TPTP Problem File: ITP029^1.p
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%------------------------------------------------------------------------------
% File : ITP029^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer BinaryTree problem prob_163__3251696_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 : BinaryTree/prob_163__3251696_1 [Des21]
% Status : ContradictoryAxioms
% Rating : 0.30 v8.2.0, 0.15 v8.1.0, 0.27 v7.5.0
% Syntax : Number of formulae : 419 ( 158 unt; 68 typ; 0 def)
% Number of atoms : 1066 ( 355 equ; 0 cnn)
% Maximal formula atoms : 10 ( 3 avg)
% Number of connectives : 2657 ( 136 ~; 30 |; 62 &;2013 @)
% ( 0 <=>; 416 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 6 avg)
% Number of types : 8 ( 7 usr)
% Number of type conns : 462 ( 462 >; 0 *; 0 +; 0 <<)
% Number of symbols : 63 ( 61 usr; 10 con; 0-4 aty)
% Number of variables : 1000 ( 102 ^; 862 !; 36 ?;1000 :)
% SPC : TH0_CAX_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:30:03.251
%------------------------------------------------------------------------------
% Could-be-implicit typings (7)
thf(ty_n_t__BinaryTree____Mirabelle____mlzyzwgbkd__OTree_It__Set__Oset_Itf__a_J_J,type,
binary594033953_set_a: $tType ).
thf(ty_n_t__BinaryTree____Mirabelle____mlzyzwgbkd__OTree_Itf__a_J,type,
binary1439146945Tree_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
set_set_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (61)
thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_OTree_OT_001t__Set__Oset_Itf__a_J,type,
binary313540327_set_a: binary594033953_set_a > set_a > binary594033953_set_a > binary594033953_set_a ).
thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_OTree_OT_001tf__a,type,
binary717961607le_T_a: binary1439146945Tree_a > a > binary1439146945Tree_a > binary1439146945Tree_a ).
thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_OTree_OTip_001tf__a,type,
binary476621312_Tip_a: binary1439146945Tree_a ).
thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_OTree_Oset__Tree_001t__Set__Oset_Itf__a_J,type,
binary1613048283_set_a: binary594033953_set_a > set_set_a ).
thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_OTree_Oset__Tree_001tf__a,type,
binary256242811Tree_a: binary1439146945Tree_a > set_a ).
thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_Oeqs_001tf__a,type,
binary504661350_eqs_a: ( a > int ) > a > set_a ).
thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_OsetOf_001t__Set__Oset_Itf__a_J,type,
binary1001944660_set_a: binary594033953_set_a > set_set_a ).
thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_OsetOf_001tf__a,type,
binary945792244etOf_a: binary1439146945Tree_a > set_a ).
thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_OsortedTree_001t__Set__Oset_Itf__a_J,type,
binary512218034_set_a: ( set_a > int ) > binary594033953_set_a > $o ).
thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_OsortedTree_001tf__a,type,
binary1721989714Tree_a: ( a > int ) > binary1439146945Tree_a > $o ).
thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_Osorted__distinct__pred_001t__Set__Oset_Itf__a_J,type,
binary1524747315_set_a: ( set_a > int ) > set_a > set_a > binary594033953_set_a > $o ).
thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_Osorted__distinct__pred_001tf__a,type,
binary670562003pred_a: ( a > int ) > a > a > binary1439146945Tree_a > $o ).
thf(sy_c_HOL_OThe_001tf__a,type,
the_a: ( a > $o ) > a ).
thf(sy_c_Lattices_Osemilattice__neutr_001_062_Itf__a_M_062_Itf__a_M_Eo_J_J,type,
semila1223476304_a_a_o: ( ( a > a > $o ) > ( a > a > $o ) > a > a > $o ) > ( a > a > $o ) > $o ).
thf(sy_c_Lattices_Osemilattice__neutr_001_062_Itf__a_M_Eo_J,type,
semila980155549tr_a_o: ( ( a > $o ) > ( a > $o ) > a > $o ) > ( a > $o ) > $o ).
thf(sy_c_Lattices_Osemilattice__neutr_001t__Set__Oset_Itf__a_J,type,
semila1409648192_set_a: ( set_a > set_a > set_a ) > set_a > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Set__Oset_Itf__a_J_M_062_It__Set__Oset_Itf__a_J_M_Eo_J_J,type,
sup_su198629954et_a_o: ( set_a > set_a > $o ) > ( set_a > set_a > $o ) > set_a > set_a > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Set__Oset_Itf__a_J_M_Eo_J,type,
sup_sup_set_a_o: ( set_a > $o ) > ( set_a > $o ) > set_a > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_Itf__a_M_062_Itf__a_M_Eo_J_J,type,
sup_sup_a_a_o: ( a > a > $o ) > ( a > a > $o ) > a > a > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_Itf__a_M_Eo_J,type,
sup_sup_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Int__Oint,type,
sup_sup_int: int > int > int ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
sup_sup_set_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_Itf__a_J_M_Eo_J,type,
bot_bot_set_a_o: set_a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_062_Itf__a_M_Eo_J_J,type,
bot_bot_a_a_o: a > a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
bot_bot_a_o: a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_Eo,type,
bot_bot_o: $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
bot_bot_set_set_a: set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Set__Oset_Itf__a_J_M_Eo_J,type,
ord_less_set_a_o: ( set_a > $o ) > ( set_a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_Itf__a_M_062_Itf__a_M_Eo_J_J,type,
ord_less_a_a_o: ( a > a > $o ) > ( a > a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_Itf__a_M_Eo_J,type,
ord_less_a_o: ( a > $o ) > ( a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_less_set_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Relation_OPowp_001t__Set__Oset_Itf__a_J,type,
powp_set_a: ( set_a > $o ) > set_set_a > $o ).
thf(sy_c_Relation_OPowp_001tf__a,type,
powp_a: ( a > $o ) > set_a > $o ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
collect_set_set_a: ( set_set_a > $o ) > set_set_set_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_OPow_001t__Set__Oset_Itf__a_J,type,
pow_set_a: set_set_a > set_set_set_a ).
thf(sy_c_Set_OPow_001tf__a,type,
pow_a: set_a > set_set_a ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__a_J,type,
insert_set_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Ois__empty_001tf__a,type,
is_empty_a: set_a > $o ).
thf(sy_c_Set_Ois__singleton_001t__Set__Oset_Itf__a_J,type,
is_singleton_set_a: set_set_a > $o ).
thf(sy_c_Set_Ois__singleton_001tf__a,type,
is_singleton_a: set_a > $o ).
thf(sy_c_Set_Othe__elem_001tf__a,type,
the_elem_a: set_a > a ).
thf(sy_c_Zorn_Ochains_001tf__a,type,
chains_a: set_set_a > set_set_set_a ).
thf(sy_c_Zorn_Opred__on_Ochain_001t__Set__Oset_Itf__a_J,type,
pred_chain_set_a: set_set_a > ( set_a > set_a > $o ) > set_set_a > $o ).
thf(sy_c_Zorn_Opred__on_Ochain_001tf__a,type,
pred_chain_a: set_a > ( a > a > $o ) > set_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
member_set_set_a: set_set_a > set_set_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_a,type,
a2: a ).
thf(sy_v_b,type,
b: a ).
thf(sy_v_h,type,
h: a > int ).
thf(sy_v_t1____,type,
t1: binary1439146945Tree_a ).
thf(sy_v_t2____,type,
t2: binary1439146945Tree_a ).
thf(sy_v_x____,type,
x: a ).
% Relevant facts (350)
thf(fact_0_hahb,axiom,
( ( h @ a2 )
= ( h @ b ) ) ).
% hahb
thf(fact_1__092_060open_062h_Ab_A_060_Ah_Aa_092_060close_062,axiom,
ord_less_int @ ( h @ b ) @ ( h @ a2 ) ).
% \<open>h b < h a\<close>
thf(fact_2__092_060open_062a_A_061_Ax_A_092_060Longrightarrow_062_Aa_A_061_Ab_092_060close_062,axiom,
( ( a2 = x )
=> ( a2 = b ) ) ).
% \<open>a = x \<Longrightarrow> a = b\<close>
thf(fact_3_adef1,axiom,
member_a @ a2 @ ( binary945792244etOf_a @ t2 ) ).
% adef1
thf(fact_4_bdef1,axiom,
member_a @ b @ ( binary945792244etOf_a @ t1 ) ).
% bdef1
thf(fact_5__092_060open_062a_A_092_060in_062_AsetOf_At1_A_092_060Longrightarrow_062_Aa_A_061_Ab_092_060close_062,axiom,
( ( member_a @ a2 @ ( binary945792244etOf_a @ t1 ) )
=> ( a2 = b ) ) ).
% \<open>a \<in> setOf t1 \<Longrightarrow> a = b\<close>
thf(fact_6_o1,axiom,
ord_less_int @ ( h @ b ) @ ( h @ x ) ).
% o1
thf(fact_7_o2,axiom,
ord_less_int @ ( h @ x ) @ ( h @ a2 ) ).
% o2
thf(fact_8_h2,axiom,
binary670562003pred_a @ h @ a2 @ b @ t2 ).
% h2
thf(fact_9_h1,axiom,
binary670562003pred_a @ h @ a2 @ b @ t1 ).
% h1
thf(fact_10__092_060open_062sorted__distinct__pred_Ah_Aa_Ab_ATip_092_060close_062,axiom,
binary670562003pred_a @ h @ a2 @ b @ binary476621312_Tip_a ).
% \<open>sorted_distinct_pred h a b Tip\<close>
thf(fact_11_calculation,axiom,
( ( member_a @ b @ ( binary945792244etOf_a @ t1 ) )
| ( b = x )
| ( member_a @ b @ ( binary945792244etOf_a @ t2 ) ) ) ).
% calculation
thf(fact_12__092_060open_062a_A_092_060in_062_AsetOf_At1_A_092_060or_062_Aa_A_061_Ax_A_092_060or_062_Aa_A_092_060in_062_AsetOf_At2_092_060close_062,axiom,
( ( member_a @ a2 @ ( binary945792244etOf_a @ t1 ) )
| ( a2 = x )
| ( member_a @ a2 @ ( binary945792244etOf_a @ t2 ) ) ) ).
% \<open>a \<in> setOf t1 \<or> a = x \<or> a \<in> setOf t2\<close>
thf(fact_13_bdef,axiom,
member_a @ b @ ( binary945792244etOf_a @ ( binary717961607le_T_a @ t1 @ x @ t2 ) ) ).
% bdef
thf(fact_14_Tree_Oinject,axiom,
! [X21: binary1439146945Tree_a,X22: a,X23: binary1439146945Tree_a,Y21: binary1439146945Tree_a,Y22: a,Y23: binary1439146945Tree_a] :
( ( ( binary717961607le_T_a @ X21 @ X22 @ X23 )
= ( binary717961607le_T_a @ Y21 @ Y22 @ Y23 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 )
& ( X23 = Y23 ) ) ) ).
% Tree.inject
thf(fact_15_adef,axiom,
member_a @ a2 @ ( binary945792244etOf_a @ ( binary717961607le_T_a @ t1 @ x @ t2 ) ) ).
% adef
thf(fact_16_s,axiom,
binary1721989714Tree_a @ h @ ( binary717961607le_T_a @ t1 @ x @ t2 ) ).
% s
thf(fact_17_Tree_Odistinct_I1_J,axiom,
! [X21: binary1439146945Tree_a,X22: a,X23: binary1439146945Tree_a] :
( binary476621312_Tip_a
!= ( binary717961607le_T_a @ X21 @ X22 @ X23 ) ) ).
% Tree.distinct(1)
thf(fact_18_Tree_Oinduct,axiom,
! [P: binary1439146945Tree_a > $o,Tree: binary1439146945Tree_a] :
( ( P @ binary476621312_Tip_a )
=> ( ! [X1: binary1439146945Tree_a,X2: a,X3: binary1439146945Tree_a] :
( ( P @ X1 )
=> ( ( P @ X3 )
=> ( P @ ( binary717961607le_T_a @ X1 @ X2 @ X3 ) ) ) )
=> ( P @ Tree ) ) ) ).
% Tree.induct
thf(fact_19_Tree_Oexhaust,axiom,
! [Y: binary1439146945Tree_a] :
( ( Y != binary476621312_Tip_a )
=> ~ ! [X212: binary1439146945Tree_a,X222: a,X232: binary1439146945Tree_a] :
( Y
!= ( binary717961607le_T_a @ X212 @ X222 @ X232 ) ) ) ).
% Tree.exhaust
thf(fact_20_s2,axiom,
binary1721989714Tree_a @ h @ t2 ).
% s2
thf(fact_21_s1,axiom,
binary1721989714Tree_a @ h @ t1 ).
% s1
thf(fact_22_sortedTree_Osimps_I2_J,axiom,
! [H: a > int,T1: binary1439146945Tree_a,X: a,T2: binary1439146945Tree_a] :
( ( binary1721989714Tree_a @ H @ ( binary717961607le_T_a @ T1 @ X @ T2 ) )
= ( ( binary1721989714Tree_a @ H @ T1 )
& ! [X4: a] :
( ( member_a @ X4 @ ( binary945792244etOf_a @ T1 ) )
=> ( ord_less_int @ ( H @ X4 ) @ ( H @ X ) ) )
& ! [X4: a] :
( ( member_a @ X4 @ ( binary945792244etOf_a @ T2 ) )
=> ( ord_less_int @ ( H @ X ) @ ( H @ X4 ) ) )
& ( binary1721989714Tree_a @ H @ T2 ) ) ) ).
% sortedTree.simps(2)
thf(fact_23_sorted__distinct__pred__def,axiom,
( binary1524747315_set_a
= ( ^ [H2: set_a > int,A: set_a,B: set_a,T: binary594033953_set_a] :
( ( ( binary512218034_set_a @ H2 @ T )
& ( member_set_a @ A @ ( binary1001944660_set_a @ T ) )
& ( member_set_a @ B @ ( binary1001944660_set_a @ T ) )
& ( ( H2 @ A )
= ( H2 @ B ) ) )
=> ( A = B ) ) ) ) ).
% sorted_distinct_pred_def
thf(fact_24_sorted__distinct__pred__def,axiom,
( binary670562003pred_a
= ( ^ [H2: a > int,A: a,B: a,T: binary1439146945Tree_a] :
( ( ( binary1721989714Tree_a @ H2 @ T )
& ( member_a @ A @ ( binary945792244etOf_a @ T ) )
& ( member_a @ B @ ( binary945792244etOf_a @ T ) )
& ( ( H2 @ A )
= ( H2 @ B ) ) )
=> ( A = B ) ) ) ) ).
% sorted_distinct_pred_def
thf(fact_25_eqs__def,axiom,
( binary504661350_eqs_a
= ( ^ [H2: a > int,X4: a] :
( collect_a
@ ^ [Y2: a] :
( ( H2 @ Y2 )
= ( H2 @ X4 ) ) ) ) ) ).
% eqs_def
thf(fact_26_minf_I7_J,axiom,
! [T3: int] :
? [Z: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z )
=> ~ ( ord_less_int @ T3 @ X5 ) ) ).
% minf(7)
thf(fact_27_minf_I5_J,axiom,
! [T3: int] :
? [Z: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z )
=> ( ord_less_int @ X5 @ T3 ) ) ).
% minf(5)
thf(fact_28_minf_I4_J,axiom,
! [T3: int] :
? [Z: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z )
=> ( X5 != T3 ) ) ).
% minf(4)
thf(fact_29_minf_I3_J,axiom,
! [T3: int] :
? [Z: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z )
=> ( X5 != T3 ) ) ).
% minf(3)
thf(fact_30_minf_I2_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z2 )
=> ( ( P @ X6 )
= ( P2 @ X6 ) ) )
=> ( ? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z2 )
=> ( ( Q @ X6 )
= ( Q2 @ X6 ) ) )
=> ? [Z: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P2 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_31_sortLemmaR,axiom,
! [H: a > int,T1: binary1439146945Tree_a,X: a,T2: binary1439146945Tree_a] :
( ( binary1721989714Tree_a @ H @ ( binary717961607le_T_a @ T1 @ X @ T2 ) )
=> ( binary1721989714Tree_a @ H @ T2 ) ) ).
% sortLemmaR
thf(fact_32_sortLemmaL,axiom,
! [H: a > int,T1: binary1439146945Tree_a,X: a,T2: binary1439146945Tree_a] :
( ( binary1721989714Tree_a @ H @ ( binary717961607le_T_a @ T1 @ X @ T2 ) )
=> ( binary1721989714Tree_a @ H @ T1 ) ) ).
% sortLemmaL
thf(fact_33_sortedTree_Osimps_I1_J,axiom,
! [H: a > int] : ( binary1721989714Tree_a @ H @ binary476621312_Tip_a ) ).
% sortedTree.simps(1)
thf(fact_34_pinf_I1_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ Z2 @ X6 )
=> ( ( P @ X6 )
= ( P2 @ X6 ) ) )
=> ( ? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ Z2 @ X6 )
=> ( ( Q @ X6 )
= ( Q2 @ X6 ) ) )
=> ? [Z: int] :
! [X5: int] :
( ( ord_less_int @ Z @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P2 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_35_pinf_I2_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ Z2 @ X6 )
=> ( ( P @ X6 )
= ( P2 @ X6 ) ) )
=> ( ? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ Z2 @ X6 )
=> ( ( Q @ X6 )
= ( Q2 @ X6 ) ) )
=> ? [Z: int] :
! [X5: int] :
( ( ord_less_int @ Z @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P2 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_36_pinf_I3_J,axiom,
! [T3: int] :
? [Z: int] :
! [X5: int] :
( ( ord_less_int @ Z @ X5 )
=> ( X5 != T3 ) ) ).
% pinf(3)
thf(fact_37_pinf_I4_J,axiom,
! [T3: int] :
? [Z: int] :
! [X5: int] :
( ( ord_less_int @ Z @ X5 )
=> ( X5 != T3 ) ) ).
% pinf(4)
thf(fact_38_pinf_I5_J,axiom,
! [T3: int] :
? [Z: int] :
! [X5: int] :
( ( ord_less_int @ Z @ X5 )
=> ~ ( ord_less_int @ X5 @ T3 ) ) ).
% pinf(5)
thf(fact_39_pinf_I7_J,axiom,
! [T3: int] :
? [Z: int] :
! [X5: int] :
( ( ord_less_int @ Z @ X5 )
=> ( ord_less_int @ T3 @ X5 ) ) ).
% pinf(7)
thf(fact_40_minf_I1_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z2 )
=> ( ( P @ X6 )
= ( P2 @ X6 ) ) )
=> ( ? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z2 )
=> ( ( Q @ X6 )
= ( Q2 @ X6 ) ) )
=> ? [Z: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P2 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_41_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_42_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_43_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: a > $o,A2: a > $o] :
( ( ord_less_a_o @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_44_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_45_mem__Collect__eq,axiom,
! [A2: set_a,P: set_a > $o] :
( ( member_set_a @ A2 @ ( collect_set_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_47_Collect__mem__eq,axiom,
! [A3: set_set_a] :
( ( collect_set_a
@ ^ [X4: set_a] : ( member_set_a @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_48_Collect__mem__eq,axiom,
! [A3: set_a] :
( ( collect_a
@ ^ [X4: a] : ( member_a @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_49_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X6: a] :
( ( P @ X6 )
= ( Q @ X6 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_50_order_Ostrict__implies__not__eq,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_51_order_Ostrict__implies__not__eq,axiom,
! [A2: a > $o,B2: a > $o] :
( ( ord_less_a_o @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_52_order_Ostrict__implies__not__eq,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_53_not__less__iff__gr__or__eq,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ( ord_less_int @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_54_dual__order_Ostrict__trans,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
=> ( ( ord_less_set_a @ C @ B2 )
=> ( ord_less_set_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_55_dual__order_Ostrict__trans,axiom,
! [B2: a > $o,A2: a > $o,C: a > $o] :
( ( ord_less_a_o @ B2 @ A2 )
=> ( ( ord_less_a_o @ C @ B2 )
=> ( ord_less_a_o @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_56_dual__order_Ostrict__trans,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ( ord_less_int @ C @ B2 )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_57_ord__eq__less__subst,axiom,
! [A2: set_a,F: int > set_a,B2: int,C: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X6: int,Y3: int] :
( ( ord_less_int @ X6 @ Y3 )
=> ( ord_less_set_a @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_58_ord__eq__less__subst,axiom,
! [A2: a > $o,F: int > a > $o,B2: int,C: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X6: int,Y3: int] :
( ( ord_less_int @ X6 @ Y3 )
=> ( ord_less_a_o @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_59_ord__eq__less__subst,axiom,
! [A2: int,F: set_a > int,B2: set_a,C: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X6: set_a,Y3: set_a] :
( ( ord_less_set_a @ X6 @ Y3 )
=> ( ord_less_int @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_60_ord__eq__less__subst,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X6: set_a,Y3: set_a] :
( ( ord_less_set_a @ X6 @ Y3 )
=> ( ord_less_set_a @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_61_ord__eq__less__subst,axiom,
! [A2: a > $o,F: set_a > a > $o,B2: set_a,C: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X6: set_a,Y3: set_a] :
( ( ord_less_set_a @ X6 @ Y3 )
=> ( ord_less_a_o @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_62_ord__eq__less__subst,axiom,
! [A2: int,F: ( a > $o ) > int,B2: a > $o,C: a > $o] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_a_o @ B2 @ C )
=> ( ! [X6: a > $o,Y3: a > $o] :
( ( ord_less_a_o @ X6 @ Y3 )
=> ( ord_less_int @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_63_ord__eq__less__subst,axiom,
! [A2: set_a,F: ( a > $o ) > set_a,B2: a > $o,C: a > $o] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_a_o @ B2 @ C )
=> ( ! [X6: a > $o,Y3: a > $o] :
( ( ord_less_a_o @ X6 @ Y3 )
=> ( ord_less_set_a @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_64_ord__eq__less__subst,axiom,
! [A2: a > $o,F: ( a > $o ) > a > $o,B2: a > $o,C: a > $o] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_a_o @ B2 @ C )
=> ( ! [X6: a > $o,Y3: a > $o] :
( ( ord_less_a_o @ X6 @ Y3 )
=> ( ord_less_a_o @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_65_ord__eq__less__subst,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X6: int,Y3: int] :
( ( ord_less_int @ X6 @ Y3 )
=> ( ord_less_int @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_66_ord__less__eq__subst,axiom,
! [A2: int,B2: int,F: int > set_a,C: set_a] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X6: int,Y3: int] :
( ( ord_less_int @ X6 @ Y3 )
=> ( ord_less_set_a @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_67_ord__less__eq__subst,axiom,
! [A2: int,B2: int,F: int > a > $o,C: a > $o] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X6: int,Y3: int] :
( ( ord_less_int @ X6 @ Y3 )
=> ( ord_less_a_o @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_68_ord__less__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > int,C: int] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X6: set_a,Y3: set_a] :
( ( ord_less_set_a @ X6 @ Y3 )
=> ( ord_less_int @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_69_ord__less__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X6: set_a,Y3: set_a] :
( ( ord_less_set_a @ X6 @ Y3 )
=> ( ord_less_set_a @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_70_ord__less__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > a > $o,C: a > $o] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X6: set_a,Y3: set_a] :
( ( ord_less_set_a @ X6 @ Y3 )
=> ( ord_less_a_o @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_71_ord__less__eq__subst,axiom,
! [A2: a > $o,B2: a > $o,F: ( a > $o ) > int,C: int] :
( ( ord_less_a_o @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X6: a > $o,Y3: a > $o] :
( ( ord_less_a_o @ X6 @ Y3 )
=> ( ord_less_int @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_72_ord__less__eq__subst,axiom,
! [A2: a > $o,B2: a > $o,F: ( a > $o ) > set_a,C: set_a] :
( ( ord_less_a_o @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X6: a > $o,Y3: a > $o] :
( ( ord_less_a_o @ X6 @ Y3 )
=> ( ord_less_set_a @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_73_ord__less__eq__subst,axiom,
! [A2: a > $o,B2: a > $o,F: ( a > $o ) > a > $o,C: a > $o] :
( ( ord_less_a_o @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X6: a > $o,Y3: a > $o] :
( ( ord_less_a_o @ X6 @ Y3 )
=> ( ord_less_a_o @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_74_ord__less__eq__subst,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X6: int,Y3: int] :
( ( ord_less_int @ X6 @ Y3 )
=> ( ord_less_int @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_75_order__less__subst1,axiom,
! [A2: int,F: set_a > int,B2: set_a,C: set_a] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X6: set_a,Y3: set_a] :
( ( ord_less_set_a @ X6 @ Y3 )
=> ( ord_less_int @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_76_order__less__subst1,axiom,
! [A2: int,F: ( a > $o ) > int,B2: a > $o,C: a > $o] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_a_o @ B2 @ C )
=> ( ! [X6: a > $o,Y3: a > $o] :
( ( ord_less_a_o @ X6 @ Y3 )
=> ( ord_less_int @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_77_order__less__subst1,axiom,
! [A2: set_a,F: int > set_a,B2: int,C: int] :
( ( ord_less_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X6: int,Y3: int] :
( ( ord_less_int @ X6 @ Y3 )
=> ( ord_less_set_a @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_78_order__less__subst1,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X6: set_a,Y3: set_a] :
( ( ord_less_set_a @ X6 @ Y3 )
=> ( ord_less_set_a @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_79_order__less__subst1,axiom,
! [A2: set_a,F: ( a > $o ) > set_a,B2: a > $o,C: a > $o] :
( ( ord_less_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_a_o @ B2 @ C )
=> ( ! [X6: a > $o,Y3: a > $o] :
( ( ord_less_a_o @ X6 @ Y3 )
=> ( ord_less_set_a @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_80_order__less__subst1,axiom,
! [A2: a > $o,F: int > a > $o,B2: int,C: int] :
( ( ord_less_a_o @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X6: int,Y3: int] :
( ( ord_less_int @ X6 @ Y3 )
=> ( ord_less_a_o @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_81_order__less__subst1,axiom,
! [A2: a > $o,F: set_a > a > $o,B2: set_a,C: set_a] :
( ( ord_less_a_o @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X6: set_a,Y3: set_a] :
( ( ord_less_set_a @ X6 @ Y3 )
=> ( ord_less_a_o @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_82_order__less__subst1,axiom,
! [A2: a > $o,F: ( a > $o ) > a > $o,B2: a > $o,C: a > $o] :
( ( ord_less_a_o @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_a_o @ B2 @ C )
=> ( ! [X6: a > $o,Y3: a > $o] :
( ( ord_less_a_o @ X6 @ Y3 )
=> ( ord_less_a_o @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_83_order__less__subst1,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X6: int,Y3: int] :
( ( ord_less_int @ X6 @ Y3 )
=> ( ord_less_int @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_84_order__less__subst2,axiom,
! [A2: int,B2: int,F: int > set_a,C: set_a] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_set_a @ ( F @ B2 ) @ C )
=> ( ! [X6: int,Y3: int] :
( ( ord_less_int @ X6 @ Y3 )
=> ( ord_less_set_a @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_85_order__less__subst2,axiom,
! [A2: int,B2: int,F: int > a > $o,C: a > $o] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_a_o @ ( F @ B2 ) @ C )
=> ( ! [X6: int,Y3: int] :
( ( ord_less_int @ X6 @ Y3 )
=> ( ord_less_a_o @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_86_order__less__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > int,C: int] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X6: set_a,Y3: set_a] :
( ( ord_less_set_a @ X6 @ Y3 )
=> ( ord_less_int @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_87_order__less__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ ( F @ B2 ) @ C )
=> ( ! [X6: set_a,Y3: set_a] :
( ( ord_less_set_a @ X6 @ Y3 )
=> ( ord_less_set_a @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_88_order__less__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > a > $o,C: a > $o] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_a_o @ ( F @ B2 ) @ C )
=> ( ! [X6: set_a,Y3: set_a] :
( ( ord_less_set_a @ X6 @ Y3 )
=> ( ord_less_a_o @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_89_order__less__subst2,axiom,
! [A2: a > $o,B2: a > $o,F: ( a > $o ) > int,C: int] :
( ( ord_less_a_o @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X6: a > $o,Y3: a > $o] :
( ( ord_less_a_o @ X6 @ Y3 )
=> ( ord_less_int @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_90_order__less__subst2,axiom,
! [A2: a > $o,B2: a > $o,F: ( a > $o ) > set_a,C: set_a] :
( ( ord_less_a_o @ A2 @ B2 )
=> ( ( ord_less_set_a @ ( F @ B2 ) @ C )
=> ( ! [X6: a > $o,Y3: a > $o] :
( ( ord_less_a_o @ X6 @ Y3 )
=> ( ord_less_set_a @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_91_order__less__subst2,axiom,
! [A2: a > $o,B2: a > $o,F: ( a > $o ) > a > $o,C: a > $o] :
( ( ord_less_a_o @ A2 @ B2 )
=> ( ( ord_less_a_o @ ( F @ B2 ) @ C )
=> ( ! [X6: a > $o,Y3: a > $o] :
( ( ord_less_a_o @ X6 @ Y3 )
=> ( ord_less_a_o @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_92_order__less__subst2,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X6: int,Y3: int] :
( ( ord_less_int @ X6 @ Y3 )
=> ( ord_less_int @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_93_lt__ex,axiom,
! [X: int] :
? [Y3: int] : ( ord_less_int @ Y3 @ X ) ).
% lt_ex
thf(fact_94_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_95_neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% neqE
thf(fact_96_neq__iff,axiom,
! [X: int,Y: int] :
( ( X != Y )
= ( ( ord_less_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ) ).
% neq_iff
thf(fact_97_order_Oasym,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ~ ( ord_less_set_a @ B2 @ A2 ) ) ).
% order.asym
thf(fact_98_order_Oasym,axiom,
! [A2: a > $o,B2: a > $o] :
( ( ord_less_a_o @ A2 @ B2 )
=> ~ ( ord_less_a_o @ B2 @ A2 ) ) ).
% order.asym
thf(fact_99_order_Oasym,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ~ ( ord_less_int @ B2 @ A2 ) ) ).
% order.asym
thf(fact_100_less__imp__neq,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_101_less__imp__neq,axiom,
! [X: a > $o,Y: a > $o] :
( ( ord_less_a_o @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_102_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_103_less__asym,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ~ ( ord_less_set_a @ Y @ X ) ) ).
% less_asym
thf(fact_104_less__asym,axiom,
! [X: a > $o,Y: a > $o] :
( ( ord_less_a_o @ X @ Y )
=> ~ ( ord_less_a_o @ Y @ X ) ) ).
% less_asym
thf(fact_105_less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% less_asym
thf(fact_106_less__asym_H,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ~ ( ord_less_set_a @ B2 @ A2 ) ) ).
% less_asym'
thf(fact_107_less__asym_H,axiom,
! [A2: a > $o,B2: a > $o] :
( ( ord_less_a_o @ A2 @ B2 )
=> ~ ( ord_less_a_o @ B2 @ A2 ) ) ).
% less_asym'
thf(fact_108_less__asym_H,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ~ ( ord_less_int @ B2 @ A2 ) ) ).
% less_asym'
thf(fact_109_less__trans,axiom,
! [X: set_a,Y: set_a,Z3: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( ( ord_less_set_a @ Y @ Z3 )
=> ( ord_less_set_a @ X @ Z3 ) ) ) ).
% less_trans
thf(fact_110_less__trans,axiom,
! [X: a > $o,Y: a > $o,Z3: a > $o] :
( ( ord_less_a_o @ X @ Y )
=> ( ( ord_less_a_o @ Y @ Z3 )
=> ( ord_less_a_o @ X @ Z3 ) ) ) ).
% less_trans
thf(fact_111_less__trans,axiom,
! [X: int,Y: int,Z3: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z3 )
=> ( ord_less_int @ X @ Z3 ) ) ) ).
% less_trans
thf(fact_112_less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
% less_linear
thf(fact_113_less__irrefl,axiom,
! [X: set_a] :
~ ( ord_less_set_a @ X @ X ) ).
% less_irrefl
thf(fact_114_less__irrefl,axiom,
! [X: a > $o] :
~ ( ord_less_a_o @ X @ X ) ).
% less_irrefl
thf(fact_115_less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% less_irrefl
thf(fact_116_ord__eq__less__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( A2 = B2 )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_117_ord__eq__less__trans,axiom,
! [A2: a > $o,B2: a > $o,C: a > $o] :
( ( A2 = B2 )
=> ( ( ord_less_a_o @ B2 @ C )
=> ( ord_less_a_o @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_118_ord__eq__less__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( A2 = B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_119_ord__less__eq__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_120_ord__less__eq__trans,axiom,
! [A2: a > $o,B2: a > $o,C: a > $o] :
( ( ord_less_a_o @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_a_o @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_121_ord__less__eq__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_122_dual__order_Oasym,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
=> ~ ( ord_less_set_a @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_123_dual__order_Oasym,axiom,
! [B2: a > $o,A2: a > $o] :
( ( ord_less_a_o @ B2 @ A2 )
=> ~ ( ord_less_a_o @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_124_dual__order_Oasym,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
=> ~ ( ord_less_int @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_125_less__imp__not__eq,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_not_eq
thf(fact_126_less__imp__not__eq,axiom,
! [X: a > $o,Y: a > $o] :
( ( ord_less_a_o @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_not_eq
thf(fact_127_less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_not_eq
thf(fact_128_less__not__sym,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ~ ( ord_less_set_a @ Y @ X ) ) ).
% less_not_sym
thf(fact_129_less__not__sym,axiom,
! [X: a > $o,Y: a > $o] :
( ( ord_less_a_o @ X @ Y )
=> ~ ( ord_less_a_o @ Y @ X ) ) ).
% less_not_sym
thf(fact_130_less__not__sym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% less_not_sym
thf(fact_131_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_132_less__imp__not__eq2,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( Y != X ) ) ).
% less_imp_not_eq2
thf(fact_133_less__imp__not__eq2,axiom,
! [X: a > $o,Y: a > $o] :
( ( ord_less_a_o @ X @ Y )
=> ( Y != X ) ) ).
% less_imp_not_eq2
thf(fact_134_less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% less_imp_not_eq2
thf(fact_135_less__imp__triv,axiom,
! [X: set_a,Y: set_a,P: $o] :
( ( ord_less_set_a @ X @ Y )
=> ( ( ord_less_set_a @ Y @ X )
=> P ) ) ).
% less_imp_triv
thf(fact_136_less__imp__triv,axiom,
! [X: a > $o,Y: a > $o,P: $o] :
( ( ord_less_a_o @ X @ Y )
=> ( ( ord_less_a_o @ Y @ X )
=> P ) ) ).
% less_imp_triv
thf(fact_137_less__imp__triv,axiom,
! [X: int,Y: int,P: $o] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ X )
=> P ) ) ).
% less_imp_triv
thf(fact_138_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_139_dual__order_Oirrefl,axiom,
! [A2: set_a] :
~ ( ord_less_set_a @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_140_dual__order_Oirrefl,axiom,
! [A2: a > $o] :
~ ( ord_less_a_o @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_141_dual__order_Oirrefl,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_142_order_Ostrict__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_143_order_Ostrict__trans,axiom,
! [A2: a > $o,B2: a > $o,C: a > $o] :
( ( ord_less_a_o @ A2 @ B2 )
=> ( ( ord_less_a_o @ B2 @ C )
=> ( ord_less_a_o @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_144_order_Ostrict__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_145_less__imp__not__less,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ~ ( ord_less_set_a @ Y @ X ) ) ).
% less_imp_not_less
thf(fact_146_less__imp__not__less,axiom,
! [X: a > $o,Y: a > $o] :
( ( ord_less_a_o @ X @ Y )
=> ~ ( ord_less_a_o @ Y @ X ) ) ).
% less_imp_not_less
thf(fact_147_less__imp__not__less,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% less_imp_not_less
thf(fact_148_linorder__less__wlog,axiom,
! [P: int > int > $o,A2: int,B2: int] :
( ! [A4: int,B3: int] :
( ( ord_less_int @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: int] : ( P @ A4 @ A4 )
=> ( ! [A4: int,B3: int] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_149_verit__comp__simplify1_I1_J,axiom,
! [A2: set_a] :
~ ( ord_less_set_a @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_150_verit__comp__simplify1_I1_J,axiom,
! [A2: a > $o] :
~ ( ord_less_a_o @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_151_verit__comp__simplify1_I1_J,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_152_setOf_Osimps_I1_J,axiom,
( ( binary945792244etOf_a @ binary476621312_Tip_a )
= bot_bot_set_a ) ).
% setOf.simps(1)
thf(fact_153_bot__apply,axiom,
( bot_bot_a_o
= ( ^ [X4: a] : bot_bot_o ) ) ).
% bot_apply
thf(fact_154_bot__fun__def,axiom,
( bot_bot_a_o
= ( ^ [X4: a] : bot_bot_o ) ) ).
% bot_fun_def
thf(fact_155_bot_Oextremum__strict,axiom,
! [A2: set_a] :
~ ( ord_less_set_a @ A2 @ bot_bot_set_a ) ).
% bot.extremum_strict
thf(fact_156_bot_Oextremum__strict,axiom,
! [A2: a > $o] :
~ ( ord_less_a_o @ A2 @ bot_bot_a_o ) ).
% bot.extremum_strict
thf(fact_157_bot_Onot__eq__extremum,axiom,
! [A2: set_a] :
( ( A2 != bot_bot_set_a )
= ( ord_less_set_a @ bot_bot_set_a @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_158_bot_Onot__eq__extremum,axiom,
! [A2: a > $o] :
( ( A2 != bot_bot_a_o )
= ( ord_less_a_o @ bot_bot_a_o @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_159_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X4: a] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_160_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X4: a] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_161_all__not__in__conv,axiom,
! [A3: set_set_a] :
( ( ! [X4: set_a] :
~ ( member_set_a @ X4 @ A3 ) )
= ( A3 = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_162_all__not__in__conv,axiom,
! [A3: set_a] :
( ( ! [X4: a] :
~ ( member_a @ X4 @ A3 ) )
= ( A3 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_163_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_164_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_165_empty__def,axiom,
( bot_bot_set_a
= ( collect_a
@ ^ [X4: a] : $false ) ) ).
% empty_def
thf(fact_166_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_167_not__psubset__empty,axiom,
! [A3: set_a] :
~ ( ord_less_set_a @ A3 @ bot_bot_set_a ) ).
% not_psubset_empty
thf(fact_168_emptyE,axiom,
! [A2: set_a] :
~ ( member_set_a @ A2 @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_169_emptyE,axiom,
! [A2: a] :
~ ( member_a @ A2 @ bot_bot_set_a ) ).
% emptyE
thf(fact_170_equals0D,axiom,
! [A3: set_set_a,A2: set_a] :
( ( A3 = bot_bot_set_set_a )
=> ~ ( member_set_a @ A2 @ A3 ) ) ).
% equals0D
thf(fact_171_equals0D,axiom,
! [A3: set_a,A2: a] :
( ( A3 = bot_bot_set_a )
=> ~ ( member_a @ A2 @ A3 ) ) ).
% equals0D
thf(fact_172_equals0I,axiom,
! [A3: set_set_a] :
( ! [Y3: set_a] :
~ ( member_set_a @ Y3 @ A3 )
=> ( A3 = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_173_equals0I,axiom,
! [A3: set_a] :
( ! [Y3: a] :
~ ( member_a @ Y3 @ A3 )
=> ( A3 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_174_ex__in__conv,axiom,
! [A3: set_set_a] :
( ( ? [X4: set_a] : ( member_set_a @ X4 @ A3 ) )
= ( A3 != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_175_ex__in__conv,axiom,
! [A3: set_a] :
( ( ? [X4: a] : ( member_a @ X4 @ A3 ) )
= ( A3 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_176_Set_Ois__empty__def,axiom,
( is_empty_a
= ( ^ [A5: set_a] : A5 = bot_bot_set_a ) ) ).
% Set.is_empty_def
thf(fact_177_Collect__empty__eq__bot,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( P = bot_bot_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_178_bot__empty__eq,axiom,
( bot_bot_set_a_o
= ( ^ [X4: set_a] : ( member_set_a @ X4 @ bot_bot_set_set_a ) ) ) ).
% bot_empty_eq
thf(fact_179_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X4: a] : ( member_a @ X4 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_180_Tree_Osimps_I14_J,axiom,
( ( binary256242811Tree_a @ binary476621312_Tip_a )
= bot_bot_set_a ) ).
% Tree.simps(14)
thf(fact_181_is__singletonI_H,axiom,
! [A3: set_set_a] :
( ( A3 != bot_bot_set_set_a )
=> ( ! [X6: set_a,Y3: set_a] :
( ( member_set_a @ X6 @ A3 )
=> ( ( member_set_a @ Y3 @ A3 )
=> ( X6 = Y3 ) ) )
=> ( is_singleton_set_a @ A3 ) ) ) ).
% is_singletonI'
thf(fact_182_is__singletonI_H,axiom,
! [A3: set_a] :
( ( A3 != bot_bot_set_a )
=> ( ! [X6: a,Y3: a] :
( ( member_a @ X6 @ A3 )
=> ( ( member_a @ Y3 @ A3 )
=> ( X6 = Y3 ) ) )
=> ( is_singleton_a @ A3 ) ) ) ).
% is_singletonI'
thf(fact_183_psubsetD,axiom,
! [A3: set_set_a,B4: set_set_a,C: set_a] :
( ( ord_less_set_set_a @ A3 @ B4 )
=> ( ( member_set_a @ C @ A3 )
=> ( member_set_a @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_184_psubsetD,axiom,
! [A3: set_a,B4: set_a,C: a] :
( ( ord_less_set_a @ A3 @ B4 )
=> ( ( member_a @ C @ A3 )
=> ( member_a @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_185_less__set__def,axiom,
( ord_less_set_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
( ord_less_set_a_o
@ ^ [X4: set_a] : ( member_set_a @ X4 @ A5 )
@ ^ [X4: set_a] : ( member_set_a @ X4 @ B5 ) ) ) ) ).
% less_set_def
thf(fact_186_less__set__def,axiom,
( ord_less_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ord_less_a_o
@ ^ [X4: a] : ( member_a @ X4 @ A5 )
@ ^ [X4: a] : ( member_a @ X4 @ B5 ) ) ) ) ).
% less_set_def
thf(fact_187_psubset__trans,axiom,
! [A3: set_a,B4: set_a,C2: set_a] :
( ( ord_less_set_a @ A3 @ B4 )
=> ( ( ord_less_set_a @ B4 @ C2 )
=> ( ord_less_set_a @ A3 @ C2 ) ) ) ).
% psubset_trans
thf(fact_188_Tree_Oset__cases,axiom,
! [E: set_a,A2: binary594033953_set_a] :
( ( member_set_a @ E @ ( binary1613048283_set_a @ A2 ) )
=> ( ! [Z1: binary594033953_set_a] :
( ? [Z22: set_a,Z32: binary594033953_set_a] :
( A2
= ( binary313540327_set_a @ Z1 @ Z22 @ Z32 ) )
=> ~ ( member_set_a @ E @ ( binary1613048283_set_a @ Z1 ) ) )
=> ( ! [Z1: binary594033953_set_a,Z32: binary594033953_set_a] :
( A2
!= ( binary313540327_set_a @ Z1 @ E @ Z32 ) )
=> ~ ! [Z1: binary594033953_set_a,Z22: set_a,Z32: binary594033953_set_a] :
( ( A2
= ( binary313540327_set_a @ Z1 @ Z22 @ Z32 ) )
=> ~ ( member_set_a @ E @ ( binary1613048283_set_a @ Z32 ) ) ) ) ) ) ).
% Tree.set_cases
thf(fact_189_Tree_Oset__cases,axiom,
! [E: a,A2: binary1439146945Tree_a] :
( ( member_a @ E @ ( binary256242811Tree_a @ A2 ) )
=> ( ! [Z1: binary1439146945Tree_a] :
( ? [Z22: a,Z32: binary1439146945Tree_a] :
( A2
= ( binary717961607le_T_a @ Z1 @ Z22 @ Z32 ) )
=> ~ ( member_a @ E @ ( binary256242811Tree_a @ Z1 ) ) )
=> ( ! [Z1: binary1439146945Tree_a,Z32: binary1439146945Tree_a] :
( A2
!= ( binary717961607le_T_a @ Z1 @ E @ Z32 ) )
=> ~ ! [Z1: binary1439146945Tree_a,Z22: a,Z32: binary1439146945Tree_a] :
( ( A2
= ( binary717961607le_T_a @ Z1 @ Z22 @ Z32 ) )
=> ~ ( member_a @ E @ ( binary256242811Tree_a @ Z32 ) ) ) ) ) ) ).
% Tree.set_cases
thf(fact_190_Tree_Oset__intros_I1_J,axiom,
! [Y: set_a,X21: binary594033953_set_a,X22: set_a,X23: binary594033953_set_a] :
( ( member_set_a @ Y @ ( binary1613048283_set_a @ X21 ) )
=> ( member_set_a @ Y @ ( binary1613048283_set_a @ ( binary313540327_set_a @ X21 @ X22 @ X23 ) ) ) ) ).
% Tree.set_intros(1)
thf(fact_191_Tree_Oset__intros_I1_J,axiom,
! [Y: a,X21: binary1439146945Tree_a,X22: a,X23: binary1439146945Tree_a] :
( ( member_a @ Y @ ( binary256242811Tree_a @ X21 ) )
=> ( member_a @ Y @ ( binary256242811Tree_a @ ( binary717961607le_T_a @ X21 @ X22 @ X23 ) ) ) ) ).
% Tree.set_intros(1)
thf(fact_192_Tree_Oset__intros_I2_J,axiom,
! [X22: set_a,X21: binary594033953_set_a,X23: binary594033953_set_a] : ( member_set_a @ X22 @ ( binary1613048283_set_a @ ( binary313540327_set_a @ X21 @ X22 @ X23 ) ) ) ).
% Tree.set_intros(2)
thf(fact_193_Tree_Oset__intros_I2_J,axiom,
! [X22: a,X21: binary1439146945Tree_a,X23: binary1439146945Tree_a] : ( member_a @ X22 @ ( binary256242811Tree_a @ ( binary717961607le_T_a @ X21 @ X22 @ X23 ) ) ) ).
% Tree.set_intros(2)
thf(fact_194_Tree_Oset__intros_I3_J,axiom,
! [Ya: set_a,X23: binary594033953_set_a,X21: binary594033953_set_a,X22: set_a] :
( ( member_set_a @ Ya @ ( binary1613048283_set_a @ X23 ) )
=> ( member_set_a @ Ya @ ( binary1613048283_set_a @ ( binary313540327_set_a @ X21 @ X22 @ X23 ) ) ) ) ).
% Tree.set_intros(3)
thf(fact_195_Tree_Oset__intros_I3_J,axiom,
! [Ya: a,X23: binary1439146945Tree_a,X21: binary1439146945Tree_a,X22: a] :
( ( member_a @ Ya @ ( binary256242811Tree_a @ X23 ) )
=> ( member_a @ Ya @ ( binary256242811Tree_a @ ( binary717961607le_T_a @ X21 @ X22 @ X23 ) ) ) ) ).
% Tree.set_intros(3)
thf(fact_196_is__singletonI,axiom,
! [X: a] : ( is_singleton_a @ ( insert_a @ X @ bot_bot_set_a ) ) ).
% is_singletonI
thf(fact_197_is__singleton__def,axiom,
( is_singleton_a
= ( ^ [A5: set_a] :
? [X4: a] :
( A5
= ( insert_a @ X4 @ bot_bot_set_a ) ) ) ) ).
% is_singleton_def
thf(fact_198_is__singletonE,axiom,
! [A3: set_a] :
( ( is_singleton_a @ A3 )
=> ~ ! [X6: a] :
( A3
!= ( insert_a @ X6 @ bot_bot_set_a ) ) ) ).
% is_singletonE
thf(fact_199_insert__absorb2,axiom,
! [X: a,A3: set_a] :
( ( insert_a @ X @ ( insert_a @ X @ A3 ) )
= ( insert_a @ X @ A3 ) ) ).
% insert_absorb2
thf(fact_200_insert__iff,axiom,
! [A2: set_a,B2: set_a,A3: set_set_a] :
( ( member_set_a @ A2 @ ( insert_set_a @ B2 @ A3 ) )
= ( ( A2 = B2 )
| ( member_set_a @ A2 @ A3 ) ) ) ).
% insert_iff
thf(fact_201_insert__iff,axiom,
! [A2: a,B2: a,A3: set_a] :
( ( member_a @ A2 @ ( insert_a @ B2 @ A3 ) )
= ( ( A2 = B2 )
| ( member_a @ A2 @ A3 ) ) ) ).
% insert_iff
thf(fact_202_insertCI,axiom,
! [A2: set_a,B4: set_set_a,B2: set_a] :
( ( ~ ( member_set_a @ A2 @ B4 )
=> ( A2 = B2 ) )
=> ( member_set_a @ A2 @ ( insert_set_a @ B2 @ B4 ) ) ) ).
% insertCI
thf(fact_203_insertCI,axiom,
! [A2: a,B4: set_a,B2: a] :
( ( ~ ( member_a @ A2 @ B4 )
=> ( A2 = B2 ) )
=> ( member_a @ A2 @ ( insert_a @ B2 @ B4 ) ) ) ).
% insertCI
thf(fact_204_singletonI,axiom,
! [A2: set_a] : ( member_set_a @ A2 @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) ) ).
% singletonI
thf(fact_205_singletonI,axiom,
! [A2: a] : ( member_a @ A2 @ ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_206_singleton__conv,axiom,
! [A2: a] :
( ( collect_a
@ ^ [X4: a] : X4 = A2 )
= ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% singleton_conv
thf(fact_207_singleton__conv2,axiom,
! [A2: a] :
( ( collect_a
@ ( ^ [Y4: a,Z4: a] : Y4 = Z4
@ A2 ) )
= ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% singleton_conv2
thf(fact_208_singleton__inject,axiom,
! [A2: a,B2: a] :
( ( ( insert_a @ A2 @ bot_bot_set_a )
= ( insert_a @ B2 @ bot_bot_set_a ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_209_insert__not__empty,axiom,
! [A2: a,A3: set_a] :
( ( insert_a @ A2 @ A3 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_210_doubleton__eq__iff,axiom,
! [A2: a,B2: a,C: a,D: a] :
( ( ( insert_a @ A2 @ ( insert_a @ B2 @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A2 = C )
& ( B2 = D ) )
| ( ( A2 = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_211_singleton__iff,axiom,
! [B2: set_a,A2: set_a] :
( ( member_set_a @ B2 @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_212_singleton__iff,axiom,
! [B2: a,A2: a] :
( ( member_a @ B2 @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_213_singletonD,axiom,
! [B2: set_a,A2: set_a] :
( ( member_set_a @ B2 @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_214_singletonD,axiom,
! [B2: a,A2: a] :
( ( member_a @ B2 @ ( insert_a @ A2 @ bot_bot_set_a ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_215_insert__compr,axiom,
( insert_set_a
= ( ^ [A: set_a,B5: set_set_a] :
( collect_set_a
@ ^ [X4: set_a] :
( ( X4 = A )
| ( member_set_a @ X4 @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_216_insert__compr,axiom,
( insert_a
= ( ^ [A: a,B5: set_a] :
( collect_a
@ ^ [X4: a] :
( ( X4 = A )
| ( member_a @ X4 @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_217_insert__Collect,axiom,
! [A2: a,P: a > $o] :
( ( insert_a @ A2 @ ( collect_a @ P ) )
= ( collect_a
@ ^ [U: a] :
( ( U != A2 )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_218_mk__disjoint__insert,axiom,
! [A2: set_a,A3: set_set_a] :
( ( member_set_a @ A2 @ A3 )
=> ? [B6: set_set_a] :
( ( A3
= ( insert_set_a @ A2 @ B6 ) )
& ~ ( member_set_a @ A2 @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_219_mk__disjoint__insert,axiom,
! [A2: a,A3: set_a] :
( ( member_a @ A2 @ A3 )
=> ? [B6: set_a] :
( ( A3
= ( insert_a @ A2 @ B6 ) )
& ~ ( member_a @ A2 @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_220_insert__commute,axiom,
! [X: a,Y: a,A3: set_a] :
( ( insert_a @ X @ ( insert_a @ Y @ A3 ) )
= ( insert_a @ Y @ ( insert_a @ X @ A3 ) ) ) ).
% insert_commute
thf(fact_221_insert__eq__iff,axiom,
! [A2: set_a,A3: set_set_a,B2: set_a,B4: set_set_a] :
( ~ ( member_set_a @ A2 @ A3 )
=> ( ~ ( member_set_a @ B2 @ B4 )
=> ( ( ( insert_set_a @ A2 @ A3 )
= ( insert_set_a @ B2 @ B4 ) )
= ( ( ( A2 = B2 )
=> ( A3 = B4 ) )
& ( ( A2 != B2 )
=> ? [C3: set_set_a] :
( ( A3
= ( insert_set_a @ B2 @ C3 ) )
& ~ ( member_set_a @ B2 @ C3 )
& ( B4
= ( insert_set_a @ A2 @ C3 ) )
& ~ ( member_set_a @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_222_insert__eq__iff,axiom,
! [A2: a,A3: set_a,B2: a,B4: set_a] :
( ~ ( member_a @ A2 @ A3 )
=> ( ~ ( member_a @ B2 @ B4 )
=> ( ( ( insert_a @ A2 @ A3 )
= ( insert_a @ B2 @ B4 ) )
= ( ( ( A2 = B2 )
=> ( A3 = B4 ) )
& ( ( A2 != B2 )
=> ? [C3: set_a] :
( ( A3
= ( insert_a @ B2 @ C3 ) )
& ~ ( member_a @ B2 @ C3 )
& ( B4
= ( insert_a @ A2 @ C3 ) )
& ~ ( member_a @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_223_insert__absorb,axiom,
! [A2: set_a,A3: set_set_a] :
( ( member_set_a @ A2 @ A3 )
=> ( ( insert_set_a @ A2 @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_224_insert__absorb,axiom,
! [A2: a,A3: set_a] :
( ( member_a @ A2 @ A3 )
=> ( ( insert_a @ A2 @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_225_insert__ident,axiom,
! [X: set_a,A3: set_set_a,B4: set_set_a] :
( ~ ( member_set_a @ X @ A3 )
=> ( ~ ( member_set_a @ X @ B4 )
=> ( ( ( insert_set_a @ X @ A3 )
= ( insert_set_a @ X @ B4 ) )
= ( A3 = B4 ) ) ) ) ).
% insert_ident
thf(fact_226_insert__ident,axiom,
! [X: a,A3: set_a,B4: set_a] :
( ~ ( member_a @ X @ A3 )
=> ( ~ ( member_a @ X @ B4 )
=> ( ( ( insert_a @ X @ A3 )
= ( insert_a @ X @ B4 ) )
= ( A3 = B4 ) ) ) ) ).
% insert_ident
thf(fact_227_Set_Oset__insert,axiom,
! [X: set_a,A3: set_set_a] :
( ( member_set_a @ X @ A3 )
=> ~ ! [B6: set_set_a] :
( ( A3
= ( insert_set_a @ X @ B6 ) )
=> ( member_set_a @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_228_Set_Oset__insert,axiom,
! [X: a,A3: set_a] :
( ( member_a @ X @ A3 )
=> ~ ! [B6: set_a] :
( ( A3
= ( insert_a @ X @ B6 ) )
=> ( member_a @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_229_insertI2,axiom,
! [A2: set_a,B4: set_set_a,B2: set_a] :
( ( member_set_a @ A2 @ B4 )
=> ( member_set_a @ A2 @ ( insert_set_a @ B2 @ B4 ) ) ) ).
% insertI2
thf(fact_230_insertI2,axiom,
! [A2: a,B4: set_a,B2: a] :
( ( member_a @ A2 @ B4 )
=> ( member_a @ A2 @ ( insert_a @ B2 @ B4 ) ) ) ).
% insertI2
thf(fact_231_insertI1,axiom,
! [A2: set_a,B4: set_set_a] : ( member_set_a @ A2 @ ( insert_set_a @ A2 @ B4 ) ) ).
% insertI1
thf(fact_232_insertI1,axiom,
! [A2: a,B4: set_a] : ( member_a @ A2 @ ( insert_a @ A2 @ B4 ) ) ).
% insertI1
thf(fact_233_insertE,axiom,
! [A2: set_a,B2: set_a,A3: set_set_a] :
( ( member_set_a @ A2 @ ( insert_set_a @ B2 @ A3 ) )
=> ( ( A2 != B2 )
=> ( member_set_a @ A2 @ A3 ) ) ) ).
% insertE
thf(fact_234_insertE,axiom,
! [A2: a,B2: a,A3: set_a] :
( ( member_a @ A2 @ ( insert_a @ B2 @ A3 ) )
=> ( ( A2 != B2 )
=> ( member_a @ A2 @ A3 ) ) ) ).
% insertE
thf(fact_235_Collect__conv__if,axiom,
! [P: a > $o,A2: a] :
( ( ( P @ A2 )
=> ( ( collect_a
@ ^ [X4: a] :
( ( X4 = A2 )
& ( P @ X4 ) ) )
= ( insert_a @ A2 @ bot_bot_set_a ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect_a
@ ^ [X4: a] :
( ( X4 = A2 )
& ( P @ X4 ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if
thf(fact_236_Collect__conv__if2,axiom,
! [P: a > $o,A2: a] :
( ( ( P @ A2 )
=> ( ( collect_a
@ ^ [X4: a] :
( ( A2 = X4 )
& ( P @ X4 ) ) )
= ( insert_a @ A2 @ bot_bot_set_a ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect_a
@ ^ [X4: a] :
( ( A2 = X4 )
& ( P @ X4 ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if2
thf(fact_237_is__singleton__the__elem,axiom,
( is_singleton_a
= ( ^ [A5: set_a] :
( A5
= ( insert_a @ ( the_elem_a @ A5 ) @ bot_bot_set_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_238_the__elem__eq,axiom,
! [X: a] :
( ( the_elem_a @ ( insert_a @ X @ bot_bot_set_a ) )
= X ) ).
% the_elem_eq
thf(fact_239_setOf_Osimps_I2_J,axiom,
! [T1: binary1439146945Tree_a,X: a,T2: binary1439146945Tree_a] :
( ( binary945792244etOf_a @ ( binary717961607le_T_a @ T1 @ X @ T2 ) )
= ( sup_sup_set_a @ ( sup_sup_set_a @ ( binary945792244etOf_a @ T1 ) @ ( binary945792244etOf_a @ T2 ) ) @ ( insert_a @ X @ bot_bot_set_a ) ) ) ).
% setOf.simps(2)
thf(fact_240_Tree_Osimps_I15_J,axiom,
! [X21: binary1439146945Tree_a,X22: a,X23: binary1439146945Tree_a] :
( ( binary256242811Tree_a @ ( binary717961607le_T_a @ X21 @ X22 @ X23 ) )
= ( sup_sup_set_a @ ( sup_sup_set_a @ ( binary256242811Tree_a @ X21 ) @ ( insert_a @ X22 @ bot_bot_set_a ) ) @ ( binary256242811Tree_a @ X23 ) ) ) ).
% Tree.simps(15)
thf(fact_241_UnCI,axiom,
! [C: set_a,B4: set_set_a,A3: set_set_a] :
( ( ~ ( member_set_a @ C @ B4 )
=> ( member_set_a @ C @ A3 ) )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A3 @ B4 ) ) ) ).
% UnCI
thf(fact_242_UnCI,axiom,
! [C: a,B4: set_a,A3: set_a] :
( ( ~ ( member_a @ C @ B4 )
=> ( member_a @ C @ A3 ) )
=> ( member_a @ C @ ( sup_sup_set_a @ A3 @ B4 ) ) ) ).
% UnCI
thf(fact_243_Un__iff,axiom,
! [C: set_a,A3: set_set_a,B4: set_set_a] :
( ( member_set_a @ C @ ( sup_sup_set_set_a @ A3 @ B4 ) )
= ( ( member_set_a @ C @ A3 )
| ( member_set_a @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_244_Un__iff,axiom,
! [C: a,A3: set_a,B4: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A3 @ B4 ) )
= ( ( member_a @ C @ A3 )
| ( member_a @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_245_Un__empty,axiom,
! [A3: set_a,B4: set_a] :
( ( ( sup_sup_set_a @ A3 @ B4 )
= bot_bot_set_a )
= ( ( A3 = bot_bot_set_a )
& ( B4 = bot_bot_set_a ) ) ) ).
% Un_empty
thf(fact_246_Un__insert__right,axiom,
! [A3: set_a,A2: a,B4: set_a] :
( ( sup_sup_set_a @ A3 @ ( insert_a @ A2 @ B4 ) )
= ( insert_a @ A2 @ ( sup_sup_set_a @ A3 @ B4 ) ) ) ).
% Un_insert_right
thf(fact_247_Un__insert__left,axiom,
! [A2: a,B4: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( insert_a @ A2 @ B4 ) @ C2 )
= ( insert_a @ A2 @ ( sup_sup_set_a @ B4 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_248_UnE,axiom,
! [C: set_a,A3: set_set_a,B4: set_set_a] :
( ( member_set_a @ C @ ( sup_sup_set_set_a @ A3 @ B4 ) )
=> ( ~ ( member_set_a @ C @ A3 )
=> ( member_set_a @ C @ B4 ) ) ) ).
% UnE
thf(fact_249_UnE,axiom,
! [C: a,A3: set_a,B4: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A3 @ B4 ) )
=> ( ~ ( member_a @ C @ A3 )
=> ( member_a @ C @ B4 ) ) ) ).
% UnE
thf(fact_250_UnI1,axiom,
! [C: set_a,A3: set_set_a,B4: set_set_a] :
( ( member_set_a @ C @ A3 )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A3 @ B4 ) ) ) ).
% UnI1
thf(fact_251_UnI1,axiom,
! [C: a,A3: set_a,B4: set_a] :
( ( member_a @ C @ A3 )
=> ( member_a @ C @ ( sup_sup_set_a @ A3 @ B4 ) ) ) ).
% UnI1
thf(fact_252_UnI2,axiom,
! [C: set_a,B4: set_set_a,A3: set_set_a] :
( ( member_set_a @ C @ B4 )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A3 @ B4 ) ) ) ).
% UnI2
thf(fact_253_UnI2,axiom,
! [C: a,B4: set_a,A3: set_a] :
( ( member_a @ C @ B4 )
=> ( member_a @ C @ ( sup_sup_set_a @ A3 @ B4 ) ) ) ).
% UnI2
thf(fact_254_bex__Un,axiom,
! [A3: set_a,B4: set_a,P: a > $o] :
( ( ? [X4: a] :
( ( member_a @ X4 @ ( sup_sup_set_a @ A3 @ B4 ) )
& ( P @ X4 ) ) )
= ( ? [X4: a] :
( ( member_a @ X4 @ A3 )
& ( P @ X4 ) )
| ? [X4: a] :
( ( member_a @ X4 @ B4 )
& ( P @ X4 ) ) ) ) ).
% bex_Un
thf(fact_255_ball__Un,axiom,
! [A3: set_a,B4: set_a,P: a > $o] :
( ( ! [X4: a] :
( ( member_a @ X4 @ ( sup_sup_set_a @ A3 @ B4 ) )
=> ( P @ X4 ) ) )
= ( ! [X4: a] :
( ( member_a @ X4 @ A3 )
=> ( P @ X4 ) )
& ! [X4: a] :
( ( member_a @ X4 @ B4 )
=> ( P @ X4 ) ) ) ) ).
% ball_Un
thf(fact_256_Un__assoc,axiom,
! [A3: set_a,B4: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A3 @ B4 ) @ C2 )
= ( sup_sup_set_a @ A3 @ ( sup_sup_set_a @ B4 @ C2 ) ) ) ).
% Un_assoc
thf(fact_257_Un__absorb,axiom,
! [A3: set_a] :
( ( sup_sup_set_a @ A3 @ A3 )
= A3 ) ).
% Un_absorb
thf(fact_258_Un__commute,axiom,
( sup_sup_set_a
= ( ^ [A5: set_a,B5: set_a] : ( sup_sup_set_a @ B5 @ A5 ) ) ) ).
% Un_commute
thf(fact_259_Un__left__absorb,axiom,
! [A3: set_a,B4: set_a] :
( ( sup_sup_set_a @ A3 @ ( sup_sup_set_a @ A3 @ B4 ) )
= ( sup_sup_set_a @ A3 @ B4 ) ) ).
% Un_left_absorb
thf(fact_260_Un__left__commute,axiom,
! [A3: set_a,B4: set_a,C2: set_a] :
( ( sup_sup_set_a @ A3 @ ( sup_sup_set_a @ B4 @ C2 ) )
= ( sup_sup_set_a @ B4 @ ( sup_sup_set_a @ A3 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_261_Collect__disj__eq,axiom,
! [P: a > $o,Q: a > $o] :
( ( collect_a
@ ^ [X4: a] :
( ( P @ X4 )
| ( Q @ X4 ) ) )
= ( sup_sup_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_262_Un__def,axiom,
( sup_sup_set_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
( collect_set_a
@ ^ [X4: set_a] :
( ( member_set_a @ X4 @ A5 )
| ( member_set_a @ X4 @ B5 ) ) ) ) ) ).
% Un_def
thf(fact_263_Un__def,axiom,
( sup_sup_set_a
= ( ^ [A5: set_a,B5: set_a] :
( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ A5 )
| ( member_a @ X4 @ B5 ) ) ) ) ) ).
% Un_def
thf(fact_264_Un__empty__left,axiom,
! [B4: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ B4 )
= B4 ) ).
% Un_empty_left
thf(fact_265_Un__empty__right,axiom,
! [A3: set_a] :
( ( sup_sup_set_a @ A3 @ bot_bot_set_a )
= A3 ) ).
% Un_empty_right
thf(fact_266_insert__def,axiom,
( insert_a
= ( ^ [A: a] :
( sup_sup_set_a
@ ( collect_a
@ ^ [X4: a] : X4 = A ) ) ) ) ).
% insert_def
thf(fact_267_insert__is__Un,axiom,
( insert_a
= ( ^ [A: a] : ( sup_sup_set_a @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% insert_is_Un
thf(fact_268_Un__singleton__iff,axiom,
! [A3: set_a,B4: set_a,X: a] :
( ( ( sup_sup_set_a @ A3 @ B4 )
= ( insert_a @ X @ bot_bot_set_a ) )
= ( ( ( A3 = bot_bot_set_a )
& ( B4
= ( insert_a @ X @ bot_bot_set_a ) ) )
| ( ( A3
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B4 = bot_bot_set_a ) )
| ( ( A3
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B4
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_269_singleton__Un__iff,axiom,
! [X: a,A3: set_a,B4: set_a] :
( ( ( insert_a @ X @ bot_bot_set_a )
= ( sup_sup_set_a @ A3 @ B4 ) )
= ( ( ( A3 = bot_bot_set_a )
& ( B4
= ( insert_a @ X @ bot_bot_set_a ) ) )
| ( ( A3
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B4 = bot_bot_set_a ) )
| ( ( A3
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B4
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_270_sup__bot__left,axiom,
! [X: a > a > $o] :
( ( sup_sup_a_a_o @ bot_bot_a_a_o @ X )
= X ) ).
% sup_bot_left
thf(fact_271_sup__bot__left,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ X )
= X ) ).
% sup_bot_left
thf(fact_272_sup__bot__left,axiom,
! [X: a > $o] :
( ( sup_sup_a_o @ bot_bot_a_o @ X )
= X ) ).
% sup_bot_left
thf(fact_273_sup__bot__right,axiom,
! [X: a > a > $o] :
( ( sup_sup_a_a_o @ X @ bot_bot_a_a_o )
= X ) ).
% sup_bot_right
thf(fact_274_sup__bot__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ bot_bot_set_a )
= X ) ).
% sup_bot_right
thf(fact_275_sup__bot__right,axiom,
! [X: a > $o] :
( ( sup_sup_a_o @ X @ bot_bot_a_o )
= X ) ).
% sup_bot_right
thf(fact_276_bot__eq__sup__iff,axiom,
! [X: a > a > $o,Y: a > a > $o] :
( ( bot_bot_a_a_o
= ( sup_sup_a_a_o @ X @ Y ) )
= ( ( X = bot_bot_a_a_o )
& ( Y = bot_bot_a_a_o ) ) ) ).
% bot_eq_sup_iff
thf(fact_277_bot__eq__sup__iff,axiom,
! [X: set_a,Y: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ X @ Y ) )
= ( ( X = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_278_bot__eq__sup__iff,axiom,
! [X: a > $o,Y: a > $o] :
( ( bot_bot_a_o
= ( sup_sup_a_o @ X @ Y ) )
= ( ( X = bot_bot_a_o )
& ( Y = bot_bot_a_o ) ) ) ).
% bot_eq_sup_iff
thf(fact_279_sup__bot_Oright__neutral,axiom,
! [A2: a > a > $o] :
( ( sup_sup_a_a_o @ A2 @ bot_bot_a_a_o )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_280_sup__bot_Oright__neutral,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_281_sup__bot_Oright__neutral,axiom,
! [A2: a > $o] :
( ( sup_sup_a_o @ A2 @ bot_bot_a_o )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_282_sup__bot_Oneutr__eq__iff,axiom,
! [A2: a > a > $o,B2: a > a > $o] :
( ( bot_bot_a_a_o
= ( sup_sup_a_a_o @ A2 @ B2 ) )
= ( ( A2 = bot_bot_a_a_o )
& ( B2 = bot_bot_a_a_o ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_283_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_a )
& ( B2 = bot_bot_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_284_sup__bot_Oneutr__eq__iff,axiom,
! [A2: a > $o,B2: a > $o] :
( ( bot_bot_a_o
= ( sup_sup_a_o @ A2 @ B2 ) )
= ( ( A2 = bot_bot_a_o )
& ( B2 = bot_bot_a_o ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_285_sup__bot_Oleft__neutral,axiom,
! [A2: a > a > $o] :
( ( sup_sup_a_a_o @ bot_bot_a_a_o @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_286_sup__bot_Oleft__neutral,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_287_sup__bot_Oleft__neutral,axiom,
! [A2: a > $o] :
( ( sup_sup_a_o @ bot_bot_a_o @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_288_sup__bot_Oeq__neutr__iff,axiom,
! [A2: a > a > $o,B2: a > a > $o] :
( ( ( sup_sup_a_a_o @ A2 @ B2 )
= bot_bot_a_a_o )
= ( ( A2 = bot_bot_a_a_o )
& ( B2 = bot_bot_a_a_o ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_289_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ( sup_sup_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ( A2 = bot_bot_set_a )
& ( B2 = bot_bot_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_290_sup__bot_Oeq__neutr__iff,axiom,
! [A2: a > $o,B2: a > $o] :
( ( ( sup_sup_a_o @ A2 @ B2 )
= bot_bot_a_o )
= ( ( A2 = bot_bot_a_o )
& ( B2 = bot_bot_a_o ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_291_sup__eq__bot__iff,axiom,
! [X: a > a > $o,Y: a > a > $o] :
( ( ( sup_sup_a_a_o @ X @ Y )
= bot_bot_a_a_o )
= ( ( X = bot_bot_a_a_o )
& ( Y = bot_bot_a_a_o ) ) ) ).
% sup_eq_bot_iff
thf(fact_292_sup__eq__bot__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ( sup_sup_set_a @ X @ Y )
= bot_bot_set_a )
= ( ( X = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_293_sup__eq__bot__iff,axiom,
! [X: a > $o,Y: a > $o] :
( ( ( sup_sup_a_o @ X @ Y )
= bot_bot_a_o )
= ( ( X = bot_bot_a_o )
& ( Y = bot_bot_a_o ) ) ) ).
% sup_eq_bot_iff
thf(fact_294_sup__set__def,axiom,
( sup_sup_set_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
( collect_set_a
@ ( sup_sup_set_a_o
@ ^ [X4: set_a] : ( member_set_a @ X4 @ A5 )
@ ^ [X4: set_a] : ( member_set_a @ X4 @ B5 ) ) ) ) ) ).
% sup_set_def
thf(fact_295_sup__set__def,axiom,
( sup_sup_set_a
= ( ^ [A5: set_a,B5: set_a] :
( collect_a
@ ( sup_sup_a_o
@ ^ [X4: a] : ( member_a @ X4 @ A5 )
@ ^ [X4: a] : ( member_a @ X4 @ B5 ) ) ) ) ) ).
% sup_set_def
thf(fact_296_sup__Un__eq,axiom,
! [R: set_set_a,S: set_set_a] :
( ( sup_sup_set_a_o
@ ^ [X4: set_a] : ( member_set_a @ X4 @ R )
@ ^ [X4: set_a] : ( member_set_a @ X4 @ S ) )
= ( ^ [X4: set_a] : ( member_set_a @ X4 @ ( sup_sup_set_set_a @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_297_sup__Un__eq,axiom,
! [R: set_a,S: set_a] :
( ( sup_sup_a_o
@ ^ [X4: a] : ( member_a @ X4 @ R )
@ ^ [X4: a] : ( member_a @ X4 @ S ) )
= ( ^ [X4: a] : ( member_a @ X4 @ ( sup_sup_set_a @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_298_sup_Ostrict__coboundedI2,axiom,
! [C: a > a > $o,B2: a > a > $o,A2: a > a > $o] :
( ( ord_less_a_a_o @ C @ B2 )
=> ( ord_less_a_a_o @ C @ ( sup_sup_a_a_o @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_299_sup_Ostrict__coboundedI2,axiom,
! [C: set_a,B2: set_a,A2: set_a] :
( ( ord_less_set_a @ C @ B2 )
=> ( ord_less_set_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_300_sup_Ostrict__coboundedI2,axiom,
! [C: a > $o,B2: a > $o,A2: a > $o] :
( ( ord_less_a_o @ C @ B2 )
=> ( ord_less_a_o @ C @ ( sup_sup_a_o @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_301_sup_Ostrict__coboundedI2,axiom,
! [C: int,B2: int,A2: int] :
( ( ord_less_int @ C @ B2 )
=> ( ord_less_int @ C @ ( sup_sup_int @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_302_sup_Ostrict__coboundedI1,axiom,
! [C: a > a > $o,A2: a > a > $o,B2: a > a > $o] :
( ( ord_less_a_a_o @ C @ A2 )
=> ( ord_less_a_a_o @ C @ ( sup_sup_a_a_o @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_303_sup_Ostrict__coboundedI1,axiom,
! [C: set_a,A2: set_a,B2: set_a] :
( ( ord_less_set_a @ C @ A2 )
=> ( ord_less_set_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_304_sup_Ostrict__coboundedI1,axiom,
! [C: a > $o,A2: a > $o,B2: a > $o] :
( ( ord_less_a_o @ C @ A2 )
=> ( ord_less_a_o @ C @ ( sup_sup_a_o @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_305_sup_Ostrict__coboundedI1,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_int @ C @ A2 )
=> ( ord_less_int @ C @ ( sup_sup_int @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_306_sup_Ostrict__order__iff,axiom,
( ord_less_a_a_o
= ( ^ [B: a > a > $o,A: a > a > $o] :
( ( A
= ( sup_sup_a_a_o @ A @ B ) )
& ( A != B ) ) ) ) ).
% sup.strict_order_iff
thf(fact_307_sup_Ostrict__order__iff,axiom,
( ord_less_set_a
= ( ^ [B: set_a,A: set_a] :
( ( A
= ( sup_sup_set_a @ A @ B ) )
& ( A != B ) ) ) ) ).
% sup.strict_order_iff
thf(fact_308_sup_Ostrict__order__iff,axiom,
( ord_less_a_o
= ( ^ [B: a > $o,A: a > $o] :
( ( A
= ( sup_sup_a_o @ A @ B ) )
& ( A != B ) ) ) ) ).
% sup.strict_order_iff
thf(fact_309_sup_Ostrict__order__iff,axiom,
( ord_less_int
= ( ^ [B: int,A: int] :
( ( A
= ( sup_sup_int @ A @ B ) )
& ( A != B ) ) ) ) ).
% sup.strict_order_iff
thf(fact_310_sup_Ostrict__boundedE,axiom,
! [B2: a > a > $o,C: a > a > $o,A2: a > a > $o] :
( ( ord_less_a_a_o @ ( sup_sup_a_a_o @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_a_a_o @ B2 @ A2 )
=> ~ ( ord_less_a_a_o @ C @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_311_sup_Ostrict__boundedE,axiom,
! [B2: set_a,C: set_a,A2: set_a] :
( ( ord_less_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_set_a @ B2 @ A2 )
=> ~ ( ord_less_set_a @ C @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_312_sup_Ostrict__boundedE,axiom,
! [B2: a > $o,C: a > $o,A2: a > $o] :
( ( ord_less_a_o @ ( sup_sup_a_o @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_a_o @ B2 @ A2 )
=> ~ ( ord_less_a_o @ C @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_313_sup_Ostrict__boundedE,axiom,
! [B2: int,C: int,A2: int] :
( ( ord_less_int @ ( sup_sup_int @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_int @ B2 @ A2 )
=> ~ ( ord_less_int @ C @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_314_less__supI2,axiom,
! [X: a > a > $o,B2: a > a > $o,A2: a > a > $o] :
( ( ord_less_a_a_o @ X @ B2 )
=> ( ord_less_a_a_o @ X @ ( sup_sup_a_a_o @ A2 @ B2 ) ) ) ).
% less_supI2
thf(fact_315_less__supI2,axiom,
! [X: set_a,B2: set_a,A2: set_a] :
( ( ord_less_set_a @ X @ B2 )
=> ( ord_less_set_a @ X @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% less_supI2
thf(fact_316_less__supI2,axiom,
! [X: a > $o,B2: a > $o,A2: a > $o] :
( ( ord_less_a_o @ X @ B2 )
=> ( ord_less_a_o @ X @ ( sup_sup_a_o @ A2 @ B2 ) ) ) ).
% less_supI2
thf(fact_317_less__supI2,axiom,
! [X: int,B2: int,A2: int] :
( ( ord_less_int @ X @ B2 )
=> ( ord_less_int @ X @ ( sup_sup_int @ A2 @ B2 ) ) ) ).
% less_supI2
thf(fact_318_less__supI1,axiom,
! [X: a > a > $o,A2: a > a > $o,B2: a > a > $o] :
( ( ord_less_a_a_o @ X @ A2 )
=> ( ord_less_a_a_o @ X @ ( sup_sup_a_a_o @ A2 @ B2 ) ) ) ).
% less_supI1
thf(fact_319_less__supI1,axiom,
! [X: set_a,A2: set_a,B2: set_a] :
( ( ord_less_set_a @ X @ A2 )
=> ( ord_less_set_a @ X @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% less_supI1
thf(fact_320_less__supI1,axiom,
! [X: a > $o,A2: a > $o,B2: a > $o] :
( ( ord_less_a_o @ X @ A2 )
=> ( ord_less_a_o @ X @ ( sup_sup_a_o @ A2 @ B2 ) ) ) ).
% less_supI1
thf(fact_321_less__supI1,axiom,
! [X: int,A2: int,B2: int] :
( ( ord_less_int @ X @ A2 )
=> ( ord_less_int @ X @ ( sup_sup_int @ A2 @ B2 ) ) ) ).
% less_supI1
thf(fact_322_the__elem__def,axiom,
( the_elem_a
= ( ^ [X7: set_a] :
( the_a
@ ^ [X4: a] :
( X7
= ( insert_a @ X4 @ bot_bot_set_a ) ) ) ) ) ).
% the_elem_def
thf(fact_323_pred__on_Ochain__extend,axiom,
! [A3: set_set_a,P: set_a > set_a > $o,C2: set_set_a,Z3: set_a] :
( ( pred_chain_set_a @ A3 @ P @ C2 )
=> ( ( member_set_a @ Z3 @ A3 )
=> ( ! [X6: set_a] :
( ( member_set_a @ X6 @ C2 )
=> ( sup_su198629954et_a_o @ P
@ ^ [Y4: set_a,Z4: set_a] : Y4 = Z4
@ X6
@ Z3 ) )
=> ( pred_chain_set_a @ A3 @ P @ ( sup_sup_set_set_a @ ( insert_set_a @ Z3 @ bot_bot_set_set_a ) @ C2 ) ) ) ) ) ).
% pred_on.chain_extend
thf(fact_324_pred__on_Ochain__extend,axiom,
! [A3: set_a,P: a > a > $o,C2: set_a,Z3: a] :
( ( pred_chain_a @ A3 @ P @ C2 )
=> ( ( member_a @ Z3 @ A3 )
=> ( ! [X6: a] :
( ( member_a @ X6 @ C2 )
=> ( sup_sup_a_a_o @ P
@ ^ [Y4: a,Z4: a] : Y4 = Z4
@ X6
@ Z3 ) )
=> ( pred_chain_a @ A3 @ P @ ( sup_sup_set_a @ ( insert_a @ Z3 @ bot_bot_set_a ) @ C2 ) ) ) ) ) ).
% pred_on.chain_extend
thf(fact_325_sup__bot_Osemilattice__neutr__axioms,axiom,
semila1223476304_a_a_o @ sup_sup_a_a_o @ bot_bot_a_a_o ).
% sup_bot.semilattice_neutr_axioms
thf(fact_326_sup__bot_Osemilattice__neutr__axioms,axiom,
semila1409648192_set_a @ sup_sup_set_a @ bot_bot_set_a ).
% sup_bot.semilattice_neutr_axioms
thf(fact_327_sup__bot_Osemilattice__neutr__axioms,axiom,
semila980155549tr_a_o @ sup_sup_a_o @ bot_bot_a_o ).
% sup_bot.semilattice_neutr_axioms
thf(fact_328_subset_Ochain__extend,axiom,
! [A3: set_set_a,C2: set_set_a,Z3: set_a] :
( ( pred_chain_set_a @ A3 @ ord_less_set_a @ C2 )
=> ( ( member_set_a @ Z3 @ A3 )
=> ( ! [X6: set_a] :
( ( member_set_a @ X6 @ C2 )
=> ( sup_su198629954et_a_o @ ord_less_set_a
@ ^ [Y4: set_a,Z4: set_a] : Y4 = Z4
@ X6
@ Z3 ) )
=> ( pred_chain_set_a @ A3 @ ord_less_set_a @ ( sup_sup_set_set_a @ ( insert_set_a @ Z3 @ bot_bot_set_set_a ) @ C2 ) ) ) ) ) ).
% subset.chain_extend
thf(fact_329_subset_Ochain__total,axiom,
! [A3: set_set_a,C2: set_set_a,X: set_a,Y: set_a] :
( ( pred_chain_set_a @ A3 @ ord_less_set_a @ C2 )
=> ( ( member_set_a @ X @ C2 )
=> ( ( member_set_a @ Y @ C2 )
=> ( ( sup_su198629954et_a_o @ ord_less_set_a
@ ^ [Y4: set_a,Z4: set_a] : Y4 = Z4
@ X
@ Y )
| ( sup_su198629954et_a_o @ ord_less_set_a
@ ^ [Y4: set_a,Z4: set_a] : Y4 = Z4
@ Y
@ X ) ) ) ) ) ).
% subset.chain_total
thf(fact_330_pred__on_Ochain__total,axiom,
! [A3: set_set_a,P: set_a > set_a > $o,C2: set_set_a,X: set_a,Y: set_a] :
( ( pred_chain_set_a @ A3 @ P @ C2 )
=> ( ( member_set_a @ X @ C2 )
=> ( ( member_set_a @ Y @ C2 )
=> ( ( sup_su198629954et_a_o @ P
@ ^ [Y4: set_a,Z4: set_a] : Y4 = Z4
@ X
@ Y )
| ( sup_su198629954et_a_o @ P
@ ^ [Y4: set_a,Z4: set_a] : Y4 = Z4
@ Y
@ X ) ) ) ) ) ).
% pred_on.chain_total
thf(fact_331_pred__on_Ochain__total,axiom,
! [A3: set_a,P: a > a > $o,C2: set_a,X: a,Y: a] :
( ( pred_chain_a @ A3 @ P @ C2 )
=> ( ( member_a @ X @ C2 )
=> ( ( member_a @ Y @ C2 )
=> ( ( sup_sup_a_a_o @ P
@ ^ [Y4: a,Z4: a] : Y4 = Z4
@ X
@ Y )
| ( sup_sup_a_a_o @ P
@ ^ [Y4: a,Z4: a] : Y4 = Z4
@ Y
@ X ) ) ) ) ) ).
% pred_on.chain_total
thf(fact_332_subset_Ochain__empty,axiom,
! [A3: set_set_a] : ( pred_chain_set_a @ A3 @ ord_less_set_a @ bot_bot_set_set_a ) ).
% subset.chain_empty
thf(fact_333_pred__on_Ochain__empty,axiom,
! [A3: set_a,P: a > a > $o] : ( pred_chain_a @ A3 @ P @ bot_bot_set_a ) ).
% pred_on.chain_empty
thf(fact_334_chain__mono,axiom,
! [A3: set_set_a,P: set_a > set_a > $o,Q: set_a > set_a > $o,C2: set_set_a] :
( ! [X6: set_a,Y3: set_a] :
( ( member_set_a @ X6 @ A3 )
=> ( ( member_set_a @ Y3 @ A3 )
=> ( ( P @ X6 @ Y3 )
=> ( Q @ X6 @ Y3 ) ) ) )
=> ( ( pred_chain_set_a @ A3 @ P @ C2 )
=> ( pred_chain_set_a @ A3 @ Q @ C2 ) ) ) ).
% chain_mono
thf(fact_335_chain__mono,axiom,
! [A3: set_a,P: a > a > $o,Q: a > a > $o,C2: set_a] :
( ! [X6: a,Y3: a] :
( ( member_a @ X6 @ A3 )
=> ( ( member_a @ Y3 @ A3 )
=> ( ( P @ X6 @ Y3 )
=> ( Q @ X6 @ Y3 ) ) ) )
=> ( ( pred_chain_a @ A3 @ P @ C2 )
=> ( pred_chain_a @ A3 @ Q @ C2 ) ) ) ).
% chain_mono
thf(fact_336_pred__on_Ochain_Ocong,axiom,
pred_chain_a = pred_chain_a ).
% pred_on.chain.cong
thf(fact_337_chains__alt__def,axiom,
( chains_a
= ( ^ [A5: set_set_a] : ( collect_set_set_a @ ( pred_chain_set_a @ A5 @ ord_less_set_a ) ) ) ) ).
% chains_alt_def
thf(fact_338_Pow__empty,axiom,
( ( pow_a @ bot_bot_set_a )
= ( insert_set_a @ bot_bot_set_a @ bot_bot_set_set_a ) ) ).
% Pow_empty
thf(fact_339_Pow__singleton__iff,axiom,
! [X8: set_a,Y5: set_a] :
( ( ( pow_a @ X8 )
= ( insert_set_a @ Y5 @ bot_bot_set_set_a ) )
= ( ( X8 = bot_bot_set_a )
& ( Y5 = bot_bot_set_a ) ) ) ).
% Pow_singleton_iff
thf(fact_340_Pow__not__empty,axiom,
! [A3: set_a] :
( ( pow_a @ A3 )
!= bot_bot_set_set_a ) ).
% Pow_not_empty
thf(fact_341_Pow__bottom,axiom,
! [B4: set_a] : ( member_set_a @ bot_bot_set_a @ ( pow_a @ B4 ) ) ).
% Pow_bottom
thf(fact_342_Pow__top,axiom,
! [A3: set_a] : ( member_set_a @ A3 @ ( pow_a @ A3 ) ) ).
% Pow_top
thf(fact_343_chains__extend,axiom,
! [C: set_set_a,S: set_set_a,Z3: set_a] :
( ( member_set_set_a @ C @ ( chains_a @ S ) )
=> ( ( member_set_a @ Z3 @ S )
=> ( ! [X6: set_a] :
( ( member_set_a @ X6 @ C )
=> ( ord_less_eq_set_a @ X6 @ Z3 ) )
=> ( member_set_set_a @ ( sup_sup_set_set_a @ ( insert_set_a @ Z3 @ bot_bot_set_set_a ) @ C ) @ ( chains_a @ S ) ) ) ) ) ).
% chains_extend
thf(fact_344_Powp__Pow__eq,axiom,
! [A3: set_set_a] :
( ( powp_set_a
@ ^ [X4: set_a] : ( member_set_a @ X4 @ A3 ) )
= ( ^ [X4: set_set_a] : ( member_set_set_a @ X4 @ ( pow_set_a @ A3 ) ) ) ) ).
% Powp_Pow_eq
thf(fact_345_Powp__Pow__eq,axiom,
! [A3: set_a] :
( ( powp_a
@ ^ [X4: a] : ( member_a @ X4 @ A3 ) )
= ( ^ [X4: set_a] : ( member_set_a @ X4 @ ( pow_a @ A3 ) ) ) ) ).
% Powp_Pow_eq
thf(fact_346_order__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% order_refl
thf(fact_347_subset__antisym,axiom,
! [A3: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ A3 )
=> ( A3 = B4 ) ) ) ).
% subset_antisym
thf(fact_348_subsetI,axiom,
! [A3: set_a,B4: set_a] :
( ! [X6: a] :
( ( member_a @ X6 @ A3 )
=> ( member_a @ X6 @ B4 ) )
=> ( ord_less_eq_set_a @ A3 @ B4 ) ) ).
% subsetI
thf(fact_349_insert__subset,axiom,
! [X: a,A3: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A3 ) @ B4 )
= ( ( member_a @ X @ B4 )
& ( ord_less_eq_set_a @ A3 @ B4 ) ) ) ).
% insert_subset
% Conjectures (1)
thf(conj_0,conjecture,
a2 = b ).
%------------------------------------------------------------------------------