SET007 Axioms: SET007+76.ax
%------------------------------------------------------------------------------
% File : SET007+76 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : A First Order Language
% 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 : qc_lang1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 121 ( 54 unt; 0 def)
% Number of atoms : 402 ( 79 equ)
% Maximal formula atoms : 19 ( 3 avg)
% Number of connectives : 322 ( 41 ~; 0 |; 111 &)
% ( 21 <=>; 149 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 4 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 24 ( 22 usr; 1 prp; 0-3 aty)
% Number of functors : 62 ( 62 usr; 21 con; 0-4 aty)
% Number of variables : 146 ( 119 !; 27 ?)
% 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_qc_lang1,axiom,
~ v1_xboole_0(k1_qc_lang1) ).
fof(fc2_qc_lang1,axiom,
~ v1_xboole_0(k2_qc_lang1) ).
fof(fc3_qc_lang1,axiom,
~ v1_xboole_0(k3_qc_lang1) ).
fof(fc4_qc_lang1,axiom,
~ v1_xboole_0(k4_qc_lang1) ).
fof(fc5_qc_lang1,axiom,
~ v1_xboole_0(k5_qc_lang1) ).
fof(fc6_qc_lang1,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ~ v1_xboole_0(k7_qc_lang1(A)) ) ).
fof(t1_qc_lang1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B,C] :
( m1_subset_1(C,A)
=> r1_tarski(k2_zfmisc_1(k1_tarski(C),B),k2_zfmisc_1(A,B)) ) ) ).
fof(t2_qc_lang1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B,C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,A)
=> r1_tarski(k2_zfmisc_1(k1_enumset1(C,D,E),B),k2_zfmisc_1(A,B)) ) ) ) ) ).
fof(d1_qc_lang1,axiom,
k1_qc_lang1 = k2_zfmisc_1(k1_enumset1(np__4,np__5,np__6),k5_numbers) ).
fof(t3_qc_lang1,axiom,
$true ).
fof(t4_qc_lang1,axiom,
r1_tarski(k1_qc_lang1,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers)) ).
fof(d2_qc_lang1,axiom,
k2_qc_lang1 = k2_zfmisc_1(k1_tarski(np__4),k5_numbers) ).
fof(d3_qc_lang1,axiom,
k3_qc_lang1 = k2_zfmisc_1(k1_tarski(np__5),k5_numbers) ).
fof(d4_qc_lang1,axiom,
k4_qc_lang1 = k2_zfmisc_1(k1_tarski(np__6),k5_numbers) ).
fof(t5_qc_lang1,axiom,
$true ).
fof(t6_qc_lang1,axiom,
$true ).
fof(t7_qc_lang1,axiom,
$true ).
fof(t8_qc_lang1,axiom,
$true ).
fof(t9_qc_lang1,axiom,
$true ).
fof(t10_qc_lang1,axiom,
r1_tarski(k5_qc_lang1,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers)) ).
fof(d6_qc_lang1,axiom,
! [A] :
( m1_subset_1(A,k5_qc_lang1)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( B = k6_qc_lang1(A)
<=> k1_mcart_1(A) = k1_nat_1(np__7,B) ) ) ) ).
fof(d8_qc_lang1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_1(B,k1_qc_lang1)
=> ( m1_qc_lang1(B,A)
<=> k3_finseq_1(B) = A ) ) ) ).
fof(d9_qc_lang1,axiom,
! [A] :
( v1_qc_lang1(A)
<=> ( m1_subset_1(A,k1_zfmisc_1(k13_finseq_1(k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))))
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k5_qc_lang1,k7_qc_lang1(B))
=> ! [D] :
( m1_qc_lang1(D,B)
=> r2_hidden(k7_finseq_1(k12_finseq_1(k7_qc_lang1(B),C),D),A) ) ) )
& r2_hidden(k9_finseq_1(k4_tarski(np__0,np__0)),A)
& ! [B] :
( m2_finseq_1(B,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))
=> ( r2_hidden(B,A)
=> r2_hidden(k7_finseq_1(k9_finseq_1(k4_tarski(np__1,np__0)),B),A) ) )
& ! [B] :
( m2_finseq_1(B,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))
=> ! [C] :
( m2_finseq_1(C,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))
=> ( ( r2_hidden(B,A)
& r2_hidden(C,A) )
=> r2_hidden(k7_finseq_1(k7_finseq_1(k9_finseq_1(k4_tarski(np__2,np__0)),B),C),A) ) ) )
& ! [B] :
( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1)
=> ! [C] :
( m2_finseq_1(C,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))
=> ( r2_hidden(C,A)
=> r2_hidden(k7_finseq_1(k7_finseq_1(k9_finseq_1(k4_tarski(np__3,np__0)),k12_finseq_1(k2_qc_lang1,B)),C),A) ) ) ) ) ) ).
fof(d10_qc_lang1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( A = k8_qc_lang1
<=> ( v1_qc_lang1(A)
& ! [B] :
( ~ v1_xboole_0(B)
=> ( v1_qc_lang1(B)
=> r1_tarski(A,B) ) ) ) ) ) ).
fof(t11_qc_lang1,axiom,
$true ).
fof(t12_qc_lang1,axiom,
$true ).
fof(t13_qc_lang1,axiom,
$true ).
fof(t14_qc_lang1,axiom,
$true ).
fof(t15_qc_lang1,axiom,
$true ).
fof(t16_qc_lang1,axiom,
$true ).
fof(t17_qc_lang1,axiom,
$true ).
fof(t18_qc_lang1,axiom,
$true ).
fof(t19_qc_lang1,axiom,
$true ).
fof(t20_qc_lang1,axiom,
$true ).
fof(t21_qc_lang1,axiom,
v1_qc_lang1(k8_qc_lang1) ).
fof(d11_qc_lang1,axiom,
! [A] :
( m1_subset_1(A,k5_qc_lang1)
=> ! [B] :
( m2_finseq_1(B,k1_qc_lang1)
=> ( k6_qc_lang1(A) = k3_finseq_1(B)
=> k9_qc_lang1(A,B) = k7_finseq_1(k12_finseq_1(k5_qc_lang1,A),B) ) ) ) ).
fof(t22_qc_lang1,axiom,
$true ).
fof(t23_qc_lang1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k5_qc_lang1,k7_qc_lang1(A))
=> ! [C] :
( m1_qc_lang1(C,A)
=> k9_qc_lang1(B,C) = k7_finseq_1(k12_finseq_1(k7_qc_lang1(A),B),C) ) ) ) ).
fof(d12_qc_lang1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> k10_qc_lang1(A) = A ) ).
fof(d13_qc_lang1,axiom,
k11_qc_lang1 = k9_finseq_1(k4_tarski(np__0,np__0)) ).
fof(d14_qc_lang1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> k12_qc_lang1(A) = k7_finseq_1(k9_finseq_1(k4_tarski(np__1,np__0)),k10_qc_lang1(A)) ) ).
fof(d15_qc_lang1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> k13_qc_lang1(A,B) = k7_finseq_1(k7_finseq_1(k9_finseq_1(k4_tarski(np__2,np__0)),k10_qc_lang1(A)),k10_qc_lang1(B)) ) ) ).
fof(d16_qc_lang1,axiom,
! [A] :
( m2_subset_1(A,k1_qc_lang1,k2_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> k14_qc_lang1(A,B) = k7_finseq_1(k7_finseq_1(k9_finseq_1(k4_tarski(np__3,np__0)),k12_finseq_1(k2_qc_lang1,A)),k10_qc_lang1(B)) ) ) ).
fof(d17_qc_lang1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ( v2_qc_lang1(A)
<=> ? [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
& ? [C] :
( m2_subset_1(C,k5_qc_lang1,k7_qc_lang1(B))
& ? [D] :
( m1_qc_lang1(D,B)
& A = k9_qc_lang1(C,D) ) ) ) ) ) ).
fof(d18_qc_lang1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ( v3_qc_lang1(A)
<=> ? [B] :
( m1_subset_1(B,k8_qc_lang1)
& A = k12_qc_lang1(B) ) ) ) ).
fof(d19_qc_lang1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ( v4_qc_lang1(A)
<=> ? [B] :
( m1_subset_1(B,k8_qc_lang1)
& ? [C] :
( m1_subset_1(C,k8_qc_lang1)
& A = k13_qc_lang1(B,C) ) ) ) ) ).
fof(d20_qc_lang1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ( v5_qc_lang1(A)
<=> ? [B] :
( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1)
& ? [C] :
( m1_subset_1(C,k8_qc_lang1)
& A = k14_qc_lang1(B,C) ) ) ) ) ).
fof(t24_qc_lang1,axiom,
$true ).
fof(t25_qc_lang1,axiom,
$true ).
fof(t26_qc_lang1,axiom,
$true ).
fof(t27_qc_lang1,axiom,
$true ).
fof(t28_qc_lang1,axiom,
$true ).
fof(t29_qc_lang1,axiom,
$true ).
fof(t30_qc_lang1,axiom,
$true ).
fof(t31_qc_lang1,axiom,
$true ).
fof(t32_qc_lang1,axiom,
$true ).
fof(t33_qc_lang1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ~ ( A != k11_qc_lang1
& ~ v2_qc_lang1(A)
& ~ v3_qc_lang1(A)
& ~ v4_qc_lang1(A)
& ~ v5_qc_lang1(A) ) ) ).
fof(t34_qc_lang1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> r1_xreal_0(np__1,k3_finseq_1(k10_qc_lang1(A))) ) ).
fof(t35_qc_lang1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k5_qc_lang1,k7_qc_lang1(A))
=> k6_qc_lang1(B) = A ) ) ).
fof(t36_qc_lang1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ( ( k1_mcart_1(k1_funct_1(k10_qc_lang1(A),np__1)) = np__0
=> A = k11_qc_lang1 )
& ( k1_mcart_1(k1_funct_1(k10_qc_lang1(A),np__1)) = np__1
=> v3_qc_lang1(A) )
& ( k1_mcart_1(k1_funct_1(k10_qc_lang1(A),np__1)) = np__2
=> v4_qc_lang1(A) )
& ( k1_mcart_1(k1_funct_1(k10_qc_lang1(A),np__1)) = np__3
=> v5_qc_lang1(A) )
& ( ? [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
& m2_subset_1(k1_funct_1(k10_qc_lang1(A),np__1),k5_qc_lang1,k7_qc_lang1(B)) )
=> v2_qc_lang1(A) ) ) ) ).
fof(t37_qc_lang1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ( k10_qc_lang1(A) = k7_finseq_1(k10_qc_lang1(B),C)
=> k10_qc_lang1(A) = k10_qc_lang1(B) ) ) ) ) ).
fof(d21_qc_lang1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ( v2_qc_lang1(A)
=> ! [B] :
( m1_subset_1(B,k5_qc_lang1)
=> ( B = k15_qc_lang1(A)
<=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& ? [D] :
( m1_qc_lang1(D,C)
& ? [E] :
( m2_subset_1(E,k5_qc_lang1,k7_qc_lang1(C))
& B = E
& A = k9_qc_lang1(E,D) ) ) ) ) ) ) ) ).
fof(d22_qc_lang1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ( v2_qc_lang1(A)
=> ! [B] :
( m2_finseq_1(B,k1_qc_lang1)
=> ( B = k16_qc_lang1(A)
<=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& ? [D] :
( m2_subset_1(D,k5_qc_lang1,k7_qc_lang1(C))
& ? [E] :
( m1_qc_lang1(E,C)
& B = E
& A = k9_qc_lang1(D,E) ) ) ) ) ) ) ) ).
fof(d23_qc_lang1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ( v3_qc_lang1(A)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ( B = k17_qc_lang1(A)
<=> A = k12_qc_lang1(B) ) ) ) ) ).
fof(d24_qc_lang1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ( v4_qc_lang1(A)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ( B = k18_qc_lang1(A)
<=> ? [C] :
( m1_subset_1(C,k8_qc_lang1)
& A = k13_qc_lang1(B,C) ) ) ) ) ) ).
fof(d25_qc_lang1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ( v4_qc_lang1(A)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ( B = k19_qc_lang1(A)
<=> ? [C] :
( m1_subset_1(C,k8_qc_lang1)
& A = k13_qc_lang1(C,B) ) ) ) ) ) ).
fof(d26_qc_lang1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ( v5_qc_lang1(A)
=> ! [B] :
( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1)
=> ( B = k20_qc_lang1(A)
<=> ? [C] :
( m1_subset_1(C,k8_qc_lang1)
& A = k14_qc_lang1(B,C) ) ) ) ) ) ).
fof(d27_qc_lang1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ( v5_qc_lang1(A)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ( B = k21_qc_lang1(A)
<=> ? [C] :
( m2_subset_1(C,k1_qc_lang1,k2_qc_lang1)
& A = k14_qc_lang1(C,B) ) ) ) ) ) ).
fof(t38_qc_lang1,axiom,
$true ).
fof(t39_qc_lang1,axiom,
$true ).
fof(t40_qc_lang1,axiom,
$true ).
fof(t41_qc_lang1,axiom,
$true ).
fof(t42_qc_lang1,axiom,
$true ).
fof(t43_qc_lang1,axiom,
$true ).
fof(t44_qc_lang1,axiom,
$true ).
fof(t45_qc_lang1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ~ ( v3_qc_lang1(A)
& r1_xreal_0(k3_finseq_1(k10_qc_lang1(A)),k3_finseq_1(k10_qc_lang1(k17_qc_lang1(A)))) ) ) ).
fof(t46_qc_lang1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ( v4_qc_lang1(A)
=> ( ~ r1_xreal_0(k3_finseq_1(k10_qc_lang1(A)),k3_finseq_1(k10_qc_lang1(k18_qc_lang1(A))))
& ~ r1_xreal_0(k3_finseq_1(k10_qc_lang1(A)),k3_finseq_1(k10_qc_lang1(k19_qc_lang1(A)))) ) ) ) ).
fof(t47_qc_lang1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ~ ( v5_qc_lang1(A)
& r1_xreal_0(k3_finseq_1(k10_qc_lang1(A)),k3_finseq_1(k10_qc_lang1(k21_qc_lang1(A)))) ) ) ).
fof(t48_qc_lang1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k5_qc_lang1,k7_qc_lang1(A))
=> ( k1_mcart_1(B) != np__0
& k1_mcart_1(B) != np__1
& k1_mcart_1(B) != np__2
& k1_mcart_1(B) != np__3 ) ) ) ).
fof(t49_qc_lang1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ( k1_mcart_1(k1_funct_1(k10_qc_lang1(k11_qc_lang1),np__1)) = np__0
& ~ ( v2_qc_lang1(A)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ m2_subset_1(k1_funct_1(k10_qc_lang1(A),np__1),k5_qc_lang1,k7_qc_lang1(B)) ) )
& ( v3_qc_lang1(A)
=> k1_mcart_1(k1_funct_1(k10_qc_lang1(A),np__1)) = np__1 )
& ( v4_qc_lang1(A)
=> k1_mcart_1(k1_funct_1(k10_qc_lang1(A),np__1)) = np__2 )
& ( v5_qc_lang1(A)
=> k1_mcart_1(k1_funct_1(k10_qc_lang1(A),np__1)) = np__3 ) ) ) ).
fof(t50_qc_lang1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ( v2_qc_lang1(A)
=> ( k1_mcart_1(k1_funct_1(k10_qc_lang1(A),np__1)) != np__0
& k1_mcart_1(k1_funct_1(k10_qc_lang1(A),np__1)) != np__1
& k1_mcart_1(k1_funct_1(k10_qc_lang1(A),np__1)) != np__2
& k1_mcart_1(k1_funct_1(k10_qc_lang1(A),np__1)) != np__3 ) ) ) ).
fof(t51_qc_lang1,axiom,
( ~ v2_qc_lang1(k11_qc_lang1)
& ~ v3_qc_lang1(k11_qc_lang1)
& ~ v4_qc_lang1(k11_qc_lang1)
& ~ v5_qc_lang1(k11_qc_lang1)
& ! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ( ~ ( v2_qc_lang1(A)
& v3_qc_lang1(A) )
& ~ ( v2_qc_lang1(A)
& v4_qc_lang1(A) )
& ~ ( v2_qc_lang1(A)
& v5_qc_lang1(A) )
& ~ ( v3_qc_lang1(A)
& v4_qc_lang1(A) )
& ~ ( v3_qc_lang1(A)
& v5_qc_lang1(A) )
& ~ ( v4_qc_lang1(A)
& v5_qc_lang1(A) ) ) ) ) ).
fof(d30_qc_lang1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ( v6_qc_lang1(A)
<=> k24_qc_lang1(A) = k1_xboole_0 ) ) ).
fof(s1_qc_lang1,axiom,
( ( ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k5_qc_lang1,k7_qc_lang1(A))
=> ! [C] :
( m1_qc_lang1(C,A)
=> p1_s1_qc_lang1(k9_qc_lang1(B,C)) ) ) )
& p1_s1_qc_lang1(k11_qc_lang1)
& ! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ( p1_s1_qc_lang1(A)
=> p1_s1_qc_lang1(k12_qc_lang1(A)) ) )
& ! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ( ( p1_s1_qc_lang1(A)
& p1_s1_qc_lang1(B) )
=> p1_s1_qc_lang1(k13_qc_lang1(A,B)) ) ) )
& ! [A] :
( m2_subset_1(A,k1_qc_lang1,k2_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ( p1_s1_qc_lang1(B)
=> p1_s1_qc_lang1(k14_qc_lang1(A,B)) ) ) ) )
=> ! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> p1_s1_qc_lang1(A) ) ) ).
fof(s2_qc_lang1,axiom,
( ! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ( ( v2_qc_lang1(A)
=> p1_s2_qc_lang1(A) )
& p1_s2_qc_lang1(k11_qc_lang1)
& ( ( v3_qc_lang1(A)
& p1_s2_qc_lang1(k17_qc_lang1(A)) )
=> p1_s2_qc_lang1(A) )
& ( ( v4_qc_lang1(A)
& p1_s2_qc_lang1(k18_qc_lang1(A))
& p1_s2_qc_lang1(k19_qc_lang1(A)) )
=> p1_s2_qc_lang1(A) )
& ( ( v5_qc_lang1(A)
& p1_s2_qc_lang1(k21_qc_lang1(A)) )
=> p1_s2_qc_lang1(A) ) ) )
=> ! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> p1_s2_qc_lang1(A) ) ) ).
fof(s3_qc_lang1,axiom,
? [A] :
( v1_funct_1(A)
& v1_funct_2(A,k8_qc_lang1,f1_s3_qc_lang1)
& m2_relset_1(A,k8_qc_lang1,f1_s3_qc_lang1)
& k8_funct_2(k8_qc_lang1,f1_s3_qc_lang1,A,k11_qc_lang1) = f2_s3_qc_lang1
& ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ( ( v2_qc_lang1(B)
=> k8_funct_2(k8_qc_lang1,f1_s3_qc_lang1,A,B) = f3_s3_qc_lang1(B) )
& ( v3_qc_lang1(B)
=> k8_funct_2(k8_qc_lang1,f1_s3_qc_lang1,A,B) = f4_s3_qc_lang1(k8_funct_2(k8_qc_lang1,f1_s3_qc_lang1,A,k17_qc_lang1(B))) )
& ( v4_qc_lang1(B)
=> k8_funct_2(k8_qc_lang1,f1_s3_qc_lang1,A,B) = f5_s3_qc_lang1(k8_funct_2(k8_qc_lang1,f1_s3_qc_lang1,A,k18_qc_lang1(B)),k8_funct_2(k8_qc_lang1,f1_s3_qc_lang1,A,k19_qc_lang1(B))) )
& ( v5_qc_lang1(B)
=> k8_funct_2(k8_qc_lang1,f1_s3_qc_lang1,A,B) = f6_s3_qc_lang1(B,k8_funct_2(k8_qc_lang1,f1_s3_qc_lang1,A,k21_qc_lang1(B))) ) ) ) ) ).
fof(dt_m1_qc_lang1,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ! [B] :
( m1_qc_lang1(B,A)
=> m2_finseq_1(B,k1_qc_lang1) ) ) ).
fof(existence_m1_qc_lang1,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ? [B] : m1_qc_lang1(B,A) ) ).
fof(dt_k1_qc_lang1,axiom,
$true ).
fof(dt_k2_qc_lang1,axiom,
m1_subset_1(k2_qc_lang1,k1_zfmisc_1(k1_qc_lang1)) ).
fof(dt_k3_qc_lang1,axiom,
m1_subset_1(k3_qc_lang1,k1_zfmisc_1(k1_qc_lang1)) ).
fof(dt_k4_qc_lang1,axiom,
m1_subset_1(k4_qc_lang1,k1_zfmisc_1(k1_qc_lang1)) ).
fof(dt_k5_qc_lang1,axiom,
$true ).
fof(dt_k6_qc_lang1,axiom,
! [A] :
( m1_subset_1(A,k5_qc_lang1)
=> m2_subset_1(k6_qc_lang1(A),k1_numbers,k5_numbers) ) ).
fof(dt_k7_qc_lang1,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> m1_subset_1(k7_qc_lang1(A),k1_zfmisc_1(k5_qc_lang1)) ) ).
fof(dt_k8_qc_lang1,axiom,
~ v1_xboole_0(k8_qc_lang1) ).
fof(dt_k9_qc_lang1,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_qc_lang1)
& m1_finseq_1(B,k1_qc_lang1) )
=> m1_subset_1(k9_qc_lang1(A,B),k8_qc_lang1) ) ).
fof(dt_k10_qc_lang1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> m2_finseq_1(k10_qc_lang1(A),k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers)) ) ).
fof(dt_k11_qc_lang1,axiom,
m1_subset_1(k11_qc_lang1,k8_qc_lang1) ).
fof(dt_k12_qc_lang1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> m1_subset_1(k12_qc_lang1(A),k8_qc_lang1) ) ).
fof(dt_k13_qc_lang1,axiom,
! [A,B] :
( ( m1_subset_1(A,k8_qc_lang1)
& m1_subset_1(B,k8_qc_lang1) )
=> m1_subset_1(k13_qc_lang1(A,B),k8_qc_lang1) ) ).
fof(dt_k14_qc_lang1,axiom,
! [A,B] :
( ( m1_subset_1(A,k2_qc_lang1)
& m1_subset_1(B,k8_qc_lang1) )
=> m1_subset_1(k14_qc_lang1(A,B),k8_qc_lang1) ) ).
fof(dt_k15_qc_lang1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> m1_subset_1(k15_qc_lang1(A),k5_qc_lang1) ) ).
fof(dt_k16_qc_lang1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> m2_finseq_1(k16_qc_lang1(A),k1_qc_lang1) ) ).
fof(dt_k17_qc_lang1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> m1_subset_1(k17_qc_lang1(A),k8_qc_lang1) ) ).
fof(dt_k18_qc_lang1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> m1_subset_1(k18_qc_lang1(A),k8_qc_lang1) ) ).
fof(dt_k19_qc_lang1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> m1_subset_1(k19_qc_lang1(A),k8_qc_lang1) ) ).
fof(dt_k20_qc_lang1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> m2_subset_1(k20_qc_lang1(A),k1_qc_lang1,k2_qc_lang1) ) ).
fof(dt_k21_qc_lang1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> m1_subset_1(k21_qc_lang1(A),k8_qc_lang1) ) ).
fof(dt_k22_qc_lang1,axiom,
! [A] :
( m1_finseq_1(A,k1_qc_lang1)
=> m1_subset_1(k22_qc_lang1(A),k1_zfmisc_1(k2_qc_lang1)) ) ).
fof(dt_k23_qc_lang1,axiom,
! [A] :
( m1_subset_1(A,k2_qc_lang1)
=> m1_subset_1(k23_qc_lang1(A),k1_zfmisc_1(k2_qc_lang1)) ) ).
fof(redefinition_k23_qc_lang1,axiom,
! [A] :
( m1_subset_1(A,k2_qc_lang1)
=> k23_qc_lang1(A) = k1_tarski(A) ) ).
fof(dt_k24_qc_lang1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> m1_subset_1(k24_qc_lang1(A),k1_zfmisc_1(k2_qc_lang1)) ) ).
fof(d5_qc_lang1,axiom,
k5_qc_lang1 = a_0_0_qc_lang1 ).
fof(d7_qc_lang1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k7_qc_lang1(A) = a_1_0_qc_lang1(A) ) ).
fof(d28_qc_lang1,axiom,
! [A] :
( m2_finseq_1(A,k1_qc_lang1)
=> k22_qc_lang1(A) = a_1_1_qc_lang1(A) ) ).
fof(d29_qc_lang1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k2_qc_lang1))
=> ( B = k24_qc_lang1(A)
<=> ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,k8_qc_lang1,k1_zfmisc_1(k2_qc_lang1))
& m2_relset_1(C,k8_qc_lang1,k1_zfmisc_1(k2_qc_lang1))
& B = k8_funct_2(k8_qc_lang1,k1_zfmisc_1(k2_qc_lang1),C,A)
& ! [D] :
( m1_subset_1(D,k8_qc_lang1)
=> ( k8_funct_2(k8_qc_lang1,k1_zfmisc_1(k2_qc_lang1),C,k11_qc_lang1) = k1_xboole_0
& ( v2_qc_lang1(D)
=> k8_funct_2(k8_qc_lang1,k1_zfmisc_1(k2_qc_lang1),C,D) = a_1_2_qc_lang1(D) )
& ( v3_qc_lang1(D)
=> k8_funct_2(k8_qc_lang1,k1_zfmisc_1(k2_qc_lang1),C,D) = k8_funct_2(k8_qc_lang1,k1_zfmisc_1(k2_qc_lang1),C,k17_qc_lang1(D)) )
& ( v4_qc_lang1(D)
=> k8_funct_2(k8_qc_lang1,k1_zfmisc_1(k2_qc_lang1),C,D) = k4_subset_1(k2_qc_lang1,k8_funct_2(k8_qc_lang1,k1_zfmisc_1(k2_qc_lang1),C,k18_qc_lang1(D)),k8_funct_2(k8_qc_lang1,k1_zfmisc_1(k2_qc_lang1),C,k19_qc_lang1(D))) )
& ( v5_qc_lang1(D)
=> k8_funct_2(k8_qc_lang1,k1_zfmisc_1(k2_qc_lang1),C,D) = k6_subset_1(k2_qc_lang1,k8_funct_2(k8_qc_lang1,k1_zfmisc_1(k2_qc_lang1),C,k21_qc_lang1(D)),k23_qc_lang1(k20_qc_lang1(D))) ) ) ) ) ) ) ) ).
fof(fraenkel_a_0_0_qc_lang1,axiom,
! [A] :
( r2_hidden(A,a_0_0_qc_lang1)
<=> ? [B,C] :
( m2_subset_1(B,k1_numbers,k5_numbers)
& m2_subset_1(C,k1_numbers,k5_numbers)
& A = k4_tarski(B,C)
& r1_xreal_0(np__7,B) ) ) ).
fof(fraenkel_a_1_0_qc_lang1,axiom,
! [A,B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r2_hidden(A,a_1_0_qc_lang1(B))
<=> ? [C] :
( m1_subset_1(C,k5_qc_lang1)
& A = C
& k6_qc_lang1(C) = B ) ) ) ).
fof(fraenkel_a_1_1_qc_lang1,axiom,
! [A,B] :
( m2_finseq_1(B,k1_qc_lang1)
=> ( r2_hidden(A,a_1_1_qc_lang1(B))
<=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& A = k1_funct_1(B,C)
& r1_xreal_0(np__1,C)
& r1_xreal_0(C,k3_finseq_1(B))
& r2_hidden(k1_funct_1(B,C),k2_qc_lang1) ) ) ) ).
fof(fraenkel_a_1_2_qc_lang1,axiom,
! [A,B] :
( m1_subset_1(B,k8_qc_lang1)
=> ( r2_hidden(A,a_1_2_qc_lang1(B))
<=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& A = k1_funct_1(k16_qc_lang1(B),C)
& r1_xreal_0(np__1,C)
& r1_xreal_0(C,k3_finseq_1(k16_qc_lang1(B)))
& r2_hidden(k1_funct_1(k16_qc_lang1(B),C),k2_qc_lang1) ) ) ) ).
%------------------------------------------------------------------------------