SET007 Axioms: SET007+68.ax
%------------------------------------------------------------------------------
% File : SET007+68 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Universal Classes
% 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 : classes2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 134 ( 22 unt; 0 def)
% Number of atoms : 515 ( 56 equ)
% Maximal formula atoms : 16 ( 3 avg)
% Number of connectives : 477 ( 96 ~; 5 |; 229 &)
% ( 5 <=>; 142 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 5 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 21 ( 19 usr; 1 prp; 0-2 aty)
% Number of functors : 44 ( 44 usr; 6 con; 0-3 aty)
% Number of variables : 225 ( 222 !; 3 ?)
% 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(cc1_classes2,axiom,
! [A] :
( v2_classes1(A)
=> v1_classes1(A) ) ).
fof(fc1_classes2,axiom,
! [A] :
( ~ v1_xboole_0(k1_classes1(A))
& v1_classes1(k1_classes1(A))
& v2_classes1(k1_classes1(A)) ) ).
fof(cc2_classes2,axiom,
! [A] :
( v1_classes2(A)
=> ( v1_ordinal1(A)
& v1_classes1(A)
& v2_classes1(A) ) ) ).
fof(cc3_classes2,axiom,
! [A] :
( ( v1_ordinal1(A)
& v2_classes1(A) )
=> v1_classes2(A) ) ).
fof(rc1_classes2,axiom,
? [A] :
( v1_ordinal1(A)
& ~ v1_xboole_0(A)
& v1_classes1(A)
& v2_classes1(A)
& v1_classes2(A) ) ).
fof(fc2_classes2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ( v1_ordinal1(k2_ordinal2(A))
& v2_ordinal1(k2_ordinal2(A))
& v3_ordinal1(k2_ordinal2(A)) ) ) ).
fof(fc3_classes2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ( v1_ordinal1(k1_classes1(A))
& ~ v1_xboole_0(k1_classes1(A))
& v1_classes1(k1_classes1(A))
& v2_classes1(k1_classes1(A))
& v1_classes2(k1_classes1(A)) ) ) ).
fof(fc4_classes2,axiom,
! [A] :
( v3_ordinal1(A)
=> ( v1_ordinal1(k1_classes1(A))
& ~ v1_xboole_0(k1_classes1(A))
& v1_classes1(k1_classes1(A))
& v2_classes1(k1_classes1(A))
& v1_classes2(k1_classes1(A)) ) ) ).
fof(rc2_classes2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ? [B] :
( m1_subset_1(B,A)
& ~ v1_xboole_0(B) ) ) ).
fof(fc5_classes2,axiom,
! [A] : v1_ordinal1(k5_classes1(A)) ).
fof(fc6_classes2,axiom,
! [A] :
( v1_ordinal1(A)
=> ( v1_ordinal1(k1_classes1(A))
& ~ v1_xboole_0(k1_classes1(A))
& v1_classes1(k1_classes1(A))
& v2_classes1(k1_classes1(A))
& v1_classes2(k1_classes1(A)) ) ) ).
fof(fc7_classes2,axiom,
! [A] :
( v3_ordinal1(A)
=> v1_ordinal1(k4_classes1(A)) ) ).
fof(fc8_classes2,axiom,
! [A] :
( v3_ordinal1(A)
=> ( v1_ordinal1(k16_classes2(A))
& ~ v1_xboole_0(k16_classes2(A))
& v1_classes1(k16_classes2(A))
& v2_classes1(k16_classes2(A))
& v1_classes2(k16_classes2(A)) ) ) ).
fof(t1_classes2,axiom,
! [A,B] :
( ( v1_classes1(A)
& r2_hidden(B,A) )
=> ( ~ r2_wellord2(B,A)
& r2_hidden(k1_card_1(B),k1_card_1(A)) ) ) ).
fof(t2_classes2,axiom,
$true ).
fof(t3_classes2,axiom,
! [A,B,C] :
( ( v2_classes1(A)
& r2_hidden(B,A)
& r2_hidden(C,A) )
=> ( r2_hidden(k1_tarski(B),A)
& r2_hidden(k2_tarski(B,C),A) ) ) ).
fof(t4_classes2,axiom,
! [A,B,C] :
( ( v2_classes1(A)
& r2_hidden(B,A)
& r2_hidden(C,A) )
=> r2_hidden(k4_tarski(B,C),A) ) ).
fof(t5_classes2,axiom,
! [A,B] :
( ( v2_classes1(A)
& r2_hidden(B,A) )
=> r1_tarski(k1_classes1(B),A) ) ).
fof(t6_classes2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v2_classes1(B)
& r2_hidden(A,B) )
=> ( r2_hidden(k1_ordinal1(A),B)
& r1_tarski(A,B) ) ) ) ).
fof(t7_classes2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( r2_hidden(A,k1_classes1(B))
=> ( r2_hidden(k1_ordinal1(A),k1_classes1(B))
& r1_tarski(A,k1_classes1(B)) ) ) ) ).
fof(t8_classes2,axiom,
! [A,B] :
( ( v1_classes1(A)
& v1_ordinal1(B)
& r2_hidden(B,A) )
=> r1_tarski(B,A) ) ).
fof(t9_classes2,axiom,
! [A,B] :
( ( v1_ordinal1(A)
& r2_hidden(A,k1_classes1(B)) )
=> r1_tarski(A,k1_classes1(B)) ) ).
fof(t10_classes2,axiom,
! [A] :
( v2_classes1(A)
=> k2_ordinal2(A) = k1_card_1(A) ) ).
fof(t11_classes2,axiom,
! [A] : k2_ordinal2(k1_classes1(A)) = k1_card_1(k1_classes1(A)) ).
fof(t12_classes2,axiom,
! [A,B] :
( ( v2_classes1(A)
& r2_hidden(B,A) )
=> r2_hidden(k1_card_1(B),A) ) ).
fof(t13_classes2,axiom,
! [A,B] :
( r2_hidden(A,k1_classes1(B))
=> r2_hidden(k1_card_1(A),k1_classes1(B)) ) ).
fof(t14_classes2,axiom,
! [A,B] :
( ( v2_classes1(A)
& r2_hidden(B,k1_card_1(A)) )
=> r2_hidden(B,A) ) ).
fof(t15_classes2,axiom,
! [A,B] :
( r2_hidden(A,k1_card_1(k1_classes1(B)))
=> r2_hidden(A,k1_classes1(B)) ) ).
fof(t16_classes2,axiom,
! [A] :
( v1_card_1(A)
=> ! [B] :
( ( v2_classes1(B)
& r2_hidden(A,k1_card_1(B)) )
=> r2_hidden(A,B) ) ) ).
fof(t17_classes2,axiom,
! [A] :
( v1_card_1(A)
=> ! [B] :
( r2_hidden(A,k1_card_1(k1_classes1(B)))
=> r2_hidden(A,k1_classes1(B)) ) ) ).
fof(t18_classes2,axiom,
! [A] :
( v1_card_1(A)
=> ! [B] :
( ( v2_classes1(B)
& r2_hidden(A,B) )
=> r1_tarski(A,B) ) ) ).
fof(t19_classes2,axiom,
! [A] :
( v1_card_1(A)
=> ! [B] :
( r2_hidden(A,k1_classes1(B))
=> r1_tarski(A,k1_classes1(B)) ) ) ).
fof(t20_classes2,axiom,
! [A] :
( v2_classes1(A)
=> v4_ordinal1(k1_card_1(A)) ) ).
fof(t21_classes2,axiom,
! [A] :
( v2_classes1(A)
=> ( A = k1_xboole_0
| ( k1_card_1(A) != np__0
& k1_card_1(A) != k1_xboole_0
& v4_ordinal1(k1_card_1(A)) ) ) ) ).
fof(t22_classes2,axiom,
! [A] :
( k1_card_1(k1_classes1(A)) != np__0
& k1_card_1(k1_classes1(A)) != k1_xboole_0
& v4_ordinal1(k1_card_1(k1_classes1(A))) ) ).
fof(t23_classes2,axiom,
! [A,B] :
( v2_classes1(A)
=> ( ( ~ ( r2_hidden(B,A)
& v1_ordinal1(A) )
& ~ ( r2_hidden(B,A)
& r1_tarski(B,A) )
& ~ ( r2_hidden(k1_card_1(B),k1_card_1(A))
& r1_tarski(B,A) ) )
| r1_tarski(k1_funct_2(B,A),A) ) ) ).
fof(t24_classes2,axiom,
! [A,B] :
( ~ ( ~ ( r2_hidden(A,k1_classes1(B))
& v1_ordinal1(B) )
& ~ ( r2_hidden(A,k1_classes1(B))
& r1_tarski(A,k1_classes1(B)) )
& ~ ( r2_hidden(k1_card_1(A),k1_card_1(k1_classes1(B)))
& r1_tarski(A,k1_classes1(B)) ) )
=> r1_tarski(k1_funct_2(A,k1_classes1(B)),k1_classes1(B)) ) ).
fof(t25_classes2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A) )
=> ( ( v4_ordinal1(k1_relat_1(A))
& ! [B] :
( v3_ordinal1(B)
=> ( r2_hidden(B,k1_relat_1(A))
=> k1_funct_1(A,B) = k4_classes1(B) ) ) )
=> k4_classes1(k1_relat_1(A)) = k3_card_3(A) ) ) ).
fof(t26_classes2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v2_classes1(B)
& r2_hidden(A,k2_ordinal2(B)) )
=> ( r2_hidden(k1_card_1(k4_classes1(A)),k1_card_1(B))
& r2_hidden(k4_classes1(A),B) ) ) ) ).
fof(t27_classes2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( r2_hidden(A,k2_ordinal2(k1_classes1(B)))
=> ( r2_hidden(k1_card_1(k4_classes1(A)),k1_card_1(k1_classes1(B)))
& r2_hidden(k4_classes1(A),k1_classes1(B)) ) ) ) ).
fof(t28_classes2,axiom,
! [A] :
( v2_classes1(A)
=> r1_tarski(k4_classes1(k1_card_1(A)),A) ) ).
fof(t29_classes2,axiom,
! [A] : r1_tarski(k4_classes1(k1_card_1(k1_classes1(A))),k1_classes1(A)) ).
fof(t30_classes2,axiom,
! [A,B] :
( ( v2_classes1(A)
& v1_ordinal1(A)
& r2_hidden(B,A) )
=> r2_hidden(k6_classes1(B),A) ) ).
fof(t31_classes2,axiom,
! [A] :
( ( v2_classes1(A)
& v1_ordinal1(A) )
=> r1_tarski(A,k4_classes1(k1_card_1(A))) ) ).
fof(t32_classes2,axiom,
! [A] :
( ( v2_classes1(A)
& v1_ordinal1(A) )
=> k4_classes1(k1_card_1(A)) = A ) ).
fof(t33_classes2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v2_classes1(B)
& r2_hidden(A,k2_ordinal2(B)) )
=> r1_tarski(k1_card_1(k4_classes1(A)),k1_card_1(B)) ) ) ).
fof(t34_classes2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( r2_hidden(A,k2_ordinal2(k1_classes1(B)))
=> r1_tarski(k1_card_1(k4_classes1(A)),k1_card_1(k1_classes1(B))) ) ) ).
fof(t35_classes2,axiom,
! [A] :
( v2_classes1(A)
=> k1_card_1(A) = k1_card_1(k4_classes1(k1_card_1(A))) ) ).
fof(t36_classes2,axiom,
! [A] : k1_card_1(k1_classes1(A)) = k1_card_1(k4_classes1(k1_card_1(k1_classes1(A)))) ).
fof(t37_classes2,axiom,
! [A,B] :
~ ( v2_classes1(A)
& r1_tarski(B,k4_classes1(k1_card_1(A)))
& ~ r2_wellord2(B,k4_classes1(k1_card_1(A)))
& ~ r2_hidden(B,k4_classes1(k1_card_1(A))) ) ).
fof(t38_classes2,axiom,
! [A,B] :
~ ( r1_tarski(A,k4_classes1(k1_card_1(k1_classes1(B))))
& ~ r2_wellord2(A,k4_classes1(k1_card_1(k1_classes1(B))))
& ~ r2_hidden(A,k4_classes1(k1_card_1(k1_classes1(B)))) ) ).
fof(t39_classes2,axiom,
! [A] :
( v2_classes1(A)
=> v2_classes1(k4_classes1(k1_card_1(A))) ) ).
fof(t40_classes2,axiom,
! [A] : v2_classes1(k4_classes1(k1_card_1(k1_classes1(A)))) ).
fof(t41_classes2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
~ ( v1_ordinal1(B)
& r2_hidden(A,k6_classes1(B))
& ! [C] :
~ ( r2_hidden(C,B)
& k6_classes1(C) = A ) ) ) ).
fof(t42_classes2,axiom,
! [A] :
( v1_ordinal1(A)
=> r1_tarski(k1_card_1(k6_classes1(A)),k1_card_1(A)) ) ).
fof(t43_classes2,axiom,
! [A,B] :
( ( v2_classes1(A)
& v1_ordinal1(B)
& r2_hidden(B,A) )
=> r2_hidden(B,k4_classes1(k1_card_1(A))) ) ).
fof(t44_classes2,axiom,
! [A,B] :
( ( v1_ordinal1(A)
& r2_hidden(A,k1_classes1(B)) )
=> r2_hidden(A,k4_classes1(k1_card_1(k1_classes1(B)))) ) ).
fof(t45_classes2,axiom,
! [A] :
( v1_ordinal1(A)
=> r1_classes1(A,k4_classes1(k1_card_1(k1_classes1(A)))) ) ).
fof(t46_classes2,axiom,
! [A] :
( v1_ordinal1(A)
=> k4_classes1(k1_card_1(k1_classes1(A))) = k1_classes1(A) ) ).
fof(d1_classes2,axiom,
! [A] :
( v1_classes2(A)
<=> ( v1_ordinal1(A)
& v2_classes1(A) ) ) ).
fof(t47_classes2,axiom,
$true ).
fof(t48_classes2,axiom,
$true ).
fof(t49_classes2,axiom,
$true ).
fof(t50_classes2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> v3_ordinal1(k2_ordinal2(A)) ) ).
fof(t51_classes2,axiom,
! [A] :
( v1_ordinal1(A)
=> v1_classes2(k1_classes1(A)) ) ).
fof(t52_classes2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ( ~ v1_xboole_0(k1_classes1(A))
& v1_classes2(k1_classes1(A)) ) ) ).
fof(t53_classes2,axiom,
! [A] :
( v3_ordinal1(A)
=> v1_classes2(k1_classes1(A)) ) ).
fof(t54_classes2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> A = k4_classes1(k2_ordinal2(A)) ) ).
fof(t55_classes2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ( k2_ordinal2(A) != k1_xboole_0
& v4_ordinal1(k2_ordinal2(A)) ) ) ).
fof(t56_classes2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_classes2(B) )
=> ~ ( ~ r2_hidden(A,B)
& A != B
& ~ r2_hidden(B,A) ) ) ) ).
fof(t57_classes2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_classes2(B) )
=> ( r1_tarski(A,B)
| r2_hidden(B,A) ) ) ) ).
fof(t58_classes2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_classes2(B) )
=> r3_xboole_0(A,B) ) ) ).
fof(t59_classes2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_classes2(B) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v1_classes2(C) )
=> ( ( r2_hidden(A,B)
& r2_hidden(B,C) )
=> r2_hidden(A,C) ) ) ) ) ).
fof(t60_classes2,axiom,
$true ).
fof(t61_classes2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_classes2(B) )
=> ( ~ v1_xboole_0(k2_xboole_0(A,B))
& v1_classes2(k2_xboole_0(A,B))
& ~ v1_xboole_0(k3_xboole_0(A,B))
& v1_classes2(k3_xboole_0(A,B)) ) ) ) ).
fof(t62_classes2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> r2_hidden(k1_xboole_0,A) ) ).
fof(t63_classes2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v1_classes2(B) )
=> ( r2_hidden(A,B)
=> r2_hidden(k1_tarski(A),B) ) ) ).
fof(t64_classes2,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(C)
& v1_classes2(C) )
=> ( ( r2_hidden(A,C)
& r2_hidden(B,C) )
=> ( r2_hidden(k2_tarski(A,B),C)
& r2_hidden(k4_tarski(A,B),C) ) ) ) ).
fof(t65_classes2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v1_classes2(B) )
=> ( r2_hidden(A,B)
=> ( r2_hidden(k1_zfmisc_1(A),B)
& r2_hidden(k3_tarski(A),B)
& r2_hidden(k1_setfam_1(A),B) ) ) ) ).
fof(t66_classes2,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(C)
& v1_classes2(C) )
=> ( ( r2_hidden(A,C)
& r2_hidden(B,C) )
=> ( r2_hidden(k2_xboole_0(A,B),C)
& r2_hidden(k3_xboole_0(A,B),C)
& r2_hidden(k4_xboole_0(A,B),C)
& r2_hidden(k5_xboole_0(A,B),C) ) ) ) ).
fof(t67_classes2,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(C)
& v1_classes2(C) )
=> ( ( r2_hidden(A,C)
& r2_hidden(B,C) )
=> ( r2_hidden(k2_zfmisc_1(A,B),C)
& r2_hidden(k1_funct_2(A,B),C) ) ) ) ).
fof(d2_classes2,axiom,
k13_classes2 = k1_classes1(k1_xboole_0) ).
fof(t68_classes2,axiom,
$true ).
fof(t69_classes2,axiom,
k1_card_1(k4_classes1(k5_ordinal2)) = k1_card_1(k5_ordinal2) ).
fof(t70_classes2,axiom,
v2_classes1(k4_classes1(k5_ordinal2)) ).
fof(t71_classes2,axiom,
k13_classes2 = k4_classes1(k5_ordinal2) ).
fof(d3_classes2,axiom,
k14_classes2 = k1_classes1(k13_classes2) ).
fof(d4_classes2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v1_classes2(B) )
=> ( B = k15_classes2(A)
<=> ( r1_tarski(A,B)
& ! [C] :
( ( ~ v1_xboole_0(C)
& v1_classes2(C) )
=> ( r1_tarski(A,C)
=> r1_tarski(B,C) ) ) ) ) ) ).
fof(d5_classes2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( B = k16_classes2(A)
<=> ? [C] :
( v1_relat_1(C)
& v1_funct_1(C)
& v5_ordinal1(C)
& B = k1_ordinal2(C)
& k1_relat_1(C) = k1_ordinal1(A)
& k1_funct_1(C,k1_xboole_0) = k13_classes2
& ! [D] :
( v3_ordinal1(D)
=> ( r2_hidden(k1_ordinal1(D),k1_ordinal1(A))
=> k1_funct_1(C,k1_ordinal1(D)) = k1_classes1(k1_funct_1(C,D)) ) )
& ! [D] :
( v3_ordinal1(D)
=> ( ( r2_hidden(D,k1_ordinal1(A))
& v4_ordinal1(D) )
=> ( D = k1_xboole_0
| k1_funct_1(C,D) = k15_classes2(k3_card_3(k2_ordinal1(C,D))) ) ) ) ) ) ) ).
fof(t72_classes2,axiom,
$true ).
fof(t73_classes2,axiom,
$true ).
fof(t74_classes2,axiom,
$true ).
fof(t75_classes2,axiom,
k16_classes2(k1_xboole_0) = k13_classes2 ).
fof(t76_classes2,axiom,
! [A] :
( v3_ordinal1(A)
=> k16_classes2(k1_ordinal1(A)) = k1_classes1(k16_classes2(A)) ) ).
fof(t77_classes2,axiom,
k16_classes2(k4_ordinal2) = k14_classes2 ).
fof(t78_classes2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B) )
=> ( ( v4_ordinal1(A)
& k1_relat_1(B) = A
& ! [C] :
( v3_ordinal1(C)
=> ( r2_hidden(C,A)
=> k1_funct_1(B,C) = k16_classes2(C) ) ) )
=> ( A = k1_xboole_0
| k16_classes2(A) = k15_classes2(k3_card_3(B)) ) ) ) ) ).
fof(t79_classes2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ( r1_tarski(k13_classes2,A)
& r1_tarski(k1_classes1(k1_xboole_0),A)
& r1_tarski(k16_classes2(k1_xboole_0),A) ) ) ).
fof(t80_classes2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ( r2_hidden(A,B)
<=> r2_hidden(k16_classes2(A),k16_classes2(B)) ) ) ) ).
fof(t81_classes2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ( k16_classes2(A) = k16_classes2(B)
=> A = B ) ) ) ).
fof(t82_classes2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ( r1_ordinal1(A,B)
<=> r1_tarski(k16_classes2(A),k16_classes2(B)) ) ) ) ).
fof(dt_k1_classes2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& m1_subset_1(B,A) )
=> m1_subset_1(k1_classes2(A,B),A) ) ).
fof(redefinition_k1_classes2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& m1_subset_1(B,A) )
=> k1_classes2(A,B) = k1_tarski(B) ) ).
fof(dt_k2_classes2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& m1_subset_1(B,A) )
=> ( ~ v1_xboole_0(k2_classes2(A,B))
& m1_subset_1(k2_classes2(A,B),A) ) ) ).
fof(redefinition_k2_classes2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& m1_subset_1(B,A) )
=> k2_classes2(A,B) = k1_zfmisc_1(B) ) ).
fof(dt_k3_classes2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& m1_subset_1(B,A) )
=> m1_subset_1(k3_classes2(A,B),A) ) ).
fof(redefinition_k3_classes2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& m1_subset_1(B,A) )
=> k3_classes2(A,B) = k3_tarski(B) ) ).
fof(dt_k4_classes2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& m1_subset_1(B,A) )
=> m1_subset_1(k4_classes2(A,B),A) ) ).
fof(redefinition_k4_classes2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& m1_subset_1(B,A) )
=> k4_classes2(A,B) = k1_setfam_1(B) ) ).
fof(dt_k5_classes2,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& m1_subset_1(B,A)
& m1_subset_1(C,A) )
=> m1_subset_1(k5_classes2(A,B,C),A) ) ).
fof(commutativity_k5_classes2,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& m1_subset_1(B,A)
& m1_subset_1(C,A) )
=> k5_classes2(A,B,C) = k5_classes2(A,C,B) ) ).
fof(redefinition_k5_classes2,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& m1_subset_1(B,A)
& m1_subset_1(C,A) )
=> k5_classes2(A,B,C) = k2_tarski(B,C) ) ).
fof(dt_k6_classes2,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& m1_subset_1(B,A)
& m1_subset_1(C,A) )
=> m1_subset_1(k6_classes2(A,B,C),A) ) ).
fof(redefinition_k6_classes2,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& m1_subset_1(B,A)
& m1_subset_1(C,A) )
=> k6_classes2(A,B,C) = k4_tarski(B,C) ) ).
fof(dt_k7_classes2,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& m1_subset_1(B,A)
& m1_subset_1(C,A) )
=> m1_subset_1(k7_classes2(A,B,C),A) ) ).
fof(commutativity_k7_classes2,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& m1_subset_1(B,A)
& m1_subset_1(C,A) )
=> k7_classes2(A,B,C) = k7_classes2(A,C,B) ) ).
fof(idempotence_k7_classes2,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& m1_subset_1(B,A)
& m1_subset_1(C,A) )
=> k7_classes2(A,B,B) = B ) ).
fof(redefinition_k7_classes2,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& m1_subset_1(B,A)
& m1_subset_1(C,A) )
=> k7_classes2(A,B,C) = k2_xboole_0(B,C) ) ).
fof(dt_k8_classes2,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& m1_subset_1(B,A)
& m1_subset_1(C,A) )
=> m1_subset_1(k8_classes2(A,B,C),A) ) ).
fof(commutativity_k8_classes2,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& m1_subset_1(B,A)
& m1_subset_1(C,A) )
=> k8_classes2(A,B,C) = k8_classes2(A,C,B) ) ).
fof(idempotence_k8_classes2,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& m1_subset_1(B,A)
& m1_subset_1(C,A) )
=> k8_classes2(A,B,B) = B ) ).
fof(redefinition_k8_classes2,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& m1_subset_1(B,A)
& m1_subset_1(C,A) )
=> k8_classes2(A,B,C) = k3_xboole_0(B,C) ) ).
fof(dt_k9_classes2,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& m1_subset_1(B,A)
& m1_subset_1(C,A) )
=> m1_subset_1(k9_classes2(A,B,C),A) ) ).
fof(redefinition_k9_classes2,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& m1_subset_1(B,A)
& m1_subset_1(C,A) )
=> k9_classes2(A,B,C) = k4_xboole_0(B,C) ) ).
fof(dt_k10_classes2,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& m1_subset_1(B,A)
& m1_subset_1(C,A) )
=> m1_subset_1(k10_classes2(A,B,C),A) ) ).
fof(commutativity_k10_classes2,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& m1_subset_1(B,A)
& m1_subset_1(C,A) )
=> k10_classes2(A,B,C) = k10_classes2(A,C,B) ) ).
fof(redefinition_k10_classes2,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& m1_subset_1(B,A)
& m1_subset_1(C,A) )
=> k10_classes2(A,B,C) = k5_xboole_0(B,C) ) ).
fof(dt_k11_classes2,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& m1_subset_1(B,A)
& m1_subset_1(C,A) )
=> m1_subset_1(k11_classes2(A,B,C),A) ) ).
fof(redefinition_k11_classes2,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& m1_subset_1(B,A)
& m1_subset_1(C,A) )
=> k11_classes2(A,B,C) = k2_zfmisc_1(B,C) ) ).
fof(dt_k12_classes2,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& m1_subset_1(B,A)
& m1_subset_1(C,A) )
=> m1_subset_1(k12_classes2(A,B,C),A) ) ).
fof(redefinition_k12_classes2,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& m1_subset_1(B,A)
& m1_subset_1(C,A) )
=> k12_classes2(A,B,C) = k1_funct_2(B,C) ) ).
fof(dt_k13_classes2,axiom,
( ~ v1_xboole_0(k13_classes2)
& v1_classes2(k13_classes2) ) ).
fof(dt_k14_classes2,axiom,
( ~ v1_xboole_0(k14_classes2)
& v1_classes2(k14_classes2) ) ).
fof(dt_k15_classes2,axiom,
! [A] :
( ~ v1_xboole_0(k15_classes2(A))
& v1_classes2(k15_classes2(A)) ) ).
fof(dt_k16_classes2,axiom,
$true ).
%------------------------------------------------------------------------------