TPTP Problem File: COM196^1.p
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%------------------------------------------------------------------------------
% File : COM196^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Computing Theory
% Problem : Grammars and languages 1357
% Version : [Bla16] axioms : Especial.
% English :
% Refs : [BH+14] Blanchette et al. (2014), Truly Modular (Co)datatypes
% : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% Source : [Bla16]
% Names : gram_lang__1357.p [Bla16]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.25 v7.5.0, 0.33 v7.2.0, 0.50 v7.1.0
% Syntax : Number of formulae : 326 ( 79 unt; 41 typ; 0 def)
% Number of atoms : 913 ( 176 equ; 0 cnn)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 3609 ( 87 ~; 16 |; 48 &;2971 @)
% ( 0 <=>; 487 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 8 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 166 ( 166 >; 0 *; 0 +; 0 <<)
% Number of symbols : 40 ( 39 usr; 1 con; 0-4 aty)
% Number of variables : 999 ( 50 ^; 872 !; 43 ?; 999 :)
% ( 34 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:48:39.061
%------------------------------------------------------------------------------
%----Could-be-implicit typings (4)
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_DTree_OT,type,
t: $tType ).
thf(ty_t_DTree_ON,type,
n: $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
%----Explicit typings (37)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Otop,type,
top:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Ono__bot,type,
no_bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Ono__top,type,
no_top:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder__bot,type,
order_bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder__top,type,
order_top:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Odense__order,type,
dense_order:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
comple187826305attice:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Complete__Lattices_Ocomplete__linorder,type,
comple1035589618norder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
condit1656338222tinuum:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__lattice,type,
condit378418413attice:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
condit1037483654norder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_Complete__Lattices_OSup__class_OSup,type,
complete_Sup_Sup:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OL,type,
gram_L1451583628elle_L: ( set @ n ) > n > ( set @ ( set @ t ) ) ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OLL,type,
gram_L861977280lle_LL: ( set @ n ) > n > ( set @ ( set @ t ) ) ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OLr,type,
gram_L861977318lle_Lr: ( set @ n ) > n > ( set @ ( set @ t ) ) ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_Oleqv,type,
gram_L1456083582e_leqv:
!>[A: $tType] : ( ( set @ ( set @ A ) ) > ( set @ ( set @ A ) ) > $o ) ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_Osubs,type,
gram_L608943123e_subs:
!>[A: $tType] : ( ( set @ ( set @ A ) ) > ( set @ ( set @ A ) ) > $o ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oorder__class_Oantimono,type,
order_antimono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Orderings_Oordering__top,type,
ordering_top:
!>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > A > $o ) ).
thf(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_OPow,type,
pow:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Set_Ois__empty,type,
is_empty:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_ts,type,
ts: n ).
%----Relevant facts (256)
thf(fact_0_Lr__leqv__L,axiom,
! [Ts: n] : ( gram_L1456083582e_leqv @ t @ ( gram_L861977318lle_Lr @ ( top_top @ ( set @ n ) ) @ Ts ) @ ( gram_L1451583628elle_L @ ( top_top @ ( set @ n ) ) @ Ts ) ) ).
% Lr_leqv_L
thf(fact_1_LL__leqv__L,axiom,
! [Ts: n] : ( gram_L1456083582e_leqv @ t @ ( gram_L861977280lle_LL @ ( top_top @ ( set @ n ) ) @ Ts ) @ ( gram_L1451583628elle_L @ ( top_top @ ( set @ n ) ) @ Ts ) ) ).
% LL_leqv_L
thf(fact_2_leqv__Sym,axiom,
! [A: $tType] :
( ( gram_L1456083582e_leqv @ A )
= ( ^ [L1: set @ ( set @ A ),L2: set @ ( set @ A )] : ( gram_L1456083582e_leqv @ A @ L2 @ L1 ) ) ) ).
% leqv_Sym
thf(fact_3_leqv__sym,axiom,
! [A: $tType,L12: set @ ( set @ A ),L22: set @ ( set @ A )] :
( ( gram_L1456083582e_leqv @ A @ L12 @ L22 )
=> ( gram_L1456083582e_leqv @ A @ L22 @ L12 ) ) ).
% leqv_sym
thf(fact_4_leqv__refl,axiom,
! [A: $tType,L12: set @ ( set @ A )] : ( gram_L1456083582e_leqv @ A @ L12 @ L12 ) ).
% leqv_refl
thf(fact_5_leqv__trans,axiom,
! [A: $tType,L12: set @ ( set @ A ),L22: set @ ( set @ A ),L3: set @ ( set @ A )] :
( ( gram_L1456083582e_leqv @ A @ L12 @ L22 )
=> ( ( gram_L1456083582e_leqv @ A @ L22 @ L3 )
=> ( gram_L1456083582e_leqv @ A @ L12 @ L3 ) ) ) ).
% leqv_trans
thf(fact_6_UNIV__I,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_7_iso__tuple__UNIV__I,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% iso_tuple_UNIV_I
thf(fact_8_top__apply,axiom,
! [C: $tType,D: $tType] :
( ( top @ C @ ( type2 @ C ) )
=> ( ( top_top @ ( D > C ) )
= ( ^ [X2: D] : ( top_top @ C ) ) ) ) ).
% top_apply
thf(fact_9_Lr__subs__L,axiom,
! [Ts: n] : ( gram_L608943123e_subs @ t @ ( gram_L861977318lle_Lr @ ( top_top @ ( set @ n ) ) @ Ts ) @ ( gram_L1451583628elle_L @ ( top_top @ ( set @ n ) ) @ Ts ) ) ).
% Lr_subs_L
thf(fact_10_UNIV__eq__I,axiom,
! [A: $tType,A2: set @ A] :
( ! [X3: A] : ( member @ A @ X3 @ A2 )
=> ( ( top_top @ ( set @ A ) )
= A2 ) ) ).
% UNIV_eq_I
thf(fact_11_UNIV__witness,axiom,
! [A: $tType] :
? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_12_Lr__LL,axiom,
! [Ns: set @ n,N: n] : ( ord_less_eq @ ( set @ ( set @ t ) ) @ ( gram_L861977318lle_Lr @ Ns @ N ) @ ( gram_L861977280lle_LL @ Ns @ N ) ) ).
% Lr_LL
thf(fact_13_leqv__def,axiom,
! [A: $tType] :
( ( gram_L1456083582e_leqv @ A )
= ( ^ [L1: set @ ( set @ A ),L2: set @ ( set @ A )] :
( ( gram_L608943123e_subs @ A @ L1 @ L2 )
& ( gram_L608943123e_subs @ A @ L2 @ L1 ) ) ) ) ).
% leqv_def
thf(fact_14_subs__leqv,axiom,
! [A: $tType,L12: set @ ( set @ A ),L22: set @ ( set @ A )] :
( ( gram_L1456083582e_leqv @ A @ L12 @ L22 )
=> ( gram_L608943123e_subs @ A @ L12 @ L22 ) ) ).
% subs_leqv
thf(fact_15_subs__leqv__sym,axiom,
! [A: $tType,L12: set @ ( set @ A ),L22: set @ ( set @ A )] :
( ( gram_L1456083582e_leqv @ A @ L12 @ L22 )
=> ( gram_L608943123e_subs @ A @ L22 @ L12 ) ) ).
% subs_leqv_sym
thf(fact_16_Sup__UNIV,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ( ( complete_Sup_Sup @ A @ ( top_top @ ( set @ A ) ) )
= ( top_top @ A ) ) ) ).
% Sup_UNIV
thf(fact_17_LL__L,axiom,
! [Ns: set @ n,N: n] : ( ord_less_eq @ ( set @ ( set @ t ) ) @ ( gram_L861977280lle_LL @ Ns @ N ) @ ( gram_L1451583628elle_L @ Ns @ N ) ) ).
% LL_L
thf(fact_18_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_19_subsetI,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ( member @ A @ X3 @ B2 ) )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% subsetI
thf(fact_20_subset__antisym,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_21_set__mp,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ X @ A2 )
=> ( member @ A @ X @ B2 ) ) ) ).
% set_mp
thf(fact_22_in__mono,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ X @ A2 )
=> ( member @ A @ X @ B2 ) ) ) ).
% in_mono
thf(fact_23_subsetD,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ C2 @ A2 )
=> ( member @ A @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_24_subsetCE,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ C2 @ A2 )
=> ( member @ A @ C2 @ B2 ) ) ) ).
% subsetCE
thf(fact_25_equalityE,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_26_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A3: set @ A,B3: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( member @ A @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_27_equalityD1,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_28_equalityD2,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_29_set__rev__mp,axiom,
! [A: $tType,X: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ X @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( member @ A @ X @ B2 ) ) ) ).
% set_rev_mp
thf(fact_30_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A3: set @ A,B3: set @ A] :
! [T: A] :
( ( member @ A @ T @ A3 )
=> ( member @ A @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_31_rev__subsetD,axiom,
! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( member @ A @ C2 @ B2 ) ) ) ).
% rev_subsetD
thf(fact_32_subset__refl,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% subset_refl
thf(fact_33_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_34_subset__trans,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_35_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funD
thf(fact_36_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funE
thf(fact_37_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B] :
( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_38_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y: set @ A,Z: set @ A] : Y = Z )
= ( ^ [A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
& ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_39_contra__subsetD,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ~ ( member @ A @ C2 @ B2 )
=> ~ ( member @ A @ C2 @ A2 ) ) ) ).
% contra_subsetD
thf(fact_40_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B] :
! [X2: A] : ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% le_fun_def
thf(fact_41_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_42_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,F: B > A,B4: B,C2: B] :
( ( ord_less_eq @ A @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_eq @ B @ B4 @ C2 )
=> ( ! [X3: B,Y2: B] :
( ( ord_less_eq @ B @ X3 @ Y2 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_43_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C @ ( type2 @ C ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,B4: A,F: A > C,C2: C] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ C @ ( F @ B4 ) @ C2 )
=> ( ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ C @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_44_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A4: A,F: B > A,B4: B,C2: B] :
( ( A4
= ( F @ B4 ) )
=> ( ( ord_less_eq @ B @ B4 @ C2 )
=> ( ! [X3: B,Y2: B] :
( ( ord_less_eq @ B @ X3 @ Y2 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A4: A,P: A > $o] :
( ( member @ A @ A4 @ ( collect @ A @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A4: A,B4: A,F: A > B,C2: B] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ B @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_50_Union__mono,axiom,
! [A: $tType,A2: set @ ( set @ A ),B2: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ A2 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B2 ) ) ) ).
% Union_mono
thf(fact_51_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y: A,Z: A] : Y = Z )
= ( ^ [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
& ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ) ) ).
% eq_iff
thf(fact_52_Union__least,axiom,
! [A: $tType,A2: set @ ( set @ A ),C3: set @ A] :
( ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ A2 )
=> ( ord_less_eq @ ( set @ A ) @ X4 @ C3 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) @ C3 ) ) ).
% Union_least
thf(fact_53_Union__upper,axiom,
! [A: $tType,B2: set @ A,A2: set @ ( set @ A )] :
( ( member @ ( set @ A ) @ B2 @ A2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) ) ) ).
% Union_upper
thf(fact_54_antisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ( ord_less_eq @ A @ X @ Y4 )
=> ( ( ord_less_eq @ A @ Y4 @ X )
=> ( X = Y4 ) ) ) ) ).
% antisym
thf(fact_55_linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ( ord_less_eq @ A @ X @ Y4 )
| ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% linear
thf(fact_56_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ( X = Y4 )
=> ( ord_less_eq @ A @ X @ Y4 ) ) ) ).
% eq_refl
thf(fact_57_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ~ ( ord_less_eq @ A @ X @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% le_cases
thf(fact_58_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ A @ B4 @ C2 )
=> ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% order.trans
thf(fact_59_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A,Z2: A] :
( ( ( ord_less_eq @ A @ X @ Y4 )
=> ~ ( ord_less_eq @ A @ Y4 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y4 @ X )
=> ~ ( ord_less_eq @ A @ X @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y4 ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y4 )
=> ~ ( ord_less_eq @ A @ Y4 @ X ) )
=> ( ( ( ord_less_eq @ A @ Y4 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X )
=> ~ ( ord_less_eq @ A @ X @ Y4 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_60_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [Y4: A,X: A] :
( ( ord_less_eq @ A @ Y4 @ X )
=> ( ( ord_less_eq @ A @ X @ Y4 )
= ( X = Y4 ) ) ) ) ).
% antisym_conv
thf(fact_61_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,C2: A] :
( ( A4 = B4 )
=> ( ( ord_less_eq @ A @ B4 @ C2 )
=> ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_62_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( B4 = C2 )
=> ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_63_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ A @ B4 @ A4 )
=> ( A4 = B4 ) ) ) ) ).
% order_class.order.antisym
thf(fact_64_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A,Z2: A] :
( ( ord_less_eq @ A @ X @ Y4 )
=> ( ( ord_less_eq @ A @ Y4 @ Z2 )
=> ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% order_trans
thf(fact_65_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A] : ( ord_less_eq @ A @ A4 @ A4 ) ) ).
% dual_order.refl
thf(fact_66_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,A4: A,B4: A] :
( ! [A5: A,B5: A] :
( ( ord_less_eq @ A @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: A,B5: A] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A4 @ B4 ) ) ) ) ).
% linorder_wlog
thf(fact_67_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B4: A,A4: A,C2: A] :
( ( ord_less_eq @ A @ B4 @ A4 )
=> ( ( ord_less_eq @ A @ C2 @ B4 )
=> ( ord_less_eq @ A @ C2 @ A4 ) ) ) ) ).
% dual_order.trans
thf(fact_68_subs__def,axiom,
! [A: $tType] :
( ( gram_L608943123e_subs @ A )
= ( ^ [L1: set @ ( set @ A ),L2: set @ ( set @ A )] :
! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ L2 )
=> ? [Y3: set @ A] :
( ( member @ ( set @ A ) @ Y3 @ L1 )
& ( ord_less_eq @ ( set @ A ) @ Y3 @ X2 ) ) ) ) ) ).
% subs_def
thf(fact_69_incl__subs,axiom,
! [A: $tType,L22: set @ ( set @ A ),L12: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ L22 @ L12 )
=> ( gram_L608943123e_subs @ A @ L12 @ L22 ) ) ).
% incl_subs
thf(fact_70_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B4: A,A4: A] :
( ( ord_less_eq @ A @ B4 @ A4 )
=> ( ( ord_less_eq @ A @ A4 @ B4 )
=> ( A4 = B4 ) ) ) ) ).
% dual_order.antisym
thf(fact_71_Sup__eqI,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [A2: set @ A,X: A] :
( ! [Y2: A] :
( ( member @ A @ Y2 @ A2 )
=> ( ord_less_eq @ A @ Y2 @ X ) )
=> ( ! [Y2: A] :
( ! [Z3: A] :
( ( member @ A @ Z3 @ A2 )
=> ( ord_less_eq @ A @ Z3 @ Y2 ) )
=> ( ord_less_eq @ A @ X @ Y2 ) )
=> ( ( complete_Sup_Sup @ A @ A2 )
= X ) ) ) ) ).
% Sup_eqI
thf(fact_72_Sup__mono,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [A2: set @ A,B2: set @ A] :
( ! [A5: A] :
( ( member @ A @ A5 @ A2 )
=> ? [X5: A] :
( ( member @ A @ X5 @ B2 )
& ( ord_less_eq @ A @ A5 @ X5 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ).
% Sup_mono
thf(fact_73_Sup__least,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [A2: set @ A,Z2: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ( ord_less_eq @ A @ X3 @ Z2 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ Z2 ) ) ) ).
% Sup_least
thf(fact_74_Sup__upper,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [X: A,A2: set @ A] :
( ( member @ A @ X @ A2 )
=> ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ A2 ) ) ) ) ).
% Sup_upper
thf(fact_75_Sup__le__iff,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [A2: set @ A,B4: A] :
( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ B4 )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A2 )
=> ( ord_less_eq @ A @ X2 @ B4 ) ) ) ) ) ).
% Sup_le_iff
thf(fact_76_Sup__upper2,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [U: A,A2: set @ A,V: A] :
( ( member @ A @ U @ A2 )
=> ( ( ord_less_eq @ A @ V @ U )
=> ( ord_less_eq @ A @ V @ ( complete_Sup_Sup @ A @ A2 ) ) ) ) ) ).
% Sup_upper2
thf(fact_77_Sup__subset__mono,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ).
% Sup_subset_mono
thf(fact_78_subs__trans,axiom,
! [A: $tType,L12: set @ ( set @ A ),L22: set @ ( set @ A ),L3: set @ ( set @ A )] :
( ( gram_L608943123e_subs @ A @ L12 @ L22 )
=> ( ( gram_L608943123e_subs @ A @ L22 @ L3 )
=> ( gram_L608943123e_subs @ A @ L12 @ L3 ) ) ) ).
% subs_trans
thf(fact_79_subs__refl,axiom,
! [A: $tType,L12: set @ ( set @ A )] : ( gram_L608943123e_subs @ A @ L12 @ L12 ) ).
% subs_refl
thf(fact_80_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_81_Union__UNIV,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Union_UNIV
thf(fact_82_top_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_top @ A @ ( type2 @ A ) )
=> ! [A4: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A4 )
=> ( A4
= ( top_top @ A ) ) ) ) ).
% top.extremum_uniqueI
thf(fact_83_top_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_top @ A @ ( type2 @ A ) )
=> ! [A4: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A4 )
= ( A4
= ( top_top @ A ) ) ) ) ).
% top.extremum_unique
thf(fact_84_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A @ ( type2 @ A ) )
=> ! [A4: A] : ( ord_less_eq @ A @ A4 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_85_subset__UNIV,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) ) ).
% subset_UNIV
thf(fact_86_Lr__incl__L,axiom,
! [Ns: set @ n,Ts: n] : ( ord_less_eq @ ( set @ ( set @ t ) ) @ ( gram_L861977318lle_Lr @ Ns @ Ts ) @ ( gram_L1451583628elle_L @ Ns @ Ts ) ) ).
% Lr_incl_L
thf(fact_87_cSup__eq,axiom,
! [A: $tType] :
( ( ( condit378418413attice @ A @ ( type2 @ A ) )
& ( no_bot @ A @ ( type2 @ A ) ) )
=> ! [X6: set @ A,A4: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ( ord_less_eq @ A @ X3 @ A4 ) )
=> ( ! [Y2: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ X6 )
=> ( ord_less_eq @ A @ X5 @ Y2 ) )
=> ( ord_less_eq @ A @ A4 @ Y2 ) )
=> ( ( complete_Sup_Sup @ A @ X6 )
= A4 ) ) ) ) ).
% cSup_eq
thf(fact_88_cSup__eq__maximum,axiom,
! [A: $tType] :
( ( condit378418413attice @ A @ ( type2 @ A ) )
=> ! [Z2: A,X6: set @ A] :
( ( member @ A @ Z2 @ X6 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ( ord_less_eq @ A @ X3 @ Z2 ) )
=> ( ( complete_Sup_Sup @ A @ X6 )
= Z2 ) ) ) ) ).
% cSup_eq_maximum
thf(fact_89_top__empty__eq,axiom,
! [A: $tType] :
( ( top_top @ ( A > $o ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% top_empty_eq
thf(fact_90_Sup__eq__top__iff,axiom,
! [A: $tType] :
( ( comple1035589618norder @ A @ ( type2 @ A ) )
=> ! [A2: set @ A] :
( ( ( complete_Sup_Sup @ A @ A2 )
= ( top_top @ A ) )
= ( ! [X2: A] :
( ( ord_less @ A @ X2 @ ( top_top @ A ) )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ A2 )
& ( ord_less @ A @ X2 @ Y3 ) ) ) ) ) ) ).
% Sup_eq_top_iff
thf(fact_91_antimonoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ! [F: A > B,X: A,Y4: A] :
( ( order_antimono @ A @ B @ F )
=> ( ( ord_less_eq @ A @ X @ Y4 )
=> ( ord_less_eq @ B @ ( F @ Y4 ) @ ( F @ X ) ) ) ) ) ).
% antimonoD
thf(fact_92_antimonoE,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ! [F: A > B,X: A,Y4: A] :
( ( order_antimono @ A @ B @ F )
=> ( ( ord_less_eq @ A @ X @ Y4 )
=> ( ord_less_eq @ B @ ( F @ Y4 ) @ ( F @ X ) ) ) ) ) ).
% antimonoE
thf(fact_93_antimonoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ! [F: A > B] :
( ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ B @ ( F @ Y2 ) @ ( F @ X3 ) ) )
=> ( order_antimono @ A @ B @ F ) ) ) ).
% antimonoI
thf(fact_94_antimono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ( ( order_antimono @ A @ B )
= ( ^ [F2: A > B] :
! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ Y3 ) @ ( F2 @ X2 ) ) ) ) ) ) ).
% antimono_def
thf(fact_95_Pow__mono,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pow @ A @ A2 ) @ ( pow @ A @ B2 ) ) ) ).
% Pow_mono
thf(fact_96_less__eq__Sup,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [A2: set @ A,U: A] :
( ! [V2: A] :
( ( member @ A @ V2 @ A2 )
=> ( ord_less_eq @ A @ U @ V2 ) )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ A2 ) ) ) ) ) ).
% less_eq_Sup
thf(fact_97_bot__apply,axiom,
! [C: $tType,D: $tType] :
( ( bot @ C @ ( type2 @ C ) )
=> ( ( bot_bot @ ( D > C ) )
= ( ^ [X2: D] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_98_empty__iff,axiom,
! [A: $tType,C2: A] :
~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_99_all__not__in__conv,axiom,
! [A: $tType,A2: set @ A] :
( ( ! [X2: A] :
~ ( member @ A @ X2 @ A2 ) )
= ( A2
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_100_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X2: A] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_101_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X2: A] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_102_psubsetI,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_103_UN__ball__bex__simps_I3_J,axiom,
! [D: $tType,A2: set @ ( set @ D ),P: D > $o] :
( ( ? [X2: D] :
( ( member @ D @ X2 @ ( complete_Sup_Sup @ ( set @ D ) @ A2 ) )
& ( P @ X2 ) ) )
= ( ? [X2: set @ D] :
( ( member @ ( set @ D ) @ X2 @ A2 )
& ? [Y3: D] :
( ( member @ D @ Y3 @ X2 )
& ( P @ Y3 ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_104_UN__ball__bex__simps_I1_J,axiom,
! [A: $tType,A2: set @ ( set @ A ),P: A > $o] :
( ( ! [X2: A] :
( ( member @ A @ X2 @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) )
=> ( P @ X2 ) ) )
= ( ! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ A2 )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ X2 )
=> ( P @ Y3 ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_105_UnionI,axiom,
! [A: $tType,X6: set @ A,C3: set @ ( set @ A ),A2: A] :
( ( member @ ( set @ A ) @ X6 @ C3 )
=> ( ( member @ A @ A2 @ X6 )
=> ( member @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ C3 ) ) ) ) ).
% UnionI
thf(fact_106_Union__iff,axiom,
! [A: $tType,A2: A,C3: set @ ( set @ A )] :
( ( member @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ C3 ) )
= ( ? [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ C3 )
& ( member @ A @ A2 @ X2 ) ) ) ) ).
% Union_iff
thf(fact_107_top1I,axiom,
! [A: $tType,X: A] : ( top_top @ ( A > $o ) @ X ) ).
% top1I
thf(fact_108_Sup__bot__conv_I1_J,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [A2: set @ A] :
( ( ( complete_Sup_Sup @ A @ A2 )
= ( bot_bot @ A ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A2 )
=> ( X2
= ( bot_bot @ A ) ) ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_109_Sup__bot__conv_I2_J,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [A2: set @ A] :
( ( ( bot_bot @ A )
= ( complete_Sup_Sup @ A @ A2 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A2 )
=> ( X2
= ( bot_bot @ A ) ) ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_110_empty__subsetI,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A2 ) ).
% empty_subsetI
thf(fact_111_subset__empty,axiom,
! [A: $tType,A2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
= ( A2
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_112_Pow__iff,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( member @ ( set @ A ) @ A2 @ ( pow @ A @ B2 ) )
= ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% Pow_iff
thf(fact_113_PowI,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( member @ ( set @ A ) @ A2 @ ( pow @ A @ B2 ) ) ) ).
% PowI
thf(fact_114_Union__Pow__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( pow @ A @ A2 ) )
= A2 ) ).
% Union_Pow_eq
thf(fact_115_Sup__empty,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ( ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ A ) ) ) ).
% Sup_empty
thf(fact_116_Pow__UNIV,axiom,
! [A: $tType] :
( ( pow @ A @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ ( set @ A ) ) ) ) ).
% Pow_UNIV
thf(fact_117_emptyE,axiom,
! [A: $tType,A4: A] :
~ ( member @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_118_Pow__top,axiom,
! [A: $tType,A2: set @ A] : ( member @ ( set @ A ) @ A2 @ ( pow @ A @ A2 ) ) ).
% Pow_top
thf(fact_119_equals0D,axiom,
! [A: $tType,A2: set @ A,A4: A] :
( ( A2
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A4 @ A2 ) ) ).
% equals0D
thf(fact_120_equals0I,axiom,
! [A: $tType,A2: set @ A] :
( ! [Y2: A] :
~ ( member @ A @ Y2 @ A2 )
=> ( A2
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_121_Pow__bottom,axiom,
! [A: $tType,B2: set @ A] : ( member @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( pow @ A @ B2 ) ) ).
% Pow_bottom
thf(fact_122_ex__in__conv,axiom,
! [A: $tType,A2: set @ A] :
( ( ? [X2: A] : ( member @ A @ X2 @ A2 ) )
= ( A2
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_123_Pow__not__empty,axiom,
! [A: $tType,A2: set @ A] :
( ( pow @ A @ A2 )
!= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% Pow_not_empty
thf(fact_124_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B @ ( type2 @ B ) )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X2: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_125_not__psubset__empty,axiom,
! [A: $tType,A2: set @ A] :
~ ( ord_less @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% not_psubset_empty
thf(fact_126_less__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
& ~ ( ord_less_eq @ ( A > B ) @ G2 @ F2 ) ) ) ) ) ).
% less_fun_def
thf(fact_127_UnionE,axiom,
! [A: $tType,A2: A,C3: set @ ( set @ A )] :
( ( member @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ C3 ) )
=> ~ ! [X4: set @ A] :
( ( member @ A @ A2 @ X4 )
=> ~ ( member @ ( set @ A ) @ X4 @ C3 ) ) ) ).
% UnionE
thf(fact_128_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A4: A,F: B > A,B4: B,C2: B] :
( ( A4
= ( F @ B4 ) )
=> ( ( ord_less @ B @ B4 @ C2 )
=> ( ! [X3: B,Y2: B] :
( ( ord_less @ B @ X3 @ Y2 )
=> ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_129_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A4: A,B4: A,F: A > B,C2: B] :
( ( ord_less @ A @ A4 @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ord_less @ B @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ B @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_130_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,F: B > A,B4: B,C2: B] :
( ( ord_less @ A @ A4 @ ( F @ B4 ) )
=> ( ( ord_less @ B @ B4 @ C2 )
=> ( ! [X3: B,Y2: B] :
( ( ord_less @ B @ X3 @ Y2 )
=> ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_131_order__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C @ ( type2 @ C ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,B4: A,F: A > C,C2: C] :
( ( ord_less @ A @ A4 @ B4 )
=> ( ( ord_less @ C @ ( F @ B4 ) @ C2 )
=> ( ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ord_less @ C @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ C @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% order_less_subst2
thf(fact_132_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A @ ( type2 @ A ) )
=> ! [X: A] :
? [Y2: A] : ( ord_less @ A @ Y2 @ X ) ) ).
% lt_ex
thf(fact_133_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A @ ( type2 @ A ) )
=> ! [X: A] :
? [X1: A] : ( ord_less @ A @ X @ X1 ) ) ).
% gt_ex
thf(fact_134_Union__empty,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Union_empty
thf(fact_135_neqE,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ( X != Y4 )
=> ( ~ ( ord_less @ A @ X @ Y4 )
=> ( ord_less @ A @ Y4 @ X ) ) ) ) ).
% neqE
thf(fact_136_neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ( X != Y4 )
= ( ( ord_less @ A @ X @ Y4 )
| ( ord_less @ A @ Y4 @ X ) ) ) ) ).
% neq_iff
thf(fact_137_order_Oasym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ~ ( ord_less @ A @ B4 @ A4 ) ) ) ).
% order.asym
thf(fact_138_dense,axiom,
! [A: $tType] :
( ( dense_order @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ( ord_less @ A @ X @ Y4 )
=> ? [Z4: A] :
( ( ord_less @ A @ X @ Z4 )
& ( ord_less @ A @ Z4 @ Y4 ) ) ) ) ).
% dense
thf(fact_139_Union__empty__conv,axiom,
! [A: $tType,A2: set @ ( set @ A )] :
( ( ( complete_Sup_Sup @ ( set @ A ) @ A2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ A2 )
=> ( X2
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Union_empty_conv
thf(fact_140_empty__Union__conv,axiom,
! [A: $tType,A2: set @ ( set @ A )] :
( ( ( bot_bot @ ( set @ A ) )
= ( complete_Sup_Sup @ ( set @ A ) @ A2 ) )
= ( ! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ A2 )
=> ( X2
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% empty_Union_conv
thf(fact_141_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ( ord_less @ A @ X @ Y4 )
=> ( X != Y4 ) ) ) ).
% less_imp_neq
thf(fact_142_less__asym,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ( ord_less @ A @ X @ Y4 )
=> ~ ( ord_less @ A @ Y4 @ X ) ) ) ).
% less_asym
thf(fact_143_less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ~ ( ord_less @ A @ B4 @ A4 ) ) ) ).
% less_asym'
thf(fact_144_less__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A,Z2: A] :
( ( ord_less @ A @ X @ Y4 )
=> ( ( ord_less @ A @ Y4 @ Z2 )
=> ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% less_trans
thf(fact_145_less__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ( ord_less @ A @ X @ Y4 )
| ( X = Y4 )
| ( ord_less @ A @ Y4 @ X ) ) ) ).
% less_linear
thf(fact_146_less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] :
~ ( ord_less @ A @ X @ X ) ) ).
% less_irrefl
thf(fact_147_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,C2: A] :
( ( A4 = B4 )
=> ( ( ord_less @ A @ B4 @ C2 )
=> ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% ord_eq_less_trans
thf(fact_148_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,C2: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ( ( B4 = C2 )
=> ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% ord_less_eq_trans
thf(fact_149_dual__order_Oasym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B4: A,A4: A] :
( ( ord_less @ A @ B4 @ A4 )
=> ~ ( ord_less @ A @ A4 @ B4 ) ) ) ).
% dual_order.asym
thf(fact_150_less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ( ord_less @ A @ X @ Y4 )
=> ( X != Y4 ) ) ) ).
% less_imp_not_eq
thf(fact_151_less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ( ord_less @ A @ X @ Y4 )
=> ~ ( ord_less @ A @ Y4 @ X ) ) ) ).
% less_not_sym
thf(fact_152_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,A4: A] :
( ! [X3: A] :
( ! [Y5: A] :
( ( ord_less @ A @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A4 ) ) ) ).
% less_induct
thf(fact_153_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y4: A,X: A] :
( ~ ( ord_less @ A @ Y4 @ X )
=> ( ( ~ ( ord_less @ A @ X @ Y4 ) )
= ( X = Y4 ) ) ) ) ).
% antisym_conv3
thf(fact_154_less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ( ord_less @ A @ X @ Y4 )
=> ( Y4 != X ) ) ) ).
% less_imp_not_eq2
thf(fact_155_less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A,P: $o] :
( ( ord_less @ A @ X @ Y4 )
=> ( ( ord_less @ A @ Y4 @ X )
=> P ) ) ) ).
% less_imp_triv
thf(fact_156_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ~ ( ord_less @ A @ X @ Y4 )
=> ( ( X != Y4 )
=> ( ord_less @ A @ Y4 @ X ) ) ) ) ).
% linorder_cases
thf(fact_157_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A] :
~ ( ord_less @ A @ A4 @ A4 ) ) ).
% dual_order.irrefl
thf(fact_158_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,C2: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ( ( ord_less @ A @ B4 @ C2 )
=> ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% order.strict_trans
thf(fact_159_less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ( ord_less @ A @ X @ Y4 )
=> ~ ( ord_less @ A @ Y4 @ X ) ) ) ).
% less_imp_not_less
thf(fact_160_bot_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A4: A] :
~ ( ord_less @ A @ A4 @ ( bot_bot @ A ) ) ) ).
% bot.extremum_strict
thf(fact_161_bot_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A4: A] :
( ( A4
!= ( bot_bot @ A ) )
= ( ord_less @ A @ ( bot_bot @ A ) @ A4 ) ) ) ).
% bot.not_eq_extremum
thf(fact_162_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B4: A,A4: A,C2: A] :
( ( ord_less @ A @ B4 @ A4 )
=> ( ( ord_less @ A @ C2 @ B4 )
=> ( ord_less @ A @ C2 @ A4 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_163_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ( ~ ( ord_less @ A @ X @ Y4 ) )
= ( ( ord_less @ A @ Y4 @ X )
| ( X = Y4 ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_164_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ( A4 != B4 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_165_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B4: A,A4: A] :
( ( ord_less @ A @ B4 @ A4 )
=> ( A4 != B4 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_166_less__cSupD,axiom,
! [A: $tType] :
( ( condit1037483654norder @ A @ ( type2 @ A ) )
=> ! [X6: set @ A,Z2: A] :
( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ Z2 @ ( complete_Sup_Sup @ A @ X6 ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ X6 )
& ( ord_less @ A @ Z2 @ X3 ) ) ) ) ) ).
% less_cSupD
thf(fact_167_less__cSupE,axiom,
! [A: $tType] :
( ( condit1037483654norder @ A @ ( type2 @ A ) )
=> ! [Y4: A,X6: set @ A] :
( ( ord_less @ A @ Y4 @ ( complete_Sup_Sup @ A @ X6 ) )
=> ( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ~ ( ord_less @ A @ Y4 @ X3 ) ) ) ) ) ).
% less_cSupE
thf(fact_168_ex__gt__or__lt,axiom,
! [A: $tType] :
( ( condit1656338222tinuum @ A @ ( type2 @ A ) )
=> ! [A4: A] :
? [B5: A] :
( ( ord_less @ A @ A4 @ B5 )
| ( ord_less @ A @ B5 @ A4 ) ) ) ).
% ex_gt_or_lt
thf(fact_169_complete__interval,axiom,
! [A: $tType] :
( ( condit1037483654norder @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,P: A > $o] :
( ( ord_less @ A @ A4 @ B4 )
=> ( ( P @ A4 )
=> ( ~ ( P @ B4 )
=> ? [C4: A] :
( ( ord_less_eq @ A @ A4 @ C4 )
& ( ord_less_eq @ A @ C4 @ B4 )
& ! [X5: A] :
( ( ( ord_less_eq @ A @ A4 @ X5 )
& ( ord_less @ A @ X5 @ C4 ) )
=> ( P @ X5 ) )
& ! [D2: A] :
( ! [X3: A] :
( ( ( ord_less_eq @ A @ A4 @ X3 )
& ( ord_less @ A @ X3 @ D2 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq @ A @ D2 @ C4 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_170_order_Onot__eq__order__implies__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A] :
( ( A4 != B4 )
=> ( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ord_less @ A @ A4 @ B4 ) ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_171_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B4: A,A4: A] :
( ( ord_less @ A @ B4 @ A4 )
=> ( ord_less_eq @ A @ B4 @ A4 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_172_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [B6: A,A6: A] :
( ( ord_less_eq @ A @ B6 @ A6 )
& ( A6 != B6 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_173_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [B6: A,A6: A] :
( ( ord_less @ A @ B6 @ A6 )
| ( A6 = B6 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_174_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ( ord_less_eq @ A @ A4 @ B4 ) ) ) ).
% order.strict_implies_order
thf(fact_175_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A,Z2: A] :
( ( ord_less @ A @ X @ Y4 )
=> ( ! [W: A] :
( ( ord_less @ A @ X @ W )
=> ( ( ord_less @ A @ W @ Y4 )
=> ( ord_less_eq @ A @ W @ Z2 ) ) )
=> ( ord_less_eq @ A @ Y4 @ Z2 ) ) ) ) ).
% dense_le_bounded
thf(fact_176_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Z2: A,X: A,Y4: A] :
( ( ord_less @ A @ Z2 @ X )
=> ( ! [W: A] :
( ( ord_less @ A @ Z2 @ W )
=> ( ( ord_less @ A @ W @ X )
=> ( ord_less_eq @ A @ Y4 @ W ) ) )
=> ( ord_less_eq @ A @ Y4 @ Z2 ) ) ) ) ).
% dense_ge_bounded
thf(fact_177_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B4: A,A4: A,C2: A] :
( ( ord_less @ A @ B4 @ A4 )
=> ( ( ord_less_eq @ A @ C2 @ B4 )
=> ( ord_less @ A @ C2 @ A4 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_178_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B4: A,A4: A,C2: A] :
( ( ord_less_eq @ A @ B4 @ A4 )
=> ( ( ord_less @ A @ C2 @ B4 )
=> ( ord_less @ A @ C2 @ A4 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_179_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [A6: A,B6: A] :
( ( ord_less_eq @ A @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_180_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [A6: A,B6: A] :
( ( ord_less @ A @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_181_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,C2: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ A @ B4 @ C2 )
=> ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% order.strict_trans2
thf(fact_182_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ord_less @ A @ B4 @ C2 )
=> ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% order.strict_trans1
thf(fact_183_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y4: A,X: A] :
( ~ ( ord_less_eq @ A @ Y4 @ X )
=> ( ord_less @ A @ X @ Y4 ) ) ) ).
% not_le_imp_less
thf(fact_184_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
& ~ ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ) ) ).
% less_le_not_le
thf(fact_185_le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ( ord_less_eq @ A @ X @ Y4 )
=> ( ( ord_less @ A @ X @ Y4 )
| ( X = Y4 ) ) ) ) ).
% le_imp_less_or_eq
thf(fact_186_le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ( ord_less_eq @ A @ X @ Y4 )
| ( ord_less @ A @ Y4 @ X ) ) ) ).
% le_less_linear
thf(fact_187_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Y4: A,Z2: A] :
( ! [X3: A] :
( ( ord_less @ A @ X3 @ Y4 )
=> ( ord_less_eq @ A @ X3 @ Z2 ) )
=> ( ord_less_eq @ A @ Y4 @ Z2 ) ) ) ).
% dense_le
thf(fact_188_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Z2: A,Y4: A] :
( ! [X3: A] :
( ( ord_less @ A @ Z2 @ X3 )
=> ( ord_less_eq @ A @ Y4 @ X3 ) )
=> ( ord_less_eq @ A @ Y4 @ Z2 ) ) ) ).
% dense_ge
thf(fact_189_less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A,Z2: A] :
( ( ord_less @ A @ X @ Y4 )
=> ( ( ord_less_eq @ A @ Y4 @ Z2 )
=> ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% less_le_trans
thf(fact_190_le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A,Z2: A] :
( ( ord_less_eq @ A @ X @ Y4 )
=> ( ( ord_less @ A @ Y4 @ Z2 )
=> ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% le_less_trans
thf(fact_191_antisym__conv2,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ( ord_less_eq @ A @ X @ Y4 )
=> ( ( ~ ( ord_less @ A @ X @ Y4 ) )
= ( X = Y4 ) ) ) ) ).
% antisym_conv2
thf(fact_192_antisym__conv1,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ~ ( ord_less @ A @ X @ Y4 )
=> ( ( ord_less_eq @ A @ X @ Y4 )
= ( X = Y4 ) ) ) ) ).
% antisym_conv1
thf(fact_193_less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ( ord_less @ A @ X @ Y4 )
=> ( ord_less_eq @ A @ X @ Y4 ) ) ) ).
% less_imp_le
thf(fact_194_le__neq__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( A4 != B4 )
=> ( ord_less @ A @ A4 @ B4 ) ) ) ) ).
% le_neq_trans
thf(fact_195_not__less,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ( ~ ( ord_less @ A @ X @ Y4 ) )
= ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% not_less
thf(fact_196_not__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ( ~ ( ord_less_eq @ A @ X @ Y4 ) )
= ( ord_less @ A @ Y4 @ X ) ) ) ).
% not_le
thf(fact_197_order__less__le__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C @ ( type2 @ C ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,B4: A,F: A > C,C2: C] :
( ( ord_less @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ C @ ( F @ B4 ) @ C2 )
=> ( ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ord_less @ C @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ C @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_198_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,F: B > A,B4: B,C2: B] :
( ( ord_less @ A @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_eq @ B @ B4 @ C2 )
=> ( ! [X3: B,Y2: B] :
( ( ord_less_eq @ B @ X3 @ Y2 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_199_order__le__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C @ ( type2 @ C ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,B4: A,F: A > C,C2: C] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ord_less @ C @ ( F @ B4 ) @ C2 )
=> ( ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ C @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_200_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,F: B > A,B4: B,C2: B] :
( ( ord_less_eq @ A @ A4 @ ( F @ B4 ) )
=> ( ( ord_less @ B @ B4 @ C2 )
=> ( ! [X3: B,Y2: B] :
( ( ord_less @ B @ X3 @ Y2 )
=> ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_201_less__le,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ) ).
% less_le
thf(fact_202_le__less,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [X2: A,Y3: A] :
( ( ord_less @ A @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ) ).
% le_less
thf(fact_203_leI,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ~ ( ord_less @ A @ X @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% leI
thf(fact_204_leD,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y4: A,X: A] :
( ( ord_less_eq @ A @ Y4 @ X )
=> ~ ( ord_less @ A @ X @ Y4 ) ) ) ).
% leD
thf(fact_205_bot_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A4: A] :
( ( ord_less_eq @ A @ A4 @ ( bot_bot @ A ) )
=> ( A4
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_uniqueI
thf(fact_206_bot_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A4: A] :
( ( ord_less_eq @ A @ A4 @ ( bot_bot @ A ) )
= ( A4
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_unique
thf(fact_207_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A4: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A4 ) ) ).
% bot.extremum
thf(fact_208_top_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_top @ A @ ( type2 @ A ) )
=> ! [A4: A] :
~ ( ord_less @ A @ ( top_top @ A ) @ A4 ) ) ).
% top.extremum_strict
thf(fact_209_top_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_top @ A @ ( type2 @ A ) )
=> ! [A4: A] :
( ( A4
!= ( top_top @ A ) )
= ( ord_less @ A @ A4 @ ( top_top @ A ) ) ) ) ).
% top.not_eq_extremum
thf(fact_210_subset__iff__psubset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A3: set @ A,B3: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_211_subset__psubset__trans,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less @ ( set @ A ) @ B2 @ C3 )
=> ( ord_less @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% subset_psubset_trans
thf(fact_212_subset__not__subset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
& ~ ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_213_psubset__subset__trans,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
( ( ord_less @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
=> ( ord_less @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% psubset_subset_trans
thf(fact_214_psubset__imp__subset,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_215_psubset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_216_psubsetE,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less @ ( set @ A ) @ A2 @ B2 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% psubsetE
thf(fact_217_less__Sup__iff,axiom,
! [A: $tType] :
( ( comple1035589618norder @ A @ ( type2 @ A ) )
=> ! [A4: A,S: set @ A] :
( ( ord_less @ A @ A4 @ ( complete_Sup_Sup @ A @ S ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ S )
& ( ord_less @ A @ A4 @ X2 ) ) ) ) ) ).
% less_Sup_iff
thf(fact_218_PowD,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( member @ ( set @ A ) @ A2 @ ( pow @ A @ B2 ) )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% PowD
thf(fact_219_empty__not__UNIV,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
!= ( top_top @ ( set @ A ) ) ) ).
% empty_not_UNIV
thf(fact_220_cSup__eq__non__empty,axiom,
! [A: $tType] :
( ( condit378418413attice @ A @ ( type2 @ A ) )
=> ! [X6: set @ A,A4: A] :
( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ( ord_less_eq @ A @ X3 @ A4 ) )
=> ( ! [Y2: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ X6 )
=> ( ord_less_eq @ A @ X5 @ Y2 ) )
=> ( ord_less_eq @ A @ A4 @ Y2 ) )
=> ( ( complete_Sup_Sup @ A @ X6 )
= A4 ) ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_221_cSup__least,axiom,
! [A: $tType] :
( ( condit378418413attice @ A @ ( type2 @ A ) )
=> ! [X6: set @ A,Z2: A] :
( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ( ord_less_eq @ A @ X3 @ Z2 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ X6 ) @ Z2 ) ) ) ) ).
% cSup_least
thf(fact_222_subset__Pow__Union,axiom,
! [A: $tType,A2: set @ ( set @ A )] : ( ord_less_eq @ ( set @ ( set @ A ) ) @ A2 @ ( pow @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) ) ) ).
% subset_Pow_Union
thf(fact_223_le__Sup__iff,axiom,
! [A: $tType] :
( ( comple1035589618norder @ A @ ( type2 @ A ) )
=> ! [X: A,A2: set @ A] :
( ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ A2 ) )
= ( ! [Y3: A] :
( ( ord_less @ A @ Y3 @ X )
=> ? [X2: A] :
( ( member @ A @ X2 @ A2 )
& ( ord_less @ A @ Y3 @ X2 ) ) ) ) ) ) ).
% le_Sup_iff
thf(fact_224_subset__emptyI,axiom,
! [A: $tType,A2: set @ A] :
( ! [X3: A] :
~ ( member @ A @ X3 @ A2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_emptyI
thf(fact_225_pinf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ Z4 @ X5 )
=> ~ ( ord_less_eq @ A @ X5 @ T2 ) ) ) ).
% pinf(6)
thf(fact_226_pinf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ Z4 @ X5 )
=> ( ord_less_eq @ A @ T2 @ X5 ) ) ) ).
% pinf(8)
thf(fact_227_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z4 )
=> ( ord_less_eq @ A @ X5 @ T2 ) ) ) ).
% minf(6)
thf(fact_228_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z4 )
=> ~ ( ord_less_eq @ A @ T2 @ X5 ) ) ) ).
% minf(8)
thf(fact_229_psubsetD,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C2: A] :
( ( ord_less @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ C2 @ A2 )
=> ( member @ A @ C2 @ B2 ) ) ) ).
% psubsetD
thf(fact_230_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_231_psubset__trans,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
( ( ord_less @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less @ ( set @ A ) @ B2 @ C3 )
=> ( ord_less @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% psubset_trans
thf(fact_232_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_233_minf_I11_J,axiom,
! [C: $tType,D: $tType] :
( ( ord @ C @ ( type2 @ C ) )
=> ! [F3: D] :
? [Z4: C] :
! [X5: C] :
( ( ord_less @ C @ X5 @ Z4 )
=> ( F3 = F3 ) ) ) ).
% minf(11)
thf(fact_234_minf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z4 )
=> ~ ( ord_less @ A @ T2 @ X5 ) ) ) ).
% minf(7)
thf(fact_235_minf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z4 )
=> ( ord_less @ A @ X5 @ T2 ) ) ) ).
% minf(5)
thf(fact_236_minf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z4 )
=> ( X5 != T2 ) ) ) ).
% minf(4)
thf(fact_237_minf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z4 )
=> ( X5 != T2 ) ) ) ).
% minf(3)
thf(fact_238_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z3: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z3: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z4 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P2 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_239_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z3: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z3: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z4 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P2 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_240_pinf_I11_J,axiom,
! [C: $tType,D: $tType] :
( ( ord @ C @ ( type2 @ C ) )
=> ! [F3: D] :
? [Z4: C] :
! [X5: C] :
( ( ord_less @ C @ Z4 @ X5 )
=> ( F3 = F3 ) ) ) ).
% pinf(11)
thf(fact_241_pinf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ Z4 @ X5 )
=> ( ord_less @ A @ T2 @ X5 ) ) ) ).
% pinf(7)
thf(fact_242_pinf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ Z4 @ X5 )
=> ~ ( ord_less @ A @ X5 @ T2 ) ) ) ).
% pinf(5)
thf(fact_243_pinf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ Z4 @ X5 )
=> ( X5 != T2 ) ) ) ).
% pinf(4)
thf(fact_244_pinf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ Z4 @ X5 )
=> ( X5 != T2 ) ) ) ).
% pinf(3)
thf(fact_245_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z3: A] :
! [X3: A] :
( ( ord_less @ A @ Z3 @ X3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z3: A] :
! [X3: A] :
( ( ord_less @ A @ Z3 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ Z4 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P2 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_246_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z3: A] :
! [X3: A] :
( ( ord_less @ A @ Z3 @ X3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z3: A] :
! [X3: A] :
( ( ord_less @ A @ Z3 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ Z4 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P2 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_247_top_Oordering__top__axioms,axiom,
! [A: $tType] :
( ( order_top @ A @ ( type2 @ A ) )
=> ( ordering_top @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) @ ( top_top @ A ) ) ) ).
% top.ordering_top_axioms
thf(fact_248_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A3: set @ A] :
( A3
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_249_ordering__top_Oextremum,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A4: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ( Less_eq @ A4 @ Top ) ) ).
% ordering_top.extremum
thf(fact_250_ordering__top_Oextremum__strict,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A4: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ~ ( Less @ Top @ A4 ) ) ).
% ordering_top.extremum_strict
thf(fact_251_ordering__top_Oextremum__unique,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A4: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A4 )
= ( A4 = Top ) ) ) ).
% ordering_top.extremum_unique
thf(fact_252_ordering__top_Onot__eq__extremum,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A4: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ( ( A4 != Top )
= ( Less @ A4 @ Top ) ) ) ).
% ordering_top.not_eq_extremum
thf(fact_253_ordering__top_Oextremum__uniqueI,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A4: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A4 )
=> ( A4 = Top ) ) ) ).
% ordering_top.extremum_uniqueI
thf(fact_254_Collect__empty__eq__bot,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( P
= ( bot_bot @ ( A > $o ) ) ) ) ).
% Collect_empty_eq_bot
thf(fact_255_top__conj_I2_J,axiom,
! [A: $tType,P: $o,X: A] :
( ( P
& ( top_top @ ( A > $o ) @ X ) )
= P ) ).
% top_conj(2)
%----Type constructors (28)
thf(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
! [A7: $tType,A8: $tType] :
( ( comple187826305attice @ A8 @ ( type2 @ A8 ) )
=> ( condit378418413attice @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [A7: $tType,A8: $tType] :
( ( comple187826305attice @ A8 @ ( type2 @ A8 ) )
=> ( comple187826305attice @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A7: $tType,A8: $tType] :
( ( order_top @ A8 @ ( type2 @ A8 ) )
=> ( order_top @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A7: $tType,A8: $tType] :
( ( order_bot @ A8 @ ( type2 @ A8 ) )
=> ( order_bot @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A7: $tType,A8: $tType] :
( ( preorder @ A8 @ ( type2 @ A8 ) )
=> ( preorder @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A7: $tType,A8: $tType] :
( ( order @ A8 @ ( type2 @ A8 ) )
=> ( order @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A7: $tType,A8: $tType] :
( ( top @ A8 @ ( type2 @ A8 ) )
=> ( top @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A7: $tType,A8: $tType] :
( ( ord @ A8 @ ( type2 @ A8 ) )
=> ( ord @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A7: $tType,A8: $tType] :
( ( bot @ A8 @ ( type2 @ A8 ) )
=> ( bot @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_1,axiom,
! [A7: $tType] : ( condit378418413attice @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_2,axiom,
! [A7: $tType] : ( comple187826305attice @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_3,axiom,
! [A7: $tType] : ( order_top @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_4,axiom,
! [A7: $tType] : ( order_bot @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_5,axiom,
! [A7: $tType] : ( preorder @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_6,axiom,
! [A7: $tType] : ( order @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_7,axiom,
! [A7: $tType] : ( top @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_8,axiom,
! [A7: $tType] : ( ord @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_9,axiom,
! [A7: $tType] : ( bot @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_10,axiom,
condit378418413attice @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_11,axiom,
comple187826305attice @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_12,axiom,
order_top @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_13,axiom,
order_bot @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_14,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_15,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Otop_16,axiom,
top @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_17,axiom,
ord @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Obot_18,axiom,
bot @ $o @ ( type2 @ $o ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
gram_L1456083582e_leqv @ t @ ( gram_L861977280lle_LL @ ( top_top @ ( set @ n ) ) @ ts ) @ ( gram_L861977318lle_Lr @ ( top_top @ ( set @ n ) ) @ ts ) ).
%------------------------------------------------------------------------------