TPTP Problem File: SET967+1.p
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%------------------------------------------------------------------------------
% File : SET967+1 : TPTP v8.2.0. Released v3.2.0.
% Domain : Set theory
% Problem : Basic properties of sets, theorem 120
% Version : [Urb06] axioms : Especial.
% English :
% Refs : [Byl90] Bylinski (1990), Some Basic Properties of Sets
% : [Urb06] Urban (2006), Email to G. Sutcliffe
% Source : [Urb06]
% Names : zfmisc_1__t120_zfmisc_1 [Urb06]
% Status : Theorem
% Rating : 0.97 v8.2.0, 1.00 v7.4.0, 0.90 v7.3.0, 0.86 v7.1.0, 0.83 v7.0.0, 0.93 v6.4.0, 0.92 v6.3.0, 0.96 v6.2.0, 1.00 v3.2.0
% Syntax : Number of formulae : 16 ( 7 unt; 0 def)
% Number of atoms : 33 ( 11 equ)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 30 ( 13 ~; 1 |; 7 &)
% ( 4 <=>; 5 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 5 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 3 ( 2 usr; 0 prp; 1-2 aty)
% Number of functors : 5 ( 5 usr; 0 con; 1-2 aty)
% Number of variables : 48 ( 46 !; 2 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : Translated by MPTP 0.2 from the original problem in the Mizar
% library, www.mizar.org
%------------------------------------------------------------------------------
fof(antisymmetry_r2_hidden,axiom,
! [A,B] :
( in(A,B)
=> ~ in(B,A) ) ).
fof(commutativity_k2_tarski,axiom,
! [A,B] : unordered_pair(A,B) = unordered_pair(B,A) ).
fof(commutativity_k2_xboole_0,axiom,
! [A,B] : set_union2(A,B) = set_union2(B,A) ).
fof(d2_xboole_0,axiom,
! [A,B,C] :
( C = set_union2(A,B)
<=> ! [D] :
( in(D,C)
<=> ( in(D,A)
| in(D,B) ) ) ) ).
fof(d5_tarski,axiom,
! [A,B] : ordered_pair(A,B) = unordered_pair(unordered_pair(A,B),singleton(A)) ).
fof(fc1_zfmisc_1,axiom,
! [A,B] : ~ empty(ordered_pair(A,B)) ).
fof(fc2_xboole_0,axiom,
! [A,B] :
( ~ empty(A)
=> ~ empty(set_union2(A,B)) ) ).
fof(fc3_xboole_0,axiom,
! [A,B] :
( ~ empty(A)
=> ~ empty(set_union2(B,A)) ) ).
fof(idempotence_k2_xboole_0,axiom,
! [A,B] : set_union2(A,A) = A ).
fof(l55_zfmisc_1,axiom,
! [A,B,C,D] :
( in(ordered_pair(A,B),cartesian_product2(C,D))
<=> ( in(A,C)
& in(B,D) ) ) ).
fof(rc1_xboole_0,axiom,
? [A] : empty(A) ).
fof(rc2_xboole_0,axiom,
? [A] : ~ empty(A) ).
fof(t102_zfmisc_1,axiom,
! [A,B,C] :
~ ( in(A,cartesian_product2(B,C))
& ! [D,E] : ordered_pair(D,E) != A ) ).
fof(t107_zfmisc_1,axiom,
! [A,B,C,D] :
( in(ordered_pair(A,B),cartesian_product2(C,D))
=> in(ordered_pair(B,A),cartesian_product2(D,C)) ) ).
fof(t112_zfmisc_1,axiom,
! [A,B] :
( ( ! [C] :
~ ( in(C,A)
& ! [D,E] : C != ordered_pair(D,E) )
& ! [C] :
~ ( in(C,B)
& ! [D,E] : C != ordered_pair(D,E) )
& ! [C,D] :
( in(ordered_pair(C,D),A)
<=> in(ordered_pair(C,D),B) ) )
=> A = B ) ).
fof(t120_zfmisc_1,conjecture,
! [A,B,C] :
( cartesian_product2(set_union2(A,B),C) = set_union2(cartesian_product2(A,C),cartesian_product2(B,C))
& cartesian_product2(C,set_union2(A,B)) = set_union2(cartesian_product2(C,A),cartesian_product2(C,B)) ) ).
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