SET007 Axioms: SET007+58.ax
%------------------------------------------------------------------------------
% File : SET007+58 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Contraction Lemma
% 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 : zf_colla [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 21 ( 3 unt; 0 def)
% Number of atoms : 132 ( 20 equ)
% Maximal formula atoms : 16 ( 6 avg)
% Number of connectives : 140 ( 29 ~; 0 |; 52 &)
% ( 9 <=>; 50 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 9 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 16 ( 14 usr; 1 prp; 0-3 aty)
% Number of functors : 15 ( 15 usr; 2 con; 0-3 aty)
% Number of variables : 71 ( 60 !; 11 ?)
% 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(t2_zf_colla,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ( ! [C] :
( m1_subset_1(C,A)
=> ~ r2_hidden(C,B) )
<=> r2_hidden(B,k1_zf_colla(A,k1_xboole_0)) ) ) ) ).
fof(t3_zf_colla,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( m1_subset_1(C,A)
=> ( r1_tarski(k3_xboole_0(C,A),k1_zf_colla(A,B))
<=> r2_hidden(C,k1_zf_colla(A,k1_ordinal1(B))) ) ) ) ) ).
fof(t4_zf_colla,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> ( r1_ordinal1(B,C)
=> r1_tarski(k1_zf_colla(A,B),k1_zf_colla(A,C)) ) ) ) ) ).
fof(t5_zf_colla,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ? [C] :
( v3_ordinal1(C)
& r2_hidden(B,k1_zf_colla(A,C)) ) ) ) ).
fof(t6_zf_colla,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> ( ( r2_hidden(C,D)
& r2_hidden(D,k1_zf_colla(A,B)) )
=> ( r2_hidden(C,k1_zf_colla(A,B))
& ? [E] :
( v3_ordinal1(E)
& r2_hidden(E,B)
& r2_hidden(C,k1_zf_colla(A,E)) ) ) ) ) ) ) ) ).
fof(t7_zf_colla,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( v3_ordinal1(B)
=> r1_tarski(k1_zf_colla(A,B),A) ) ) ).
fof(t8_zf_colla,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ? [B] :
( v3_ordinal1(B)
& A = k1_zf_colla(A,B) ) ) ).
fof(t9_zf_colla,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ? [B] :
( v1_relat_1(B)
& v1_funct_1(B)
& k1_relat_1(B) = A
& ! [C] :
( m1_subset_1(C,A)
=> k1_funct_1(B,C) = k9_relat_1(B,C) ) ) ) ).
fof(d2_zf_colla,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B,C] :
( r1_zf_colla(A,B,C)
<=> ( k1_relat_1(A) = B
& k2_relat_1(A) = C
& v2_funct_1(A)
& ! [D,E] :
( ( r2_hidden(D,B)
& r2_hidden(E,B) )
=> ( ~ ( ? [F] :
( F = E
& r2_hidden(D,F) )
& ! [F] :
~ ( k1_funct_1(A,E) = F
& r2_hidden(k1_funct_1(A,D),F) ) )
& ~ ( ? [F] :
( k1_funct_1(A,E) = F
& r2_hidden(k1_funct_1(A,D),F) )
& ! [F] :
~ ( F = E
& r2_hidden(D,F) ) ) ) ) ) ) ) ).
fof(d3_zf_colla,axiom,
! [A,B] :
( r2_zf_colla(A,B)
<=> ? [C] :
( v1_relat_1(C)
& v1_funct_1(C)
& r1_zf_colla(C,A,B) ) ) ).
fof(t10_zf_colla,axiom,
$true ).
fof(t11_zf_colla,axiom,
$true ).
fof(t12_zf_colla,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( ( k1_relat_1(B) = A
& ! [C] :
( m1_subset_1(C,A)
=> k1_funct_1(B,C) = k9_relat_1(B,C) ) )
=> v1_ordinal1(k2_relat_1(B)) ) ) ) ).
fof(t13_zf_colla,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( r2_zf_model(A,k6_zf_model)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> ( ! [D] :
( m1_subset_1(D,A)
=> ( r2_hidden(D,B)
<=> r2_hidden(D,C) ) )
=> B = C ) ) ) ) ) ).
fof(t14_zf_colla,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ~ ( r2_zf_model(A,k6_zf_model)
& ! [B] :
~ ( v1_ordinal1(B)
& r2_zf_colla(A,B) ) ) ) ).
fof(dt_k1_zf_colla,axiom,
$true ).
fof(d1_zf_colla,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( C = k1_zf_colla(A,B)
<=> ? [D] :
( v1_relat_1(D)
& v1_funct_1(D)
& v5_ordinal1(D)
& C = a_2_0_zf_colla(A,D)
& k1_relat_1(D) = B
& ! [E] :
( v3_ordinal1(E)
=> ( r2_hidden(E,B)
=> k1_funct_1(D,E) = a_3_0_zf_colla(A,D,E) ) ) ) ) ) ) ).
fof(t1_zf_colla,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( v3_ordinal1(B)
=> k1_zf_colla(A,B) = a_2_1_zf_colla(A,B) ) ) ).
fof(fraenkel_a_2_0_zf_colla,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& v1_relat_1(C)
& v1_funct_1(C)
& v5_ordinal1(C) )
=> ( r2_hidden(A,a_2_0_zf_colla(B,C))
<=> ? [D] :
( m1_subset_1(D,B)
& A = D
& ! [E] :
( m1_subset_1(E,B)
=> ~ ( r2_hidden(E,D)
& ! [F] :
( v3_ordinal1(F)
=> ~ ( r2_hidden(F,k1_relat_1(C))
& r2_hidden(E,k3_tarski(k1_tarski(k1_funct_1(C,F)))) ) ) ) ) ) ) ) ).
fof(fraenkel_a_3_0_zf_colla,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(B)
& v1_relat_1(C)
& v1_funct_1(C)
& v5_ordinal1(C)
& v3_ordinal1(D) )
=> ( r2_hidden(A,a_3_0_zf_colla(B,C,D))
<=> ? [E] :
( m1_subset_1(E,B)
& A = E
& ! [F] :
( m1_subset_1(F,B)
=> ~ ( r2_hidden(F,E)
& ! [G] :
( v3_ordinal1(G)
=> ~ ( r2_hidden(G,k1_relat_1(k2_ordinal1(C,D)))
& r2_hidden(F,k3_tarski(k1_tarski(k1_funct_1(k2_ordinal1(C,D),G)))) ) ) ) ) ) ) ) ).
fof(fraenkel_a_2_1_zf_colla,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& v3_ordinal1(C) )
=> ( r2_hidden(A,a_2_1_zf_colla(B,C))
<=> ? [D] :
( m1_subset_1(D,B)
& A = D
& ! [E] :
( m1_subset_1(E,B)
=> ~ ( r2_hidden(E,D)
& ! [F] :
( v3_ordinal1(F)
=> ~ ( r2_hidden(F,C)
& r2_hidden(E,k1_zf_colla(B,F)) ) ) ) ) ) ) ) ).
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