SET007 Axioms: SET007+118.ax
%------------------------------------------------------------------------------
% File : SET007+118 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Introduction to Probability
% Version : [Urb08] axioms.
% English :
% Refs : [Mat90] Matuszewski (1990), Formalized Mathematics
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : rpr_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 100 ( 32 unt; 0 def)
% Number of atoms : 475 ( 71 equ)
% Maximal formula atoms : 14 ( 4 avg)
% Number of connectives : 500 ( 125 ~; 21 |; 139 &)
% ( 3 <=>; 212 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 6 avg)
% Maximal term depth : 7 ( 1 avg)
% Number of predicates : 16 ( 14 usr; 1 prp; 0-3 aty)
% Number of functors : 21 ( 21 usr; 4 con; 0-3 aty)
% Number of variables : 193 ( 189 !; 4 ?)
% SPC :
% Comments : The individual reference can be found in [Mat90] by looking for
% the name provided by [Urb08].
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : These set theory axioms are used in encodings of problems in
% various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(rc1_rpr_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
& ~ v1_xboole_0(B)
& v1_realset1(B) ) ) ).
fof(cc1_rpr_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ( ( ~ v1_xboole_0(B)
& v1_realset1(B) )
=> ( ~ v1_xboole_0(B)
& v1_realset1(B)
& v1_finset_1(B) ) ) ) ) ).
fof(t1_rpr_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> ( ( ~ v1_xboole_0(B)
& v1_realset1(B)
& m1_subset_1(B,k1_zfmisc_1(A)) )
<=> ! [C] :
( r1_tarski(C,B)
<=> ( C = k1_xboole_0
| C = B ) ) ) ) ) ).
fof(t2_rpr_1,axiom,
$true ).
fof(t3_rpr_1,axiom,
$true ).
fof(t4_rpr_1,axiom,
$true ).
fof(t5_rpr_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( ( ~ v1_xboole_0(D)
& v1_realset1(D)
& m1_subset_1(D,k1_zfmisc_1(A)) )
=> ~ ( D = k4_subset_1(A,B,C)
& B != C
& ~ ( B = k1_xboole_0
& C = D )
& ~ ( B = D
& C = k1_xboole_0 ) ) ) ) ) ) ).
fof(t6_rpr_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( ( ~ v1_xboole_0(D)
& v1_realset1(D)
& m1_subset_1(D,k1_zfmisc_1(A)) )
=> ~ ( D = k4_subset_1(A,B,C)
& ~ ( B = D
& C = D )
& ~ ( B = D
& C = k1_xboole_0 )
& ~ ( B = k1_xboole_0
& C = D ) ) ) ) ) ) ).
fof(t7_rpr_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ( ~ v1_xboole_0(k6_domain_1(A,B))
& v1_realset1(k6_domain_1(A,B))
& m1_subset_1(k6_domain_1(A,B),k1_zfmisc_1(A)) ) ) ) ).
fof(t8_rpr_1,axiom,
$true ).
fof(t9_rpr_1,axiom,
$true ).
fof(t10_rpr_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_realset1(B)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v1_realset1(C)
& m1_subset_1(C,k1_zfmisc_1(A)) )
=> ( r1_tarski(B,C)
=> B = C ) ) ) ) ).
fof(t11_rpr_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_realset1(B)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> ? [C] :
( m1_subset_1(C,A)
& r2_hidden(C,A)
& B = k6_domain_1(A,C) ) ) ) ).
fof(t12_rpr_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ? [B] :
( ~ v1_xboole_0(B)
& v1_realset1(B)
& m1_subset_1(B,k1_zfmisc_1(A))
& ~ v1_xboole_0(B)
& v1_realset1(B)
& m1_subset_1(B,k1_zfmisc_1(A)) ) ) ).
fof(t13_rpr_1,axiom,
$true ).
fof(t14_rpr_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_realset1(B)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> ? [C] :
( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C)
& m2_finseq_1(C,A)
& k2_relat_1(C) = B
& k3_finseq_1(C) = np__1 ) ) ) ).
fof(t15_rpr_1,axiom,
$true ).
fof(t16_rpr_1,axiom,
$true ).
fof(t17_rpr_1,axiom,
$true ).
fof(t18_rpr_1,axiom,
$true ).
fof(t19_rpr_1,axiom,
$true ).
fof(t20_rpr_1,axiom,
$true ).
fof(t21_rpr_1,axiom,
$true ).
fof(t22_rpr_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_realset1(B)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( r1_xboole_0(B,C)
| k5_subset_1(A,B,C) = B ) ) ) ) ).
fof(t23_rpr_1,axiom,
$true ).
fof(t24_rpr_1,axiom,
$true ).
fof(t25_rpr_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ~ ( B != k1_xboole_0
& ! [C] :
( ( ~ v1_xboole_0(C)
& v1_realset1(C)
& m1_subset_1(C,k1_zfmisc_1(A)) )
=> ~ r1_tarski(C,B) ) ) ) ) ).
fof(t26_rpr_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_realset1(B)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ~ ( r1_tarski(B,k4_subset_1(A,C,k3_subset_1(A,C)))
& ~ r1_tarski(B,C)
& ~ r1_tarski(B,k3_subset_1(A,C)) ) ) ) ) ).
fof(t27_rpr_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_realset1(B)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v1_realset1(C)
& m1_subset_1(C,k1_zfmisc_1(A)) )
=> ( B = C
| r1_subset_1(B,C) ) ) ) ) ).
fof(t28_rpr_1,axiom,
$true ).
fof(t29_rpr_1,axiom,
$true ).
fof(t30_rpr_1,axiom,
$true ).
fof(t31_rpr_1,axiom,
$true ).
fof(t32_rpr_1,axiom,
$true ).
fof(t33_rpr_1,axiom,
$true ).
fof(t34_rpr_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> r1_xboole_0(k5_subset_1(A,B,C),k5_subset_1(A,B,k3_subset_1(A,C))) ) ) ) ).
fof(d1_rpr_1,axiom,
$true ).
fof(d2_rpr_1,axiom,
$true ).
fof(d3_rpr_1,axiom,
$true ).
fof(d4_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> k1_rpr_1(A,B) = k6_real_1(k4_card_1(B),k4_card_1(A)) ) ) ).
fof(t35_rpr_1,axiom,
$true ).
fof(t36_rpr_1,axiom,
$true ).
fof(t37_rpr_1,axiom,
$true ).
fof(t38_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_realset1(B)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> k1_rpr_1(A,B) = k6_real_1(np__1,k4_card_1(A)) ) ) ).
fof(t39_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> k1_rpr_1(A,k2_subset_1(A)) = np__1 ) ).
fof(t40_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> k1_rpr_1(A,k1_subset_1(A)) = np__0 ) ).
fof(t41_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( r1_xboole_0(B,C)
=> k1_rpr_1(A,k5_subset_1(A,B,C)) = np__0 ) ) ) ) ).
fof(t42_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> r1_xreal_0(k1_rpr_1(A,B),np__1) ) ) ).
fof(t43_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> r1_xreal_0(np__0,k1_rpr_1(A,B)) ) ) ).
fof(t44_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( r1_tarski(B,C)
=> r1_xreal_0(k1_rpr_1(A,B),k1_rpr_1(A,C)) ) ) ) ) ).
fof(t45_rpr_1,axiom,
$true ).
fof(t46_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> k1_rpr_1(A,k4_subset_1(A,B,C)) = k5_real_1(k3_real_1(k1_rpr_1(A,B),k1_rpr_1(A,C)),k1_rpr_1(A,k5_subset_1(A,B,C))) ) ) ) ).
fof(t47_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( r1_xboole_0(B,C)
=> k1_rpr_1(A,k4_subset_1(A,B,C)) = k3_real_1(k1_rpr_1(A,B),k1_rpr_1(A,C)) ) ) ) ) ).
fof(t48_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ( k1_rpr_1(A,B) = k5_real_1(np__1,k1_rpr_1(A,k3_subset_1(A,B)))
& k1_rpr_1(A,k3_subset_1(A,B)) = k5_real_1(np__1,k1_rpr_1(A,B)) ) ) ) ).
fof(t49_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> k1_rpr_1(A,k6_subset_1(A,B,C)) = k5_real_1(k1_rpr_1(A,B),k1_rpr_1(A,k5_subset_1(A,B,C))) ) ) ) ).
fof(t50_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( r1_tarski(C,B)
=> k1_rpr_1(A,k6_subset_1(A,B,C)) = k5_real_1(k1_rpr_1(A,B),k1_rpr_1(A,C)) ) ) ) ) ).
fof(t51_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> r1_xreal_0(k1_rpr_1(A,k4_subset_1(A,B,C)),k3_real_1(k1_rpr_1(A,B),k1_rpr_1(A,C))) ) ) ) ).
fof(t52_rpr_1,axiom,
$true ).
fof(t53_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> k1_rpr_1(A,B) = k3_real_1(k1_rpr_1(A,k5_subset_1(A,B,C)),k1_rpr_1(A,k5_subset_1(A,B,k3_subset_1(A,C)))) ) ) ) ).
fof(t54_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> k1_rpr_1(A,B) = k5_real_1(k1_rpr_1(A,k4_subset_1(A,B,C)),k1_rpr_1(A,k6_subset_1(A,C,B))) ) ) ) ).
fof(t55_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> k3_real_1(k1_rpr_1(A,B),k1_rpr_1(A,k5_subset_1(A,k3_subset_1(A,B),C))) = k3_real_1(k1_rpr_1(A,C),k1_rpr_1(A,k5_subset_1(A,k3_subset_1(A,C),B))) ) ) ) ).
fof(t56_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> k1_rpr_1(A,k4_subset_1(A,k4_subset_1(A,B,C),D)) = k3_real_1(k5_real_1(k3_real_1(k3_real_1(k1_rpr_1(A,B),k1_rpr_1(A,C)),k1_rpr_1(A,D)),k3_real_1(k3_real_1(k1_rpr_1(A,k5_subset_1(A,B,C)),k1_rpr_1(A,k5_subset_1(A,B,D))),k1_rpr_1(A,k5_subset_1(A,C,D)))),k1_rpr_1(A,k5_subset_1(A,k5_subset_1(A,B,C),D))) ) ) ) ) ).
fof(t57_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ( ( r1_xboole_0(B,C)
& r1_xboole_0(B,D)
& r1_xboole_0(C,D) )
=> k1_rpr_1(A,k4_subset_1(A,k4_subset_1(A,B,C),D)) = k3_real_1(k3_real_1(k1_rpr_1(A,B),k1_rpr_1(A,C)),k1_rpr_1(A,D)) ) ) ) ) ) ).
fof(t58_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> r1_xreal_0(k5_real_1(k1_rpr_1(A,B),k1_rpr_1(A,C)),k1_rpr_1(A,k6_subset_1(A,B,C))) ) ) ) ).
fof(d5_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> k2_rpr_1(A,B,C) = k6_real_1(k1_rpr_1(A,k5_subset_1(A,C,B)),k1_rpr_1(A,B)) ) ) ) ).
fof(t59_rpr_1,axiom,
$true ).
fof(t60_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( ~ r1_xreal_0(k1_rpr_1(A,C),np__0)
=> k1_rpr_1(A,k5_subset_1(A,B,C)) = k4_real_1(k2_rpr_1(A,C,B),k1_rpr_1(A,C)) ) ) ) ) ).
fof(t61_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> k2_rpr_1(A,k2_subset_1(A),B) = k1_rpr_1(A,B) ) ) ).
fof(t62_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> k2_rpr_1(A,k2_subset_1(A),k2_subset_1(A)) = np__1 ) ).
fof(t63_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> k2_rpr_1(A,k2_subset_1(A),k1_subset_1(A)) = np__0 ) ).
fof(t64_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( ~ r1_xreal_0(k1_rpr_1(A,C),np__0)
=> r1_xreal_0(k2_rpr_1(A,C,B),np__1) ) ) ) ) ).
fof(t65_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( ~ r1_xreal_0(k1_rpr_1(A,C),np__0)
=> r1_xreal_0(np__0,k2_rpr_1(A,C,B)) ) ) ) ) ).
fof(t66_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( ~ r1_xreal_0(k1_rpr_1(A,C),np__0)
=> k2_rpr_1(A,C,B) = k5_real_1(np__1,k6_real_1(k1_rpr_1(A,k6_subset_1(A,C,B)),k1_rpr_1(A,C))) ) ) ) ) ).
fof(t67_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( r1_tarski(B,C)
=> ( r1_xreal_0(k1_rpr_1(A,C),np__0)
| k2_rpr_1(A,C,B) = k6_real_1(k1_rpr_1(A,B),k1_rpr_1(A,C)) ) ) ) ) ) ).
fof(t68_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( r1_xboole_0(B,C)
=> k2_rpr_1(A,C,B) = np__0 ) ) ) ) ).
fof(t69_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ~ ( ~ r1_xreal_0(k1_rpr_1(A,B),np__0)
& ~ r1_xreal_0(k1_rpr_1(A,C),np__0)
& k4_real_1(k1_rpr_1(A,B),k2_rpr_1(A,B,C)) != k4_real_1(k1_rpr_1(A,C),k2_rpr_1(A,C,B)) ) ) ) ) ).
fof(t70_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( ~ r1_xreal_0(k1_rpr_1(A,C),np__0)
=> ( k2_rpr_1(A,C,B) = k5_real_1(np__1,k2_rpr_1(A,C,k3_subset_1(A,B)))
& k2_rpr_1(A,C,k3_subset_1(A,B)) = k5_real_1(np__1,k2_rpr_1(A,C,B)) ) ) ) ) ) ).
fof(t71_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( r1_tarski(C,B)
=> ( r1_xreal_0(k1_rpr_1(A,C),np__0)
| k2_rpr_1(A,C,B) = np__1 ) ) ) ) ) ).
fof(t72_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ( ~ r1_xreal_0(k1_rpr_1(A,B),np__0)
=> k2_rpr_1(A,B,k2_subset_1(A)) = np__1 ) ) ) ).
fof(t73_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ( ~ r1_xreal_0(k1_rpr_1(A,B),np__0)
=> k2_rpr_1(A,B,k3_subset_1(A,B)) = np__0 ) ) ) ).
fof(t74_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ( ~ r1_xreal_0(np__1,k1_rpr_1(A,B))
=> k2_rpr_1(A,k3_subset_1(A,B),B) = np__0 ) ) ) ).
fof(t75_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( r1_xboole_0(B,C)
=> ( r1_xreal_0(k1_rpr_1(A,C),np__0)
| k2_rpr_1(A,C,k3_subset_1(A,B)) = np__1 ) ) ) ) ) ).
fof(t76_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( r1_xboole_0(B,C)
=> ( r1_xreal_0(k1_rpr_1(A,B),np__0)
| r1_xreal_0(np__1,k1_rpr_1(A,C))
| k2_rpr_1(A,k3_subset_1(A,C),B) = k6_real_1(k1_rpr_1(A,B),k5_real_1(np__1,k1_rpr_1(A,C))) ) ) ) ) ) ).
fof(t77_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( r1_xboole_0(B,C)
=> ( r1_xreal_0(k1_rpr_1(A,B),np__0)
| r1_xreal_0(np__1,k1_rpr_1(A,C))
| k2_rpr_1(A,k3_subset_1(A,C),k3_subset_1(A,B)) = k5_real_1(np__1,k6_real_1(k1_rpr_1(A,B),k5_real_1(np__1,k1_rpr_1(A,C)))) ) ) ) ) ) ).
fof(t78_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ~ ( ~ r1_xreal_0(k1_rpr_1(A,k5_subset_1(A,C,D)),np__0)
& ~ r1_xreal_0(k1_rpr_1(A,D),np__0)
& k1_rpr_1(A,k5_subset_1(A,k5_subset_1(A,B,C),D)) != k4_real_1(k4_real_1(k2_rpr_1(A,k5_subset_1(A,C,D),B),k2_rpr_1(A,D,C)),k1_rpr_1(A,D)) ) ) ) ) ) ).
fof(t79_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ~ ( ~ r1_xreal_0(k1_rpr_1(A,C),np__0)
& ~ r1_xreal_0(np__1,k1_rpr_1(A,C))
& k1_rpr_1(A,B) != k3_real_1(k4_real_1(k2_rpr_1(A,C,B),k1_rpr_1(A,C)),k4_real_1(k2_rpr_1(A,k3_subset_1(A,C),B),k1_rpr_1(A,k3_subset_1(A,C)))) ) ) ) ) ).
fof(t80_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ( ( k4_subset_1(A,C,D) = A
& r1_xboole_0(C,D) )
=> ( r1_xreal_0(k1_rpr_1(A,C),np__0)
| r1_xreal_0(k1_rpr_1(A,D),np__0)
| k1_rpr_1(A,B) = k3_real_1(k4_real_1(k2_rpr_1(A,C,B),k1_rpr_1(A,C)),k4_real_1(k2_rpr_1(A,D,B),k1_rpr_1(A,D))) ) ) ) ) ) ) ).
fof(t81_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(A))
=> ( ( k4_subset_1(A,k4_subset_1(A,C,D),E) = A
& r1_xboole_0(C,D)
& r1_xboole_0(C,E)
& r1_xboole_0(D,E) )
=> ( r1_xreal_0(k1_rpr_1(A,C),np__0)
| r1_xreal_0(k1_rpr_1(A,D),np__0)
| r1_xreal_0(k1_rpr_1(A,E),np__0)
| k1_rpr_1(A,B) = k3_real_1(k3_real_1(k4_real_1(k2_rpr_1(A,C,B),k1_rpr_1(A,C)),k4_real_1(k2_rpr_1(A,D,B),k1_rpr_1(A,D))),k4_real_1(k2_rpr_1(A,E,B),k1_rpr_1(A,E))) ) ) ) ) ) ) ) ).
fof(t82_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ( ( k4_subset_1(A,C,D) = A
& r1_xboole_0(C,D) )
=> ( r1_xreal_0(k1_rpr_1(A,C),np__0)
| r1_xreal_0(k1_rpr_1(A,D),np__0)
| k2_rpr_1(A,B,C) = k6_real_1(k4_real_1(k2_rpr_1(A,C,B),k1_rpr_1(A,C)),k3_real_1(k4_real_1(k2_rpr_1(A,C,B),k1_rpr_1(A,C)),k4_real_1(k2_rpr_1(A,D,B),k1_rpr_1(A,D)))) ) ) ) ) ) ) ).
fof(t83_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(A))
=> ( ( k4_subset_1(A,k4_subset_1(A,C,D),E) = A
& r1_xboole_0(C,D)
& r1_xboole_0(C,E)
& r1_xboole_0(D,E) )
=> ( r1_xreal_0(k1_rpr_1(A,C),np__0)
| r1_xreal_0(k1_rpr_1(A,D),np__0)
| r1_xreal_0(k1_rpr_1(A,E),np__0)
| k2_rpr_1(A,B,C) = k6_real_1(k4_real_1(k2_rpr_1(A,C,B),k1_rpr_1(A,C)),k3_real_1(k3_real_1(k4_real_1(k2_rpr_1(A,C,B),k1_rpr_1(A,C)),k4_real_1(k2_rpr_1(A,D,B),k1_rpr_1(A,D))),k4_real_1(k2_rpr_1(A,E,B),k1_rpr_1(A,E)))) ) ) ) ) ) ) ) ).
fof(d6_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( r1_rpr_1(A,B,C)
<=> k1_rpr_1(A,k5_subset_1(A,B,C)) = k4_real_1(k1_rpr_1(A,B),k1_rpr_1(A,C)) ) ) ) ) ).
fof(t84_rpr_1,axiom,
$true ).
fof(t85_rpr_1,axiom,
$true ).
fof(t86_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( r1_rpr_1(A,B,C)
=> ( r1_xreal_0(k1_rpr_1(A,C),np__0)
| k2_rpr_1(A,C,B) = k1_rpr_1(A,B) ) ) ) ) ) ).
fof(t87_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( k1_rpr_1(A,C) = np__0
=> r1_rpr_1(A,B,C) ) ) ) ) ).
fof(t88_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( r1_rpr_1(A,B,C)
=> r1_rpr_1(A,k3_subset_1(A,B),C) ) ) ) ) ).
fof(t89_rpr_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ~ ( r1_xboole_0(B,C)
& r1_rpr_1(A,B,C)
& k1_rpr_1(A,B) != np__0
& k1_rpr_1(A,C) != np__0 ) ) ) ) ).
fof(symmetry_r1_rpr_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& m1_subset_1(B,k1_zfmisc_1(A))
& m1_subset_1(C,k1_zfmisc_1(A)) )
=> ( r1_rpr_1(A,B,C)
=> r1_rpr_1(A,C,B) ) ) ).
fof(dt_k1_rpr_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> m1_subset_1(k1_rpr_1(A,B),k1_numbers) ) ).
fof(dt_k2_rpr_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& m1_subset_1(B,k1_zfmisc_1(A))
& m1_subset_1(C,k1_zfmisc_1(A)) )
=> m1_subset_1(k2_rpr_1(A,B,C),k1_numbers) ) ).
%------------------------------------------------------------------------------