SET007 Axioms: SET007+59.ax
%------------------------------------------------------------------------------
% File : SET007+59 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Partially Ordered Sets
% 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 : orders_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 215 ( 108 unt; 0 def)
% Number of atoms : 635 ( 40 equ)
% Maximal formula atoms : 28 ( 2 avg)
% Number of connectives : 495 ( 75 ~; 4 |; 202 &)
% ( 22 <=>; 192 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 4 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 39 ( 37 usr; 1 prp; 0-3 aty)
% Number of functors : 21 ( 21 usr; 3 con; 0-3 aty)
% Number of variables : 260 ( 254 !; 6 ?)
% 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(fc1_orders_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ~ v1_xboole_0(k1_orders_1(A)) ) ).
fof(d1_orders_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( ~ r2_hidden(k1_xboole_0,A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k3_tarski(A))
& m2_relset_1(B,A,k3_tarski(A)) )
=> ( m1_orders_1(B,A)
<=> ! [C] :
( r2_hidden(C,A)
=> r2_hidden(k1_funct_1(B,C),C) ) ) ) ) ) ).
fof(d2_orders_1,axiom,
! [A] : k1_orders_1(A) = k4_xboole_0(k1_zfmisc_1(A),k1_tarski(k1_xboole_0)) ).
fof(t1_orders_1,axiom,
$true ).
fof(t2_orders_1,axiom,
$true ).
fof(t3_orders_1,axiom,
$true ).
fof(t4_orders_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ~ r2_hidden(k1_xboole_0,k1_orders_1(A)) ) ).
fof(t5_orders_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ( r1_tarski(A,B)
<=> r2_hidden(A,k1_orders_1(B)) ) ) ) ).
fof(t6_orders_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ( m1_subset_1(A,k1_zfmisc_1(B))
<=> r2_hidden(A,k1_orders_1(B)) ) ) ) ).
fof(t7_orders_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> r2_hidden(A,k1_orders_1(A)) ) ).
fof(t8_orders_1,axiom,
$true ).
fof(t9_orders_1,axiom,
$true ).
fof(t10_orders_1,axiom,
$true ).
fof(t11_orders_1,axiom,
$true ).
fof(t12_orders_1,axiom,
! [A,B,C] :
( ( v1_relat_2(C)
& v4_relat_2(C)
& v8_relat_2(C)
& v1_partfun1(C,A,A)
& m2_relset_1(C,A,A) )
=> ( r2_hidden(B,A)
=> r2_hidden(k4_tarski(B,B),C) ) ) ).
fof(t13_orders_1,axiom,
! [A,B,C,D] :
( ( v1_relat_2(D)
& v4_relat_2(D)
& v8_relat_2(D)
& v1_partfun1(D,A,A)
& m2_relset_1(D,A,A) )
=> ( ( r2_hidden(B,A)
& r2_hidden(C,A)
& r2_hidden(k4_tarski(B,C),D)
& r2_hidden(k4_tarski(C,B),D) )
=> B = C ) ) ).
fof(t14_orders_1,axiom,
! [A,B,C,D,E] :
( ( v1_relat_2(E)
& v4_relat_2(E)
& v8_relat_2(E)
& v1_partfun1(E,A,A)
& m2_relset_1(E,A,A) )
=> ( ( r2_hidden(B,A)
& r2_hidden(C,A)
& r2_hidden(D,A)
& r2_hidden(k4_tarski(B,C),E)
& r2_hidden(k4_tarski(C,D),E) )
=> r2_hidden(k4_tarski(B,D),E) ) ) ).
fof(t15_orders_1,axiom,
$true ).
fof(t16_orders_1,axiom,
$true ).
fof(t17_orders_1,axiom,
$true ).
fof(t18_orders_1,axiom,
$true ).
fof(t19_orders_1,axiom,
$true ).
fof(t20_orders_1,axiom,
$true ).
fof(t21_orders_1,axiom,
$true ).
fof(t22_orders_1,axiom,
$true ).
fof(t23_orders_1,axiom,
$true ).
fof(t24_orders_1,axiom,
$true ).
fof(t25_orders_1,axiom,
$true ).
fof(t26_orders_1,axiom,
$true ).
fof(t27_orders_1,axiom,
$true ).
fof(t28_orders_1,axiom,
$true ).
fof(t29_orders_1,axiom,
$true ).
fof(t30_orders_1,axiom,
$true ).
fof(t31_orders_1,axiom,
$true ).
fof(t32_orders_1,axiom,
$true ).
fof(t33_orders_1,axiom,
$true ).
fof(t34_orders_1,axiom,
$true ).
fof(t35_orders_1,axiom,
$true ).
fof(t36_orders_1,axiom,
$true ).
fof(t37_orders_1,axiom,
$true ).
fof(t38_orders_1,axiom,
$true ).
fof(t39_orders_1,axiom,
$true ).
fof(t40_orders_1,axiom,
$true ).
fof(t41_orders_1,axiom,
$true ).
fof(t42_orders_1,axiom,
$true ).
fof(t43_orders_1,axiom,
$true ).
fof(t44_orders_1,axiom,
$true ).
fof(t45_orders_1,axiom,
$true ).
fof(t46_orders_1,axiom,
$true ).
fof(t47_orders_1,axiom,
$true ).
fof(t48_orders_1,axiom,
$true ).
fof(t49_orders_1,axiom,
$true ).
fof(t50_orders_1,axiom,
$true ).
fof(t51_orders_1,axiom,
$true ).
fof(t52_orders_1,axiom,
$true ).
fof(t53_orders_1,axiom,
$true ).
fof(t54_orders_1,axiom,
$true ).
fof(t55_orders_1,axiom,
$true ).
fof(t56_orders_1,axiom,
$true ).
fof(t57_orders_1,axiom,
$true ).
fof(t58_orders_1,axiom,
$true ).
fof(t59_orders_1,axiom,
$true ).
fof(t60_orders_1,axiom,
$true ).
fof(t61_orders_1,axiom,
$true ).
fof(t62_orders_1,axiom,
$true ).
fof(t63_orders_1,axiom,
$true ).
fof(t64_orders_1,axiom,
$true ).
fof(t65_orders_1,axiom,
$true ).
fof(t66_orders_1,axiom,
$true ).
fof(t67_orders_1,axiom,
$true ).
fof(t68_orders_1,axiom,
$true ).
fof(t69_orders_1,axiom,
$true ).
fof(t70_orders_1,axiom,
$true ).
fof(t71_orders_1,axiom,
$true ).
fof(t72_orders_1,axiom,
$true ).
fof(t73_orders_1,axiom,
$true ).
fof(t74_orders_1,axiom,
$true ).
fof(t75_orders_1,axiom,
$true ).
fof(t76_orders_1,axiom,
$true ).
fof(t77_orders_1,axiom,
$true ).
fof(t78_orders_1,axiom,
$true ).
fof(t79_orders_1,axiom,
$true ).
fof(t80_orders_1,axiom,
$true ).
fof(t81_orders_1,axiom,
$true ).
fof(t82_orders_1,axiom,
$true ).
fof(t83_orders_1,axiom,
$true ).
fof(t84_orders_1,axiom,
$true ).
fof(t85_orders_1,axiom,
$true ).
fof(t86_orders_1,axiom,
$true ).
fof(t87_orders_1,axiom,
$true ).
fof(t88_orders_1,axiom,
$true ).
fof(t89_orders_1,axiom,
$true ).
fof(t90_orders_1,axiom,
$true ).
fof(t91_orders_1,axiom,
! [A] :
( ~ ( ? [B] :
( B != k1_xboole_0
& r2_hidden(B,A) )
& k3_tarski(A) = k1_xboole_0 )
& ~ ( k3_tarski(A) != k1_xboole_0
& ! [B] :
~ ( B != k1_xboole_0
& r2_hidden(B,A) ) ) ) ).
fof(t92_orders_1,axiom,
! [A,B] :
( v1_relat_1(B)
=> ( r7_relat_2(B,A)
<=> ( r1_relat_2(B,A)
& r6_relat_2(B,A) ) ) ) ).
fof(t93_orders_1,axiom,
! [A,B,C] :
( v1_relat_1(C)
=> ( ( r1_relat_2(C,A)
& r1_tarski(B,A) )
=> r1_relat_2(C,B) ) ) ).
fof(t94_orders_1,axiom,
! [A,B,C] :
( v1_relat_1(C)
=> ( ( r4_relat_2(C,A)
& r1_tarski(B,A) )
=> r4_relat_2(C,B) ) ) ).
fof(t95_orders_1,axiom,
! [A,B,C] :
( v1_relat_1(C)
=> ( ( r8_relat_2(C,A)
& r1_tarski(B,A) )
=> r8_relat_2(C,B) ) ) ).
fof(t96_orders_1,axiom,
! [A,B,C] :
( v1_relat_1(C)
=> ( ( r7_relat_2(C,A)
& r1_tarski(B,A) )
=> r7_relat_2(C,B) ) ) ).
fof(t97_orders_1,axiom,
! [A,B] :
( ( v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> k3_relat_1(B) = A ) ).
fof(t98_orders_1,axiom,
! [A,B] :
( m2_relset_1(B,A,A)
=> ( r1_relat_2(B,A)
=> ( k4_relset_1(A,A,B) = A
& k3_relat_1(B) = A ) ) ) ).
fof(t99_orders_1,axiom,
! [A,B] :
( ( v1_relat_2(B)
& v4_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ( k4_relset_1(A,A,B) = A
& k5_relset_1(A,A,B) = A ) ) ).
fof(t100_orders_1,axiom,
! [A,B] :
( ( v1_relat_2(B)
& v4_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> k3_relat_1(B) = A ) ).
fof(d3_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ( v1_orders_1(A)
<=> ( v1_relat_2(A)
& v8_relat_2(A) ) ) ) ).
fof(d4_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ( v2_orders_1(A)
<=> ( v1_relat_2(A)
& v8_relat_2(A)
& v4_relat_2(A) ) ) ) ).
fof(d5_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ( v3_orders_1(A)
<=> ( v1_relat_2(A)
& v8_relat_2(A)
& v4_relat_2(A)
& v6_relat_2(A) ) ) ) ).
fof(t101_orders_1,axiom,
$true ).
fof(t102_orders_1,axiom,
$true ).
fof(t103_orders_1,axiom,
$true ).
fof(t104_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ( v1_orders_1(A)
=> v1_orders_1(k4_relat_1(A)) ) ) ).
fof(t105_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ( v2_orders_1(A)
=> v2_orders_1(k4_relat_1(A)) ) ) ).
fof(t106_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ( v3_orders_1(A)
=> v3_orders_1(k4_relat_1(A)) ) ) ).
fof(t107_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ( v2_wellord1(A)
=> ( v1_orders_1(A)
& v2_orders_1(A)
& v3_orders_1(A) ) ) ) ).
fof(t108_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ( v3_orders_1(A)
=> ( v1_orders_1(A)
& v2_orders_1(A) ) ) ) ).
fof(t109_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ( v2_orders_1(A)
=> v1_orders_1(A) ) ) ).
fof(t110_orders_1,axiom,
! [A,B] :
( ( v1_relat_2(B)
& v4_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> v2_orders_1(B) ) ).
fof(t111_orders_1,axiom,
! [A,B] :
( ( v1_relat_2(B)
& v4_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> v1_orders_1(B) ) ).
fof(t112_orders_1,axiom,
! [A,B] :
( ( v1_relat_2(B)
& v4_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ( v6_relat_2(B)
=> v3_orders_1(B) ) ) ).
fof(t113_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( v1_orders_1(A)
=> v1_orders_1(k2_wellord1(A,B)) ) ) ).
fof(t114_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( v2_orders_1(A)
=> v2_orders_1(k2_wellord1(A,B)) ) ) ).
fof(t115_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( v3_orders_1(A)
=> v3_orders_1(k2_wellord1(A,B)) ) ) ).
fof(t116_orders_1,axiom,
$true ).
fof(t117_orders_1,axiom,
$true ).
fof(t118_orders_1,axiom,
$true ).
fof(t119_orders_1,axiom,
( v1_orders_1(k1_xboole_0)
& v2_orders_1(k1_xboole_0)
& v3_orders_1(k1_xboole_0)
& v2_wellord1(k1_xboole_0) ) ).
fof(t120_orders_1,axiom,
! [A] :
( v1_orders_1(k6_partfun1(A))
& v2_orders_1(k6_partfun1(A)) ) ).
fof(d6_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r1_orders_1(A,B)
<=> ( r1_relat_2(A,B)
& r8_relat_2(A,B) ) ) ) ).
fof(d7_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r2_orders_1(A,B)
<=> ( r1_relat_2(A,B)
& r8_relat_2(A,B)
& r4_relat_2(A,B) ) ) ) ).
fof(d8_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r3_orders_1(A,B)
<=> ( r1_relat_2(A,B)
& r8_relat_2(A,B)
& r4_relat_2(A,B)
& r6_relat_2(A,B) ) ) ) ).
fof(t121_orders_1,axiom,
$true ).
fof(t122_orders_1,axiom,
$true ).
fof(t123_orders_1,axiom,
$true ).
fof(t124_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r2_wellord1(A,B)
=> ( r1_orders_1(A,B)
& r2_orders_1(A,B)
& r3_orders_1(A,B) ) ) ) ).
fof(t125_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r3_orders_1(A,B)
=> ( r1_orders_1(A,B)
& r2_orders_1(A,B) ) ) ) ).
fof(t126_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r2_orders_1(A,B)
=> r1_orders_1(A,B) ) ) ).
fof(t127_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ( v1_orders_1(A)
=> r1_orders_1(A,k3_relat_1(A)) ) ) ).
fof(t128_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B,C] :
( ( r1_orders_1(A,B)
& r1_tarski(C,B) )
=> r1_orders_1(A,C) ) ) ).
fof(t129_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r1_orders_1(A,B)
=> v1_orders_1(k2_wellord1(A,B)) ) ) ).
fof(t130_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ( v2_orders_1(A)
=> r2_orders_1(A,k3_relat_1(A)) ) ) ).
fof(t131_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B,C] :
( ( r2_orders_1(A,B)
& r1_tarski(C,B) )
=> r2_orders_1(A,C) ) ) ).
fof(t132_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r2_orders_1(A,B)
=> v2_orders_1(k2_wellord1(A,B)) ) ) ).
fof(t133_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ( v3_orders_1(A)
=> r3_orders_1(A,k3_relat_1(A)) ) ) ).
fof(t134_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B,C] :
( ( r3_orders_1(A,B)
& r1_tarski(C,B) )
=> r3_orders_1(A,C) ) ) ).
fof(t135_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r3_orders_1(A,B)
=> v3_orders_1(k2_wellord1(A,B)) ) ) ).
fof(t136_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r1_orders_1(A,B)
=> r1_orders_1(k4_relat_1(A),B) ) ) ).
fof(t137_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r2_orders_1(A,B)
=> r2_orders_1(k4_relat_1(A),B) ) ) ).
fof(t138_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r3_orders_1(A,B)
=> r3_orders_1(k4_relat_1(A),B) ) ) ).
fof(t139_orders_1,axiom,
! [A,B] :
( ( v1_relat_2(B)
& v4_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> r1_orders_1(B,A) ) ).
fof(t140_orders_1,axiom,
! [A,B] :
( ( v1_relat_2(B)
& v4_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> r2_orders_1(B,A) ) ).
fof(t141_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r2_orders_1(A,B)
=> ( v1_relat_2(k2_wellord1(A,B))
& v4_relat_2(k2_wellord1(A,B))
& v8_relat_2(k2_wellord1(A,B))
& v1_partfun1(k2_wellord1(A,B),B,B)
& m2_relset_1(k2_wellord1(A,B),B,B) ) ) ) ).
fof(t142_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r3_orders_1(A,B)
=> ( v1_relat_2(k2_wellord1(A,B))
& v4_relat_2(k2_wellord1(A,B))
& v8_relat_2(k2_wellord1(A,B))
& v1_partfun1(k2_wellord1(A,B),B,B)
& m2_relset_1(k2_wellord1(A,B),B,B) ) ) ) ).
fof(t143_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r2_wellord1(A,B)
=> ( v1_relat_2(k2_wellord1(A,B))
& v4_relat_2(k2_wellord1(A,B))
& v8_relat_2(k2_wellord1(A,B))
& v1_partfun1(k2_wellord1(A,B),B,B)
& m2_relset_1(k2_wellord1(A,B),B,B) ) ) ) ).
fof(t144_orders_1,axiom,
$true ).
fof(t145_orders_1,axiom,
$true ).
fof(t146_orders_1,axiom,
! [A] :
( r1_orders_1(k6_partfun1(A),A)
& r2_orders_1(k6_partfun1(A),A) ) ).
fof(d9_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r4_orders_1(A,B)
<=> ! [C] :
~ ( r1_tarski(C,B)
& v3_orders_1(k2_wellord1(A,C))
& ! [D] :
~ ( r2_hidden(D,B)
& ! [E] :
( r2_hidden(E,C)
=> r2_hidden(k4_tarski(E,D),A) ) ) ) ) ) ).
fof(d10_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r5_orders_1(A,B)
<=> ! [C] :
~ ( r1_tarski(C,B)
& v3_orders_1(k2_wellord1(A,C))
& ! [D] :
~ ( r2_hidden(D,B)
& ! [E] :
( r2_hidden(E,C)
=> r2_hidden(k4_tarski(D,E),A) ) ) ) ) ) ).
fof(t147_orders_1,axiom,
$true ).
fof(t148_orders_1,axiom,
$true ).
fof(t149_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
~ ( r4_orders_1(A,B)
& B = k1_xboole_0 ) ) ).
fof(t150_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
~ ( r5_orders_1(A,B)
& B = k1_xboole_0 ) ) ).
fof(t151_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r4_orders_1(A,B)
<=> r5_orders_1(k4_relat_1(A),B) ) ) ).
fof(t152_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r4_orders_1(k4_relat_1(A),B)
<=> r5_orders_1(A,B) ) ) ).
fof(d11_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r6_orders_1(A,B)
<=> ( r2_hidden(B,k3_relat_1(A))
& ! [C] :
~ ( r2_hidden(C,k3_relat_1(A))
& C != B
& r2_hidden(k4_tarski(B,C),A) ) ) ) ) ).
fof(d12_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r7_orders_1(A,B)
<=> ( r2_hidden(B,k3_relat_1(A))
& ! [C] :
~ ( r2_hidden(C,k3_relat_1(A))
& C != B
& r2_hidden(k4_tarski(C,B),A) ) ) ) ) ).
fof(d13_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r8_orders_1(A,B)
<=> ( r2_hidden(B,k3_relat_1(A))
& ! [C] :
( r2_hidden(C,k3_relat_1(A))
=> ( C = B
| r2_hidden(k4_tarski(C,B),A) ) ) ) ) ) ).
fof(d14_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r9_orders_1(A,B)
<=> ( r2_hidden(B,k3_relat_1(A))
& ! [C] :
( r2_hidden(C,k3_relat_1(A))
=> ( C = B
| r2_hidden(k4_tarski(B,C),A) ) ) ) ) ) ).
fof(t153_orders_1,axiom,
$true ).
fof(t154_orders_1,axiom,
$true ).
fof(t155_orders_1,axiom,
$true ).
fof(t156_orders_1,axiom,
$true ).
fof(t157_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( ( r9_orders_1(A,B)
& v4_relat_2(A) )
=> r7_orders_1(A,B) ) ) ).
fof(t158_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( ( r8_orders_1(A,B)
& v4_relat_2(A) )
=> r6_orders_1(A,B) ) ) ).
fof(t159_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( ( r7_orders_1(A,B)
& v6_relat_2(A) )
=> r9_orders_1(A,B) ) ) ).
fof(t160_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( ( r6_orders_1(A,B)
& v6_relat_2(A) )
=> r8_orders_1(A,B) ) ) ).
fof(t161_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B,C] :
( ( r2_hidden(B,C)
& r8_orders_1(A,B)
& r1_tarski(C,k3_relat_1(A))
& v1_relat_2(A) )
=> r4_orders_1(A,C) ) ) ).
fof(t162_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B,C] :
( ( r2_hidden(B,C)
& r9_orders_1(A,B)
& r1_tarski(C,k3_relat_1(A))
& v1_relat_2(A) )
=> r5_orders_1(A,C) ) ) ).
fof(t163_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r7_orders_1(A,B)
<=> r6_orders_1(k4_relat_1(A),B) ) ) ).
fof(t164_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r7_orders_1(k4_relat_1(A),B)
<=> r6_orders_1(A,B) ) ) ).
fof(t165_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r9_orders_1(A,B)
<=> r8_orders_1(k4_relat_1(A),B) ) ) ).
fof(t166_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r9_orders_1(k4_relat_1(A),B)
<=> r8_orders_1(A,B) ) ) ).
fof(t167_orders_1,axiom,
$true ).
fof(t168_orders_1,axiom,
$true ).
fof(t169_orders_1,axiom,
$true ).
fof(t170_orders_1,axiom,
$true ).
fof(t171_orders_1,axiom,
$true ).
fof(t172_orders_1,axiom,
$true ).
fof(t173_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
~ ( r2_orders_1(A,B)
& k3_relat_1(A) = B
& r4_orders_1(A,B)
& ! [C] : ~ r6_orders_1(A,C) ) ) ).
fof(t174_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
~ ( r2_orders_1(A,B)
& k3_relat_1(A) = B
& r5_orders_1(A,B)
& ! [C] : ~ r7_orders_1(A,C) ) ) ).
fof(t175_orders_1,axiom,
! [A] :
~ ( A != k1_xboole_0
& ! [B] :
~ ( r1_tarski(B,A)
& v6_ordinal1(B)
& ! [C] :
~ ( r2_hidden(C,A)
& ! [D] :
( r2_hidden(D,B)
=> r1_tarski(D,C) ) ) )
& ! [B] :
~ ( r2_hidden(B,A)
& ! [C] :
~ ( r2_hidden(C,A)
& C != B
& r1_tarski(B,C) ) ) ) ).
fof(t176_orders_1,axiom,
! [A] :
~ ( A != k1_xboole_0
& ! [B] :
~ ( r1_tarski(B,A)
& v6_ordinal1(B)
& ! [C] :
~ ( r2_hidden(C,A)
& ! [D] :
( r2_hidden(D,B)
=> r1_tarski(C,D) ) ) )
& ! [B] :
~ ( r2_hidden(B,A)
& ! [C] :
~ ( r2_hidden(C,A)
& C != B
& r1_tarski(C,B) ) ) ) ).
fof(t177_orders_1,axiom,
! [A] :
~ ( A != k1_xboole_0
& ! [B] :
( ( r1_tarski(B,A)
& v6_ordinal1(B) )
=> ( B = k1_xboole_0
| r2_hidden(k3_tarski(B),A) ) )
& ! [B] :
~ ( r2_hidden(B,A)
& ! [C] :
~ ( r2_hidden(C,A)
& C != B
& r1_tarski(B,C) ) ) ) ).
fof(t178_orders_1,axiom,
! [A] :
~ ( A != k1_xboole_0
& ! [B] :
( ( r1_tarski(B,A)
& v6_ordinal1(B) )
=> ( B = k1_xboole_0
| r2_hidden(k1_setfam_1(B),A) ) )
& ! [B] :
~ ( r2_hidden(B,A)
& ! [C] :
~ ( r2_hidden(C,A)
& C != B
& r1_tarski(C,B) ) ) ) ).
fof(t179_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
~ ( r2_orders_1(A,B)
& k3_relat_1(A) = B
& ! [C] :
( v1_relat_1(C)
=> ~ ( r1_tarski(A,C)
& r3_orders_1(C,B)
& k3_relat_1(C) = B ) ) ) ) ).
fof(t180_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> r1_tarski(A,k2_zfmisc_1(k3_relat_1(A),k3_relat_1(A))) ) ).
fof(t181_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( ( v1_relat_2(A)
& r1_tarski(B,k3_relat_1(A)) )
=> k3_relat_1(k2_wellord1(A,B)) = B ) ) ).
fof(t182_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r1_relat_2(A,B)
=> v1_relat_2(k2_wellord1(A,B)) ) ) ).
fof(t183_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r8_relat_2(A,B)
=> v8_relat_2(k2_wellord1(A,B)) ) ) ).
fof(t184_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r4_relat_2(A,B)
=> v4_relat_2(k2_wellord1(A,B)) ) ) ).
fof(t185_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r6_relat_2(A,B)
=> v6_relat_2(k2_wellord1(A,B)) ) ) ).
fof(t186_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B,C] :
( ( r6_relat_2(A,B)
& r1_tarski(C,B) )
=> r6_relat_2(A,C) ) ) ).
fof(t187_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B,C] :
( ( r2_wellord1(A,B)
& r1_tarski(C,B) )
=> r2_wellord1(A,C) ) ) ).
fof(t188_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ( v6_relat_2(A)
=> v6_relat_2(k4_relat_1(A)) ) ) ).
fof(t189_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r1_relat_2(A,B)
=> r1_relat_2(k4_relat_1(A),B) ) ) ).
fof(t190_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r8_relat_2(A,B)
=> r8_relat_2(k4_relat_1(A),B) ) ) ).
fof(t191_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r4_relat_2(A,B)
=> r4_relat_2(k4_relat_1(A),B) ) ) ).
fof(t192_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r6_relat_2(A,B)
=> r6_relat_2(k4_relat_1(A),B) ) ) ).
fof(t193_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] : k4_relat_1(k2_wellord1(A,B)) = k2_wellord1(k4_relat_1(A),B) ) ).
fof(t194_orders_1,axiom,
! [A] :
( v1_relat_1(A)
=> k2_wellord1(A,k1_xboole_0) = k1_xboole_0 ) ).
fof(t195_orders_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
~ ( v1_finset_1(B)
& r1_tarski(B,k2_relat_1(A))
& ! [C] :
~ ( r1_tarski(C,k1_relat_1(A))
& v1_finset_1(C)
& k9_relat_1(A,C) = B ) ) ) ).
fof(s1_orders_1,axiom,
( ( ! [A] :
( m1_subset_1(A,f1_s1_orders_1)
=> p1_s1_orders_1(A,A) )
& ! [A] :
( m1_subset_1(A,f1_s1_orders_1)
=> ! [B] :
( m1_subset_1(B,f1_s1_orders_1)
=> ( ( p1_s1_orders_1(A,B)
& p1_s1_orders_1(B,A) )
=> A = B ) ) )
& ! [A] :
( m1_subset_1(A,f1_s1_orders_1)
=> ! [B] :
( m1_subset_1(B,f1_s1_orders_1)
=> ! [C] :
( m1_subset_1(C,f1_s1_orders_1)
=> ( ( p1_s1_orders_1(A,B)
& p1_s1_orders_1(B,C) )
=> p1_s1_orders_1(A,C) ) ) ) )
& ! [A] :
~ ( r1_tarski(A,f1_s1_orders_1)
& ! [B] :
( m1_subset_1(B,f1_s1_orders_1)
=> ! [C] :
( m1_subset_1(C,f1_s1_orders_1)
=> ~ ( r2_hidden(B,A)
& r2_hidden(C,A)
& ~ p1_s1_orders_1(B,C)
& ~ p1_s1_orders_1(C,B) ) ) )
& ! [B] :
( m1_subset_1(B,f1_s1_orders_1)
=> ? [C] :
( m1_subset_1(C,f1_s1_orders_1)
& r2_hidden(C,A)
& ~ p1_s1_orders_1(C,B) ) ) ) )
=> ? [A] :
( m1_subset_1(A,f1_s1_orders_1)
& ! [B] :
( m1_subset_1(B,f1_s1_orders_1)
=> ~ ( A != B
& p1_s1_orders_1(A,B) ) ) ) ) ).
fof(s2_orders_1,axiom,
( ( ! [A] :
( m1_subset_1(A,f1_s2_orders_1)
=> p1_s2_orders_1(A,A) )
& ! [A] :
( m1_subset_1(A,f1_s2_orders_1)
=> ! [B] :
( m1_subset_1(B,f1_s2_orders_1)
=> ( ( p1_s2_orders_1(A,B)
& p1_s2_orders_1(B,A) )
=> A = B ) ) )
& ! [A] :
( m1_subset_1(A,f1_s2_orders_1)
=> ! [B] :
( m1_subset_1(B,f1_s2_orders_1)
=> ! [C] :
( m1_subset_1(C,f1_s2_orders_1)
=> ( ( p1_s2_orders_1(A,B)
& p1_s2_orders_1(B,C) )
=> p1_s2_orders_1(A,C) ) ) ) )
& ! [A] :
~ ( r1_tarski(A,f1_s2_orders_1)
& ! [B] :
( m1_subset_1(B,f1_s2_orders_1)
=> ! [C] :
( m1_subset_1(C,f1_s2_orders_1)
=> ~ ( r2_hidden(B,A)
& r2_hidden(C,A)
& ~ p1_s2_orders_1(B,C)
& ~ p1_s2_orders_1(C,B) ) ) )
& ! [B] :
( m1_subset_1(B,f1_s2_orders_1)
=> ? [C] :
( m1_subset_1(C,f1_s2_orders_1)
& r2_hidden(C,A)
& ~ p1_s2_orders_1(B,C) ) ) ) )
=> ? [A] :
( m1_subset_1(A,f1_s2_orders_1)
& ! [B] :
( m1_subset_1(B,f1_s2_orders_1)
=> ~ ( A != B
& p1_s2_orders_1(B,A) ) ) ) ) ).
fof(dt_m1_orders_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_orders_1(B,A)
=> ( v1_funct_1(B)
& v1_funct_2(B,A,k3_tarski(A))
& m2_relset_1(B,A,k3_tarski(A)) ) ) ) ).
fof(existence_m1_orders_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ? [B] : m1_orders_1(B,A) ) ).
fof(dt_k1_orders_1,axiom,
$true ).
%------------------------------------------------------------------------------