SET007 Axioms: SET007+6.ax
%------------------------------------------------------------------------------
% File : SET007+6 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Some Basic Properties of 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 : zfmisc_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 150 ( 52 unt; 0 def)
% Number of atoms : 343 ( 159 equ)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 281 ( 88 ~; 17 |; 90 &)
% ( 25 <=>; 61 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 5 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 8 ( 6 usr; 1 prp; 0-2 aty)
% Number of functors : 13 ( 13 usr; 1 con; 0-4 aty)
% Number of variables : 371 ( 368 !; 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(fc1_zfmisc_1,axiom,
! [A,B] : ~ v1_xboole_0(k4_tarski(A,B)) ).
fof(d1_zfmisc_1,axiom,
! [A,B] :
( B = k1_zfmisc_1(A)
<=> ! [C] :
( r2_hidden(C,B)
<=> r1_tarski(C,A) ) ) ).
fof(d2_zfmisc_1,axiom,
! [A,B,C] :
( C = k2_zfmisc_1(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> ? [E,F] :
( r2_hidden(E,A)
& r2_hidden(F,B)
& D = k4_tarski(E,F) ) ) ) ).
fof(d3_zfmisc_1,axiom,
! [A,B,C] : k3_zfmisc_1(A,B,C) = k2_zfmisc_1(k2_zfmisc_1(A,B),C) ).
fof(d4_zfmisc_1,axiom,
! [A,B,C,D] : k4_zfmisc_1(A,B,C,D) = k2_zfmisc_1(k3_zfmisc_1(A,B,C),D) ).
fof(t1_zfmisc_1,axiom,
k1_zfmisc_1(k1_xboole_0) = k1_tarski(k1_xboole_0) ).
fof(t2_zfmisc_1,axiom,
k3_tarski(k1_xboole_0) = k1_xboole_0 ).
fof(t3_zfmisc_1,axiom,
$true ).
fof(t4_zfmisc_1,axiom,
$true ).
fof(t5_zfmisc_1,axiom,
$true ).
fof(t6_zfmisc_1,axiom,
! [A,B] :
( r1_tarski(k1_tarski(A),k1_tarski(B))
=> A = B ) ).
fof(t7_zfmisc_1,axiom,
$true ).
fof(t8_zfmisc_1,axiom,
! [A,B,C] :
( k1_tarski(A) = k2_tarski(B,C)
=> A = B ) ).
fof(t9_zfmisc_1,axiom,
! [A,B,C] :
( k1_tarski(A) = k2_tarski(B,C)
=> B = C ) ).
fof(t10_zfmisc_1,axiom,
! [A,B,C,D] :
~ ( k2_tarski(A,B) = k2_tarski(C,D)
& A != C
& A != D ) ).
fof(t11_zfmisc_1,axiom,
$true ).
fof(t12_zfmisc_1,axiom,
! [A,B] : r1_tarski(k1_tarski(A),k2_tarski(A,B)) ).
fof(t13_zfmisc_1,axiom,
! [A,B] :
( k2_xboole_0(k1_tarski(A),k1_tarski(B)) = k1_tarski(A)
=> A = B ) ).
fof(t14_zfmisc_1,axiom,
! [A,B] : k2_xboole_0(k1_tarski(A),k2_tarski(A,B)) = k2_tarski(A,B) ).
fof(t15_zfmisc_1,axiom,
$true ).
fof(t16_zfmisc_1,axiom,
! [A,B] :
~ ( r1_xboole_0(k1_tarski(A),k1_tarski(B))
& A = B ) ).
fof(t17_zfmisc_1,axiom,
! [A,B] :
( A != B
=> r1_xboole_0(k1_tarski(A),k1_tarski(B)) ) ).
fof(t18_zfmisc_1,axiom,
! [A,B] :
( k3_xboole_0(k1_tarski(A),k1_tarski(B)) = k1_tarski(A)
=> A = B ) ).
fof(t19_zfmisc_1,axiom,
! [A,B] : k3_xboole_0(k1_tarski(A),k2_tarski(A,B)) = k1_tarski(A) ).
fof(t20_zfmisc_1,axiom,
! [A,B] :
( k4_xboole_0(k1_tarski(A),k1_tarski(B)) = k1_tarski(A)
<=> A != B ) ).
fof(t21_zfmisc_1,axiom,
! [A,B] :
( k4_xboole_0(k1_tarski(A),k1_tarski(B)) = k1_xboole_0
=> A = B ) ).
fof(t22_zfmisc_1,axiom,
! [A,B] : k4_xboole_0(k1_tarski(A),k2_tarski(A,B)) = k1_xboole_0 ).
fof(t23_zfmisc_1,axiom,
! [A,B] :
( A != B
=> k4_xboole_0(k2_tarski(A,B),k1_tarski(B)) = k1_tarski(A) ) ).
fof(t24_zfmisc_1,axiom,
! [A,B] :
( r1_tarski(k1_tarski(A),k1_tarski(B))
=> A = B ) ).
fof(t25_zfmisc_1,axiom,
! [A,B,C] :
~ ( r1_tarski(k1_tarski(A),k2_tarski(B,C))
& A != B
& A != C ) ).
fof(t26_zfmisc_1,axiom,
! [A,B,C] :
( r1_tarski(k2_tarski(A,B),k1_tarski(C))
=> A = C ) ).
fof(t27_zfmisc_1,axiom,
! [A,B,C] :
( r1_tarski(k2_tarski(A,B),k1_tarski(C))
=> k2_tarski(A,B) = k1_tarski(C) ) ).
fof(t28_zfmisc_1,axiom,
! [A,B,C,D] :
~ ( r1_tarski(k2_tarski(A,B),k2_tarski(C,D))
& A != C
& A != D ) ).
fof(t29_zfmisc_1,axiom,
! [A,B] :
( A != B
=> k5_xboole_0(k1_tarski(A),k1_tarski(B)) = k2_tarski(A,B) ) ).
fof(t30_zfmisc_1,axiom,
! [A] : k1_zfmisc_1(k1_tarski(A)) = k2_tarski(k1_xboole_0,k1_tarski(A)) ).
fof(t31_zfmisc_1,axiom,
! [A] : k3_tarski(k1_tarski(A)) = A ).
fof(t32_zfmisc_1,axiom,
! [A,B] : k3_tarski(k2_tarski(k1_tarski(A),k1_tarski(B))) = k2_tarski(A,B) ).
fof(t33_zfmisc_1,axiom,
! [A,B,C,D] :
( k4_tarski(A,B) = k4_tarski(C,D)
=> ( A = C
& B = D ) ) ).
fof(t34_zfmisc_1,axiom,
! [A,B,C,D] :
( r2_hidden(k4_tarski(A,B),k2_zfmisc_1(k1_tarski(C),k1_tarski(D)))
<=> ( A = C
& B = D ) ) ).
fof(t35_zfmisc_1,axiom,
! [A,B] : k2_zfmisc_1(k1_tarski(A),k1_tarski(B)) = k1_tarski(k4_tarski(A,B)) ).
fof(t36_zfmisc_1,axiom,
! [A,B,C] :
( k2_zfmisc_1(k1_tarski(A),k2_tarski(B,C)) = k2_tarski(k4_tarski(A,B),k4_tarski(A,C))
& k2_zfmisc_1(k2_tarski(A,B),k1_tarski(C)) = k2_tarski(k4_tarski(A,C),k4_tarski(B,C)) ) ).
fof(t37_zfmisc_1,axiom,
! [A,B] :
( r1_tarski(k1_tarski(A),B)
<=> r2_hidden(A,B) ) ).
fof(t38_zfmisc_1,axiom,
! [A,B,C] :
( r1_tarski(k2_tarski(A,B),C)
<=> ( r2_hidden(A,C)
& r2_hidden(B,C) ) ) ).
fof(t39_zfmisc_1,axiom,
! [A,B] :
( r1_tarski(A,k1_tarski(B))
<=> ( A = k1_xboole_0
| A = k1_tarski(B) ) ) ).
fof(t40_zfmisc_1,axiom,
! [A,B,C] :
( r1_tarski(A,B)
=> ( r2_hidden(C,A)
| r1_tarski(A,k4_xboole_0(B,k1_tarski(C))) ) ) ).
fof(t41_zfmisc_1,axiom,
! [A,B] :
~ ( A != k1_tarski(B)
& A != k1_xboole_0
& ! [C] :
~ ( r2_hidden(C,A)
& C != B ) ) ).
fof(t42_zfmisc_1,axiom,
! [A,B,C] :
( r1_tarski(A,k2_tarski(B,C))
<=> ~ ( A != k1_xboole_0
& A != k1_tarski(B)
& A != k1_tarski(C)
& A != k2_tarski(B,C) ) ) ).
fof(t43_zfmisc_1,axiom,
! [A,B,C] :
~ ( k1_tarski(A) = k2_xboole_0(B,C)
& ~ ( B = k1_tarski(A)
& C = k1_tarski(A) )
& ~ ( B = k1_xboole_0
& C = k1_tarski(A) )
& ~ ( B = k1_tarski(A)
& C = k1_xboole_0 ) ) ).
fof(t44_zfmisc_1,axiom,
! [A,B,C] :
~ ( k1_tarski(A) = k2_xboole_0(B,C)
& B != C
& B != k1_xboole_0
& C != k1_xboole_0 ) ).
fof(t45_zfmisc_1,axiom,
! [A,B] :
( r1_tarski(k2_xboole_0(k1_tarski(A),B),B)
=> r2_hidden(A,B) ) ).
fof(t46_zfmisc_1,axiom,
! [A,B] :
( r2_hidden(A,B)
=> k2_xboole_0(k1_tarski(A),B) = B ) ).
fof(t47_zfmisc_1,axiom,
! [A,B,C] :
( r1_tarski(k2_xboole_0(k2_tarski(A,B),C),C)
=> r2_hidden(A,C) ) ).
fof(t48_zfmisc_1,axiom,
! [A,B,C] :
( ( r2_hidden(A,B)
& r2_hidden(C,B) )
=> k2_xboole_0(k2_tarski(A,C),B) = B ) ).
fof(t49_zfmisc_1,axiom,
! [A,B] : k2_xboole_0(k1_tarski(A),B) != k1_xboole_0 ).
fof(t50_zfmisc_1,axiom,
! [A,B,C] : k2_xboole_0(k2_tarski(A,B),C) != k1_xboole_0 ).
fof(t51_zfmisc_1,axiom,
! [A,B] :
( k3_xboole_0(A,k1_tarski(B)) = k1_tarski(B)
=> r2_hidden(B,A) ) ).
fof(t52_zfmisc_1,axiom,
! [A,B] :
( r2_hidden(A,B)
=> k3_xboole_0(B,k1_tarski(A)) = k1_tarski(A) ) ).
fof(t53_zfmisc_1,axiom,
! [A,B,C] :
( ( r2_hidden(A,B)
& r2_hidden(C,B) )
=> k3_xboole_0(k2_tarski(A,C),B) = k2_tarski(A,C) ) ).
fof(t54_zfmisc_1,axiom,
! [A,B] :
~ ( r1_xboole_0(k1_tarski(A),B)
& r2_hidden(A,B) ) ).
fof(t55_zfmisc_1,axiom,
! [A,B,C] :
~ ( r1_xboole_0(k2_tarski(A,B),C)
& r2_hidden(A,C) ) ).
fof(t56_zfmisc_1,axiom,
! [A,B] :
( ~ r2_hidden(A,B)
=> r1_xboole_0(k1_tarski(A),B) ) ).
fof(t57_zfmisc_1,axiom,
! [A,B,C] :
~ ( ~ r2_hidden(A,B)
& ~ r2_hidden(C,B)
& ~ r1_xboole_0(k2_tarski(A,C),B) ) ).
fof(t58_zfmisc_1,axiom,
! [A,B] :
( r1_xboole_0(k1_tarski(A),B)
| k3_xboole_0(k1_tarski(A),B) = k1_tarski(A) ) ).
fof(t59_zfmisc_1,axiom,
! [A,B,C] :
~ ( k3_xboole_0(k2_tarski(A,B),C) = k1_tarski(A)
& r2_hidden(B,C)
& A != B ) ).
fof(t60_zfmisc_1,axiom,
! [A,B,C] :
( r2_hidden(A,B)
=> ( ( r2_hidden(C,B)
& A != C )
| k3_xboole_0(k2_tarski(A,C),B) = k1_tarski(A) ) ) ).
fof(t61_zfmisc_1,axiom,
$true ).
fof(t62_zfmisc_1,axiom,
$true ).
fof(t63_zfmisc_1,axiom,
! [A,B,C] :
( k3_xboole_0(k2_tarski(A,B),C) = k2_tarski(A,B)
=> r2_hidden(A,C) ) ).
fof(t64_zfmisc_1,axiom,
! [A,B,C] :
( r2_hidden(A,k4_xboole_0(B,k1_tarski(C)))
<=> ( r2_hidden(A,B)
& A != C ) ) ).
fof(t65_zfmisc_1,axiom,
! [A,B] :
( k4_xboole_0(A,k1_tarski(B)) = A
<=> ~ r2_hidden(B,A) ) ).
fof(t66_zfmisc_1,axiom,
! [A,B] :
~ ( k4_xboole_0(A,k1_tarski(B)) = k1_xboole_0
& A != k1_xboole_0
& A != k1_tarski(B) ) ).
fof(t67_zfmisc_1,axiom,
! [A,B] :
( k4_xboole_0(k1_tarski(A),B) = k1_tarski(A)
<=> ~ r2_hidden(A,B) ) ).
fof(t68_zfmisc_1,axiom,
! [A,B] :
( k4_xboole_0(k1_tarski(A),B) = k1_xboole_0
<=> r2_hidden(A,B) ) ).
fof(t69_zfmisc_1,axiom,
! [A,B] :
( k4_xboole_0(k1_tarski(A),B) = k1_xboole_0
| k4_xboole_0(k1_tarski(A),B) = k1_tarski(A) ) ).
fof(t70_zfmisc_1,axiom,
! [A,B,C] :
( k4_xboole_0(k2_tarski(A,B),C) = k1_tarski(A)
<=> ( ~ r2_hidden(A,C)
& ( r2_hidden(B,C)
| A = B ) ) ) ).
fof(t71_zfmisc_1,axiom,
$true ).
fof(t72_zfmisc_1,axiom,
! [A,B,C] :
( k4_xboole_0(k2_tarski(A,B),C) = k2_tarski(A,B)
<=> ( ~ r2_hidden(A,C)
& ~ r2_hidden(B,C) ) ) ).
fof(t73_zfmisc_1,axiom,
! [A,B,C] :
( k4_xboole_0(k2_tarski(A,B),C) = k1_xboole_0
<=> ( r2_hidden(A,C)
& r2_hidden(B,C) ) ) ).
fof(t74_zfmisc_1,axiom,
! [A,B,C] :
~ ( k4_xboole_0(k2_tarski(A,B),C) != k1_xboole_0
& k4_xboole_0(k2_tarski(A,B),C) != k1_tarski(A)
& k4_xboole_0(k2_tarski(A,B),C) != k1_tarski(B)
& k4_xboole_0(k2_tarski(A,B),C) != k2_tarski(A,B) ) ).
fof(t75_zfmisc_1,axiom,
! [A,B,C] :
( k4_xboole_0(A,k2_tarski(B,C)) = k1_xboole_0
<=> ~ ( A != k1_xboole_0
& A != k1_tarski(B)
& A != k1_tarski(C)
& A != k2_tarski(B,C) ) ) ).
fof(t76_zfmisc_1,axiom,
$true ).
fof(t77_zfmisc_1,axiom,
$true ).
fof(t78_zfmisc_1,axiom,
$true ).
fof(t79_zfmisc_1,axiom,
! [A,B] :
( r1_tarski(A,B)
=> r1_tarski(k1_zfmisc_1(A),k1_zfmisc_1(B)) ) ).
fof(t80_zfmisc_1,axiom,
! [A] : r1_tarski(k1_tarski(A),k1_zfmisc_1(A)) ).
fof(t81_zfmisc_1,axiom,
! [A,B] : r1_tarski(k2_xboole_0(k1_zfmisc_1(A),k1_zfmisc_1(B)),k1_zfmisc_1(k2_xboole_0(A,B))) ).
fof(t82_zfmisc_1,axiom,
! [A,B] :
( k2_xboole_0(k1_zfmisc_1(A),k1_zfmisc_1(B)) = k1_zfmisc_1(k2_xboole_0(A,B))
=> r3_xboole_0(A,B) ) ).
fof(t83_zfmisc_1,axiom,
! [A,B] : k1_zfmisc_1(k3_xboole_0(A,B)) = k3_xboole_0(k1_zfmisc_1(A),k1_zfmisc_1(B)) ).
fof(t84_zfmisc_1,axiom,
! [A,B] : r1_tarski(k1_zfmisc_1(k4_xboole_0(A,B)),k2_xboole_0(k1_tarski(k1_xboole_0),k4_xboole_0(k1_zfmisc_1(A),k1_zfmisc_1(B)))) ).
fof(t85_zfmisc_1,axiom,
$true ).
fof(t86_zfmisc_1,axiom,
! [A,B] : r1_tarski(k2_xboole_0(k1_zfmisc_1(k4_xboole_0(A,B)),k1_zfmisc_1(k4_xboole_0(B,A))),k1_zfmisc_1(k5_xboole_0(A,B))) ).
fof(t87_zfmisc_1,axiom,
$true ).
fof(t88_zfmisc_1,axiom,
$true ).
fof(t89_zfmisc_1,axiom,
$true ).
fof(t90_zfmisc_1,axiom,
$true ).
fof(t91_zfmisc_1,axiom,
$true ).
fof(t92_zfmisc_1,axiom,
! [A,B] :
( r2_hidden(A,B)
=> r1_tarski(A,k3_tarski(B)) ) ).
fof(t93_zfmisc_1,axiom,
! [A,B] : k3_tarski(k2_tarski(A,B)) = k2_xboole_0(A,B) ).
fof(t94_zfmisc_1,axiom,
! [A,B] :
( ! [C] :
( r2_hidden(C,A)
=> r1_tarski(C,B) )
=> r1_tarski(k3_tarski(A),B) ) ).
fof(t95_zfmisc_1,axiom,
! [A,B] :
( r1_tarski(A,B)
=> r1_tarski(k3_tarski(A),k3_tarski(B)) ) ).
fof(t96_zfmisc_1,axiom,
! [A,B] : k3_tarski(k2_xboole_0(A,B)) = k2_xboole_0(k3_tarski(A),k3_tarski(B)) ).
fof(t97_zfmisc_1,axiom,
! [A,B] : r1_tarski(k3_tarski(k3_xboole_0(A,B)),k3_xboole_0(k3_tarski(A),k3_tarski(B))) ).
fof(t98_zfmisc_1,axiom,
! [A,B] :
( ! [C] :
( r2_hidden(C,A)
=> r1_xboole_0(C,B) )
=> r1_xboole_0(k3_tarski(A),B) ) ).
fof(t99_zfmisc_1,axiom,
! [A] : k3_tarski(k1_zfmisc_1(A)) = A ).
fof(t100_zfmisc_1,axiom,
! [A] : r1_tarski(A,k1_zfmisc_1(k3_tarski(A))) ).
fof(t101_zfmisc_1,axiom,
! [A,B] :
( ! [C,D] :
( ( r2_hidden(C,k2_xboole_0(A,B))
& r2_hidden(D,k2_xboole_0(A,B)) )
=> ( C = D
| r1_xboole_0(C,D) ) )
=> k3_tarski(k3_xboole_0(A,B)) = k3_xboole_0(k3_tarski(A),k3_tarski(B)) ) ).
fof(t102_zfmisc_1,axiom,
! [A,B,C] :
~ ( r2_hidden(A,k2_zfmisc_1(B,C))
& ! [D,E] : k4_tarski(D,E) != A ) ).
fof(t103_zfmisc_1,axiom,
! [A,B,C,D] :
~ ( r1_tarski(A,k2_zfmisc_1(B,C))
& r2_hidden(D,A)
& ! [E,F] :
~ ( r2_hidden(E,B)
& r2_hidden(F,C)
& D = k4_tarski(E,F) ) ) ).
fof(t104_zfmisc_1,axiom,
! [A,B,C,D,E] :
~ ( r2_hidden(A,k3_xboole_0(k2_zfmisc_1(B,C),k2_zfmisc_1(D,E)))
& ! [F,G] :
~ ( A = k4_tarski(F,G)
& r2_hidden(F,k3_xboole_0(B,D))
& r2_hidden(G,k3_xboole_0(C,E)) ) ) ).
fof(t105_zfmisc_1,axiom,
! [A,B] : r1_tarski(k2_zfmisc_1(A,B),k1_zfmisc_1(k1_zfmisc_1(k2_xboole_0(A,B)))) ).
fof(t106_zfmisc_1,axiom,
! [A,B,C,D] :
( r2_hidden(k4_tarski(A,B),k2_zfmisc_1(C,D))
<=> ( r2_hidden(A,C)
& r2_hidden(B,D) ) ) ).
fof(t107_zfmisc_1,axiom,
! [A,B,C,D] :
( r2_hidden(k4_tarski(A,B),k2_zfmisc_1(C,D))
=> r2_hidden(k4_tarski(B,A),k2_zfmisc_1(D,C)) ) ).
fof(t108_zfmisc_1,axiom,
! [A,B,C,D] :
( ! [E,F] :
( r2_hidden(k4_tarski(E,F),k2_zfmisc_1(A,B))
<=> r2_hidden(k4_tarski(E,F),k2_zfmisc_1(C,D)) )
=> k2_zfmisc_1(A,B) = k2_zfmisc_1(C,D) ) ).
fof(t109_zfmisc_1,axiom,
! [A,B,C,D] :
( ( r1_tarski(A,k2_zfmisc_1(B,C))
& ! [E,F] :
( r2_hidden(k4_tarski(E,F),A)
=> r2_hidden(k4_tarski(E,F),D) ) )
=> r1_tarski(A,D) ) ).
fof(t110_zfmisc_1,axiom,
! [A,B,C,D,E,F] :
( ( r1_tarski(A,k2_zfmisc_1(B,C))
& r1_tarski(D,k2_zfmisc_1(E,F))
& ! [G,H] :
( r2_hidden(k4_tarski(G,H),A)
<=> r2_hidden(k4_tarski(G,H),D) ) )
=> A = D ) ).
fof(t111_zfmisc_1,axiom,
! [A,B] :
( ( ! [C] :
~ ( r2_hidden(C,A)
& ! [D,E] : C != k4_tarski(D,E) )
& ! [C,D] :
( r2_hidden(k4_tarski(C,D),A)
=> r2_hidden(k4_tarski(C,D),B) ) )
=> r1_tarski(A,B) ) ).
fof(t112_zfmisc_1,axiom,
! [A,B] :
( ( ! [C] :
~ ( r2_hidden(C,A)
& ! [D,E] : C != k4_tarski(D,E) )
& ! [C] :
~ ( r2_hidden(C,B)
& ! [D,E] : C != k4_tarski(D,E) )
& ! [C,D] :
( r2_hidden(k4_tarski(C,D),A)
<=> r2_hidden(k4_tarski(C,D),B) ) )
=> A = B ) ).
fof(t113_zfmisc_1,axiom,
! [A,B] :
( k2_zfmisc_1(A,B) = k1_xboole_0
<=> ( A = k1_xboole_0
| B = k1_xboole_0 ) ) ).
fof(t114_zfmisc_1,axiom,
! [A,B] :
( k2_zfmisc_1(A,B) = k2_zfmisc_1(B,A)
=> ( A = k1_xboole_0
| B = k1_xboole_0
| A = B ) ) ).
fof(t115_zfmisc_1,axiom,
! [A,B] :
( k2_zfmisc_1(A,A) = k2_zfmisc_1(B,B)
=> A = B ) ).
fof(t116_zfmisc_1,axiom,
! [A] :
( r1_tarski(A,k2_zfmisc_1(A,A))
=> A = k1_xboole_0 ) ).
fof(t117_zfmisc_1,axiom,
! [A,B,C] :
~ ( A != k1_xboole_0
& ( r1_tarski(k2_zfmisc_1(B,A),k2_zfmisc_1(C,A))
| r1_tarski(k2_zfmisc_1(A,B),k2_zfmisc_1(A,C)) )
& ~ r1_tarski(B,C) ) ).
fof(t118_zfmisc_1,axiom,
! [A,B,C] :
( r1_tarski(A,B)
=> ( r1_tarski(k2_zfmisc_1(A,C),k2_zfmisc_1(B,C))
& r1_tarski(k2_zfmisc_1(C,A),k2_zfmisc_1(C,B)) ) ) ).
fof(t119_zfmisc_1,axiom,
! [A,B,C,D] :
( ( r1_tarski(A,B)
& r1_tarski(C,D) )
=> r1_tarski(k2_zfmisc_1(A,C),k2_zfmisc_1(B,D)) ) ).
fof(t120_zfmisc_1,axiom,
! [A,B,C] :
( k2_zfmisc_1(k2_xboole_0(A,B),C) = k2_xboole_0(k2_zfmisc_1(A,C),k2_zfmisc_1(B,C))
& k2_zfmisc_1(C,k2_xboole_0(A,B)) = k2_xboole_0(k2_zfmisc_1(C,A),k2_zfmisc_1(C,B)) ) ).
fof(t121_zfmisc_1,axiom,
! [A,B,C,D] : k2_zfmisc_1(k2_xboole_0(A,B),k2_xboole_0(C,D)) = k2_xboole_0(k2_xboole_0(k2_xboole_0(k2_zfmisc_1(A,C),k2_zfmisc_1(A,D)),k2_zfmisc_1(B,C)),k2_zfmisc_1(B,D)) ).
fof(t122_zfmisc_1,axiom,
! [A,B,C] :
( k2_zfmisc_1(k3_xboole_0(A,B),C) = k3_xboole_0(k2_zfmisc_1(A,C),k2_zfmisc_1(B,C))
& k2_zfmisc_1(C,k3_xboole_0(A,B)) = k3_xboole_0(k2_zfmisc_1(C,A),k2_zfmisc_1(C,B)) ) ).
fof(t123_zfmisc_1,axiom,
! [A,B,C,D] : k2_zfmisc_1(k3_xboole_0(A,B),k3_xboole_0(C,D)) = k3_xboole_0(k2_zfmisc_1(A,C),k2_zfmisc_1(B,D)) ).
fof(t124_zfmisc_1,axiom,
! [A,B,C,D] :
( ( r1_tarski(A,B)
& r1_tarski(C,D) )
=> k3_xboole_0(k2_zfmisc_1(A,D),k2_zfmisc_1(B,C)) = k2_zfmisc_1(A,C) ) ).
fof(t125_zfmisc_1,axiom,
! [A,B,C] :
( k2_zfmisc_1(k4_xboole_0(A,B),C) = k4_xboole_0(k2_zfmisc_1(A,C),k2_zfmisc_1(B,C))
& k2_zfmisc_1(C,k4_xboole_0(A,B)) = k4_xboole_0(k2_zfmisc_1(C,A),k2_zfmisc_1(C,B)) ) ).
fof(t126_zfmisc_1,axiom,
! [A,B,C,D] : k4_xboole_0(k2_zfmisc_1(A,B),k2_zfmisc_1(C,D)) = k2_xboole_0(k2_zfmisc_1(k4_xboole_0(A,C),B),k2_zfmisc_1(A,k4_xboole_0(B,D))) ).
fof(t127_zfmisc_1,axiom,
! [A,B,C,D] :
( ( r1_xboole_0(A,B)
| r1_xboole_0(C,D) )
=> r1_xboole_0(k2_zfmisc_1(A,C),k2_zfmisc_1(B,D)) ) ).
fof(t128_zfmisc_1,axiom,
! [A,B,C,D] :
( r2_hidden(k4_tarski(A,B),k2_zfmisc_1(k1_tarski(C),D))
<=> ( A = C
& r2_hidden(B,D) ) ) ).
fof(t129_zfmisc_1,axiom,
! [A,B,C,D] :
( r2_hidden(k4_tarski(A,B),k2_zfmisc_1(C,k1_tarski(D)))
<=> ( r2_hidden(A,C)
& B = D ) ) ).
fof(t130_zfmisc_1,axiom,
! [A,B] :
( A != k1_xboole_0
=> ( k2_zfmisc_1(k1_tarski(B),A) != k1_xboole_0
& k2_zfmisc_1(A,k1_tarski(B)) != k1_xboole_0 ) ) ).
fof(t131_zfmisc_1,axiom,
! [A,B,C,D] :
( A != B
=> ( r1_xboole_0(k2_zfmisc_1(k1_tarski(A),C),k2_zfmisc_1(k1_tarski(B),D))
& r1_xboole_0(k2_zfmisc_1(C,k1_tarski(A)),k2_zfmisc_1(D,k1_tarski(B))) ) ) ).
fof(t132_zfmisc_1,axiom,
! [A,B,C] :
( k2_zfmisc_1(k2_tarski(A,B),C) = k2_xboole_0(k2_zfmisc_1(k1_tarski(A),C),k2_zfmisc_1(k1_tarski(B),C))
& k2_zfmisc_1(C,k2_tarski(A,B)) = k2_xboole_0(k2_zfmisc_1(C,k1_tarski(A)),k2_zfmisc_1(C,k1_tarski(B))) ) ).
fof(t133_zfmisc_1,axiom,
$true ).
fof(t134_zfmisc_1,axiom,
! [A,B,C,D] :
( k2_zfmisc_1(A,B) = k2_zfmisc_1(C,D)
=> ( A = k1_xboole_0
| B = k1_xboole_0
| ( A = C
& B = D ) ) ) ).
fof(t135_zfmisc_1,axiom,
! [A,B] :
( ( r1_tarski(A,k2_zfmisc_1(A,B))
| r1_tarski(A,k2_zfmisc_1(B,A)) )
=> A = k1_xboole_0 ) ).
fof(t136_zfmisc_1,axiom,
! [A] :
? [B] :
( r2_hidden(A,B)
& ! [C,D] :
( ( r2_hidden(C,B)
& r1_tarski(D,C) )
=> r2_hidden(D,B) )
& ! [C] :
( r2_hidden(C,B)
=> r2_hidden(k1_zfmisc_1(C),B) )
& ! [C] :
~ ( r1_tarski(C,B)
& ~ r2_tarski(C,B)
& ~ r2_hidden(C,B) ) ) ).
fof(t137_zfmisc_1,axiom,
! [A,B,C,D,E] :
( ( r2_hidden(A,k2_zfmisc_1(B,C))
& r2_hidden(A,k2_zfmisc_1(D,E)) )
=> r2_hidden(A,k2_zfmisc_1(k3_xboole_0(B,D),k3_xboole_0(C,E))) ) ).
fof(t138_zfmisc_1,axiom,
! [A,B,C,D] :
( r1_tarski(k2_zfmisc_1(A,B),k2_zfmisc_1(C,D))
=> ( k2_zfmisc_1(A,B) = k1_xboole_0
| ( r1_tarski(A,C)
& r1_tarski(B,D) ) ) ) ).
fof(t139_zfmisc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B,C,D] :
( ( r1_tarski(k2_zfmisc_1(A,B),k2_zfmisc_1(C,D))
| r1_tarski(k2_zfmisc_1(B,A),k2_zfmisc_1(D,C)) )
=> r1_tarski(B,D) ) ) ).
fof(t140_zfmisc_1,axiom,
! [A,B] :
( r2_hidden(A,B)
=> k2_xboole_0(k4_xboole_0(B,k1_tarski(A)),k1_tarski(A)) = B ) ).
fof(t141_zfmisc_1,axiom,
! [A,B] :
( ~ r2_hidden(A,B)
=> k4_xboole_0(k2_xboole_0(B,k1_tarski(A)),k1_tarski(A)) = B ) ).
fof(dt_k1_zfmisc_1,axiom,
$true ).
fof(dt_k2_zfmisc_1,axiom,
$true ).
fof(dt_k3_zfmisc_1,axiom,
$true ).
fof(dt_k4_zfmisc_1,axiom,
$true ).
%------------------------------------------------------------------------------