SET007 Axioms: SET007+4.ax
%------------------------------------------------------------------------------
% File : SET007+4 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Boolean Properties of Sets - Theorems
% 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 : xboole_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 114 ( 50 unt; 0 def)
% Number of atoms : 225 ( 63 equ)
% Maximal formula atoms : 6 ( 1 avg)
% Number of connectives : 140 ( 29 ~; 0 |; 54 &)
% ( 4 <=>; 53 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 5 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 6 ( 5 usr; 0 prp; 1-2 aty)
% Number of functors : 5 ( 5 usr; 1 con; 0-2 aty)
% Number of variables : 297 ( 297 !; 0 ?)
% 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(t1_xboole_1,axiom,
! [A,B,C] :
( ( r1_tarski(A,B)
& r1_tarski(B,C) )
=> r1_tarski(A,C) ) ).
fof(t2_xboole_1,axiom,
! [A] : r1_tarski(k1_xboole_0,A) ).
fof(t3_xboole_1,axiom,
! [A] :
( r1_tarski(A,k1_xboole_0)
=> A = k1_xboole_0 ) ).
fof(t4_xboole_1,axiom,
! [A,B,C] : k2_xboole_0(k2_xboole_0(A,B),C) = k2_xboole_0(A,k2_xboole_0(B,C)) ).
fof(t5_xboole_1,axiom,
! [A,B,C] : k2_xboole_0(k2_xboole_0(A,B),C) = k2_xboole_0(k2_xboole_0(A,C),k2_xboole_0(B,C)) ).
fof(t6_xboole_1,axiom,
! [A,B] : k2_xboole_0(A,k2_xboole_0(A,B)) = k2_xboole_0(A,B) ).
fof(t7_xboole_1,axiom,
! [A,B] : r1_tarski(A,k2_xboole_0(A,B)) ).
fof(t8_xboole_1,axiom,
! [A,B,C] :
( ( r1_tarski(A,B)
& r1_tarski(C,B) )
=> r1_tarski(k2_xboole_0(A,C),B) ) ).
fof(t9_xboole_1,axiom,
! [A,B,C] :
( r1_tarski(A,B)
=> r1_tarski(k2_xboole_0(A,C),k2_xboole_0(B,C)) ) ).
fof(t10_xboole_1,axiom,
! [A,B,C] :
( r1_tarski(A,B)
=> r1_tarski(A,k2_xboole_0(C,B)) ) ).
fof(t11_xboole_1,axiom,
! [A,B,C] :
( r1_tarski(k2_xboole_0(A,B),C)
=> r1_tarski(A,C) ) ).
fof(t12_xboole_1,axiom,
! [A,B] :
( r1_tarski(A,B)
=> k2_xboole_0(A,B) = B ) ).
fof(t13_xboole_1,axiom,
! [A,B,C,D] :
( ( r1_tarski(A,B)
& r1_tarski(C,D) )
=> r1_tarski(k2_xboole_0(A,C),k2_xboole_0(B,D)) ) ).
fof(t14_xboole_1,axiom,
! [A,B,C] :
( ( r1_tarski(A,B)
& r1_tarski(C,B)
& ! [D] :
( ( r1_tarski(A,D)
& r1_tarski(C,D) )
=> r1_tarski(B,D) ) )
=> B = k2_xboole_0(A,C) ) ).
fof(t15_xboole_1,axiom,
! [A,B] :
( k2_xboole_0(A,B) = k1_xboole_0
=> A = k1_xboole_0 ) ).
fof(t16_xboole_1,axiom,
! [A,B,C] : k3_xboole_0(k3_xboole_0(A,B),C) = k3_xboole_0(A,k3_xboole_0(B,C)) ).
fof(t17_xboole_1,axiom,
! [A,B] : r1_tarski(k3_xboole_0(A,B),A) ).
fof(t18_xboole_1,axiom,
! [A,B,C] :
( r1_tarski(A,k3_xboole_0(B,C))
=> r1_tarski(A,B) ) ).
fof(t19_xboole_1,axiom,
! [A,B,C] :
( ( r1_tarski(A,B)
& r1_tarski(A,C) )
=> r1_tarski(A,k3_xboole_0(B,C)) ) ).
fof(t20_xboole_1,axiom,
! [A,B,C] :
( ( r1_tarski(A,B)
& r1_tarski(A,C)
& ! [D] :
( ( r1_tarski(D,B)
& r1_tarski(D,C) )
=> r1_tarski(D,A) ) )
=> A = k3_xboole_0(B,C) ) ).
fof(t21_xboole_1,axiom,
! [A,B] : k3_xboole_0(A,k2_xboole_0(A,B)) = A ).
fof(t22_xboole_1,axiom,
! [A,B] : k2_xboole_0(A,k3_xboole_0(A,B)) = A ).
fof(t23_xboole_1,axiom,
! [A,B,C] : k3_xboole_0(A,k2_xboole_0(B,C)) = k2_xboole_0(k3_xboole_0(A,B),k3_xboole_0(A,C)) ).
fof(t24_xboole_1,axiom,
! [A,B,C] : k2_xboole_0(A,k3_xboole_0(B,C)) = k3_xboole_0(k2_xboole_0(A,B),k2_xboole_0(A,C)) ).
fof(t25_xboole_1,axiom,
! [A,B,C] : k2_xboole_0(k2_xboole_0(k3_xboole_0(A,B),k3_xboole_0(B,C)),k3_xboole_0(C,A)) = k3_xboole_0(k3_xboole_0(k2_xboole_0(A,B),k2_xboole_0(B,C)),k2_xboole_0(C,A)) ).
fof(t26_xboole_1,axiom,
! [A,B,C] :
( r1_tarski(A,B)
=> r1_tarski(k3_xboole_0(A,C),k3_xboole_0(B,C)) ) ).
fof(t27_xboole_1,axiom,
! [A,B,C,D] :
( ( r1_tarski(A,B)
& r1_tarski(C,D) )
=> r1_tarski(k3_xboole_0(A,C),k3_xboole_0(B,D)) ) ).
fof(t28_xboole_1,axiom,
! [A,B] :
( r1_tarski(A,B)
=> k3_xboole_0(A,B) = A ) ).
fof(t29_xboole_1,axiom,
! [A,B,C] : r1_tarski(k3_xboole_0(A,B),k2_xboole_0(A,C)) ).
fof(t30_xboole_1,axiom,
! [A,B,C] :
( r1_tarski(A,B)
=> k2_xboole_0(A,k3_xboole_0(C,B)) = k3_xboole_0(k2_xboole_0(A,C),B) ) ).
fof(t31_xboole_1,axiom,
! [A,B,C] : r1_tarski(k2_xboole_0(k3_xboole_0(A,B),k3_xboole_0(A,C)),k2_xboole_0(B,C)) ).
fof(t32_xboole_1,axiom,
! [A,B] :
( k4_xboole_0(A,B) = k4_xboole_0(B,A)
=> A = B ) ).
fof(t33_xboole_1,axiom,
! [A,B,C] :
( r1_tarski(A,B)
=> r1_tarski(k4_xboole_0(A,C),k4_xboole_0(B,C)) ) ).
fof(t34_xboole_1,axiom,
! [A,B,C] :
( r1_tarski(A,B)
=> r1_tarski(k4_xboole_0(C,B),k4_xboole_0(C,A)) ) ).
fof(t35_xboole_1,axiom,
! [A,B,C,D] :
( ( r1_tarski(A,B)
& r1_tarski(C,D) )
=> r1_tarski(k4_xboole_0(A,D),k4_xboole_0(B,C)) ) ).
fof(t36_xboole_1,axiom,
! [A,B] : r1_tarski(k4_xboole_0(A,B),A) ).
fof(t37_xboole_1,axiom,
! [A,B] :
( k4_xboole_0(A,B) = k1_xboole_0
<=> r1_tarski(A,B) ) ).
fof(t38_xboole_1,axiom,
! [A,B] :
( r1_tarski(A,k4_xboole_0(B,A))
=> A = k1_xboole_0 ) ).
fof(t39_xboole_1,axiom,
! [A,B] : k2_xboole_0(A,k4_xboole_0(B,A)) = k2_xboole_0(A,B) ).
fof(t40_xboole_1,axiom,
! [A,B] : k4_xboole_0(k2_xboole_0(A,B),B) = k4_xboole_0(A,B) ).
fof(t41_xboole_1,axiom,
! [A,B,C] : k4_xboole_0(k4_xboole_0(A,B),C) = k4_xboole_0(A,k2_xboole_0(B,C)) ).
fof(t42_xboole_1,axiom,
! [A,B,C] : k4_xboole_0(k2_xboole_0(A,B),C) = k2_xboole_0(k4_xboole_0(A,C),k4_xboole_0(B,C)) ).
fof(t43_xboole_1,axiom,
! [A,B,C] :
( r1_tarski(A,k2_xboole_0(B,C))
=> r1_tarski(k4_xboole_0(A,B),C) ) ).
fof(t44_xboole_1,axiom,
! [A,B,C] :
( r1_tarski(k4_xboole_0(A,B),C)
=> r1_tarski(A,k2_xboole_0(B,C)) ) ).
fof(t45_xboole_1,axiom,
! [A,B] :
( r1_tarski(A,B)
=> B = k2_xboole_0(A,k4_xboole_0(B,A)) ) ).
fof(t46_xboole_1,axiom,
! [A,B] : k4_xboole_0(A,k2_xboole_0(A,B)) = k1_xboole_0 ).
fof(t47_xboole_1,axiom,
! [A,B] : k4_xboole_0(A,k3_xboole_0(A,B)) = k4_xboole_0(A,B) ).
fof(t48_xboole_1,axiom,
! [A,B] : k4_xboole_0(A,k4_xboole_0(A,B)) = k3_xboole_0(A,B) ).
fof(t49_xboole_1,axiom,
! [A,B,C] : k3_xboole_0(A,k4_xboole_0(B,C)) = k4_xboole_0(k3_xboole_0(A,B),C) ).
fof(t50_xboole_1,axiom,
! [A,B,C] : k3_xboole_0(A,k4_xboole_0(B,C)) = k4_xboole_0(k3_xboole_0(A,B),k3_xboole_0(A,C)) ).
fof(t51_xboole_1,axiom,
! [A,B] : k2_xboole_0(k3_xboole_0(A,B),k4_xboole_0(A,B)) = A ).
fof(t52_xboole_1,axiom,
! [A,B,C] : k4_xboole_0(A,k4_xboole_0(B,C)) = k2_xboole_0(k4_xboole_0(A,B),k3_xboole_0(A,C)) ).
fof(t53_xboole_1,axiom,
! [A,B,C] : k4_xboole_0(A,k2_xboole_0(B,C)) = k3_xboole_0(k4_xboole_0(A,B),k4_xboole_0(A,C)) ).
fof(t54_xboole_1,axiom,
! [A,B,C] : k4_xboole_0(A,k3_xboole_0(B,C)) = k2_xboole_0(k4_xboole_0(A,B),k4_xboole_0(A,C)) ).
fof(t55_xboole_1,axiom,
! [A,B] : k4_xboole_0(k2_xboole_0(A,B),k3_xboole_0(A,B)) = k2_xboole_0(k4_xboole_0(A,B),k4_xboole_0(B,A)) ).
fof(t56_xboole_1,axiom,
! [A,B,C] :
( ( r2_xboole_0(A,B)
& r2_xboole_0(B,C) )
=> r2_xboole_0(A,C) ) ).
fof(t57_xboole_1,axiom,
! [A,B] :
~ ( r2_xboole_0(A,B)
& r2_xboole_0(B,A) ) ).
fof(t58_xboole_1,axiom,
! [A,B,C] :
( ( r2_xboole_0(A,B)
& r1_tarski(B,C) )
=> r2_xboole_0(A,C) ) ).
fof(t59_xboole_1,axiom,
! [A,B,C] :
( ( r1_tarski(A,B)
& r2_xboole_0(B,C) )
=> r2_xboole_0(A,C) ) ).
fof(t60_xboole_1,axiom,
! [A,B] :
~ ( r1_tarski(A,B)
& r2_xboole_0(B,A) ) ).
fof(t61_xboole_1,axiom,
! [A] :
( A != k1_xboole_0
=> r2_xboole_0(k1_xboole_0,A) ) ).
fof(t62_xboole_1,axiom,
! [A] : ~ r2_xboole_0(A,k1_xboole_0) ).
fof(t63_xboole_1,axiom,
! [A,B,C] :
( ( r1_tarski(A,B)
& r1_xboole_0(B,C) )
=> r1_xboole_0(A,C) ) ).
fof(t64_xboole_1,axiom,
! [A,B,C,D] :
( ( r1_tarski(A,B)
& r1_tarski(C,D)
& r1_xboole_0(B,D) )
=> r1_xboole_0(A,C) ) ).
fof(t65_xboole_1,axiom,
! [A] : r1_xboole_0(A,k1_xboole_0) ).
fof(t66_xboole_1,axiom,
! [A] :
( ~ ( ~ r1_xboole_0(A,A)
& A = k1_xboole_0 )
& ~ ( A != k1_xboole_0
& r1_xboole_0(A,A) ) ) ).
fof(t67_xboole_1,axiom,
! [A,B,C] :
( ( r1_tarski(A,B)
& r1_tarski(A,C)
& r1_xboole_0(B,C) )
=> A = k1_xboole_0 ) ).
fof(t68_xboole_1,axiom,
! [A,B,C] :
( ~ v1_xboole_0(C)
=> ~ ( r1_tarski(C,A)
& r1_tarski(C,B)
& r1_xboole_0(A,B) ) ) ).
fof(t69_xboole_1,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ~ ( r1_tarski(B,A)
& r1_xboole_0(B,A) ) ) ).
fof(t70_xboole_1,axiom,
! [A,B,C] :
( ~ ( ~ r1_xboole_0(A,k2_xboole_0(B,C))
& r1_xboole_0(A,B)
& r1_xboole_0(A,C) )
& ~ ( ~ ( r1_xboole_0(A,B)
& r1_xboole_0(A,C) )
& r1_xboole_0(A,k2_xboole_0(B,C)) ) ) ).
fof(t71_xboole_1,axiom,
! [A,B,C] :
( ( k2_xboole_0(A,B) = k2_xboole_0(C,B)
& r1_xboole_0(A,B)
& r1_xboole_0(C,B) )
=> A = C ) ).
fof(t72_xboole_1,axiom,
! [A,B,C,D] :
( ( k2_xboole_0(A,B) = k2_xboole_0(C,D)
& r1_xboole_0(C,A)
& r1_xboole_0(D,B) )
=> C = B ) ).
fof(t73_xboole_1,axiom,
! [A,B,C] :
( ( r1_tarski(A,k2_xboole_0(B,C))
& r1_xboole_0(A,C) )
=> r1_tarski(A,B) ) ).
fof(t74_xboole_1,axiom,
! [A,B,C] :
~ ( ~ r1_xboole_0(A,k3_xboole_0(B,C))
& r1_xboole_0(A,B) ) ).
fof(t75_xboole_1,axiom,
! [A,B] :
~ ( ~ r1_xboole_0(A,B)
& r1_xboole_0(k3_xboole_0(A,B),B) ) ).
fof(t76_xboole_1,axiom,
! [A,B,C] :
( r1_xboole_0(A,B)
=> r1_xboole_0(k3_xboole_0(C,A),k3_xboole_0(C,B)) ) ).
fof(t77_xboole_1,axiom,
! [A,B,C] :
~ ( ~ r1_xboole_0(A,B)
& r1_tarski(A,C)
& r1_xboole_0(A,k3_xboole_0(B,C)) ) ).
fof(t78_xboole_1,axiom,
! [A,B,C] :
( r1_xboole_0(A,B)
=> k3_xboole_0(A,k2_xboole_0(B,C)) = k3_xboole_0(A,C) ) ).
fof(t79_xboole_1,axiom,
! [A,B] : r1_xboole_0(k4_xboole_0(A,B),B) ).
fof(t80_xboole_1,axiom,
! [A,B,C] :
( r1_xboole_0(A,B)
=> r1_xboole_0(A,k4_xboole_0(B,C)) ) ).
fof(t81_xboole_1,axiom,
! [A,B,C] :
( r1_xboole_0(A,k4_xboole_0(B,C))
=> r1_xboole_0(B,k4_xboole_0(A,C)) ) ).
fof(t82_xboole_1,axiom,
! [A,B] : r1_xboole_0(k4_xboole_0(A,B),k4_xboole_0(B,A)) ).
fof(t83_xboole_1,axiom,
! [A,B] :
( r1_xboole_0(A,B)
<=> k4_xboole_0(A,B) = A ) ).
fof(t84_xboole_1,axiom,
! [A,B,C] :
~ ( ~ r1_xboole_0(A,B)
& r1_xboole_0(A,C)
& r1_xboole_0(A,k4_xboole_0(B,C)) ) ).
fof(t85_xboole_1,axiom,
! [A,B,C] :
( r1_tarski(A,B)
=> r1_xboole_0(A,k4_xboole_0(C,B)) ) ).
fof(t86_xboole_1,axiom,
! [A,B,C] :
( ( r1_tarski(A,B)
& r1_xboole_0(A,C) )
=> r1_tarski(A,k4_xboole_0(B,C)) ) ).
fof(t87_xboole_1,axiom,
! [A,B,C] :
( r1_xboole_0(A,B)
=> k2_xboole_0(k4_xboole_0(C,A),B) = k4_xboole_0(k2_xboole_0(C,B),A) ) ).
fof(t88_xboole_1,axiom,
! [A,B] :
( r1_xboole_0(A,B)
=> k4_xboole_0(k2_xboole_0(A,B),B) = A ) ).
fof(t89_xboole_1,axiom,
! [A,B] : r1_xboole_0(k3_xboole_0(A,B),k4_xboole_0(A,B)) ).
fof(t90_xboole_1,axiom,
! [A,B] : r1_xboole_0(k4_xboole_0(A,k3_xboole_0(A,B)),B) ).
fof(t91_xboole_1,axiom,
! [A,B,C] : k5_xboole_0(k5_xboole_0(A,B),C) = k5_xboole_0(A,k5_xboole_0(B,C)) ).
fof(t92_xboole_1,axiom,
! [A] : k5_xboole_0(A,A) = k1_xboole_0 ).
fof(t93_xboole_1,axiom,
! [A,B] : k2_xboole_0(A,B) = k2_xboole_0(k5_xboole_0(A,B),k3_xboole_0(A,B)) ).
fof(t94_xboole_1,axiom,
! [A,B] : k2_xboole_0(A,B) = k5_xboole_0(k5_xboole_0(A,B),k3_xboole_0(A,B)) ).
fof(t95_xboole_1,axiom,
! [A,B] : k3_xboole_0(A,B) = k5_xboole_0(k5_xboole_0(A,B),k2_xboole_0(A,B)) ).
fof(t96_xboole_1,axiom,
! [A,B] : r1_tarski(k4_xboole_0(A,B),k5_xboole_0(A,B)) ).
fof(t97_xboole_1,axiom,
! [A,B,C] :
( ( r1_tarski(k4_xboole_0(A,B),C)
& r1_tarski(k4_xboole_0(B,A),C) )
=> r1_tarski(k5_xboole_0(A,B),C) ) ).
fof(t98_xboole_1,axiom,
! [A,B] : k2_xboole_0(A,B) = k5_xboole_0(A,k4_xboole_0(B,A)) ).
fof(t99_xboole_1,axiom,
! [A,B,C] : k4_xboole_0(k5_xboole_0(A,B),C) = k2_xboole_0(k4_xboole_0(A,k2_xboole_0(B,C)),k4_xboole_0(B,k2_xboole_0(A,C))) ).
fof(t100_xboole_1,axiom,
! [A,B] : k4_xboole_0(A,B) = k5_xboole_0(A,k3_xboole_0(A,B)) ).
fof(t101_xboole_1,axiom,
! [A,B] : k5_xboole_0(A,B) = k4_xboole_0(k2_xboole_0(A,B),k3_xboole_0(A,B)) ).
fof(t102_xboole_1,axiom,
! [A,B,C] : k4_xboole_0(A,k5_xboole_0(B,C)) = k2_xboole_0(k4_xboole_0(A,k2_xboole_0(B,C)),k3_xboole_0(k3_xboole_0(A,B),C)) ).
fof(t103_xboole_1,axiom,
! [A,B] : r1_xboole_0(k3_xboole_0(A,B),k5_xboole_0(A,B)) ).
fof(t104_xboole_1,axiom,
! [A,B] :
( ~ ( ~ r2_xboole_0(A,B)
& A != B
& ~ r2_xboole_0(B,A) )
<=> r3_xboole_0(A,B) ) ).
fof(t105_xboole_1,axiom,
! [A,B] :
~ ( r2_xboole_0(A,B)
& k4_xboole_0(B,A) = k1_xboole_0 ) ).
fof(t106_xboole_1,axiom,
! [A,B,C] :
( r1_tarski(A,k4_xboole_0(B,C))
=> ( r1_tarski(A,B)
& r1_xboole_0(A,C) ) ) ).
fof(t107_xboole_1,axiom,
! [A,B,C] :
( r1_tarski(A,k5_xboole_0(B,C))
<=> ( r1_tarski(A,k2_xboole_0(B,C))
& r1_xboole_0(A,k3_xboole_0(B,C)) ) ) ).
fof(t108_xboole_1,axiom,
! [A,B,C] :
( r1_tarski(A,B)
=> r1_tarski(k3_xboole_0(A,C),B) ) ).
fof(t109_xboole_1,axiom,
! [A,B,C] :
( r1_tarski(A,B)
=> r1_tarski(k4_xboole_0(A,C),B) ) ).
fof(t110_xboole_1,axiom,
! [A,B,C] :
( ( r1_tarski(A,B)
& r1_tarski(C,B) )
=> r1_tarski(k5_xboole_0(A,C),B) ) ).
fof(t111_xboole_1,axiom,
! [A,B,C] : k4_xboole_0(k3_xboole_0(A,B),k3_xboole_0(C,B)) = k3_xboole_0(k4_xboole_0(A,C),B) ).
fof(t112_xboole_1,axiom,
! [A,B,C] : k5_xboole_0(k3_xboole_0(A,B),k3_xboole_0(C,B)) = k3_xboole_0(k5_xboole_0(A,C),B) ).
fof(t113_xboole_1,axiom,
! [A,B,C,D] : k2_xboole_0(k2_xboole_0(k2_xboole_0(A,B),C),D) = k2_xboole_0(A,k2_xboole_0(k2_xboole_0(B,C),D)) ).
fof(t114_xboole_1,axiom,
! [A,B,C,D] :
( ( r1_xboole_0(A,D)
& r1_xboole_0(B,D)
& r1_xboole_0(C,D) )
=> r1_xboole_0(k2_xboole_0(k2_xboole_0(A,B),C),D) ) ).
%------------------------------------------------------------------------------