TPTP Problem File: ITP079^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP079^1 : TPTP v9.0.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Irreducible problem prob_209__6623758_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Irreducible/prob_209__6623758_1 [Des21]
% Status : Theorem
% Rating : 0.12 v9.0.0, 0.30 v8.2.0, 0.23 v8.1.0, 0.27 v7.5.0
% Syntax : Number of formulae : 437 ( 125 unt; 77 typ; 0 def)
% Number of atoms : 1207 ( 255 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 3030 ( 50 ~; 19 |; 37 &;2359 @)
% ( 0 <=>; 565 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 7 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 567 ( 567 >; 0 *; 0 +; 0 <<)
% Number of symbols : 76 ( 73 usr; 4 con; 0-4 aty)
% Number of variables : 1151 ( 131 ^;1020 !; 0 ?;1151 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:36:10.728
%------------------------------------------------------------------------------
% Could-be-implicit typings (4)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__val_J_J_J,type,
set_set_set_val: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__val_J_J,type,
set_set_val: $tType ).
thf(ty_n_t__Set__Oset_Itf__val_J,type,
set_val: $tType ).
thf(ty_n_tf__val,type,
val: $tType ).
% Explicit typings (73)
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_Itf__val_J,type,
uminus1961393973et_val: set_val > set_val ).
thf(sy_c_If_001_062_I_Eo_Mt__Set__Oset_Itf__val_J_J,type,
if_o_set_val: $o > ( $o > set_val ) > ( $o > set_val ) > $o > set_val ).
thf(sy_c_If_001t__Set__Oset_It__Set__Oset_Itf__val_J_J,type,
if_set_set_val: $o > set_set_val > set_set_val > set_set_val ).
thf(sy_c_If_001t__Set__Oset_Itf__val_J,type,
if_set_val: $o > set_val > set_val > set_val ).
thf(sy_c_If_001tf__val,type,
if_val: $o > val > val > val ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__val_J,type,
sup_sup_set_val: set_val > set_val > set_val ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_Itf__val_J_J,type,
ord_less_set_set_val: set_set_val > set_set_val > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__val_J,type,
ord_less_set_val: set_val > set_val > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001tf__val,type,
ord_less_val: val > val > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_062_I_Eo_Mt__Set__Oset_Itf__val_J_J_J,type,
ord_le1441829300et_val: ( $o > $o > set_val ) > ( $o > $o > set_val ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_It__Set__Oset_Itf__val_J_J_J,type,
ord_le936627147et_val: ( $o > set_set_val ) > ( $o > set_set_val ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_Itf__val_J_J,type,
ord_le994826475et_val: ( $o > set_val ) > ( $o > set_val ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mtf__val_J,type,
ord_less_eq_o_val: ( $o > val ) > ( $o > val ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__val_J_J_J,type,
ord_le1135847966et_val: set_set_set_val > set_set_set_val > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__val_J_J,type,
ord_le1742111550et_val: set_set_val > set_set_val > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__val_J,type,
ord_less_eq_set_val: set_val > set_val > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001tf__val,type,
ord_less_eq_val: val > val > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001_062_I_Eo_Mt__Set__Oset_Itf__val_J_J,type,
order_985126002et_val: ( ( $o > set_val ) > $o ) > $o > set_val ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__Set__Oset_Itf__val_J_J,type,
order_544136439et_val: ( set_set_val > $o ) > set_set_val ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_Itf__val_J,type,
order_1960465943et_val: ( set_val > $o ) > set_val ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001tf__val,type,
order_Greatest_val: ( val > $o ) > val ).
thf(sy_c_Orderings_Oorder__class_Oantimono_001_062_I_Eo_Mt__Set__Oset_Itf__val_J_J_001tf__val,type,
order_23236447al_val: ( ( $o > set_val ) > val ) > $o ).
thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Set__Oset_It__Set__Oset_Itf__val_J_J_001t__Set__Oset_Itf__val_J,type,
order_650823122et_val: ( set_set_val > set_val ) > $o ).
thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Set__Oset_It__Set__Oset_Itf__val_J_J_001tf__val,type,
order_1520822002al_val: ( set_set_val > val ) > $o ).
thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Set__Oset_Itf__val_J_001t__Set__Oset_It__Set__Oset_Itf__val_J_J,type,
order_1957703570et_val: ( set_val > set_set_val ) > $o ).
thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Set__Oset_Itf__val_J_001t__Set__Oset_Itf__val_J,type,
order_1719375026et_val: ( set_val > set_val ) > $o ).
thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Set__Oset_Itf__val_J_001tf__val,type,
order_1375876562al_val: ( set_val > val ) > $o ).
thf(sy_c_Orderings_Oorder__class_Oantimono_001tf__val_001_062_I_Eo_Mt__Set__Oset_Itf__val_J_J,type,
order_255079095et_val: ( val > $o > set_val ) > $o ).
thf(sy_c_Orderings_Oorder__class_Oantimono_001tf__val_001t__Set__Oset_It__Set__Oset_Itf__val_J_J,type,
order_1945059954et_val: ( val > set_set_val ) > $o ).
thf(sy_c_Orderings_Oorder__class_Oantimono_001tf__val_001t__Set__Oset_Itf__val_J,type,
order_1484138386et_val: ( val > set_val ) > $o ).
thf(sy_c_Orderings_Oorder__class_Oantimono_001tf__val_001tf__val,type,
order_1689798834al_val: ( val > val ) > $o ).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001tf__val_001_062_I_Eo_Mt__Set__Oset_Itf__val_J_J,type,
order_1205732848et_val: ( val > $o > set_val ) > $o ).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001tf__val_001t__Set__Oset_It__Set__Oset_Itf__val_J_J,type,
order_68692473et_val: ( val > set_set_val ) > $o ).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001tf__val_001t__Set__Oset_Itf__val_J,type,
order_99134233et_val: ( val > set_val ) > $o ).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001tf__val_001tf__val,type,
order_1139042361al_val: ( val > val ) > $o ).
thf(sy_c_RefineG__Transfer_OREFINEG__TRANSFER__ALIGN_001_062_I_Eo_Mt__Set__Oset_Itf__val_J_J_001_062_I_Eo_Mt__Set__Oset_Itf__val_J_J,type,
refine1593386065et_val: ( $o > set_val ) > ( $o > set_val ) > $o ).
thf(sy_c_RefineG__Transfer_OREFINEG__TRANSFER__ALIGN_001t__Set__Oset_It__Set__Oset_Itf__val_J_J_001t__Set__Oset_It__Set__Oset_Itf__val_J_J,type,
refine1034950713et_val: set_set_val > set_set_val > $o ).
thf(sy_c_RefineG__Transfer_OREFINEG__TRANSFER__ALIGN_001t__Set__Oset_Itf__val_J_001t__Set__Oset_Itf__val_J,type,
refine654574649et_val: set_val > set_val > $o ).
thf(sy_c_RefineG__Transfer_OREFINEG__TRANSFER__ALIGN_001tf__val_001tf__val,type,
refine2021912633al_val: val > val > $o ).
thf(sy_c_RefineG__Transfer_OREFINEG__TRANSFER__POST__SIMP_001_062_I_Eo_Mt__Set__Oset_Itf__val_J_J,type,
refine511606705et_val: ( $o > set_val ) > ( $o > set_val ) > $o ).
thf(sy_c_RefineG__Transfer_OREFINEG__TRANSFER__POST__SIMP_001t__Set__Oset_It__Set__Oset_Itf__val_J_J,type,
refine1908789496et_val: set_set_val > set_set_val > $o ).
thf(sy_c_RefineG__Transfer_OREFINEG__TRANSFER__POST__SIMP_001t__Set__Oset_Itf__val_J,type,
refine1893322264et_val: set_val > set_val > $o ).
thf(sy_c_RefineG__Transfer_OREFINEG__TRANSFER__POST__SIMP_001tf__val,type,
refine403244856MP_val: val > val > $o ).
thf(sy_c_Refine__Misc_Ogalois__connection_001_062_I_Eo_Mt__Set__Oset_Itf__val_J_J_001_062_I_Eo_Mt__Set__Oset_Itf__val_J_J,type,
refine1924899106et_val: ( ( $o > set_val ) > $o > set_val ) > ( ( $o > set_val ) > $o > set_val ) > $o ).
thf(sy_c_Refine__Misc_Ogalois__connection_001_062_I_Eo_Mt__Set__Oset_Itf__val_J_J_001t__Set__Oset_It__Set__Oset_Itf__val_J_J,type,
refine1169396807et_val: ( ( $o > set_val ) > set_set_val ) > ( set_set_val > $o > set_val ) > $o ).
thf(sy_c_Refine__Misc_Ogalois__connection_001_062_I_Eo_Mt__Set__Oset_Itf__val_J_J_001t__Set__Oset_Itf__val_J,type,
refine1948536167et_val: ( ( $o > set_val ) > set_val ) > ( set_val > $o > set_val ) > $o ).
thf(sy_c_Refine__Misc_Ogalois__connection_001t__Set__Oset_It__Set__Oset_Itf__val_J_J_001_062_I_Eo_Mt__Set__Oset_Itf__val_J_J,type,
refine86398879et_val: ( set_set_val > $o > set_val ) > ( ( $o > set_val ) > set_set_val ) > $o ).
thf(sy_c_Refine__Misc_Ogalois__connection_001t__Set__Oset_It__Set__Oset_Itf__val_J_J_001t__Set__Oset_It__Set__Oset_Itf__val_J_J,type,
refine855985034et_val: ( set_set_val > set_set_val ) > ( set_set_val > set_set_val ) > $o ).
thf(sy_c_Refine__Misc_Ogalois__connection_001t__Set__Oset_It__Set__Oset_Itf__val_J_J_001t__Set__Oset_Itf__val_J,type,
refine8163498et_val: ( set_set_val > set_val ) > ( set_val > set_set_val ) > $o ).
thf(sy_c_Refine__Misc_Ogalois__connection_001t__Set__Oset_Itf__val_J_001_062_I_Eo_Mt__Set__Oset_Itf__val_J_J,type,
refine51544255et_val: ( set_val > $o > set_val ) > ( ( $o > set_val ) > set_val ) > $o ).
thf(sy_c_Refine__Misc_Ogalois__connection_001t__Set__Oset_Itf__val_J_001t__Set__Oset_It__Set__Oset_Itf__val_J_J,type,
refine1315043946et_val: ( set_val > set_set_val ) > ( set_set_val > set_val ) > $o ).
thf(sy_c_Refine__Misc_Ogalois__connection_001t__Set__Oset_Itf__val_J_001t__Set__Oset_Itf__val_J,type,
refine1929691018et_val: ( set_val > set_val ) > ( set_val > set_val ) > $o ).
thf(sy_c_Refine__Misc_Oinf__distrib_001t__Set__Oset_Itf__val_J_001t__Set__Oset_Itf__val_J,type,
refine2057434761et_val: ( set_val > set_val ) > $o ).
thf(sy_c_Refine__Mono__Prover_Omono__setup__loc_001_062_I_Eo_Mt__Set__Oset_Itf__val_J_J,type,
refine16560100et_val: ( ( $o > set_val ) > ( $o > set_val ) > $o ) > $o ).
thf(sy_c_Refine__Mono__Prover_Omono__setup__loc_001t__Set__Oset_It__Set__Oset_Itf__val_J_J,type,
refine1506227973et_val: ( set_set_val > set_set_val > $o ) > $o ).
thf(sy_c_Refine__Mono__Prover_Omono__setup__loc_001t__Set__Oset_Itf__val_J,type,
refine275900453et_val: ( set_val > set_val > $o ) > $o ).
thf(sy_c_Refine__Mono__Prover_Omono__setup__loc_001tf__val,type,
refine67622213oc_val: ( val > val > $o ) > $o ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__val_J,type,
collect_set_val: ( set_val > $o ) > set_set_val ).
thf(sy_c_Set_OCollect_001tf__val,type,
collect_val: ( val > $o ) > set_val ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_Itf__val_J,type,
set_or1554624499et_val: set_val > set_val > set_set_val ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001tf__val,type,
set_or18208019st_val: val > val > set_val ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Set__Oset_It__Set__Oset_Itf__val_J_J,type,
set_or1337784553et_val: set_set_val > set_set_set_val ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Set__Oset_Itf__val_J,type,
set_or650197513et_val: set_val > set_set_val ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001tf__val,type,
set_ord_atLeast_val: val > set_val ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_Itf__val_J,type,
set_or867701765et_val: set_val > set_set_val ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001tf__val,type,
set_ord_atMost_val: val > set_val ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001tf__val,type,
set_or2126567818an_val: val > val > set_val ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001tf__val,type,
set_or1288365508an_val: val > set_val ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001tf__val,type,
set_ord_lessThan_val: val > set_val ).
thf(sy_c_member_001t__Set__Oset_Itf__val_J,type,
member_set_val: set_val > set_set_val > $o ).
thf(sy_c_member_001tf__val,type,
member_val: val > set_val > $o ).
thf(sy_v_P,type,
p: set_val ).
thf(sy_v_scc,type,
scc: set_val ).
% Relevant facts (350)
thf(fact_0__092_060open_062_092_060And_062x_O_Ax_A_092_060in_062_Ascc_A_092_060Longrightarrow_062_Ax_A_092_060in_062_AP_092_060close_062,axiom,
! [X: val] :
( ( member_val @ X @ scc )
=> ( member_val @ X @ p ) ) ).
% \<open>\<And>x. x \<in> scc \<Longrightarrow> x \<in> P\<close>
thf(fact_1_subsetI,axiom,
! [A: set_set_val,B: set_set_val] :
( ! [X2: set_val] :
( ( member_set_val @ X2 @ A )
=> ( member_set_val @ X2 @ B ) )
=> ( ord_le1742111550et_val @ A @ B ) ) ).
% subsetI
thf(fact_2_subsetI,axiom,
! [A: set_val,B: set_val] :
( ! [X2: val] :
( ( member_val @ X2 @ A )
=> ( member_val @ X2 @ B ) )
=> ( ord_less_eq_set_val @ A @ B ) ) ).
% subsetI
thf(fact_3_subset__antisym,axiom,
! [A: set_set_val,B: set_set_val] :
( ( ord_le1742111550et_val @ A @ B )
=> ( ( ord_le1742111550et_val @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_4_subset__antisym,axiom,
! [A: set_val,B: set_val] :
( ( ord_less_eq_set_val @ A @ B )
=> ( ( ord_less_eq_set_val @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_5_order__mono__setup_Orefl,axiom,
! [X: $o > set_val] : ( ord_le994826475et_val @ X @ X ) ).
% order_mono_setup.refl
thf(fact_6_order__mono__setup_Orefl,axiom,
! [X: val] : ( ord_less_eq_val @ X @ X ) ).
% order_mono_setup.refl
thf(fact_7_order__mono__setup_Orefl,axiom,
! [X: set_set_val] : ( ord_le1742111550et_val @ X @ X ) ).
% order_mono_setup.refl
thf(fact_8_order__mono__setup_Orefl,axiom,
! [X: set_val] : ( ord_less_eq_set_val @ X @ X ) ).
% order_mono_setup.refl
thf(fact_9_relprop__triggers_I6_J,axiom,
! [R: set_set_val,R2: set_set_val] :
( ( ord_le1742111550et_val @ R @ R2 )
=> ( ord_le1742111550et_val @ R @ R2 ) ) ).
% relprop_triggers(6)
thf(fact_10_relprop__triggers_I6_J,axiom,
! [R: set_val,R2: set_val] :
( ( ord_less_eq_set_val @ R @ R2 )
=> ( ord_less_eq_set_val @ R @ R2 ) ) ).
% relprop_triggers(6)
thf(fact_11_in__mono,axiom,
! [A: set_set_val,B: set_set_val,X: set_val] :
( ( ord_le1742111550et_val @ A @ B )
=> ( ( member_set_val @ X @ A )
=> ( member_set_val @ X @ B ) ) ) ).
% in_mono
thf(fact_12_in__mono,axiom,
! [A: set_val,B: set_val,X: val] :
( ( ord_less_eq_set_val @ A @ B )
=> ( ( member_val @ X @ A )
=> ( member_val @ X @ B ) ) ) ).
% in_mono
thf(fact_13_subsetD,axiom,
! [A: set_set_val,B: set_set_val,C: set_val] :
( ( ord_le1742111550et_val @ A @ B )
=> ( ( member_set_val @ C @ A )
=> ( member_set_val @ C @ B ) ) ) ).
% subsetD
thf(fact_14_subsetD,axiom,
! [A: set_val,B: set_val,C: val] :
( ( ord_less_eq_set_val @ A @ B )
=> ( ( member_val @ C @ A )
=> ( member_val @ C @ B ) ) ) ).
% subsetD
thf(fact_15_equalityE,axiom,
! [A: set_set_val,B: set_set_val] :
( ( A = B )
=> ~ ( ( ord_le1742111550et_val @ A @ B )
=> ~ ( ord_le1742111550et_val @ B @ A ) ) ) ).
% equalityE
thf(fact_16_equalityE,axiom,
! [A: set_val,B: set_val] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_val @ A @ B )
=> ~ ( ord_less_eq_set_val @ B @ A ) ) ) ).
% equalityE
thf(fact_17_subset__eq,axiom,
( ord_le1742111550et_val
= ( ^ [A2: set_set_val,B2: set_set_val] :
! [X3: set_val] :
( ( member_set_val @ X3 @ A2 )
=> ( member_set_val @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_18_subset__eq,axiom,
( ord_less_eq_set_val
= ( ^ [A2: set_val,B2: set_val] :
! [X3: val] :
( ( member_val @ X3 @ A2 )
=> ( member_val @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_19_equalityD1,axiom,
! [A: set_set_val,B: set_set_val] :
( ( A = B )
=> ( ord_le1742111550et_val @ A @ B ) ) ).
% equalityD1
thf(fact_20_equalityD1,axiom,
! [A: set_val,B: set_val] :
( ( A = B )
=> ( ord_less_eq_set_val @ A @ B ) ) ).
% equalityD1
thf(fact_21_equalityD2,axiom,
! [A: set_set_val,B: set_set_val] :
( ( A = B )
=> ( ord_le1742111550et_val @ B @ A ) ) ).
% equalityD2
thf(fact_22_equalityD2,axiom,
! [A: set_val,B: set_val] :
( ( A = B )
=> ( ord_less_eq_set_val @ B @ A ) ) ).
% equalityD2
thf(fact_23_subset__iff,axiom,
( ord_le1742111550et_val
= ( ^ [A2: set_set_val,B2: set_set_val] :
! [T: set_val] :
( ( member_set_val @ T @ A2 )
=> ( member_set_val @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_24_subset__iff,axiom,
( ord_less_eq_set_val
= ( ^ [A2: set_val,B2: set_val] :
! [T: val] :
( ( member_val @ T @ A2 )
=> ( member_val @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_25_subset__refl,axiom,
! [A: set_set_val] : ( ord_le1742111550et_val @ A @ A ) ).
% subset_refl
thf(fact_26_subset__refl,axiom,
! [A: set_val] : ( ord_less_eq_set_val @ A @ A ) ).
% subset_refl
thf(fact_27_order__mono__setup_Omono__if,axiom,
! [T2: $o > set_val,T3: $o > set_val,E: $o > set_val,E2: $o > set_val,B3: $o] :
( ( ord_le994826475et_val @ T2 @ T3 )
=> ( ( ord_le994826475et_val @ E @ E2 )
=> ( ord_le994826475et_val @ ( if_o_set_val @ B3 @ T2 @ E ) @ ( if_o_set_val @ B3 @ T3 @ E2 ) ) ) ) ).
% order_mono_setup.mono_if
thf(fact_28_order__mono__setup_Omono__if,axiom,
! [T2: val,T3: val,E: val,E2: val,B3: $o] :
( ( ord_less_eq_val @ T2 @ T3 )
=> ( ( ord_less_eq_val @ E @ E2 )
=> ( ord_less_eq_val @ ( if_val @ B3 @ T2 @ E ) @ ( if_val @ B3 @ T3 @ E2 ) ) ) ) ).
% order_mono_setup.mono_if
thf(fact_29_order__mono__setup_Omono__if,axiom,
! [T2: set_set_val,T3: set_set_val,E: set_set_val,E2: set_set_val,B3: $o] :
( ( ord_le1742111550et_val @ T2 @ T3 )
=> ( ( ord_le1742111550et_val @ E @ E2 )
=> ( ord_le1742111550et_val @ ( if_set_set_val @ B3 @ T2 @ E ) @ ( if_set_set_val @ B3 @ T3 @ E2 ) ) ) ) ).
% order_mono_setup.mono_if
thf(fact_30_order__mono__setup_Omono__if,axiom,
! [T2: set_val,T3: set_val,E: set_val,E2: set_val,B3: $o] :
( ( ord_less_eq_set_val @ T2 @ T3 )
=> ( ( ord_less_eq_set_val @ E @ E2 )
=> ( ord_less_eq_set_val @ ( if_set_val @ B3 @ T2 @ E ) @ ( if_set_val @ B3 @ T3 @ E2 ) ) ) ) ).
% order_mono_setup.mono_if
thf(fact_31_Collect__mono__iff,axiom,
! [P: set_val > $o,Q: set_val > $o] :
( ( ord_le1742111550et_val @ ( collect_set_val @ P ) @ ( collect_set_val @ Q ) )
= ( ! [X3: set_val] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_32_Collect__mono__iff,axiom,
! [P: val > $o,Q: val > $o] :
( ( ord_less_eq_set_val @ ( collect_val @ P ) @ ( collect_val @ Q ) )
= ( ! [X3: val] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_33_set__eq__subset,axiom,
( ( ^ [Y: set_set_val,Z: set_set_val] : ( Y = Z ) )
= ( ^ [A2: set_set_val,B2: set_set_val] :
( ( ord_le1742111550et_val @ A2 @ B2 )
& ( ord_le1742111550et_val @ B2 @ A2 ) ) ) ) ).
% set_eq_subset
thf(fact_34_set__eq__subset,axiom,
( ( ^ [Y: set_val,Z: set_val] : ( Y = Z ) )
= ( ^ [A2: set_val,B2: set_val] :
( ( ord_less_eq_set_val @ A2 @ B2 )
& ( ord_less_eq_set_val @ B2 @ A2 ) ) ) ) ).
% set_eq_subset
thf(fact_35_subset__trans,axiom,
! [A: set_set_val,B: set_set_val,C2: set_set_val] :
( ( ord_le1742111550et_val @ A @ B )
=> ( ( ord_le1742111550et_val @ B @ C2 )
=> ( ord_le1742111550et_val @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_36_subset__trans,axiom,
! [A: set_val,B: set_val,C2: set_val] :
( ( ord_less_eq_set_val @ A @ B )
=> ( ( ord_less_eq_set_val @ B @ C2 )
=> ( ord_less_eq_set_val @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_37_Collect__mono,axiom,
! [P: set_val > $o,Q: set_val > $o] :
( ! [X2: set_val] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le1742111550et_val @ ( collect_set_val @ P ) @ ( collect_set_val @ Q ) ) ) ).
% Collect_mono
thf(fact_38_Collect__mono,axiom,
! [P: val > $o,Q: val > $o] :
( ! [X2: val] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_val @ ( collect_val @ P ) @ ( collect_val @ Q ) ) ) ).
% Collect_mono
thf(fact_39_subset__Collect__conv,axiom,
! [S: set_set_val,P: set_val > $o] :
( ( ord_le1742111550et_val @ S @ ( collect_set_val @ P ) )
= ( ! [X3: set_val] :
( ( member_set_val @ X3 @ S )
=> ( P @ X3 ) ) ) ) ).
% subset_Collect_conv
thf(fact_40_subset__Collect__conv,axiom,
! [S: set_val,P: val > $o] :
( ( ord_less_eq_set_val @ S @ ( collect_val @ P ) )
= ( ! [X3: val] :
( ( member_val @ X3 @ S )
=> ( P @ X3 ) ) ) ) ).
% subset_Collect_conv
thf(fact_41_dual__order_Oantisym,axiom,
! [B3: $o > set_val,A3: $o > set_val] :
( ( ord_le994826475et_val @ B3 @ A3 )
=> ( ( ord_le994826475et_val @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_42_dual__order_Oantisym,axiom,
! [B3: val,A3: val] :
( ( ord_less_eq_val @ B3 @ A3 )
=> ( ( ord_less_eq_val @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_43_dual__order_Oantisym,axiom,
! [B3: set_set_val,A3: set_set_val] :
( ( ord_le1742111550et_val @ B3 @ A3 )
=> ( ( ord_le1742111550et_val @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_44_dual__order_Oantisym,axiom,
! [B3: set_val,A3: set_val] :
( ( ord_less_eq_set_val @ B3 @ A3 )
=> ( ( ord_less_eq_set_val @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_45_dual__order_Oeq__iff,axiom,
( ( ^ [Y: $o > set_val,Z: $o > set_val] : ( Y = Z ) )
= ( ^ [A4: $o > set_val,B4: $o > set_val] :
( ( ord_le994826475et_val @ B4 @ A4 )
& ( ord_le994826475et_val @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_46_dual__order_Oeq__iff,axiom,
( ( ^ [Y: val,Z: val] : ( Y = Z ) )
= ( ^ [A4: val,B4: val] :
( ( ord_less_eq_val @ B4 @ A4 )
& ( ord_less_eq_val @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_47_dual__order_Oeq__iff,axiom,
( ( ^ [Y: set_set_val,Z: set_set_val] : ( Y = Z ) )
= ( ^ [A4: set_set_val,B4: set_set_val] :
( ( ord_le1742111550et_val @ B4 @ A4 )
& ( ord_le1742111550et_val @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_48_dual__order_Oeq__iff,axiom,
( ( ^ [Y: set_val,Z: set_val] : ( Y = Z ) )
= ( ^ [A4: set_val,B4: set_val] :
( ( ord_less_eq_set_val @ B4 @ A4 )
& ( ord_less_eq_set_val @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_49_dual__order_Otrans,axiom,
! [B3: $o > set_val,A3: $o > set_val,C: $o > set_val] :
( ( ord_le994826475et_val @ B3 @ A3 )
=> ( ( ord_le994826475et_val @ C @ B3 )
=> ( ord_le994826475et_val @ C @ A3 ) ) ) ).
% dual_order.trans
thf(fact_50_dual__order_Otrans,axiom,
! [B3: val,A3: val,C: val] :
( ( ord_less_eq_val @ B3 @ A3 )
=> ( ( ord_less_eq_val @ C @ B3 )
=> ( ord_less_eq_val @ C @ A3 ) ) ) ).
% dual_order.trans
thf(fact_51_dual__order_Otrans,axiom,
! [B3: set_set_val,A3: set_set_val,C: set_set_val] :
( ( ord_le1742111550et_val @ B3 @ A3 )
=> ( ( ord_le1742111550et_val @ C @ B3 )
=> ( ord_le1742111550et_val @ C @ A3 ) ) ) ).
% dual_order.trans
thf(fact_52_dual__order_Otrans,axiom,
! [B3: set_val,A3: set_val,C: set_val] :
( ( ord_less_eq_set_val @ B3 @ A3 )
=> ( ( ord_less_eq_set_val @ C @ B3 )
=> ( ord_less_eq_set_val @ C @ A3 ) ) ) ).
% dual_order.trans
thf(fact_53_linorder__wlog,axiom,
! [P: val > val > $o,A3: val,B3: val] :
( ! [A5: val,B5: val] :
( ( ord_less_eq_val @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: val,B5: val] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A3 @ B3 ) ) ) ).
% linorder_wlog
thf(fact_54_dual__order_Orefl,axiom,
! [A3: $o > set_val] : ( ord_le994826475et_val @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_55_dual__order_Orefl,axiom,
! [A3: val] : ( ord_less_eq_val @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_56_dual__order_Orefl,axiom,
! [A3: set_set_val] : ( ord_le1742111550et_val @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_57_dual__order_Orefl,axiom,
! [A3: set_val] : ( ord_less_eq_set_val @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_58_order__trans,axiom,
! [X: $o > set_val,Y2: $o > set_val,Z2: $o > set_val] :
( ( ord_le994826475et_val @ X @ Y2 )
=> ( ( ord_le994826475et_val @ Y2 @ Z2 )
=> ( ord_le994826475et_val @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_59_order__trans,axiom,
! [X: val,Y2: val,Z2: val] :
( ( ord_less_eq_val @ X @ Y2 )
=> ( ( ord_less_eq_val @ Y2 @ Z2 )
=> ( ord_less_eq_val @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_60_order__trans,axiom,
! [X: set_set_val,Y2: set_set_val,Z2: set_set_val] :
( ( ord_le1742111550et_val @ X @ Y2 )
=> ( ( ord_le1742111550et_val @ Y2 @ Z2 )
=> ( ord_le1742111550et_val @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_61_order__trans,axiom,
! [X: set_val,Y2: set_val,Z2: set_val] :
( ( ord_less_eq_set_val @ X @ Y2 )
=> ( ( ord_less_eq_set_val @ Y2 @ Z2 )
=> ( ord_less_eq_set_val @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_62_order__class_Oorder_Oantisym,axiom,
! [A3: $o > set_val,B3: $o > set_val] :
( ( ord_le994826475et_val @ A3 @ B3 )
=> ( ( ord_le994826475et_val @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% order_class.order.antisym
thf(fact_63_order__class_Oorder_Oantisym,axiom,
! [A3: val,B3: val] :
( ( ord_less_eq_val @ A3 @ B3 )
=> ( ( ord_less_eq_val @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% order_class.order.antisym
thf(fact_64_order__class_Oorder_Oantisym,axiom,
! [A3: set_set_val,B3: set_set_val] :
( ( ord_le1742111550et_val @ A3 @ B3 )
=> ( ( ord_le1742111550et_val @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% order_class.order.antisym
thf(fact_65_order__class_Oorder_Oantisym,axiom,
! [A3: set_val,B3: set_val] :
( ( ord_less_eq_set_val @ A3 @ B3 )
=> ( ( ord_less_eq_set_val @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% order_class.order.antisym
thf(fact_66_ord__le__eq__trans,axiom,
! [A3: $o > set_val,B3: $o > set_val,C: $o > set_val] :
( ( ord_le994826475et_val @ A3 @ B3 )
=> ( ( B3 = C )
=> ( ord_le994826475et_val @ A3 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_67_ord__le__eq__trans,axiom,
! [A3: val,B3: val,C: val] :
( ( ord_less_eq_val @ A3 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_val @ A3 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_68_ord__le__eq__trans,axiom,
! [A3: set_set_val,B3: set_set_val,C: set_set_val] :
( ( ord_le1742111550et_val @ A3 @ B3 )
=> ( ( B3 = C )
=> ( ord_le1742111550et_val @ A3 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_69_ord__le__eq__trans,axiom,
! [A3: set_val,B3: set_val,C: set_val] :
( ( ord_less_eq_set_val @ A3 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_set_val @ A3 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_70_le__funD,axiom,
! [F: $o > set_val,G: $o > set_val,X: $o] :
( ( ord_le994826475et_val @ F @ G )
=> ( ord_less_eq_set_val @ ( F @ X ) @ ( G @ X ) ) ) ).
% le_funD
thf(fact_71_le__funE,axiom,
! [F: $o > set_val,G: $o > set_val,X: $o] :
( ( ord_le994826475et_val @ F @ G )
=> ( ord_less_eq_set_val @ ( F @ X ) @ ( G @ X ) ) ) ).
% le_funE
thf(fact_72_le__funI,axiom,
! [F: $o > set_val,G: $o > set_val] :
( ! [X2: $o] : ( ord_less_eq_set_val @ ( F @ X2 ) @ ( G @ X2 ) )
=> ( ord_le994826475et_val @ F @ G ) ) ).
% le_funI
thf(fact_73_le__fun__def,axiom,
( ord_le994826475et_val
= ( ^ [F2: $o > set_val,G2: $o > set_val] :
! [X3: $o] : ( ord_less_eq_set_val @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% le_fun_def
thf(fact_74_order__subst1,axiom,
! [A3: set_val,F: set_val > set_val,B3: set_val,C: set_val] :
( ( ord_less_eq_set_val @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_val @ B3 @ C )
=> ( ! [X2: set_val,Y3: set_val] :
( ( ord_less_eq_set_val @ X2 @ Y3 )
=> ( ord_less_eq_set_val @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_val @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_75_order__subst1,axiom,
! [A3: val,F: val > val,B3: val,C: val] :
( ( ord_less_eq_val @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_val @ B3 @ C )
=> ( ! [X2: val,Y3: val] :
( ( ord_less_eq_val @ X2 @ Y3 )
=> ( ord_less_eq_val @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_val @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_76_order__subst1,axiom,
! [A3: set_val,F: val > set_val,B3: val,C: val] :
( ( ord_less_eq_set_val @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_val @ B3 @ C )
=> ( ! [X2: val,Y3: val] :
( ( ord_less_eq_val @ X2 @ Y3 )
=> ( ord_less_eq_set_val @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_val @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_77_order__subst1,axiom,
! [A3: val,F: set_val > val,B3: set_val,C: set_val] :
( ( ord_less_eq_val @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_val @ B3 @ C )
=> ( ! [X2: set_val,Y3: set_val] :
( ( ord_less_eq_set_val @ X2 @ Y3 )
=> ( ord_less_eq_val @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_val @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_78_order__subst1,axiom,
! [A3: val,F: set_set_val > val,B3: set_set_val,C: set_set_val] :
( ( ord_less_eq_val @ A3 @ ( F @ B3 ) )
=> ( ( ord_le1742111550et_val @ B3 @ C )
=> ( ! [X2: set_set_val,Y3: set_set_val] :
( ( ord_le1742111550et_val @ X2 @ Y3 )
=> ( ord_less_eq_val @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_val @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_79_order__subst1,axiom,
! [A3: set_set_val,F: val > set_set_val,B3: val,C: val] :
( ( ord_le1742111550et_val @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_val @ B3 @ C )
=> ( ! [X2: val,Y3: val] :
( ( ord_less_eq_val @ X2 @ Y3 )
=> ( ord_le1742111550et_val @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le1742111550et_val @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_80_order__subst1,axiom,
! [A3: set_val,F: set_set_val > set_val,B3: set_set_val,C: set_set_val] :
( ( ord_less_eq_set_val @ A3 @ ( F @ B3 ) )
=> ( ( ord_le1742111550et_val @ B3 @ C )
=> ( ! [X2: set_set_val,Y3: set_set_val] :
( ( ord_le1742111550et_val @ X2 @ Y3 )
=> ( ord_less_eq_set_val @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_val @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_81_order__subst1,axiom,
! [A3: $o > set_val,F: val > $o > set_val,B3: val,C: val] :
( ( ord_le994826475et_val @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_val @ B3 @ C )
=> ( ! [X2: val,Y3: val] :
( ( ord_less_eq_val @ X2 @ Y3 )
=> ( ord_le994826475et_val @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le994826475et_val @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_82_order__subst1,axiom,
! [A3: val,F: ( $o > set_val ) > val,B3: $o > set_val,C: $o > set_val] :
( ( ord_less_eq_val @ A3 @ ( F @ B3 ) )
=> ( ( ord_le994826475et_val @ B3 @ C )
=> ( ! [X2: $o > set_val,Y3: $o > set_val] :
( ( ord_le994826475et_val @ X2 @ Y3 )
=> ( ord_less_eq_val @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_val @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_83_order__subst1,axiom,
! [A3: set_set_val,F: set_val > set_set_val,B3: set_val,C: set_val] :
( ( ord_le1742111550et_val @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_val @ B3 @ C )
=> ( ! [X2: set_val,Y3: set_val] :
( ( ord_less_eq_set_val @ X2 @ Y3 )
=> ( ord_le1742111550et_val @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le1742111550et_val @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_84_order__subst2,axiom,
! [A3: set_val,B3: set_val,F: set_val > set_val,C: set_val] :
( ( ord_less_eq_set_val @ A3 @ B3 )
=> ( ( ord_less_eq_set_val @ ( F @ B3 ) @ C )
=> ( ! [X2: set_val,Y3: set_val] :
( ( ord_less_eq_set_val @ X2 @ Y3 )
=> ( ord_less_eq_set_val @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_val @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_85_order__subst2,axiom,
! [A3: val,B3: val,F: val > val,C: val] :
( ( ord_less_eq_val @ A3 @ B3 )
=> ( ( ord_less_eq_val @ ( F @ B3 ) @ C )
=> ( ! [X2: val,Y3: val] :
( ( ord_less_eq_val @ X2 @ Y3 )
=> ( ord_less_eq_val @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_val @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_86_order__subst2,axiom,
! [A3: set_val,B3: set_val,F: set_val > val,C: val] :
( ( ord_less_eq_set_val @ A3 @ B3 )
=> ( ( ord_less_eq_val @ ( F @ B3 ) @ C )
=> ( ! [X2: set_val,Y3: set_val] :
( ( ord_less_eq_set_val @ X2 @ Y3 )
=> ( ord_less_eq_val @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_val @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_87_order__subst2,axiom,
! [A3: val,B3: val,F: val > set_val,C: set_val] :
( ( ord_less_eq_val @ A3 @ B3 )
=> ( ( ord_less_eq_set_val @ ( F @ B3 ) @ C )
=> ( ! [X2: val,Y3: val] :
( ( ord_less_eq_val @ X2 @ Y3 )
=> ( ord_less_eq_set_val @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_val @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_88_order__subst2,axiom,
! [A3: val,B3: val,F: val > set_set_val,C: set_set_val] :
( ( ord_less_eq_val @ A3 @ B3 )
=> ( ( ord_le1742111550et_val @ ( F @ B3 ) @ C )
=> ( ! [X2: val,Y3: val] :
( ( ord_less_eq_val @ X2 @ Y3 )
=> ( ord_le1742111550et_val @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le1742111550et_val @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_89_order__subst2,axiom,
! [A3: set_set_val,B3: set_set_val,F: set_set_val > val,C: val] :
( ( ord_le1742111550et_val @ A3 @ B3 )
=> ( ( ord_less_eq_val @ ( F @ B3 ) @ C )
=> ( ! [X2: set_set_val,Y3: set_set_val] :
( ( ord_le1742111550et_val @ X2 @ Y3 )
=> ( ord_less_eq_val @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_val @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_90_order__subst2,axiom,
! [A3: set_val,B3: set_val,F: set_val > set_set_val,C: set_set_val] :
( ( ord_less_eq_set_val @ A3 @ B3 )
=> ( ( ord_le1742111550et_val @ ( F @ B3 ) @ C )
=> ( ! [X2: set_val,Y3: set_val] :
( ( ord_less_eq_set_val @ X2 @ Y3 )
=> ( ord_le1742111550et_val @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le1742111550et_val @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_91_order__subst2,axiom,
! [A3: $o > set_val,B3: $o > set_val,F: ( $o > set_val ) > val,C: val] :
( ( ord_le994826475et_val @ A3 @ B3 )
=> ( ( ord_less_eq_val @ ( F @ B3 ) @ C )
=> ( ! [X2: $o > set_val,Y3: $o > set_val] :
( ( ord_le994826475et_val @ X2 @ Y3 )
=> ( ord_less_eq_val @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_val @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_92_order__subst2,axiom,
! [A3: val,B3: val,F: val > $o > set_val,C: $o > set_val] :
( ( ord_less_eq_val @ A3 @ B3 )
=> ( ( ord_le994826475et_val @ ( F @ B3 ) @ C )
=> ( ! [X2: val,Y3: val] :
( ( ord_less_eq_val @ X2 @ Y3 )
=> ( ord_le994826475et_val @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le994826475et_val @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_93_order__subst2,axiom,
! [A3: set_set_val,B3: set_set_val,F: set_set_val > set_val,C: set_val] :
( ( ord_le1742111550et_val @ A3 @ B3 )
=> ( ( ord_less_eq_set_val @ ( F @ B3 ) @ C )
=> ( ! [X2: set_set_val,Y3: set_set_val] :
( ( ord_le1742111550et_val @ X2 @ Y3 )
=> ( ord_less_eq_set_val @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_val @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_94_ord__eq__le__eq__trans,axiom,
! [A3: $o > set_val,B3: $o > set_val,C: $o > set_val,D: $o > set_val] :
( ( A3 = B3 )
=> ( ( ord_le994826475et_val @ B3 @ C )
=> ( ( C = D )
=> ( ord_le994826475et_val @ A3 @ D ) ) ) ) ).
% ord_eq_le_eq_trans
thf(fact_95_ord__eq__le__eq__trans,axiom,
! [A3: val,B3: val,C: val,D: val] :
( ( A3 = B3 )
=> ( ( ord_less_eq_val @ B3 @ C )
=> ( ( C = D )
=> ( ord_less_eq_val @ A3 @ D ) ) ) ) ).
% ord_eq_le_eq_trans
thf(fact_96_ord__eq__le__eq__trans,axiom,
! [A3: set_set_val,B3: set_set_val,C: set_set_val,D: set_set_val] :
( ( A3 = B3 )
=> ( ( ord_le1742111550et_val @ B3 @ C )
=> ( ( C = D )
=> ( ord_le1742111550et_val @ A3 @ D ) ) ) ) ).
% ord_eq_le_eq_trans
thf(fact_97_ord__eq__le__eq__trans,axiom,
! [A3: set_val,B3: set_val,C: set_val,D: set_val] :
( ( A3 = B3 )
=> ( ( ord_less_eq_set_val @ B3 @ C )
=> ( ( C = D )
=> ( ord_less_eq_set_val @ A3 @ D ) ) ) ) ).
% ord_eq_le_eq_trans
thf(fact_98_ord__eq__le__subst,axiom,
! [A3: set_val,F: set_val > set_val,B3: set_val,C: set_val] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_val @ B3 @ C )
=> ( ! [X2: set_val,Y3: set_val] :
( ( ord_less_eq_set_val @ X2 @ Y3 )
=> ( ord_less_eq_set_val @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_val @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_99_ord__eq__le__subst,axiom,
! [A3: val,F: val > val,B3: val,C: val] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_val @ B3 @ C )
=> ( ! [X2: val,Y3: val] :
( ( ord_less_eq_val @ X2 @ Y3 )
=> ( ord_less_eq_val @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_val @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_100_ord__eq__le__subst,axiom,
! [A3: val,F: set_val > val,B3: set_val,C: set_val] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_val @ B3 @ C )
=> ( ! [X2: set_val,Y3: set_val] :
( ( ord_less_eq_set_val @ X2 @ Y3 )
=> ( ord_less_eq_val @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_val @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_101_ord__eq__le__subst,axiom,
! [A3: set_val,F: val > set_val,B3: val,C: val] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_val @ B3 @ C )
=> ( ! [X2: val,Y3: val] :
( ( ord_less_eq_val @ X2 @ Y3 )
=> ( ord_less_eq_set_val @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_val @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_102_ord__eq__le__subst,axiom,
! [A3: set_set_val,F: val > set_set_val,B3: val,C: val] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_val @ B3 @ C )
=> ( ! [X2: val,Y3: val] :
( ( ord_less_eq_val @ X2 @ Y3 )
=> ( ord_le1742111550et_val @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le1742111550et_val @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_103_ord__eq__le__subst,axiom,
! [A3: val,F: set_set_val > val,B3: set_set_val,C: set_set_val] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_le1742111550et_val @ B3 @ C )
=> ( ! [X2: set_set_val,Y3: set_set_val] :
( ( ord_le1742111550et_val @ X2 @ Y3 )
=> ( ord_less_eq_val @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_val @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_104_ord__eq__le__subst,axiom,
! [A3: set_set_val,F: set_val > set_set_val,B3: set_val,C: set_val] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_val @ B3 @ C )
=> ( ! [X2: set_val,Y3: set_val] :
( ( ord_less_eq_set_val @ X2 @ Y3 )
=> ( ord_le1742111550et_val @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le1742111550et_val @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_105_ord__eq__le__subst,axiom,
! [A3: val,F: ( $o > set_val ) > val,B3: $o > set_val,C: $o > set_val] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_le994826475et_val @ B3 @ C )
=> ( ! [X2: $o > set_val,Y3: $o > set_val] :
( ( ord_le994826475et_val @ X2 @ Y3 )
=> ( ord_less_eq_val @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_val @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_106_ord__eq__le__subst,axiom,
! [A3: $o > set_val,F: val > $o > set_val,B3: val,C: val] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_val @ B3 @ C )
=> ( ! [X2: val,Y3: val] :
( ( ord_less_eq_val @ X2 @ Y3 )
=> ( ord_le994826475et_val @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le994826475et_val @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_107_ord__eq__le__subst,axiom,
! [A3: set_val,F: set_set_val > set_val,B3: set_set_val,C: set_set_val] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_le1742111550et_val @ B3 @ C )
=> ( ! [X2: set_set_val,Y3: set_set_val] :
( ( ord_le1742111550et_val @ X2 @ Y3 )
=> ( ord_less_eq_set_val @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_val @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_108_ord__le__eq__subst,axiom,
! [A3: set_val,B3: set_val,F: set_val > set_val,C: set_val] :
( ( ord_less_eq_set_val @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: set_val,Y3: set_val] :
( ( ord_less_eq_set_val @ X2 @ Y3 )
=> ( ord_less_eq_set_val @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_val @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_109_ord__le__eq__subst,axiom,
! [A3: val,B3: val,F: val > val,C: val] :
( ( ord_less_eq_val @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: val,Y3: val] :
( ( ord_less_eq_val @ X2 @ Y3 )
=> ( ord_less_eq_val @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_val @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_110_ord__le__eq__subst,axiom,
! [A3: set_val,B3: set_val,F: set_val > val,C: val] :
( ( ord_less_eq_set_val @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: set_val,Y3: set_val] :
( ( ord_less_eq_set_val @ X2 @ Y3 )
=> ( ord_less_eq_val @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_val @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_111_ord__le__eq__subst,axiom,
! [A3: val,B3: val,F: val > set_val,C: set_val] :
( ( ord_less_eq_val @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: val,Y3: val] :
( ( ord_less_eq_val @ X2 @ Y3 )
=> ( ord_less_eq_set_val @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_val @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_112_ord__le__eq__subst,axiom,
! [A3: val,B3: val,F: val > set_set_val,C: set_set_val] :
( ( ord_less_eq_val @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: val,Y3: val] :
( ( ord_less_eq_val @ X2 @ Y3 )
=> ( ord_le1742111550et_val @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le1742111550et_val @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_113_ord__le__eq__subst,axiom,
! [A3: set_set_val,B3: set_set_val,F: set_set_val > val,C: val] :
( ( ord_le1742111550et_val @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: set_set_val,Y3: set_set_val] :
( ( ord_le1742111550et_val @ X2 @ Y3 )
=> ( ord_less_eq_val @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_val @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_114_ord__le__eq__subst,axiom,
! [A3: set_val,B3: set_val,F: set_val > set_set_val,C: set_set_val] :
( ( ord_less_eq_set_val @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: set_val,Y3: set_val] :
( ( ord_less_eq_set_val @ X2 @ Y3 )
=> ( ord_le1742111550et_val @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le1742111550et_val @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_115_ord__le__eq__subst,axiom,
! [A3: $o > set_val,B3: $o > set_val,F: ( $o > set_val ) > val,C: val] :
( ( ord_le994826475et_val @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: $o > set_val,Y3: $o > set_val] :
( ( ord_le994826475et_val @ X2 @ Y3 )
=> ( ord_less_eq_val @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_val @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_116_ord__le__eq__subst,axiom,
! [A3: val,B3: val,F: val > $o > set_val,C: $o > set_val] :
( ( ord_less_eq_val @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: val,Y3: val] :
( ( ord_less_eq_val @ X2 @ Y3 )
=> ( ord_le994826475et_val @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le994826475et_val @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_117_ord__le__eq__subst,axiom,
! [A3: set_set_val,B3: set_set_val,F: set_set_val > set_val,C: set_val] :
( ( ord_le1742111550et_val @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: set_set_val,Y3: set_set_val] :
( ( ord_le1742111550et_val @ X2 @ Y3 )
=> ( ord_less_eq_set_val @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_val @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_118_eq__iff,axiom,
( ( ^ [Y: $o > set_val,Z: $o > set_val] : ( Y = Z ) )
= ( ^ [X3: $o > set_val,Y4: $o > set_val] :
( ( ord_le994826475et_val @ X3 @ Y4 )
& ( ord_le994826475et_val @ Y4 @ X3 ) ) ) ) ).
% eq_iff
thf(fact_119_eq__iff,axiom,
( ( ^ [Y: val,Z: val] : ( Y = Z ) )
= ( ^ [X3: val,Y4: val] :
( ( ord_less_eq_val @ X3 @ Y4 )
& ( ord_less_eq_val @ Y4 @ X3 ) ) ) ) ).
% eq_iff
thf(fact_120_eq__iff,axiom,
( ( ^ [Y: set_set_val,Z: set_set_val] : ( Y = Z ) )
= ( ^ [X3: set_set_val,Y4: set_set_val] :
( ( ord_le1742111550et_val @ X3 @ Y4 )
& ( ord_le1742111550et_val @ Y4 @ X3 ) ) ) ) ).
% eq_iff
thf(fact_121_eq__iff,axiom,
( ( ^ [Y: set_val,Z: set_val] : ( Y = Z ) )
= ( ^ [X3: set_val,Y4: set_val] :
( ( ord_less_eq_set_val @ X3 @ Y4 )
& ( ord_less_eq_set_val @ Y4 @ X3 ) ) ) ) ).
% eq_iff
thf(fact_122_antisym,axiom,
! [X: $o > set_val,Y2: $o > set_val] :
( ( ord_le994826475et_val @ X @ Y2 )
=> ( ( ord_le994826475et_val @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% antisym
thf(fact_123_antisym,axiom,
! [X: val,Y2: val] :
( ( ord_less_eq_val @ X @ Y2 )
=> ( ( ord_less_eq_val @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% antisym
thf(fact_124_antisym,axiom,
! [X: set_set_val,Y2: set_set_val] :
( ( ord_le1742111550et_val @ X @ Y2 )
=> ( ( ord_le1742111550et_val @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% antisym
thf(fact_125_antisym,axiom,
! [X: set_val,Y2: set_val] :
( ( ord_less_eq_set_val @ X @ Y2 )
=> ( ( ord_less_eq_set_val @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% antisym
thf(fact_126_linear,axiom,
! [X: val,Y2: val] :
( ( ord_less_eq_val @ X @ Y2 )
| ( ord_less_eq_val @ Y2 @ X ) ) ).
% linear
thf(fact_127_eq__refl,axiom,
! [X: $o > set_val,Y2: $o > set_val] :
( ( X = Y2 )
=> ( ord_le994826475et_val @ X @ Y2 ) ) ).
% eq_refl
thf(fact_128_eq__refl,axiom,
! [X: val,Y2: val] :
( ( X = Y2 )
=> ( ord_less_eq_val @ X @ Y2 ) ) ).
% eq_refl
thf(fact_129_eq__refl,axiom,
! [X: set_set_val,Y2: set_set_val] :
( ( X = Y2 )
=> ( ord_le1742111550et_val @ X @ Y2 ) ) ).
% eq_refl
thf(fact_130_eq__refl,axiom,
! [X: set_val,Y2: set_val] :
( ( X = Y2 )
=> ( ord_less_eq_set_val @ X @ Y2 ) ) ).
% eq_refl
thf(fact_131_le__cases,axiom,
! [X: val,Y2: val] :
( ~ ( ord_less_eq_val @ X @ Y2 )
=> ( ord_less_eq_val @ Y2 @ X ) ) ).
% le_cases
thf(fact_132_order_Otrans,axiom,
! [A3: $o > set_val,B3: $o > set_val,C: $o > set_val] :
( ( ord_le994826475et_val @ A3 @ B3 )
=> ( ( ord_le994826475et_val @ B3 @ C )
=> ( ord_le994826475et_val @ A3 @ C ) ) ) ).
% order.trans
thf(fact_133_order_Otrans,axiom,
! [A3: val,B3: val,C: val] :
( ( ord_less_eq_val @ A3 @ B3 )
=> ( ( ord_less_eq_val @ B3 @ C )
=> ( ord_less_eq_val @ A3 @ C ) ) ) ).
% order.trans
thf(fact_134_order_Otrans,axiom,
! [A3: set_set_val,B3: set_set_val,C: set_set_val] :
( ( ord_le1742111550et_val @ A3 @ B3 )
=> ( ( ord_le1742111550et_val @ B3 @ C )
=> ( ord_le1742111550et_val @ A3 @ C ) ) ) ).
% order.trans
thf(fact_135_order_Otrans,axiom,
! [A3: set_val,B3: set_val,C: set_val] :
( ( ord_less_eq_set_val @ A3 @ B3 )
=> ( ( ord_less_eq_set_val @ B3 @ C )
=> ( ord_less_eq_set_val @ A3 @ C ) ) ) ).
% order.trans
thf(fact_136_mem__Collect__eq,axiom,
! [A3: set_val,P: set_val > $o] :
( ( member_set_val @ A3 @ ( collect_set_val @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_137_mem__Collect__eq,axiom,
! [A3: val,P: val > $o] :
( ( member_val @ A3 @ ( collect_val @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_138_Collect__mem__eq,axiom,
! [A: set_set_val] :
( ( collect_set_val
@ ^ [X3: set_val] : ( member_set_val @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_139_Collect__mem__eq,axiom,
! [A: set_val] :
( ( collect_val
@ ^ [X3: val] : ( member_val @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_140_Collect__cong,axiom,
! [P: val > $o,Q: val > $o] :
( ! [X2: val] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_val @ P )
= ( collect_val @ Q ) ) ) ).
% Collect_cong
thf(fact_141_le__cases3,axiom,
! [X: val,Y2: val,Z2: val] :
( ( ( ord_less_eq_val @ X @ Y2 )
=> ~ ( ord_less_eq_val @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_val @ Y2 @ X )
=> ~ ( ord_less_eq_val @ X @ Z2 ) )
=> ( ( ( ord_less_eq_val @ X @ Z2 )
=> ~ ( ord_less_eq_val @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_val @ Z2 @ Y2 )
=> ~ ( ord_less_eq_val @ Y2 @ X ) )
=> ( ( ( ord_less_eq_val @ Y2 @ Z2 )
=> ~ ( ord_less_eq_val @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_val @ Z2 @ X )
=> ~ ( ord_less_eq_val @ X @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_142_antisym__conv,axiom,
! [Y2: $o > set_val,X: $o > set_val] :
( ( ord_le994826475et_val @ Y2 @ X )
=> ( ( ord_le994826475et_val @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv
thf(fact_143_antisym__conv,axiom,
! [Y2: val,X: val] :
( ( ord_less_eq_val @ Y2 @ X )
=> ( ( ord_less_eq_val @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv
thf(fact_144_antisym__conv,axiom,
! [Y2: set_set_val,X: set_set_val] :
( ( ord_le1742111550et_val @ Y2 @ X )
=> ( ( ord_le1742111550et_val @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv
thf(fact_145_antisym__conv,axiom,
! [Y2: set_val,X: set_val] :
( ( ord_less_eq_set_val @ Y2 @ X )
=> ( ( ord_less_eq_set_val @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv
thf(fact_146_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y: $o > set_val,Z: $o > set_val] : ( Y = Z ) )
= ( ^ [A4: $o > set_val,B4: $o > set_val] :
( ( ord_le994826475et_val @ A4 @ B4 )
& ( ord_le994826475et_val @ B4 @ A4 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_147_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y: val,Z: val] : ( Y = Z ) )
= ( ^ [A4: val,B4: val] :
( ( ord_less_eq_val @ A4 @ B4 )
& ( ord_less_eq_val @ B4 @ A4 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_148_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y: set_set_val,Z: set_set_val] : ( Y = Z ) )
= ( ^ [A4: set_set_val,B4: set_set_val] :
( ( ord_le1742111550et_val @ A4 @ B4 )
& ( ord_le1742111550et_val @ B4 @ A4 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_149_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y: set_val,Z: set_val] : ( Y = Z ) )
= ( ^ [A4: set_val,B4: set_val] :
( ( ord_less_eq_set_val @ A4 @ B4 )
& ( ord_less_eq_set_val @ B4 @ A4 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_150_ord__eq__le__trans,axiom,
! [A3: $o > set_val,B3: $o > set_val,C: $o > set_val] :
( ( A3 = B3 )
=> ( ( ord_le994826475et_val @ B3 @ C )
=> ( ord_le994826475et_val @ A3 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_151_ord__eq__le__trans,axiom,
! [A3: val,B3: val,C: val] :
( ( A3 = B3 )
=> ( ( ord_less_eq_val @ B3 @ C )
=> ( ord_less_eq_val @ A3 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_152_ord__eq__le__trans,axiom,
! [A3: set_set_val,B3: set_set_val,C: set_set_val] :
( ( A3 = B3 )
=> ( ( ord_le1742111550et_val @ B3 @ C )
=> ( ord_le1742111550et_val @ A3 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_153_ord__eq__le__trans,axiom,
! [A3: set_val,B3: set_val,C: set_val] :
( ( A3 = B3 )
=> ( ( ord_less_eq_set_val @ B3 @ C )
=> ( ord_less_eq_set_val @ A3 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_154_GreatestI2__order,axiom,
! [P: ( $o > set_val ) > $o,X: $o > set_val,Q: ( $o > set_val ) > $o] :
( ( P @ X )
=> ( ! [Y3: $o > set_val] :
( ( P @ Y3 )
=> ( ord_le994826475et_val @ Y3 @ X ) )
=> ( ! [X2: $o > set_val] :
( ( P @ X2 )
=> ( ! [Y5: $o > set_val] :
( ( P @ Y5 )
=> ( ord_le994826475et_val @ Y5 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_985126002et_val @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_155_GreatestI2__order,axiom,
! [P: val > $o,X: val,Q: val > $o] :
( ( P @ X )
=> ( ! [Y3: val] :
( ( P @ Y3 )
=> ( ord_less_eq_val @ Y3 @ X ) )
=> ( ! [X2: val] :
( ( P @ X2 )
=> ( ! [Y5: val] :
( ( P @ Y5 )
=> ( ord_less_eq_val @ Y5 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_Greatest_val @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_156_GreatestI2__order,axiom,
! [P: set_set_val > $o,X: set_set_val,Q: set_set_val > $o] :
( ( P @ X )
=> ( ! [Y3: set_set_val] :
( ( P @ Y3 )
=> ( ord_le1742111550et_val @ Y3 @ X ) )
=> ( ! [X2: set_set_val] :
( ( P @ X2 )
=> ( ! [Y5: set_set_val] :
( ( P @ Y5 )
=> ( ord_le1742111550et_val @ Y5 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_544136439et_val @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_157_GreatestI2__order,axiom,
! [P: set_val > $o,X: set_val,Q: set_val > $o] :
( ( P @ X )
=> ( ! [Y3: set_val] :
( ( P @ Y3 )
=> ( ord_less_eq_set_val @ Y3 @ X ) )
=> ( ! [X2: set_val] :
( ( P @ X2 )
=> ( ! [Y5: set_val] :
( ( P @ Y5 )
=> ( ord_less_eq_set_val @ Y5 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_1960465943et_val @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_158_Greatest__equality,axiom,
! [P: ( $o > set_val ) > $o,X: $o > set_val] :
( ( P @ X )
=> ( ! [Y3: $o > set_val] :
( ( P @ Y3 )
=> ( ord_le994826475et_val @ Y3 @ X ) )
=> ( ( order_985126002et_val @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_159_Greatest__equality,axiom,
! [P: val > $o,X: val] :
( ( P @ X )
=> ( ! [Y3: val] :
( ( P @ Y3 )
=> ( ord_less_eq_val @ Y3 @ X ) )
=> ( ( order_Greatest_val @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_160_Greatest__equality,axiom,
! [P: set_set_val > $o,X: set_set_val] :
( ( P @ X )
=> ( ! [Y3: set_set_val] :
( ( P @ Y3 )
=> ( ord_le1742111550et_val @ Y3 @ X ) )
=> ( ( order_544136439et_val @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_161_Greatest__equality,axiom,
! [P: set_val > $o,X: set_val] :
( ( P @ X )
=> ( ! [Y3: set_val] :
( ( P @ Y3 )
=> ( ord_less_eq_set_val @ Y3 @ X ) )
=> ( ( order_1960465943et_val @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_162_Refine__Misc_Oif__mono,axiom,
! [B3: $o,M1: $o > set_val,M12: $o > set_val,M2: $o > set_val,M22: $o > set_val] :
( ( B3
=> ( ord_le994826475et_val @ M1 @ M12 ) )
=> ( ( ~ B3
=> ( ord_le994826475et_val @ M2 @ M22 ) )
=> ( ord_le994826475et_val @ ( if_o_set_val @ B3 @ M1 @ M2 ) @ ( if_o_set_val @ B3 @ M12 @ M22 ) ) ) ) ).
% Refine_Misc.if_mono
thf(fact_163_Refine__Misc_Oif__mono,axiom,
! [B3: $o,M1: val,M12: val,M2: val,M22: val] :
( ( B3
=> ( ord_less_eq_val @ M1 @ M12 ) )
=> ( ( ~ B3
=> ( ord_less_eq_val @ M2 @ M22 ) )
=> ( ord_less_eq_val @ ( if_val @ B3 @ M1 @ M2 ) @ ( if_val @ B3 @ M12 @ M22 ) ) ) ) ).
% Refine_Misc.if_mono
thf(fact_164_Refine__Misc_Oif__mono,axiom,
! [B3: $o,M1: set_set_val,M12: set_set_val,M2: set_set_val,M22: set_set_val] :
( ( B3
=> ( ord_le1742111550et_val @ M1 @ M12 ) )
=> ( ( ~ B3
=> ( ord_le1742111550et_val @ M2 @ M22 ) )
=> ( ord_le1742111550et_val @ ( if_set_set_val @ B3 @ M1 @ M2 ) @ ( if_set_set_val @ B3 @ M12 @ M22 ) ) ) ) ).
% Refine_Misc.if_mono
thf(fact_165_Refine__Misc_Oif__mono,axiom,
! [B3: $o,M1: set_val,M12: set_val,M2: set_val,M22: set_val] :
( ( B3
=> ( ord_less_eq_set_val @ M1 @ M12 ) )
=> ( ( ~ B3
=> ( ord_less_eq_set_val @ M2 @ M22 ) )
=> ( ord_less_eq_set_val @ ( if_set_val @ B3 @ M1 @ M2 ) @ ( if_set_val @ B3 @ M12 @ M22 ) ) ) ) ).
% Refine_Misc.if_mono
thf(fact_166_lhs__step__If,axiom,
! [B3: $o,T2: $o > set_val,M: $o > set_val,E: $o > set_val] :
( ( B3
=> ( ord_le994826475et_val @ T2 @ M ) )
=> ( ( ~ B3
=> ( ord_le994826475et_val @ E @ M ) )
=> ( ord_le994826475et_val @ ( if_o_set_val @ B3 @ T2 @ E ) @ M ) ) ) ).
% lhs_step_If
thf(fact_167_lhs__step__If,axiom,
! [B3: $o,T2: val,M: val,E: val] :
( ( B3
=> ( ord_less_eq_val @ T2 @ M ) )
=> ( ( ~ B3
=> ( ord_less_eq_val @ E @ M ) )
=> ( ord_less_eq_val @ ( if_val @ B3 @ T2 @ E ) @ M ) ) ) ).
% lhs_step_If
thf(fact_168_lhs__step__If,axiom,
! [B3: $o,T2: set_set_val,M: set_set_val,E: set_set_val] :
( ( B3
=> ( ord_le1742111550et_val @ T2 @ M ) )
=> ( ( ~ B3
=> ( ord_le1742111550et_val @ E @ M ) )
=> ( ord_le1742111550et_val @ ( if_set_set_val @ B3 @ T2 @ E ) @ M ) ) ) ).
% lhs_step_If
thf(fact_169_lhs__step__If,axiom,
! [B3: $o,T2: set_val,M: set_val,E: set_val] :
( ( B3
=> ( ord_less_eq_set_val @ T2 @ M ) )
=> ( ( ~ B3
=> ( ord_less_eq_set_val @ E @ M ) )
=> ( ord_less_eq_set_val @ ( if_set_val @ B3 @ T2 @ E ) @ M ) ) ) ).
% lhs_step_If
thf(fact_170_verit__la__disequality,axiom,
! [A3: val,B3: val] :
( ( A3 = B3 )
| ~ ( ord_less_eq_val @ A3 @ B3 )
| ~ ( ord_less_eq_val @ B3 @ A3 ) ) ).
% verit_la_disequality
thf(fact_171_le__rel__bool__arg__iff,axiom,
( ord_le1441829300et_val
= ( ^ [X4: $o > $o > set_val,Y6: $o > $o > set_val] :
( ( ord_le994826475et_val @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_le994826475et_val @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_172_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_val
= ( ^ [X4: $o > val,Y6: $o > val] :
( ( ord_less_eq_val @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_val @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_173_le__rel__bool__arg__iff,axiom,
( ord_le936627147et_val
= ( ^ [X4: $o > set_set_val,Y6: $o > set_set_val] :
( ( ord_le1742111550et_val @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_le1742111550et_val @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_174_le__rel__bool__arg__iff,axiom,
( ord_le994826475et_val
= ( ^ [X4: $o > set_val,Y6: $o > set_val] :
( ( ord_less_eq_set_val @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_set_val @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_175_antimonoD,axiom,
! [F: set_val > set_val,X: set_val,Y2: set_val] :
( ( order_1719375026et_val @ F )
=> ( ( ord_less_eq_set_val @ X @ Y2 )
=> ( ord_less_eq_set_val @ ( F @ Y2 ) @ ( F @ X ) ) ) ) ).
% antimonoD
thf(fact_176_antimonoD,axiom,
! [F: val > val,X: val,Y2: val] :
( ( order_1689798834al_val @ F )
=> ( ( ord_less_eq_val @ X @ Y2 )
=> ( ord_less_eq_val @ ( F @ Y2 ) @ ( F @ X ) ) ) ) ).
% antimonoD
thf(fact_177_antimonoD,axiom,
! [F: set_val > val,X: set_val,Y2: set_val] :
( ( order_1375876562al_val @ F )
=> ( ( ord_less_eq_set_val @ X @ Y2 )
=> ( ord_less_eq_val @ ( F @ Y2 ) @ ( F @ X ) ) ) ) ).
% antimonoD
thf(fact_178_antimonoD,axiom,
! [F: val > set_val,X: val,Y2: val] :
( ( order_1484138386et_val @ F )
=> ( ( ord_less_eq_val @ X @ Y2 )
=> ( ord_less_eq_set_val @ ( F @ Y2 ) @ ( F @ X ) ) ) ) ).
% antimonoD
thf(fact_179_antimonoD,axiom,
! [F: val > set_set_val,X: val,Y2: val] :
( ( order_1945059954et_val @ F )
=> ( ( ord_less_eq_val @ X @ Y2 )
=> ( ord_le1742111550et_val @ ( F @ Y2 ) @ ( F @ X ) ) ) ) ).
% antimonoD
thf(fact_180_antimonoD,axiom,
! [F: set_set_val > val,X: set_set_val,Y2: set_set_val] :
( ( order_1520822002al_val @ F )
=> ( ( ord_le1742111550et_val @ X @ Y2 )
=> ( ord_less_eq_val @ ( F @ Y2 ) @ ( F @ X ) ) ) ) ).
% antimonoD
thf(fact_181_antimonoD,axiom,
! [F: set_val > set_set_val,X: set_val,Y2: set_val] :
( ( order_1957703570et_val @ F )
=> ( ( ord_less_eq_set_val @ X @ Y2 )
=> ( ord_le1742111550et_val @ ( F @ Y2 ) @ ( F @ X ) ) ) ) ).
% antimonoD
thf(fact_182_antimonoD,axiom,
! [F: ( $o > set_val ) > val,X: $o > set_val,Y2: $o > set_val] :
( ( order_23236447al_val @ F )
=> ( ( ord_le994826475et_val @ X @ Y2 )
=> ( ord_less_eq_val @ ( F @ Y2 ) @ ( F @ X ) ) ) ) ).
% antimonoD
thf(fact_183_antimonoD,axiom,
! [F: val > $o > set_val,X: val,Y2: val] :
( ( order_255079095et_val @ F )
=> ( ( ord_less_eq_val @ X @ Y2 )
=> ( ord_le994826475et_val @ ( F @ Y2 ) @ ( F @ X ) ) ) ) ).
% antimonoD
thf(fact_184_antimonoD,axiom,
! [F: set_set_val > set_val,X: set_set_val,Y2: set_set_val] :
( ( order_650823122et_val @ F )
=> ( ( ord_le1742111550et_val @ X @ Y2 )
=> ( ord_less_eq_set_val @ ( F @ Y2 ) @ ( F @ X ) ) ) ) ).
% antimonoD
thf(fact_185_antimonoE,axiom,
! [F: set_val > set_val,X: set_val,Y2: set_val] :
( ( order_1719375026et_val @ F )
=> ( ( ord_less_eq_set_val @ X @ Y2 )
=> ( ord_less_eq_set_val @ ( F @ Y2 ) @ ( F @ X ) ) ) ) ).
% antimonoE
thf(fact_186_antimonoE,axiom,
! [F: val > val,X: val,Y2: val] :
( ( order_1689798834al_val @ F )
=> ( ( ord_less_eq_val @ X @ Y2 )
=> ( ord_less_eq_val @ ( F @ Y2 ) @ ( F @ X ) ) ) ) ).
% antimonoE
thf(fact_187_antimonoE,axiom,
! [F: set_val > val,X: set_val,Y2: set_val] :
( ( order_1375876562al_val @ F )
=> ( ( ord_less_eq_set_val @ X @ Y2 )
=> ( ord_less_eq_val @ ( F @ Y2 ) @ ( F @ X ) ) ) ) ).
% antimonoE
thf(fact_188_antimonoE,axiom,
! [F: val > set_val,X: val,Y2: val] :
( ( order_1484138386et_val @ F )
=> ( ( ord_less_eq_val @ X @ Y2 )
=> ( ord_less_eq_set_val @ ( F @ Y2 ) @ ( F @ X ) ) ) ) ).
% antimonoE
thf(fact_189_antimonoE,axiom,
! [F: val > set_set_val,X: val,Y2: val] :
( ( order_1945059954et_val @ F )
=> ( ( ord_less_eq_val @ X @ Y2 )
=> ( ord_le1742111550et_val @ ( F @ Y2 ) @ ( F @ X ) ) ) ) ).
% antimonoE
thf(fact_190_antimonoE,axiom,
! [F: set_set_val > val,X: set_set_val,Y2: set_set_val] :
( ( order_1520822002al_val @ F )
=> ( ( ord_le1742111550et_val @ X @ Y2 )
=> ( ord_less_eq_val @ ( F @ Y2 ) @ ( F @ X ) ) ) ) ).
% antimonoE
thf(fact_191_antimonoE,axiom,
! [F: set_val > set_set_val,X: set_val,Y2: set_val] :
( ( order_1957703570et_val @ F )
=> ( ( ord_less_eq_set_val @ X @ Y2 )
=> ( ord_le1742111550et_val @ ( F @ Y2 ) @ ( F @ X ) ) ) ) ).
% antimonoE
thf(fact_192_antimonoE,axiom,
! [F: ( $o > set_val ) > val,X: $o > set_val,Y2: $o > set_val] :
( ( order_23236447al_val @ F )
=> ( ( ord_le994826475et_val @ X @ Y2 )
=> ( ord_less_eq_val @ ( F @ Y2 ) @ ( F @ X ) ) ) ) ).
% antimonoE
thf(fact_193_antimonoE,axiom,
! [F: val > $o > set_val,X: val,Y2: val] :
( ( order_255079095et_val @ F )
=> ( ( ord_less_eq_val @ X @ Y2 )
=> ( ord_le994826475et_val @ ( F @ Y2 ) @ ( F @ X ) ) ) ) ).
% antimonoE
thf(fact_194_antimonoE,axiom,
! [F: set_set_val > set_val,X: set_set_val,Y2: set_set_val] :
( ( order_650823122et_val @ F )
=> ( ( ord_le1742111550et_val @ X @ Y2 )
=> ( ord_less_eq_set_val @ ( F @ Y2 ) @ ( F @ X ) ) ) ) ).
% antimonoE
thf(fact_195_antimono__def,axiom,
( order_1719375026et_val
= ( ^ [F2: set_val > set_val] :
! [X3: set_val,Y4: set_val] :
( ( ord_less_eq_set_val @ X3 @ Y4 )
=> ( ord_less_eq_set_val @ ( F2 @ Y4 ) @ ( F2 @ X3 ) ) ) ) ) ).
% antimono_def
thf(fact_196_antimono__def,axiom,
( order_1689798834al_val
= ( ^ [F2: val > val] :
! [X3: val,Y4: val] :
( ( ord_less_eq_val @ X3 @ Y4 )
=> ( ord_less_eq_val @ ( F2 @ Y4 ) @ ( F2 @ X3 ) ) ) ) ) ).
% antimono_def
thf(fact_197_antimono__def,axiom,
( order_1375876562al_val
= ( ^ [F2: set_val > val] :
! [X3: set_val,Y4: set_val] :
( ( ord_less_eq_set_val @ X3 @ Y4 )
=> ( ord_less_eq_val @ ( F2 @ Y4 ) @ ( F2 @ X3 ) ) ) ) ) ).
% antimono_def
thf(fact_198_antimono__def,axiom,
( order_1484138386et_val
= ( ^ [F2: val > set_val] :
! [X3: val,Y4: val] :
( ( ord_less_eq_val @ X3 @ Y4 )
=> ( ord_less_eq_set_val @ ( F2 @ Y4 ) @ ( F2 @ X3 ) ) ) ) ) ).
% antimono_def
thf(fact_199_antimono__def,axiom,
( order_1945059954et_val
= ( ^ [F2: val > set_set_val] :
! [X3: val,Y4: val] :
( ( ord_less_eq_val @ X3 @ Y4 )
=> ( ord_le1742111550et_val @ ( F2 @ Y4 ) @ ( F2 @ X3 ) ) ) ) ) ).
% antimono_def
thf(fact_200_antimono__def,axiom,
( order_1520822002al_val
= ( ^ [F2: set_set_val > val] :
! [X3: set_set_val,Y4: set_set_val] :
( ( ord_le1742111550et_val @ X3 @ Y4 )
=> ( ord_less_eq_val @ ( F2 @ Y4 ) @ ( F2 @ X3 ) ) ) ) ) ).
% antimono_def
thf(fact_201_antimono__def,axiom,
( order_1957703570et_val
= ( ^ [F2: set_val > set_set_val] :
! [X3: set_val,Y4: set_val] :
( ( ord_less_eq_set_val @ X3 @ Y4 )
=> ( ord_le1742111550et_val @ ( F2 @ Y4 ) @ ( F2 @ X3 ) ) ) ) ) ).
% antimono_def
thf(fact_202_antimono__def,axiom,
( order_23236447al_val
= ( ^ [F2: ( $o > set_val ) > val] :
! [X3: $o > set_val,Y4: $o > set_val] :
( ( ord_le994826475et_val @ X3 @ Y4 )
=> ( ord_less_eq_val @ ( F2 @ Y4 ) @ ( F2 @ X3 ) ) ) ) ) ).
% antimono_def
thf(fact_203_antimono__def,axiom,
( order_255079095et_val
= ( ^ [F2: val > $o > set_val] :
! [X3: val,Y4: val] :
( ( ord_less_eq_val @ X3 @ Y4 )
=> ( ord_le994826475et_val @ ( F2 @ Y4 ) @ ( F2 @ X3 ) ) ) ) ) ).
% antimono_def
thf(fact_204_antimono__def,axiom,
( order_650823122et_val
= ( ^ [F2: set_set_val > set_val] :
! [X3: set_set_val,Y4: set_set_val] :
( ( ord_le1742111550et_val @ X3 @ Y4 )
=> ( ord_less_eq_set_val @ ( F2 @ Y4 ) @ ( F2 @ X3 ) ) ) ) ) ).
% antimono_def
thf(fact_205_antimonoI,axiom,
! [F: set_val > set_val] :
( ! [X2: set_val,Y3: set_val] :
( ( ord_less_eq_set_val @ X2 @ Y3 )
=> ( ord_less_eq_set_val @ ( F @ Y3 ) @ ( F @ X2 ) ) )
=> ( order_1719375026et_val @ F ) ) ).
% antimonoI
thf(fact_206_antimonoI,axiom,
! [F: val > val] :
( ! [X2: val,Y3: val] :
( ( ord_less_eq_val @ X2 @ Y3 )
=> ( ord_less_eq_val @ ( F @ Y3 ) @ ( F @ X2 ) ) )
=> ( order_1689798834al_val @ F ) ) ).
% antimonoI
thf(fact_207_antimonoI,axiom,
! [F: set_val > val] :
( ! [X2: set_val,Y3: set_val] :
( ( ord_less_eq_set_val @ X2 @ Y3 )
=> ( ord_less_eq_val @ ( F @ Y3 ) @ ( F @ X2 ) ) )
=> ( order_1375876562al_val @ F ) ) ).
% antimonoI
thf(fact_208_antimonoI,axiom,
! [F: val > set_val] :
( ! [X2: val,Y3: val] :
( ( ord_less_eq_val @ X2 @ Y3 )
=> ( ord_less_eq_set_val @ ( F @ Y3 ) @ ( F @ X2 ) ) )
=> ( order_1484138386et_val @ F ) ) ).
% antimonoI
thf(fact_209_antimonoI,axiom,
! [F: val > set_set_val] :
( ! [X2: val,Y3: val] :
( ( ord_less_eq_val @ X2 @ Y3 )
=> ( ord_le1742111550et_val @ ( F @ Y3 ) @ ( F @ X2 ) ) )
=> ( order_1945059954et_val @ F ) ) ).
% antimonoI
thf(fact_210_antimonoI,axiom,
! [F: set_set_val > val] :
( ! [X2: set_set_val,Y3: set_set_val] :
( ( ord_le1742111550et_val @ X2 @ Y3 )
=> ( ord_less_eq_val @ ( F @ Y3 ) @ ( F @ X2 ) ) )
=> ( order_1520822002al_val @ F ) ) ).
% antimonoI
thf(fact_211_antimonoI,axiom,
! [F: set_val > set_set_val] :
( ! [X2: set_val,Y3: set_val] :
( ( ord_less_eq_set_val @ X2 @ Y3 )
=> ( ord_le1742111550et_val @ ( F @ Y3 ) @ ( F @ X2 ) ) )
=> ( order_1957703570et_val @ F ) ) ).
% antimonoI
thf(fact_212_antimonoI,axiom,
! [F: ( $o > set_val ) > val] :
( ! [X2: $o > set_val,Y3: $o > set_val] :
( ( ord_le994826475et_val @ X2 @ Y3 )
=> ( ord_less_eq_val @ ( F @ Y3 ) @ ( F @ X2 ) ) )
=> ( order_23236447al_val @ F ) ) ).
% antimonoI
thf(fact_213_antimonoI,axiom,
! [F: val > $o > set_val] :
( ! [X2: val,Y3: val] :
( ( ord_less_eq_val @ X2 @ Y3 )
=> ( ord_le994826475et_val @ ( F @ Y3 ) @ ( F @ X2 ) ) )
=> ( order_255079095et_val @ F ) ) ).
% antimonoI
thf(fact_214_antimonoI,axiom,
! [F: set_set_val > set_val] :
( ! [X2: set_set_val,Y3: set_set_val] :
( ( ord_le1742111550et_val @ X2 @ Y3 )
=> ( ord_less_eq_set_val @ ( F @ Y3 ) @ ( F @ X2 ) ) )
=> ( order_650823122et_val @ F ) ) ).
% antimonoI
thf(fact_215_order__mono__setup_Omono__setup__loc__axioms,axiom,
refine16560100et_val @ ord_le994826475et_val ).
% order_mono_setup.mono_setup_loc_axioms
thf(fact_216_order__mono__setup_Omono__setup__loc__axioms,axiom,
refine67622213oc_val @ ord_less_eq_val ).
% order_mono_setup.mono_setup_loc_axioms
thf(fact_217_order__mono__setup_Omono__setup__loc__axioms,axiom,
refine1506227973et_val @ ord_le1742111550et_val ).
% order_mono_setup.mono_setup_loc_axioms
thf(fact_218_order__mono__setup_Omono__setup__loc__axioms,axiom,
refine275900453et_val @ ord_less_eq_set_val ).
% order_mono_setup.mono_setup_loc_axioms
thf(fact_219_galois__connection_O_092_060gamma_062_092_060alpha_062__infl,axiom,
! [Alpha: set_val > set_val,Gamma: set_val > set_val,X: set_val] :
( ( refine1929691018et_val @ Alpha @ Gamma )
=> ( ord_less_eq_set_val @ X @ ( Gamma @ ( Alpha @ X ) ) ) ) ).
% galois_connection.\<gamma>\<alpha>_infl
thf(fact_220_galois__connection_O_092_060alpha_062_092_060gamma_062__defl,axiom,
! [Alpha: set_val > set_val,Gamma: set_val > set_val,X: set_val] :
( ( refine1929691018et_val @ Alpha @ Gamma )
=> ( ord_less_eq_set_val @ ( Alpha @ ( Gamma @ X ) ) @ X ) ) ).
% galois_connection.\<alpha>\<gamma>_defl
thf(fact_221_galois__connection_Ogalois,axiom,
! [Alpha: set_val > $o > set_val,Gamma: ( $o > set_val ) > set_val,C: set_val,A3: $o > set_val] :
( ( refine51544255et_val @ Alpha @ Gamma )
=> ( ( ord_less_eq_set_val @ C @ ( Gamma @ A3 ) )
= ( ord_le994826475et_val @ ( Alpha @ C ) @ A3 ) ) ) ).
% galois_connection.galois
thf(fact_222_galois__connection_Ogalois,axiom,
! [Alpha: set_val > set_set_val,Gamma: set_set_val > set_val,C: set_val,A3: set_set_val] :
( ( refine1315043946et_val @ Alpha @ Gamma )
=> ( ( ord_less_eq_set_val @ C @ ( Gamma @ A3 ) )
= ( ord_le1742111550et_val @ ( Alpha @ C ) @ A3 ) ) ) ).
% galois_connection.galois
thf(fact_223_galois__connection_Ogalois,axiom,
! [Alpha: ( $o > set_val ) > set_val,Gamma: set_val > $o > set_val,C: $o > set_val,A3: set_val] :
( ( refine1948536167et_val @ Alpha @ Gamma )
=> ( ( ord_le994826475et_val @ C @ ( Gamma @ A3 ) )
= ( ord_less_eq_set_val @ ( Alpha @ C ) @ A3 ) ) ) ).
% galois_connection.galois
thf(fact_224_galois__connection_Ogalois,axiom,
! [Alpha: ( $o > set_val ) > $o > set_val,Gamma: ( $o > set_val ) > $o > set_val,C: $o > set_val,A3: $o > set_val] :
( ( refine1924899106et_val @ Alpha @ Gamma )
=> ( ( ord_le994826475et_val @ C @ ( Gamma @ A3 ) )
= ( ord_le994826475et_val @ ( Alpha @ C ) @ A3 ) ) ) ).
% galois_connection.galois
thf(fact_225_galois__connection_Ogalois,axiom,
! [Alpha: ( $o > set_val ) > set_set_val,Gamma: set_set_val > $o > set_val,C: $o > set_val,A3: set_set_val] :
( ( refine1169396807et_val @ Alpha @ Gamma )
=> ( ( ord_le994826475et_val @ C @ ( Gamma @ A3 ) )
= ( ord_le1742111550et_val @ ( Alpha @ C ) @ A3 ) ) ) ).
% galois_connection.galois
thf(fact_226_galois__connection_Ogalois,axiom,
! [Alpha: set_set_val > set_val,Gamma: set_val > set_set_val,C: set_set_val,A3: set_val] :
( ( refine8163498et_val @ Alpha @ Gamma )
=> ( ( ord_le1742111550et_val @ C @ ( Gamma @ A3 ) )
= ( ord_less_eq_set_val @ ( Alpha @ C ) @ A3 ) ) ) ).
% galois_connection.galois
thf(fact_227_galois__connection_Ogalois,axiom,
! [Alpha: set_set_val > $o > set_val,Gamma: ( $o > set_val ) > set_set_val,C: set_set_val,A3: $o > set_val] :
( ( refine86398879et_val @ Alpha @ Gamma )
=> ( ( ord_le1742111550et_val @ C @ ( Gamma @ A3 ) )
= ( ord_le994826475et_val @ ( Alpha @ C ) @ A3 ) ) ) ).
% galois_connection.galois
thf(fact_228_galois__connection_Ogalois,axiom,
! [Alpha: set_set_val > set_set_val,Gamma: set_set_val > set_set_val,C: set_set_val,A3: set_set_val] :
( ( refine855985034et_val @ Alpha @ Gamma )
=> ( ( ord_le1742111550et_val @ C @ ( Gamma @ A3 ) )
= ( ord_le1742111550et_val @ ( Alpha @ C ) @ A3 ) ) ) ).
% galois_connection.galois
thf(fact_229_galois__connection_Ogalois,axiom,
! [Alpha: set_val > set_val,Gamma: set_val > set_val,C: set_val,A3: set_val] :
( ( refine1929691018et_val @ Alpha @ Gamma )
=> ( ( ord_less_eq_set_val @ C @ ( Gamma @ A3 ) )
= ( ord_less_eq_set_val @ ( Alpha @ C ) @ A3 ) ) ) ).
% galois_connection.galois
thf(fact_230_galois__connection_Ointro,axiom,
! [Gamma: ( $o > set_val ) > set_val,Alpha: set_val > $o > set_val] :
( ! [C3: set_val,A5: $o > set_val] :
( ( ord_less_eq_set_val @ C3 @ ( Gamma @ A5 ) )
= ( ord_le994826475et_val @ ( Alpha @ C3 ) @ A5 ) )
=> ( refine51544255et_val @ Alpha @ Gamma ) ) ).
% galois_connection.intro
thf(fact_231_galois__connection_Ointro,axiom,
! [Gamma: set_set_val > set_val,Alpha: set_val > set_set_val] :
( ! [C3: set_val,A5: set_set_val] :
( ( ord_less_eq_set_val @ C3 @ ( Gamma @ A5 ) )
= ( ord_le1742111550et_val @ ( Alpha @ C3 ) @ A5 ) )
=> ( refine1315043946et_val @ Alpha @ Gamma ) ) ).
% galois_connection.intro
thf(fact_232_galois__connection_Ointro,axiom,
! [Gamma: set_val > $o > set_val,Alpha: ( $o > set_val ) > set_val] :
( ! [C3: $o > set_val,A5: set_val] :
( ( ord_le994826475et_val @ C3 @ ( Gamma @ A5 ) )
= ( ord_less_eq_set_val @ ( Alpha @ C3 ) @ A5 ) )
=> ( refine1948536167et_val @ Alpha @ Gamma ) ) ).
% galois_connection.intro
thf(fact_233_galois__connection_Ointro,axiom,
! [Gamma: ( $o > set_val ) > $o > set_val,Alpha: ( $o > set_val ) > $o > set_val] :
( ! [C3: $o > set_val,A5: $o > set_val] :
( ( ord_le994826475et_val @ C3 @ ( Gamma @ A5 ) )
= ( ord_le994826475et_val @ ( Alpha @ C3 ) @ A5 ) )
=> ( refine1924899106et_val @ Alpha @ Gamma ) ) ).
% galois_connection.intro
thf(fact_234_galois__connection_Ointro,axiom,
! [Gamma: set_set_val > $o > set_val,Alpha: ( $o > set_val ) > set_set_val] :
( ! [C3: $o > set_val,A5: set_set_val] :
( ( ord_le994826475et_val @ C3 @ ( Gamma @ A5 ) )
= ( ord_le1742111550et_val @ ( Alpha @ C3 ) @ A5 ) )
=> ( refine1169396807et_val @ Alpha @ Gamma ) ) ).
% galois_connection.intro
thf(fact_235_galois__connection_Ointro,axiom,
! [Gamma: set_val > set_set_val,Alpha: set_set_val > set_val] :
( ! [C3: set_set_val,A5: set_val] :
( ( ord_le1742111550et_val @ C3 @ ( Gamma @ A5 ) )
= ( ord_less_eq_set_val @ ( Alpha @ C3 ) @ A5 ) )
=> ( refine8163498et_val @ Alpha @ Gamma ) ) ).
% galois_connection.intro
thf(fact_236_galois__connection_Ointro,axiom,
! [Gamma: ( $o > set_val ) > set_set_val,Alpha: set_set_val > $o > set_val] :
( ! [C3: set_set_val,A5: $o > set_val] :
( ( ord_le1742111550et_val @ C3 @ ( Gamma @ A5 ) )
= ( ord_le994826475et_val @ ( Alpha @ C3 ) @ A5 ) )
=> ( refine86398879et_val @ Alpha @ Gamma ) ) ).
% galois_connection.intro
thf(fact_237_galois__connection_Ointro,axiom,
! [Gamma: set_set_val > set_set_val,Alpha: set_set_val > set_set_val] :
( ! [C3: set_set_val,A5: set_set_val] :
( ( ord_le1742111550et_val @ C3 @ ( Gamma @ A5 ) )
= ( ord_le1742111550et_val @ ( Alpha @ C3 ) @ A5 ) )
=> ( refine855985034et_val @ Alpha @ Gamma ) ) ).
% galois_connection.intro
thf(fact_238_galois__connection_Ointro,axiom,
! [Gamma: set_val > set_val,Alpha: set_val > set_val] :
( ! [C3: set_val,A5: set_val] :
( ( ord_less_eq_set_val @ C3 @ ( Gamma @ A5 ) )
= ( ord_less_eq_set_val @ ( Alpha @ C3 ) @ A5 ) )
=> ( refine1929691018et_val @ Alpha @ Gamma ) ) ).
% galois_connection.intro
thf(fact_239_galois__connection__def,axiom,
( refine51544255et_val
= ( ^ [Alpha2: set_val > $o > set_val,Gamma2: ( $o > set_val ) > set_val] :
! [C4: set_val,A4: $o > set_val] :
( ( ord_less_eq_set_val @ C4 @ ( Gamma2 @ A4 ) )
= ( ord_le994826475et_val @ ( Alpha2 @ C4 ) @ A4 ) ) ) ) ).
% galois_connection_def
thf(fact_240_galois__connection__def,axiom,
( refine1315043946et_val
= ( ^ [Alpha2: set_val > set_set_val,Gamma2: set_set_val > set_val] :
! [C4: set_val,A4: set_set_val] :
( ( ord_less_eq_set_val @ C4 @ ( Gamma2 @ A4 ) )
= ( ord_le1742111550et_val @ ( Alpha2 @ C4 ) @ A4 ) ) ) ) ).
% galois_connection_def
thf(fact_241_galois__connection__def,axiom,
( refine1948536167et_val
= ( ^ [Alpha2: ( $o > set_val ) > set_val,Gamma2: set_val > $o > set_val] :
! [C4: $o > set_val,A4: set_val] :
( ( ord_le994826475et_val @ C4 @ ( Gamma2 @ A4 ) )
= ( ord_less_eq_set_val @ ( Alpha2 @ C4 ) @ A4 ) ) ) ) ).
% galois_connection_def
thf(fact_242_galois__connection__def,axiom,
( refine1924899106et_val
= ( ^ [Alpha2: ( $o > set_val ) > $o > set_val,Gamma2: ( $o > set_val ) > $o > set_val] :
! [C4: $o > set_val,A4: $o > set_val] :
( ( ord_le994826475et_val @ C4 @ ( Gamma2 @ A4 ) )
= ( ord_le994826475et_val @ ( Alpha2 @ C4 ) @ A4 ) ) ) ) ).
% galois_connection_def
thf(fact_243_galois__connection__def,axiom,
( refine1169396807et_val
= ( ^ [Alpha2: ( $o > set_val ) > set_set_val,Gamma2: set_set_val > $o > set_val] :
! [C4: $o > set_val,A4: set_set_val] :
( ( ord_le994826475et_val @ C4 @ ( Gamma2 @ A4 ) )
= ( ord_le1742111550et_val @ ( Alpha2 @ C4 ) @ A4 ) ) ) ) ).
% galois_connection_def
thf(fact_244_galois__connection__def,axiom,
( refine8163498et_val
= ( ^ [Alpha2: set_set_val > set_val,Gamma2: set_val > set_set_val] :
! [C4: set_set_val,A4: set_val] :
( ( ord_le1742111550et_val @ C4 @ ( Gamma2 @ A4 ) )
= ( ord_less_eq_set_val @ ( Alpha2 @ C4 ) @ A4 ) ) ) ) ).
% galois_connection_def
thf(fact_245_galois__connection__def,axiom,
( refine86398879et_val
= ( ^ [Alpha2: set_set_val > $o > set_val,Gamma2: ( $o > set_val ) > set_set_val] :
! [C4: set_set_val,A4: $o > set_val] :
( ( ord_le1742111550et_val @ C4 @ ( Gamma2 @ A4 ) )
= ( ord_le994826475et_val @ ( Alpha2 @ C4 ) @ A4 ) ) ) ) ).
% galois_connection_def
thf(fact_246_galois__connection__def,axiom,
( refine855985034et_val
= ( ^ [Alpha2: set_set_val > set_set_val,Gamma2: set_set_val > set_set_val] :
! [C4: set_set_val,A4: set_set_val] :
( ( ord_le1742111550et_val @ C4 @ ( Gamma2 @ A4 ) )
= ( ord_le1742111550et_val @ ( Alpha2 @ C4 ) @ A4 ) ) ) ) ).
% galois_connection_def
thf(fact_247_galois__connection__def,axiom,
( refine1929691018et_val
= ( ^ [Alpha2: set_val > set_val,Gamma2: set_val > set_val] :
! [C4: set_val,A4: set_val] :
( ( ord_less_eq_set_val @ C4 @ ( Gamma2 @ A4 ) )
= ( ord_less_eq_set_val @ ( Alpha2 @ C4 ) @ A4 ) ) ) ) ).
% galois_connection_def
thf(fact_248_mono__setup__loc__def,axiom,
( refine275900453et_val
= ( ^ [Le: set_val > set_val > $o] :
! [X3: set_val] : ( Le @ X3 @ X3 ) ) ) ).
% mono_setup_loc_def
thf(fact_249_mono__setup__loc_Orefl,axiom,
! [Le2: set_val > set_val > $o,X: set_val] :
( ( refine275900453et_val @ Le2 )
=> ( Le2 @ X @ X ) ) ).
% mono_setup_loc.refl
thf(fact_250_mono__setup__loc_Ointro,axiom,
! [Le2: set_val > set_val > $o] :
( ! [X2: set_val] : ( Le2 @ X2 @ X2 )
=> ( refine275900453et_val @ Le2 ) ) ).
% mono_setup_loc.intro
thf(fact_251_mono__setup__loc_Omono__if,axiom,
! [Le2: set_val > set_val > $o,T2: set_val,T3: set_val,E: set_val,E2: set_val,B3: $o] :
( ( refine275900453et_val @ Le2 )
=> ( ( Le2 @ T2 @ T3 )
=> ( ( Le2 @ E @ E2 )
=> ( Le2 @ ( if_set_val @ B3 @ T2 @ E ) @ ( if_set_val @ B3 @ T3 @ E2 ) ) ) ) ) ).
% mono_setup_loc.mono_if
thf(fact_252_galois__connection_Oinf__dist___092_060alpha_062,axiom,
! [Alpha: set_val > set_val,Gamma: set_val > set_val] :
( ( refine1929691018et_val @ Alpha @ Gamma )
=> ( refine2057434761et_val @ Alpha ) ) ).
% galois_connection.inf_dist_\<alpha>
thf(fact_253_strict__mono__less__eq,axiom,
! [F: val > set_val,X: val,Y2: val] :
( ( order_99134233et_val @ F )
=> ( ( ord_less_eq_set_val @ ( F @ X ) @ ( F @ Y2 ) )
= ( ord_less_eq_val @ X @ Y2 ) ) ) ).
% strict_mono_less_eq
thf(fact_254_strict__mono__less__eq,axiom,
! [F: val > $o > set_val,X: val,Y2: val] :
( ( order_1205732848et_val @ F )
=> ( ( ord_le994826475et_val @ ( F @ X ) @ ( F @ Y2 ) )
= ( ord_less_eq_val @ X @ Y2 ) ) ) ).
% strict_mono_less_eq
thf(fact_255_strict__mono__less__eq,axiom,
! [F: val > val,X: val,Y2: val] :
( ( order_1139042361al_val @ F )
=> ( ( ord_less_eq_val @ ( F @ X ) @ ( F @ Y2 ) )
= ( ord_less_eq_val @ X @ Y2 ) ) ) ).
% strict_mono_less_eq
thf(fact_256_strict__mono__less__eq,axiom,
! [F: val > set_set_val,X: val,Y2: val] :
( ( order_68692473et_val @ F )
=> ( ( ord_le1742111550et_val @ ( F @ X ) @ ( F @ Y2 ) )
= ( ord_less_eq_val @ X @ Y2 ) ) ) ).
% strict_mono_less_eq
thf(fact_257_greaterThan__subset__iff,axiom,
! [X: val,Y2: val] :
( ( ord_less_eq_set_val @ ( set_or1288365508an_val @ X ) @ ( set_or1288365508an_val @ Y2 ) )
= ( ord_less_eq_val @ Y2 @ X ) ) ).
% greaterThan_subset_iff
thf(fact_258_START__REFINEG__TRANSFER,axiom,
! [D: $o > set_val,C: $o > set_val,A3: $o > set_val] :
( ( refine1593386065et_val @ D @ C )
=> ( ( ord_le994826475et_val @ C @ A3 )
=> ( ( refine511606705et_val @ C @ D )
=> ( ord_le994826475et_val @ D @ A3 ) ) ) ) ).
% START_REFINEG_TRANSFER
thf(fact_259_START__REFINEG__TRANSFER,axiom,
! [D: val,C: val,A3: val] :
( ( refine2021912633al_val @ D @ C )
=> ( ( ord_less_eq_val @ C @ A3 )
=> ( ( refine403244856MP_val @ C @ D )
=> ( ord_less_eq_val @ D @ A3 ) ) ) ) ).
% START_REFINEG_TRANSFER
thf(fact_260_START__REFINEG__TRANSFER,axiom,
! [D: set_set_val,C: set_set_val,A3: set_set_val] :
( ( refine1034950713et_val @ D @ C )
=> ( ( ord_le1742111550et_val @ C @ A3 )
=> ( ( refine1908789496et_val @ C @ D )
=> ( ord_le1742111550et_val @ D @ A3 ) ) ) ) ).
% START_REFINEG_TRANSFER
thf(fact_261_START__REFINEG__TRANSFER,axiom,
! [D: set_val,C: set_val,A3: set_val] :
( ( refine654574649et_val @ D @ C )
=> ( ( ord_less_eq_set_val @ C @ A3 )
=> ( ( refine1893322264et_val @ C @ D )
=> ( ord_less_eq_set_val @ D @ A3 ) ) ) ) ).
% START_REFINEG_TRANSFER
thf(fact_262_lessThan__subset__iff,axiom,
! [X: val,Y2: val] :
( ( ord_less_eq_set_val @ ( set_ord_lessThan_val @ X ) @ ( set_ord_lessThan_val @ Y2 ) )
= ( ord_less_eq_val @ X @ Y2 ) ) ).
% lessThan_subset_iff
thf(fact_263_atLeast__subset__iff,axiom,
! [X: set_set_val,Y2: set_set_val] :
( ( ord_le1135847966et_val @ ( set_or1337784553et_val @ X ) @ ( set_or1337784553et_val @ Y2 ) )
= ( ord_le1742111550et_val @ Y2 @ X ) ) ).
% atLeast_subset_iff
thf(fact_264_atLeast__subset__iff,axiom,
! [X: set_val,Y2: set_val] :
( ( ord_le1742111550et_val @ ( set_or650197513et_val @ X ) @ ( set_or650197513et_val @ Y2 ) )
= ( ord_less_eq_set_val @ Y2 @ X ) ) ).
% atLeast_subset_iff
thf(fact_265_atLeast__subset__iff,axiom,
! [X: val,Y2: val] :
( ( ord_less_eq_set_val @ ( set_ord_atLeast_val @ X ) @ ( set_ord_atLeast_val @ Y2 ) )
= ( ord_less_eq_val @ Y2 @ X ) ) ).
% atLeast_subset_iff
thf(fact_266_atLeast__iff,axiom,
! [I: val,K: val] :
( ( member_val @ I @ ( set_ord_atLeast_val @ K ) )
= ( ord_less_eq_val @ K @ I ) ) ).
% atLeast_iff
thf(fact_267_atLeast__iff,axiom,
! [I: set_val,K: set_val] :
( ( member_set_val @ I @ ( set_or650197513et_val @ K ) )
= ( ord_less_eq_set_val @ K @ I ) ) ).
% atLeast_iff
thf(fact_268_Ioi__le__Ico,axiom,
! [A3: val] : ( ord_less_eq_set_val @ ( set_or1288365508an_val @ A3 ) @ ( set_ord_atLeast_val @ A3 ) ) ).
% Ioi_le_Ico
thf(fact_269_Ici__subset__Ioi__iff,axiom,
! [A3: val,B3: val] :
( ( ord_less_eq_set_val @ ( set_ord_atLeast_val @ A3 ) @ ( set_or1288365508an_val @ B3 ) )
= ( ord_less_val @ B3 @ A3 ) ) ).
% Ici_subset_Ioi_iff
thf(fact_270_Icc__subset__Ici__iff,axiom,
! [L: set_val,H: set_val,L2: set_val] :
( ( ord_le1742111550et_val @ ( set_or1554624499et_val @ L @ H ) @ ( set_or650197513et_val @ L2 ) )
= ( ~ ( ord_less_eq_set_val @ L @ H )
| ( ord_less_eq_set_val @ L2 @ L ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_271_Icc__subset__Ici__iff,axiom,
! [L: val,H: val,L2: val] :
( ( ord_less_eq_set_val @ ( set_or18208019st_val @ L @ H ) @ ( set_ord_atLeast_val @ L2 ) )
= ( ~ ( ord_less_eq_val @ L @ H )
| ( ord_less_eq_val @ L2 @ L ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_272_atMost__subset__iff,axiom,
! [X: set_val,Y2: set_val] :
( ( ord_le1742111550et_val @ ( set_or867701765et_val @ X ) @ ( set_or867701765et_val @ Y2 ) )
= ( ord_less_eq_set_val @ X @ Y2 ) ) ).
% atMost_subset_iff
thf(fact_273_atMost__subset__iff,axiom,
! [X: val,Y2: val] :
( ( ord_less_eq_set_val @ ( set_ord_atMost_val @ X ) @ ( set_ord_atMost_val @ Y2 ) )
= ( ord_less_eq_val @ X @ Y2 ) ) ).
% atMost_subset_iff
thf(fact_274_psubsetI,axiom,
! [A: set_val,B: set_val] :
( ( ord_less_eq_set_val @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_val @ A @ B ) ) ) ).
% psubsetI
thf(fact_275_ComplI,axiom,
! [C: val,A: set_val] :
( ~ ( member_val @ C @ A )
=> ( member_val @ C @ ( uminus1961393973et_val @ A ) ) ) ).
% ComplI
thf(fact_276_Compl__iff,axiom,
! [C: val,A: set_val] :
( ( member_val @ C @ ( uminus1961393973et_val @ A ) )
= ( ~ ( member_val @ C @ A ) ) ) ).
% Compl_iff
thf(fact_277_atLeastAtMost__iff,axiom,
! [I: val,L: val,U: val] :
( ( member_val @ I @ ( set_or18208019st_val @ L @ U ) )
= ( ( ord_less_eq_val @ L @ I )
& ( ord_less_eq_val @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_278_atLeastAtMost__iff,axiom,
! [I: set_val,L: set_val,U: set_val] :
( ( member_set_val @ I @ ( set_or1554624499et_val @ L @ U ) )
= ( ( ord_less_eq_set_val @ L @ I )
& ( ord_less_eq_set_val @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_279_Icc__eq__Icc,axiom,
! [L: set_val,H: set_val,L2: set_val,H2: set_val] :
( ( ( set_or1554624499et_val @ L @ H )
= ( set_or1554624499et_val @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_set_val @ L @ H )
& ~ ( ord_less_eq_set_val @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_280_atMost__iff,axiom,
! [I: val,K: val] :
( ( member_val @ I @ ( set_ord_atMost_val @ K ) )
= ( ord_less_eq_val @ I @ K ) ) ).
% atMost_iff
thf(fact_281_atMost__iff,axiom,
! [I: set_val,K: set_val] :
( ( member_set_val @ I @ ( set_or867701765et_val @ K ) )
= ( ord_less_eq_set_val @ I @ K ) ) ).
% atMost_iff
thf(fact_282_lessThan__iff,axiom,
! [I: val,K: val] :
( ( member_val @ I @ ( set_ord_lessThan_val @ K ) )
= ( ord_less_val @ I @ K ) ) ).
% lessThan_iff
thf(fact_283_Compl__anti__mono,axiom,
! [A: set_val,B: set_val] :
( ( ord_less_eq_set_val @ A @ B )
=> ( ord_less_eq_set_val @ ( uminus1961393973et_val @ B ) @ ( uminus1961393973et_val @ A ) ) ) ).
% Compl_anti_mono
thf(fact_284_Compl__subset__Compl__iff,axiom,
! [A: set_val,B: set_val] :
( ( ord_less_eq_set_val @ ( uminus1961393973et_val @ A ) @ ( uminus1961393973et_val @ B ) )
= ( ord_less_eq_set_val @ B @ A ) ) ).
% Compl_subset_Compl_iff
thf(fact_285_greaterThan__iff,axiom,
! [I: val,K: val] :
( ( member_val @ I @ ( set_or1288365508an_val @ K ) )
= ( ord_less_val @ K @ I ) ) ).
% greaterThan_iff
thf(fact_286_atLeastatMost__subset__iff,axiom,
! [A3: set_val,B3: set_val,C: set_val,D: set_val] :
( ( ord_le1742111550et_val @ ( set_or1554624499et_val @ A3 @ B3 ) @ ( set_or1554624499et_val @ C @ D ) )
= ( ~ ( ord_less_eq_set_val @ A3 @ B3 )
| ( ( ord_less_eq_set_val @ C @ A3 )
& ( ord_less_eq_set_val @ B3 @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_287_atLeastatMost__subset__iff,axiom,
! [A3: val,B3: val,C: val,D: val] :
( ( ord_less_eq_set_val @ ( set_or18208019st_val @ A3 @ B3 ) @ ( set_or18208019st_val @ C @ D ) )
= ( ~ ( ord_less_eq_val @ A3 @ B3 )
| ( ( ord_less_eq_val @ C @ A3 )
& ( ord_less_eq_val @ B3 @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_288_Icc__subset__Iic__iff,axiom,
! [L: set_val,H: set_val,H2: set_val] :
( ( ord_le1742111550et_val @ ( set_or1554624499et_val @ L @ H ) @ ( set_or867701765et_val @ H2 ) )
= ( ~ ( ord_less_eq_set_val @ L @ H )
| ( ord_less_eq_set_val @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_289_Icc__subset__Iic__iff,axiom,
! [L: val,H: val,H2: val] :
( ( ord_less_eq_set_val @ ( set_or18208019st_val @ L @ H ) @ ( set_ord_atMost_val @ H2 ) )
= ( ~ ( ord_less_eq_val @ L @ H )
| ( ord_less_eq_val @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_290_atLeastatMost__psubset__iff,axiom,
! [A3: set_val,B3: set_val,C: set_val,D: set_val] :
( ( ord_less_set_set_val @ ( set_or1554624499et_val @ A3 @ B3 ) @ ( set_or1554624499et_val @ C @ D ) )
= ( ( ~ ( ord_less_eq_set_val @ A3 @ B3 )
| ( ( ord_less_eq_set_val @ C @ A3 )
& ( ord_less_eq_set_val @ B3 @ D )
& ( ( ord_less_set_val @ C @ A3 )
| ( ord_less_set_val @ B3 @ D ) ) ) )
& ( ord_less_eq_set_val @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_291_ComplD,axiom,
! [C: val,A: set_val] :
( ( member_val @ C @ ( uminus1961393973et_val @ A ) )
=> ~ ( member_val @ C @ A ) ) ).
% ComplD
thf(fact_292_Iic__subset__Iio__iff,axiom,
! [A3: val,B3: val] :
( ( ord_less_eq_set_val @ ( set_ord_atMost_val @ A3 ) @ ( set_ord_lessThan_val @ B3 ) )
= ( ord_less_val @ A3 @ B3 ) ) ).
% Iic_subset_Iio_iff
thf(fact_293_leD,axiom,
! [Y2: set_val,X: set_val] :
( ( ord_less_eq_set_val @ Y2 @ X )
=> ~ ( ord_less_set_val @ X @ Y2 ) ) ).
% leD
thf(fact_294_le__less,axiom,
( ord_less_eq_set_val
= ( ^ [X3: set_val,Y4: set_val] :
( ( ord_less_set_val @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% le_less
thf(fact_295_less__le,axiom,
( ord_less_set_val
= ( ^ [X3: set_val,Y4: set_val] :
( ( ord_less_eq_set_val @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% less_le
thf(fact_296_order__le__less__subst2,axiom,
! [A3: set_val,B3: set_val,F: set_val > set_val,C: set_val] :
( ( ord_less_eq_set_val @ A3 @ B3 )
=> ( ( ord_less_set_val @ ( F @ B3 ) @ C )
=> ( ! [X2: set_val,Y3: set_val] :
( ( ord_less_eq_set_val @ X2 @ Y3 )
=> ( ord_less_eq_set_val @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_val @ ( F @ A3 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_297_order__less__le__subst1,axiom,
! [A3: set_val,F: set_val > set_val,B3: set_val,C: set_val] :
( ( ord_less_set_val @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_val @ B3 @ C )
=> ( ! [X2: set_val,Y3: set_val] :
( ( ord_less_eq_set_val @ X2 @ Y3 )
=> ( ord_less_eq_set_val @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_val @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_298_le__neq__trans,axiom,
! [A3: set_val,B3: set_val] :
( ( ord_less_eq_set_val @ A3 @ B3 )
=> ( ( A3 != B3 )
=> ( ord_less_set_val @ A3 @ B3 ) ) ) ).
% le_neq_trans
thf(fact_299_antisym__conv1,axiom,
! [X: set_val,Y2: set_val] :
( ~ ( ord_less_set_val @ X @ Y2 )
=> ( ( ord_less_eq_set_val @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_300_antisym__conv2,axiom,
! [X: set_val,Y2: set_val] :
( ( ord_less_eq_set_val @ X @ Y2 )
=> ( ( ~ ( ord_less_set_val @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_301_less__imp__le,axiom,
! [X: set_val,Y2: set_val] :
( ( ord_less_set_val @ X @ Y2 )
=> ( ord_less_eq_set_val @ X @ Y2 ) ) ).
% less_imp_le
thf(fact_302_le__less__trans,axiom,
! [X: set_val,Y2: set_val,Z2: set_val] :
( ( ord_less_eq_set_val @ X @ Y2 )
=> ( ( ord_less_set_val @ Y2 @ Z2 )
=> ( ord_less_set_val @ X @ Z2 ) ) ) ).
% le_less_trans
thf(fact_303_less__le__trans,axiom,
! [X: set_val,Y2: set_val,Z2: set_val] :
( ( ord_less_set_val @ X @ Y2 )
=> ( ( ord_less_eq_set_val @ Y2 @ Z2 )
=> ( ord_less_set_val @ X @ Z2 ) ) ) ).
% less_le_trans
thf(fact_304_le__imp__less__or__eq,axiom,
! [X: set_val,Y2: set_val] :
( ( ord_less_eq_set_val @ X @ Y2 )
=> ( ( ord_less_set_val @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% le_imp_less_or_eq
thf(fact_305_less__le__not__le,axiom,
( ord_less_set_val
= ( ^ [X3: set_val,Y4: set_val] :
( ( ord_less_eq_set_val @ X3 @ Y4 )
& ~ ( ord_less_eq_set_val @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_306_order_Ostrict__trans1,axiom,
! [A3: set_val,B3: set_val,C: set_val] :
( ( ord_less_eq_set_val @ A3 @ B3 )
=> ( ( ord_less_set_val @ B3 @ C )
=> ( ord_less_set_val @ A3 @ C ) ) ) ).
% order.strict_trans1
thf(fact_307_order_Ostrict__trans2,axiom,
! [A3: set_val,B3: set_val,C: set_val] :
( ( ord_less_set_val @ A3 @ B3 )
=> ( ( ord_less_eq_set_val @ B3 @ C )
=> ( ord_less_set_val @ A3 @ C ) ) ) ).
% order.strict_trans2
thf(fact_308_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_val
= ( ^ [A4: set_val,B4: set_val] :
( ( ord_less_set_val @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_309_order_Ostrict__iff__order,axiom,
( ord_less_set_val
= ( ^ [A4: set_val,B4: set_val] :
( ( ord_less_eq_set_val @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_310_dual__order_Ostrict__trans1,axiom,
! [B3: set_val,A3: set_val,C: set_val] :
( ( ord_less_eq_set_val @ B3 @ A3 )
=> ( ( ord_less_set_val @ C @ B3 )
=> ( ord_less_set_val @ C @ A3 ) ) ) ).
% dual_order.strict_trans1
thf(fact_311_dual__order_Ostrict__trans2,axiom,
! [B3: set_val,A3: set_val,C: set_val] :
( ( ord_less_set_val @ B3 @ A3 )
=> ( ( ord_less_eq_set_val @ C @ B3 )
=> ( ord_less_set_val @ C @ A3 ) ) ) ).
% dual_order.strict_trans2
thf(fact_312_order_Ostrict__implies__order,axiom,
! [A3: set_val,B3: set_val] :
( ( ord_less_set_val @ A3 @ B3 )
=> ( ord_less_eq_set_val @ A3 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_313_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_val
= ( ^ [B4: set_val,A4: set_val] :
( ( ord_less_set_val @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_314_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_val
= ( ^ [B4: set_val,A4: set_val] :
( ( ord_less_eq_set_val @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_315_dual__order_Ostrict__implies__order,axiom,
! [B3: set_val,A3: set_val] :
( ( ord_less_set_val @ B3 @ A3 )
=> ( ord_less_eq_set_val @ B3 @ A3 ) ) ).
% dual_order.strict_implies_order
thf(fact_316_order_Onot__eq__order__implies__strict,axiom,
! [A3: set_val,B3: set_val] :
( ( A3 != B3 )
=> ( ( ord_less_eq_set_val @ A3 @ B3 )
=> ( ord_less_set_val @ A3 @ B3 ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_317_psubsetE,axiom,
! [A: set_val,B: set_val] :
( ( ord_less_set_val @ A @ B )
=> ~ ( ( ord_less_eq_set_val @ A @ B )
=> ( ord_less_eq_set_val @ B @ A ) ) ) ).
% psubsetE
thf(fact_318_psubset__eq,axiom,
( ord_less_set_val
= ( ^ [A2: set_val,B2: set_val] :
( ( ord_less_eq_set_val @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% psubset_eq
thf(fact_319_psubset__imp__subset,axiom,
! [A: set_val,B: set_val] :
( ( ord_less_set_val @ A @ B )
=> ( ord_less_eq_set_val @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_320_psubset__subset__trans,axiom,
! [A: set_val,B: set_val,C2: set_val] :
( ( ord_less_set_val @ A @ B )
=> ( ( ord_less_eq_set_val @ B @ C2 )
=> ( ord_less_set_val @ A @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_321_subset__not__subset__eq,axiom,
( ord_less_set_val
= ( ^ [A2: set_val,B2: set_val] :
( ( ord_less_eq_set_val @ A2 @ B2 )
& ~ ( ord_less_eq_set_val @ B2 @ A2 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_322_subset__psubset__trans,axiom,
! [A: set_val,B: set_val,C2: set_val] :
( ( ord_less_eq_set_val @ A @ B )
=> ( ( ord_less_set_val @ B @ C2 )
=> ( ord_less_set_val @ A @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_323_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_val
= ( ^ [A2: set_val,B2: set_val] :
( ( ord_less_set_val @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_324_compl__le__compl__iff,axiom,
! [X: set_val,Y2: set_val] :
( ( ord_less_eq_set_val @ ( uminus1961393973et_val @ X ) @ ( uminus1961393973et_val @ Y2 ) )
= ( ord_less_eq_set_val @ Y2 @ X ) ) ).
% compl_le_compl_iff
thf(fact_325_compl__mono,axiom,
! [X: set_val,Y2: set_val] :
( ( ord_less_eq_set_val @ X @ Y2 )
=> ( ord_less_eq_set_val @ ( uminus1961393973et_val @ Y2 ) @ ( uminus1961393973et_val @ X ) ) ) ).
% compl_mono
thf(fact_326_compl__le__swap1,axiom,
! [Y2: set_val,X: set_val] :
( ( ord_less_eq_set_val @ Y2 @ ( uminus1961393973et_val @ X ) )
=> ( ord_less_eq_set_val @ X @ ( uminus1961393973et_val @ Y2 ) ) ) ).
% compl_le_swap1
thf(fact_327_compl__le__swap2,axiom,
! [Y2: set_val,X: set_val] :
( ( ord_less_eq_set_val @ ( uminus1961393973et_val @ Y2 ) @ X )
=> ( ord_less_eq_set_val @ ( uminus1961393973et_val @ X ) @ Y2 ) ) ).
% compl_le_swap2
thf(fact_328_psubsetD,axiom,
! [A: set_val,B: set_val,C: val] :
( ( ord_less_set_val @ A @ B )
=> ( ( member_val @ C @ A )
=> ( member_val @ C @ B ) ) ) ).
% psubsetD
thf(fact_329_UnCI,axiom,
! [C: val,B: set_val,A: set_val] :
( ( ~ ( member_val @ C @ B )
=> ( member_val @ C @ A ) )
=> ( member_val @ C @ ( sup_sup_set_val @ A @ B ) ) ) ).
% UnCI
thf(fact_330_Un__iff,axiom,
! [C: val,A: set_val,B: set_val] :
( ( member_val @ C @ ( sup_sup_set_val @ A @ B ) )
= ( ( member_val @ C @ A )
| ( member_val @ C @ B ) ) ) ).
% Un_iff
thf(fact_331_le__sup__iff,axiom,
! [X: set_val,Y2: set_val,Z2: set_val] :
( ( ord_less_eq_set_val @ ( sup_sup_set_val @ X @ Y2 ) @ Z2 )
= ( ( ord_less_eq_set_val @ X @ Z2 )
& ( ord_less_eq_set_val @ Y2 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_332_sup_Obounded__iff,axiom,
! [B3: set_val,C: set_val,A3: set_val] :
( ( ord_less_eq_set_val @ ( sup_sup_set_val @ B3 @ C ) @ A3 )
= ( ( ord_less_eq_set_val @ B3 @ A3 )
& ( ord_less_eq_set_val @ C @ A3 ) ) ) ).
% sup.bounded_iff
thf(fact_333_Un__subset__iff,axiom,
! [A: set_val,B: set_val,C2: set_val] :
( ( ord_less_eq_set_val @ ( sup_sup_set_val @ A @ B ) @ C2 )
= ( ( ord_less_eq_set_val @ A @ C2 )
& ( ord_less_eq_set_val @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_334_greaterThanLessThan__iff,axiom,
! [I: val,L: val,U: val] :
( ( member_val @ I @ ( set_or2126567818an_val @ L @ U ) )
= ( ( ord_less_val @ L @ I )
& ( ord_less_val @ I @ U ) ) ) ).
% greaterThanLessThan_iff
thf(fact_335_sup_OcoboundedI2,axiom,
! [C: set_val,B3: set_val,A3: set_val] :
( ( ord_less_eq_set_val @ C @ B3 )
=> ( ord_less_eq_set_val @ C @ ( sup_sup_set_val @ A3 @ B3 ) ) ) ).
% sup.coboundedI2
thf(fact_336_sup_OcoboundedI1,axiom,
! [C: set_val,A3: set_val,B3: set_val] :
( ( ord_less_eq_set_val @ C @ A3 )
=> ( ord_less_eq_set_val @ C @ ( sup_sup_set_val @ A3 @ B3 ) ) ) ).
% sup.coboundedI1
thf(fact_337_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_val
= ( ^ [A4: set_val,B4: set_val] :
( ( sup_sup_set_val @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_338_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_val
= ( ^ [B4: set_val,A4: set_val] :
( ( sup_sup_set_val @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_339_sup_Ocobounded2,axiom,
! [B3: set_val,A3: set_val] : ( ord_less_eq_set_val @ B3 @ ( sup_sup_set_val @ A3 @ B3 ) ) ).
% sup.cobounded2
thf(fact_340_sup_Ocobounded1,axiom,
! [A3: set_val,B3: set_val] : ( ord_less_eq_set_val @ A3 @ ( sup_sup_set_val @ A3 @ B3 ) ) ).
% sup.cobounded1
thf(fact_341_sup_Oorder__iff,axiom,
( ord_less_eq_set_val
= ( ^ [B4: set_val,A4: set_val] :
( A4
= ( sup_sup_set_val @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_342_sup_OboundedI,axiom,
! [B3: set_val,A3: set_val,C: set_val] :
( ( ord_less_eq_set_val @ B3 @ A3 )
=> ( ( ord_less_eq_set_val @ C @ A3 )
=> ( ord_less_eq_set_val @ ( sup_sup_set_val @ B3 @ C ) @ A3 ) ) ) ).
% sup.boundedI
thf(fact_343_sup_OboundedE,axiom,
! [B3: set_val,C: set_val,A3: set_val] :
( ( ord_less_eq_set_val @ ( sup_sup_set_val @ B3 @ C ) @ A3 )
=> ~ ( ( ord_less_eq_set_val @ B3 @ A3 )
=> ~ ( ord_less_eq_set_val @ C @ A3 ) ) ) ).
% sup.boundedE
thf(fact_344_sup__absorb2,axiom,
! [X: set_val,Y2: set_val] :
( ( ord_less_eq_set_val @ X @ Y2 )
=> ( ( sup_sup_set_val @ X @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_345_sup__absorb1,axiom,
! [Y2: set_val,X: set_val] :
( ( ord_less_eq_set_val @ Y2 @ X )
=> ( ( sup_sup_set_val @ X @ Y2 )
= X ) ) ).
% sup_absorb1
thf(fact_346_sup_Oabsorb2,axiom,
! [A3: set_val,B3: set_val] :
( ( ord_less_eq_set_val @ A3 @ B3 )
=> ( ( sup_sup_set_val @ A3 @ B3 )
= B3 ) ) ).
% sup.absorb2
thf(fact_347_sup_Oabsorb1,axiom,
! [B3: set_val,A3: set_val] :
( ( ord_less_eq_set_val @ B3 @ A3 )
=> ( ( sup_sup_set_val @ A3 @ B3 )
= A3 ) ) ).
% sup.absorb1
thf(fact_348_sup__unique,axiom,
! [F: set_val > set_val > set_val,X: set_val,Y2: set_val] :
( ! [X2: set_val,Y3: set_val] : ( ord_less_eq_set_val @ X2 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: set_val,Y3: set_val] : ( ord_less_eq_set_val @ Y3 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: set_val,Y3: set_val,Z3: set_val] :
( ( ord_less_eq_set_val @ Y3 @ X2 )
=> ( ( ord_less_eq_set_val @ Z3 @ X2 )
=> ( ord_less_eq_set_val @ ( F @ Y3 @ Z3 ) @ X2 ) ) )
=> ( ( sup_sup_set_val @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_349_sup_OorderI,axiom,
! [A3: set_val,B3: set_val] :
( ( A3
= ( sup_sup_set_val @ A3 @ B3 ) )
=> ( ord_less_eq_set_val @ B3 @ A3 ) ) ).
% sup.orderI
% Helper facts (9)
thf(help_If_2_1_If_001tf__val_T,axiom,
! [X: val,Y2: val] :
( ( if_val @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001tf__val_T,axiom,
! [X: val,Y2: val] :
( ( if_val @ $true @ X @ Y2 )
= X ) ).
thf(help_If_2_1_If_001t__Set__Oset_Itf__val_J_T,axiom,
! [X: set_val,Y2: set_val] :
( ( if_set_val @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Set__Oset_Itf__val_J_T,axiom,
! [X: set_val,Y2: set_val] :
( ( if_set_val @ $true @ X @ Y2 )
= X ) ).
thf(help_If_2_1_If_001_062_I_Eo_Mt__Set__Oset_Itf__val_J_J_T,axiom,
! [X: $o > set_val,Y2: $o > set_val] :
( ( if_o_set_val @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001_062_I_Eo_Mt__Set__Oset_Itf__val_J_J_T,axiom,
! [X: $o > set_val,Y2: $o > set_val] :
( ( if_o_set_val @ $true @ X @ Y2 )
= X ) ).
thf(help_If_3_1_If_001t__Set__Oset_It__Set__Oset_Itf__val_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Set__Oset_Itf__val_J_J_T,axiom,
! [X: set_set_val,Y2: set_set_val] :
( ( if_set_set_val @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Set__Oset_Itf__val_J_J_T,axiom,
! [X: set_set_val,Y2: set_set_val] :
( ( if_set_set_val @ $true @ X @ Y2 )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_eq_set_val @ scc @ p ).
%------------------------------------------------------------------------------