TPTP Problem File: ITP028^1.p
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%------------------------------------------------------------------------------
% File : ITP028^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Algebra8 problem prob_962__6421504_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 : Algebra8/prob_962__6421504_1 [Des21]
% Status : Theorem
% Rating : 0.30 v8.2.0, 0.15 v8.1.0, 0.18 v7.5.0
% Syntax : Number of formulae : 456 ( 120 unt; 102 typ; 0 def)
% Number of atoms : 1002 ( 288 equ; 0 cnn)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 2874 ( 78 ~; 10 |; 54 &;2255 @)
% ( 0 <=>; 477 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 7 avg)
% Number of types : 16 ( 15 usr)
% Number of type conns : 618 ( 618 >; 0 *; 0 +; 0 <<)
% Number of symbols : 90 ( 87 usr; 12 con; 0-5 aty)
% Number of variables : 1103 ( 97 ^; 962 !; 44 ?;1103 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:45:55.322
%------------------------------------------------------------------------------
% Could-be-implicit typings (15)
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thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001_062_Itf__a_Mtf__e_J,type,
image_set_a_a_e: ( set_a > a > e ) > set_set_a > set_a_e ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_M_062_Itf__a_Mtf__e_J_J,type,
image_210900495_a_a_e: ( set_a > produc715398134_a_a_e ) > set_set_a > set_Pr204296108_a_a_e ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
image_1930509541_set_a: ( set_a > set_set_a ) > set_set_a > set_set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
image_set_a_set_a: ( set_a > set_a ) > set_set_a > set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001tf__a,type,
image_set_a_a: ( set_a > a ) > set_set_a > set_a ).
thf(sy_c_Set_Oimage_001tf__a_001_062_Itf__a_Mtf__e_J,type,
image_a_a_e: ( a > a > e ) > set_a > set_a_e ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
image_a_set_set_a: ( a > set_set_a ) > set_a > set_set_set_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_Itf__a_J,type,
image_a_set_a: ( a > set_a ) > set_a > set_set_a ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_member_001_062_Itf__a_Mtf__e_J,type,
member_a_e: ( a > e ) > set_a_e > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_Itf__a_Mtf__e_J_Mt__Set__Oset_Itf__a_J_J,type,
member360729153_set_a: produc1965228714_set_a > set_Pr305929312_set_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_M_062_Itf__a_Mtf__e_J_J,type,
member1258382221_a_a_e: produc715398134_a_a_e > set_Pr204296108_a_a_e > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_Itf__a_J_M_062_Itf__a_Mtf__e_J_J_J,type,
member1437121347_a_a_e: set_Pr204296108_a_a_e > set_se821189730_a_a_e > $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_C,type,
c: set_Pr204296108_a_a_e ).
thf(sy_v_a1,type,
a1: set_a ).
thf(sy_v_a2,type,
a2: set_a ).
thf(sy_v_a3,type,
a3: set_a ).
thf(sy_v_b1,type,
b1: a > e ).
thf(sy_v_b2,type,
b2: a > e ).
thf(sy_v_b3,type,
b3: a > e ).
thf(sy_v_thesis____,type,
thesis: $o ).
% Relevant facts (352)
thf(fact_0__092_060open_062_092_060forall_062a_092_060in_062C_O_A_092_060forall_062b_092_060in_062C_O_Afst_Aa_A_092_060subseteq_062_Afst_Ab_A_092_060or_062_Afst_Ab_A_092_060subseteq_062_Afst_Aa_092_060close_062,axiom,
! [X: produc715398134_a_a_e] :
( ( member1258382221_a_a_e @ X @ c )
=> ! [Xa: produc715398134_a_a_e] :
( ( member1258382221_a_a_e @ Xa @ c )
=> ( ( ord_less_eq_set_a @ ( produc1040928388_a_a_e @ X ) @ ( produc1040928388_a_a_e @ Xa ) )
| ( ord_less_eq_set_a @ ( produc1040928388_a_a_e @ Xa ) @ ( produc1040928388_a_a_e @ X ) ) ) ) ) ).
% \<open>\<forall>a\<in>C. \<forall>b\<in>C. fst a \<subseteq> fst b \<or> fst b \<subseteq> fst a\<close>
thf(fact_1__092_060open_062_092_060exists_062a_092_060in_062fst_A_096_AC_O_Aa1_A_092_060subseteq_062_Aa_A_092_060and_062_Aa2_A_092_060subseteq_062_Aa_A_092_060and_062_Aa3_A_092_060subseteq_062_Aa_092_060close_062,axiom,
? [X2: set_a] :
( ( member_set_a @ X2 @ ( image_64240033_set_a @ produc1040928388_a_a_e @ c ) )
& ( ord_less_eq_set_a @ a1 @ X2 )
& ( ord_less_eq_set_a @ a2 @ X2 )
& ( ord_less_eq_set_a @ a3 @ X2 ) ) ).
% \<open>\<exists>a\<in>fst ` C. a1 \<subseteq> a \<and> a2 \<subseteq> a \<and> a3 \<subseteq> a\<close>
thf(fact_2__092_060open_062a1_A_092_060subseteq_062_Aa2_A_092_060or_062_Aa2_A_092_060subseteq_062_Aa1_092_060close_062,axiom,
( ( ord_less_eq_set_a @ a1 @ a2 )
| ( ord_less_eq_set_a @ a2 @ a1 ) ) ).
% \<open>a1 \<subseteq> a2 \<or> a2 \<subseteq> a1\<close>
thf(fact_3__092_060open_062a1_A_092_060subseteq_062_Aa3_A_092_060or_062_Aa3_A_092_060subseteq_062_Aa1_092_060close_062,axiom,
( ( ord_less_eq_set_a @ a1 @ a3 )
| ( ord_less_eq_set_a @ a3 @ a1 ) ) ).
% \<open>a1 \<subseteq> a3 \<or> a3 \<subseteq> a1\<close>
thf(fact_4__092_060open_062a2_A_092_060subseteq_062_Aa3_A_092_060or_062_Aa3_A_092_060subseteq_062_Aa2_092_060close_062,axiom,
( ( ord_less_eq_set_a @ a2 @ a3 )
| ( ord_less_eq_set_a @ a3 @ a2 ) ) ).
% \<open>a2 \<subseteq> a3 \<or> a3 \<subseteq> a2\<close>
thf(fact_5_A_I3_J,axiom,
member_set_a @ a3 @ ( image_64240033_set_a @ produc1040928388_a_a_e @ c ) ).
% A(3)
thf(fact_6_A_I2_J,axiom,
member_set_a @ a2 @ ( image_64240033_set_a @ produc1040928388_a_a_e @ c ) ).
% A(2)
thf(fact_7_A_I1_J,axiom,
member_set_a @ a1 @ ( image_64240033_set_a @ produc1040928388_a_a_e @ c ) ).
% A(1)
thf(fact_8__092_060open_062_Ia1_M_Ab1_J_A_092_060in_062_AC_092_060close_062,axiom,
member1258382221_a_a_e @ ( produc1134456688_a_a_e @ a1 @ b1 ) @ c ).
% \<open>(a1, b1) \<in> C\<close>
thf(fact_9__092_060open_062_Ia2_M_Ab2_J_A_092_060in_062_AC_092_060close_062,axiom,
member1258382221_a_a_e @ ( produc1134456688_a_a_e @ a2 @ b2 ) @ c ).
% \<open>(a2, b2) \<in> C\<close>
thf(fact_10__092_060open_062_Ia3_M_Ab3_J_A_092_060in_062_AC_092_060close_062,axiom,
member1258382221_a_a_e @ ( produc1134456688_a_a_e @ a3 @ b3 ) @ c ).
% \<open>(a3, b3) \<in> C\<close>
thf(fact_11_subsetI,axiom,
! [A: set_a_e,B: set_a_e] :
( ! [X2: a > e] :
( ( member_a_e @ X2 @ A )
=> ( member_a_e @ X2 @ B ) )
=> ( ord_less_eq_set_a_e @ A @ B ) ) ).
% subsetI
thf(fact_12_subsetI,axiom,
! [A: set_Pr305929312_set_a,B: set_Pr305929312_set_a] :
( ! [X2: produc1965228714_set_a] :
( ( member360729153_set_a @ X2 @ A )
=> ( member360729153_set_a @ X2 @ B ) )
=> ( ord_le802356416_set_a @ A @ B ) ) ).
% subsetI
thf(fact_13_subsetI,axiom,
! [A: set_set_set_a,B: set_set_set_a] :
( ! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ A )
=> ( member_set_set_a @ X2 @ B ) )
=> ( ord_le2103863998_set_a @ A @ B ) ) ).
% subsetI
thf(fact_14_subsetI,axiom,
! [A: set_set_a,B: set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( member_set_a @ X2 @ B ) )
=> ( ord_le318720350_set_a @ A @ B ) ) ).
% subsetI
thf(fact_15_subsetI,axiom,
! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e] :
( ! [X2: produc715398134_a_a_e] :
( ( member1258382221_a_a_e @ X2 @ A )
=> ( member1258382221_a_a_e @ X2 @ B ) )
=> ( ord_le700723212_a_a_e @ A @ B ) ) ).
% subsetI
thf(fact_16_subsetI,axiom,
! [A: set_a,B: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_a @ X2 @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% subsetI
thf(fact_17_subset__antisym,axiom,
! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e] :
( ( ord_le700723212_a_a_e @ A @ B )
=> ( ( ord_le700723212_a_a_e @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_18_subset__antisym,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le318720350_set_a @ A @ B )
=> ( ( ord_le318720350_set_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_19_subset__antisym,axiom,
! [A: set_set_set_a,B: set_set_set_a] :
( ( ord_le2103863998_set_a @ A @ B )
=> ( ( ord_le2103863998_set_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_20_subset__antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_21_image__eqI,axiom,
! [B2: set_a,F: set_a > set_a,X3: set_a,A: set_set_a] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_set_a @ X3 @ A )
=> ( member_set_a @ B2 @ ( image_set_a_set_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_22_image__eqI,axiom,
! [B2: produc715398134_a_a_e,F: set_a > produc715398134_a_a_e,X3: set_a,A: set_set_a] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_set_a @ X3 @ A )
=> ( member1258382221_a_a_e @ B2 @ ( image_210900495_a_a_e @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_23_image__eqI,axiom,
! [B2: set_a,F: produc715398134_a_a_e > set_a,X3: produc715398134_a_a_e,A: set_Pr204296108_a_a_e] :
( ( B2
= ( F @ X3 ) )
=> ( ( member1258382221_a_a_e @ X3 @ A )
=> ( member_set_a @ B2 @ ( image_64240033_set_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_24_image__eqI,axiom,
! [B2: produc715398134_a_a_e,F: produc715398134_a_a_e > produc715398134_a_a_e,X3: produc715398134_a_a_e,A: set_Pr204296108_a_a_e] :
( ( B2
= ( F @ X3 ) )
=> ( ( member1258382221_a_a_e @ X3 @ A )
=> ( member1258382221_a_a_e @ B2 @ ( image_625620211_a_a_e @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_25_image__eqI,axiom,
! [B2: a,F: a > a,X3: a,A: set_a] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_a @ X3 @ A )
=> ( member_a @ B2 @ ( image_a_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_26_image__eqI,axiom,
! [B2: a,F: set_a > a,X3: set_a,A: set_set_a] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_set_a @ X3 @ A )
=> ( member_a @ B2 @ ( image_set_a_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_27_image__eqI,axiom,
! [B2: set_a,F: a > set_a,X3: a,A: set_a] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_a @ X3 @ A )
=> ( member_set_a @ B2 @ ( image_a_set_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_28_image__eqI,axiom,
! [B2: a > e,F: a > a > e,X3: a,A: set_a] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_a @ X3 @ A )
=> ( member_a_e @ B2 @ ( image_a_a_e @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_29_image__eqI,axiom,
! [B2: a,F: ( a > e ) > a,X3: a > e,A: set_a_e] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_a_e @ X3 @ A )
=> ( member_a @ B2 @ ( image_a_e_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_30_image__eqI,axiom,
! [B2: a > e,F: set_a > a > e,X3: set_a,A: set_set_a] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_set_a @ X3 @ A )
=> ( member_a_e @ B2 @ ( image_set_a_a_e @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_31_order__refl,axiom,
! [X3: set_Pr204296108_a_a_e] : ( ord_le700723212_a_a_e @ X3 @ X3 ) ).
% order_refl
thf(fact_32_order__refl,axiom,
! [X3: set_set_a] : ( ord_le318720350_set_a @ X3 @ X3 ) ).
% order_refl
thf(fact_33_order__refl,axiom,
! [X3: set_set_set_a] : ( ord_le2103863998_set_a @ X3 @ X3 ) ).
% order_refl
thf(fact_34_order__refl,axiom,
! [X3: $o > set_a] : ( ord_less_eq_o_set_a @ X3 @ X3 ) ).
% order_refl
thf(fact_35_order__refl,axiom,
! [X3: set_a] : ( ord_less_eq_set_a @ X3 @ X3 ) ).
% order_refl
thf(fact_36_image__mono,axiom,
! [A: set_a,B: set_a,F: a > a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ ( image_a_a @ F @ B ) ) ) ).
% image_mono
thf(fact_37_image__mono,axiom,
! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e,F: produc715398134_a_a_e > set_a] :
( ( ord_le700723212_a_a_e @ A @ B )
=> ( ord_le318720350_set_a @ ( image_64240033_set_a @ F @ A ) @ ( image_64240033_set_a @ F @ B ) ) ) ).
% image_mono
thf(fact_38_image__mono,axiom,
! [A: set_a,B: set_a,F: a > set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_le318720350_set_a @ ( image_a_set_a @ F @ A ) @ ( image_a_set_a @ F @ B ) ) ) ).
% image_mono
thf(fact_39_image__mono,axiom,
! [A: set_set_a,B: set_set_a,F: set_a > a] :
( ( ord_le318720350_set_a @ A @ B )
=> ( ord_less_eq_set_a @ ( image_set_a_a @ F @ A ) @ ( image_set_a_a @ F @ B ) ) ) ).
% image_mono
thf(fact_40_image__mono,axiom,
! [A: set_a,B: set_a,F: a > set_set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_le2103863998_set_a @ ( image_a_set_set_a @ F @ A ) @ ( image_a_set_set_a @ F @ B ) ) ) ).
% image_mono
thf(fact_41_image__mono,axiom,
! [A: set_set_a,B: set_set_a,F: set_a > set_a] :
( ( ord_le318720350_set_a @ A @ B )
=> ( ord_le318720350_set_a @ ( image_set_a_set_a @ F @ A ) @ ( image_set_a_set_a @ F @ B ) ) ) ).
% image_mono
thf(fact_42_image__mono,axiom,
! [A: set_set_set_a,B: set_set_set_a,F: set_set_a > a] :
( ( ord_le2103863998_set_a @ A @ B )
=> ( ord_less_eq_set_a @ ( image_set_set_a_a @ F @ A ) @ ( image_set_set_a_a @ F @ B ) ) ) ).
% image_mono
thf(fact_43_image__mono,axiom,
! [A: set_set_a,B: set_set_a,F: set_a > set_set_a] :
( ( ord_le318720350_set_a @ A @ B )
=> ( ord_le2103863998_set_a @ ( image_1930509541_set_a @ F @ A ) @ ( image_1930509541_set_a @ F @ B ) ) ) ).
% image_mono
thf(fact_44_image__mono,axiom,
! [A: set_set_set_a,B: set_set_set_a,F: set_set_a > set_a] :
( ( ord_le2103863998_set_a @ A @ B )
=> ( ord_le318720350_set_a @ ( image_402591269_set_a @ F @ A ) @ ( image_402591269_set_a @ F @ B ) ) ) ).
% image_mono
thf(fact_45_image__mono,axiom,
! [A: set_set_set_a,B: set_set_set_a,F: set_set_a > set_set_a] :
( ( ord_le2103863998_set_a @ A @ B )
=> ( ord_le2103863998_set_a @ ( image_1956910981_set_a @ F @ A ) @ ( image_1956910981_set_a @ F @ B ) ) ) ).
% image_mono
thf(fact_46_image__subsetI,axiom,
! [A: set_set_a,F: set_a > a,B: set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_set_a_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_47_image__subsetI,axiom,
! [A: set_set_a,F: set_a > set_a,B: set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( member_set_a @ ( F @ X2 ) @ B ) )
=> ( ord_le318720350_set_a @ ( image_set_a_set_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_48_image__subsetI,axiom,
! [A: set_Pr204296108_a_a_e,F: produc715398134_a_a_e > a,B: set_a] :
( ! [X2: produc715398134_a_a_e] :
( ( member1258382221_a_a_e @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_2116492353_a_e_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_49_image__subsetI,axiom,
! [A: set_set_a,F: set_a > produc715398134_a_a_e,B: set_Pr204296108_a_a_e] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( member1258382221_a_a_e @ ( F @ X2 ) @ B ) )
=> ( ord_le700723212_a_a_e @ ( image_210900495_a_a_e @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_50_image__subsetI,axiom,
! [A: set_Pr204296108_a_a_e,F: produc715398134_a_a_e > set_a,B: set_set_a] :
( ! [X2: produc715398134_a_a_e] :
( ( member1258382221_a_a_e @ X2 @ A )
=> ( member_set_a @ ( F @ X2 ) @ B ) )
=> ( ord_le318720350_set_a @ ( image_64240033_set_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_51_image__subsetI,axiom,
! [A: set_Pr204296108_a_a_e,F: produc715398134_a_a_e > produc715398134_a_a_e,B: set_Pr204296108_a_a_e] :
( ! [X2: produc715398134_a_a_e] :
( ( member1258382221_a_a_e @ X2 @ A )
=> ( member1258382221_a_a_e @ ( F @ X2 ) @ B ) )
=> ( ord_le700723212_a_a_e @ ( image_625620211_a_a_e @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_52_image__subsetI,axiom,
! [A: set_a,F: a > a,B: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_53_image__subsetI,axiom,
! [A: set_a,F: a > set_a,B: set_set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_set_a @ ( F @ X2 ) @ B ) )
=> ( ord_le318720350_set_a @ ( image_a_set_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_54_image__subsetI,axiom,
! [A: set_a,F: a > a > e,B: set_a_e] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_a_e @ ( F @ X2 ) @ B ) )
=> ( ord_less_eq_set_a_e @ ( image_a_a_e @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_55_image__subsetI,axiom,
! [A: set_a_e,F: ( a > e ) > a,B: set_a] :
( ! [X2: a > e] :
( ( member_a_e @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_a_e_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_56_subset__imageE,axiom,
! [B: set_a,F: a > a,A: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
=> ~ ! [C: set_a] :
( ( ord_less_eq_set_a @ C @ A )
=> ( B
!= ( image_a_a @ F @ C ) ) ) ) ).
% subset_imageE
thf(fact_57_subset__imageE,axiom,
! [B: set_set_a,F: produc715398134_a_a_e > set_a,A: set_Pr204296108_a_a_e] :
( ( ord_le318720350_set_a @ B @ ( image_64240033_set_a @ F @ A ) )
=> ~ ! [C: set_Pr204296108_a_a_e] :
( ( ord_le700723212_a_a_e @ C @ A )
=> ( B
!= ( image_64240033_set_a @ F @ C ) ) ) ) ).
% subset_imageE
thf(fact_58_subset__imageE,axiom,
! [B: set_a,F: set_a > a,A: set_set_a] :
( ( ord_less_eq_set_a @ B @ ( image_set_a_a @ F @ A ) )
=> ~ ! [C: set_set_a] :
( ( ord_le318720350_set_a @ C @ A )
=> ( B
!= ( image_set_a_a @ F @ C ) ) ) ) ).
% subset_imageE
thf(fact_59_subset__imageE,axiom,
! [B: set_set_a,F: a > set_a,A: set_a] :
( ( ord_le318720350_set_a @ B @ ( image_a_set_a @ F @ A ) )
=> ~ ! [C: set_a] :
( ( ord_less_eq_set_a @ C @ A )
=> ( B
!= ( image_a_set_a @ F @ C ) ) ) ) ).
% subset_imageE
thf(fact_60_subset__imageE,axiom,
! [B: set_a,F: set_set_a > a,A: set_set_set_a] :
( ( ord_less_eq_set_a @ B @ ( image_set_set_a_a @ F @ A ) )
=> ~ ! [C: set_set_set_a] :
( ( ord_le2103863998_set_a @ C @ A )
=> ( B
!= ( image_set_set_a_a @ F @ C ) ) ) ) ).
% subset_imageE
thf(fact_61_subset__imageE,axiom,
! [B: set_set_a,F: set_a > set_a,A: set_set_a] :
( ( ord_le318720350_set_a @ B @ ( image_set_a_set_a @ F @ A ) )
=> ~ ! [C: set_set_a] :
( ( ord_le318720350_set_a @ C @ A )
=> ( B
!= ( image_set_a_set_a @ F @ C ) ) ) ) ).
% subset_imageE
thf(fact_62_subset__imageE,axiom,
! [B: set_set_set_a,F: a > set_set_a,A: set_a] :
( ( ord_le2103863998_set_a @ B @ ( image_a_set_set_a @ F @ A ) )
=> ~ ! [C: set_a] :
( ( ord_less_eq_set_a @ C @ A )
=> ( B
!= ( image_a_set_set_a @ F @ C ) ) ) ) ).
% subset_imageE
thf(fact_63_subset__imageE,axiom,
! [B: set_set_a,F: set_set_a > set_a,A: set_set_set_a] :
( ( ord_le318720350_set_a @ B @ ( image_402591269_set_a @ F @ A ) )
=> ~ ! [C: set_set_set_a] :
( ( ord_le2103863998_set_a @ C @ A )
=> ( B
!= ( image_402591269_set_a @ F @ C ) ) ) ) ).
% subset_imageE
thf(fact_64_subset__imageE,axiom,
! [B: set_set_set_a,F: set_a > set_set_a,A: set_set_a] :
( ( ord_le2103863998_set_a @ B @ ( image_1930509541_set_a @ F @ A ) )
=> ~ ! [C: set_set_a] :
( ( ord_le318720350_set_a @ C @ A )
=> ( B
!= ( image_1930509541_set_a @ F @ C ) ) ) ) ).
% subset_imageE
thf(fact_65_subset__imageE,axiom,
! [B: set_set_set_a,F: set_set_a > set_set_a,A: set_set_set_a] :
( ( ord_le2103863998_set_a @ B @ ( image_1956910981_set_a @ F @ A ) )
=> ~ ! [C: set_set_set_a] :
( ( ord_le2103863998_set_a @ C @ A )
=> ( B
!= ( image_1956910981_set_a @ F @ C ) ) ) ) ).
% subset_imageE
thf(fact_66_image__subset__iff,axiom,
! [F: set_a > a,A: set_set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( image_set_a_a @ F @ A ) @ B )
= ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A )
=> ( member_a @ ( F @ X4 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_67_image__subset__iff,axiom,
! [F: a > a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B )
= ( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( member_a @ ( F @ X4 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_68_image__subset__iff,axiom,
! [F: set_a > produc715398134_a_a_e,A: set_set_a,B: set_Pr204296108_a_a_e] :
( ( ord_le700723212_a_a_e @ ( image_210900495_a_a_e @ F @ A ) @ B )
= ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A )
=> ( member1258382221_a_a_e @ ( F @ X4 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_69_image__subset__iff,axiom,
! [F: set_a > set_a,A: set_set_a,B: set_set_a] :
( ( ord_le318720350_set_a @ ( image_set_a_set_a @ F @ A ) @ B )
= ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A )
=> ( member_set_a @ ( F @ X4 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_70_image__subset__iff,axiom,
! [F: set_Pr204296108_a_a_e > set_set_a,A: set_se821189730_a_a_e,B: set_set_set_a] :
( ( ord_le2103863998_set_a @ ( image_1441218763_set_a @ F @ A ) @ B )
= ( ! [X4: set_Pr204296108_a_a_e] :
( ( member1437121347_a_a_e @ X4 @ A )
=> ( member_set_set_a @ ( F @ X4 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_71_image__subset__iff,axiom,
! [F: produc715398134_a_a_e > set_a,A: set_Pr204296108_a_a_e,B: set_set_a] :
( ( ord_le318720350_set_a @ ( image_64240033_set_a @ F @ A ) @ B )
= ( ! [X4: produc715398134_a_a_e] :
( ( member1258382221_a_a_e @ X4 @ A )
=> ( member_set_a @ ( F @ X4 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_72_subset__image__iff,axiom,
! [B: set_a,F: a > a,A: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B
= ( image_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_73_subset__image__iff,axiom,
! [B: set_set_a,F: produc715398134_a_a_e > set_a,A: set_Pr204296108_a_a_e] :
( ( ord_le318720350_set_a @ B @ ( image_64240033_set_a @ F @ A ) )
= ( ? [AA: set_Pr204296108_a_a_e] :
( ( ord_le700723212_a_a_e @ AA @ A )
& ( B
= ( image_64240033_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_74_subset__image__iff,axiom,
! [B: set_a,F: set_a > a,A: set_set_a] :
( ( ord_less_eq_set_a @ B @ ( image_set_a_a @ F @ A ) )
= ( ? [AA: set_set_a] :
( ( ord_le318720350_set_a @ AA @ A )
& ( B
= ( image_set_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_75_subset__image__iff,axiom,
! [B: set_set_a,F: a > set_a,A: set_a] :
( ( ord_le318720350_set_a @ B @ ( image_a_set_a @ F @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B
= ( image_a_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_76_subset__image__iff,axiom,
! [B: set_a,F: set_set_a > a,A: set_set_set_a] :
( ( ord_less_eq_set_a @ B @ ( image_set_set_a_a @ F @ A ) )
= ( ? [AA: set_set_set_a] :
( ( ord_le2103863998_set_a @ AA @ A )
& ( B
= ( image_set_set_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_77_subset__image__iff,axiom,
! [B: set_set_a,F: set_a > set_a,A: set_set_a] :
( ( ord_le318720350_set_a @ B @ ( image_set_a_set_a @ F @ A ) )
= ( ? [AA: set_set_a] :
( ( ord_le318720350_set_a @ AA @ A )
& ( B
= ( image_set_a_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_78_subset__image__iff,axiom,
! [B: set_set_set_a,F: a > set_set_a,A: set_a] :
( ( ord_le2103863998_set_a @ B @ ( image_a_set_set_a @ F @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B
= ( image_a_set_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_79_subset__image__iff,axiom,
! [B: set_set_a,F: set_set_a > set_a,A: set_set_set_a] :
( ( ord_le318720350_set_a @ B @ ( image_402591269_set_a @ F @ A ) )
= ( ? [AA: set_set_set_a] :
( ( ord_le2103863998_set_a @ AA @ A )
& ( B
= ( image_402591269_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_80_subset__image__iff,axiom,
! [B: set_set_set_a,F: set_a > set_set_a,A: set_set_a] :
( ( ord_le2103863998_set_a @ B @ ( image_1930509541_set_a @ F @ A ) )
= ( ? [AA: set_set_a] :
( ( ord_le318720350_set_a @ AA @ A )
& ( B
= ( image_1930509541_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_81_subset__image__iff,axiom,
! [B: set_set_set_a,F: set_set_a > set_set_a,A: set_set_set_a] :
( ( ord_le2103863998_set_a @ B @ ( image_1956910981_set_a @ F @ A ) )
= ( ? [AA: set_set_set_a] :
( ( ord_le2103863998_set_a @ AA @ A )
& ( B
= ( image_1956910981_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_82_PairE__lemma,axiom,
! [P: produc1965228714_set_a] :
? [X2: a > e,Y: set_a] :
( P
= ( produc479206876_set_a @ X2 @ Y ) ) ).
% PairE_lemma
thf(fact_83_PairE__lemma,axiom,
! [P: produc715398134_a_a_e] :
? [X2: set_a,Y: a > e] :
( P
= ( produc1134456688_a_a_e @ X2 @ Y ) ) ).
% PairE_lemma
thf(fact_84_dual__order_Oantisym,axiom,
! [B2: set_Pr204296108_a_a_e,A2: set_Pr204296108_a_a_e] :
( ( ord_le700723212_a_a_e @ B2 @ A2 )
=> ( ( ord_le700723212_a_a_e @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_85_dual__order_Oantisym,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( ord_le318720350_set_a @ B2 @ A2 )
=> ( ( ord_le318720350_set_a @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_86_dual__order_Oantisym,axiom,
! [B2: set_set_set_a,A2: set_set_set_a] :
( ( ord_le2103863998_set_a @ B2 @ A2 )
=> ( ( ord_le2103863998_set_a @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_87_dual__order_Oantisym,axiom,
! [B2: $o > set_a,A2: $o > set_a] :
( ( ord_less_eq_o_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_o_set_a @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_88_dual__order_Oantisym,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_89_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_Pr204296108_a_a_e,Z: set_Pr204296108_a_a_e] : Y2 = Z )
= ( ^ [A3: set_Pr204296108_a_a_e,B3: set_Pr204296108_a_a_e] :
( ( ord_le700723212_a_a_e @ B3 @ A3 )
& ( ord_le700723212_a_a_e @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_90_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_set_a,Z: set_set_a] : Y2 = Z )
= ( ^ [A3: set_set_a,B3: set_set_a] :
( ( ord_le318720350_set_a @ B3 @ A3 )
& ( ord_le318720350_set_a @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_91_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_set_set_a,Z: set_set_set_a] : Y2 = Z )
= ( ^ [A3: set_set_set_a,B3: set_set_set_a] :
( ( ord_le2103863998_set_a @ B3 @ A3 )
& ( ord_le2103863998_set_a @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_92_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: $o > set_a,Z: $o > set_a] : Y2 = Z )
= ( ^ [A3: $o > set_a,B3: $o > set_a] :
( ( ord_less_eq_o_set_a @ B3 @ A3 )
& ( ord_less_eq_o_set_a @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_93_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_a,Z: set_a] : Y2 = Z )
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
& ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_94_dual__order_Otrans,axiom,
! [B2: set_Pr204296108_a_a_e,A2: set_Pr204296108_a_a_e,C2: set_Pr204296108_a_a_e] :
( ( ord_le700723212_a_a_e @ B2 @ A2 )
=> ( ( ord_le700723212_a_a_e @ C2 @ B2 )
=> ( ord_le700723212_a_a_e @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_95_dual__order_Otrans,axiom,
! [B2: set_set_a,A2: set_set_a,C2: set_set_a] :
( ( ord_le318720350_set_a @ B2 @ A2 )
=> ( ( ord_le318720350_set_a @ C2 @ B2 )
=> ( ord_le318720350_set_a @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_96_dual__order_Otrans,axiom,
! [B2: set_set_set_a,A2: set_set_set_a,C2: set_set_set_a] :
( ( ord_le2103863998_set_a @ B2 @ A2 )
=> ( ( ord_le2103863998_set_a @ C2 @ B2 )
=> ( ord_le2103863998_set_a @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_97_dual__order_Otrans,axiom,
! [B2: $o > set_a,A2: $o > set_a,C2: $o > set_a] :
( ( ord_less_eq_o_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_o_set_a @ C2 @ B2 )
=> ( ord_less_eq_o_set_a @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_98_dual__order_Otrans,axiom,
! [B2: set_a,A2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ C2 @ B2 )
=> ( ord_less_eq_set_a @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_99_dual__order_Orefl,axiom,
! [A2: set_Pr204296108_a_a_e] : ( ord_le700723212_a_a_e @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_100_dual__order_Orefl,axiom,
! [A2: set_set_a] : ( ord_le318720350_set_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_101_dual__order_Orefl,axiom,
! [A2: set_set_set_a] : ( ord_le2103863998_set_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_102_dual__order_Orefl,axiom,
! [A2: $o > set_a] : ( ord_less_eq_o_set_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_103_dual__order_Orefl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_104_order__trans,axiom,
! [X3: set_Pr204296108_a_a_e,Y3: set_Pr204296108_a_a_e,Z2: set_Pr204296108_a_a_e] :
( ( ord_le700723212_a_a_e @ X3 @ Y3 )
=> ( ( ord_le700723212_a_a_e @ Y3 @ Z2 )
=> ( ord_le700723212_a_a_e @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_105_order__trans,axiom,
! [X3: set_set_a,Y3: set_set_a,Z2: set_set_a] :
( ( ord_le318720350_set_a @ X3 @ Y3 )
=> ( ( ord_le318720350_set_a @ Y3 @ Z2 )
=> ( ord_le318720350_set_a @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_106_order__trans,axiom,
! [X3: set_set_set_a,Y3: set_set_set_a,Z2: set_set_set_a] :
( ( ord_le2103863998_set_a @ X3 @ Y3 )
=> ( ( ord_le2103863998_set_a @ Y3 @ Z2 )
=> ( ord_le2103863998_set_a @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_107_order__trans,axiom,
! [X3: $o > set_a,Y3: $o > set_a,Z2: $o > set_a] :
( ( ord_less_eq_o_set_a @ X3 @ Y3 )
=> ( ( ord_less_eq_o_set_a @ Y3 @ Z2 )
=> ( ord_less_eq_o_set_a @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_108_order__trans,axiom,
! [X3: set_a,Y3: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ( ord_less_eq_set_a @ Y3 @ Z2 )
=> ( ord_less_eq_set_a @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_109_order__class_Oorder_Oantisym,axiom,
! [A2: set_Pr204296108_a_a_e,B2: set_Pr204296108_a_a_e] :
( ( ord_le700723212_a_a_e @ A2 @ B2 )
=> ( ( ord_le700723212_a_a_e @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% order_class.order.antisym
thf(fact_110_order__class_Oorder_Oantisym,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le318720350_set_a @ A2 @ B2 )
=> ( ( ord_le318720350_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% order_class.order.antisym
thf(fact_111_order__class_Oorder_Oantisym,axiom,
! [A2: set_set_set_a,B2: set_set_set_a] :
( ( ord_le2103863998_set_a @ A2 @ B2 )
=> ( ( ord_le2103863998_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% order_class.order.antisym
thf(fact_112_order__class_Oorder_Oantisym,axiom,
! [A2: $o > set_a,B2: $o > set_a] :
( ( ord_less_eq_o_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_o_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% order_class.order.antisym
thf(fact_113_order__class_Oorder_Oantisym,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% order_class.order.antisym
thf(fact_114_ord__le__eq__trans,axiom,
! [A2: set_Pr204296108_a_a_e,B2: set_Pr204296108_a_a_e,C2: set_Pr204296108_a_a_e] :
( ( ord_le700723212_a_a_e @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_le700723212_a_a_e @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_115_ord__le__eq__trans,axiom,
! [A2: set_set_a,B2: set_set_a,C2: set_set_a] :
( ( ord_le318720350_set_a @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_le318720350_set_a @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_116_ord__le__eq__trans,axiom,
! [A2: set_set_set_a,B2: set_set_set_a,C2: set_set_set_a] :
( ( ord_le2103863998_set_a @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_le2103863998_set_a @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_117_ord__le__eq__trans,axiom,
! [A2: $o > set_a,B2: $o > set_a,C2: $o > set_a] :
( ( ord_less_eq_o_set_a @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_o_set_a @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_118_ord__le__eq__trans,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_119_ord__eq__le__trans,axiom,
! [A2: set_Pr204296108_a_a_e,B2: set_Pr204296108_a_a_e,C2: set_Pr204296108_a_a_e] :
( ( A2 = B2 )
=> ( ( ord_le700723212_a_a_e @ B2 @ C2 )
=> ( ord_le700723212_a_a_e @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_120_ord__eq__le__trans,axiom,
! [A2: set_set_a,B2: set_set_a,C2: set_set_a] :
( ( A2 = B2 )
=> ( ( ord_le318720350_set_a @ B2 @ C2 )
=> ( ord_le318720350_set_a @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_121_ord__eq__le__trans,axiom,
! [A2: set_set_set_a,B2: set_set_set_a,C2: set_set_set_a] :
( ( A2 = B2 )
=> ( ( ord_le2103863998_set_a @ B2 @ C2 )
=> ( ord_le2103863998_set_a @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_122_ord__eq__le__trans,axiom,
! [A2: $o > set_a,B2: $o > set_a,C2: $o > set_a] :
( ( A2 = B2 )
=> ( ( ord_less_eq_o_set_a @ B2 @ C2 )
=> ( ord_less_eq_o_set_a @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_123_ord__eq__le__trans,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( A2 = B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_124_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y2: set_Pr204296108_a_a_e,Z: set_Pr204296108_a_a_e] : Y2 = Z )
= ( ^ [A3: set_Pr204296108_a_a_e,B3: set_Pr204296108_a_a_e] :
( ( ord_le700723212_a_a_e @ A3 @ B3 )
& ( ord_le700723212_a_a_e @ B3 @ A3 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_125_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y2: set_set_a,Z: set_set_a] : Y2 = Z )
= ( ^ [A3: set_set_a,B3: set_set_a] :
( ( ord_le318720350_set_a @ A3 @ B3 )
& ( ord_le318720350_set_a @ B3 @ A3 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_126_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y2: set_set_set_a,Z: set_set_set_a] : Y2 = Z )
= ( ^ [A3: set_set_set_a,B3: set_set_set_a] :
( ( ord_le2103863998_set_a @ A3 @ B3 )
& ( ord_le2103863998_set_a @ B3 @ A3 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_127_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y2: $o > set_a,Z: $o > set_a] : Y2 = Z )
= ( ^ [A3: $o > set_a,B3: $o > set_a] :
( ( ord_less_eq_o_set_a @ A3 @ B3 )
& ( ord_less_eq_o_set_a @ B3 @ A3 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_128_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y2: set_a,Z: set_a] : Y2 = Z )
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_129_antisym__conv,axiom,
! [Y3: set_Pr204296108_a_a_e,X3: set_Pr204296108_a_a_e] :
( ( ord_le700723212_a_a_e @ Y3 @ X3 )
=> ( ( ord_le700723212_a_a_e @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% antisym_conv
thf(fact_130_antisym__conv,axiom,
! [Y3: set_set_a,X3: set_set_a] :
( ( ord_le318720350_set_a @ Y3 @ X3 )
=> ( ( ord_le318720350_set_a @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% antisym_conv
thf(fact_131_antisym__conv,axiom,
! [Y3: set_set_set_a,X3: set_set_set_a] :
( ( ord_le2103863998_set_a @ Y3 @ X3 )
=> ( ( ord_le2103863998_set_a @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% antisym_conv
thf(fact_132_antisym__conv,axiom,
! [Y3: $o > set_a,X3: $o > set_a] :
( ( ord_less_eq_o_set_a @ Y3 @ X3 )
=> ( ( ord_less_eq_o_set_a @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% antisym_conv
thf(fact_133_antisym__conv,axiom,
! [Y3: set_a,X3: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X3 )
=> ( ( ord_less_eq_set_a @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% antisym_conv
thf(fact_134_order_Otrans,axiom,
! [A2: set_Pr204296108_a_a_e,B2: set_Pr204296108_a_a_e,C2: set_Pr204296108_a_a_e] :
( ( ord_le700723212_a_a_e @ A2 @ B2 )
=> ( ( ord_le700723212_a_a_e @ B2 @ C2 )
=> ( ord_le700723212_a_a_e @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_135_order_Otrans,axiom,
! [A2: set_set_a,B2: set_set_a,C2: set_set_a] :
( ( ord_le318720350_set_a @ A2 @ B2 )
=> ( ( ord_le318720350_set_a @ B2 @ C2 )
=> ( ord_le318720350_set_a @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_136_order_Otrans,axiom,
! [A2: set_set_set_a,B2: set_set_set_a,C2: set_set_set_a] :
( ( ord_le2103863998_set_a @ A2 @ B2 )
=> ( ( ord_le2103863998_set_a @ B2 @ C2 )
=> ( ord_le2103863998_set_a @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_137_order_Otrans,axiom,
! [A2: $o > set_a,B2: $o > set_a,C2: $o > set_a] :
( ( ord_less_eq_o_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_o_set_a @ B2 @ C2 )
=> ( ord_less_eq_o_set_a @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_138_order_Otrans,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_139_eq__refl,axiom,
! [X3: set_Pr204296108_a_a_e,Y3: set_Pr204296108_a_a_e] :
( ( X3 = Y3 )
=> ( ord_le700723212_a_a_e @ X3 @ Y3 ) ) ).
% eq_refl
thf(fact_140_eq__refl,axiom,
! [X3: set_set_a,Y3: set_set_a] :
( ( X3 = Y3 )
=> ( ord_le318720350_set_a @ X3 @ Y3 ) ) ).
% eq_refl
thf(fact_141_eq__refl,axiom,
! [X3: set_set_set_a,Y3: set_set_set_a] :
( ( X3 = Y3 )
=> ( ord_le2103863998_set_a @ X3 @ Y3 ) ) ).
% eq_refl
thf(fact_142_eq__refl,axiom,
! [X3: $o > set_a,Y3: $o > set_a] :
( ( X3 = Y3 )
=> ( ord_less_eq_o_set_a @ X3 @ Y3 ) ) ).
% eq_refl
thf(fact_143_eq__refl,axiom,
! [X3: set_a,Y3: set_a] :
( ( X3 = Y3 )
=> ( ord_less_eq_set_a @ X3 @ Y3 ) ) ).
% eq_refl
thf(fact_144_antisym,axiom,
! [X3: set_Pr204296108_a_a_e,Y3: set_Pr204296108_a_a_e] :
( ( ord_le700723212_a_a_e @ X3 @ Y3 )
=> ( ( ord_le700723212_a_a_e @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ).
% antisym
thf(fact_145_antisym,axiom,
! [X3: set_set_a,Y3: set_set_a] :
( ( ord_le318720350_set_a @ X3 @ Y3 )
=> ( ( ord_le318720350_set_a @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ).
% antisym
thf(fact_146_antisym,axiom,
! [X3: set_set_set_a,Y3: set_set_set_a] :
( ( ord_le2103863998_set_a @ X3 @ Y3 )
=> ( ( ord_le2103863998_set_a @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ).
% antisym
thf(fact_147_antisym,axiom,
! [X3: $o > set_a,Y3: $o > set_a] :
( ( ord_less_eq_o_set_a @ X3 @ Y3 )
=> ( ( ord_less_eq_o_set_a @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ).
% antisym
thf(fact_148_antisym,axiom,
! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ( ord_less_eq_set_a @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ).
% antisym
thf(fact_149_eq__iff,axiom,
( ( ^ [Y2: set_Pr204296108_a_a_e,Z: set_Pr204296108_a_a_e] : Y2 = Z )
= ( ^ [X4: set_Pr204296108_a_a_e,Y4: set_Pr204296108_a_a_e] :
( ( ord_le700723212_a_a_e @ X4 @ Y4 )
& ( ord_le700723212_a_a_e @ Y4 @ X4 ) ) ) ) ).
% eq_iff
thf(fact_150_eq__iff,axiom,
( ( ^ [Y2: set_set_a,Z: set_set_a] : Y2 = Z )
= ( ^ [X4: set_set_a,Y4: set_set_a] :
( ( ord_le318720350_set_a @ X4 @ Y4 )
& ( ord_le318720350_set_a @ Y4 @ X4 ) ) ) ) ).
% eq_iff
thf(fact_151_eq__iff,axiom,
( ( ^ [Y2: set_set_set_a,Z: set_set_set_a] : Y2 = Z )
= ( ^ [X4: set_set_set_a,Y4: set_set_set_a] :
( ( ord_le2103863998_set_a @ X4 @ Y4 )
& ( ord_le2103863998_set_a @ Y4 @ X4 ) ) ) ) ).
% eq_iff
thf(fact_152_eq__iff,axiom,
( ( ^ [Y2: $o > set_a,Z: $o > set_a] : Y2 = Z )
= ( ^ [X4: $o > set_a,Y4: $o > set_a] :
( ( ord_less_eq_o_set_a @ X4 @ Y4 )
& ( ord_less_eq_o_set_a @ Y4 @ X4 ) ) ) ) ).
% eq_iff
thf(fact_153_eq__iff,axiom,
( ( ^ [Y2: set_a,Z: set_a] : Y2 = Z )
= ( ^ [X4: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y4 )
& ( ord_less_eq_set_a @ Y4 @ X4 ) ) ) ) ).
% eq_iff
thf(fact_154_ord__le__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_155_ord__le__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_set_a,C2: set_set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ord_le318720350_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_le318720350_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_156_ord__le__eq__subst,axiom,
! [A2: set_set_a,B2: set_set_a,F: set_set_a > set_a,C2: set_a] :
( ( ord_le318720350_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: set_set_a,Y: set_set_a] :
( ( ord_le318720350_set_a @ X2 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_157_ord__le__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_set_set_a,C2: set_set_set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ord_le2103863998_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_le2103863998_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_158_ord__le__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > $o > set_a,C2: $o > set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ord_less_eq_o_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_o_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_159_ord__le__eq__subst,axiom,
! [A2: set_set_a,B2: set_set_a,F: set_set_a > set_set_a,C2: set_set_a] :
( ( ord_le318720350_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: set_set_a,Y: set_set_a] :
( ( ord_le318720350_set_a @ X2 @ Y )
=> ( ord_le318720350_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_le318720350_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_160_ord__le__eq__subst,axiom,
! [A2: set_set_set_a,B2: set_set_set_a,F: set_set_set_a > set_a,C2: set_a] :
( ( ord_le2103863998_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: set_set_set_a,Y: set_set_set_a] :
( ( ord_le2103863998_set_a @ X2 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_161_ord__le__eq__subst,axiom,
! [A2: $o > set_a,B2: $o > set_a,F: ( $o > set_a ) > set_a,C2: set_a] :
( ( ord_less_eq_o_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: $o > set_a,Y: $o > set_a] :
( ( ord_less_eq_o_set_a @ X2 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_162_ord__le__eq__subst,axiom,
! [A2: set_set_a,B2: set_set_a,F: set_set_a > set_set_set_a,C2: set_set_set_a] :
( ( ord_le318720350_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: set_set_a,Y: set_set_a] :
( ( ord_le318720350_set_a @ X2 @ Y )
=> ( ord_le2103863998_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_le2103863998_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_163_ord__le__eq__subst,axiom,
! [A2: set_set_a,B2: set_set_a,F: set_set_a > $o > set_a,C2: $o > set_a] :
( ( ord_le318720350_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: set_set_a,Y: set_set_a] :
( ( ord_le318720350_set_a @ X2 @ Y )
=> ( ord_less_eq_o_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_o_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_164_ord__eq__le__subst,axiom,
! [A2: set_Pr204296108_a_a_e,F: ( $o > set_a ) > set_Pr204296108_a_a_e,B2: $o > set_a,C2: $o > set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_o_set_a @ B2 @ C2 )
=> ( ! [X2: $o > set_a,Y: $o > set_a] :
( ( ord_less_eq_o_set_a @ X2 @ Y )
=> ( ord_le700723212_a_a_e @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_le700723212_a_a_e @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_165_ord__eq__le__subst,axiom,
! [A2: set_set_a,F: ( $o > set_a ) > set_set_a,B2: $o > set_a,C2: $o > set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_o_set_a @ B2 @ C2 )
=> ( ! [X2: $o > set_a,Y: $o > set_a] :
( ( ord_less_eq_o_set_a @ X2 @ Y )
=> ( ord_le318720350_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_le318720350_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_166_ord__eq__le__subst,axiom,
! [A2: set_set_set_a,F: ( $o > set_a ) > set_set_set_a,B2: $o > set_a,C2: $o > set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_o_set_a @ B2 @ C2 )
=> ( ! [X2: $o > set_a,Y: $o > set_a] :
( ( ord_less_eq_o_set_a @ X2 @ Y )
=> ( ord_le2103863998_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_le2103863998_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_167_ord__eq__le__subst,axiom,
! [A2: $o > set_a,F: ( $o > set_a ) > $o > set_a,B2: $o > set_a,C2: $o > set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_o_set_a @ B2 @ C2 )
=> ( ! [X2: $o > set_a,Y: $o > set_a] :
( ( ord_less_eq_o_set_a @ X2 @ Y )
=> ( ord_less_eq_o_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_o_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_168_ord__eq__le__subst,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_a,C2: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_169_order__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C2 )
=> ( ! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_170_order__subst1,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_171_mem__Collect__eq,axiom,
! [A2: set_a,P2: set_a > $o] :
( ( member_set_a @ A2 @ ( collect_set_a @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_172_mem__Collect__eq,axiom,
! [A2: produc715398134_a_a_e,P2: produc715398134_a_a_e > $o] :
( ( member1258382221_a_a_e @ A2 @ ( collec1369693387_a_a_e @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_173_Collect__mem__eq,axiom,
! [A: set_set_a] :
( ( collect_set_a
@ ^ [X4: set_a] : ( member_set_a @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_174_Collect__mem__eq,axiom,
! [A: set_Pr204296108_a_a_e] :
( ( collec1369693387_a_a_e
@ ^ [X4: produc715398134_a_a_e] : ( member1258382221_a_a_e @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_175_rev__image__eqI,axiom,
! [X3: set_a,A: set_set_a,B2: set_a,F: set_a > set_a] :
( ( member_set_a @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_set_a @ B2 @ ( image_set_a_set_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_176_rev__image__eqI,axiom,
! [X3: set_a,A: set_set_a,B2: produc715398134_a_a_e,F: set_a > produc715398134_a_a_e] :
( ( member_set_a @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member1258382221_a_a_e @ B2 @ ( image_210900495_a_a_e @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_177_rev__image__eqI,axiom,
! [X3: produc715398134_a_a_e,A: set_Pr204296108_a_a_e,B2: set_a,F: produc715398134_a_a_e > set_a] :
( ( member1258382221_a_a_e @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_set_a @ B2 @ ( image_64240033_set_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_178_rev__image__eqI,axiom,
! [X3: produc715398134_a_a_e,A: set_Pr204296108_a_a_e,B2: produc715398134_a_a_e,F: produc715398134_a_a_e > produc715398134_a_a_e] :
( ( member1258382221_a_a_e @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member1258382221_a_a_e @ B2 @ ( image_625620211_a_a_e @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_179_ball__imageD,axiom,
! [F: produc715398134_a_a_e > set_a,A: set_Pr204296108_a_a_e,P2: set_a > $o] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ ( image_64240033_set_a @ F @ A ) )
=> ( P2 @ X2 ) )
=> ! [X: produc715398134_a_a_e] :
( ( member1258382221_a_a_e @ X @ A )
=> ( P2 @ ( F @ X ) ) ) ) ).
% ball_imageD
thf(fact_180_image__cong,axiom,
! [M: set_Pr204296108_a_a_e,N: set_Pr204296108_a_a_e,F: produc715398134_a_a_e > set_a,G: produc715398134_a_a_e > set_a] :
( ( M = N )
=> ( ! [X2: produc715398134_a_a_e] :
( ( member1258382221_a_a_e @ X2 @ N )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_64240033_set_a @ F @ M )
= ( image_64240033_set_a @ G @ N ) ) ) ) ).
% image_cong
thf(fact_181_bex__imageD,axiom,
! [F: produc715398134_a_a_e > set_a,A: set_Pr204296108_a_a_e,P2: set_a > $o] :
( ? [X: set_a] :
( ( member_set_a @ X @ ( image_64240033_set_a @ F @ A ) )
& ( P2 @ X ) )
=> ? [X2: produc715398134_a_a_e] :
( ( member1258382221_a_a_e @ X2 @ A )
& ( P2 @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_182_image__iff,axiom,
! [Z2: set_a,F: produc715398134_a_a_e > set_a,A: set_Pr204296108_a_a_e] :
( ( member_set_a @ Z2 @ ( image_64240033_set_a @ F @ A ) )
= ( ? [X4: produc715398134_a_a_e] :
( ( member1258382221_a_a_e @ X4 @ A )
& ( Z2
= ( F @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_183_imageI,axiom,
! [X3: set_a,A: set_set_a,F: set_a > set_a] :
( ( member_set_a @ X3 @ A )
=> ( member_set_a @ ( F @ X3 ) @ ( image_set_a_set_a @ F @ A ) ) ) ).
% imageI
thf(fact_184_imageI,axiom,
! [X3: set_a,A: set_set_a,F: set_a > produc715398134_a_a_e] :
( ( member_set_a @ X3 @ A )
=> ( member1258382221_a_a_e @ ( F @ X3 ) @ ( image_210900495_a_a_e @ F @ A ) ) ) ).
% imageI
thf(fact_185_imageI,axiom,
! [X3: produc715398134_a_a_e,A: set_Pr204296108_a_a_e,F: produc715398134_a_a_e > set_a] :
( ( member1258382221_a_a_e @ X3 @ A )
=> ( member_set_a @ ( F @ X3 ) @ ( image_64240033_set_a @ F @ A ) ) ) ).
% imageI
thf(fact_186_imageI,axiom,
! [X3: produc715398134_a_a_e,A: set_Pr204296108_a_a_e,F: produc715398134_a_a_e > produc715398134_a_a_e] :
( ( member1258382221_a_a_e @ X3 @ A )
=> ( member1258382221_a_a_e @ ( F @ X3 ) @ ( image_625620211_a_a_e @ F @ A ) ) ) ).
% imageI
thf(fact_187_Collect__mono__iff,axiom,
! [P2: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q ) )
= ( ! [X4: a] :
( ( P2 @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_188_set__eq__subset,axiom,
( ( ^ [Y2: set_a,Z: set_a] : Y2 = Z )
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_189_subset__trans,axiom,
! [A: set_a,B: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C3 )
=> ( ord_less_eq_set_a @ A @ C3 ) ) ) ).
% subset_trans
thf(fact_190_Collect__mono,axiom,
! [P2: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P2 @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_191_subset__iff,axiom,
( ord_le318720350_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] :
! [T: set_a] :
( ( member_set_a @ T @ A4 )
=> ( member_set_a @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_192_subset__iff,axiom,
( ord_le700723212_a_a_e
= ( ^ [A4: set_Pr204296108_a_a_e,B4: set_Pr204296108_a_a_e] :
! [T: produc715398134_a_a_e] :
( ( member1258382221_a_a_e @ T @ A4 )
=> ( member1258382221_a_a_e @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_193_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
! [T: a] :
( ( member_a @ T @ A4 )
=> ( member_a @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_194_equalityD2,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% equalityD2
thf(fact_195_subset__eq,axiom,
( ord_le318720350_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] :
! [X4: set_a] :
( ( member_set_a @ X4 @ A4 )
=> ( member_set_a @ X4 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_196_subset__eq,axiom,
( ord_le700723212_a_a_e
= ( ^ [A4: set_Pr204296108_a_a_e,B4: set_Pr204296108_a_a_e] :
! [X4: produc715398134_a_a_e] :
( ( member1258382221_a_a_e @ X4 @ A4 )
=> ( member1258382221_a_a_e @ X4 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_197_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
! [X4: a] :
( ( member_a @ X4 @ A4 )
=> ( member_a @ X4 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_198_equalityE,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ~ ( ord_less_eq_set_a @ B @ A ) ) ) ).
% equalityE
thf(fact_199_subsetD,axiom,
! [A: set_set_a,B: set_set_a,C2: set_a] :
( ( ord_le318720350_set_a @ A @ B )
=> ( ( member_set_a @ C2 @ A )
=> ( member_set_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_200_subsetD,axiom,
! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e,C2: produc715398134_a_a_e] :
( ( ord_le700723212_a_a_e @ A @ B )
=> ( ( member1258382221_a_a_e @ C2 @ A )
=> ( member1258382221_a_a_e @ C2 @ B ) ) ) ).
% subsetD
thf(fact_201_subsetD,axiom,
! [A: set_a,B: set_a,C2: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ C2 @ A )
=> ( member_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_202_in__mono,axiom,
! [A: set_set_a,B: set_set_a,X3: set_a] :
( ( ord_le318720350_set_a @ A @ B )
=> ( ( member_set_a @ X3 @ A )
=> ( member_set_a @ X3 @ B ) ) ) ).
% in_mono
thf(fact_203_in__mono,axiom,
! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e,X3: produc715398134_a_a_e] :
( ( ord_le700723212_a_a_e @ A @ B )
=> ( ( member1258382221_a_a_e @ X3 @ A )
=> ( member1258382221_a_a_e @ X3 @ B ) ) ) ).
% in_mono
thf(fact_204_in__mono,axiom,
! [A: set_a,B: set_a,X3: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ X3 @ A )
=> ( member_a @ X3 @ B ) ) ) ).
% in_mono
thf(fact_205_prod_Oinject,axiom,
! [X1: set_a,X22: a > e,Y1: set_a,Y22: a > e] :
( ( ( produc1134456688_a_a_e @ X1 @ X22 )
= ( produc1134456688_a_a_e @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_206_old_Oprod_Oinject,axiom,
! [A2: set_a,B2: a > e,A5: set_a,B5: a > e] :
( ( ( produc1134456688_a_a_e @ A2 @ B2 )
= ( produc1134456688_a_a_e @ A5 @ B5 ) )
= ( ( A2 = A5 )
& ( B2 = B5 ) ) ) ).
% old.prod.inject
thf(fact_207_eq__fst__iff,axiom,
! [A2: set_a,P: produc715398134_a_a_e] :
( ( A2
= ( produc1040928388_a_a_e @ P ) )
= ( ? [B3: a > e] :
( P
= ( produc1134456688_a_a_e @ A2 @ B3 ) ) ) ) ).
% eq_fst_iff
thf(fact_208_fst__conv,axiom,
! [X1: set_a,X22: a > e] :
( ( produc1040928388_a_a_e @ ( produc1134456688_a_a_e @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_209_fst__eqD,axiom,
! [X3: set_a,Y3: a > e,A2: set_a] :
( ( ( produc1040928388_a_a_e @ ( produc1134456688_a_a_e @ X3 @ Y3 ) )
= A2 )
=> ( X3 = A2 ) ) ).
% fst_eqD
thf(fact_210_fstI,axiom,
! [X3: produc715398134_a_a_e,Y3: set_a,Z2: a > e] :
( ( X3
= ( produc1134456688_a_a_e @ Y3 @ Z2 ) )
=> ( ( produc1040928388_a_a_e @ X3 )
= Y3 ) ) ).
% fstI
thf(fact_211_all__subset__image,axiom,
! [F: produc715398134_a_a_e > set_a,A: set_Pr204296108_a_a_e,P2: set_set_a > $o] :
( ( ! [B4: set_set_a] :
( ( ord_le318720350_set_a @ B4 @ ( image_64240033_set_a @ F @ A ) )
=> ( P2 @ B4 ) ) )
= ( ! [B4: set_Pr204296108_a_a_e] :
( ( ord_le700723212_a_a_e @ B4 @ A )
=> ( P2 @ ( image_64240033_set_a @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_212_all__subset__image,axiom,
! [F: a > a,A: set_a,P2: set_a > $o] :
( ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ ( image_a_a @ F @ A ) )
=> ( P2 @ B4 ) ) )
= ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( P2 @ ( image_a_a @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_213_Greatest__equality,axiom,
! [P2: set_a > $o,X3: set_a] :
( ( P2 @ X3 )
=> ( ! [Y: set_a] :
( ( P2 @ Y )
=> ( ord_less_eq_set_a @ Y @ X3 ) )
=> ( ( order_Greatest_set_a @ P2 )
= X3 ) ) ) ).
% Greatest_equality
thf(fact_214_GreatestI2__order,axiom,
! [P2: set_a > $o,X3: set_a,Q: set_a > $o] :
( ( P2 @ X3 )
=> ( ! [Y: set_a] :
( ( P2 @ Y )
=> ( ord_less_eq_set_a @ Y @ X3 ) )
=> ( ! [X2: set_a] :
( ( P2 @ X2 )
=> ( ! [Y5: set_a] :
( ( P2 @ Y5 )
=> ( ord_less_eq_set_a @ Y5 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_Greatest_set_a @ P2 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_215_old_Oprod_Oinducts,axiom,
! [P2: produc715398134_a_a_e > $o,Prod: produc715398134_a_a_e] :
( ! [A6: set_a,B6: a > e] : ( P2 @ ( produc1134456688_a_a_e @ A6 @ B6 ) )
=> ( P2 @ Prod ) ) ).
% old.prod.inducts
thf(fact_216_old_Oprod_Oexhaust,axiom,
! [Y3: produc715398134_a_a_e] :
~ ! [A6: set_a,B6: a > e] :
( Y3
!= ( produc1134456688_a_a_e @ A6 @ B6 ) ) ).
% old.prod.exhaust
thf(fact_217_Pair__inject,axiom,
! [A2: set_a,B2: a > e,A5: set_a,B5: a > e] :
( ( ( produc1134456688_a_a_e @ A2 @ B2 )
= ( produc1134456688_a_a_e @ A5 @ B5 ) )
=> ~ ( ( A2 = A5 )
=> ( B2 != B5 ) ) ) ).
% Pair_inject
thf(fact_218_prod__cases,axiom,
! [P2: produc715398134_a_a_e > $o,P: produc715398134_a_a_e] :
( ! [A6: set_a,B6: a > e] : ( P2 @ ( produc1134456688_a_a_e @ A6 @ B6 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_219_image__Fpow__mono,axiom,
! [F: produc715398134_a_a_e > set_a,A: set_Pr204296108_a_a_e,B: set_set_a] :
( ( ord_le318720350_set_a @ ( image_64240033_set_a @ F @ A ) @ B )
=> ( ord_le2103863998_set_a @ ( image_1441218763_set_a @ ( image_64240033_set_a @ F ) @ ( finite33065488_a_a_e @ A ) ) @ ( finite_Fpow_set_a @ B ) ) ) ).
% image_Fpow_mono
thf(fact_220_not__sub,axiom,
! [A: set_set_a,B: set_set_a] :
( ~ ( ord_le318720350_set_a @ A @ B )
=> ? [A6: set_a] :
( ( member_set_a @ A6 @ A )
& ~ ( member_set_a @ A6 @ B ) ) ) ).
% not_sub
thf(fact_221_not__sub,axiom,
! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e] :
( ~ ( ord_le700723212_a_a_e @ A @ B )
=> ? [A6: produc715398134_a_a_e] :
( ( member1258382221_a_a_e @ A6 @ A )
& ~ ( member1258382221_a_a_e @ A6 @ B ) ) ) ).
% not_sub
thf(fact_222_not__sub,axiom,
! [A: set_a,B: set_a] :
( ~ ( ord_less_eq_set_a @ A @ B )
=> ? [A6: a] :
( ( member_a @ A6 @ A )
& ~ ( member_a @ A6 @ B ) ) ) ).
% not_sub
thf(fact_223_eqsets__sub,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% eqsets_sub
thf(fact_224_sub__which1,axiom,
! [A: set_set_a,B: set_set_a,X3: set_a] :
( ( ( ord_le318720350_set_a @ A @ B )
| ( ord_le318720350_set_a @ B @ A ) )
=> ( ( member_set_a @ X3 @ A )
=> ( ~ ( member_set_a @ X3 @ B )
=> ( ord_le318720350_set_a @ B @ A ) ) ) ) ).
% sub_which1
thf(fact_225_sub__which1,axiom,
! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e,X3: produc715398134_a_a_e] :
( ( ( ord_le700723212_a_a_e @ A @ B )
| ( ord_le700723212_a_a_e @ B @ A ) )
=> ( ( member1258382221_a_a_e @ X3 @ A )
=> ( ~ ( member1258382221_a_a_e @ X3 @ B )
=> ( ord_le700723212_a_a_e @ B @ A ) ) ) ) ).
% sub_which1
thf(fact_226_sub__which1,axiom,
! [A: set_a,B: set_a,X3: a] :
( ( ( ord_less_eq_set_a @ A @ B )
| ( ord_less_eq_set_a @ B @ A ) )
=> ( ( member_a @ X3 @ A )
=> ( ~ ( member_a @ X3 @ B )
=> ( ord_less_eq_set_a @ B @ A ) ) ) ) ).
% sub_which1
thf(fact_227_Fpow__mono,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_le318720350_set_a @ ( finite_Fpow_a @ A ) @ ( finite_Fpow_a @ B ) ) ) ).
% Fpow_mono
thf(fact_228_le__convert,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( A2 = B2 )
=> ( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ord_less_eq_set_a @ B2 @ C2 ) ) ) ).
% le_convert
thf(fact_229_ge__convert,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( A2 = B2 )
=> ( ( ord_less_eq_set_a @ C2 @ A2 )
=> ( ord_less_eq_set_a @ C2 @ B2 ) ) ) ).
% ge_convert
thf(fact_230_mem__in__image3,axiom,
! [B2: set_a,F: produc715398134_a_a_e > set_a,A: set_Pr204296108_a_a_e] :
( ( member_set_a @ B2 @ ( image_64240033_set_a @ F @ A ) )
=> ? [X2: produc715398134_a_a_e] :
( ( member1258382221_a_a_e @ X2 @ A )
& ( B2
= ( F @ X2 ) ) ) ) ).
% mem_in_image3
thf(fact_231_mem__in__image2,axiom,
! [A2: set_a,A: set_set_a,F: set_a > set_a] :
( ( member_set_a @ A2 @ A )
=> ( member_set_a @ ( F @ A2 ) @ ( image_set_a_set_a @ F @ A ) ) ) ).
% mem_in_image2
thf(fact_232_mem__in__image2,axiom,
! [A2: set_a,A: set_set_a,F: set_a > produc715398134_a_a_e] :
( ( member_set_a @ A2 @ A )
=> ( member1258382221_a_a_e @ ( F @ A2 ) @ ( image_210900495_a_a_e @ F @ A ) ) ) ).
% mem_in_image2
thf(fact_233_mem__in__image2,axiom,
! [A2: produc715398134_a_a_e,A: set_Pr204296108_a_a_e,F: produc715398134_a_a_e > set_a] :
( ( member1258382221_a_a_e @ A2 @ A )
=> ( member_set_a @ ( F @ A2 ) @ ( image_64240033_set_a @ F @ A ) ) ) ).
% mem_in_image2
thf(fact_234_mem__in__image2,axiom,
! [A2: produc715398134_a_a_e,A: set_Pr204296108_a_a_e,F: produc715398134_a_a_e > produc715398134_a_a_e] :
( ( member1258382221_a_a_e @ A2 @ A )
=> ( member1258382221_a_a_e @ ( F @ A2 ) @ ( image_625620211_a_a_e @ F @ A ) ) ) ).
% mem_in_image2
thf(fact_235_mem__in__image1,axiom,
! [A: set_set_a,F: set_a > set_a,B: set_set_a,A2: set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( member_set_a @ ( F @ X2 ) @ B ) )
=> ( ( member_set_a @ A2 @ A )
=> ( member_set_a @ ( F @ A2 ) @ ( image_set_a_set_a @ F @ A ) ) ) ) ).
% mem_in_image1
thf(fact_236_mem__in__image1,axiom,
! [A: set_Pr204296108_a_a_e,F: produc715398134_a_a_e > set_a,B: set_set_a,A2: produc715398134_a_a_e] :
( ! [X2: produc715398134_a_a_e] :
( ( member1258382221_a_a_e @ X2 @ A )
=> ( member_set_a @ ( F @ X2 ) @ B ) )
=> ( ( member1258382221_a_a_e @ A2 @ A )
=> ( member_set_a @ ( F @ A2 ) @ ( image_64240033_set_a @ F @ A ) ) ) ) ).
% mem_in_image1
thf(fact_237_mem__in__image1,axiom,
! [A: set_set_a,F: set_a > produc715398134_a_a_e,B: set_Pr204296108_a_a_e,A2: set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( member1258382221_a_a_e @ ( F @ X2 ) @ B ) )
=> ( ( member_set_a @ A2 @ A )
=> ( member1258382221_a_a_e @ ( F @ A2 ) @ ( image_210900495_a_a_e @ F @ A ) ) ) ) ).
% mem_in_image1
thf(fact_238_mem__in__image1,axiom,
! [A: set_Pr204296108_a_a_e,F: produc715398134_a_a_e > produc715398134_a_a_e,B: set_Pr204296108_a_a_e,A2: produc715398134_a_a_e] :
( ! [X2: produc715398134_a_a_e] :
( ( member1258382221_a_a_e @ X2 @ A )
=> ( member1258382221_a_a_e @ ( F @ X2 ) @ B ) )
=> ( ( member1258382221_a_a_e @ A2 @ A )
=> ( member1258382221_a_a_e @ ( F @ A2 ) @ ( image_625620211_a_a_e @ F @ A ) ) ) ) ).
% mem_in_image1
thf(fact_239_proper__subset,axiom,
! [A: set_set_a,B: set_set_a,X3: set_a] :
( ( ord_le318720350_set_a @ A @ B )
=> ( ~ ( member_set_a @ X3 @ A )
=> ( ( member_set_a @ X3 @ B )
=> ( A != B ) ) ) ) ).
% proper_subset
thf(fact_240_proper__subset,axiom,
! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e,X3: produc715398134_a_a_e] :
( ( ord_le700723212_a_a_e @ A @ B )
=> ( ~ ( member1258382221_a_a_e @ X3 @ A )
=> ( ( member1258382221_a_a_e @ X3 @ B )
=> ( A != B ) ) ) ) ).
% proper_subset
thf(fact_241_proper__subset,axiom,
! [A: set_a,B: set_a,X3: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ~ ( member_a @ X3 @ A )
=> ( ( member_a @ X3 @ B )
=> ( A != B ) ) ) ) ).
% proper_subset
thf(fact_242_not__subseteq,axiom,
! [A: set_set_a,B: set_set_a] :
( ~ ( ord_le318720350_set_a @ A @ B )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ~ ( member_set_a @ X2 @ B ) ) ) ).
% not_subseteq
thf(fact_243_not__subseteq,axiom,
! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e] :
( ~ ( ord_le700723212_a_a_e @ A @ B )
=> ? [X2: produc715398134_a_a_e] :
( ( member1258382221_a_a_e @ X2 @ A )
& ~ ( member1258382221_a_a_e @ X2 @ B ) ) ) ).
% not_subseteq
thf(fact_244_not__subseteq,axiom,
! [A: set_a,B: set_a] :
( ~ ( ord_less_eq_set_a @ A @ B )
=> ? [X2: a] :
( ( member_a @ X2 @ A )
& ~ ( member_a @ X2 @ B ) ) ) ).
% not_subseteq
thf(fact_245_subset__self,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% subset_self
thf(fact_246_sets__not__eq,axiom,
! [A: set_set_a,B: set_set_a] :
( ( A != B )
=> ( ( ord_le318720350_set_a @ B @ A )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ~ ( member_set_a @ X2 @ B ) ) ) ) ).
% sets_not_eq
thf(fact_247_sets__not__eq,axiom,
! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e] :
( ( A != B )
=> ( ( ord_le700723212_a_a_e @ B @ A )
=> ? [X2: produc715398134_a_a_e] :
( ( member1258382221_a_a_e @ X2 @ A )
& ~ ( member1258382221_a_a_e @ X2 @ B ) ) ) ) ).
% sets_not_eq
thf(fact_248_sets__not__eq,axiom,
! [A: set_a,B: set_a] :
( ( A != B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ? [X2: a] :
( ( member_a @ X2 @ A )
& ~ ( member_a @ X2 @ B ) ) ) ) ).
% sets_not_eq
thf(fact_249_sub__which2,axiom,
! [A: set_set_a,B: set_set_a,X3: set_a] :
( ( ( ord_le318720350_set_a @ A @ B )
| ( ord_le318720350_set_a @ B @ A ) )
=> ( ~ ( member_set_a @ X3 @ A )
=> ( ( member_set_a @ X3 @ B )
=> ( ord_le318720350_set_a @ A @ B ) ) ) ) ).
% sub_which2
thf(fact_250_sub__which2,axiom,
! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e,X3: produc715398134_a_a_e] :
( ( ( ord_le700723212_a_a_e @ A @ B )
| ( ord_le700723212_a_a_e @ B @ A ) )
=> ( ~ ( member1258382221_a_a_e @ X3 @ A )
=> ( ( member1258382221_a_a_e @ X3 @ B )
=> ( ord_le700723212_a_a_e @ A @ B ) ) ) ) ).
% sub_which2
thf(fact_251_sub__which2,axiom,
! [A: set_a,B: set_a,X3: a] :
( ( ( ord_less_eq_set_a @ A @ B )
| ( ord_less_eq_set_a @ B @ A ) )
=> ( ~ ( member_a @ X3 @ A )
=> ( ( member_a @ X3 @ B )
=> ( ord_less_eq_set_a @ A @ B ) ) ) ) ).
% sub_which2
thf(fact_252_image__Pow__mono,axiom,
! [F: produc715398134_a_a_e > set_a,A: set_Pr204296108_a_a_e,B: set_set_a] :
( ( ord_le318720350_set_a @ ( image_64240033_set_a @ F @ A ) @ B )
=> ( ord_le2103863998_set_a @ ( image_1441218763_set_a @ ( image_64240033_set_a @ F ) @ ( pow_Pr328302361_a_a_e @ A ) ) @ ( pow_set_a @ B ) ) ) ).
% image_Pow_mono
thf(fact_253_Sup_OSUP__cong,axiom,
! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e,C3: produc715398134_a_a_e > set_a,D: produc715398134_a_a_e > set_a,Sup: set_set_a > set_a] :
( ( A = B )
=> ( ! [X2: produc715398134_a_a_e] :
( ( member1258382221_a_a_e @ X2 @ B )
=> ( ( C3 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_64240033_set_a @ C3 @ A ) )
= ( Sup @ ( image_64240033_set_a @ D @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_254_Inf_OINF__cong,axiom,
! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e,C3: produc715398134_a_a_e > set_a,D: produc715398134_a_a_e > set_a,Inf: set_set_a > set_a] :
( ( A = B )
=> ( ! [X2: produc715398134_a_a_e] :
( ( member1258382221_a_a_e @ X2 @ B )
=> ( ( C3 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_64240033_set_a @ C3 @ A ) )
= ( Inf @ ( image_64240033_set_a @ D @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_255_subrelI,axiom,
! [R: set_Pr204296108_a_a_e,S: set_Pr204296108_a_a_e] :
( ! [X2: set_a,Y: a > e] :
( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ X2 @ Y ) @ R )
=> ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ X2 @ Y ) @ S ) )
=> ( ord_le700723212_a_a_e @ R @ S ) ) ).
% subrelI
thf(fact_256_ssubst__Pair__rhs,axiom,
! [R: set_a,S: a > e,R2: set_Pr204296108_a_a_e,S2: a > e] :
( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ R @ S ) @ R2 )
=> ( ( S2 = S )
=> ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ R @ S2 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_257_PowI,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( member_set_a @ A @ ( pow_a @ B ) ) ) ).
% PowI
thf(fact_258_Pow__iff,axiom,
! [A: set_a,B: set_a] :
( ( member_set_a @ A @ ( pow_a @ B ) )
= ( ord_less_eq_set_a @ A @ B ) ) ).
% Pow_iff
thf(fact_259_Pow__top,axiom,
! [A: set_a] : ( member_set_a @ A @ ( pow_a @ A ) ) ).
% Pow_top
thf(fact_260_PowD,axiom,
! [A: set_a,B: set_a] :
( ( member_set_a @ A @ ( pow_a @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% PowD
thf(fact_261_Pow__mono,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_le318720350_set_a @ ( pow_a @ A ) @ ( pow_a @ B ) ) ) ).
% Pow_mono
thf(fact_262_image__Pow__surj,axiom,
! [F: produc715398134_a_a_e > set_a,A: set_Pr204296108_a_a_e,B: set_set_a] :
( ( ( image_64240033_set_a @ F @ A )
= B )
=> ( ( image_1441218763_set_a @ ( image_64240033_set_a @ F ) @ ( pow_Pr328302361_a_a_e @ A ) )
= ( pow_set_a @ B ) ) ) ).
% image_Pow_surj
thf(fact_263_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_set_a
= ( ^ [X5: $o > set_a,Y6: $o > set_a] :
( ( ord_less_eq_set_a @ ( X5 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_set_a @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_264_curryI,axiom,
! [F: produc715398134_a_a_e > $o,A2: set_a,B2: a > e] :
( ( F @ ( produc1134456688_a_a_e @ A2 @ B2 ) )
=> ( produc952037978_a_e_o @ F @ A2 @ B2 ) ) ).
% curryI
thf(fact_265_curryD,axiom,
! [F: produc715398134_a_a_e > $o,A2: set_a,B2: a > e] :
( ( produc952037978_a_e_o @ F @ A2 @ B2 )
=> ( F @ ( produc1134456688_a_a_e @ A2 @ B2 ) ) ) ).
% curryD
thf(fact_266_curryE,axiom,
! [F: produc715398134_a_a_e > $o,A2: set_a,B2: a > e] :
( ( produc952037978_a_e_o @ F @ A2 @ B2 )
=> ( F @ ( produc1134456688_a_a_e @ A2 @ B2 ) ) ) ).
% curryE
thf(fact_267_fst__eq__Domain,axiom,
! [R2: set_Pr204296108_a_a_e] :
( ( image_64240033_set_a @ produc1040928388_a_a_e @ R2 )
= ( domain_set_a_a_e @ R2 ) ) ).
% fst_eq_Domain
thf(fact_268_Domain__fst,axiom,
( domain_set_a_a_e
= ( image_64240033_set_a @ produc1040928388_a_a_e ) ) ).
% Domain_fst
thf(fact_269_RangeE,axiom,
! [B2: a > e,R: set_Pr204296108_a_a_e] :
( ( member_a_e @ B2 @ ( range_set_a_a_e @ R ) )
=> ~ ! [A6: set_a] :
~ ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A6 @ B2 ) @ R ) ) ).
% RangeE
thf(fact_270_Range__iff,axiom,
! [A2: a > e,R: set_Pr204296108_a_a_e] :
( ( member_a_e @ A2 @ ( range_set_a_a_e @ R ) )
= ( ? [Y4: set_a] : ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ Y4 @ A2 ) @ R ) ) ) ).
% Range_iff
thf(fact_271_Range_Ocases,axiom,
! [A2: a > e,R: set_Pr204296108_a_a_e] :
( ( member_a_e @ A2 @ ( range_set_a_a_e @ R ) )
=> ~ ! [A6: set_a] :
~ ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A6 @ A2 ) @ R ) ) ).
% Range.cases
thf(fact_272_Range_Osimps,axiom,
! [A2: a > e,R: set_Pr204296108_a_a_e] :
( ( member_a_e @ A2 @ ( range_set_a_a_e @ R ) )
= ( ? [A3: set_a,B3: a > e] :
( ( A2 = B3 )
& ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A3 @ B3 ) @ R ) ) ) ) ).
% Range.simps
thf(fact_273_Range_Ointros,axiom,
! [A2: set_a,B2: a > e,R: set_Pr204296108_a_a_e] :
( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A2 @ B2 ) @ R )
=> ( member_a_e @ B2 @ ( range_set_a_a_e @ R ) ) ) ).
% Range.intros
thf(fact_274_Range_Oinducts,axiom,
! [X3: a > e,R: set_Pr204296108_a_a_e,P2: ( a > e ) > $o] :
( ( member_a_e @ X3 @ ( range_set_a_a_e @ R ) )
=> ( ! [A6: set_a,B6: a > e] :
( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A6 @ B6 ) @ R )
=> ( P2 @ B6 ) )
=> ( P2 @ X3 ) ) ) ).
% Range.inducts
thf(fact_275_DomainE,axiom,
! [A2: set_a,R: set_Pr204296108_a_a_e] :
( ( member_set_a @ A2 @ ( domain_set_a_a_e @ R ) )
=> ~ ! [B6: a > e] :
~ ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A2 @ B6 ) @ R ) ) ).
% DomainE
thf(fact_276_Domain__iff,axiom,
! [A2: set_a,R: set_Pr204296108_a_a_e] :
( ( member_set_a @ A2 @ ( domain_set_a_a_e @ R ) )
= ( ? [Y4: a > e] : ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A2 @ Y4 ) @ R ) ) ) ).
% Domain_iff
thf(fact_277_Domain_Ocases,axiom,
! [A2: set_a,R: set_Pr204296108_a_a_e] :
( ( member_set_a @ A2 @ ( domain_set_a_a_e @ R ) )
=> ~ ! [B6: a > e] :
~ ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A2 @ B6 ) @ R ) ) ).
% Domain.cases
thf(fact_278_Domain_Osimps,axiom,
! [A2: set_a,R: set_Pr204296108_a_a_e] :
( ( member_set_a @ A2 @ ( domain_set_a_a_e @ R ) )
= ( ? [A3: set_a,B3: a > e] :
( ( A2 = A3 )
& ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A3 @ B3 ) @ R ) ) ) ) ).
% Domain.simps
thf(fact_279_Domain_ODomainI,axiom,
! [A2: set_a,B2: a > e,R: set_Pr204296108_a_a_e] :
( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A2 @ B2 ) @ R )
=> ( member_set_a @ A2 @ ( domain_set_a_a_e @ R ) ) ) ).
% Domain.DomainI
thf(fact_280_Domain_Oinducts,axiom,
! [X3: set_a,R: set_Pr204296108_a_a_e,P2: set_a > $o] :
( ( member_set_a @ X3 @ ( domain_set_a_a_e @ R ) )
=> ( ! [A6: set_a,B6: a > e] :
( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A6 @ B6 ) @ R )
=> ( P2 @ A6 ) )
=> ( P2 @ X3 ) ) ) ).
% Domain.inducts
thf(fact_281_pair__in__swap__image,axiom,
! [Y3: a > e,X3: set_a,A: set_Pr204296108_a_a_e] :
( ( member360729153_set_a @ ( produc479206876_set_a @ Y3 @ X3 ) @ ( image_1875450791_set_a @ produc755714512_a_a_e @ A ) )
= ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ X3 @ Y3 ) @ A ) ) ).
% pair_in_swap_image
thf(fact_282_pair__in__swap__image,axiom,
! [Y3: set_a,X3: a > e,A: set_Pr305929312_set_a] :
( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ Y3 @ X3 ) @ ( image_376979367_a_a_e @ produc100464700_set_a @ A ) )
= ( member360729153_set_a @ ( produc479206876_set_a @ X3 @ Y3 ) @ A ) ) ).
% pair_in_swap_image
thf(fact_283_swap__simp,axiom,
! [X3: a > e,Y3: set_a] :
( ( produc100464700_set_a @ ( produc479206876_set_a @ X3 @ Y3 ) )
= ( produc1134456688_a_a_e @ Y3 @ X3 ) ) ).
% swap_simp
thf(fact_284_swap__simp,axiom,
! [X3: set_a,Y3: a > e] :
( ( produc755714512_a_a_e @ ( produc1134456688_a_a_e @ X3 @ Y3 ) )
= ( produc479206876_set_a @ Y3 @ X3 ) ) ).
% swap_simp
thf(fact_285_converse__iff,axiom,
! [A2: a > e,B2: set_a,R: set_Pr204296108_a_a_e] :
( ( member360729153_set_a @ ( produc479206876_set_a @ A2 @ B2 ) @ ( converse_set_a_a_e @ R ) )
= ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ B2 @ A2 ) @ R ) ) ).
% converse_iff
thf(fact_286_converse__iff,axiom,
! [A2: set_a,B2: a > e,R: set_Pr305929312_set_a] :
( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A2 @ B2 ) @ ( converse_a_e_set_a @ R ) )
= ( member360729153_set_a @ ( produc479206876_set_a @ B2 @ A2 ) @ R ) ) ).
% converse_iff
thf(fact_287_converse_Oinducts,axiom,
! [X1: a > e,X22: set_a,R: set_Pr204296108_a_a_e,P2: ( a > e ) > set_a > $o] :
( ( member360729153_set_a @ ( produc479206876_set_a @ X1 @ X22 ) @ ( converse_set_a_a_e @ R ) )
=> ( ! [A6: set_a,B6: a > e] :
( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A6 @ B6 ) @ R )
=> ( P2 @ B6 @ A6 ) )
=> ( P2 @ X1 @ X22 ) ) ) ).
% converse.inducts
thf(fact_288_converse_Oinducts,axiom,
! [X1: set_a,X22: a > e,R: set_Pr305929312_set_a,P2: set_a > ( a > e ) > $o] :
( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ X1 @ X22 ) @ ( converse_a_e_set_a @ R ) )
=> ( ! [A6: a > e,B6: set_a] :
( ( member360729153_set_a @ ( produc479206876_set_a @ A6 @ B6 ) @ R )
=> ( P2 @ B6 @ A6 ) )
=> ( P2 @ X1 @ X22 ) ) ) ).
% converse.inducts
thf(fact_289_converse_Ointros,axiom,
! [A2: a > e,B2: set_a,R: set_Pr305929312_set_a] :
( ( member360729153_set_a @ ( produc479206876_set_a @ A2 @ B2 ) @ R )
=> ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ B2 @ A2 ) @ ( converse_a_e_set_a @ R ) ) ) ).
% converse.intros
thf(fact_290_converse_Ointros,axiom,
! [A2: set_a,B2: a > e,R: set_Pr204296108_a_a_e] :
( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A2 @ B2 ) @ R )
=> ( member360729153_set_a @ ( produc479206876_set_a @ B2 @ A2 ) @ ( converse_set_a_a_e @ R ) ) ) ).
% converse.intros
thf(fact_291_converse_Osimps,axiom,
! [A1: a > e,A22: set_a,R: set_Pr204296108_a_a_e] :
( ( member360729153_set_a @ ( produc479206876_set_a @ A1 @ A22 ) @ ( converse_set_a_a_e @ R ) )
= ( ? [A3: set_a,B3: a > e] :
( ( A1 = B3 )
& ( A22 = A3 )
& ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A3 @ B3 ) @ R ) ) ) ) ).
% converse.simps
thf(fact_292_converse_Osimps,axiom,
! [A1: set_a,A22: a > e,R: set_Pr305929312_set_a] :
( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A1 @ A22 ) @ ( converse_a_e_set_a @ R ) )
= ( ? [A3: a > e,B3: set_a] :
( ( A1 = B3 )
& ( A22 = A3 )
& ( member360729153_set_a @ ( produc479206876_set_a @ A3 @ B3 ) @ R ) ) ) ) ).
% converse.simps
thf(fact_293_converse_Ocases,axiom,
! [A1: a > e,A22: set_a,R: set_Pr204296108_a_a_e] :
( ( member360729153_set_a @ ( produc479206876_set_a @ A1 @ A22 ) @ ( converse_set_a_a_e @ R ) )
=> ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A22 @ A1 ) @ R ) ) ).
% converse.cases
thf(fact_294_converse_Ocases,axiom,
! [A1: set_a,A22: a > e,R: set_Pr305929312_set_a] :
( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A1 @ A22 ) @ ( converse_a_e_set_a @ R ) )
=> ( member360729153_set_a @ ( produc479206876_set_a @ A22 @ A1 ) @ R ) ) ).
% converse.cases
thf(fact_295_converseE,axiom,
! [Yx: produc1965228714_set_a,R: set_Pr204296108_a_a_e] :
( ( member360729153_set_a @ Yx @ ( converse_set_a_a_e @ R ) )
=> ~ ! [X2: set_a,Y: a > e] :
( ( Yx
= ( produc479206876_set_a @ Y @ X2 ) )
=> ~ ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ X2 @ Y ) @ R ) ) ) ).
% converseE
thf(fact_296_converseE,axiom,
! [Yx: produc715398134_a_a_e,R: set_Pr305929312_set_a] :
( ( member1258382221_a_a_e @ Yx @ ( converse_a_e_set_a @ R ) )
=> ~ ! [X2: a > e,Y: set_a] :
( ( Yx
= ( produc1134456688_a_a_e @ Y @ X2 ) )
=> ~ ( member360729153_set_a @ ( produc479206876_set_a @ X2 @ Y ) @ R ) ) ) ).
% converseE
thf(fact_297_converseD,axiom,
! [A2: a > e,B2: set_a,R: set_Pr204296108_a_a_e] :
( ( member360729153_set_a @ ( produc479206876_set_a @ A2 @ B2 ) @ ( converse_set_a_a_e @ R ) )
=> ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ B2 @ A2 ) @ R ) ) ).
% converseD
thf(fact_298_converseD,axiom,
! [A2: set_a,B2: a > e,R: set_Pr305929312_set_a] :
( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A2 @ B2 ) @ ( converse_a_e_set_a @ R ) )
=> ( member360729153_set_a @ ( produc479206876_set_a @ B2 @ A2 ) @ R ) ) ).
% converseD
thf(fact_299_prod_Oswap__def,axiom,
( produc100464700_set_a
= ( ^ [P3: produc1965228714_set_a] : ( produc1134456688_a_a_e @ ( produc1809800494_set_a @ P3 ) @ ( produc385678576_set_a @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_300_prod_Oswap__def,axiom,
( produc755714512_a_a_e
= ( ^ [P3: produc715398134_a_a_e] : ( produc479206876_set_a @ ( produc317566658_a_a_e @ P3 ) @ ( produc1040928388_a_a_e @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_301_map__prod__imageI,axiom,
! [A2: set_a,B2: a > e,R2: set_Pr204296108_a_a_e,F: set_a > set_a,G: ( a > e ) > a > e] :
( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A2 @ B2 ) @ R2 )
=> ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ ( F @ A2 ) @ ( G @ B2 ) ) @ ( image_625620211_a_a_e @ ( produc1443219402_e_a_e @ F @ G ) @ R2 ) ) ) ).
% map_prod_imageI
thf(fact_302_UNIV__I,axiom,
! [X3: set_a] : ( member_set_a @ X3 @ top_top_set_set_a ) ).
% UNIV_I
thf(fact_303_UNIV__I,axiom,
! [X3: produc715398134_a_a_e] : ( member1258382221_a_a_e @ X3 @ top_to1601855068_a_a_e ) ).
% UNIV_I
thf(fact_304_map__prod__simp,axiom,
! [F: set_a > set_a,G: ( a > e ) > a > e,A2: set_a,B2: a > e] :
( ( produc1443219402_e_a_e @ F @ G @ ( produc1134456688_a_a_e @ A2 @ B2 ) )
= ( produc1134456688_a_a_e @ ( F @ A2 ) @ ( G @ B2 ) ) ) ).
% map_prod_simp
thf(fact_305_fst__map__prod,axiom,
! [F: set_a > set_a,G: ( a > e ) > a > e,X3: produc715398134_a_a_e] :
( ( produc1040928388_a_a_e @ ( produc1443219402_e_a_e @ F @ G @ X3 ) )
= ( F @ ( produc1040928388_a_a_e @ X3 ) ) ) ).
% fst_map_prod
thf(fact_306_prod_Ocollapse,axiom,
! [Prod: produc715398134_a_a_e] :
( ( produc1134456688_a_a_e @ ( produc1040928388_a_a_e @ Prod ) @ ( produc317566658_a_a_e @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_307_fst__swap,axiom,
! [X3: produc1965228714_set_a] :
( ( produc1040928388_a_a_e @ ( produc100464700_set_a @ X3 ) )
= ( produc1809800494_set_a @ X3 ) ) ).
% fst_swap
thf(fact_308_snd__swap,axiom,
! [X3: produc715398134_a_a_e] :
( ( produc1809800494_set_a @ ( produc755714512_a_a_e @ X3 ) )
= ( produc1040928388_a_a_e @ X3 ) ) ).
% snd_swap
thf(fact_309_range__fst,axiom,
( ( image_64240033_set_a @ produc1040928388_a_a_e @ top_to1601855068_a_a_e )
= top_top_set_set_a ) ).
% range_fst
thf(fact_310_rangeI,axiom,
! [F: produc715398134_a_a_e > set_a,X3: produc715398134_a_a_e] : ( member_set_a @ ( F @ X3 ) @ ( image_64240033_set_a @ F @ top_to1601855068_a_a_e ) ) ).
% rangeI
thf(fact_311_range__eqI,axiom,
! [B2: set_a,F: produc715398134_a_a_e > set_a,X3: produc715398134_a_a_e] :
( ( B2
= ( F @ X3 ) )
=> ( member_set_a @ B2 @ ( image_64240033_set_a @ F @ top_to1601855068_a_a_e ) ) ) ).
% range_eqI
thf(fact_312_surj__def,axiom,
! [F: produc715398134_a_a_e > set_a] :
( ( ( image_64240033_set_a @ F @ top_to1601855068_a_a_e )
= top_top_set_set_a )
= ( ! [Y4: set_a] :
? [X4: produc715398134_a_a_e] :
( Y4
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_313_surjI,axiom,
! [G: produc715398134_a_a_e > set_a,F: set_a > produc715398134_a_a_e] :
( ! [X2: set_a] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_64240033_set_a @ G @ top_to1601855068_a_a_e )
= top_top_set_set_a ) ) ).
% surjI
thf(fact_314_surjE,axiom,
! [F: produc715398134_a_a_e > set_a,Y3: set_a] :
( ( ( image_64240033_set_a @ F @ top_to1601855068_a_a_e )
= top_top_set_set_a )
=> ~ ! [X2: produc715398134_a_a_e] :
( Y3
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_315_surjD,axiom,
! [F: produc715398134_a_a_e > set_a,Y3: set_a] :
( ( ( image_64240033_set_a @ F @ top_to1601855068_a_a_e )
= top_top_set_set_a )
=> ? [X2: produc715398134_a_a_e] :
( Y3
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_316_map__prod__surj,axiom,
! [F: produc715398134_a_a_e > set_a,G: produc715398134_a_a_e > set_a] :
( ( ( image_64240033_set_a @ F @ top_to1601855068_a_a_e )
= top_top_set_set_a )
=> ( ( ( image_64240033_set_a @ G @ top_to1601855068_a_a_e )
= top_top_set_set_a )
=> ( ( image_2101504031_set_a @ ( produc387505142_set_a @ F @ G ) @ top_to970499581_a_a_e )
= top_to144004695_set_a ) ) ) ).
% map_prod_surj
thf(fact_317_top_Oextremum__uniqueI,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A2 )
=> ( A2 = top_top_set_a ) ) ).
% top.extremum_uniqueI
thf(fact_318_top_Oextremum__unique,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A2 )
= ( A2 = top_top_set_a ) ) ).
% top.extremum_unique
thf(fact_319_top__greatest,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ top_top_set_a ) ).
% top_greatest
thf(fact_320_subset__UNIV,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ top_top_set_a ) ).
% subset_UNIV
thf(fact_321_sndI,axiom,
! [X3: produc715398134_a_a_e,Y3: set_a,Z2: a > e] :
( ( X3
= ( produc1134456688_a_a_e @ Y3 @ Z2 ) )
=> ( ( produc317566658_a_a_e @ X3 )
= Z2 ) ) ).
% sndI
thf(fact_322_snd__eqD,axiom,
! [X3: set_a,Y3: a > e,A2: a > e] :
( ( ( produc317566658_a_a_e @ ( produc1134456688_a_a_e @ X3 @ Y3 ) )
= A2 )
=> ( Y3 = A2 ) ) ).
% snd_eqD
thf(fact_323_snd__conv,axiom,
! [X1: set_a,X22: a > e] :
( ( produc317566658_a_a_e @ ( produc1134456688_a_a_e @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_324_eq__snd__iff,axiom,
! [B2: a > e,P: produc715398134_a_a_e] :
( ( B2
= ( produc317566658_a_a_e @ P ) )
= ( ? [A3: set_a] :
( P
= ( produc1134456688_a_a_e @ A3 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_325_prod__eq__iff,axiom,
( ( ^ [Y2: produc715398134_a_a_e,Z: produc715398134_a_a_e] : Y2 = Z )
= ( ^ [S3: produc715398134_a_a_e,T: produc715398134_a_a_e] :
( ( ( produc1040928388_a_a_e @ S3 )
= ( produc1040928388_a_a_e @ T ) )
& ( ( produc317566658_a_a_e @ S3 )
= ( produc317566658_a_a_e @ T ) ) ) ) ) ).
% prod_eq_iff
thf(fact_326_prod_Oexpand,axiom,
! [Prod: produc715398134_a_a_e,Prod2: produc715398134_a_a_e] :
( ( ( ( produc1040928388_a_a_e @ Prod )
= ( produc1040928388_a_a_e @ Prod2 ) )
& ( ( produc317566658_a_a_e @ Prod )
= ( produc317566658_a_a_e @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_327_prod__eqI,axiom,
! [P: produc715398134_a_a_e,Q2: produc715398134_a_a_e] :
( ( ( produc1040928388_a_a_e @ P )
= ( produc1040928388_a_a_e @ Q2 ) )
=> ( ( ( produc317566658_a_a_e @ P )
= ( produc317566658_a_a_e @ Q2 ) )
=> ( P = Q2 ) ) ) ).
% prod_eqI
thf(fact_328_UNIV__eq__I,axiom,
! [A: set_set_a] :
( ! [X2: set_a] : ( member_set_a @ X2 @ A )
=> ( top_top_set_set_a = A ) ) ).
% UNIV_eq_I
thf(fact_329_UNIV__eq__I,axiom,
! [A: set_Pr204296108_a_a_e] :
( ! [X2: produc715398134_a_a_e] : ( member1258382221_a_a_e @ X2 @ A )
=> ( top_to1601855068_a_a_e = A ) ) ).
% UNIV_eq_I
thf(fact_330_UNIV__witness,axiom,
? [X2: set_a] : ( member_set_a @ X2 @ top_top_set_set_a ) ).
% UNIV_witness
thf(fact_331_UNIV__witness,axiom,
? [X2: produc715398134_a_a_e] : ( member1258382221_a_a_e @ X2 @ top_to1601855068_a_a_e ) ).
% UNIV_witness
thf(fact_332_surjective__pairing,axiom,
! [T2: produc715398134_a_a_e] :
( T2
= ( produc1134456688_a_a_e @ ( produc1040928388_a_a_e @ T2 ) @ ( produc317566658_a_a_e @ T2 ) ) ) ).
% surjective_pairing
thf(fact_333_prod_Oexhaust__sel,axiom,
! [Prod: produc715398134_a_a_e] :
( Prod
= ( produc1134456688_a_a_e @ ( produc1040928388_a_a_e @ Prod ) @ ( produc317566658_a_a_e @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_334_prod__fun__imageE,axiom,
! [C2: produc715398134_a_a_e,F: set_a > set_a,G: ( a > e ) > a > e,R2: set_Pr204296108_a_a_e] :
( ( member1258382221_a_a_e @ C2 @ ( image_625620211_a_a_e @ ( produc1443219402_e_a_e @ F @ G ) @ R2 ) )
=> ~ ! [X2: set_a,Y: a > e] :
( ( C2
= ( produc1134456688_a_a_e @ ( F @ X2 ) @ ( G @ Y ) ) )
=> ~ ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ X2 @ Y ) @ R2 ) ) ) ).
% prod_fun_imageE
thf(fact_335_exI__realizer,axiom,
! [P2: ( a > e ) > set_a > $o,Y3: a > e,X3: set_a] :
( ( P2 @ Y3 @ X3 )
=> ( P2 @ ( produc317566658_a_a_e @ ( produc1134456688_a_a_e @ X3 @ Y3 ) ) @ ( produc1040928388_a_a_e @ ( produc1134456688_a_a_e @ X3 @ Y3 ) ) ) ) ).
% exI_realizer
thf(fact_336_conjI__realizer,axiom,
! [P2: set_a > $o,P: set_a,Q: ( a > e ) > $o,Q2: a > e] :
( ( P2 @ P )
=> ( ( Q @ Q2 )
=> ( ( P2 @ ( produc1040928388_a_a_e @ ( produc1134456688_a_a_e @ P @ Q2 ) ) )
& ( Q @ ( produc317566658_a_a_e @ ( produc1134456688_a_a_e @ P @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_337_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P2: set_a > ( a > e ) > $o,X3: set_a,Y3: a > e,A2: produc715398134_a_a_e] :
( ( P2 @ X3 @ Y3 )
=> ( ( A2
= ( produc1134456688_a_a_e @ X3 @ Y3 ) )
=> ( P2 @ ( produc1040928388_a_a_e @ A2 ) @ ( produc317566658_a_a_e @ A2 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_338_top__empty__eq,axiom,
( top_top_set_a_o
= ( ^ [X4: set_a] : ( member_set_a @ X4 @ top_top_set_set_a ) ) ) ).
% top_empty_eq
thf(fact_339_top__empty__eq,axiom,
( top_to2000253225_a_e_o
= ( ^ [X4: produc715398134_a_a_e] : ( member1258382221_a_a_e @ X4 @ top_to1601855068_a_a_e ) ) ) ).
% top_empty_eq
thf(fact_340_exE__realizer_H,axiom,
! [P2: ( a > e ) > set_a > $o,P: produc715398134_a_a_e] :
( ( P2 @ ( produc317566658_a_a_e @ P ) @ ( produc1040928388_a_a_e @ P ) )
=> ~ ! [X2: set_a,Y: a > e] :
~ ( P2 @ Y @ X2 ) ) ).
% exE_realizer'
thf(fact_341_Collect__split__mono__strong,axiom,
! [X6: set_set_a,A: set_Pr204296108_a_a_e,Y7: set_a_e,P2: set_a > ( a > e ) > $o,Q: set_a > ( a > e ) > $o] :
( ( X6
= ( image_64240033_set_a @ produc1040928388_a_a_e @ A ) )
=> ( ( Y7
= ( image_1149879910_e_a_e @ produc317566658_a_a_e @ A ) )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ X6 )
=> ! [Xa2: a > e] :
( ( member_a_e @ Xa2 @ Y7 )
=> ( ( P2 @ X2 @ Xa2 )
=> ( Q @ X2 @ Xa2 ) ) ) )
=> ( ( ord_le700723212_a_a_e @ A @ ( collec1369693387_a_a_e @ ( produc1341862587_a_e_o @ P2 ) ) )
=> ( ord_le700723212_a_a_e @ A @ ( collec1369693387_a_a_e @ ( produc1341862587_a_e_o @ Q ) ) ) ) ) ) ) ).
% Collect_split_mono_strong
thf(fact_342_sndOp__def,axiom,
( bNF_sn375433437_a_a_e
= ( ^ [P4: set_a > set_a > $o,Q3: set_a > ( a > e ) > $o,Ac: produc715398134_a_a_e] : ( produc1134456688_a_a_e @ ( bNF_pi929741261_a_a_e @ P4 @ Q3 @ ( produc1040928388_a_a_e @ Ac ) @ ( produc317566658_a_a_e @ Ac ) ) @ ( produc317566658_a_a_e @ Ac ) ) ) ) ).
% sndOp_def
thf(fact_343_Product__Type_OCollect__case__prodD,axiom,
! [X3: produc715398134_a_a_e,A: set_a > ( a > e ) > $o] :
( ( member1258382221_a_a_e @ X3 @ ( collec1369693387_a_a_e @ ( produc1341862587_a_e_o @ A ) ) )
=> ( A @ ( produc1040928388_a_a_e @ X3 ) @ ( produc317566658_a_a_e @ X3 ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_344_fstOp__def,axiom,
( bNF_fs1024561536_e_a_e
= ( ^ [P4: set_a > ( a > e ) > $o,Q3: ( a > e ) > ( a > e ) > $o,Ac: produc715398134_a_a_e] : ( produc1134456688_a_a_e @ ( produc1040928388_a_a_e @ Ac ) @ ( bNF_pi1175744974_e_a_e @ P4 @ Q3 @ ( produc1040928388_a_a_e @ Ac ) @ ( produc317566658_a_a_e @ Ac ) ) ) ) ) ).
% fstOp_def
thf(fact_345_mem__case__prodE,axiom,
! [Z2: set_a,C2: set_a > ( a > e ) > set_set_a,P: produc715398134_a_a_e] :
( ( member_set_a @ Z2 @ ( produc1620938593_set_a @ C2 @ P ) )
=> ~ ! [X2: set_a,Y: a > e] :
( ( P
= ( produc1134456688_a_a_e @ X2 @ Y ) )
=> ~ ( member_set_a @ Z2 @ ( C2 @ X2 @ Y ) ) ) ) ).
% mem_case_prodE
thf(fact_346_mem__case__prodE,axiom,
! [Z2: produc715398134_a_a_e,C2: set_a > ( a > e ) > set_Pr204296108_a_a_e,P: produc715398134_a_a_e] :
( ( member1258382221_a_a_e @ Z2 @ ( produc1096319049_a_a_e @ C2 @ P ) )
=> ~ ! [X2: set_a,Y: a > e] :
( ( P
= ( produc1134456688_a_a_e @ X2 @ Y ) )
=> ~ ( member1258382221_a_a_e @ Z2 @ ( C2 @ X2 @ Y ) ) ) ) ).
% mem_case_prodE
thf(fact_347_in__rel__def,axiom,
( fun_in_rel_set_a_a_e
= ( ^ [R3: set_Pr204296108_a_a_e,X4: set_a,Y4: a > e] : ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ X4 @ Y4 ) @ R3 ) ) ) ).
% in_rel_def
thf(fact_348_fstOp__in,axiom,
! [Ac2: produc715398134_a_a_e,P2: set_a > ( a > e ) > $o,Q: ( a > e ) > ( a > e ) > $o] :
( ( member1258382221_a_a_e @ Ac2 @ ( collec1369693387_a_a_e @ ( produc1341862587_a_e_o @ ( relcom658603713_e_a_e @ P2 @ Q ) ) ) )
=> ( member1258382221_a_a_e @ ( bNF_fs1024561536_e_a_e @ P2 @ Q @ Ac2 ) @ ( collec1369693387_a_a_e @ ( produc1341862587_a_e_o @ P2 ) ) ) ) ).
% fstOp_in
thf(fact_349_sndOp__in,axiom,
! [Ac2: produc715398134_a_a_e,P2: set_a > set_a > $o,Q: set_a > ( a > e ) > $o] :
( ( member1258382221_a_a_e @ Ac2 @ ( collec1369693387_a_a_e @ ( produc1341862587_a_e_o @ ( relcom779348378_a_a_e @ P2 @ Q ) ) ) )
=> ( member1258382221_a_a_e @ ( bNF_sn375433437_a_a_e @ P2 @ Q @ Ac2 ) @ ( collec1369693387_a_a_e @ ( produc1341862587_a_e_o @ Q ) ) ) ) ).
% sndOp_in
thf(fact_350_image2__eqI,axiom,
! [B2: set_a,F: set_a > set_a,X3: set_a,C2: a > e,G: set_a > a > e,A: set_set_a] :
( ( B2
= ( F @ X3 ) )
=> ( ( C2
= ( G @ X3 ) )
=> ( ( member_set_a @ X3 @ A )
=> ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ B2 @ C2 ) @ ( bNF_Gr1872575811_a_a_e @ A @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_351_image2__eqI,axiom,
! [B2: set_a,F: produc715398134_a_a_e > set_a,X3: produc715398134_a_a_e,C2: a > e,G: produc715398134_a_a_e > a > e,A: set_Pr204296108_a_a_e] :
( ( B2
= ( F @ X3 ) )
=> ( ( C2
= ( G @ X3 ) )
=> ( ( member1258382221_a_a_e @ X3 @ A )
=> ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ B2 @ C2 ) @ ( bNF_Gr2072183031_a_a_e @ A @ F @ G ) ) ) ) ) ).
% image2_eqI
% Conjectures (2)
thf(conj_0,hypothesis,
! [A7: set_a] :
( ( member_set_a @ A7 @ ( image_64240033_set_a @ produc1040928388_a_a_e @ c ) )
=> ( ( ord_less_eq_set_a @ a1 @ A7 )
=> ( ( ord_less_eq_set_a @ a2 @ A7 )
=> ( ( ord_less_eq_set_a @ a3 @ A7 )
=> thesis ) ) ) ) ).
thf(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------