] :
( ( sup_sup_a_o @ ( sup_sup_a_o @ X @ Y ) @ Z2 )
= ( sup_sup_a_o @ X @ ( sup_sup_a_o @ Y @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_459_inf__sup__aci_I6_J,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z2 )
= ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_460_inf__sup__aci_I5_J,axiom,
( sup_sup_set_set_a
= ( ^ [X3: set_set_a,Y3: set_set_a] : ( sup_sup_set_set_a @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_461_inf__sup__aci_I5_J,axiom,
( sup_sup_a_o
= ( ^ [X3: a > $o,Y3: a > $o] : ( sup_sup_a_o @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_462_inf__sup__aci_I5_J,axiom,
( sup_sup_set_a
= ( ^ [X3: set_a,Y3: set_a] : ( sup_sup_set_a @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_463_sup_Oassoc,axiom,
! [A: set_set_a,B4: set_set_a,C2: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ A @ B4 ) @ C2 )
= ( sup_sup_set_set_a @ A @ ( sup_sup_set_set_a @ B4 @ C2 ) ) ) ).
% sup.assoc
thf(fact_464_sup_Oassoc,axiom,
! [A: a > $o,B4: a > $o,C2: a > $o] :
( ( sup_sup_a_o @ ( sup_sup_a_o @ A @ B4 ) @ C2 )
= ( sup_sup_a_o @ A @ ( sup_sup_a_o @ B4 @ C2 ) ) ) ).
% sup.assoc
thf(fact_465_sup_Oassoc,axiom,
! [A: set_a,B4: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B4 ) @ C2 )
= ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B4 @ C2 ) ) ) ).
% sup.assoc
thf(fact_466_sup__assoc,axiom,
! [X: set_set_a,Y: set_set_a,Z2: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ X @ Y ) @ Z2 )
= ( sup_sup_set_set_a @ X @ ( sup_sup_set_set_a @ Y @ Z2 ) ) ) ).
% sup_assoc
thf(fact_467_sup__assoc,axiom,
! [X: a > $o,Y: a > $o,Z2: a > $o] :
( ( sup_sup_a_o @ ( sup_sup_a_o @ X @ Y ) @ Z2 )
= ( sup_sup_a_o @ X @ ( sup_sup_a_o @ Y @ Z2 ) ) ) ).
% sup_assoc
thf(fact_468_sup__assoc,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z2 )
= ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z2 ) ) ) ).
% sup_assoc
thf(fact_469_sup_Ocommute,axiom,
( sup_sup_set_set_a
= ( ^ [A4: set_set_a,B3: set_set_a] : ( sup_sup_set_set_a @ B3 @ A4 ) ) ) ).
% sup.commute
thf(fact_470_sup_Ocommute,axiom,
( sup_sup_a_o
= ( ^ [A4: a > $o,B3: a > $o] : ( sup_sup_a_o @ B3 @ A4 ) ) ) ).
% sup.commute
thf(fact_471_sup_Ocommute,axiom,
( sup_sup_set_a
= ( ^ [A4: set_a,B3: set_a] : ( sup_sup_set_a @ B3 @ A4 ) ) ) ).
% sup.commute
thf(fact_472_sup__commute,axiom,
( sup_sup_set_set_a
= ( ^ [X3: set_set_a,Y3: set_set_a] : ( sup_sup_set_set_a @ Y3 @ X3 ) ) ) ).
% sup_commute
thf(fact_473_sup__commute,axiom,
( sup_sup_a_o
= ( ^ [X3: a > $o,Y3: a > $o] : ( sup_sup_a_o @ Y3 @ X3 ) ) ) ).
% sup_commute
thf(fact_474_sup__commute,axiom,
( sup_sup_set_a
= ( ^ [X3: set_a,Y3: set_a] : ( sup_sup_set_a @ Y3 @ X3 ) ) ) ).
% sup_commute
thf(fact_475_boolean__algebra__cancel_Osup1,axiom,
! [A2: set_set_a,K: set_set_a,A: set_set_a,B4: set_set_a] :
( ( A2
= ( sup_sup_set_set_a @ K @ A ) )
=> ( ( sup_sup_set_set_a @ A2 @ B4 )
= ( sup_sup_set_set_a @ K @ ( sup_sup_set_set_a @ A @ B4 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_476_boolean__algebra__cancel_Osup1,axiom,
! [A2: a > $o,K: a > $o,A: a > $o,B4: a > $o] :
( ( A2
= ( sup_sup_a_o @ K @ A ) )
=> ( ( sup_sup_a_o @ A2 @ B4 )
= ( sup_sup_a_o @ K @ ( sup_sup_a_o @ A @ B4 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_477_boolean__algebra__cancel_Osup1,axiom,
! [A2: set_a,K: set_a,A: set_a,B4: set_a] :
( ( A2
= ( sup_sup_set_a @ K @ A ) )
=> ( ( sup_sup_set_a @ A2 @ B4 )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A @ B4 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_478_boolean__algebra__cancel_Osup2,axiom,
! [B: set_set_a,K: set_set_a,B4: set_set_a,A: set_set_a] :
( ( B
= ( sup_sup_set_set_a @ K @ B4 ) )
=> ( ( sup_sup_set_set_a @ A @ B )
= ( sup_sup_set_set_a @ K @ ( sup_sup_set_set_a @ A @ B4 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_479_boolean__algebra__cancel_Osup2,axiom,
! [B: a > $o,K: a > $o,B4: a > $o,A: a > $o] :
( ( B
= ( sup_sup_a_o @ K @ B4 ) )
=> ( ( sup_sup_a_o @ A @ B )
= ( sup_sup_a_o @ K @ ( sup_sup_a_o @ A @ B4 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_480_boolean__algebra__cancel_Osup2,axiom,
! [B: set_a,K: set_a,B4: set_a,A: set_a] :
( ( B
= ( sup_sup_set_a @ K @ B4 ) )
=> ( ( sup_sup_set_a @ A @ B )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A @ B4 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_481_sup_Oleft__commute,axiom,
! [B4: set_set_a,A: set_set_a,C2: set_set_a] :
( ( sup_sup_set_set_a @ B4 @ ( sup_sup_set_set_a @ A @ C2 ) )
= ( sup_sup_set_set_a @ A @ ( sup_sup_set_set_a @ B4 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_482_sup_Oleft__commute,axiom,
! [B4: a > $o,A: a > $o,C2: a > $o] :
( ( sup_sup_a_o @ B4 @ ( sup_sup_a_o @ A @ C2 ) )
= ( sup_sup_a_o @ A @ ( sup_sup_a_o @ B4 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_483_sup_Oleft__commute,axiom,
! [B4: set_a,A: set_a,C2: set_a] :
( ( sup_sup_set_a @ B4 @ ( sup_sup_set_a @ A @ C2 ) )
= ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B4 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_484_sup__left__commute,axiom,
! [X: set_set_a,Y: set_set_a,Z2: set_set_a] :
( ( sup_sup_set_set_a @ X @ ( sup_sup_set_set_a @ Y @ Z2 ) )
= ( sup_sup_set_set_a @ Y @ ( sup_sup_set_set_a @ X @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_485_sup__left__commute,axiom,
! [X: a > $o,Y: a > $o,Z2: a > $o] :
( ( sup_sup_a_o @ X @ ( sup_sup_a_o @ Y @ Z2 ) )
= ( sup_sup_a_o @ Y @ ( sup_sup_a_o @ X @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_486_sup__left__commute,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z2 ) )
= ( sup_sup_set_a @ Y @ ( sup_sup_set_a @ X @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_487_sup__fun__def,axiom,
( sup_sup_a_o
= ( ^ [F2: a > $o,G2: a > $o,X3: a] : ( sup_sup_o @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% sup_fun_def
thf(fact_488_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_489_inf__sup__ord_I2_J,axiom,
! [X: set_set_set_a,Y: set_set_set_a] : ( ord_le5722252365846178494_set_a @ ( inf_in1205276777018777868_set_a @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_490_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_491_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_492_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_493_inf__sup__ord_I1_J,axiom,
! [X: set_set_set_a,Y: set_set_set_a] : ( ord_le5722252365846178494_set_a @ ( inf_in1205276777018777868_set_a @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_494_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_495_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_496_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_497_inf__le1,axiom,
! [X: set_set_set_a,Y: set_set_set_a] : ( ord_le5722252365846178494_set_a @ ( inf_in1205276777018777868_set_a @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_498_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_499_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_500_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_501_inf__le2,axiom,
! [X: set_set_set_a,Y: set_set_set_a] : ( ord_le5722252365846178494_set_a @ ( inf_in1205276777018777868_set_a @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_502_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_503_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_504_le__infE,axiom,
! [X: set_set_a,A: set_set_a,B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ ( inf_inf_set_set_a @ A @ B4 ) )
=> ~ ( ( ord_le3724670747650509150_set_a @ X @ A )
=> ~ ( ord_le3724670747650509150_set_a @ X @ B4 ) ) ) ).
% le_infE
thf(fact_505_le__infE,axiom,
! [X: set_set_set_a,A: set_set_set_a,B4: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ X @ ( inf_in1205276777018777868_set_a @ A @ B4 ) )
=> ~ ( ( ord_le5722252365846178494_set_a @ X @ A )
=> ~ ( ord_le5722252365846178494_set_a @ X @ B4 ) ) ) ).
% le_infE
thf(fact_506_le__infE,axiom,
! [X: a > $o,A: a > $o,B4: a > $o] :
( ( ord_less_eq_a_o @ X @ ( inf_inf_a_o @ A @ B4 ) )
=> ~ ( ( ord_less_eq_a_o @ X @ A )
=> ~ ( ord_less_eq_a_o @ X @ B4 ) ) ) ).
% le_infE
thf(fact_507_le__infE,axiom,
! [X: set_a,A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A @ B4 ) )
=> ~ ( ( ord_less_eq_set_a @ X @ A )
=> ~ ( ord_less_eq_set_a @ X @ B4 ) ) ) ).
% le_infE
thf(fact_508_le__infI,axiom,
! [X: set_a,A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ X @ A )
=> ( ( ord_less_eq_set_a @ X @ B4 )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A @ B4 ) ) ) ) ).
% le_infI
thf(fact_509_inf__mono,axiom,
! [A: set_a,C2: set_a,B4: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B4 @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B4 ) @ ( inf_inf_set_a @ C2 @ D2 ) ) ) ) ).
% inf_mono
thf(fact_510_le__infI1,axiom,
! [A: set_a,X: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B4 ) @ X ) ) ).
% le_infI1
thf(fact_511_le__infI2,axiom,
! [B4: set_a,X: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B4 @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B4 ) @ X ) ) ).
% le_infI2
thf(fact_512_inf_OorderE,axiom,
! [A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( A
= ( inf_inf_set_a @ A @ B4 ) ) ) ).
% inf.orderE
thf(fact_513_inf_OorderI,axiom,
! [A: set_a,B4: set_a] :
( ( A
= ( inf_inf_set_a @ A @ B4 ) )
=> ( ord_less_eq_set_a @ A @ B4 ) ) ).
% inf.orderI
thf(fact_514_inf__unique,axiom,
! [F3: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X2: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( F3 @ X2 @ Y2 ) @ X2 )
=> ( ! [X2: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( F3 @ X2 @ Y2 ) @ Y2 )
=> ( ! [X2: set_a,Y2: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ( ord_less_eq_set_a @ X2 @ Z3 )
=> ( ord_less_eq_set_a @ X2 @ ( F3 @ Y2 @ Z3 ) ) ) )
=> ( ( inf_inf_set_a @ X @ Y )
= ( F3 @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_515_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_516_inf_Oabsorb1,axiom,
! [A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( inf_inf_set_a @ A @ B4 )
= A ) ) ).
% inf.absorb1
thf(fact_517_inf_Oabsorb2,axiom,
! [B4: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( ( inf_inf_set_a @ A @ B4 )
= B4 ) ) ).
% inf.absorb2
thf(fact_518_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_519_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_520_inf_OboundedE,axiom,
! [A: set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B4 @ C2 ) )
=> ~ ( ( ord_less_eq_set_a @ A @ B4 )
=> ~ ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% inf.boundedE
thf(fact_521_inf_OboundedI,axiom,
! [A: set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ord_less_eq_set_a @ A @ C2 )
=> ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B4 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_522_inf__greatest,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ X @ Z2 )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_523_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
( A4
= ( inf_inf_set_a @ A4 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_524_inf_Ocobounded1,axiom,
! [A: set_a,B4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B4 ) @ A ) ).
% inf.cobounded1
thf(fact_525_inf_Ocobounded2,axiom,
! [A: set_a,B4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B4 ) @ B4 ) ).
% inf.cobounded2
thf(fact_526_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( inf_inf_set_a @ A4 @ B3 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_527_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A4: set_a] :
( ( inf_inf_set_a @ A4 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_528_inf_OcoboundedI1,axiom,
! [A: set_a,C2: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B4 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_529_inf_OcoboundedI2,axiom,
! [B4: set_a,C2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B4 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_530_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_531_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_532_le__supE,axiom,
! [A: set_a,B4: set_a,X: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B4 ) @ X )
=> ~ ( ( ord_less_eq_set_a @ A @ X )
=> ~ ( ord_less_eq_set_a @ B4 @ X ) ) ) ).
% le_supE
thf(fact_533_le__supI,axiom,
! [A: set_a,X: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ X )
=> ( ( ord_less_eq_set_a @ B4 @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B4 ) @ X ) ) ) ).
% le_supI
thf(fact_534_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_535_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_536_le__supI1,axiom,
! [X: set_a,A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ X @ A )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A @ B4 ) ) ) ).
% le_supI1
thf(fact_537_le__supI2,axiom,
! [X: set_a,B4: set_a,A: set_a] :
( ( ord_less_eq_set_a @ X @ B4 )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A @ B4 ) ) ) ).
% le_supI2
thf(fact_538_sup_Omono,axiom,
! [C2: set_a,A: set_a,D2: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ C2 @ A )
=> ( ( ord_less_eq_set_a @ D2 @ B4 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ C2 @ D2 ) @ ( sup_sup_set_a @ A @ B4 ) ) ) ) ).
% sup.mono
thf(fact_539_sup__mono,axiom,
! [A: set_a,C2: set_a,B4: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B4 @ D2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B4 ) @ ( sup_sup_set_a @ C2 @ D2 ) ) ) ) ).
% sup_mono
thf(fact_540_sup__least,axiom,
! [Y: set_a,X: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ Z2 @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_541_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_542_sup_OorderE,axiom,
! [B4: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( A
= ( sup_sup_set_a @ A @ B4 ) ) ) ).
% sup.orderE
thf(fact_543_sup_OorderI,axiom,
! [A: set_a,B4: set_a] :
( ( A
= ( sup_sup_set_a @ A @ B4 ) )
=> ( ord_less_eq_set_a @ B4 @ A ) ) ).
% sup.orderI
thf(fact_544_sup__unique,axiom,
! [F3: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X2: set_a,Y2: set_a] : ( ord_less_eq_set_a @ X2 @ ( F3 @ X2 @ Y2 ) )
=> ( ! [X2: set_a,Y2: set_a] : ( ord_less_eq_set_a @ Y2 @ ( F3 @ X2 @ Y2 ) )
=> ( ! [X2: set_a,Y2: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X2 )
=> ( ( ord_less_eq_set_a @ Z3 @ X2 )
=> ( ord_less_eq_set_a @ ( F3 @ Y2 @ Z3 ) @ X2 ) ) )
=> ( ( sup_sup_set_a @ X @ Y )
= ( F3 @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_545_sup_Oabsorb1,axiom,
! [B4: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( ( sup_sup_set_a @ A @ B4 )
= A ) ) ).
% sup.absorb1
thf(fact_546_sup_Oabsorb2,axiom,
! [A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( sup_sup_set_a @ A @ B4 )
= B4 ) ) ).
% sup.absorb2
thf(fact_547_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_548_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_549_sup_OboundedE,axiom,
! [B4: set_a,C2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B4 @ C2 ) @ A )
=> ~ ( ( ord_less_eq_set_a @ B4 @ A )
=> ~ ( ord_less_eq_set_a @ C2 @ A ) ) ) ).
% sup.boundedE
thf(fact_550_sup_OboundedI,axiom,
! [B4: set_a,A: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( ( ord_less_eq_set_a @ C2 @ A )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ B4 @ C2 ) @ A ) ) ) ).
% sup.boundedI
thf(fact_551_sup_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A4: set_a] :
( A4
= ( sup_sup_set_a @ A4 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_552_sup_Ocobounded1,axiom,
! [A: set_a,B4: set_a] : ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ A @ B4 ) ) ).
% sup.cobounded1
thf(fact_553_sup_Ocobounded2,axiom,
! [B4: set_a,A: set_a] : ( ord_less_eq_set_a @ B4 @ ( sup_sup_set_a @ A @ B4 ) ) ).
% sup.cobounded2
thf(fact_554_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A4: set_a] :
( ( sup_sup_set_a @ A4 @ B3 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_555_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( sup_sup_set_a @ A4 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_556_sup_OcoboundedI1,axiom,
! [C2: set_a,A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ C2 @ A )
=> ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A @ B4 ) ) ) ).
% sup.coboundedI1
thf(fact_557_sup_OcoboundedI2,axiom,
! [C2: set_a,B4: set_a,A: set_a] :
( ( ord_less_eq_set_a @ C2 @ B4 )
=> ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A @ B4 ) ) ) ).
% sup.coboundedI2
thf(fact_558_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ bot_bot_set_a )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_559_distrib__imp1,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ! [X2: set_a,Y2: set_a,Z3: set_a] :
( ( inf_inf_set_a @ X2 @ ( sup_sup_set_a @ Y2 @ Z3 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X2 @ Y2 ) @ ( inf_inf_set_a @ X2 @ Z3 ) ) )
=> ( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z2 ) ) ) ) ).
% distrib_imp1
thf(fact_560_distrib__imp2,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ! [X2: set_a,Y2: set_a,Z3: set_a] :
( ( sup_sup_set_a @ X2 @ ( inf_inf_set_a @ Y2 @ Z3 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X2 @ Y2 ) @ ( sup_sup_set_a @ X2 @ Z3 ) ) )
=> ( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z2 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z2 ) ) ) ) ).
% distrib_imp2
thf(fact_561_inf__sup__distrib1,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z2 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z2 ) ) ) ).
% inf_sup_distrib1
thf(fact_562_inf__sup__distrib2,axiom,
! [Y: set_a,Z2: set_a,X: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ Z2 ) @ X )
= ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ X ) @ ( inf_inf_set_a @ Z2 @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_563_sup__inf__distrib1,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z2 ) ) ) ).
% sup_inf_distrib1
thf(fact_564_sup__inf__distrib2,axiom,
! [Y: set_a,Z2: set_a,X: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ Z2 ) @ X )
= ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ X ) @ ( sup_sup_set_a @ Z2 @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_565_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z2 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z2 ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_566_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z2 ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_567_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_a,Z2: set_a,X: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ Z2 ) @ X )
= ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ X ) @ ( inf_inf_set_a @ Z2 @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_568_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_a,Z2: set_a,X: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ Z2 ) @ X )
= ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ X ) @ ( sup_sup_set_a @ Z2 @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_569_distrib__inf__le,axiom,
! [X: set_a,Y: set_a,Z2: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z2 ) ) @ ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_570_distrib__sup__le,axiom,
! [X: set_a,Y: set_a,Z2: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) ) @ ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_571_sumset__subset__insert_I1_J,axiom,
! [A2: set_a,B: set_a,X: a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ X @ B ) ) ) ).
% sumset_subset_insert(1)
thf(fact_572_sumset__subset__insert_I2_J,axiom,
! [A2: set_a,B: set_a,X: a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ ( insert_a @ X @ A2 ) @ B ) ) ).
% sumset_subset_insert(2)
thf(fact_573_order__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% order_refl
thf(fact_574_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_575_sumset__insert2,axiom,
! [B: set_a,A2: set_a,X: a] :
( ( nO_MATCH_set_a_set_a @ bot_bot_set_a @ B )
=> ( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ X @ B ) )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ) ) ).
% sumset_insert2
thf(fact_576_sumset__insert1,axiom,
! [A2: set_a,X: a,B: set_a] :
( ( nO_MATCH_set_a_set_a @ bot_bot_set_a @ A2 )
=> ( ( pluenn3038260743871226533mset_a @ g @ addition @ ( insert_a @ X @ A2 ) @ B )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ ( insert_a @ X @ bot_bot_set_a ) @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ) ) ).
% sumset_insert1
thf(fact_577_subset__emptyI,axiom,
! [A2: set_a] :
( ! [X2: a] :
~ ( member_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ A2 @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_578_bot_Oextremum,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% bot.extremum
thf(fact_579_insertCI,axiom,
! [A: a,B: set_a,B4: a] :
( ( ~ ( member_a @ A @ B )
=> ( A = B4 ) )
=> ( member_a @ A @ ( insert_a @ B4 @ B ) ) ) ).
% insertCI
thf(fact_580_insert__iff,axiom,
! [A: a,B4: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B4 @ A2 ) )
= ( ( A = B4 )
| ( member_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_581_insert__absorb2,axiom,
! [X: a,A2: set_a] :
( ( insert_a @ X @ ( insert_a @ X @ A2 ) )
= ( insert_a @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_582_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_583_finite__insert,axiom,
! [A: a,A2: set_a] :
( ( finite_finite_a @ ( insert_a @ A @ A2 ) )
= ( finite_finite_a @ A2 ) ) ).
% finite_insert
thf(fact_584_insert__subset,axiom,
! [X: a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A2 ) @ B )
= ( ( member_a @ X @ B )
& ( ord_less_eq_set_a @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_585_Int__insert__right__if1,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_586_Int__insert__right__if0,axiom,
! [A: a,A2: set_a,B: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( inf_inf_set_a @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_587_insert__inter__insert,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ ( insert_a @ A @ B ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B ) ) ) ).
% insert_inter_insert
thf(fact_588_Int__insert__left__if1,axiom,
! [A: a,C: set_a,B: set_a] :
( ( member_a @ A @ C )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C )
= ( insert_a @ A @ ( inf_inf_set_a @ B @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_589_Int__insert__left__if0,axiom,
! [A: a,C: set_a,B: set_a] :
( ~ ( member_a @ A @ C )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C )
= ( inf_inf_set_a @ B @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_590_Un__insert__right,axiom,
! [A2: set_a,A: a,B: set_a] :
( ( sup_sup_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( insert_a @ A @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_591_Un__insert__left,axiom,
! [A: a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ ( insert_a @ A @ B ) @ C )
= ( insert_a @ A @ ( sup_sup_set_a @ B @ C ) ) ) ).
% Un_insert_left
thf(fact_592_singleton__insert__inj__eq,axiom,
! [B4: a,A: a,A2: set_a] :
( ( ( insert_a @ B4 @ bot_bot_set_a )
= ( insert_a @ A @ A2 ) )
= ( ( A = B4 )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B4 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_593_singleton__insert__inj__eq_H,axiom,
! [A: a,A2: set_a,B4: a] :
( ( ( insert_a @ A @ A2 )
= ( insert_a @ B4 @ bot_bot_set_a ) )
= ( ( A = B4 )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B4 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_594_disjoint__insert_I2_J,axiom,
! [A2: set_a,B4: a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ ( insert_a @ B4 @ B ) ) )
= ( ~ ( member_a @ B4 @ A2 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_595_disjoint__insert_I1_J,axiom,
! [B: set_a,A: a,A2: set_a] :
( ( ( inf_inf_set_a @ B @ ( insert_a @ A @ A2 ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B )
& ( ( inf_inf_set_a @ B @ A2 )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_596_insert__disjoint_I2_J,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B ) )
= ( ~ ( member_a @ A @ B )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_597_insert__disjoint_I1_J,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B )
& ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_598_insert__subsetI,axiom,
! [X: a,A2: set_a,X4: set_a] :
( ( member_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ X4 @ A2 )
=> ( ord_less_eq_set_a @ ( insert_a @ X @ X4 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_599_insertE,axiom,
! [A: a,B4: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B4 @ A2 ) )
=> ( ( A != B4 )
=> ( member_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_600_insertI1,axiom,
! [A: a,B: set_a] : ( member_a @ A @ ( insert_a @ A @ B ) ) ).
% insertI1
thf(fact_601_insertI2,axiom,
! [A: a,B: set_a,B4: a] :
( ( member_a @ A @ B )
=> ( member_a @ A @ ( insert_a @ B4 @ B ) ) ) ).
% insertI2
thf(fact_602_Set_Oset__insert,axiom,
! [X: a,A2: set_a] :
( ( member_a @ X @ A2 )
=> ~ ! [B8: set_a] :
( ( A2
= ( insert_a @ X @ B8 ) )
=> ( member_a @ X @ B8 ) ) ) ).
% Set.set_insert
thf(fact_603_insert__ident,axiom,
! [X: a,A2: set_a,B: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ~ ( member_a @ X @ B )
=> ( ( ( insert_a @ X @ A2 )
= ( insert_a @ X @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_604_insert__absorb,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_605_insert__eq__iff,axiom,
! [A: a,A2: set_a,B4: a,B: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ~ ( member_a @ B4 @ B )
=> ( ( ( insert_a @ A @ A2 )
= ( insert_a @ B4 @ B ) )
= ( ( ( A = B4 )
=> ( A2 = B ) )
& ( ( A != B4 )
=> ? [C3: set_a] :
( ( A2
= ( insert_a @ B4 @ C3 ) )
& ~ ( member_a @ B4 @ C3 )
& ( B
= ( insert_a @ A @ C3 ) )
& ~ ( member_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_606_insert__commute,axiom,
! [X: a,Y: a,A2: set_a] :
( ( insert_a @ X @ ( insert_a @ Y @ A2 ) )
= ( insert_a @ Y @ ( insert_a @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_607_mk__disjoint__insert,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ? [B8: set_a] :
( ( A2
= ( insert_a @ A @ B8 ) )
& ~ ( member_a @ A @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_608_singleton__inject,axiom,
! [A: a,B4: a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B4 @ bot_bot_set_a ) )
=> ( A = B4 ) ) ).
% singleton_inject
thf(fact_609_insert__not__empty,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ A2 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_610_doubleton__eq__iff,axiom,
! [A: a,B4: a,C2: a,D2: a] :
( ( ( insert_a @ A @ ( insert_a @ B4 @ bot_bot_set_a ) )
= ( insert_a @ C2 @ ( insert_a @ D2 @ bot_bot_set_a ) ) )
= ( ( ( A = C2 )
& ( B4 = D2 ) )
| ( ( A = D2 )
& ( B4 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_611_singleton__iff,axiom,
! [B4: a,A: a] :
( ( member_a @ B4 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B4 = A ) ) ).
% singleton_iff
thf(fact_612_singletonD,axiom,
! [B4: a,A: a] :
( ( member_a @ B4 @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B4 = A ) ) ).
% singletonD
thf(fact_613_finite_OinsertI,axiom,
! [A2: set_a,A: a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( insert_a @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_614_subset__insertI2,axiom,
! [A2: set_a,B: set_a,B4: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ B4 @ B ) ) ) ).
% subset_insertI2
thf(fact_615_subset__insertI,axiom,
! [B: set_a,A: a] : ( ord_less_eq_set_a @ B @ ( insert_a @ A @ B ) ) ).
% subset_insertI
thf(fact_616_subset__insert,axiom,
! [X: a,A2: set_a,B: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B ) )
= ( ord_less_eq_set_a @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_617_insert__mono,axiom,
! [C: set_a,D: set_a,A: a] :
( ( ord_less_eq_set_a @ C @ D )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C ) @ ( insert_a @ A @ D ) ) ) ).
% insert_mono
thf(fact_618_Int__insert__right,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B ) ) ) )
& ( ~ ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_619_Int__insert__left,axiom,
! [A: a,C: set_a,B: set_a] :
( ( ( member_a @ A @ C )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C )
= ( insert_a @ A @ ( inf_inf_set_a @ B @ C ) ) ) )
& ( ~ ( member_a @ A @ C )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C )
= ( inf_inf_set_a @ B @ C ) ) ) ) ).
% Int_insert_left
thf(fact_620_infinite__finite__induct,axiom,
! [P: set_a > $o,A2: set_a] :
( ! [A8: set_a] :
( ~ ( finite_finite_a @ A8 )
=> ( P @ A8 ) )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X2: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ~ ( member_a @ X2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ X2 @ F4 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_621_finite__ne__induct,axiom,
! [F: set_a,P: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( F != bot_bot_set_a )
=> ( ! [X2: a] : ( P @ ( insert_a @ X2 @ bot_bot_set_a ) )
=> ( ! [X2: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ( F4 != bot_bot_set_a )
=> ( ~ ( member_a @ X2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ X2 @ F4 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_ne_induct
thf(fact_622_finite__induct,axiom,
! [F: set_a,P: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X2: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ~ ( member_a @ X2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ X2 @ F4 ) ) ) ) )
=> ( P @ F ) ) ) ) ).
% finite_induct
thf(fact_623_finite_Osimps,axiom,
( finite_finite_a
= ( ^ [A4: set_a] :
( ( A4 = bot_bot_set_a )
| ? [A6: set_a,B3: a] :
( ( A4
= ( insert_a @ B3 @ A6 ) )
& ( finite_finite_a @ A6 ) ) ) ) ) ).
% finite.simps
thf(fact_624_finite_Ocases,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( ( A != bot_bot_set_a )
=> ~ ! [A8: set_a] :
( ? [A3: a] :
( A
= ( insert_a @ A3 @ A8 ) )
=> ~ ( finite_finite_a @ A8 ) ) ) ) ).
% finite.cases
thf(fact_625_subset__singletonD,axiom,
! [A2: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) )
=> ( ( A2 = bot_bot_set_a )
| ( A2
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_626_subset__singleton__iff,axiom,
! [X4: set_a,A: a] :
( ( ord_less_eq_set_a @ X4 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( ( X4 = bot_bot_set_a )
| ( X4
= ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_627_insert__is__Un,axiom,
( insert_a
= ( ^ [A4: a] : ( sup_sup_set_a @ ( insert_a @ A4 @ bot_bot_set_a ) ) ) ) ).
% insert_is_Un
thf(fact_628_Un__singleton__iff,axiom,
! [A2: set_a,B: set_a,X: a] :
( ( ( sup_sup_set_a @ A2 @ B )
= ( insert_a @ X @ bot_bot_set_a ) )
= ( ( ( A2 = bot_bot_set_a )
& ( B
= ( insert_a @ X @ bot_bot_set_a ) ) )
| ( ( A2
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B = bot_bot_set_a ) )
| ( ( A2
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_629_singleton__Un__iff,axiom,
! [X: a,A2: set_a,B: set_a] :
( ( ( insert_a @ X @ bot_bot_set_a )
= ( sup_sup_set_a @ A2 @ B ) )
= ( ( ( A2 = bot_bot_set_a )
& ( B
= ( insert_a @ X @ bot_bot_set_a ) ) )
| ( ( A2
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B = bot_bot_set_a ) )
| ( ( A2
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_630_finite__subset__induct_H,axiom,
! [F: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( ord_less_eq_set_a @ F @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A3: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ( member_a @ A3 @ A2 )
=> ( ( ord_less_eq_set_a @ F4 @ A2 )
=> ( ~ ( member_a @ A3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ A3 @ F4 ) ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_631_finite__subset__induct,axiom,
! [F: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( ord_less_eq_set_a @ F @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A3: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ( member_a @ A3 @ A2 )
=> ( ~ ( member_a @ A3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ A3 @ F4 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct
thf(fact_632_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_633_ord__le__eq__subst,axiom,
! [A: set_a,B4: set_a,F3: set_a > set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ( F3 @ B4 )
= C2 )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F3 @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_634_ord__eq__le__subst,axiom,
! [A: set_a,F3: set_a > set_a,B4: set_a,C2: set_a] :
( ( A
= ( F3 @ B4 ) )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F3 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_635_order__eq__refl,axiom,
! [X: set_a,Y: set_a] :
( ( X = Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_636_order__subst2,axiom,
! [A: set_a,B4: set_a,F3: set_a > set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ord_less_eq_set_a @ ( F3 @ B4 ) @ C2 )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F3 @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_637_order__subst1,axiom,
! [A: set_a,F3: set_a > set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ ( F3 @ B4 ) )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F3 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_638_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_a,Z: set_a] : ( Y4 = Z ) )
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_639_antisym,axiom,
! [A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ A )
=> ( A = B4 ) ) ) ).
% antisym
thf(fact_640_dual__order_Otrans,axiom,
! [B4: set_a,A: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( ( ord_less_eq_set_a @ C2 @ B4 )
=> ( ord_less_eq_set_a @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_641_dual__order_Oantisym,axiom,
! [B4: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( ( ord_less_eq_set_a @ A @ B4 )
=> ( A = B4 ) ) ) ).
% dual_order.antisym
thf(fact_642_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_a,Z: set_a] : ( Y4 = Z ) )
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A4 )
& ( ord_less_eq_set_a @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_643_order__trans,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z2 )
=> ( ord_less_eq_set_a @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_644_order_Otrans,axiom,
! [A: set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% order.trans
thf(fact_645_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_646_ord__le__eq__trans,axiom,
! [A: set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( B4 = C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_647_ord__eq__le__trans,axiom,
! [A: set_a,B4: set_a,C2: set_a] :
( ( A = B4 )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_648_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_a,Z: set_a] : ( Y4 = Z ) )
= ( ^ [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_649_bot_Oextremum__uniqueI,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
=> ( A = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_650_bot_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_651_the__elem__eq,axiom,
! [X: a] :
( ( the_elem_a @ ( insert_a @ X @ bot_bot_set_a ) )
= X ) ).
% the_elem_eq
thf(fact_652_Collect__empty__eq__bot,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( P = bot_bot_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_653_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X3: a] : ( member_a @ X3 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_654_is__singletonI,axiom,
! [X: a] : ( is_singleton_a @ ( insert_a @ X @ bot_bot_set_a ) ) ).
% is_singletonI
thf(fact_655_additive__abelian__group_Osumset__insert2,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,B: set_a,A2: set_a,X: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( nO_MATCH_set_a_set_a @ bot_bot_set_a @ B )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( insert_a @ X @ B ) )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ) ) ).
% additive_abelian_group.sumset_insert2
thf(fact_656_additive__abelian__group_Osumset__insert1,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,X: a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( nO_MATCH_set_a_set_a @ bot_bot_set_a @ A2 )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( insert_a @ X @ A2 ) @ B )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ ( insert_a @ X @ bot_bot_set_a ) @ B ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ) ) ).
% additive_abelian_group.sumset_insert1
thf(fact_657_additive__abelian__group_Osumset__commute,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ B @ A2 ) ) ) ).
% additive_abelian_group.sumset_commute
thf(fact_658_additive__abelian__group_Osumset_OsumsetI,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: a,A2: set_a,B4: a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( member_a @ A @ A2 )
=> ( ( member_a @ A @ G )
=> ( ( member_a @ B4 @ B )
=> ( ( member_a @ B4 @ G )
=> ( member_a @ ( Addition @ A @ B4 ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ) ) ) ) ).
% additive_abelian_group.sumset.sumsetI
thf(fact_659_additive__abelian__group_Osumset__assoc,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a,C: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) @ C )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( pluenn3038260743871226533mset_a @ G @ Addition @ B @ C ) ) ) ) ).
% additive_abelian_group.sumset_assoc
thf(fact_660_additive__abelian__group_Osumset_Osimps,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( member_a @ A @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) )
= ( ? [A4: a,B3: a] :
( ( A
= ( Addition @ A4 @ B3 ) )
& ( member_a @ A4 @ A2 )
& ( member_a @ A4 @ G )
& ( member_a @ B3 @ B )
& ( member_a @ B3 @ G ) ) ) ) ) ).
% additive_abelian_group.sumset.simps
thf(fact_661_additive__abelian__group_Osumset_Ocases,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( member_a @ A @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) )
=> ~ ! [A3: a,B2: a] :
( ( A
= ( Addition @ A3 @ B2 ) )
=> ( ( member_a @ A3 @ A2 )
=> ( ( member_a @ A3 @ G )
=> ( ( member_a @ B2 @ B )
=> ~ ( member_a @ B2 @ G ) ) ) ) ) ) ) ).
% additive_abelian_group.sumset.cases
thf(fact_662_additive__abelian__group_Osumsetp_Ocases,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: a > $o,B: a > $o,A: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn895083305082786853setp_a @ G @ Addition @ A2 @ B @ A )
=> ~ ! [A3: a,B2: a] :
( ( A
= ( Addition @ A3 @ B2 ) )
=> ( ( A2 @ A3 )
=> ( ( member_a @ A3 @ G )
=> ( ( B @ B2 )
=> ~ ( member_a @ B2 @ G ) ) ) ) ) ) ) ).
% additive_abelian_group.sumsetp.cases
thf(fact_663_additive__abelian__group_Osumsetp_Osimps,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: a > $o,B: a > $o,A: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn895083305082786853setp_a @ G @ Addition @ A2 @ B @ A )
= ( ? [A4: a,B3: a] :
( ( A
= ( Addition @ A4 @ B3 ) )
& ( A2 @ A4 )
& ( member_a @ A4 @ G )
& ( B @ B3 )
& ( member_a @ B3 @ G ) ) ) ) ) ).
% additive_abelian_group.sumsetp.simps
thf(fact_664_additive__abelian__group_Osumsetp_OsumsetI,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: a > $o,A: a,B: a > $o,B4: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( A2 @ A )
=> ( ( member_a @ A @ G )
=> ( ( B @ B4 )
=> ( ( member_a @ B4 @ G )
=> ( pluenn895083305082786853setp_a @ G @ Addition @ A2 @ B @ ( Addition @ A @ B4 ) ) ) ) ) ) ) ).
% additive_abelian_group.sumsetp.sumsetI
thf(fact_665_is__singleton__the__elem,axiom,
( is_singleton_a
= ( ^ [A6: set_a] :
( A6
= ( insert_a @ ( the_elem_a @ A6 ) @ bot_bot_set_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_666_additive__abelian__group_Osumset__empty_I1_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ bot_bot_set_a )
= bot_bot_set_a ) ) ).
% additive_abelian_group.sumset_empty(1)
thf(fact_667_additive__abelian__group_Osumset__empty_I2_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ bot_bot_set_a @ A2 )
= bot_bot_set_a ) ) ).
% additive_abelian_group.sumset_empty(2)
thf(fact_668_additive__abelian__group_Ofinite__sumset,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ) ) ).
% additive_abelian_group.finite_sumset
thf(fact_669_additive__abelian__group_Osumset__subset__carrier,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) @ G ) ) ).
% additive_abelian_group.sumset_subset_carrier
thf(fact_670_additive__abelian__group_Osumset__mono,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A5: set_a,A2: set_a,B5: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ord_less_eq_set_a @ A5 @ A2 )
=> ( ( ord_less_eq_set_a @ B5 @ B )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A5 @ B5 ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ) ) ).
% additive_abelian_group.sumset_mono
thf(fact_671_additive__abelian__group_Osumset__Int__carrier,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( inf_inf_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) @ G )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ).
% additive_abelian_group.sumset_Int_carrier
thf(fact_672_additive__abelian__group_Osumset__Int__carrier__eq_I1_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( inf_inf_set_a @ B @ G ) )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ).
% additive_abelian_group.sumset_Int_carrier_eq(1)
thf(fact_673_additive__abelian__group_Osumset__Int__carrier__eq_I2_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( inf_inf_set_a @ A2 @ G ) @ B )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ).
% additive_abelian_group.sumset_Int_carrier_eq(2)
thf(fact_674_additive__abelian__group_Osumset__subset__Un2,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a,B5: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( sup_sup_set_a @ B @ B5 ) )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B5 ) ) ) ) ).
% additive_abelian_group.sumset_subset_Un2
thf(fact_675_additive__abelian__group_Osumset__subset__Un1,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,A5: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( sup_sup_set_a @ A2 @ A5 ) @ B )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A5 @ B ) ) ) ) ).
% additive_abelian_group.sumset_subset_Un1
thf(fact_676_is__singletonI_H,axiom,
! [A2: set_a] :
( ( A2 != bot_bot_set_a )
=> ( ! [X2: a,Y2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( member_a @ Y2 @ A2 )
=> ( X2 = Y2 ) ) )
=> ( is_singleton_a @ A2 ) ) ) ).
% is_singletonI'
thf(fact_677_additive__abelian__group_Osumset__subset__insert_I2_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a,X: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ ( insert_a @ X @ A2 ) @ B ) ) ) ).
% additive_abelian_group.sumset_subset_insert(2)
thf(fact_678_additive__abelian__group_Osumset__subset__insert_I1_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a,X: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( insert_a @ X @ B ) ) ) ) ).
% additive_abelian_group.sumset_subset_insert(1)
thf(fact_679_additive__abelian__group_Osumset__is__empty__iff,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B )
= bot_bot_set_a )
= ( ( ( inf_inf_set_a @ A2 @ G )
= bot_bot_set_a )
| ( ( inf_inf_set_a @ B @ G )
= bot_bot_set_a ) ) ) ) ).
% additive_abelian_group.sumset_is_empty_iff
thf(fact_680_additive__abelian__group_Osumset__empty_H_I1_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ( inf_inf_set_a @ A2 @ G )
= bot_bot_set_a )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ B @ A2 )
= bot_bot_set_a ) ) ) ).
% additive_abelian_group.sumset_empty'(1)
thf(fact_681_additive__abelian__group_Osumset__empty_H_I2_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ( inf_inf_set_a @ A2 @ G )
= bot_bot_set_a )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B )
= bot_bot_set_a ) ) ) ).
% additive_abelian_group.sumset_empty'(2)
thf(fact_682_additive__abelian__group_Osumset__subset__Un_I2_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a,C: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ ( sup_sup_set_a @ A2 @ C ) @ B ) ) ) ).
% additive_abelian_group.sumset_subset_Un(2)
thf(fact_683_additive__abelian__group_Osumset__subset__Un_I1_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a,C: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( sup_sup_set_a @ B @ C ) ) ) ) ).
% additive_abelian_group.sumset_subset_Un(1)
thf(fact_684_additive__abelian__group_Osumset__D_I2_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( insert_a @ Zero @ bot_bot_set_a ) @ A2 )
= ( inf_inf_set_a @ A2 @ G ) ) ) ).
% additive_abelian_group.sumset_D(2)
thf(fact_685_additive__abelian__group_Osumset__D_I1_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( insert_a @ Zero @ bot_bot_set_a ) )
= ( inf_inf_set_a @ A2 @ G ) ) ) ).
% additive_abelian_group.sumset_D(1)
thf(fact_686_is__singletonE,axiom,
! [A2: set_a] :
( ( is_singleton_a @ A2 )
=> ~ ! [X2: a] :
( A2
!= ( insert_a @ X2 @ bot_bot_set_a ) ) ) ).
% is_singletonE
thf(fact_687_is__singleton__def,axiom,
( is_singleton_a
= ( ^ [A6: set_a] :
? [X3: a] :
( A6
= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ).
% is_singleton_def
thf(fact_688_additive__abelian__group__axioms,axiom,
pluenn1164192988769422572roup_a @ g @ addition @ zero ).
% additive_abelian_group_axioms
thf(fact_689_sumset__D_I1_J,axiom,
! [A2: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ zero @ bot_bot_set_a ) )
= ( inf_inf_set_a @ A2 @ g ) ) ).
% sumset_D(1)
thf(fact_690_sumset__D_I2_J,axiom,
! [A2: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( insert_a @ zero @ bot_bot_set_a ) @ A2 )
= ( inf_inf_set_a @ A2 @ g ) ) ).
% sumset_D(2)
thf(fact_691_Sup__fin_Oinsert,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ( lattic2918178356826803221_set_a @ ( insert_set_a @ X @ A2 ) )
= ( sup_sup_set_a @ X @ ( lattic2918178356826803221_set_a @ A2 ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_692_local_Oinverse__unique,axiom,
! [U: a,V: a,V2: a] :
( ( ( addition @ U @ V )
= zero )
=> ( ( ( addition @ V2 @ U )
= zero )
=> ( ( member_a @ U @ g )
=> ( ( member_a @ V2 @ g )
=> ( ( member_a @ V @ g )
=> ( V2 = V ) ) ) ) ) ) ).
% local.inverse_unique
thf(fact_693_inf__Sup__absorb,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A @ ( lattic2918178356826803221_set_a @ A2 ) )
= A ) ) ) ).
% inf_Sup_absorb
thf(fact_694_unit__closed,axiom,
member_a @ zero @ g ).
% unit_closed
thf(fact_695_left__unit,axiom,
! [A: a] :
( ( member_a @ A @ g )
=> ( ( addition @ zero @ A )
= A ) ) ).
% left_unit
thf(fact_696_right__unit,axiom,
! [A: a] :
( ( member_a @ A @ g )
=> ( ( addition @ A @ zero )
= A ) ) ).
% right_unit
thf(fact_697_commutative__monoid__axioms,axiom,
group_4866109990395492029noid_a @ g @ addition @ zero ).
% commutative_monoid_axioms
thf(fact_698_Sup__fin_OcoboundedI,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ( ord_less_eq_set_a @ A @ ( lattic2918178356826803221_set_a @ A2 ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_699_Sup__fin_Oin__idem,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ X @ A2 )
=> ( ( sup_sup_set_a @ X @ ( lattic2918178356826803221_set_a @ A2 ) )
= ( lattic2918178356826803221_set_a @ A2 ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_700_Sup__fin_Obounded__iff,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ( ord_less_eq_set_a @ ( lattic2918178356826803221_set_a @ A2 ) @ X )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( ord_less_eq_set_a @ X3 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_701_Sup__fin_OboundedI,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ! [A3: set_a] :
( ( member_set_a @ A3 @ A2 )
=> ( ord_less_eq_set_a @ A3 @ X ) )
=> ( ord_less_eq_set_a @ ( lattic2918178356826803221_set_a @ A2 ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_702_Sup__fin_OboundedE,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ( ord_less_eq_set_a @ ( lattic2918178356826803221_set_a @ A2 ) @ X )
=> ! [A9: set_a] :
( ( member_set_a @ A9 @ A2 )
=> ( ord_less_eq_set_a @ A9 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_703_Sup__fin_Osubset__imp,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ( finite_finite_set_a @ B )
=> ( ord_less_eq_set_a @ ( lattic2918178356826803221_set_a @ A2 ) @ ( lattic2918178356826803221_set_a @ B ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_704_Sup__fin_Osubset,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( B != bot_bot_set_set_a )
=> ( ( ord_le3724670747650509150_set_a @ B @ A2 )
=> ( ( sup_sup_set_a @ ( lattic2918178356826803221_set_a @ B ) @ ( lattic2918178356826803221_set_a @ A2 ) )
= ( lattic2918178356826803221_set_a @ A2 ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_705_Sup__fin_Oinsert__not__elem,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ~ ( member_set_a @ X @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ( lattic2918178356826803221_set_a @ ( insert_set_a @ X @ A2 ) )
= ( sup_sup_set_a @ X @ ( lattic2918178356826803221_set_a @ A2 ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_706_Sup__fin_Oclosed,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ! [X2: set_a,Y2: set_a] : ( member_set_a @ ( sup_sup_set_a @ X2 @ Y2 ) @ ( insert_set_a @ X2 @ ( insert_set_a @ Y2 @ bot_bot_set_set_a ) ) )
=> ( member_set_a @ ( lattic2918178356826803221_set_a @ A2 ) @ A2 ) ) ) ) ).
% Sup_fin.closed
thf(fact_707_Sup__fin_Ounion,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ( finite_finite_set_a @ B )
=> ( ( B != bot_bot_set_set_a )
=> ( ( lattic2918178356826803221_set_a @ ( sup_sup_set_set_a @ A2 @ B ) )
= ( sup_sup_set_a @ ( lattic2918178356826803221_set_a @ A2 ) @ ( lattic2918178356826803221_set_a @ B ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_708_abelian__group__axioms,axiom,
group_201663378560352916roup_a @ g @ addition @ zero ).
% abelian_group_axioms
thf(fact_709_group__axioms,axiom,
group_group_a @ g @ addition @ zero ).
% group_axioms
thf(fact_710_invertibleE,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ! [V3: a] :
( ( ( ( addition @ U @ V3 )
= zero )
& ( ( addition @ V3 @ U )
= zero ) )
=> ~ ( member_a @ V3 @ g ) )
=> ~ ( member_a @ U @ g ) ) ) ).
% invertibleE
thf(fact_711_invertible__def,axiom,
! [U: a] :
( ( member_a @ U @ g )
=> ( ( group_invertible_a @ g @ addition @ zero @ U )
= ( ? [X3: a] :
( ( member_a @ X3 @ g )
& ( ( addition @ U @ X3 )
= zero )
& ( ( addition @ X3 @ U )
= zero ) ) ) ) ) ).
% invertible_def
thf(fact_712_unit__invertible,axiom,
group_invertible_a @ g @ addition @ zero @ zero ).
% unit_invertible
thf(fact_713_invertible__right__cancel,axiom,
! [X: a,Y: a,Z2: a] :
( ( group_invertible_a @ g @ addition @ zero @ X )
=> ( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( ( member_a @ Z2 @ g )
=> ( ( ( addition @ Y @ X )
= ( addition @ Z2 @ X ) )
= ( Y = Z2 ) ) ) ) ) ) ).
% invertible_right_cancel
thf(fact_714_invertible__left__cancel,axiom,
! [X: a,Y: a,Z2: a] :
( ( group_invertible_a @ g @ addition @ zero @ X )
=> ( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( ( member_a @ Z2 @ g )
=> ( ( ( addition @ X @ Y )
= ( addition @ X @ Z2 ) )
= ( Y = Z2 ) ) ) ) ) ) ).
% invertible_left_cancel
thf(fact_715_invertibleI,axiom,
! [U: a,V2: a] :
( ( ( addition @ U @ V2 )
= zero )
=> ( ( ( addition @ V2 @ U )
= zero )
=> ( ( member_a @ U @ g )
=> ( ( member_a @ V2 @ g )
=> ( group_invertible_a @ g @ addition @ zero @ U ) ) ) ) ) ).
% invertibleI
thf(fact_716_invertible,axiom,
! [U: a] :
( ( member_a @ U @ g )
=> ( group_invertible_a @ g @ addition @ zero @ U ) ) ).
% invertible
thf(fact_717_composition__invertible,axiom,
! [X: a,Y: a] :
( ( group_invertible_a @ g @ addition @ zero @ X )
=> ( ( group_invertible_a @ g @ addition @ zero @ Y )
=> ( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( group_invertible_a @ g @ addition @ zero @ ( addition @ X @ Y ) ) ) ) ) ) ).
% composition_invertible
thf(fact_718_additive__abelian__group_Ointro,axiom,
! [G: set_a,Addition: a > a > a,Zero: a] :
( ( group_201663378560352916roup_a @ G @ Addition @ Zero )
=> ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero ) ) ).
% additive_abelian_group.intro
thf(fact_719_additive__abelian__group_Oaxioms,axiom,
! [G: set_a,Addition: a > a > a,Zero: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( group_201663378560352916roup_a @ G @ Addition @ Zero ) ) ).
% additive_abelian_group.axioms
thf(fact_720_additive__abelian__group__def,axiom,
pluenn1164192988769422572roup_a = group_201663378560352916roup_a ).
% additive_abelian_group_def
thf(fact_721_mem__UnitsI,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( member_a @ U @ ( group_Units_a @ g @ addition @ zero ) ) ) ) ).
% mem_UnitsI
thf(fact_722_mem__UnitsD,axiom,
! [U: a] :
( ( member_a @ U @ ( group_Units_a @ g @ addition @ zero ) )
=> ( ( group_invertible_a @ g @ addition @ zero @ U )
& ( member_a @ U @ g ) ) ) ).
% mem_UnitsD
thf(fact_723_group__of__Units,axiom,
group_group_a @ ( group_Units_a @ g @ addition @ zero ) @ addition @ zero ).
% group_of_Units
thf(fact_724_inverse__composition__commute,axiom,
! [X: a,Y: a] :
( ( group_invertible_a @ g @ addition @ zero @ X )
=> ( ( group_invertible_a @ g @ addition @ zero @ Y )
=> ( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( ( group_inverse_a @ g @ addition @ zero @ ( addition @ X @ Y ) )
= ( addition @ ( group_inverse_a @ g @ addition @ zero @ Y ) @ ( group_inverse_a @ g @ addition @ zero @ X ) ) ) ) ) ) ) ).
% inverse_composition_commute
thf(fact_725_invertible__left__inverse2,axiom,
! [U: a,V2: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( ( member_a @ V2 @ g )
=> ( ( addition @ ( group_inverse_a @ g @ addition @ zero @ U ) @ ( addition @ U @ V2 ) )
= V2 ) ) ) ) ).
% invertible_left_inverse2
thf(fact_726_inverse__closed,axiom,
! [X: a] :
( ( member_a @ X @ g )
=> ( member_a @ ( group_inverse_a @ g @ addition @ zero @ X ) @ g ) ) ).
% inverse_closed
thf(fact_727_inverse__equality,axiom,
! [U: a,V2: a] :
( ( ( addition @ U @ V2 )
= zero )
=> ( ( ( addition @ V2 @ U )
= zero )
=> ( ( member_a @ U @ g )
=> ( ( member_a @ V2 @ g )
=> ( ( group_inverse_a @ g @ addition @ zero @ U )
= V2 ) ) ) ) ) ).
% inverse_equality
thf(fact_728_invertible__right__inverse2,axiom,
! [U: a,V2: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( ( member_a @ V2 @ g )
=> ( ( addition @ U @ ( addition @ ( group_inverse_a @ g @ addition @ zero @ U ) @ V2 ) )
= V2 ) ) ) ) ).
% invertible_right_inverse2
thf(fact_729_inverse__unit,axiom,
( ( group_inverse_a @ g @ addition @ zero @ zero )
= zero ) ).
% inverse_unit
thf(fact_730_invertible__right__inverse,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( ( addition @ U @ ( group_inverse_a @ g @ addition @ zero @ U ) )
= zero ) ) ) ).
% invertible_right_inverse
thf(fact_731_invertible__left__inverse,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( ( addition @ ( group_inverse_a @ g @ addition @ zero @ U ) @ U )
= zero ) ) ) ).
% invertible_left_inverse
thf(fact_732_invertible__inverse__invertible,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( group_invertible_a @ g @ addition @ zero @ ( group_inverse_a @ g @ addition @ zero @ U ) ) ) ) ).
% invertible_inverse_invertible
thf(fact_733_invertible__inverse__inverse,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( ( group_inverse_a @ g @ addition @ zero @ ( group_inverse_a @ g @ addition @ zero @ U ) )
= U ) ) ) ).
% invertible_inverse_inverse
thf(fact_734_invertible__inverse__closed,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( member_a @ ( group_inverse_a @ g @ addition @ zero @ U ) @ g ) ) ) ).
% invertible_inverse_closed
thf(fact_735_additive__abelian__group_Oinverse__closed,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,X: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( member_a @ X @ G )
=> ( member_a @ ( group_inverse_a @ G @ Addition @ Zero @ X ) @ G ) ) ) ).
% additive_abelian_group.inverse_closed
thf(fact_736_subgroupI,axiom,
! [G: set_a] :
( ( ord_less_eq_set_a @ G @ g )
=> ( ( member_a @ zero @ G )
=> ( ! [G3: a,H: a] :
( ( member_a @ G3 @ G )
=> ( ( member_a @ H @ G )
=> ( member_a @ ( addition @ G3 @ H ) @ G ) ) )
=> ( ! [G3: a] :
( ( member_a @ G3 @ G )
=> ( group_invertible_a @ g @ addition @ zero @ G3 ) )
=> ( ! [G3: a] :
( ( member_a @ G3 @ G )
=> ( member_a @ ( group_inverse_a @ g @ addition @ zero @ G3 ) @ G ) )
=> ( group_subgroup_a @ G @ g @ addition @ zero ) ) ) ) ) ) ).
% subgroupI
thf(fact_737_monoid__axioms,axiom,
group_monoid_a @ g @ addition @ zero ).
% monoid_axioms
thf(fact_738_inverse__undefined,axiom,
! [U: a] :
( ~ ( member_a @ U @ g )
=> ( ( group_inverse_a @ g @ addition @ zero @ U )
= undefined_a ) ) ).
% inverse_undefined
thf(fact_739_inverse__subgroupD,axiom,
! [H2: set_a] :
( ( group_subgroup_a @ ( image_a_a @ ( group_inverse_a @ g @ addition @ zero ) @ H2 ) @ g @ addition @ zero )
=> ( ( ord_less_eq_set_a @ H2 @ ( group_Units_a @ g @ addition @ zero ) )
=> ( group_subgroup_a @ H2 @ g @ addition @ zero ) ) ) ).
% inverse_subgroupD
thf(fact_740_Inf__fin_Oinsert,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ( lattic8209813465164889211_set_a @ ( insert_set_a @ X @ A2 ) )
= ( inf_inf_set_a @ X @ ( lattic8209813465164889211_set_a @ A2 ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_741_image__eqI,axiom,
! [B4: set_a,F3: a > set_a,X: a,A2: set_a] :
( ( B4
= ( F3 @ X ) )
=> ( ( member_a @ X @ A2 )
=> ( member_set_a @ B4 @ ( image_a_set_a @ F3 @ A2 ) ) ) ) ).
% image_eqI
thf(fact_742_image__eqI,axiom,
! [B4: a,F3: a > a,X: a,A2: set_a] :
( ( B4
= ( F3 @ X ) )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ B4 @ ( image_a_a @ F3 @ A2 ) ) ) ) ).
% image_eqI
thf(fact_743_image__empty,axiom,
! [F3: a > set_a] :
( ( image_a_set_a @ F3 @ bot_bot_set_a )
= bot_bot_set_set_a ) ).
% image_empty
thf(fact_744_image__empty,axiom,
! [F3: a > a] :
( ( image_a_a @ F3 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% image_empty
thf(fact_745_empty__is__image,axiom,
! [F3: a > set_a,A2: set_a] :
( ( bot_bot_set_set_a
= ( image_a_set_a @ F3 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_746_empty__is__image,axiom,
! [F3: a > a,A2: set_a] :
( ( bot_bot_set_a
= ( image_a_a @ F3 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_747_image__is__empty,axiom,
! [F3: a > set_a,A2: set_a] :
( ( ( image_a_set_a @ F3 @ A2 )
= bot_bot_set_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_748_image__is__empty,axiom,
! [F3: a > a,A2: set_a] :
( ( ( image_a_a @ F3 @ A2 )
= bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_749_finite__imageI,axiom,
! [F: set_a,H3: a > set_a] :
( ( finite_finite_a @ F )
=> ( finite_finite_set_a @ ( image_a_set_a @ H3 @ F ) ) ) ).
% finite_imageI
thf(fact_750_finite__imageI,axiom,
! [F: set_a,H3: a > a] :
( ( finite_finite_a @ F )
=> ( finite_finite_a @ ( image_a_a @ H3 @ F ) ) ) ).
% finite_imageI
thf(fact_751_insert__image,axiom,
! [X: a,A2: set_a,F3: a > set_a] :
( ( member_a @ X @ A2 )
=> ( ( insert_set_a @ ( F3 @ X ) @ ( image_a_set_a @ F3 @ A2 ) )
= ( image_a_set_a @ F3 @ A2 ) ) ) ).
% insert_image
thf(fact_752_insert__image,axiom,
! [X: a,A2: set_a,F3: a > a] :
( ( member_a @ X @ A2 )
=> ( ( insert_a @ ( F3 @ X ) @ ( image_a_a @ F3 @ A2 ) )
= ( image_a_a @ F3 @ A2 ) ) ) ).
% insert_image
thf(fact_753_image__insert,axiom,
! [F3: a > set_a,A: a,B: set_a] :
( ( image_a_set_a @ F3 @ ( insert_a @ A @ B ) )
= ( insert_set_a @ ( F3 @ A ) @ ( image_a_set_a @ F3 @ B ) ) ) ).
% image_insert
thf(fact_754_image__insert,axiom,
! [F3: a > a,A: a,B: set_a] :
( ( image_a_a @ F3 @ ( insert_a @ A @ B ) )
= ( insert_a @ ( F3 @ A ) @ ( image_a_a @ F3 @ B ) ) ) ).
% image_insert
thf(fact_755_inverse__subgroupI,axiom,
! [H2: set_a] :
( ( group_subgroup_a @ H2 @ g @ addition @ zero )
=> ( group_subgroup_a @ ( image_a_a @ ( group_inverse_a @ g @ addition @ zero ) @ H2 ) @ g @ addition @ zero ) ) ).
% inverse_subgroupI
thf(fact_756_sup__Inf__absorb,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ( ( sup_sup_set_a @ ( lattic8209813465164889211_set_a @ A2 ) @ A )
= A ) ) ) ).
% sup_Inf_absorb
thf(fact_757_subset__image__iff,axiom,
! [B: set_set_a,F3: a > set_a,A2: set_a] :
( ( ord_le3724670747650509150_set_a @ B @ ( image_a_set_a @ F3 @ A2 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A2 )
& ( B
= ( image_a_set_a @ F3 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_758_subset__image__iff,axiom,
! [B: set_a,F3: a > a,A2: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F3 @ A2 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A2 )
& ( B
= ( image_a_a @ F3 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_759_image__subset__iff,axiom,
! [F3: a > set_a,A2: set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F3 @ A2 ) @ B )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_set_a @ ( F3 @ X3 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_760_image__subset__iff,axiom,
! [F3: a > a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F3 @ A2 ) @ B )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_a @ ( F3 @ X3 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_761_subset__imageE,axiom,
! [B: set_set_a,F3: a > set_a,A2: set_a] :
( ( ord_le3724670747650509150_set_a @ B @ ( image_a_set_a @ F3 @ A2 ) )
=> ~ ! [C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ A2 )
=> ( B
!= ( image_a_set_a @ F3 @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_762_subset__imageE,axiom,
! [B: set_a,F3: a > a,A2: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F3 @ A2 ) )
=> ~ ! [C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ A2 )
=> ( B
!= ( image_a_a @ F3 @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_763_image__subsetI,axiom,
! [A2: set_a,F3: a > set_a,B: set_set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_set_a @ ( F3 @ X2 ) @ B ) )
=> ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F3 @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_764_image__subsetI,axiom,
! [A2: set_a,F3: a > a,B: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_a @ ( F3 @ X2 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F3 @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_765_image__mono,axiom,
! [A2: set_a,B: set_a,F3: a > set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F3 @ A2 ) @ ( image_a_set_a @ F3 @ B ) ) ) ).
% image_mono
thf(fact_766_image__mono,axiom,
! [A2: set_a,B: set_a,F3: a > a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_less_eq_set_a @ ( image_a_a @ F3 @ A2 ) @ ( image_a_a @ F3 @ B ) ) ) ).
% image_mono
thf(fact_767_all__subset__image,axiom,
! [F3: a > set_a,A2: set_a,P: set_set_a > $o] :
( ( ! [B6: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B6 @ ( image_a_set_a @ F3 @ A2 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_a] :
( ( ord_less_eq_set_a @ B6 @ A2 )
=> ( P @ ( image_a_set_a @ F3 @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_768_all__subset__image,axiom,
! [F3: a > a,A2: set_a,P: set_a > $o] :
( ( ! [B6: set_a] :
( ( ord_less_eq_set_a @ B6 @ ( image_a_a @ F3 @ A2 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_a] :
( ( ord_less_eq_set_a @ B6 @ A2 )
=> ( P @ ( image_a_a @ F3 @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_769_image__Un,axiom,
! [F3: a > set_a,A2: set_a,B: set_a] :
( ( image_a_set_a @ F3 @ ( sup_sup_set_a @ A2 @ B ) )
= ( sup_sup_set_set_a @ ( image_a_set_a @ F3 @ A2 ) @ ( image_a_set_a @ F3 @ B ) ) ) ).
% image_Un
thf(fact_770_image__Un,axiom,
! [F3: a > a,A2: set_a,B: set_a] :
( ( image_a_a @ F3 @ ( sup_sup_set_a @ A2 @ B ) )
= ( sup_sup_set_a @ ( image_a_a @ F3 @ A2 ) @ ( image_a_a @ F3 @ B ) ) ) ).
% image_Un
thf(fact_771_rev__image__eqI,axiom,
! [X: a,A2: set_a,B4: set_a,F3: a > set_a] :
( ( member_a @ X @ A2 )
=> ( ( B4
= ( F3 @ X ) )
=> ( member_set_a @ B4 @ ( image_a_set_a @ F3 @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_772_rev__image__eqI,axiom,
! [X: a,A2: set_a,B4: a,F3: a > a] :
( ( member_a @ X @ A2 )
=> ( ( B4
= ( F3 @ X ) )
=> ( member_a @ B4 @ ( image_a_a @ F3 @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_773_ball__imageD,axiom,
! [F3: a > a,A2: set_a,P: a > $o] :
( ! [X2: a] :
( ( member_a @ X2 @ ( image_a_a @ F3 @ A2 ) )
=> ( P @ X2 ) )
=> ! [X5: a] :
( ( member_a @ X5 @ A2 )
=> ( P @ ( F3 @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_774_ball__imageD,axiom,
! [F3: a > set_a,A2: set_a,P: set_a > $o] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ ( image_a_set_a @ F3 @ A2 ) )
=> ( P @ X2 ) )
=> ! [X5: a] :
( ( member_a @ X5 @ A2 )
=> ( P @ ( F3 @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_775_image__cong,axiom,
! [M: set_a,N: set_a,F3: a > a,G4: a > a] :
( ( M = N )
=> ( ! [X2: a] :
( ( member_a @ X2 @ N )
=> ( ( F3 @ X2 )
= ( G4 @ X2 ) ) )
=> ( ( image_a_a @ F3 @ M )
= ( image_a_a @ G4 @ N ) ) ) ) ).
% image_cong
thf(fact_776_image__cong,axiom,
! [M: set_a,N: set_a,F3: a > set_a,G4: a > set_a] :
( ( M = N )
=> ( ! [X2: a] :
( ( member_a @ X2 @ N )
=> ( ( F3 @ X2 )
= ( G4 @ X2 ) ) )
=> ( ( image_a_set_a @ F3 @ M )
= ( image_a_set_a @ G4 @ N ) ) ) ) ).
% image_cong
thf(fact_777_bex__imageD,axiom,
! [F3: a > a,A2: set_a,P: a > $o] :
( ? [X5: a] :
( ( member_a @ X5 @ ( image_a_a @ F3 @ A2 ) )
& ( P @ X5 ) )
=> ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ( P @ ( F3 @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_778_bex__imageD,axiom,
! [F3: a > set_a,A2: set_a,P: set_a > $o] :
( ? [X5: set_a] :
( ( member_set_a @ X5 @ ( image_a_set_a @ F3 @ A2 ) )
& ( P @ X5 ) )
=> ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ( P @ ( F3 @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_779_image__iff,axiom,
! [Z2: set_a,F3: a > set_a,A2: set_a] :
( ( member_set_a @ Z2 @ ( image_a_set_a @ F3 @ A2 ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A2 )
& ( Z2
= ( F3 @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_780_image__iff,axiom,
! [Z2: a,F3: a > a,A2: set_a] :
( ( member_a @ Z2 @ ( image_a_a @ F3 @ A2 ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A2 )
& ( Z2
= ( F3 @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_781_imageI,axiom,
! [X: a,A2: set_a,F3: a > set_a] :
( ( member_a @ X @ A2 )
=> ( member_set_a @ ( F3 @ X ) @ ( image_a_set_a @ F3 @ A2 ) ) ) ).
% imageI
thf(fact_782_imageI,axiom,
! [X: a,A2: set_a,F3: a > a] :
( ( member_a @ X @ A2 )
=> ( member_a @ ( F3 @ X ) @ ( image_a_a @ F3 @ A2 ) ) ) ).
% imageI
thf(fact_783_Inf__fin_Ohom__commute,axiom,
! [H3: set_a > set_a,N: set_set_a] :
( ! [X2: set_a,Y2: set_a] :
( ( H3 @ ( inf_inf_set_a @ X2 @ Y2 ) )
= ( inf_inf_set_a @ ( H3 @ X2 ) @ ( H3 @ Y2 ) ) )
=> ( ( finite_finite_set_a @ N )
=> ( ( N != bot_bot_set_set_a )
=> ( ( H3 @ ( lattic8209813465164889211_set_a @ N ) )
= ( lattic8209813465164889211_set_a @ ( image_set_a_set_a @ H3 @ N ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_784_finite__surj,axiom,
! [A2: set_a,B: set_set_a,F3: a > set_a] :
( ( finite_finite_a @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ B @ ( image_a_set_a @ F3 @ A2 ) )
=> ( finite_finite_set_a @ B ) ) ) ).
% finite_surj
thf(fact_785_finite__surj,axiom,
! [A2: set_a,B: set_a,F3: a > a] :
( ( finite_finite_a @ A2 )
=> ( ( ord_less_eq_set_a @ B @ ( image_a_a @ F3 @ A2 ) )
=> ( finite_finite_a @ B ) ) ) ).
% finite_surj
thf(fact_786_finite__subset__image,axiom,
! [B: set_set_a,F3: a > set_a,A2: set_a] :
( ( finite_finite_set_a @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ ( image_a_set_a @ F3 @ A2 ) )
=> ? [C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ A2 )
& ( finite_finite_a @ C4 )
& ( B
= ( image_a_set_a @ F3 @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_787_finite__subset__image,axiom,
! [B: set_a,F3: a > a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ B @ ( image_a_a @ F3 @ A2 ) )
=> ? [C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ A2 )
& ( finite_finite_a @ C4 )
& ( B
= ( image_a_a @ F3 @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_788_ex__finite__subset__image,axiom,
! [F3: a > set_a,A2: set_a,P: set_set_a > $o] :
( ( ? [B6: set_set_a] :
( ( finite_finite_set_a @ B6 )
& ( ord_le3724670747650509150_set_a @ B6 @ ( image_a_set_a @ F3 @ A2 ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_a] :
( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ A2 )
& ( P @ ( image_a_set_a @ F3 @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_789_ex__finite__subset__image,axiom,
! [F3: a > a,A2: set_a,P: set_a > $o] :
( ( ? [B6: set_a] :
( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ ( image_a_a @ F3 @ A2 ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_a] :
( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ A2 )
& ( P @ ( image_a_a @ F3 @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_790_all__finite__subset__image,axiom,
! [F3: a > set_a,A2: set_a,P: set_set_a > $o] :
( ( ! [B6: set_set_a] :
( ( ( finite_finite_set_a @ B6 )
& ( ord_le3724670747650509150_set_a @ B6 @ ( image_a_set_a @ F3 @ A2 ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_a] :
( ( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ A2 ) )
=> ( P @ ( image_a_set_a @ F3 @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_791_all__finite__subset__image,axiom,
! [F3: a > a,A2: set_a,P: set_a > $o] :
( ( ! [B6: set_a] :
( ( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ ( image_a_a @ F3 @ A2 ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_a] :
( ( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ A2 ) )
=> ( P @ ( image_a_a @ F3 @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_792_image__Int__subset,axiom,
! [F3: a > set_a,A2: set_a,B: set_a] : ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F3 @ ( inf_inf_set_a @ A2 @ B ) ) @ ( inf_inf_set_set_a @ ( image_a_set_a @ F3 @ A2 ) @ ( image_a_set_a @ F3 @ B ) ) ) ).
% image_Int_subset
thf(fact_793_image__Int__subset,axiom,
! [F3: a > a,A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( image_a_a @ F3 @ ( inf_inf_set_a @ A2 @ B ) ) @ ( inf_inf_set_a @ ( image_a_a @ F3 @ A2 ) @ ( image_a_a @ F3 @ B ) ) ) ).
% image_Int_subset
thf(fact_794_Inf__fin_OcoboundedI,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ( ord_less_eq_set_a @ ( lattic8209813465164889211_set_a @ A2 ) @ A ) ) ) ).
% Inf_fin.coboundedI
thf(fact_795_Inf__fin_Oin__idem,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ X @ A2 )
=> ( ( inf_inf_set_a @ X @ ( lattic8209813465164889211_set_a @ A2 ) )
= ( lattic8209813465164889211_set_a @ A2 ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_796_the__elem__image__unique,axiom,
! [A2: set_a,F3: a > a,X: a] :
( ( A2 != bot_bot_set_a )
=> ( ! [Y2: a] :
( ( member_a @ Y2 @ A2 )
=> ( ( F3 @ Y2 )
= ( F3 @ X ) ) )
=> ( ( the_elem_a @ ( image_a_a @ F3 @ A2 ) )
= ( F3 @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_797_the__elem__image__unique,axiom,
! [A2: set_a,F3: a > set_a,X: a] :
( ( A2 != bot_bot_set_a )
=> ( ! [Y2: a] :
( ( member_a @ Y2 @ A2 )
=> ( ( F3 @ Y2 )
= ( F3 @ X ) ) )
=> ( ( the_elem_set_a @ ( image_a_set_a @ F3 @ A2 ) )
= ( F3 @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_798_Inf__fin_OboundedE,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ( ord_less_eq_set_a @ X @ ( lattic8209813465164889211_set_a @ A2 ) )
=> ! [A9: set_a] :
( ( member_set_a @ A9 @ A2 )
=> ( ord_less_eq_set_a @ X @ A9 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_799_Inf__fin_OboundedI,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ! [A3: set_a] :
( ( member_set_a @ A3 @ A2 )
=> ( ord_less_eq_set_a @ X @ A3 ) )
=> ( ord_less_eq_set_a @ X @ ( lattic8209813465164889211_set_a @ A2 ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_800_Inf__fin_Obounded__iff,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ( ord_less_eq_set_a @ X @ ( lattic8209813465164889211_set_a @ A2 ) )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( ord_less_eq_set_a @ X @ X3 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_801_Sup__fin_Ohom__commute,axiom,
! [H3: set_a > set_a,N: set_set_a] :
( ! [X2: set_a,Y2: set_a] :
( ( H3 @ ( sup_sup_set_a @ X2 @ Y2 ) )
= ( sup_sup_set_a @ ( H3 @ X2 ) @ ( H3 @ Y2 ) ) )
=> ( ( finite_finite_set_a @ N )
=> ( ( N != bot_bot_set_set_a )
=> ( ( H3 @ ( lattic2918178356826803221_set_a @ N ) )
= ( lattic2918178356826803221_set_a @ ( image_set_a_set_a @ H3 @ N ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_802_Inf__fin_Osubset__imp,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ( finite_finite_set_a @ B )
=> ( ord_less_eq_set_a @ ( lattic8209813465164889211_set_a @ B ) @ ( lattic8209813465164889211_set_a @ A2 ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_803_Inf__fin_Osubset,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( B != bot_bot_set_set_a )
=> ( ( ord_le3724670747650509150_set_a @ B @ A2 )
=> ( ( inf_inf_set_a @ ( lattic8209813465164889211_set_a @ B ) @ ( lattic8209813465164889211_set_a @ A2 ) )
= ( lattic8209813465164889211_set_a @ A2 ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_804_Inf__fin_Oclosed,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ! [X2: set_a,Y2: set_a] : ( member_set_a @ ( inf_inf_set_a @ X2 @ Y2 ) @ ( insert_set_a @ X2 @ ( insert_set_a @ Y2 @ bot_bot_set_set_a ) ) )
=> ( member_set_a @ ( lattic8209813465164889211_set_a @ A2 ) @ A2 ) ) ) ) ).
% Inf_fin.closed
thf(fact_805_Inf__fin_Oinsert__not__elem,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ~ ( member_set_a @ X @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ( lattic8209813465164889211_set_a @ ( insert_set_a @ X @ A2 ) )
= ( inf_inf_set_a @ X @ ( lattic8209813465164889211_set_a @ A2 ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_806_Inf__fin_Ounion,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ( finite_finite_set_a @ B )
=> ( ( B != bot_bot_set_set_a )
=> ( ( lattic8209813465164889211_set_a @ ( sup_sup_set_set_a @ A2 @ B ) )
= ( inf_inf_set_a @ ( lattic8209813465164889211_set_a @ A2 ) @ ( lattic8209813465164889211_set_a @ B ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_807_Inf__fin__le__Sup__fin,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ ( lattic8209813465164889211_set_a @ A2 ) @ ( lattic2918178356826803221_set_a @ A2 ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_808_group_Oinverse__subgroupD,axiom,
! [G: set_a,Composition: a > a > a,Unit: a,H2: set_a] :
( ( group_group_a @ G @ Composition @ Unit )
=> ( ( group_subgroup_a @ ( image_a_a @ ( group_inverse_a @ G @ Composition @ Unit ) @ H2 ) @ G @ Composition @ Unit )
=> ( ( ord_less_eq_set_a @ H2 @ ( group_Units_a @ G @ Composition @ Unit ) )
=> ( group_subgroup_a @ H2 @ G @ Composition @ Unit ) ) ) ) ).
% group.inverse_subgroupD
thf(fact_809_monoid_OsubgroupI,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,G: set_a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( ord_less_eq_set_a @ G @ M )
=> ( ( member_a @ Unit @ G )
=> ( ! [G3: a,H: a] :
( ( member_a @ G3 @ G )
=> ( ( member_a @ H @ G )
=> ( member_a @ ( Composition @ G3 @ H ) @ G ) ) )
=> ( ! [G3: a] :
( ( member_a @ G3 @ G )
=> ( group_invertible_a @ M @ Composition @ Unit @ G3 ) )
=> ( ! [G3: a] :
( ( member_a @ G3 @ G )
=> ( member_a @ ( group_inverse_a @ M @ Composition @ Unit @ G3 ) @ G ) )
=> ( group_subgroup_a @ G @ M @ Composition @ Unit ) ) ) ) ) ) ) ).
% monoid.subgroupI
thf(fact_810_Sup__fin_Oinsert__remove,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( ( ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
= bot_bot_set_set_a )
=> ( ( lattic2918178356826803221_set_a @ ( insert_set_a @ X @ A2 ) )
= X ) )
& ( ( ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
!= bot_bot_set_set_a )
=> ( ( lattic2918178356826803221_set_a @ ( insert_set_a @ X @ A2 ) )
= ( sup_sup_set_a @ X @ ( lattic2918178356826803221_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_811_Sup__fin_Oremove,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ X @ A2 )
=> ( ( ( ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
= bot_bot_set_set_a )
=> ( ( lattic2918178356826803221_set_a @ A2 )
= X ) )
& ( ( ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
!= bot_bot_set_set_a )
=> ( ( lattic2918178356826803221_set_a @ A2 )
= ( sup_sup_set_a @ X @ ( lattic2918178356826803221_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_812_Inf__fin_Oinsert__remove,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( ( ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
= bot_bot_set_set_a )
=> ( ( lattic8209813465164889211_set_a @ ( insert_set_a @ X @ A2 ) )
= X ) )
& ( ( ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
!= bot_bot_set_set_a )
=> ( ( lattic8209813465164889211_set_a @ ( insert_set_a @ X @ A2 ) )
= ( inf_inf_set_a @ X @ ( lattic8209813465164889211_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_813_DiffI,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ A2 )
=> ( ~ ( member_a @ C2 @ B )
=> ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_814_Diff__iff,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B ) )
= ( ( member_a @ C2 @ A2 )
& ~ ( member_a @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_815_Diff__cancel,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ A2 )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_816_empty__Diff,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A2 )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_817_Diff__empty,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Diff_empty
thf(fact_818_finite__Diff,axiom,
! [A2: set_a,B: set_a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B ) ) ) ).
% finite_Diff
thf(fact_819_finite__Diff2,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B ) )
= ( finite_finite_a @ A2 ) ) ) ).
% finite_Diff2
thf(fact_820_Diff__insert0,axiom,
! [X: a,A2: set_a,B: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( minus_minus_set_a @ A2 @ ( insert_a @ X @ B ) )
= ( minus_minus_set_a @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_821_insert__Diff1,axiom,
! [X: a,B: set_a,A2: set_a] :
( ( member_a @ X @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B )
= ( minus_minus_set_a @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_822_Un__Diff__cancel2,axiom,
! [B: set_a,A2: set_a] :
( ( sup_sup_set_a @ ( minus_minus_set_a @ B @ A2 ) @ A2 )
= ( sup_sup_set_a @ B @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_823_Un__Diff__cancel,axiom,
! [A2: set_a,B: set_a] :
( ( sup_sup_set_a @ A2 @ ( minus_minus_set_a @ B @ A2 ) )
= ( sup_sup_set_a @ A2 @ B ) ) ).
% Un_Diff_cancel
thf(fact_824_Diff__eq__empty__iff,axiom,
! [A2: set_a,B: set_a] :
( ( ( minus_minus_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A2 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_825_insert__Diff__single,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( insert_a @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_826_finite__Diff__insert,axiom,
! [A2: set_a,A: a,B: set_a] :
( ( finite_finite_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B ) ) )
= ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B ) ) ) ).
% finite_Diff_insert
thf(fact_827_Diff__disjoint,axiom,
! [A2: set_a,B: set_a] :
( ( inf_inf_set_a @ A2 @ ( minus_minus_set_a @ B @ A2 ) )
= bot_bot_set_a ) ).
% Diff_disjoint
thf(fact_828_in__image__insert__iff,axiom,
! [B: set_set_a,X: a,A2: set_a] :
( ! [C4: set_a] :
( ( member_set_a @ C4 @ B )
=> ~ ( member_a @ X @ C4 ) )
=> ( ( member_set_a @ A2 @ ( image_set_a_set_a @ ( insert_a @ X ) @ B ) )
= ( ( member_a @ X @ A2 )
& ( member_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_829_Diff__infinite__finite,axiom,
! [T: set_a,S: set_a] :
( ( finite_finite_a @ T )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_830_Diff__mono,axiom,
! [A2: set_a,C: set_a,D: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ D @ B )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B ) @ ( minus_minus_set_a @ C @ D ) ) ) ) ).
% Diff_mono
thf(fact_831_Diff__subset,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B ) @ A2 ) ).
% Diff_subset
thf(fact_832_double__diff,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ( minus_minus_set_a @ B @ ( minus_minus_set_a @ C @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_833_insert__Diff__if,axiom,
! [X: a,B: set_a,A2: set_a] :
( ( ( member_a @ X @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B )
= ( minus_minus_set_a @ A2 @ B ) ) )
& ( ~ ( member_a @ X @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B )
= ( insert_a @ X @ ( minus_minus_set_a @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_834_Int__Diff,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ B ) @ C )
= ( inf_inf_set_a @ A2 @ ( minus_minus_set_a @ B @ C ) ) ) ).
% Int_Diff
thf(fact_835_Diff__Int2,axiom,
! [A2: set_a,C: set_a,B: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ C ) @ ( inf_inf_set_a @ B @ C ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ C ) @ B ) ) ).
% Diff_Int2
thf(fact_836_Diff__Diff__Int,axiom,
! [A2: set_a,B: set_a] :
( ( minus_minus_set_a @ A2 @ ( minus_minus_set_a @ A2 @ B ) )
= ( inf_inf_set_a @ A2 @ B ) ) ).
% Diff_Diff_Int
thf(fact_837_Diff__Int__distrib,axiom,
! [C: set_a,A2: set_a,B: set_a] :
( ( inf_inf_set_a @ C @ ( minus_minus_set_a @ A2 @ B ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ C @ A2 ) @ ( inf_inf_set_a @ C @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_838_Diff__Int__distrib2,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ ( minus_minus_set_a @ A2 @ B ) @ C )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ C ) @ ( inf_inf_set_a @ B @ C ) ) ) ).
% Diff_Int_distrib2
thf(fact_839_Un__Diff,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( minus_minus_set_a @ ( sup_sup_set_a @ A2 @ B ) @ C )
= ( sup_sup_set_a @ ( minus_minus_set_a @ A2 @ C ) @ ( minus_minus_set_a @ B @ C ) ) ) ).
% Un_Diff
thf(fact_840_DiffE,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B ) )
=> ~ ( ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ B ) ) ) ).
% DiffE
thf(fact_841_DiffD1,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B ) )
=> ( member_a @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_842_DiffD2,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B ) )
=> ~ ( member_a @ C2 @ B ) ) ).
% DiffD2
thf(fact_843_diff__shunt__var,axiom,
! [X: set_a,Y: set_a] :
( ( ( minus_minus_set_a @ X @ Y )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_844_image__diff__subset,axiom,
! [F3: a > set_a,A2: set_a,B: set_a] : ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ ( image_a_set_a @ F3 @ A2 ) @ ( image_a_set_a @ F3 @ B ) ) @ ( image_a_set_a @ F3 @ ( minus_minus_set_a @ A2 @ B ) ) ) ).
% image_diff_subset
thf(fact_845_image__diff__subset,axiom,
! [F3: a > a,A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ ( image_a_a @ F3 @ A2 ) @ ( image_a_a @ F3 @ B ) ) @ ( image_a_a @ F3 @ ( minus_minus_set_a @ A2 @ B ) ) ) ).
% image_diff_subset
thf(fact_846_Diff__insert,axiom,
! [A2: set_a,A: a,B: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B ) @ ( insert_a @ A @ bot_bot_set_a ) ) ) ).
% Diff_insert
thf(fact_847_insert__Diff,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_848_Diff__insert2,axiom,
! [A2: set_a,A: a,B: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) @ B ) ) ).
% Diff_insert2
thf(fact_849_Diff__insert__absorb,axiom,
! [X: a,A2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ ( insert_a @ X @ bot_bot_set_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_850_subset__Diff__insert,axiom,
! [A2: set_a,B: set_a,X: a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B @ ( insert_a @ X @ C ) ) )
= ( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B @ C ) )
& ~ ( member_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_851_Diff__triv,axiom,
! [A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a )
=> ( ( minus_minus_set_a @ A2 @ B )
= A2 ) ) ).
% Diff_triv
thf(fact_852_Int__Diff__disjoint,axiom,
! [A2: set_a,B: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( minus_minus_set_a @ A2 @ B ) )
= bot_bot_set_a ) ).
% Int_Diff_disjoint
thf(fact_853_Diff__subset__conv,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B ) @ C )
= ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ B @ C ) ) ) ).
% Diff_subset_conv
thf(fact_854_Diff__partition,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( sup_sup_set_a @ A2 @ ( minus_minus_set_a @ B @ A2 ) )
= B ) ) ).
% Diff_partition
thf(fact_855_Un__Diff__Int,axiom,
! [A2: set_a,B: set_a] :
( ( sup_sup_set_a @ ( minus_minus_set_a @ A2 @ B ) @ ( inf_inf_set_a @ A2 @ B ) )
= A2 ) ).
% Un_Diff_Int
thf(fact_856_Int__Diff__Un,axiom,
! [A2: set_a,B: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( minus_minus_set_a @ A2 @ B ) )
= A2 ) ).
% Int_Diff_Un
thf(fact_857_Diff__Int,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( minus_minus_set_a @ A2 @ ( inf_inf_set_a @ B @ C ) )
= ( sup_sup_set_a @ ( minus_minus_set_a @ A2 @ B ) @ ( minus_minus_set_a @ A2 @ C ) ) ) ).
% Diff_Int
thf(fact_858_Diff__Un,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( minus_minus_set_a @ A2 @ ( sup_sup_set_a @ B @ C ) )
= ( inf_inf_set_a @ ( minus_minus_set_a @ A2 @ B ) @ ( minus_minus_set_a @ A2 @ C ) ) ) ).
% Diff_Un
thf(fact_859_infinite__remove,axiom,
! [S: set_a,A: a] :
( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% infinite_remove
thf(fact_860_infinite__coinduct,axiom,
! [X4: set_a > $o,A2: set_a] :
( ( X4 @ A2 )
=> ( ! [A8: set_a] :
( ( X4 @ A8 )
=> ? [X5: a] :
( ( member_a @ X5 @ A8 )
& ( ( X4 @ ( minus_minus_set_a @ A8 @ ( insert_a @ X5 @ bot_bot_set_a ) ) )
| ~ ( finite_finite_a @ ( minus_minus_set_a @ A8 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) ) ) )
=> ~ ( finite_finite_a @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_861_finite__empty__induct,axiom,
! [A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ A2 )
=> ( ( P @ A2 )
=> ( ! [A3: a,A8: set_a] :
( ( finite_finite_a @ A8 )
=> ( ( member_a @ A3 @ A8 )
=> ( ( P @ A8 )
=> ( P @ ( minus_minus_set_a @ A8 @ ( insert_a @ A3 @ bot_bot_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_862_subset__insert__iff,axiom,
! [A2: set_a,X: a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B ) )
= ( ( ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_863_Diff__single__insert,axiom,
! [A2: set_a,X: a,B: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B ) ) ) ).
% Diff_single_insert
thf(fact_864_finite__remove__induct,axiom,
! [B: set_a,P: set_a > $o] :
( ( finite_finite_a @ B )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A8: set_a] :
( ( finite_finite_a @ A8 )
=> ( ( A8 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A8 @ B )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A8 )
=> ( P @ ( minus_minus_set_a @ A8 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_865_remove__induct,axiom,
! [P: set_a > $o,B: set_a] :
( ( P @ bot_bot_set_a )
=> ( ( ~ ( finite_finite_a @ B )
=> ( P @ B ) )
=> ( ! [A8: set_a] :
( ( finite_finite_a @ A8 )
=> ( ( A8 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A8 @ B )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A8 )
=> ( P @ ( minus_minus_set_a @ A8 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_866_Inf__fin_Oremove,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ X @ A2 )
=> ( ( ( ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
= bot_bot_set_set_a )
=> ( ( lattic8209813465164889211_set_a @ A2 )
= X ) )
& ( ( ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
!= bot_bot_set_set_a )
=> ( ( lattic8209813465164889211_set_a @ A2 )
= ( inf_inf_set_a @ X @ ( lattic8209813465164889211_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_867_remove__def,axiom,
( remove_a
= ( ^ [X3: a,A6: set_a] : ( minus_minus_set_a @ A6 @ ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ).
% remove_def
thf(fact_868_image__Fpow__mono,axiom,
! [F3: a > set_a,A2: set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F3 @ A2 ) @ B )
=> ( ord_le5722252365846178494_set_a @ ( image_4955109552351689957_set_a @ ( image_a_set_a @ F3 ) @ ( finite_Fpow_a @ A2 ) ) @ ( finite_Fpow_set_a @ B ) ) ) ).
% image_Fpow_mono
thf(fact_869_image__Fpow__mono,axiom,
! [F3: a > a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F3 @ A2 ) @ B )
=> ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ ( image_a_a @ F3 ) @ ( finite_Fpow_a @ A2 ) ) @ ( finite_Fpow_a @ B ) ) ) ).
% image_Fpow_mono
thf(fact_870_Set_Ois__empty__def,axiom,
( is_empty_a
= ( ^ [A6: set_a] : ( A6 = bot_bot_set_a ) ) ) ).
% Set.is_empty_def
thf(fact_871_Units__def,axiom,
( ( group_Units_a @ g @ addition @ zero )
= ( collect_a
@ ^ [U2: a] :
( ( member_a @ U2 @ g )
& ( group_invertible_a @ g @ addition @ zero @ U2 ) ) ) ) ).
% Units_def
thf(fact_872_image__ident,axiom,
! [Y5: set_a] :
( ( image_a_a
@ ^ [X3: a] : X3
@ Y5 )
= Y5 ) ).
% image_ident
thf(fact_873_finite__Collect__conjI,axiom,
! [P: a > $o,Q: a > $o] :
( ( ( finite_finite_a @ ( collect_a @ P ) )
| ( finite_finite_a @ ( collect_a @ Q ) ) )
=> ( finite_finite_a
@ ( collect_a
@ ^ [X3: a] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_874_finite__Collect__disjI,axiom,
! [P: a > $o,Q: a > $o] :
( ( finite_finite_a
@ ( collect_a
@ ^ [X3: a] :
( ( P @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite_finite_a @ ( collect_a @ P ) )
& ( finite_finite_a @ ( collect_a @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_875_sumset__def,axiom,
( ( pluenn3038260743871226533mset_a @ g @ addition )
= ( ^ [A6: set_a,B6: set_a] :
( collect_a
@ ( pluenn895083305082786853setp_a @ g @ addition
@ ^ [X3: a] : ( member_a @ X3 @ A6 )
@ ^ [X3: a] : ( member_a @ X3 @ B6 ) ) ) ) ) ).
% sumset_def
thf(fact_876_sumsetp__sumset__eq,axiom,
! [A2: set_a,B: set_a] :
( ( pluenn895083305082786853setp_a @ g @ addition
@ ^ [X3: a] : ( member_a @ X3 @ A2 )
@ ^ [X3: a] : ( member_a @ X3 @ B ) )
= ( ^ [X3: a] : ( member_a @ X3 @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ) ) ).
% sumsetp_sumset_eq
thf(fact_877_member__remove,axiom,
! [X: a,Y: a,A2: set_a] :
( ( member_a @ X @ ( remove_a @ Y @ A2 ) )
= ( ( member_a @ X @ A2 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_878_finite__Collect__subsets,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_set_a
@ ( collect_set_a
@ ^ [B6: set_a] : ( ord_less_eq_set_a @ B6 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_879_singleton__conv,axiom,
! [A: a] :
( ( collect_a
@ ^ [X3: a] : ( X3 = A ) )
= ( insert_a @ A @ bot_bot_set_a ) ) ).
% singleton_conv
thf(fact_880_singleton__conv2,axiom,
! [A: a] :
( ( collect_a
@ ( ^ [Y4: a,Z: a] : ( Y4 = Z )
@ A ) )
= ( insert_a @ A @ bot_bot_set_a ) ) ).
% singleton_conv2
thf(fact_881_if__image__distrib,axiom,
! [P: a > $o,F3: a > set_a,G4: a > set_a,S: set_a] :
( ( image_a_set_a
@ ^ [X3: a] : ( if_set_a @ ( P @ X3 ) @ ( F3 @ X3 ) @ ( G4 @ X3 ) )
@ S )
= ( sup_sup_set_set_a @ ( image_a_set_a @ F3 @ ( inf_inf_set_a @ S @ ( collect_a @ P ) ) )
@ ( image_a_set_a @ G4
@ ( inf_inf_set_a @ S
@ ( collect_a
@ ^ [X3: a] :
~ ( P @ X3 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_882_if__image__distrib,axiom,
! [P: a > $o,F3: a > a,G4: a > a,S: set_a] :
( ( image_a_a
@ ^ [X3: a] : ( if_a @ ( P @ X3 ) @ ( F3 @ X3 ) @ ( G4 @ X3 ) )
@ S )
= ( sup_sup_set_a @ ( image_a_a @ F3 @ ( inf_inf_set_a @ S @ ( collect_a @ P ) ) )
@ ( image_a_a @ G4
@ ( inf_inf_set_a @ S
@ ( collect_a
@ ^ [X3: a] :
~ ( P @ X3 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_883_Compr__image__eq,axiom,
! [F3: a > set_a,A2: set_a,P: set_a > $o] :
( ( collect_set_a
@ ^ [X3: set_a] :
( ( member_set_a @ X3 @ ( image_a_set_a @ F3 @ A2 ) )
& ( P @ X3 ) ) )
= ( image_a_set_a @ F3
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A2 )
& ( P @ ( F3 @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_884_Compr__image__eq,axiom,
! [F3: a > a,A2: set_a,P: a > $o] :
( ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ ( image_a_a @ F3 @ A2 ) )
& ( P @ X3 ) ) )
= ( image_a_a @ F3
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A2 )
& ( P @ ( F3 @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_885_image__image,axiom,
! [F3: set_a > a,G4: a > set_a,A2: set_a] :
( ( image_set_a_a @ F3 @ ( image_a_set_a @ G4 @ A2 ) )
= ( image_a_a
@ ^ [X3: a] : ( F3 @ ( G4 @ X3 ) )
@ A2 ) ) ).
% image_image
thf(fact_886_image__image,axiom,
! [F3: set_a > set_a,G4: a > set_a,A2: set_a] :
( ( image_set_a_set_a @ F3 @ ( image_a_set_a @ G4 @ A2 ) )
= ( image_a_set_a
@ ^ [X3: a] : ( F3 @ ( G4 @ X3 ) )
@ A2 ) ) ).
% image_image
thf(fact_887_image__image,axiom,
! [F3: a > a,G4: a > a,A2: set_a] :
( ( image_a_a @ F3 @ ( image_a_a @ G4 @ A2 ) )
= ( image_a_a
@ ^ [X3: a] : ( F3 @ ( G4 @ X3 ) )
@ A2 ) ) ).
% image_image
thf(fact_888_image__image,axiom,
! [F3: a > set_a,G4: a > a,A2: set_a] :
( ( image_a_set_a @ F3 @ ( image_a_a @ G4 @ A2 ) )
= ( image_a_set_a
@ ^ [X3: a] : ( F3 @ ( G4 @ X3 ) )
@ A2 ) ) ).
% image_image
thf(fact_889_imageE,axiom,
! [B4: set_a,F3: a > set_a,A2: set_a] :
( ( member_set_a @ B4 @ ( image_a_set_a @ F3 @ A2 ) )
=> ~ ! [X2: a] :
( ( B4
= ( F3 @ X2 ) )
=> ~ ( member_a @ X2 @ A2 ) ) ) ).
% imageE
thf(fact_890_imageE,axiom,
! [B4: a,F3: a > a,A2: set_a] :
( ( member_a @ B4 @ ( image_a_a @ F3 @ A2 ) )
=> ~ ! [X2: a] :
( ( B4
= ( F3 @ X2 ) )
=> ~ ( member_a @ X2 @ A2 ) ) ) ).
% imageE
thf(fact_891_pigeonhole__infinite,axiom,
! [A2: set_a,F3: a > set_a] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_finite_set_a @ ( image_a_set_a @ F3 @ A2 ) )
=> ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A2 )
& ( ( F3 @ A4 )
= ( F3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_892_pigeonhole__infinite,axiom,
! [A2: set_a,F3: a > a] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ ( image_a_a @ F3 @ A2 ) )
=> ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A2 )
& ( ( F3 @ A4 )
= ( F3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_893_image__Collect__subsetI,axiom,
! [P: a > $o,F3: a > set_a,B: set_set_a] :
( ! [X2: a] :
( ( P @ X2 )
=> ( member_set_a @ ( F3 @ X2 ) @ B ) )
=> ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F3 @ ( collect_a @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_894_image__Collect__subsetI,axiom,
! [P: a > $o,F3: a > a,B: set_a] :
( ! [X2: a] :
( ( P @ X2 )
=> ( member_a @ ( F3 @ X2 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F3 @ ( collect_a @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_895_image__constant,axiom,
! [X: a,A2: set_a,C2: set_a] :
( ( member_a @ X @ A2 )
=> ( ( image_a_set_a
@ ^ [X3: a] : C2
@ A2 )
= ( insert_set_a @ C2 @ bot_bot_set_set_a ) ) ) ).
% image_constant
thf(fact_896_image__constant,axiom,
! [X: a,A2: set_a,C2: a] :
( ( member_a @ X @ A2 )
=> ( ( image_a_a
@ ^ [X3: a] : C2
@ A2 )
= ( insert_a @ C2 @ bot_bot_set_a ) ) ) ).
% image_constant
thf(fact_897_image__constant__conv,axiom,
! [A2: set_a,C2: set_a] :
( ( ( A2 = bot_bot_set_a )
=> ( ( image_a_set_a
@ ^ [X3: a] : C2
@ A2 )
= bot_bot_set_set_a ) )
& ( ( A2 != bot_bot_set_a )
=> ( ( image_a_set_a
@ ^ [X3: a] : C2
@ A2 )
= ( insert_set_a @ C2 @ bot_bot_set_set_a ) ) ) ) ).
% image_constant_conv
thf(fact_898_image__constant__conv,axiom,
! [A2: set_a,C2: a] :
( ( ( A2 = bot_bot_set_a )
=> ( ( image_a_a
@ ^ [X3: a] : C2
@ A2 )
= bot_bot_set_a ) )
& ( ( A2 != bot_bot_set_a )
=> ( ( image_a_a
@ ^ [X3: a] : C2
@ A2 )
= ( insert_a @ C2 @ bot_bot_set_a ) ) ) ) ).
% image_constant_conv
thf(fact_899_set__diff__eq,axiom,
( minus_minus_set_a
= ( ^ [A6: set_a,B6: set_a] :
( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A6 )
& ~ ( member_a @ X3 @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_900_minus__set__def,axiom,
( minus_minus_set_a
= ( ^ [A6: set_a,B6: set_a] :
( collect_a
@ ( minus_minus_a_o
@ ^ [X3: a] : ( member_a @ X3 @ A6 )
@ ^ [X3: a] : ( member_a @ X3 @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_901_additive__abelian__group_Osumsetp__sumset__eq,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn895083305082786853setp_a @ G @ Addition
@ ^ [X3: a] : ( member_a @ X3 @ A2 )
@ ^ [X3: a] : ( member_a @ X3 @ B ) )
= ( ^ [X3: a] : ( member_a @ X3 @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ) ) ).
% additive_abelian_group.sumsetp_sumset_eq
thf(fact_902_additive__abelian__group_Osumset__def,axiom,
! [G: set_a,Addition: a > a > a,Zero: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition )
= ( ^ [A6: set_a,B6: set_a] :
( collect_a
@ ( pluenn895083305082786853setp_a @ G @ Addition
@ ^ [X3: a] : ( member_a @ X3 @ A6 )
@ ^ [X3: a] : ( member_a @ X3 @ B6 ) ) ) ) ) ) ).
% additive_abelian_group.sumset_def
thf(fact_903_sup__Un__eq,axiom,
! [R: set_a,S: set_a] :
( ( sup_sup_a_o
@ ^ [X3: a] : ( member_a @ X3 @ R )
@ ^ [X3: a] : ( member_a @ X3 @ S ) )
= ( ^ [X3: a] : ( member_a @ X3 @ ( sup_sup_set_a @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_904_inf__Int__eq,axiom,
! [R: set_a,S: set_a] :
( ( inf_inf_a_o
@ ^ [X3: a] : ( member_a @ X3 @ R )
@ ^ [X3: a] : ( member_a @ X3 @ S ) )
= ( ^ [X3: a] : ( member_a @ X3 @ ( inf_inf_set_a @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_905_Collect__conv__if2,axiom,
! [P: a > $o,A: a] :
( ( ( P @ A )
=> ( ( collect_a
@ ^ [X3: a] :
( ( A = X3 )
& ( P @ X3 ) ) )
= ( insert_a @ A @ bot_bot_set_a ) ) )
& ( ~ ( P @ A )
=> ( ( collect_a
@ ^ [X3: a] :
( ( A = X3 )
& ( P @ X3 ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if2
thf(fact_906_Collect__conv__if,axiom,
! [P: a > $o,A: a] :
( ( ( P @ A )
=> ( ( collect_a
@ ^ [X3: a] :
( ( X3 = A )
& ( P @ X3 ) ) )
= ( insert_a @ A @ bot_bot_set_a ) ) )
& ( ~ ( P @ A )
=> ( ( collect_a
@ ^ [X3: a] :
( ( X3 = A )
& ( P @ X3 ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if
thf(fact_907_insert__compr,axiom,
( insert_a
= ( ^ [A4: a,B6: set_a] :
( collect_a
@ ^ [X3: a] :
( ( X3 = A4 )
| ( member_a @ X3 @ B6 ) ) ) ) ) ).
% insert_compr
thf(fact_908_insert__Collect,axiom,
! [A: a,P: a > $o] :
( ( insert_a @ A @ ( collect_a @ P ) )
= ( collect_a
@ ^ [U2: a] :
( ( U2 != A )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_909_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_910_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_911_Fpow__def,axiom,
( finite_Fpow_a
= ( ^ [A6: set_a] :
( collect_set_a
@ ^ [X6: set_a] :
( ( ord_less_eq_set_a @ X6 @ A6 )
& ( finite_finite_a @ X6 ) ) ) ) ) ).
% Fpow_def
thf(fact_912_Collect__subset,axiom,
! [A2: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_913_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
( ord_less_eq_a_o
@ ^ [X3: a] : ( member_a @ X3 @ A6 )
@ ^ [X3: a] : ( member_a @ X3 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_914_pred__subset__eq,axiom,
! [R: set_a,S: set_a] :
( ( ord_less_eq_a_o
@ ^ [X3: a] : ( member_a @ X3 @ R )
@ ^ [X3: a] : ( member_a @ X3 @ S ) )
= ( ord_less_eq_set_a @ R @ S ) ) ).
% pred_subset_eq
thf(fact_915_insert__def,axiom,
( insert_a
= ( ^ [A4: a] :
( sup_sup_set_a
@ ( collect_a
@ ^ [X3: a] : ( X3 = A4 ) ) ) ) ) ).
% insert_def
thf(fact_916_not__finite__existsD,axiom,
! [P: a > $o] :
( ~ ( finite_finite_a @ ( collect_a @ P ) )
=> ? [X_1: a] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_917_pigeonhole__infinite__rel,axiom,
! [A2: set_a,B: set_a,R: a > a > $o] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ? [Xa: a] :
( ( member_a @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ B )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A2 )
& ( R @ A4 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_918_empty__def,axiom,
( bot_bot_set_a
= ( collect_a
@ ^ [X3: a] : $false ) ) ).
% empty_def
thf(fact_919_Collect__disj__eq,axiom,
! [P: a > $o,Q: a > $o] :
( ( collect_a
@ ^ [X3: a] :
( ( P @ X3 )
| ( Q @ X3 ) ) )
= ( sup_sup_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_920_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_921_sup__set__def,axiom,
( sup_sup_set_a
= ( ^ [A6: set_a,B6: set_a] :
( collect_a
@ ( sup_sup_a_o
@ ^ [X3: a] : ( member_a @ X3 @ A6 )
@ ^ [X3: a] : ( member_a @ X3 @ B6 ) ) ) ) ) ).
% sup_set_def
thf(fact_922_inf__set__def,axiom,
( inf_inf_set_a
= ( ^ [A6: set_a,B6: set_a] :
( collect_a
@ ( inf_inf_a_o
@ ^ [X3: a] : ( member_a @ X3 @ A6 )
@ ^ [X3: a] : ( member_a @ X3 @ B6 ) ) ) ) ) ).
% inf_set_def
thf(fact_923_Int__Collect,axiom,
! [X: a,A2: set_a,P: a > $o] :
( ( member_a @ X @ ( inf_inf_set_a @ A2 @ ( collect_a @ P ) ) )
= ( ( member_a @ X @ A2 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_924_Int__def,axiom,
( inf_inf_set_a
= ( ^ [A6: set_a,B6: set_a] :
( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A6 )
& ( member_a @ X3 @ B6 ) ) ) ) ) ).
% Int_def
thf(fact_925_Un__def,axiom,
( sup_sup_set_a
= ( ^ [A6: set_a,B6: set_a] :
( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A6 )
| ( member_a @ X3 @ B6 ) ) ) ) ) ).
% Un_def
thf(fact_926_empty__in__Fpow,axiom,
! [A2: set_a] : ( member_set_a @ bot_bot_set_a @ ( finite_Fpow_a @ A2 ) ) ).
% empty_in_Fpow
thf(fact_927_Fpow__mono,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_le3724670747650509150_set_a @ ( finite_Fpow_a @ A2 ) @ ( finite_Fpow_a @ B ) ) ) ).
% Fpow_mono
thf(fact_928_sumset,axiom,
! [A2: set_a,B: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [A4: a] :
( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [B3: a] : ( insert_a @ ( addition @ A4 @ B3 ) @ bot_bot_set_a )
@ ( inf_inf_set_a @ B @ g ) ) )
@ ( inf_inf_set_a @ A2 @ g ) ) ) ) ).
% sumset
thf(fact_929_sumset__eq,axiom,
! [A2: set_a,B: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B )
= ( collect_a
@ ^ [C5: a] :
? [X3: a] :
( ( member_a @ X3 @ ( inf_inf_set_a @ A2 @ g ) )
& ? [Y3: a] :
( ( member_a @ Y3 @ ( inf_inf_set_a @ B @ g ) )
& ( C5
= ( addition @ X3 @ Y3 ) ) ) ) ) ) ).
% sumset_eq
thf(fact_930_additive__abelian__group_Osumset,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [A4: a] :
( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [B3: a] : ( insert_a @ ( Addition @ A4 @ B3 ) @ bot_bot_set_a )
@ ( inf_inf_set_a @ B @ G ) ) )
@ ( inf_inf_set_a @ A2 @ G ) ) ) ) ) ).
% additive_abelian_group.sumset
thf(fact_931_bex__empty,axiom,
! [P: a > $o] :
~ ? [X5: a] :
( ( member_a @ X5 @ bot_bot_set_a )
& ( P @ X5 ) ) ).
% bex_empty
thf(fact_932_finite__Collect__bex,axiom,
! [A2: set_a,Q: a > a > $o] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a
@ ( collect_a
@ ^ [X3: a] :
? [Y3: a] :
( ( member_a @ Y3 @ A2 )
& ( Q @ X3 @ Y3 ) ) ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( finite_finite_a
@ ( collect_a
@ ^ [Y3: a] : ( Q @ Y3 @ X3 ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_933_finite__UN,axiom,
! [A2: set_a,B: a > set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A2 ) ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( finite_finite_a @ ( B @ X3 ) ) ) ) ) ) ).
% finite_UN
thf(fact_934_finite__Union,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ! [M2: set_a] :
( ( member_set_a @ M2 @ A2 )
=> ( finite_finite_a @ M2 ) )
=> ( finite_finite_a @ ( comple2307003609928055243_set_a @ A2 ) ) ) ) ).
% finite_Union
thf(fact_935_finite__UN__I,axiom,
! [A2: set_a,B: a > set_a] :
( ( finite_finite_a @ A2 )
=> ( ! [A3: a] :
( ( member_a @ A3 @ A2 )
=> ( finite_finite_a @ ( B @ A3 ) ) )
=> ( finite_finite_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A2 ) ) ) ) ) ).
% finite_UN_I
thf(fact_936_image__def,axiom,
( image_a_set_a
= ( ^ [F2: a > set_a,A6: set_a] :
( collect_set_a
@ ^ [Y3: set_a] :
? [X3: a] :
( ( member_a @ X3 @ A6 )
& ( Y3
= ( F2 @ X3 ) ) ) ) ) ) ).
% image_def
thf(fact_937_image__def,axiom,
( image_a_a
= ( ^ [F2: a > a,A6: set_a] :
( collect_a
@ ^ [Y3: a] :
? [X3: a] :
( ( member_a @ X3 @ A6 )
& ( Y3
= ( F2 @ X3 ) ) ) ) ) ) ).
% image_def
thf(fact_938_finite__UnionD,axiom,
! [A2: set_set_a] :
( ( finite_finite_a @ ( comple2307003609928055243_set_a @ A2 ) )
=> ( finite_finite_set_a @ A2 ) ) ).
% finite_UnionD
thf(fact_939_Bex__def,axiom,
( bex_a
= ( ^ [A6: set_a,P2: a > $o] :
? [X3: a] :
( ( member_a @ X3 @ A6 )
& ( P2 @ X3 ) ) ) ) ).
% Bex_def
thf(fact_940_insert__partition,axiom,
! [X: set_a,F: set_set_a] :
( ~ ( member_set_a @ X @ F )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ ( insert_set_a @ X @ F ) )
=> ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ ( insert_set_a @ X @ F ) )
=> ( ( X2 != Xa2 )
=> ( ( inf_inf_set_a @ X2 @ Xa2 )
= bot_bot_set_a ) ) ) )
=> ( ( inf_inf_set_a @ X @ ( comple2307003609928055243_set_a @ F ) )
= bot_bot_set_a ) ) ) ).
% insert_partition
thf(fact_941_finite__Sup__in,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ! [X2: set_a,Y2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( ( member_set_a @ Y2 @ A2 )
=> ( member_set_a @ ( sup_sup_set_a @ X2 @ Y2 ) @ A2 ) ) )
=> ( member_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ A2 ) ) ) ) ).
% finite_Sup_in
thf(fact_942_Sup__fin__Sup,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ( lattic2918178356826803221_set_a @ A2 )
= ( comple2307003609928055243_set_a @ A2 ) ) ) ) ).
% Sup_fin_Sup
thf(fact_943_additive__abelian__group_Osumset__eq,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B )
= ( collect_a
@ ^ [C5: a] :
? [X3: a] :
( ( member_a @ X3 @ ( inf_inf_set_a @ A2 @ G ) )
& ? [Y3: a] :
( ( member_a @ Y3 @ ( inf_inf_set_a @ B @ G ) )
& ( C5
= ( Addition @ X3 @ Y3 ) ) ) ) ) ) ) ).
% additive_abelian_group.sumset_eq
thf(fact_944_UN__insert,axiom,
! [B: a > set_a,A: a,A2: set_a] :
( ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ ( insert_a @ A @ A2 ) ) )
= ( sup_sup_set_a @ ( B @ A ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A2 ) ) ) ) ).
% UN_insert
thf(fact_945_UN__simps_I2_J,axiom,
! [C: set_a,A2: a > set_a,B: set_a] :
( ( ( C = bot_bot_set_a )
=> ( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( sup_sup_set_a @ ( A2 @ X3 ) @ B )
@ C ) )
= bot_bot_set_a ) )
& ( ( C != bot_bot_set_a )
=> ( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( sup_sup_set_a @ ( A2 @ X3 ) @ B )
@ C ) )
= ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ A2 @ C ) ) @ B ) ) ) ) ).
% UN_simps(2)
thf(fact_946_UN__simps_I3_J,axiom,
! [C: set_a,A2: set_a,B: a > set_a] :
( ( ( C = bot_bot_set_a )
=> ( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( sup_sup_set_a @ A2 @ ( B @ X3 ) )
@ C ) )
= bot_bot_set_a ) )
& ( ( C != bot_bot_set_a )
=> ( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( sup_sup_set_a @ A2 @ ( B @ X3 ) )
@ C ) )
= ( sup_sup_set_a @ A2 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ C ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_947_Sup__bot__conv_I2_J,axiom,
! [A2: set_set_a] :
( ( bot_bot_set_a
= ( comple2307003609928055243_set_a @ A2 ) )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( X3 = bot_bot_set_a ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_948_Sup__bot__conv_I1_J,axiom,
! [A2: set_set_a] :
( ( ( comple2307003609928055243_set_a @ A2 )
= bot_bot_set_a )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( X3 = bot_bot_set_a ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_949_Union__Un__distrib,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( comple2307003609928055243_set_a @ ( sup_sup_set_set_a @ A2 @ B ) )
= ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Union_Un_distrib
thf(fact_950_Sup__empty,axiom,
( ( comple2307003609928055243_set_a @ bot_bot_set_set_a )
= bot_bot_set_a ) ).
% Sup_empty
thf(fact_951_Sup__insert,axiom,
! [A: set_a,A2: set_set_a] :
( ( comple2307003609928055243_set_a @ ( insert_set_a @ A @ A2 ) )
= ( sup_sup_set_a @ A @ ( comple2307003609928055243_set_a @ A2 ) ) ) ).
% Sup_insert
thf(fact_952_SUP__bot,axiom,
! [A2: set_a] :
( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X3: a] : bot_bot_set_a
@ A2 ) )
= bot_bot_set_a ) ).
% SUP_bot
thf(fact_953_SUP__bot__conv_I1_J,axiom,
! [B: a > set_a,A2: set_a] :
( ( ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A2 ) )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ( B @ X3 )
= bot_bot_set_a ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_954_SUP__bot__conv_I2_J,axiom,
! [B: a > set_a,A2: set_a] :
( ( bot_bot_set_a
= ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A2 ) ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ( B @ X3 )
= bot_bot_set_a ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_955_SUP__const,axiom,
! [A2: set_a,F3: set_a] :
( ( A2 != bot_bot_set_a )
=> ( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [I: a] : F3
@ A2 ) )
= F3 ) ) ).
% SUP_const
thf(fact_956_UN__constant,axiom,
! [A2: set_a,C2: set_a] :
( ( ( A2 = bot_bot_set_a )
=> ( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [Y3: a] : C2
@ A2 ) )
= bot_bot_set_a ) )
& ( ( A2 != bot_bot_set_a )
=> ( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [Y3: a] : C2
@ A2 ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_957_UN__Un,axiom,
! [M: a > set_a,A2: set_a,B: set_a] :
( ( comple2307003609928055243_set_a @ ( image_a_set_a @ M @ ( sup_sup_set_a @ A2 @ B ) ) )
= ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ M @ A2 ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ M @ B ) ) ) ) ).
% UN_Un
thf(fact_958_UN__simps_I1_J,axiom,
! [C: set_a,A: a,B: a > set_a] :
( ( ( C = bot_bot_set_a )
=> ( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( insert_a @ A @ ( B @ X3 ) )
@ C ) )
= bot_bot_set_a ) )
& ( ( C != bot_bot_set_a )
=> ( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( insert_a @ A @ ( B @ X3 ) )
@ C ) )
= ( insert_a @ A @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ C ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_959_UN__singleton,axiom,
! [A2: set_a] :
( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( insert_a @ X3 @ bot_bot_set_a )
@ A2 ) )
= A2 ) ).
% UN_singleton
thf(fact_960_SUP__Sup__eq,axiom,
! [S: set_set_a] :
( ( complete_Sup_Sup_a_o
@ ( image_set_a_a_o
@ ^ [I: set_a,X3: a] : ( member_a @ X3 @ I )
@ S ) )
= ( ^ [X3: a] : ( member_a @ X3 @ ( comple2307003609928055243_set_a @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_961_Sup__SUP__eq,axiom,
( complete_Sup_Sup_a_o
= ( ^ [S2: set_a_o,X3: a] : ( member_a @ X3 @ ( comple2307003609928055243_set_a @ ( image_a_o_set_a @ collect_a @ S2 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_962_SUP__UN__eq,axiom,
! [R2: a > set_a,S: set_a] :
( ( complete_Sup_Sup_a_o
@ ( image_a_a_o
@ ^ [I: a,X3: a] : ( member_a @ X3 @ ( R2 @ I ) )
@ S ) )
= ( ^ [X3: a] : ( member_a @ X3 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ R2 @ S ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_963_Sup__eqI,axiom,
! [A2: set_set_a,X: set_a] :
( ! [Y2: set_a] :
( ( member_set_a @ Y2 @ A2 )
=> ( ord_less_eq_set_a @ Y2 @ X ) )
=> ( ! [Y2: set_a] :
( ! [Z5: set_a] :
( ( member_set_a @ Z5 @ A2 )
=> ( ord_less_eq_set_a @ Z5 @ Y2 ) )
=> ( ord_less_eq_set_a @ X @ Y2 ) )
=> ( ( comple2307003609928055243_set_a @ A2 )
= X ) ) ) ).
% Sup_eqI
thf(fact_964_Sup__mono,axiom,
! [A2: set_set_a,B: set_set_a] :
( ! [A3: set_a] :
( ( member_set_a @ A3 @ A2 )
=> ? [X5: set_a] :
( ( member_set_a @ X5 @ B )
& ( ord_less_eq_set_a @ A3 @ X5 ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Sup_mono
thf(fact_965_Sup__least,axiom,
! [A2: set_set_a,Z2: set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ X2 @ Z2 ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ Z2 ) ) ).
% Sup_least
thf(fact_966_Sup__upper,axiom,
! [X: set_a,A2: set_set_a] :
( ( member_set_a @ X @ A2 )
=> ( ord_less_eq_set_a @ X @ ( comple2307003609928055243_set_a @ A2 ) ) ) ).
% Sup_upper
thf(fact_967_Sup__le__iff,axiom,
! [A2: set_set_a,B4: set_a] :
( ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ B4 )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( ord_less_eq_set_a @ X3 @ B4 ) ) ) ) ).
% Sup_le_iff
thf(fact_968_Sup__upper2,axiom,
! [U: set_a,A2: set_set_a,V2: set_a] :
( ( member_set_a @ U @ A2 )
=> ( ( ord_less_eq_set_a @ V2 @ U )
=> ( ord_less_eq_set_a @ V2 @ ( comple2307003609928055243_set_a @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_969_empty__Union__conv,axiom,
! [A2: set_set_a] :
( ( bot_bot_set_a
= ( comple2307003609928055243_set_a @ A2 ) )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( X3 = bot_bot_set_a ) ) ) ) ).
% empty_Union_conv
thf(fact_970_Union__empty__conv,axiom,
! [A2: set_set_a] :
( ( ( comple2307003609928055243_set_a @ A2 )
= bot_bot_set_a )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( X3 = bot_bot_set_a ) ) ) ) ).
% Union_empty_conv
thf(fact_971_Union__least,axiom,
! [A2: set_set_a,C: set_a] :
( ! [X7: set_a] :
( ( member_set_a @ X7 @ A2 )
=> ( ord_less_eq_set_a @ X7 @ C ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ C ) ) ).
% Union_least
thf(fact_972_Union__upper,axiom,
! [B: set_a,A2: set_set_a] :
( ( member_set_a @ B @ A2 )
=> ( ord_less_eq_set_a @ B @ ( comple2307003609928055243_set_a @ A2 ) ) ) ).
% Union_upper
thf(fact_973_Union__subsetI,axiom,
! [A2: set_set_a,B: set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ? [Y6: set_a] :
( ( member_set_a @ Y6 @ B )
& ( ord_less_eq_set_a @ X2 @ Y6 ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Union_subsetI
thf(fact_974_Union__insert,axiom,
! [A: set_a,B: set_set_a] :
( ( comple2307003609928055243_set_a @ ( insert_set_a @ A @ B ) )
= ( sup_sup_set_a @ A @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Union_insert
thf(fact_975_SUP__eq,axiom,
! [A2: set_a,B: set_a,F3: a > set_a,G4: a > set_a] :
( ! [I2: a] :
( ( member_a @ I2 @ A2 )
=> ? [X5: a] :
( ( member_a @ X5 @ B )
& ( ord_less_eq_set_a @ ( F3 @ I2 ) @ ( G4 @ X5 ) ) ) )
=> ( ! [J: a] :
( ( member_a @ J @ B )
=> ? [X5: a] :
( ( member_a @ X5 @ A2 )
& ( ord_less_eq_set_a @ ( G4 @ J ) @ ( F3 @ X5 ) ) ) )
=> ( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F3 @ A2 ) )
= ( comple2307003609928055243_set_a @ ( image_a_set_a @ G4 @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_976_less__eq__Sup,axiom,
! [A2: set_set_a,U: set_a] :
( ! [V3: set_a] :
( ( member_set_a @ V3 @ A2 )
=> ( ord_less_eq_set_a @ U @ V3 ) )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ U @ ( comple2307003609928055243_set_a @ A2 ) ) ) ) ).
% less_eq_Sup
thf(fact_977_Sup__subset__mono,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Sup_subset_mono
thf(fact_978_SUP__eq__const,axiom,
! [I3: set_a,F3: a > set_a,X: set_a] :
( ( I3 != bot_bot_set_a )
=> ( ! [I2: a] :
( ( member_a @ I2 @ I3 )
=> ( ( F3 @ I2 )
= X ) )
=> ( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F3 @ I3 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_979_Union__disjoint,axiom,
! [C: set_set_a,A2: set_a] :
( ( ( inf_inf_set_a @ ( comple2307003609928055243_set_a @ C ) @ A2 )
= bot_bot_set_a )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ C )
=> ( ( inf_inf_set_a @ X3 @ A2 )
= bot_bot_set_a ) ) ) ) ).
% Union_disjoint
thf(fact_980_Union__Int__subset,axiom,
! [A2: set_set_a,B: set_set_a] : ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( inf_inf_set_set_a @ A2 @ B ) ) @ ( inf_inf_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Union_Int_subset
thf(fact_981_Sup__union__distrib,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( comple2307003609928055243_set_a @ ( sup_sup_set_set_a @ A2 @ B ) )
= ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Sup_union_distrib
thf(fact_982_Union__mono,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Union_mono
thf(fact_983_Union__empty,axiom,
( ( comple2307003609928055243_set_a @ bot_bot_set_set_a )
= bot_bot_set_a ) ).
% Union_empty
thf(fact_984_SUP__eqI,axiom,
! [A2: set_a,F3: a > set_a,X: set_a] :
( ! [I2: a] :
( ( member_a @ I2 @ A2 )
=> ( ord_less_eq_set_a @ ( F3 @ I2 ) @ X ) )
=> ( ! [Y2: set_a] :
( ! [I4: a] :
( ( member_a @ I4 @ A2 )
=> ( ord_less_eq_set_a @ ( F3 @ I4 ) @ Y2 ) )
=> ( ord_less_eq_set_a @ X @ Y2 ) )
=> ( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F3 @ A2 ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_985_SUP__mono,axiom,
! [A2: set_a,B: set_a,F3: a > set_a,G4: a > set_a] :
( ! [N2: a] :
( ( member_a @ N2 @ A2 )
=> ? [X5: a] :
( ( member_a @ X5 @ B )
& ( ord_less_eq_set_a @ ( F3 @ N2 ) @ ( G4 @ X5 ) ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F3 @ A2 ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ G4 @ B ) ) ) ) ).
% SUP_mono
thf(fact_986_SUP__least,axiom,
! [A2: set_a,F3: a > set_a,U: set_a] :
( ! [I2: a] :
( ( member_a @ I2 @ A2 )
=> ( ord_less_eq_set_a @ ( F3 @ I2 ) @ U ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F3 @ A2 ) ) @ U ) ) ).
% SUP_least
thf(fact_987_SUP__mono_H,axiom,
! [F3: a > set_a,G4: a > set_a,A2: set_a] :
( ! [X2: a] : ( ord_less_eq_set_a @ ( F3 @ X2 ) @ ( G4 @ X2 ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F3 @ A2 ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ G4 @ A2 ) ) ) ) ).
% SUP_mono'
thf(fact_988_SUP__upper,axiom,
! [I5: a,A2: set_a,F3: a > set_a] :
( ( member_a @ I5 @ A2 )
=> ( ord_less_eq_set_a @ ( F3 @ I5 ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F3 @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_989_SUP__le__iff,axiom,
! [F3: a > set_a,A2: set_a,U: set_a] :
( ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F3 @ A2 ) ) @ U )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ord_less_eq_set_a @ ( F3 @ X3 ) @ U ) ) ) ) ).
% SUP_le_iff
thf(fact_990_SUP__upper2,axiom,
! [I5: a,A2: set_a,U: set_a,F3: a > set_a] :
( ( member_a @ I5 @ A2 )
=> ( ( ord_less_eq_set_a @ U @ ( F3 @ I5 ) )
=> ( ord_less_eq_set_a @ U @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F3 @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_991_complete__lattice__class_OSUP__sup__distrib,axiom,
! [F3: a > set_a,A2: set_a,G4: a > set_a] :
( ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F3 @ A2 ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ G4 @ A2 ) ) )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [A4: a] : ( sup_sup_set_a @ ( F3 @ A4 ) @ ( G4 @ A4 ) )
@ A2 ) ) ) ).
% complete_lattice_class.SUP_sup_distrib
thf(fact_992_SUP__absorb,axiom,
! [K: a,I3: set_a,A2: a > set_a] :
( ( member_a @ K @ I3 )
=> ( ( sup_sup_set_a @ ( A2 @ K ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ A2 @ I3 ) ) )
= ( comple2307003609928055243_set_a @ ( image_a_set_a @ A2 @ I3 ) ) ) ) ).
% SUP_absorb
thf(fact_993_UN__empty2,axiom,
! [A2: set_a] :
( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X3: a] : bot_bot_set_a
@ A2 ) )
= bot_bot_set_a ) ).
% UN_empty2
thf(fact_994_UN__empty,axiom,
! [B: a > set_a] :
( ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ bot_bot_set_a ) )
= bot_bot_set_a ) ).
% UN_empty
thf(fact_995_UNION__empty__conv_I1_J,axiom,
! [B: a > set_a,A2: set_a] :
( ( bot_bot_set_a
= ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A2 ) ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ( B @ X3 )
= bot_bot_set_a ) ) ) ) ).
% UNION_empty_conv(1)
thf(fact_996_UNION__empty__conv_I2_J,axiom,
! [B: a > set_a,A2: set_a] :
( ( ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A2 ) )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ( B @ X3 )
= bot_bot_set_a ) ) ) ) ).
% UNION_empty_conv(2)
thf(fact_997_UN__subset__iff,axiom,
! [A2: a > set_a,I3: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ A2 @ I3 ) ) @ B )
= ( ! [X3: a] :
( ( member_a @ X3 @ I3 )
=> ( ord_less_eq_set_a @ ( A2 @ X3 ) @ B ) ) ) ) ).
% UN_subset_iff
thf(fact_998_UN__upper,axiom,
! [A: a,A2: set_a,B: a > set_a] :
( ( member_a @ A @ A2 )
=> ( ord_less_eq_set_a @ ( B @ A ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A2 ) ) ) ) ).
% UN_upper
thf(fact_999_UN__least,axiom,
! [A2: set_a,B: a > set_a,C: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ ( B @ X2 ) @ C ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A2 ) ) @ C ) ) ).
% UN_least
thf(fact_1000_UN__mono,axiom,
! [A2: set_a,B: set_a,F3: a > set_a,G4: a > set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ ( F3 @ X2 ) @ ( G4 @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F3 @ A2 ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ G4 @ B ) ) ) ) ) ).
% UN_mono
thf(fact_1001_UN__insert__distrib,axiom,
! [U: a,A2: set_a,A: a,B: a > set_a] :
( ( member_a @ U @ A2 )
=> ( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( insert_a @ A @ ( B @ X3 ) )
@ A2 ) )
= ( insert_a @ A @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A2 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_1002_UN__extend__simps_I5_J,axiom,
! [A2: set_a,B: a > set_a,C: set_a] :
( ( inf_inf_set_a @ A2 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ C ) ) )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( inf_inf_set_a @ A2 @ ( B @ X3 ) )
@ C ) ) ) ).
% UN_extend_simps(5)
thf(fact_1003_UN__extend__simps_I4_J,axiom,
! [A2: a > set_a,C: set_a,B: set_a] :
( ( inf_inf_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ A2 @ C ) ) @ B )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( inf_inf_set_a @ ( A2 @ X3 ) @ B )
@ C ) ) ) ).
% UN_extend_simps(4)
thf(fact_1004_Int__UN__distrib,axiom,
! [B: set_a,A2: a > set_a,I3: set_a] :
( ( inf_inf_set_a @ B @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ A2 @ I3 ) ) )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [I: a] : ( inf_inf_set_a @ B @ ( A2 @ I ) )
@ I3 ) ) ) ).
% Int_UN_distrib
thf(fact_1005_Int__UN__distrib2,axiom,
! [A2: a > set_a,I3: set_a,B: a > set_a,J2: set_a] :
( ( inf_inf_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ A2 @ I3 ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ J2 ) ) )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [I: a] :
( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [J3: a] : ( inf_inf_set_a @ ( A2 @ I ) @ ( B @ J3 ) )
@ J2 ) )
@ I3 ) ) ) ).
% Int_UN_distrib2
thf(fact_1006_UN__absorb,axiom,
! [K: a,I3: set_a,A2: a > set_a] :
( ( member_a @ K @ I3 )
=> ( ( sup_sup_set_a @ ( A2 @ K ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ A2 @ I3 ) ) )
= ( comple2307003609928055243_set_a @ ( image_a_set_a @ A2 @ I3 ) ) ) ) ).
% UN_absorb
thf(fact_1007_UN__Un__distrib,axiom,
! [A2: a > set_a,B: a > set_a,I3: set_a] :
( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [I: a] : ( sup_sup_set_a @ ( A2 @ I ) @ ( B @ I ) )
@ I3 ) )
= ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ A2 @ I3 ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ I3 ) ) ) ) ).
% UN_Un_distrib
thf(fact_1008_Un__Union__image,axiom,
! [A2: a > set_a,B: a > set_a,C: set_a] :
( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( sup_sup_set_a @ ( A2 @ X3 ) @ ( B @ X3 ) )
@ C ) )
= ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ A2 @ C ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ C ) ) ) ) ).
% Un_Union_image
thf(fact_1009_Int__Union,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 ) ) ) ).
% Int_Union
thf(fact_1010_Int__Union2,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
@ ^ [C3: set_a] : ( inf_inf_set_a @ C3 @ A2 )
@ B ) ) ) ).
% Int_Union2
thf(fact_1011_SUP__eq__iff,axiom,
! [I3: set_a,C2: set_a,F3: a > set_a] :
( ( I3 != bot_bot_set_a )
=> ( ! [I2: a] :
( ( member_a @ I2 @ I3 )
=> ( ord_less_eq_set_a @ C2 @ ( F3 @ I2 ) ) )
=> ( ( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F3 @ I3 ) )
= C2 )
= ( ! [X3: a] :
( ( member_a @ X3 @ I3 )
=> ( ( F3 @ X3 )
= C2 ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1012_Sup__inter__less__eq,axiom,
! [A2: set_set_a,B: set_set_a] : ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( inf_inf_set_set_a @ A2 @ B ) ) @ ( inf_inf_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Sup_inter_less_eq
thf(fact_1013_SUP__subset__mono,axiom,
! [A2: set_a,B: set_a,F3: a > set_a,G4: a > set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ ( F3 @ X2 ) @ ( G4 @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F3 @ A2 ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ G4 @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1014_SUP__constant,axiom,
! [A2: set_a,C2: set_a] :
( ( ( A2 = bot_bot_set_a )
=> ( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [Y3: a] : C2
@ A2 ) )
= bot_bot_set_a ) )
& ( ( A2 != bot_bot_set_a )
=> ( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [Y3: a] : C2
@ A2 ) )
= C2 ) ) ) ).
% SUP_constant
thf(fact_1015_SUP__empty,axiom,
! [F3: a > set_a] :
( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F3 @ bot_bot_set_a ) )
= bot_bot_set_a ) ).
% SUP_empty
thf(fact_1016_SUP__insert,axiom,
! [F3: a > set_a,A: a,A2: set_a] :
( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F3 @ ( insert_a @ A @ A2 ) ) )
= ( sup_sup_set_a @ ( F3 @ A ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F3 @ A2 ) ) ) ) ).
% SUP_insert
thf(fact_1017_SUP__union,axiom,
! [M: a > set_a,A2: set_a,B: set_a] :
( ( comple2307003609928055243_set_a @ ( image_a_set_a @ M @ ( sup_sup_set_a @ A2 @ B ) ) )
= ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ M @ A2 ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ M @ B ) ) ) ) ).
% SUP_union
thf(fact_1018_UN__extend__simps_I1_J,axiom,
! [C: set_a,A: a,B: a > set_a] :
( ( ( C = bot_bot_set_a )
=> ( ( insert_a @ A @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ C ) ) )
= ( insert_a @ A @ bot_bot_set_a ) ) )
& ( ( C != bot_bot_set_a )
=> ( ( insert_a @ A @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ C ) ) )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( insert_a @ A @ ( B @ X3 ) )
@ C ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_1019_UN__extend__simps_I3_J,axiom,
! [C: set_a,A2: set_a,B: a > set_a] :
( ( ( C = bot_bot_set_a )
=> ( ( sup_sup_set_a @ A2 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ C ) ) )
= A2 ) )
& ( ( C != bot_bot_set_a )
=> ( ( sup_sup_set_a @ A2 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ C ) ) )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( sup_sup_set_a @ A2 @ ( B @ X3 ) )
@ C ) ) ) ) ) ).
% UN_extend_simps(3)
thf(fact_1020_UN__extend__simps_I2_J,axiom,
! [C: set_a,A2: a > set_a,B: set_a] :
( ( ( C = bot_bot_set_a )
=> ( ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ A2 @ C ) ) @ B )
= B ) )
& ( ( C != bot_bot_set_a )
=> ( ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ A2 @ C ) ) @ B )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( sup_sup_set_a @ ( A2 @ X3 ) @ B )
@ C ) ) ) ) ) ).
% UN_extend_simps(2)
thf(fact_1021_UNION__singleton__eq__range,axiom,
! [F3: a > set_a,A2: set_a] :
( ( comple3958522678809307947_set_a
@ ( image_a_set_set_a
@ ^ [X3: a] : ( insert_set_a @ ( F3 @ X3 ) @ bot_bot_set_set_a )
@ A2 ) )
= ( image_a_set_a @ F3 @ A2 ) ) ).
% UNION_singleton_eq_range
thf(fact_1022_UNION__singleton__eq__range,axiom,
! [F3: a > a,A2: set_a] :
( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( insert_a @ ( F3 @ X3 ) @ bot_bot_set_a )
@ A2 ) )
= ( image_a_a @ F3 @ A2 ) ) ).
% UNION_singleton_eq_range
thf(fact_1023_cSUP__const,axiom,
! [A2: set_a,C2: set_a] :
( ( A2 != bot_bot_set_a )
=> ( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X3: a] : C2
@ A2 ) )
= C2 ) ) ).
% cSUP_const
thf(fact_1024_ccpo__Sup__singleton,axiom,
! [X: set_a] :
( ( comple2307003609928055243_set_a @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
= X ) ).
% ccpo_Sup_singleton
thf(fact_1025_cSup__singleton,axiom,
! [X: set_a] :
( ( comple2307003609928055243_set_a @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
= X ) ).
% cSup_singleton
thf(fact_1026_cSup__eq__maximum,axiom,
! [Z2: set_a,X4: set_set_a] :
( ( member_set_a @ Z2 @ X4 )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ X4 )
=> ( ord_less_eq_set_a @ X2 @ Z2 ) )
=> ( ( comple2307003609928055243_set_a @ X4 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_1027_cSup__eq__non__empty,axiom,
! [X4: set_set_a,A: set_a] :
( ( X4 != bot_bot_set_set_a )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ X4 )
=> ( ord_less_eq_set_a @ X2 @ A ) )
=> ( ! [Y2: set_a] :
( ! [X5: set_a] :
( ( member_set_a @ X5 @ X4 )
=> ( ord_less_eq_set_a @ X5 @ Y2 ) )
=> ( ord_less_eq_set_a @ A @ Y2 ) )
=> ( ( comple2307003609928055243_set_a @ X4 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_1028_cSup__least,axiom,
! [X4: set_set_a,Z2: set_a] :
( ( X4 != bot_bot_set_set_a )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ X4 )
=> ( ord_less_eq_set_a @ X2 @ Z2 ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ X4 ) @ Z2 ) ) ) ).
% cSup_least
thf(fact_1029_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_1030_cSUP__least,axiom,
! [A2: set_a,F3: a > set_a,M: set_a] :
( ( A2 != bot_bot_set_a )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ ( F3 @ X2 ) @ M ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F3 @ A2 ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_1031_cSup__eq__Sup__fin,axiom,
! [X4: set_set_a] :
( ( finite_finite_set_a @ X4 )
=> ( ( X4 != bot_bot_set_set_a )
=> ( ( comple2307003609928055243_set_a @ X4 )
= ( lattic2918178356826803221_set_a @ X4 ) ) ) ) ).
% cSup_eq_Sup_fin
thf(fact_1032_finite__subset__Union,axiom,
! [A2: set_a,B9: set_set_a] :
( ( finite_finite_a @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ ( comple2307003609928055243_set_a @ B9 ) )
=> ~ ! [F5: set_set_a] :
( ( finite_finite_set_a @ F5 )
=> ( ( ord_le3724670747650509150_set_a @ F5 @ B9 )
=> ~ ( ord_less_eq_set_a @ A2 @ ( comple2307003609928055243_set_a @ F5 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_1033_Union__image__insert,axiom,
! [F3: a > set_a,A: a,B: set_a] :
( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F3 @ ( insert_a @ A @ B ) ) )
= ( sup_sup_set_a @ ( F3 @ A ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F3 @ B ) ) ) ) ).
% Union_image_insert
thf(fact_1034_Union__image__empty,axiom,
! [A2: set_a,F3: a > set_a] :
( ( sup_sup_set_a @ A2 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F3 @ bot_bot_set_a ) ) )
= A2 ) ).
% Union_image_empty
thf(fact_1035_conj__subset__def,axiom,
! [A2: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A2
@ ( collect_a
@ ^ [X3: a] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) )
= ( ( ord_less_eq_set_a @ A2 @ ( collect_a @ P ) )
& ( ord_less_eq_set_a @ A2 @ ( collect_a @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_1036_inf__Sup,axiom,
! [A: set_a,B: set_set_a] :
( ( inf_inf_set_a @ A @ ( comple2307003609928055243_set_a @ B ) )
= ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ ( inf_inf_set_a @ A ) @ B ) ) ) ).
% inf_Sup
thf(fact_1037_Sup__inf__eq__bot__iff,axiom,
! [B: set_set_a,A: set_a] :
( ( ( inf_inf_set_a @ ( comple2307003609928055243_set_a @ B ) @ A )
= bot_bot_set_a )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ B )
=> ( ( inf_inf_set_a @ X3 @ A )
= bot_bot_set_a ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_1038_SUP__inf__distrib2,axiom,
! [F3: a > set_a,A2: set_a,G4: a > set_a,B: set_a] :
( ( inf_inf_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F3 @ A2 ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ G4 @ B ) ) )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [A4: a] :
( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [B3: a] : ( inf_inf_set_a @ ( F3 @ A4 ) @ ( G4 @ B3 ) )
@ B ) )
@ A2 ) ) ) ).
% SUP_inf_distrib2
thf(fact_1039_inf__SUP,axiom,
! [A: set_a,F3: a > set_a,B: set_a] :
( ( inf_inf_set_a @ A @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F3 @ B ) ) )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [B3: a] : ( inf_inf_set_a @ A @ ( F3 @ B3 ) )
@ B ) ) ) ).
% inf_SUP
thf(fact_1040_Sup__inf,axiom,
! [B: set_set_a,A: set_a] :
( ( inf_inf_set_a @ ( comple2307003609928055243_set_a @ B ) @ A )
= ( comple2307003609928055243_set_a
@ ( image_set_a_set_a
@ ^ [B3: set_a] : ( inf_inf_set_a @ B3 @ A )
@ B ) ) ) ).
% Sup_inf
thf(fact_1041_SUP__inf,axiom,
! [F3: a > set_a,B: set_a,A: set_a] :
( ( inf_inf_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F3 @ B ) ) @ A )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [B3: a] : ( inf_inf_set_a @ ( F3 @ B3 ) @ A )
@ B ) ) ) ).
% SUP_inf
thf(fact_1042_UNION__fun__upd,axiom,
! [A2: a > set_a,I5: a,B: set_a,J2: set_a] :
( ( comple2307003609928055243_set_a @ ( image_a_set_a @ ( fun_upd_a_set_a @ A2 @ I5 @ B ) @ J2 ) )
= ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ A2 @ ( minus_minus_set_a @ J2 @ ( insert_a @ I5 @ bot_bot_set_a ) ) ) ) @ ( if_set_a @ ( member_a @ I5 @ J2 ) @ B @ bot_bot_set_a ) ) ) ).
% UNION_fun_upd
thf(fact_1043_comp__fun__commute__Pow__fold,axiom,
( finite312795530508511377_set_a
@ ^ [X3: a,A6: set_set_a] : ( sup_sup_set_set_a @ A6 @ ( image_set_a_set_a @ ( insert_a @ X3 ) @ A6 ) ) ) ).
% comp_fun_commute_Pow_fold
thf(fact_1044_comp__fun__commute__filter__fold,axiom,
! [P: a > $o] :
( finite3518785373051244337_set_a
@ ^ [X3: a,A10: set_a] : ( if_set_a @ ( P @ X3 ) @ ( insert_a @ X3 @ A10 ) @ A10 ) ) ).
% comp_fun_commute_filter_fold
thf(fact_1045_fun__upd__image,axiom,
! [X: a,A2: set_a,F3: a > set_a,Y: set_a] :
( ( ( member_a @ X @ A2 )
=> ( ( image_a_set_a @ ( fun_upd_a_set_a @ F3 @ X @ Y ) @ A2 )
= ( insert_set_a @ Y @ ( image_a_set_a @ F3 @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ( image_a_set_a @ ( fun_upd_a_set_a @ F3 @ X @ Y ) @ A2 )
= ( image_a_set_a @ F3 @ A2 ) ) ) ) ).
% fun_upd_image
thf(fact_1046_fun__upd__image,axiom,
! [X: a,A2: set_a,F3: a > a,Y: a] :
( ( ( member_a @ X @ A2 )
=> ( ( image_a_a @ ( fun_upd_a_a @ F3 @ X @ Y ) @ A2 )
= ( insert_a @ Y @ ( image_a_a @ F3 @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ( image_a_a @ ( fun_upd_a_a @ F3 @ X @ Y ) @ A2 )
= ( image_a_a @ F3 @ A2 ) ) ) ) ).
% fun_upd_image
thf(fact_1047_chains__extend,axiom,
! [C2: set_set_a,S: set_set_a,Z2: set_a] :
( ( member_set_set_a @ C2 @ ( chains_a @ S ) )
=> ( ( member_set_a @ Z2 @ S )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ C2 )
=> ( ord_less_eq_set_a @ X2 @ Z2 ) )
=> ( member_set_set_a @ ( sup_sup_set_set_a @ ( insert_set_a @ Z2 @ bot_bot_set_set_a ) @ C2 ) @ ( chains_a @ S ) ) ) ) ) ).
% chains_extend
thf(fact_1048_cSUP__UNION,axiom,
! [A2: set_a,B: a > set_a,F3: a > set_a] :
( ( A2 != bot_bot_set_a )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( B @ X2 )
!= bot_bot_set_a ) )
=> ( ( condit3373647341569784514_set_a
@ ( comple3958522678809307947_set_a
@ ( image_a_set_set_a
@ ^ [X3: a] : ( image_a_set_a @ F3 @ ( B @ X3 ) )
@ A2 ) ) )
=> ( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F3 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A2 ) ) ) )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( comple2307003609928055243_set_a @ ( image_a_set_a @ F3 @ ( B @ X3 ) ) )
@ A2 ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_1049_bdd__above_OI,axiom,
! [A2: set_set_a,M: set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ X2 @ M ) )
=> ( condit3373647341569784514_set_a @ A2 ) ) ).
% bdd_above.I
thf(fact_1050_bdd__above__image__sup,axiom,
! [F3: a > set_a,G4: a > set_a,A2: set_a] :
( ( condit3373647341569784514_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( sup_sup_set_a @ ( F3 @ X3 ) @ ( G4 @ X3 ) )
@ A2 ) )
= ( ( condit3373647341569784514_set_a @ ( image_a_set_a @ F3 @ A2 ) )
& ( condit3373647341569784514_set_a @ ( image_a_set_a @ G4 @ A2 ) ) ) ) ).
% bdd_above_image_sup
thf(fact_1051_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_1052_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_1053_bdd__above_OE,axiom,
! [A2: set_set_a] :
( ( condit3373647341569784514_set_a @ A2 )
=> ~ ! [M2: set_a] :
~ ! [X5: set_a] :
( ( member_set_a @ X5 @ A2 )
=> ( ord_less_eq_set_a @ X5 @ M2 ) ) ) ).
% bdd_above.E
thf(fact_1054_bdd__above_Ounfold,axiom,
( condit3373647341569784514_set_a
= ( ^ [A6: set_set_a] :
? [M3: set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ A6 )
=> ( ord_less_eq_set_a @ X3 @ M3 ) ) ) ) ).
% bdd_above.unfold
thf(fact_1055_bdd__above_OI2,axiom,
! [A2: set_a,F3: a > set_a,M: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ ( F3 @ X2 ) @ M ) )
=> ( condit3373647341569784514_set_a @ ( image_a_set_a @ F3 @ A2 ) ) ) ).
% bdd_above.I2
thf(fact_1056_chainsD,axiom,
! [C2: set_set_a,S: set_set_a,X: set_a,Y: set_a] :
( ( member_set_set_a @ C2 @ ( chains_a @ S ) )
=> ( ( member_set_a @ X @ C2 )
=> ( ( member_set_a @ Y @ C2 )
=> ( ( ord_less_eq_set_a @ X @ Y )
| ( ord_less_eq_set_a @ Y @ X ) ) ) ) ) ).
% chainsD
thf(fact_1057_Zorn__Lemma2,axiom,
! [A2: set_set_a] :
( ! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ ( chains_a @ A2 ) )
=> ? [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
& ! [Xb: set_a] :
( ( member_set_a @ Xb @ X2 )
=> ( ord_less_eq_set_a @ Xb @ Xa ) ) ) )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X2 @ Xa )
=> ( Xa = X2 ) ) ) ) ) ).
% Zorn_Lemma2
thf(fact_1058_Zorn__Lemma,axiom,
! [A2: set_set_a] :
( ! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ ( chains_a @ A2 ) )
=> ( member_set_a @ ( comple2307003609928055243_set_a @ X2 ) @ A2 ) )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X2 @ Xa )
=> ( Xa = X2 ) ) ) ) ) ).
% Zorn_Lemma
thf(fact_1059_cSUP__upper,axiom,
! [X: a,A2: set_a,F3: a > set_a] :
( ( member_a @ X @ A2 )
=> ( ( condit3373647341569784514_set_a @ ( image_a_set_a @ F3 @ A2 ) )
=> ( ord_less_eq_set_a @ ( F3 @ X ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F3 @ A2 ) ) ) ) ) ).
% cSUP_upper
thf(fact_1060_cSUP__upper2,axiom,
! [F3: a > set_a,A2: set_a,X: a,U: set_a] :
( ( condit3373647341569784514_set_a @ ( image_a_set_a @ F3 @ A2 ) )
=> ( ( member_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ U @ ( F3 @ X ) )
=> ( ord_less_eq_set_a @ U @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F3 @ A2 ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_1061_cSup__mono,axiom,
! [B: set_set_a,A2: set_set_a] :
( ( B != bot_bot_set_set_a )
=> ( ( condit3373647341569784514_set_a @ A2 )
=> ( ! [B2: set_a] :
( ( member_set_a @ B2 @ B )
=> ? [X5: set_a] :
( ( member_set_a @ X5 @ A2 )
& ( ord_less_eq_set_a @ B2 @ X5 ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ B ) @ ( comple2307003609928055243_set_a @ A2 ) ) ) ) ) ).
% cSup_mono
thf(fact_1062_cSup__le__iff,axiom,
! [S: set_set_a,A: set_a] :
( ( S != bot_bot_set_set_a )
=> ( ( condit3373647341569784514_set_a @ S )
=> ( ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ S ) @ A )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ S )
=> ( ord_less_eq_set_a @ X3 @ A ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_1063_cSUP__mono,axiom,
! [A2: set_a,G4: a > set_a,B: set_a,F3: a > set_a] :
( ( A2 != bot_bot_set_a )
=> ( ( condit3373647341569784514_set_a @ ( image_a_set_a @ G4 @ B ) )
=> ( ! [N2: a] :
( ( member_a @ N2 @ A2 )
=> ? [X5: a] :
( ( member_a @ X5 @ B )
& ( ord_less_eq_set_a @ ( F3 @ N2 ) @ ( G4 @ X5 ) ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F3 @ A2 ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ G4 @ B ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_1064_cSUP__le__iff,axiom,
! [A2: set_a,F3: a > set_a,U: set_a] :
( ( A2 != bot_bot_set_a )
=> ( ( condit3373647341569784514_set_a @ ( image_a_set_a @ F3 @ A2 ) )
=> ( ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F3 @ A2 ) ) @ U )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ord_less_eq_set_a @ ( F3 @ X3 ) @ U ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_1065_cSup__subset__mono,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( A2 != bot_bot_set_set_a )
=> ( ( condit3373647341569784514_set_a @ B )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ ( comple2307003609928055243_set_a @ B ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_1066_cSup__insert__If,axiom,
! [X4: set_set_a,A: set_a] :
( ( condit3373647341569784514_set_a @ X4 )
=> ( ( ( X4 = bot_bot_set_set_a )
=> ( ( comple2307003609928055243_set_a @ ( insert_set_a @ A @ X4 ) )
= A ) )
& ( ( X4 != bot_bot_set_set_a )
=> ( ( comple2307003609928055243_set_a @ ( insert_set_a @ A @ X4 ) )
= ( sup_sup_set_a @ A @ ( comple2307003609928055243_set_a @ X4 ) ) ) ) ) ) ).
% cSup_insert_If
thf(fact_1067_cSup__insert,axiom,
! [X4: set_set_a,A: set_a] :
( ( X4 != bot_bot_set_set_a )
=> ( ( condit3373647341569784514_set_a @ X4 )
=> ( ( comple2307003609928055243_set_a @ ( insert_set_a @ A @ X4 ) )
= ( sup_sup_set_a @ A @ ( comple2307003609928055243_set_a @ X4 ) ) ) ) ) ).
% cSup_insert
thf(fact_1068_cSup__union__distrib,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( A2 != bot_bot_set_set_a )
=> ( ( condit3373647341569784514_set_a @ A2 )
=> ( ( B != bot_bot_set_set_a )
=> ( ( condit3373647341569784514_set_a @ B )
=> ( ( comple2307003609928055243_set_a @ ( sup_sup_set_set_a @ A2 @ B ) )
= ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ ( comple2307003609928055243_set_a @ B ) ) ) ) ) ) ) ).
% cSup_union_distrib
thf(fact_1069_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
! [A2: set_a,F3: a > set_a,G4: a > set_a] :
( ( A2 != bot_bot_set_a )
=> ( ( condit3373647341569784514_set_a @ ( image_a_set_a @ F3 @ A2 ) )
=> ( ( condit3373647341569784514_set_a @ ( image_a_set_a @ G4 @ A2 ) )
=> ( ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F3 @ A2 ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ G4 @ A2 ) ) )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [A4: a] : ( sup_sup_set_a @ ( F3 @ A4 ) @ ( G4 @ A4 ) )
@ A2 ) ) ) ) ) ) ).
% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_1070_cSUP__subset__mono,axiom,
! [A2: set_a,G4: a > set_a,B: set_a,F3: a > set_a] :
( ( A2 != bot_bot_set_a )
=> ( ( condit3373647341569784514_set_a @ ( image_a_set_a @ G4 @ B ) )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ ( F3 @ X2 ) @ ( G4 @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F3 @ A2 ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ G4 @ B ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_1071_cSup__inter__less__eq,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( condit3373647341569784514_set_a @ A2 )
=> ( ( condit3373647341569784514_set_a @ B )
=> ( ( ( inf_inf_set_set_a @ A2 @ B )
!= bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( inf_inf_set_set_a @ A2 @ B ) ) @ ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ ( comple2307003609928055243_set_a @ B ) ) ) ) ) ) ).
% cSup_inter_less_eq
thf(fact_1072_cSUP__insert,axiom,
! [A2: set_a,F3: a > set_a,A: a] :
( ( A2 != bot_bot_set_a )
=> ( ( condit3373647341569784514_set_a @ ( image_a_set_a @ F3 @ A2 ) )
=> ( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F3 @ ( insert_a @ A @ A2 ) ) )
= ( sup_sup_set_a @ ( F3 @ A ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F3 @ A2 ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_1073_cSUP__union,axiom,
! [A2: set_a,F3: a > set_a,B: set_a] :
( ( A2 != bot_bot_set_a )
=> ( ( condit3373647341569784514_set_a @ ( image_a_set_a @ F3 @ A2 ) )
=> ( ( B != bot_bot_set_a )
=> ( ( condit3373647341569784514_set_a @ ( image_a_set_a @ F3 @ B ) )
=> ( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F3 @ ( sup_sup_set_a @ A2 @ B ) ) )
= ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F3 @ A2 ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F3 @ B ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_1074_subset__singleton__iff__Uniq,axiom,
! [A2: set_a] :
( ( ? [A4: a] : ( ord_less_eq_set_a @ A2 @ ( insert_a @ A4 @ bot_bot_set_a ) ) )
= ( uniq_a
@ ^ [X3: a] : ( member_a @ X3 @ A2 ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_1075_chain__subset__def,axiom,
( chain_subset_a
= ( ^ [C3: set_set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ C3 )
=> ! [Y3: set_a] :
( ( member_set_a @ Y3 @ C3 )
=> ( ( ord_less_eq_set_a @ X3 @ Y3 )
| ( ord_less_eq_set_a @ Y3 @ X3 ) ) ) ) ) ) ).
% chain_subset_def
thf(fact_1076_bind__singleton__conv__image,axiom,
! [A2: set_a,F3: a > set_a] :
( ( bind_a_set_a @ A2
@ ^ [X3: a] : ( insert_set_a @ ( F3 @ X3 ) @ bot_bot_set_set_a ) )
= ( image_a_set_a @ F3 @ A2 ) ) ).
% bind_singleton_conv_image
thf(fact_1077_bind__singleton__conv__image,axiom,
! [A2: set_a,F3: a > a] :
( ( bind_a_a @ A2
@ ^ [X3: a] : ( insert_a @ ( F3 @ X3 ) @ bot_bot_set_a ) )
= ( image_a_a @ F3 @ A2 ) ) ).
% bind_singleton_conv_image
thf(fact_1078_inter__Set__filter,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( inf_inf_set_a @ A2 @ B )
= ( filter_a
@ ^ [X3: a] : ( member_a @ X3 @ A2 )
@ B ) ) ) ).
% inter_Set_filter
thf(fact_1079_member__filter,axiom,
! [X: a,P: a > $o,A2: set_a] :
( ( member_a @ X @ ( filter_a @ P @ A2 ) )
= ( ( member_a @ X @ A2 )
& ( P @ X ) ) ) ).
% member_filter
thf(fact_1080_empty__bind,axiom,
! [F3: a > set_a] :
( ( bind_a_a @ bot_bot_set_a @ F3 )
= bot_bot_set_a ) ).
% empty_bind
thf(fact_1081_finite__filter,axiom,
! [S: set_a,P: a > $o] :
( ( finite_finite_a @ S )
=> ( finite_finite_a @ ( filter_a @ P @ S ) ) ) ).
% finite_filter
thf(fact_1082_Set_Obind__def,axiom,
( bind_a_a
= ( ^ [A6: set_a,F2: a > set_a] :
( collect_a
@ ^ [X3: a] :
? [Y3: set_a] :
( ( member_set_a @ Y3 @ ( image_a_set_a @ F2 @ A6 ) )
& ( member_a @ X3 @ Y3 ) ) ) ) ) ).
% Set.bind_def
thf(fact_1083_finite__bind,axiom,
! [S: set_a,F3: a > set_a] :
( ( finite_finite_a @ S )
=> ( ! [X2: a] :
( ( member_a @ X2 @ S )
=> ( finite_finite_a @ ( F3 @ X2 ) ) )
=> ( finite_finite_a @ ( bind_a_a @ S @ F3 ) ) ) ) ).
% finite_bind
thf(fact_1084_Set_Ofilter__def,axiom,
( filter_a
= ( ^ [P2: a > $o,A6: set_a] :
( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A6 )
& ( P2 @ A4 ) ) ) ) ) ).
% Set.filter_def
thf(fact_1085_bind__const,axiom,
! [A2: set_a,B: set_a] :
( ( ( A2 = bot_bot_set_a )
=> ( ( bind_a_a @ A2
@ ^ [Uu: a] : B )
= bot_bot_set_a ) )
& ( ( A2 != bot_bot_set_a )
=> ( ( bind_a_a @ A2
@ ^ [Uu: a] : B )
= B ) ) ) ).
% bind_const
thf(fact_1086_Set__filter__fold,axiom,
! [A2: set_a,P: a > $o] :
( ( finite_finite_a @ A2 )
=> ( ( filter_a @ P @ A2 )
= ( finite_fold_a_set_a
@ ^ [X3: a,A10: set_a] : ( if_set_a @ ( P @ X3 ) @ ( insert_a @ X3 @ A10 ) @ A10 )
@ bot_bot_set_a
@ A2 ) ) ) ).
% Set_filter_fold
thf(fact_1087_subset__CollectI,axiom,
! [B: set_a,A2: set_a,Q: a > $o,P: a > $o] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( ! [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 @ A2 )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_1088_fold__closed__eq,axiom,
! [A2: set_a,B: set_a,F3: a > a > a,G4: a > a > a,Z2: a] :
( ! [A3: a,B2: a] :
( ( member_a @ A3 @ A2 )
=> ( ( member_a @ B2 @ B )
=> ( ( F3 @ A3 @ B2 )
= ( G4 @ A3 @ B2 ) ) ) )
=> ( ! [A3: a,B2: a] :
( ( member_a @ A3 @ A2 )
=> ( ( member_a @ B2 @ B )
=> ( member_a @ ( G4 @ A3 @ B2 ) @ B ) ) )
=> ( ( member_a @ Z2 @ B )
=> ( ( finite_fold_a_a @ F3 @ Z2 @ A2 )
= ( finite_fold_a_a @ G4 @ Z2 @ A2 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_1089_union__fold__insert,axiom,
! [A2: set_a,B: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( sup_sup_set_a @ A2 @ B )
= ( finite_fold_a_set_a @ insert_a @ B @ A2 ) ) ) ).
% union_fold_insert
thf(fact_1090_sup__Sup__fold__sup,axiom,
! [A2: set_set_a,B: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ B )
= ( finite5985231929012247624_set_a @ sup_sup_set_a @ B @ A2 ) ) ) ).
% sup_Sup_fold_sup
thf(fact_1091_minus__fold__remove,axiom,
! [A2: set_a,B: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( minus_minus_set_a @ B @ A2 )
= ( finite_fold_a_set_a @ remove_a @ B @ A2 ) ) ) ).
% minus_fold_remove
thf(fact_1092_Sup__fold__sup,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( comple2307003609928055243_set_a @ A2 )
= ( finite5985231929012247624_set_a @ sup_sup_set_a @ bot_bot_set_a @ A2 ) ) ) ).
% Sup_fold_sup
thf(fact_1093_Inf__fin_Oeq__fold,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( lattic8209813465164889211_set_a @ ( insert_set_a @ X @ A2 ) )
= ( finite5985231929012247624_set_a @ inf_inf_set_a @ X @ A2 ) ) ) ).
% Inf_fin.eq_fold
thf(fact_1094_Sup__fin_Oeq__fold,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( lattic2918178356826803221_set_a @ ( insert_set_a @ X @ A2 ) )
= ( finite5985231929012247624_set_a @ sup_sup_set_a @ X @ A2 ) ) ) ).
% Sup_fin.eq_fold
thf(fact_1095_image__fold__insert,axiom,
! [A2: set_a,F3: a > set_a] :
( ( finite_finite_a @ A2 )
=> ( ( image_a_set_a @ F3 @ A2 )
= ( finite9006272623207878408_set_a
@ ^ [K2: a] : ( insert_set_a @ ( F3 @ K2 ) )
@ bot_bot_set_set_a
@ A2 ) ) ) ).
% image_fold_insert
thf(fact_1096_image__fold__insert,axiom,
! [A2: set_a,F3: a > a] :
( ( finite_finite_a @ A2 )
=> ( ( image_a_a @ F3 @ A2 )
= ( finite_fold_a_set_a
@ ^ [K2: a] : ( insert_a @ ( F3 @ K2 ) )
@ bot_bot_set_a
@ A2 ) ) ) ).
% image_fold_insert
thf(fact_1097_subset__Collect__iff,axiom,
! [B: set_a,A2: set_a,P: a > $o] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( ord_less_eq_set_a @ B
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A2 )
& ( P @ X3 ) ) ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ B )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_1098_Pow__fold,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( pow_a @ A2 )
= ( finite9006272623207878408_set_a
@ ^ [X3: a,A6: set_set_a] : ( sup_sup_set_set_a @ A6 @ ( image_set_a_set_a @ ( insert_a @ X3 ) @ A6 ) )
@ ( insert_set_a @ bot_bot_set_a @ bot_bot_set_set_a )
@ A2 ) ) ) ).
% Pow_fold
thf(fact_1099_PowI,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( member_set_a @ A2 @ ( pow_a @ B ) ) ) ).
% PowI
thf(fact_1100_Pow__iff,axiom,
! [A2: set_a,B: set_a] :
( ( member_set_a @ A2 @ ( pow_a @ B ) )
= ( ord_less_eq_set_a @ A2 @ B ) ) ).
% Pow_iff
thf(fact_1101_finite__Pow__iff,axiom,
! [A2: set_a] :
( ( finite_finite_set_a @ ( pow_a @ A2 ) )
= ( finite_finite_a @ A2 ) ) ).
% finite_Pow_iff
thf(fact_1102_Pow__Int__eq,axiom,
! [A2: set_a,B: set_a] :
( ( pow_a @ ( inf_inf_set_a @ A2 @ B ) )
= ( inf_inf_set_set_a @ ( pow_a @ A2 ) @ ( pow_a @ B ) ) ) ).
% Pow_Int_eq
thf(fact_1103_Pow__empty,axiom,
( ( pow_a @ bot_bot_set_a )
= ( insert_set_a @ bot_bot_set_a @ bot_bot_set_set_a ) ) ).
% Pow_empty
thf(fact_1104_Pow__singleton__iff,axiom,
! [X4: set_a,Y5: set_a] :
( ( ( pow_a @ X4 )
= ( insert_set_a @ Y5 @ bot_bot_set_set_a ) )
= ( ( X4 = bot_bot_set_a )
& ( Y5 = bot_bot_set_a ) ) ) ).
% Pow_singleton_iff
thf(fact_1105_subset__Pow__Union,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ A2 @ ( pow_a @ ( comple2307003609928055243_set_a @ A2 ) ) ) ).
% subset_Pow_Union
thf(fact_1106_Cantors__paradox,axiom,
! [A2: set_a] :
~ ? [F6: a > set_a] :
( ( image_a_set_a @ F6 @ A2 )
= ( pow_a @ A2 ) ) ).
% Cantors_paradox
thf(fact_1107_PowD,axiom,
! [A2: set_a,B: set_a] :
( ( member_set_a @ A2 @ ( pow_a @ B ) )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% PowD
thf(fact_1108_Pow__def,axiom,
( pow_a
= ( ^ [A6: set_a] :
( collect_set_a
@ ^ [B6: set_a] : ( ord_less_eq_set_a @ B6 @ A6 ) ) ) ) ).
% Pow_def
thf(fact_1109_Pow__bottom,axiom,
! [B: set_a] : ( member_set_a @ bot_bot_set_a @ ( pow_a @ B ) ) ).
% Pow_bottom
thf(fact_1110_Pow__mono,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_le3724670747650509150_set_a @ ( pow_a @ A2 ) @ ( pow_a @ B ) ) ) ).
% Pow_mono
thf(fact_1111_image__Pow__surj,axiom,
! [F3: a > a,A2: set_a,B: set_a] :
( ( ( image_a_a @ F3 @ A2 )
= B )
=> ( ( image_set_a_set_a @ ( image_a_a @ F3 ) @ ( pow_a @ A2 ) )
= ( pow_a @ B ) ) ) ).
% image_Pow_surj
thf(fact_1112_image__Pow__surj,axiom,
! [F3: a > set_a,A2: set_a,B: set_set_a] :
( ( ( image_a_set_a @ F3 @ A2 )
= B )
=> ( ( image_4955109552351689957_set_a @ ( image_a_set_a @ F3 ) @ ( pow_a @ A2 ) )
= ( pow_set_a @ B ) ) ) ).
% image_Pow_surj
thf(fact_1113_Bex__fold,axiom,
! [A2: set_a,P: a > $o] :
( ( finite_finite_a @ A2 )
=> ( ( ? [X3: a] :
( ( member_a @ X3 @ A2 )
& ( P @ X3 ) ) )
= ( finite_fold_a_o
@ ^ [K2: a,S3: $o] :
( S3
| ( P @ K2 ) )
@ $false
@ A2 ) ) ) ).
% Bex_fold
thf(fact_1114_Un__Pow__subset,axiom,
! [A2: set_a,B: set_a] : ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ ( pow_a @ A2 ) @ ( pow_a @ B ) ) @ ( pow_a @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% Un_Pow_subset
thf(fact_1115_Pow__insert,axiom,
! [A: a,A2: set_a] :
( ( pow_a @ ( insert_a @ A @ A2 ) )
= ( sup_sup_set_set_a @ ( pow_a @ A2 ) @ ( image_set_a_set_a @ ( insert_a @ A ) @ ( pow_a @ A2 ) ) ) ) ).
% Pow_insert
thf(fact_1116_UN__Pow__subset,axiom,
! [B: a > set_a,A2: set_a] :
( ord_le3724670747650509150_set_a
@ ( comple3958522678809307947_set_a
@ ( image_a_set_set_a
@ ^ [X3: a] : ( pow_a @ ( B @ X3 ) )
@ A2 ) )
@ ( pow_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A2 ) ) ) ) ).
% UN_Pow_subset
thf(fact_1117_Fpow__Pow__finite,axiom,
( finite_Fpow_a
= ( ^ [A6: set_a] : ( inf_inf_set_set_a @ ( pow_a @ A6 ) @ ( collect_set_a @ finite_finite_a ) ) ) ) ).
% Fpow_Pow_finite
thf(fact_1118_image__Pow__mono,axiom,
! [F3: a > set_a,A2: set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F3 @ A2 ) @ B )
=> ( ord_le5722252365846178494_set_a @ ( image_4955109552351689957_set_a @ ( image_a_set_a @ F3 ) @ ( pow_a @ A2 ) ) @ ( pow_set_a @ B ) ) ) ).
% image_Pow_mono
thf(fact_1119_image__Pow__mono,axiom,
! [F3: a > a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F3 @ A2 ) @ B )
=> ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ ( image_a_a @ F3 ) @ ( pow_a @ A2 ) ) @ ( pow_a @ B ) ) ) ).
% image_Pow_mono
thf(fact_1120_SUP__fold__sup,axiom,
! [A2: set_a,F3: a > set_a] :
( ( finite_finite_a @ A2 )
=> ( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F3 @ A2 ) )
= ( finite_fold_a_set_a @ ( comp_s1103547301056249180et_a_a @ sup_sup_set_a @ F3 ) @ bot_bot_set_a @ A2 ) ) ) ).
% SUP_fold_sup
thf(fact_1121_fold__graph__closed__lemma,axiom,
! [G4: a > a > a,Z2: a,A2: set_a,X: a,B: set_a,F3: a > a > a] :
( ( finite7874008084079289286ph_a_a @ G4 @ Z2 @ A2 @ X )
=> ( ! [A3: a,B2: a] :
( ( member_a @ A3 @ A2 )
=> ( ( member_a @ B2 @ B )
=> ( ( F3 @ A3 @ B2 )
= ( G4 @ A3 @ B2 ) ) ) )
=> ( ! [A3: a,B2: a] :
( ( member_a @ A3 @ A2 )
=> ( ( member_a @ B2 @ B )
=> ( member_a @ ( G4 @ A3 @ B2 ) @ B ) ) )
=> ( ( member_a @ Z2 @ B )
=> ( ( finite7874008084079289286ph_a_a @ F3 @ Z2 @ A2 @ X )
& ( member_a @ X @ B ) ) ) ) ) ) ).
% fold_graph_closed_lemma
thf(fact_1122_fold__graph__closed__eq,axiom,
! [A2: set_a,B: set_a,F3: a > a > a,G4: a > a > a,Z2: a] :
( ! [A3: a,B2: a] :
( ( member_a @ A3 @ A2 )
=> ( ( member_a @ B2 @ B )
=> ( ( F3 @ A3 @ B2 )
= ( G4 @ A3 @ B2 ) ) ) )
=> ( ! [A3: a,B2: a] :
( ( member_a @ A3 @ A2 )
=> ( ( member_a @ B2 @ B )
=> ( member_a @ ( G4 @ A3 @ B2 ) @ B ) ) )
=> ( ( member_a @ Z2 @ B )
=> ( ( finite7874008084079289286ph_a_a @ F3 @ Z2 @ A2 )
= ( finite7874008084079289286ph_a_a @ G4 @ Z2 @ A2 ) ) ) ) ) ).
% fold_graph_closed_eq
thf(fact_1123_sup__SUP__fold__sup,axiom,
! [A2: set_a,B: set_a,F3: a > set_a] :
( ( finite_finite_a @ A2 )
=> ( ( sup_sup_set_a @ B @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F3 @ A2 ) ) )
= ( finite_fold_a_set_a @ ( comp_s1103547301056249180et_a_a @ sup_sup_set_a @ F3 ) @ B @ A2 ) ) ) ).
% sup_SUP_fold_sup
thf(fact_1124_inf__INF__fold__inf,axiom,
! [A2: set_a,B: set_a,F3: a > set_a] :
( ( finite_finite_a @ A2 )
=> ( ( inf_inf_set_a @ B @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ F3 @ A2 ) ) )
= ( finite_fold_a_set_a @ ( comp_s1103547301056249180et_a_a @ inf_inf_set_a @ F3 ) @ B @ A2 ) ) ) ).
% inf_INF_fold_inf
thf(fact_1125_partition_Ointro,axiom,
! [P: set_set_a,S: set_a] :
( ( ord_le3724670747650509150_set_a @ P @ ( pow_a @ S ) )
=> ( ~ ( member_set_a @ bot_bot_set_a @ P )
=> ( ( ( comple2307003609928055243_set_a @ P )
= S )
=> ( ! [A8: set_a,B8: set_a] :
( ( member_set_a @ A8 @ P )
=> ( ( member_set_a @ B8 @ P )
=> ( ( A8 != B8 )
=> ( ( inf_inf_set_a @ A8 @ B8 )
= bot_bot_set_a ) ) ) )
=> ( set_partition_a @ S @ P ) ) ) ) ) ).
% partition.intro
thf(fact_1126_finite__Inter,axiom,
! [M: set_set_a] :
( ? [X5: set_a] :
( ( member_set_a @ X5 @ M )
& ( finite_finite_a @ X5 ) )
=> ( finite_finite_a @ ( comple6135023378680113637_set_a @ M ) ) ) ).
% finite_Inter
thf(fact_1127_finite__INT,axiom,
! [I3: set_a,A2: a > set_a] :
( ? [X5: a] :
( ( member_a @ X5 @ I3 )
& ( finite_finite_a @ ( A2 @ X5 ) ) )
=> ( finite_finite_a @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ A2 @ I3 ) ) ) ) ).
% finite_INT
thf(fact_1128_Inf__insert,axiom,
! [A: set_a,A2: set_set_a] :
( ( comple6135023378680113637_set_a @ ( insert_set_a @ A @ A2 ) )
= ( inf_inf_set_a @ A @ ( comple6135023378680113637_set_a @ A2 ) ) ) ).
% Inf_insert
thf(fact_1129_INF__const,axiom,
! [A2: set_a,F3: set_a] :
( ( A2 != bot_bot_set_a )
=> ( ( comple6135023378680113637_set_a
@ ( image_a_set_a
@ ^ [I: a] : F3
@ A2 ) )
= F3 ) ) ).
% INF_const
thf(fact_1130_cINF__const,axiom,
! [A2: set_a,C2: set_a] :
( ( A2 != bot_bot_set_a )
=> ( ( comple6135023378680113637_set_a
@ ( image_a_set_a
@ ^ [X3: a] : C2
@ A2 ) )
= C2 ) ) ).
% cINF_const
thf(fact_1131_INT__insert,axiom,
! [B: a > set_a,A: a,A2: set_a] :
( ( comple6135023378680113637_set_a @ ( image_a_set_a @ B @ ( insert_a @ A @ A2 ) ) )
= ( inf_inf_set_a @ ( B @ A ) @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ B @ A2 ) ) ) ) ).
% INT_insert
thf(fact_1132_INT__subset__iff,axiom,
! [B: set_a,A2: a > set_a,I3: set_a] :
( ( ord_less_eq_set_a @ B @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ A2 @ I3 ) ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ I3 )
=> ( ord_less_eq_set_a @ B @ ( A2 @ X3 ) ) ) ) ) ).
% INT_subset_iff
thf(fact_1133_INT__anti__mono,axiom,
! [A2: set_a,B: set_a,F3: a > set_a,G4: a > set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ ( F3 @ X2 ) @ ( G4 @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ F3 @ B ) ) @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ G4 @ A2 ) ) ) ) ) ).
% INT_anti_mono
thf(fact_1134_INT__greatest,axiom,
! [A2: set_a,C: set_a,B: a > set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ C @ ( B @ X2 ) ) )
=> ( ord_less_eq_set_a @ C @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ B @ A2 ) ) ) ) ).
% INT_greatest
thf(fact_1135_INT__lower,axiom,
! [A: a,A2: set_a,B: a > set_a] :
( ( member_a @ A @ A2 )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ B @ A2 ) ) @ ( B @ A ) ) ) ).
% INT_lower
thf(fact_1136_INT__insert__distrib,axiom,
! [U: a,A2: set_a,A: a,B: a > set_a] :
( ( member_a @ U @ A2 )
=> ( ( comple6135023378680113637_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( insert_a @ A @ ( B @ X3 ) )
@ A2 ) )
= ( insert_a @ A @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ B @ A2 ) ) ) ) ) ).
% INT_insert_distrib
thf(fact_1137_INT__extend__simps_I5_J,axiom,
! [A: a,B: a > set_a,C: set_a] :
( ( insert_a @ A @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ B @ C ) ) )
= ( comple6135023378680113637_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( insert_a @ A @ ( B @ X3 ) )
@ C ) ) ) ).
% INT_extend_simps(5)
thf(fact_1138_sup__Inf,axiom,
! [A: set_a,B: set_set_a] :
( ( sup_sup_set_a @ A @ ( comple6135023378680113637_set_a @ B ) )
= ( comple6135023378680113637_set_a @ ( image_set_a_set_a @ ( sup_sup_set_a @ A ) @ B ) ) ) ).
% sup_Inf
thf(fact_1139_INF__eq__const,axiom,
! [I3: set_a,F3: a > set_a,X: set_a] :
( ( I3 != bot_bot_set_a )
=> ( ! [I2: a] :
( ( member_a @ I2 @ I3 )
=> ( ( F3 @ I2 )
= X ) )
=> ( ( comple6135023378680113637_set_a @ ( image_a_set_a @ F3 @ I3 ) )
= X ) ) ) ).
% INF_eq_const
thf(fact_1140_INF__eq,axiom,
! [A2: set_a,B: set_a,G4: a > set_a,F3: a > set_a] :
( ! [I2: a] :
( ( member_a @ I2 @ A2 )
=> ? [X5: a] :
( ( member_a @ X5 @ B )
& ( ord_less_eq_set_a @ ( G4 @ X5 ) @ ( F3 @ I2 ) ) ) )
=> ( ! [J: a] :
( ( member_a @ J @ B )
=> ? [X5: a] :
( ( member_a @ X5 @ A2 )
& ( ord_less_eq_set_a @ ( F3 @ X5 ) @ ( G4 @ J ) ) ) )
=> ( ( comple6135023378680113637_set_a @ ( image_a_set_a @ F3 @ A2 ) )
= ( comple6135023378680113637_set_a @ ( image_a_set_a @ G4 @ B ) ) ) ) ) ).
% INF_eq
thf(fact_1141_INF__Int__eq,axiom,
! [S: set_set_a] :
( ( complete_Inf_Inf_a_o
@ ( image_set_a_a_o
@ ^ [I: set_a,X3: a] : ( member_a @ X3 @ I )
@ S ) )
= ( ^ [X3: a] : ( member_a @ X3 @ ( comple6135023378680113637_set_a @ S ) ) ) ) ).
% INF_Int_eq
thf(fact_1142_INF__INT__eq,axiom,
! [R2: a > set_a,S: set_a] :
( ( complete_Inf_Inf_a_o
@ ( image_a_a_o
@ ^ [I: a,X3: a] : ( member_a @ X3 @ ( R2 @ I ) )
@ S ) )
= ( ^ [X3: a] : ( member_a @ X3 @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ R2 @ S ) ) ) ) ) ).
% INF_INT_eq
thf(fact_1143_Inf__INT__eq,axiom,
( complete_Inf_Inf_a_o
= ( ^ [S2: set_a_o,X3: a] : ( member_a @ X3 @ ( comple6135023378680113637_set_a @ ( image_a_o_set_a @ collect_a @ S2 ) ) ) ) ) ).
% Inf_INT_eq
thf(fact_1144_INF__sup__distrib2,axiom,
! [F3: a > set_a,A2: set_a,G4: a > set_a,B: set_a] :
( ( sup_sup_set_a @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ F3 @ A2 ) ) @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ G4 @ B ) ) )
= ( comple6135023378680113637_set_a
@ ( image_a_set_a
@ ^ [A4: a] :
( comple6135023378680113637_set_a
@ ( image_a_set_a
@ ^ [B3: a] : ( sup_sup_set_a @ ( F3 @ A4 ) @ ( G4 @ B3 ) )
@ B ) )
@ A2 ) ) ) ).
% INF_sup_distrib2
thf(fact_1145_sup__INF,axiom,
! [A: set_a,F3: a > set_a,B: set_a] :
( ( sup_sup_set_a @ A @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ F3 @ B ) ) )
= ( comple6135023378680113637_set_a
@ ( image_a_set_a
@ ^ [B3: a] : ( sup_sup_set_a @ A @ ( F3 @ B3 ) )
@ B ) ) ) ).
% sup_INF
thf(fact_1146_Inf__sup,axiom,
! [B: set_set_a,A: set_a] :
( ( sup_sup_set_a @ ( comple6135023378680113637_set_a @ B ) @ A )
= ( comple6135023378680113637_set_a
@ ( image_set_a_set_a
@ ^ [B3: set_a] : ( sup_sup_set_a @ B3 @ A )
@ B ) ) ) ).
% Inf_sup
thf(fact_1147_INF__sup,axiom,
! [F3: a > set_a,B: set_a,A: set_a] :
( ( sup_sup_set_a @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ F3 @ B ) ) @ A )
= ( comple6135023378680113637_set_a
@ ( image_a_set_a
@ ^ [B3: a] : ( sup_sup_set_a @ ( F3 @ B3 ) @ A )
@ B ) ) ) ).
% INF_sup
thf(fact_1148_INF__absorb,axiom,
! [K: a,I3: set_a,A2: a > set_a] :
( ( member_a @ K @ I3 )
=> ( ( inf_inf_set_a @ ( A2 @ K ) @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ A2 @ I3 ) ) )
= ( comple6135023378680113637_set_a @ ( image_a_set_a @ A2 @ I3 ) ) ) ) ).
% INF_absorb
thf(fact_1149_INF__inf__distrib,axiom,
! [F3: a > set_a,A2: set_a,G4: a > set_a] :
( ( inf_inf_set_a @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ F3 @ A2 ) ) @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ G4 @ A2 ) ) )
= ( comple6135023378680113637_set_a
@ ( image_a_set_a
@ ^ [A4: a] : ( inf_inf_set_a @ ( F3 @ A4 ) @ ( G4 @ A4 ) )
@ A2 ) ) ) ).
% INF_inf_distrib
thf(fact_1150_INF__greatest,axiom,
! [A2: set_a,U: set_a,F3: a > set_a] :
( ! [I2: a] :
( ( member_a @ I2 @ A2 )
=> ( ord_less_eq_set_a @ U @ ( F3 @ I2 ) ) )
=> ( ord_less_eq_set_a @ U @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ F3 @ A2 ) ) ) ) ).
% INF_greatest
thf(fact_1151_le__INF__iff,axiom,
! [U: set_a,F3: a > set_a,A2: set_a] :
( ( ord_less_eq_set_a @ U @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ F3 @ A2 ) ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ord_less_eq_set_a @ U @ ( F3 @ X3 ) ) ) ) ) ).
% le_INF_iff
thf(fact_1152_INF__lower2,axiom,
! [I5: a,A2: set_a,F3: a > set_a,U: set_a] :
( ( member_a @ I5 @ A2 )
=> ( ( ord_less_eq_set_a @ ( F3 @ I5 ) @ U )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ F3 @ A2 ) ) @ U ) ) ) ).
% INF_lower2
thf(fact_1153_INF__mono_H,axiom,
! [F3: a > set_a,G4: a > set_a,A2: set_a] :
( ! [X2: a] : ( ord_less_eq_set_a @ ( F3 @ X2 ) @ ( G4 @ X2 ) )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ F3 @ A2 ) ) @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ G4 @ A2 ) ) ) ) ).
% INF_mono'
thf(fact_1154_INF__lower,axiom,
! [I5: a,A2: set_a,F3: a > set_a] :
( ( member_a @ I5 @ A2 )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ F3 @ A2 ) ) @ ( F3 @ I5 ) ) ) ).
% INF_lower
thf(fact_1155_INF__mono,axiom,
! [B: set_a,A2: set_a,F3: a > set_a,G4: a > set_a] :
( ! [M4: a] :
( ( member_a @ M4 @ B )
=> ? [X5: a] :
( ( member_a @ X5 @ A2 )
& ( ord_less_eq_set_a @ ( F3 @ X5 ) @ ( G4 @ M4 ) ) ) )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ F3 @ A2 ) ) @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ G4 @ B ) ) ) ) ).
% INF_mono
thf(fact_1156_INF__eqI,axiom,
! [A2: set_a,X: set_a,F3: a > set_a] :
( ! [I2: a] :
( ( member_a @ I2 @ A2 )
=> ( ord_less_eq_set_a @ X @ ( F3 @ I2 ) ) )
=> ( ! [Y2: set_a] :
( ! [I4: a] :
( ( member_a @ I4 @ A2 )
=> ( ord_less_eq_set_a @ Y2 @ ( F3 @ I4 ) ) )
=> ( ord_less_eq_set_a @ Y2 @ X ) )
=> ( ( comple6135023378680113637_set_a @ ( image_a_set_a @ F3 @ A2 ) )
= X ) ) ) ).
% INF_eqI
thf(fact_1157_INT__absorb,axiom,
! [K: a,I3: set_a,A2: a > set_a] :
( ( member_a @ K @ I3 )
=> ( ( inf_inf_set_a @ ( A2 @ K ) @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ A2 @ I3 ) ) )
= ( comple6135023378680113637_set_a @ ( image_a_set_a @ A2 @ I3 ) ) ) ) ).
% INT_absorb
thf(fact_1158_INT__Int__distrib,axiom,
! [A2: a > set_a,B: a > set_a,I3: set_a] :
( ( comple6135023378680113637_set_a
@ ( image_a_set_a
@ ^ [I: a] : ( inf_inf_set_a @ ( A2 @ I ) @ ( B @ I ) )
@ I3 ) )
= ( inf_inf_set_a @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ A2 @ I3 ) ) @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ B @ I3 ) ) ) ) ).
% INT_Int_distrib
thf(fact_1159_Int__Inter__image,axiom,
! [A2: a > set_a,B: a > set_a,C: set_a] :
( ( comple6135023378680113637_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( inf_inf_set_a @ ( A2 @ X3 ) @ ( B @ X3 ) )
@ C ) )
= ( inf_inf_set_a @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ A2 @ C ) ) @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ B @ C ) ) ) ) ).
% Int_Inter_image
thf(fact_1160_INT__extend__simps_I7_J,axiom,
! [A2: set_a,B: a > set_a,C: set_a] :
( ( sup_sup_set_a @ A2 @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ B @ C ) ) )
= ( comple6135023378680113637_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( sup_sup_set_a @ A2 @ ( B @ X3 ) )
@ C ) ) ) ).
% INT_extend_simps(7)
thf(fact_1161_INT__extend__simps_I6_J,axiom,
! [A2: a > set_a,C: set_a,B: set_a] :
( ( sup_sup_set_a @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ A2 @ C ) ) @ B )
= ( comple6135023378680113637_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( sup_sup_set_a @ ( A2 @ X3 ) @ B )
@ C ) ) ) ).
% INT_extend_simps(6)
thf(fact_1162_Un__INT__distrib,axiom,
! [B: set_a,A2: a > set_a,I3: set_a] :
( ( sup_sup_set_a @ B @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ A2 @ I3 ) ) )
= ( comple6135023378680113637_set_a
@ ( image_a_set_a
@ ^ [I: a] : ( sup_sup_set_a @ B @ ( A2 @ I ) )
@ I3 ) ) ) ).
% Un_INT_distrib
thf(fact_1163_Un__INT__distrib2,axiom,
! [A2: a > set_a,I3: set_a,B: a > set_a,J2: set_a] :
( ( sup_sup_set_a @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ A2 @ I3 ) ) @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ B @ J2 ) ) )
= ( comple6135023378680113637_set_a
@ ( image_a_set_a
@ ^ [I: a] :
( comple6135023378680113637_set_a
@ ( image_a_set_a
@ ^ [J3: a] : ( sup_sup_set_a @ ( A2 @ I ) @ ( B @ J3 ) )
@ J2 ) )
@ I3 ) ) ) ).
% Un_INT_distrib2
thf(fact_1164_Inter__insert,axiom,
! [A: set_a,B: set_set_a] :
( ( comple6135023378680113637_set_a @ ( insert_set_a @ A @ B ) )
= ( inf_inf_set_a @ A @ ( comple6135023378680113637_set_a @ B ) ) ) ).
% Inter_insert
thf(fact_1165_Inter__Un__distrib,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( comple6135023378680113637_set_a @ ( sup_sup_set_set_a @ A2 @ B ) )
= ( inf_inf_set_a @ ( comple6135023378680113637_set_a @ A2 ) @ ( comple6135023378680113637_set_a @ B ) ) ) ).
% Inter_Un_distrib
thf(fact_1166_Inter__subset,axiom,
! [A2: set_set_a,B: set_a] :
( ! [X7: set_a] :
( ( member_set_a @ X7 @ A2 )
=> ( ord_less_eq_set_a @ X7 @ B ) )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A2 ) @ B ) ) ) ).
% Inter_subset
thf(fact_1167_partition_Odisjoint,axiom,
! [S: set_a,P: set_set_a,A2: set_a,B: set_a] :
( ( set_partition_a @ S @ P )
=> ( ( member_set_a @ A2 @ P )
=> ( ( member_set_a @ B @ P )
=> ( ( A2 != B )
=> ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a ) ) ) ) ) ).
% partition.disjoint
thf(fact_1168_cInf__greatest,axiom,
! [X4: set_set_a,Z2: set_a] :
( ( X4 != bot_bot_set_set_a )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ X4 )
=> ( ord_less_eq_set_a @ Z2 @ X2 ) )
=> ( ord_less_eq_set_a @ Z2 @ ( comple6135023378680113637_set_a @ X4 ) ) ) ) ).
% cInf_greatest
thf(fact_1169_cInf__eq__non__empty,axiom,
! [X4: set_set_a,A: set_a] :
( ( X4 != bot_bot_set_set_a )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ X4 )
=> ( ord_less_eq_set_a @ A @ X2 ) )
=> ( ! [Y2: set_a] :
( ! [X5: set_a] :
( ( member_set_a @ X5 @ X4 )
=> ( ord_less_eq_set_a @ Y2 @ X5 ) )
=> ( ord_less_eq_set_a @ Y2 @ A ) )
=> ( ( comple6135023378680113637_set_a @ X4 )
= A ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_1170_cInf__le__finite,axiom,
! [X4: set_set_a,X: set_a] :
( ( finite_finite_set_a @ X4 )
=> ( ( member_set_a @ X @ X4 )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ X4 ) @ X ) ) ) ).
% cInf_le_finite
thf(fact_1171_cInf__eq__minimum,axiom,
! [Z2: set_a,X4: set_set_a] :
( ( member_set_a @ Z2 @ X4 )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ X4 )
=> ( ord_less_eq_set_a @ Z2 @ X2 ) )
=> ( ( comple6135023378680113637_set_a @ X4 )
= Z2 ) ) ) ).
% cInf_eq_minimum
thf(fact_1172_Inf__less__eq,axiom,
! [A2: set_set_a,U: set_a] :
( ! [V3: set_a] :
( ( member_set_a @ V3 @ A2 )
=> ( ord_less_eq_set_a @ V3 @ U ) )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A2 ) @ U ) ) ) ).
% Inf_less_eq
thf(fact_1173_Inf__superset__mono,axiom,
! [B: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A2 )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A2 ) @ ( comple6135023378680113637_set_a @ B ) ) ) ).
% Inf_superset_mono
thf(fact_1174_Inf__union__distrib,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( comple6135023378680113637_set_a @ ( sup_sup_set_set_a @ A2 @ B ) )
= ( inf_inf_set_a @ ( comple6135023378680113637_set_a @ A2 ) @ ( comple6135023378680113637_set_a @ B ) ) ) ).
% Inf_union_distrib
thf(fact_1175_partition_Onon__vacuous,axiom,
! [S: set_a,P: set_set_a] :
( ( set_partition_a @ S @ P )
=> ~ ( member_set_a @ bot_bot_set_a @ P ) ) ).
% partition.non_vacuous
thf(fact_1176_Inf__eqI,axiom,
! [A2: set_set_a,X: set_a] :
( ! [I2: set_a] :
( ( member_set_a @ I2 @ A2 )
=> ( ord_less_eq_set_a @ X @ I2 ) )
=> ( ! [Y2: set_a] :
( ! [I4: set_a] :
( ( member_set_a @ I4 @ A2 )
=> ( ord_less_eq_set_a @ Y2 @ I4 ) )
=> ( ord_less_eq_set_a @ Y2 @ X ) )
=> ( ( comple6135023378680113637_set_a @ A2 )
= X ) ) ) ).
% Inf_eqI
thf(fact_1177_Inf__mono,axiom,
! [B: set_set_a,A2: set_set_a] :
( ! [B2: set_a] :
( ( member_set_a @ B2 @ B )
=> ? [X5: set_a] :
( ( member_set_a @ X5 @ A2 )
& ( ord_less_eq_set_a @ X5 @ B2 ) ) )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A2 ) @ ( comple6135023378680113637_set_a @ B ) ) ) ).
% Inf_mono
thf(fact_1178_Inf__lower,axiom,
! [X: set_a,A2: set_set_a] :
( ( member_set_a @ X @ A2 )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A2 ) @ X ) ) ).
% Inf_lower
thf(fact_1179_Inf__lower2,axiom,
! [U: set_a,A2: set_set_a,V2: set_a] :
( ( member_set_a @ U @ A2 )
=> ( ( ord_less_eq_set_a @ U @ V2 )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A2 ) @ V2 ) ) ) ).
% Inf_lower2
thf(fact_1180_le__Inf__iff,axiom,
! [B4: set_a,A2: set_set_a] :
( ( ord_less_eq_set_a @ B4 @ ( comple6135023378680113637_set_a @ A2 ) )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( ord_less_eq_set_a @ B4 @ X3 ) ) ) ) ).
% le_Inf_iff
thf(fact_1181_Inf__greatest,axiom,
! [A2: set_set_a,Z2: set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ Z2 @ X2 ) )
=> ( ord_less_eq_set_a @ Z2 @ ( comple6135023378680113637_set_a @ A2 ) ) ) ).
% Inf_greatest
thf(fact_1182_Inter__lower,axiom,
! [B: set_a,A2: set_set_a] :
( ( member_set_a @ B @ A2 )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A2 ) @ B ) ) ).
% Inter_lower
thf(fact_1183_Inter__greatest,axiom,
! [A2: set_set_a,C: set_a] :
( ! [X7: set_a] :
( ( member_set_a @ X7 @ A2 )
=> ( ord_less_eq_set_a @ C @ X7 ) )
=> ( ord_less_eq_set_a @ C @ ( comple6135023378680113637_set_a @ A2 ) ) ) ).
% Inter_greatest
thf(fact_1184_Inter__anti__mono,axiom,
! [B: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A2 )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A2 ) @ ( comple6135023378680113637_set_a @ B ) ) ) ).
% Inter_anti_mono
thf(fact_1185_Un__Inter,axiom,
! [A2: set_a,B: set_set_a] :
( ( sup_sup_set_a @ A2 @ ( comple6135023378680113637_set_a @ B ) )
= ( comple6135023378680113637_set_a @ ( image_set_a_set_a @ ( sup_sup_set_a @ A2 ) @ B ) ) ) ).
% Un_Inter
thf(fact_1186_INF__eq__iff,axiom,
! [I3: set_a,F3: a > set_a,C2: set_a] :
( ( I3 != bot_bot_set_a )
=> ( ! [I2: a] :
( ( member_a @ I2 @ I3 )
=> ( ord_less_eq_set_a @ ( F3 @ I2 ) @ C2 ) )
=> ( ( ( comple6135023378680113637_set_a @ ( image_a_set_a @ F3 @ I3 ) )
= C2 )
= ( ! [X3: a] :
( ( member_a @ X3 @ I3 )
=> ( ( F3 @ X3 )
= C2 ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_1187_cINF__greatest,axiom,
! [A2: set_a,M5: set_a,F3: a > set_a] :
( ( A2 != bot_bot_set_a )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ M5 @ ( F3 @ X2 ) ) )
=> ( ord_less_eq_set_a @ M5 @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ F3 @ A2 ) ) ) ) ) ).
% cINF_greatest
thf(fact_1188_Inf__le__Sup,axiom,
! [A2: set_set_a] :
( ( A2 != bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A2 ) @ ( comple2307003609928055243_set_a @ A2 ) ) ) ).
% Inf_le_Sup
thf(fact_1189_less__eq__Inf__inter,axiom,
! [A2: set_set_a,B: set_set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( comple6135023378680113637_set_a @ A2 ) @ ( comple6135023378680113637_set_a @ B ) ) @ ( comple6135023378680113637_set_a @ ( inf_inf_set_set_a @ A2 @ B ) ) ) ).
% less_eq_Inf_inter
thf(fact_1190_finite__Inf__in,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ! [X2: set_a,Y2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( ( member_set_a @ Y2 @ A2 )
=> ( member_set_a @ ( inf_inf_set_a @ X2 @ Y2 ) @ A2 ) ) )
=> ( member_set_a @ ( comple6135023378680113637_set_a @ A2 ) @ A2 ) ) ) ) ).
% finite_Inf_in
thf(fact_1191_inf__Inf__fold__inf,axiom,
! [A2: set_set_a,B: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( inf_inf_set_a @ ( comple6135023378680113637_set_a @ A2 ) @ B )
= ( finite5985231929012247624_set_a @ inf_inf_set_a @ B @ A2 ) ) ) ).
% inf_Inf_fold_inf
thf(fact_1192_INF__superset__mono,axiom,
! [B: set_a,A2: set_a,F3: a > set_a,G4: a > set_a] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( ord_less_eq_set_a @ ( F3 @ X2 ) @ ( G4 @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ F3 @ A2 ) ) @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ G4 @ B ) ) ) ) ) ).
% INF_superset_mono
thf(fact_1193_INF__inf__const1,axiom,
! [I3: set_a,X: set_a,F3: a > set_a] :
( ( I3 != bot_bot_set_a )
=> ( ( comple6135023378680113637_set_a
@ ( image_a_set_a
@ ^ [I: a] : ( inf_inf_set_a @ X @ ( F3 @ I ) )
@ I3 ) )
= ( inf_inf_set_a @ X @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ F3 @ I3 ) ) ) ) ) ).
% INF_inf_const1
thf(fact_1194_INF__inf__const2,axiom,
! [I3: set_a,F3: a > set_a,X: set_a] :
( ( I3 != bot_bot_set_a )
=> ( ( comple6135023378680113637_set_a
@ ( image_a_set_a
@ ^ [I: a] : ( inf_inf_set_a @ ( F3 @ I ) @ X )
@ I3 ) )
= ( inf_inf_set_a @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ F3 @ I3 ) ) @ X ) ) ) ).
% INF_inf_const2
thf(fact_1195_INF__insert,axiom,
! [F3: a > set_a,A: a,A2: set_a] :
( ( comple6135023378680113637_set_a @ ( image_a_set_a @ F3 @ ( insert_a @ A @ A2 ) ) )
= ( inf_inf_set_a @ ( F3 @ A ) @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ F3 @ A2 ) ) ) ) ).
% INF_insert
thf(fact_1196_INF__union,axiom,
! [M: a > set_a,A2: set_a,B: set_a] :
( ( comple6135023378680113637_set_a @ ( image_a_set_a @ M @ ( sup_sup_set_a @ A2 @ B ) ) )
= ( inf_inf_set_a @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ M @ A2 ) ) @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ M @ B ) ) ) ) ).
% INF_union
thf(fact_1197_INT__extend__simps_I2_J,axiom,
! [C: set_a,A2: set_a,B: a > set_a] :
( ( ( C = bot_bot_set_a )
=> ( ( inf_inf_set_a @ A2 @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ B @ C ) ) )
= A2 ) )
& ( ( C != bot_bot_set_a )
=> ( ( inf_inf_set_a @ A2 @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ B @ C ) ) )
= ( comple6135023378680113637_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( inf_inf_set_a @ A2 @ ( B @ X3 ) )
@ C ) ) ) ) ) ).
% INT_extend_simps(2)
thf(fact_1198_INT__extend__simps_I1_J,axiom,
! [C: set_a,A2: a > set_a,B: set_a] :
( ( ( C = bot_bot_set_a )
=> ( ( inf_inf_set_a @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ A2 @ C ) ) @ B )
= B ) )
& ( ( C != bot_bot_set_a )
=> ( ( inf_inf_set_a @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ A2 @ C ) ) @ B )
= ( comple6135023378680113637_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( inf_inf_set_a @ ( A2 @ X3 ) @ B )
@ C ) ) ) ) ) ).
% INT_extend_simps(1)
thf(fact_1199_INT__Un,axiom,
! [M: a > set_a,A2: set_a,B: set_a] :
( ( comple6135023378680113637_set_a @ ( image_a_set_a @ M @ ( sup_sup_set_a @ A2 @ B ) ) )
= ( inf_inf_set_a @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ M @ A2 ) ) @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ M @ B ) ) ) ) ).
% INT_Un
thf(fact_1200_Inter__Un__subset,axiom,
! [A2: set_set_a,B: set_set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( comple6135023378680113637_set_a @ A2 ) @ ( comple6135023378680113637_set_a @ B ) ) @ ( comple6135023378680113637_set_a @ ( inf_inf_set_set_a @ A2 @ B ) ) ) ).
% Inter_Un_subset
thf(fact_1201_INF__le__SUP,axiom,
! [A2: set_a,F3: a > set_a] :
( ( A2 != bot_bot_set_a )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ F3 @ A2 ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F3 @ A2 ) ) ) ) ).
% INF_le_SUP
thf(fact_1202_Int__Inter__eq_I2_J,axiom,
! [B9: set_set_a,A2: set_a] :
( ( ( B9 = bot_bot_set_set_a )
=> ( ( inf_inf_set_a @ ( comple6135023378680113637_set_a @ B9 ) @ A2 )
= A2 ) )
& ( ( B9 != bot_bot_set_set_a )
=> ( ( inf_inf_set_a @ ( comple6135023378680113637_set_a @ B9 ) @ A2 )
= ( comple6135023378680113637_set_a
@ ( image_set_a_set_a
@ ^ [B6: set_a] : ( inf_inf_set_a @ B6 @ A2 )
@ B9 ) ) ) ) ) ).
% Int_Inter_eq(2)
thf(fact_1203_Int__Inter__eq_I1_J,axiom,
! [B9: set_set_a,A2: set_a] :
( ( ( B9 = bot_bot_set_set_a )
=> ( ( inf_inf_set_a @ A2 @ ( comple6135023378680113637_set_a @ B9 ) )
= A2 ) )
& ( ( B9 != bot_bot_set_set_a )
=> ( ( inf_inf_set_a @ A2 @ ( comple6135023378680113637_set_a @ B9 ) )
= ( comple6135023378680113637_set_a @ ( image_set_a_set_a @ ( inf_inf_set_a @ A2 ) @ B9 ) ) ) ) ) ).
% Int_Inter_eq(1)
thf(fact_1204_INT__extend__simps_I4_J,axiom,
! [C: set_a,A2: set_a,B: a > set_a] :
( ( ( C = bot_bot_set_a )
=> ( ( minus_minus_set_a @ A2 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ C ) ) )
= A2 ) )
& ( ( C != bot_bot_set_a )
=> ( ( minus_minus_set_a @ A2 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ C ) ) )
= ( comple6135023378680113637_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( minus_minus_set_a @ A2 @ ( B @ X3 ) )
@ C ) ) ) ) ) ).
% INT_extend_simps(4)
thf(fact_1205_Set__Theory_Opartition__def,axiom,
( set_partition_a
= ( ^ [S2: set_a,P2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ P2 @ ( pow_a @ S2 ) )
& ~ ( member_set_a @ bot_bot_set_a @ P2 )
& ( ( comple2307003609928055243_set_a @ P2 )
= S2 )
& ! [A6: set_a,B6: set_a] :
( ( member_set_a @ A6 @ P2 )
=> ( ( member_set_a @ B6 @ P2 )
=> ( ( A6 != B6 )
=> ( ( inf_inf_set_a @ A6 @ B6 )
= bot_bot_set_a ) ) ) ) ) ) ) ).
% Set_Theory.partition_def
thf(fact_1206_INF__fold__inf,axiom,
! [A2: set_a,F3: a > set_a] :
( ( finite_finite_a @ A2 )
=> ( ( comple6135023378680113637_set_a @ ( image_a_set_a @ F3 @ A2 ) )
= ( finite_fold_a_set_a @ ( comp_s1103547301056249180et_a_a @ inf_inf_set_a @ F3 ) @ top_top_set_a @ A2 ) ) ) ).
% INF_fold_inf
thf(fact_1207_INT__simps_I4_J,axiom,
! [C: set_a,A2: set_a,B: a > set_a] :
( ( ( C = bot_bot_set_a )
=> ( ( comple6135023378680113637_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( minus_minus_set_a @ A2 @ ( B @ X3 ) )
@ C ) )
= top_top_set_a ) )
& ( ( C != bot_bot_set_a )
=> ( ( comple6135023378680113637_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( minus_minus_set_a @ A2 @ ( B @ X3 ) )
@ C ) )
= ( minus_minus_set_a @ A2 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ C ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_1208_UNIV__I,axiom,
! [X: a] : ( member_a @ X @ top_top_set_a ) ).
% UNIV_I
thf(fact_1209_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_1210_inf__top_Oright__neutral,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ top_top_set_a )
= A ) ).
% inf_top.right_neutral
thf(fact_1211_inf__top_Oneutr__eq__iff,axiom,
! [A: set_a,B4: set_a] :
( ( top_top_set_a
= ( inf_inf_set_a @ A @ B4 ) )
= ( ( A = top_top_set_a )
& ( B4 = top_top_set_a ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_1212_inf__top_Oleft__neutral,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ top_top_set_a @ A )
= A ) ).
% inf_top.left_neutral
thf(fact_1213_inf__top_Oeq__neutr__iff,axiom,
! [A: set_a,B4: set_a] :
( ( ( inf_inf_set_a @ A @ B4 )
= top_top_set_a )
= ( ( A = top_top_set_a )
& ( B4 = top_top_set_a ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_1214_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_1215_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_1216_inf__top__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ top_top_set_a )
= X ) ).
% inf_top_right
thf(fact_1217_inf__top__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ top_top_set_a @ X )
= X ) ).
% inf_top_left
thf(fact_1218_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_1219_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_1220_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_1221_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_1222_Int__UNIV,axiom,
! [A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A2 @ B )
= top_top_set_a )
= ( ( A2 = top_top_set_a )
& ( B = top_top_set_a ) ) ) ).
% Int_UNIV
thf(fact_1223_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_1224_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_1225_Inf__UNIV,axiom,
( ( comple6135023378680113637_set_a @ top_top_set_set_a )
= bot_bot_set_a ) ).
% Inf_UNIV
thf(fact_1226_Diff__UNIV,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ top_top_set_a )
= bot_bot_set_a ) ).
% Diff_UNIV
thf(fact_1227_range__constant,axiom,
! [X: set_a] :
( ( image_a_set_a
@ ^ [Uu: a] : X
@ top_top_set_a )
= ( insert_set_a @ X @ bot_bot_set_set_a ) ) ).
% range_constant
thf(fact_1228_range__constant,axiom,
! [X: a] :
( ( image_a_a
@ ^ [Uu: a] : X
@ top_top_set_a )
= ( insert_a @ X @ bot_bot_set_a ) ) ).
% range_constant
thf(fact_1229_INT__constant,axiom,
! [A2: set_a,C2: set_a] :
( ( ( A2 = bot_bot_set_a )
=> ( ( comple6135023378680113637_set_a
@ ( image_a_set_a
@ ^ [Y3: a] : C2
@ A2 ) )
= top_top_set_a ) )
& ( ( A2 != bot_bot_set_a )
=> ( ( comple6135023378680113637_set_a
@ ( image_a_set_a
@ ^ [Y3: a] : C2
@ A2 ) )
= C2 ) ) ) ).
% INT_constant
thf(fact_1230_INT__simps_I1_J,axiom,
! [C: set_a,A2: a > set_a,B: set_a] :
( ( ( C = bot_bot_set_a )
=> ( ( comple6135023378680113637_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( inf_inf_set_a @ ( A2 @ X3 ) @ B )
@ C ) )
= top_top_set_a ) )
& ( ( C != bot_bot_set_a )
=> ( ( comple6135023378680113637_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( inf_inf_set_a @ ( A2 @ X3 ) @ B )
@ C ) )
= ( inf_inf_set_a @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ A2 @ C ) ) @ B ) ) ) ) ).
% INT_simps(1)
thf(fact_1231_INT__simps_I2_J,axiom,
! [C: set_a,A2: set_a,B: a > set_a] :
( ( ( C = bot_bot_set_a )
=> ( ( comple6135023378680113637_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( inf_inf_set_a @ A2 @ ( B @ X3 ) )
@ C ) )
= top_top_set_a ) )
& ( ( C != bot_bot_set_a )
=> ( ( comple6135023378680113637_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( inf_inf_set_a @ A2 @ ( B @ X3 ) )
@ C ) )
= ( inf_inf_set_a @ A2 @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ B @ C ) ) ) ) ) ) ).
% INT_simps(2)
thf(fact_1232_INT__simps_I3_J,axiom,
! [C: set_a,A2: a > set_a,B: set_a] :
( ( ( C = bot_bot_set_a )
=> ( ( comple6135023378680113637_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( minus_minus_set_a @ ( A2 @ X3 ) @ B )
@ C ) )
= top_top_set_a ) )
& ( ( C != bot_bot_set_a )
=> ( ( comple6135023378680113637_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( minus_minus_set_a @ ( A2 @ X3 ) @ B )
@ C ) )
= ( minus_minus_set_a @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ A2 @ C ) ) @ B ) ) ) ) ).
% INT_simps(3)
thf(fact_1233_INT__empty,axiom,
! [B: a > set_a] :
( ( comple6135023378680113637_set_a @ ( image_a_set_a @ B @ bot_bot_set_a ) )
= top_top_set_a ) ).
% INT_empty
thf(fact_1234_INT__bool__eq,axiom,
! [A2: $o > set_a] :
( ( comple6135023378680113637_set_a @ ( image_o_set_a @ A2 @ top_top_set_o ) )
= ( inf_inf_set_a @ ( A2 @ $true ) @ ( A2 @ $false ) ) ) ).
% INT_bool_eq
thf(fact_1235_INF__UNIV__bool__expand,axiom,
! [A2: $o > set_a] :
( ( comple6135023378680113637_set_a @ ( image_o_set_a @ A2 @ top_top_set_o ) )
= ( inf_inf_set_a @ ( A2 @ $true ) @ ( A2 @ $false ) ) ) ).
% INF_UNIV_bool_expand
thf(fact_1236_Inter__UNIV,axiom,
( ( comple6135023378680113637_set_a @ top_top_set_set_a )
= bot_bot_set_a ) ).
% Inter_UNIV
thf(fact_1237_Inf__sup__eq__top__iff,axiom,
! [B: set_set_a,A: set_a] :
( ( ( sup_sup_set_a @ ( comple6135023378680113637_set_a @ B ) @ A )
= top_top_set_a )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ B )
=> ( ( sup_sup_set_a @ X3 @ A )
= top_top_set_a ) ) ) ) ).
% Inf_sup_eq_top_iff
thf(fact_1238_finite__range__imageI,axiom,
! [G4: a > set_a,F3: set_a > set_a] :
( ( finite_finite_set_a @ ( image_a_set_a @ G4 @ top_top_set_a ) )
=> ( finite_finite_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( F3 @ ( G4 @ X3 ) )
@ top_top_set_a ) ) ) ).
% finite_range_imageI
thf(fact_1239_finite__range__imageI,axiom,
! [G4: a > set_a,F3: set_a > a] :
( ( finite_finite_set_a @ ( image_a_set_a @ G4 @ top_top_set_a ) )
=> ( finite_finite_a
@ ( image_a_a
@ ^ [X3: a] : ( F3 @ ( G4 @ X3 ) )
@ top_top_set_a ) ) ) ).
% finite_range_imageI
thf(fact_1240_finite__range__imageI,axiom,
! [G4: a > a,F3: a > set_a] :
( ( finite_finite_a @ ( image_a_a @ G4 @ top_top_set_a ) )
=> ( finite_finite_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( F3 @ ( G4 @ X3 ) )
@ top_top_set_a ) ) ) ).
% finite_range_imageI
thf(fact_1241_finite__range__imageI,axiom,
! [G4: a > a,F3: a > a] :
( ( finite_finite_a @ ( image_a_a @ G4 @ top_top_set_a ) )
=> ( finite_finite_a
@ ( image_a_a
@ ^ [X3: a] : ( F3 @ ( G4 @ X3 ) )
@ top_top_set_a ) ) ) ).
% finite_range_imageI
thf(fact_1242_SUP__UNIV__bool__expand,axiom,
! [A2: $o > set_a] :
( ( comple2307003609928055243_set_a @ ( image_o_set_a @ A2 @ top_top_set_o ) )
= ( sup_sup_set_a @ ( A2 @ $true ) @ ( A2 @ $false ) ) ) ).
% SUP_UNIV_bool_expand
thf(fact_1243_UN__bool__eq,axiom,
! [A2: $o > set_a] :
( ( comple2307003609928055243_set_a @ ( image_o_set_a @ A2 @ top_top_set_o ) )
= ( sup_sup_set_a @ ( A2 @ $true ) @ ( A2 @ $false ) ) ) ).
% UN_bool_eq
thf(fact_1244_Un__eq__UN,axiom,
( sup_sup_set_a
= ( ^ [A6: set_a,B6: set_a] :
( comple2307003609928055243_set_a
@ ( image_o_set_a
@ ^ [B3: $o] : ( if_set_a @ B3 @ A6 @ B6 )
@ top_top_set_o ) ) ) ) ).
% Un_eq_UN
thf(fact_1245_insert__UNIV,axiom,
! [X: a] :
( ( insert_a @ X @ top_top_set_a )
= top_top_set_a ) ).
% insert_UNIV
thf(fact_1246_UNIV__def,axiom,
( top_top_set_a
= ( collect_a
@ ^ [X3: a] : $true ) ) ).
% UNIV_def
thf(fact_1247_UNIV__witness,axiom,
? [X2: a] : ( member_a @ X2 @ top_top_set_a ) ).
% UNIV_witness
thf(fact_1248_top__set__def,axiom,
( top_top_set_a
= ( collect_a @ top_top_a_o ) ) ).
% top_set_def
thf(fact_1249_UNIV__eq__I,axiom,
! [A2: set_a] :
( ! [X2: a] : ( member_a @ X2 @ A2 )
=> ( top_top_set_a = A2 ) ) ).
% UNIV_eq_I
thf(fact_1250_top__empty__eq,axiom,
( top_top_a_o
= ( ^ [X3: a] : ( member_a @ X3 @ top_top_set_a ) ) ) ).
% top_empty_eq
thf(fact_1251_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_1252_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_1253_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_1254_ex__new__if__finite,axiom,
! [A2: set_a] :
( ~ ( finite_finite_a @ top_top_set_a )
=> ( ( finite_finite_a @ A2 )
=> ? [A3: a] :
~ ( member_a @ A3 @ A2 ) ) ) ).
% ex_new_if_finite
thf(fact_1255_empty__not__UNIV,axiom,
bot_bot_set_a != top_top_set_a ).
% empty_not_UNIV
thf(fact_1256_subset__UNIV,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ top_top_set_a ) ).
% subset_UNIV
thf(fact_1257_top__greatest,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ top_top_set_a ) ).
% top_greatest
thf(fact_1258_top_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A )
= ( A = top_top_set_a ) ) ).
% top.extremum_unique
thf(fact_1259_top_Oextremum__uniqueI,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A )
=> ( A = top_top_set_a ) ) ).
% top.extremum_uniqueI
thf(fact_1260_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_1261_Un__UNIV__right,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ top_top_set_a )
= top_top_set_a ) ).
% Un_UNIV_right
thf(fact_1262_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_1263_Int__UNIV__right,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ top_top_set_a )
= A2 ) ).
% Int_UNIV_right
thf(fact_1264_Int__UNIV__left,axiom,
! [B: set_a] :
( ( inf_inf_set_a @ top_top_set_a @ B )
= B ) ).
% Int_UNIV_left
thf(fact_1265_range__subsetD,axiom,
! [F3: a > set_a,B: set_set_a,I5: a] :
( ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F3 @ top_top_set_a ) @ B )
=> ( member_set_a @ ( F3 @ I5 ) @ B ) ) ).
% range_subsetD
thf(fact_1266_range__subsetD,axiom,
! [F3: a > a,B: set_a,I5: a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F3 @ top_top_set_a ) @ B )
=> ( member_a @ ( F3 @ I5 ) @ B ) ) ).
% range_subsetD
thf(fact_1267_range__composition,axiom,
! [F3: set_a > a,G4: a > set_a] :
( ( image_a_a
@ ^ [X3: a] : ( F3 @ ( G4 @ X3 ) )
@ top_top_set_a )
= ( image_set_a_a @ F3 @ ( image_a_set_a @ G4 @ top_top_set_a ) ) ) ).
% range_composition
thf(fact_1268_range__composition,axiom,
! [F3: a > a,G4: a > a] :
( ( image_a_a
@ ^ [X3: a] : ( F3 @ ( G4 @ X3 ) )
@ top_top_set_a )
= ( image_a_a @ F3 @ ( image_a_a @ G4 @ top_top_set_a ) ) ) ).
% range_composition
thf(fact_1269_range__composition,axiom,
! [F3: set_a > set_a,G4: a > set_a] :
( ( image_a_set_a
@ ^ [X3: a] : ( F3 @ ( G4 @ X3 ) )
@ top_top_set_a )
= ( image_set_a_set_a @ F3 @ ( image_a_set_a @ G4 @ top_top_set_a ) ) ) ).
% range_composition
thf(fact_1270_range__composition,axiom,
! [F3: a > set_a,G4: a > a] :
( ( image_a_set_a
@ ^ [X3: a] : ( F3 @ ( G4 @ X3 ) )
@ top_top_set_a )
= ( image_a_set_a @ F3 @ ( image_a_a @ G4 @ top_top_set_a ) ) ) ).
% range_composition
thf(fact_1271_rangeE,axiom,
! [B4: set_a,F3: a > set_a] :
( ( member_set_a @ B4 @ ( image_a_set_a @ F3 @ top_top_set_a ) )
=> ~ ! [X2: a] :
( B4
!= ( F3 @ X2 ) ) ) ).
% rangeE
thf(fact_1272_rangeE,axiom,
! [B4: a,F3: a > a] :
( ( member_a @ B4 @ ( image_a_a @ F3 @ top_top_set_a ) )
=> ~ ! [X2: a] :
( B4
!= ( F3 @ X2 ) ) ) ).
% rangeE
thf(fact_1273_range__eqI,axiom,
! [B4: set_a,F3: a > set_a,X: a] :
( ( B4
= ( F3 @ X ) )
=> ( member_set_a @ B4 @ ( image_a_set_a @ F3 @ top_top_set_a ) ) ) ).
% range_eqI
thf(fact_1274_range__eqI,axiom,
! [B4: a,F3: a > a,X: a] :
( ( B4
= ( F3 @ X ) )
=> ( member_a @ B4 @ ( image_a_a @ F3 @ top_top_set_a ) ) ) ).
% range_eqI
thf(fact_1275_rangeI,axiom,
! [F3: a > set_a,X: a] : ( member_set_a @ ( F3 @ X ) @ ( image_a_set_a @ F3 @ top_top_set_a ) ) ).
% rangeI
thf(fact_1276_rangeI,axiom,
! [F3: a > a,X: a] : ( member_a @ ( F3 @ X ) @ ( image_a_a @ F3 @ top_top_set_a ) ) ).
% rangeI
% Helper facts (5)
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X: a,Y: a] :
( ( if_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X: a,Y: a] :
( ( if_a @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Set__Oset_Itf__a_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_Itf__a_J_T,axiom,
! [X: set_a,Y: set_a] :
( ( if_set_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_Itf__a_J_T,axiom,
! [X: set_a,Y: set_a] :
( ( if_set_a @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a2 @ b ) ).
%------------------------------------------------------------------------------