SET007 Axioms: SET007+56.ax
%------------------------------------------------------------------------------
% File : SET007+56 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : A Model of ZF Set Theory 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 : zf_lang [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 193 ( 25 unt; 0 def)
% Number of atoms : 1239 ( 217 equ)
% Maximal formula atoms : 48 ( 6 avg)
% Number of connectives : 1141 ( 95 ~; 8 |; 546 &)
% ( 41 <=>; 451 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 7 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 30 ( 28 usr; 1 prp; 0-3 aty)
% Number of functors : 48 ( 48 usr; 11 con; 0-4 aty)
% Number of variables : 430 ( 367 !; 63 ?)
% 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_zf_lang,axiom,
~ v1_xboole_0(k1_zf_lang) ).
fof(rc1_zf_lang,axiom,
? [A] :
( m1_finseq_1(A,k5_numbers)
& v1_relat_1(A)
& v1_funct_1(A)
& v1_finset_1(A)
& v1_finseq_1(A)
& v1_zf_lang(A) ) ).
fof(fc2_zf_lang,axiom,
! [A,B] :
( ( m1_subset_1(A,k1_zf_lang)
& m1_subset_1(B,k1_zf_lang) )
=> ( v1_relat_1(k4_zf_lang(A,B))
& v1_funct_1(k4_zf_lang(A,B))
& v1_finset_1(k4_zf_lang(A,B))
& v1_finseq_1(k4_zf_lang(A,B))
& v1_zf_lang(k4_zf_lang(A,B)) ) ) ).
fof(fc3_zf_lang,axiom,
! [A,B] :
( ( m1_subset_1(A,k1_zf_lang)
& m1_subset_1(B,k1_zf_lang) )
=> ( v1_relat_1(k5_zf_lang(A,B))
& v1_funct_1(k5_zf_lang(A,B))
& v1_finset_1(k5_zf_lang(A,B))
& v1_finseq_1(k5_zf_lang(A,B))
& v1_zf_lang(k5_zf_lang(A,B)) ) ) ).
fof(fc4_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m1_finseq_1(A,k5_numbers) )
=> ( v1_relat_1(k6_zf_lang(A))
& v1_funct_1(k6_zf_lang(A))
& v1_finset_1(k6_zf_lang(A))
& v1_finseq_1(k6_zf_lang(A))
& v1_zf_lang(k6_zf_lang(A)) ) ) ).
fof(fc5_zf_lang,axiom,
! [A,B] :
( ( v1_zf_lang(A)
& m1_finseq_1(A,k5_numbers)
& v1_zf_lang(B)
& m1_finseq_1(B,k5_numbers) )
=> ( v1_relat_1(k7_zf_lang(A,B))
& v1_funct_1(k7_zf_lang(A,B))
& v1_finset_1(k7_zf_lang(A,B))
& v1_finseq_1(k7_zf_lang(A,B))
& v1_zf_lang(k7_zf_lang(A,B)) ) ) ).
fof(fc6_zf_lang,axiom,
! [A,B] :
( ( m1_subset_1(A,k1_zf_lang)
& v1_zf_lang(B)
& m1_finseq_1(B,k5_numbers) )
=> ( v1_relat_1(k8_zf_lang(A,B))
& v1_funct_1(k8_zf_lang(A,B))
& v1_finset_1(k8_zf_lang(A,B))
& v1_finseq_1(k8_zf_lang(A,B))
& v1_zf_lang(k8_zf_lang(A,B)) ) ) ).
fof(d2_zf_lang,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k2_zf_lang(A) = k1_nat_1(np__5,A) ) ).
fof(d3_zf_lang,axiom,
! [A] :
( m2_subset_1(A,k5_numbers,k1_zf_lang)
=> ! [B] :
( m2_subset_1(B,k5_numbers,k1_zf_lang)
=> k4_zf_lang(A,B) = k8_finseq_1(k5_numbers,k8_finseq_1(k5_numbers,k12_finseq_1(k5_numbers,np__0),k3_zf_lang(A)),k3_zf_lang(B)) ) ) ).
fof(d4_zf_lang,axiom,
! [A] :
( m2_subset_1(A,k5_numbers,k1_zf_lang)
=> ! [B] :
( m2_subset_1(B,k5_numbers,k1_zf_lang)
=> k5_zf_lang(A,B) = k8_finseq_1(k5_numbers,k8_finseq_1(k5_numbers,k12_finseq_1(k5_numbers,np__1),k3_zf_lang(A)),k3_zf_lang(B)) ) ) ).
fof(t1_zf_lang,axiom,
$true ).
fof(t2_zf_lang,axiom,
$true ).
fof(t3_zf_lang,axiom,
$true ).
fof(t4_zf_lang,axiom,
$true ).
fof(t5_zf_lang,axiom,
$true ).
fof(t6_zf_lang,axiom,
! [A] :
( m2_subset_1(A,k5_numbers,k1_zf_lang)
=> ! [B] :
( m2_subset_1(B,k5_numbers,k1_zf_lang)
=> ! [C] :
( m2_subset_1(C,k5_numbers,k1_zf_lang)
=> ! [D] :
( m2_subset_1(D,k5_numbers,k1_zf_lang)
=> ( k4_zf_lang(A,B) = k4_zf_lang(C,D)
=> ( A = C
& B = D ) ) ) ) ) ) ).
fof(t7_zf_lang,axiom,
! [A] :
( m2_subset_1(A,k5_numbers,k1_zf_lang)
=> ! [B] :
( m2_subset_1(B,k5_numbers,k1_zf_lang)
=> ! [C] :
( m2_subset_1(C,k5_numbers,k1_zf_lang)
=> ! [D] :
( m2_subset_1(D,k5_numbers,k1_zf_lang)
=> ( k5_zf_lang(A,B) = k5_zf_lang(C,D)
=> ( A = C
& B = D ) ) ) ) ) ) ).
fof(d5_zf_lang,axiom,
! [A] :
( m2_finseq_1(A,k5_numbers)
=> k6_zf_lang(A) = k8_finseq_1(k5_numbers,k12_finseq_1(k5_numbers,np__2),A) ) ).
fof(d6_zf_lang,axiom,
! [A] :
( m2_finseq_1(A,k5_numbers)
=> ! [B] :
( m2_finseq_1(B,k5_numbers)
=> k7_zf_lang(A,B) = k8_finseq_1(k5_numbers,k8_finseq_1(k5_numbers,k12_finseq_1(k5_numbers,np__3),A),B) ) ) ).
fof(t8_zf_lang,axiom,
$true ).
fof(t9_zf_lang,axiom,
$true ).
fof(t10_zf_lang,axiom,
! [A] :
( m2_finseq_1(A,k5_numbers)
=> ! [B] :
( m2_finseq_1(B,k5_numbers)
=> ( k6_zf_lang(A) = k6_zf_lang(B)
=> A = B ) ) ) ).
fof(d7_zf_lang,axiom,
! [A] :
( m2_subset_1(A,k5_numbers,k1_zf_lang)
=> ! [B] :
( m2_finseq_1(B,k5_numbers)
=> k8_zf_lang(A,B) = k8_finseq_1(k5_numbers,k8_finseq_1(k5_numbers,k12_finseq_1(k5_numbers,np__4),k3_zf_lang(A)),B) ) ) ).
fof(t11_zf_lang,axiom,
$true ).
fof(t12_zf_lang,axiom,
! [A] :
( m2_finseq_1(A,k5_numbers)
=> ! [B] :
( m2_finseq_1(B,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k5_numbers,k1_zf_lang)
=> ! [D] :
( m2_subset_1(D,k5_numbers,k1_zf_lang)
=> ( k8_zf_lang(C,A) = k8_zf_lang(D,B)
=> ( C = D
& A = B ) ) ) ) ) ) ).
fof(d8_zf_lang,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( A = k9_zf_lang
<=> ( ! [B] :
( r2_hidden(B,A)
=> m2_finseq_1(B,k5_numbers) )
& ! [B] :
( m2_subset_1(B,k5_numbers,k1_zf_lang)
=> ! [C] :
( m2_subset_1(C,k5_numbers,k1_zf_lang)
=> ( r2_hidden(k4_zf_lang(B,C),A)
& r2_hidden(k5_zf_lang(B,C),A) ) ) )
& ! [B] :
( m2_finseq_1(B,k5_numbers)
=> ( r2_hidden(B,A)
=> r2_hidden(k6_zf_lang(B),A) ) )
& ! [B] :
( m2_finseq_1(B,k5_numbers)
=> ! [C] :
( m2_finseq_1(C,k5_numbers)
=> ( ( r2_hidden(B,A)
& r2_hidden(C,A) )
=> r2_hidden(k7_zf_lang(B,C),A) ) ) )
& ! [B] :
( m2_subset_1(B,k5_numbers,k1_zf_lang)
=> ! [C] :
( m2_finseq_1(C,k5_numbers)
=> ( r2_hidden(C,A)
=> r2_hidden(k8_zf_lang(B,C),A) ) ) )
& ! [B] :
( ~ v1_xboole_0(B)
=> ( ( ! [C] :
( r2_hidden(C,B)
=> m2_finseq_1(C,k5_numbers) )
& ! [C] :
( m2_subset_1(C,k5_numbers,k1_zf_lang)
=> ! [D] :
( m2_subset_1(D,k5_numbers,k1_zf_lang)
=> ( r2_hidden(k4_zf_lang(C,D),B)
& r2_hidden(k5_zf_lang(C,D),B) ) ) )
& ! [C] :
( m2_finseq_1(C,k5_numbers)
=> ( r2_hidden(C,B)
=> r2_hidden(k6_zf_lang(C),B) ) )
& ! [C] :
( m2_finseq_1(C,k5_numbers)
=> ! [D] :
( m2_finseq_1(D,k5_numbers)
=> ( ( r2_hidden(C,B)
& r2_hidden(D,B) )
=> r2_hidden(k7_zf_lang(C,D),B) ) ) )
& ! [C] :
( m2_subset_1(C,k5_numbers,k1_zf_lang)
=> ! [D] :
( m2_finseq_1(D,k5_numbers)
=> ( r2_hidden(D,B)
=> r2_hidden(k8_zf_lang(C,D),B) ) ) ) )
=> r1_tarski(A,B) ) ) ) ) ) ).
fof(d9_zf_lang,axiom,
! [A] :
( m2_finseq_1(A,k5_numbers)
=> ( v1_zf_lang(A)
<=> m1_subset_1(A,k9_zf_lang) ) ) ).
fof(t13_zf_lang,axiom,
$true ).
fof(t14_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
<=> r2_hidden(A,k9_zf_lang) ) ).
fof(d10_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( v2_zf_lang(A)
<=> ? [B] :
( m2_subset_1(B,k5_numbers,k1_zf_lang)
& ? [C] :
( m2_subset_1(C,k5_numbers,k1_zf_lang)
& A = k4_zf_lang(B,C) ) ) ) ) ).
fof(d11_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( v3_zf_lang(A)
<=> ? [B] :
( m2_subset_1(B,k5_numbers,k1_zf_lang)
& ? [C] :
( m2_subset_1(C,k5_numbers,k1_zf_lang)
& A = k5_zf_lang(B,C) ) ) ) ) ).
fof(d12_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( v4_zf_lang(A)
<=> ? [B] :
( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers)
& A = k6_zf_lang(B) ) ) ) ).
fof(d13_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( v5_zf_lang(A)
<=> ? [B] :
( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers)
& ? [C] :
( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers)
& A = k7_zf_lang(B,C) ) ) ) ) ).
fof(d14_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( v6_zf_lang(A)
<=> ? [B] :
( m2_subset_1(B,k5_numbers,k1_zf_lang)
& ? [C] :
( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers)
& A = k8_zf_lang(B,C) ) ) ) ) ).
fof(t15_zf_lang,axiom,
$true ).
fof(t16_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( ~ ( v2_zf_lang(A)
& ! [B] :
( m2_subset_1(B,k5_numbers,k1_zf_lang)
=> ! [C] :
( m2_subset_1(C,k5_numbers,k1_zf_lang)
=> A != k4_zf_lang(B,C) ) ) )
& ( ? [B] :
( m2_subset_1(B,k5_numbers,k1_zf_lang)
& ? [C] :
( m2_subset_1(C,k5_numbers,k1_zf_lang)
& A = k4_zf_lang(B,C) ) )
=> v2_zf_lang(A) )
& ~ ( v3_zf_lang(A)
& ! [B] :
( m2_subset_1(B,k5_numbers,k1_zf_lang)
=> ! [C] :
( m2_subset_1(C,k5_numbers,k1_zf_lang)
=> A != k5_zf_lang(B,C) ) ) )
& ( ? [B] :
( m2_subset_1(B,k5_numbers,k1_zf_lang)
& ? [C] :
( m2_subset_1(C,k5_numbers,k1_zf_lang)
& A = k5_zf_lang(B,C) ) )
=> v3_zf_lang(A) )
& ~ ( v4_zf_lang(A)
& ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> A != k6_zf_lang(B) ) )
& ( ? [B] :
( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers)
& A = k6_zf_lang(B) )
=> v4_zf_lang(A) )
& ~ ( v5_zf_lang(A)
& ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> A != k7_zf_lang(B,C) ) ) )
& ( ? [B] :
( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers)
& ? [C] :
( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers)
& A = k7_zf_lang(B,C) ) )
=> v5_zf_lang(A) )
& ~ ( v6_zf_lang(A)
& ! [B] :
( m2_subset_1(B,k5_numbers,k1_zf_lang)
=> ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> A != k8_zf_lang(B,C) ) ) )
& ( ? [B] :
( m2_subset_1(B,k5_numbers,k1_zf_lang)
& ? [C] :
( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers)
& A = k8_zf_lang(B,C) ) )
=> v6_zf_lang(A) ) ) ) ).
fof(d15_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( v7_zf_lang(A)
<=> ( v2_zf_lang(A)
| v3_zf_lang(A) ) ) ) ).
fof(d16_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> k10_zf_lang(A,B) = k6_zf_lang(k7_zf_lang(k6_zf_lang(A),k6_zf_lang(B))) ) ) ).
fof(d17_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> k11_zf_lang(A,B) = k6_zf_lang(k7_zf_lang(A,k6_zf_lang(B))) ) ) ).
fof(d18_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> k12_zf_lang(A,B) = k7_zf_lang(k11_zf_lang(A,B),k11_zf_lang(B,A)) ) ) ).
fof(d19_zf_lang,axiom,
! [A] :
( m2_subset_1(A,k5_numbers,k1_zf_lang)
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> k13_zf_lang(A,B) = k6_zf_lang(k8_zf_lang(A,k6_zf_lang(B))) ) ) ).
fof(d20_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( v8_zf_lang(A)
<=> ? [B] :
( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers)
& ? [C] :
( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers)
& A = k10_zf_lang(B,C) ) ) ) ) ).
fof(d21_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( v9_zf_lang(A)
<=> ? [B] :
( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers)
& ? [C] :
( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers)
& A = k11_zf_lang(B,C) ) ) ) ) ).
fof(d22_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( v10_zf_lang(A)
<=> ? [B] :
( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers)
& ? [C] :
( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers)
& A = k12_zf_lang(B,C) ) ) ) ) ).
fof(d23_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( v11_zf_lang(A)
<=> ? [B] :
( m2_subset_1(B,k5_numbers,k1_zf_lang)
& ? [C] :
( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers)
& A = k13_zf_lang(B,C) ) ) ) ) ).
fof(t17_zf_lang,axiom,
$true ).
fof(t18_zf_lang,axiom,
$true ).
fof(t19_zf_lang,axiom,
$true ).
fof(t20_zf_lang,axiom,
$true ).
fof(t21_zf_lang,axiom,
$true ).
fof(t22_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( ~ ( v8_zf_lang(A)
& ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> A != k10_zf_lang(B,C) ) ) )
& ( ? [B] :
( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers)
& ? [C] :
( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers)
& A = k10_zf_lang(B,C) ) )
=> v8_zf_lang(A) )
& ~ ( v9_zf_lang(A)
& ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> A != k11_zf_lang(B,C) ) ) )
& ( ? [B] :
( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers)
& ? [C] :
( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers)
& A = k11_zf_lang(B,C) ) )
=> v9_zf_lang(A) )
& ~ ( v10_zf_lang(A)
& ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> A != k12_zf_lang(B,C) ) ) )
& ( ? [B] :
( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers)
& ? [C] :
( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers)
& A = k12_zf_lang(B,C) ) )
=> v10_zf_lang(A) )
& ~ ( v11_zf_lang(A)
& ! [B] :
( m2_subset_1(B,k5_numbers,k1_zf_lang)
=> ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> A != k13_zf_lang(B,C) ) ) )
& ( ? [B] :
( m2_subset_1(B,k5_numbers,k1_zf_lang)
& ? [C] :
( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers)
& A = k13_zf_lang(B,C) ) )
=> v11_zf_lang(A) ) ) ) ).
fof(d24_zf_lang,axiom,
! [A] :
( m2_subset_1(A,k5_numbers,k1_zf_lang)
=> ! [B] :
( m2_subset_1(B,k5_numbers,k1_zf_lang)
=> ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> k14_zf_lang(A,B,C) = k8_zf_lang(A,k8_zf_lang(B,C)) ) ) ) ).
fof(d25_zf_lang,axiom,
! [A] :
( m2_subset_1(A,k5_numbers,k1_zf_lang)
=> ! [B] :
( m2_subset_1(B,k5_numbers,k1_zf_lang)
=> ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> k15_zf_lang(A,B,C) = k13_zf_lang(A,k13_zf_lang(B,C)) ) ) ) ).
fof(t23_zf_lang,axiom,
! [A] :
( m2_subset_1(A,k5_numbers,k1_zf_lang)
=> ! [B] :
( m2_subset_1(B,k5_numbers,k1_zf_lang)
=> ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> ( k14_zf_lang(A,B,C) = k8_zf_lang(A,k8_zf_lang(B,C))
& k15_zf_lang(A,B,C) = k13_zf_lang(A,k13_zf_lang(B,C)) ) ) ) ) ).
fof(d26_zf_lang,axiom,
! [A] :
( m2_subset_1(A,k5_numbers,k1_zf_lang)
=> ! [B] :
( m2_subset_1(B,k5_numbers,k1_zf_lang)
=> ! [C] :
( m2_subset_1(C,k5_numbers,k1_zf_lang)
=> ! [D] :
( ( v1_zf_lang(D)
& m2_finseq_1(D,k5_numbers) )
=> k16_zf_lang(A,B,C,D) = k8_zf_lang(A,k14_zf_lang(B,C,D)) ) ) ) ) ).
fof(d27_zf_lang,axiom,
! [A] :
( m2_subset_1(A,k5_numbers,k1_zf_lang)
=> ! [B] :
( m2_subset_1(B,k5_numbers,k1_zf_lang)
=> ! [C] :
( m2_subset_1(C,k5_numbers,k1_zf_lang)
=> ! [D] :
( ( v1_zf_lang(D)
& m2_finseq_1(D,k5_numbers) )
=> k17_zf_lang(A,B,C,D) = k13_zf_lang(A,k15_zf_lang(B,C,D)) ) ) ) ) ).
fof(t24_zf_lang,axiom,
! [A] :
( m2_subset_1(A,k5_numbers,k1_zf_lang)
=> ! [B] :
( m2_subset_1(B,k5_numbers,k1_zf_lang)
=> ! [C] :
( m2_subset_1(C,k5_numbers,k1_zf_lang)
=> ! [D] :
( ( v1_zf_lang(D)
& m2_finseq_1(D,k5_numbers) )
=> ( k16_zf_lang(A,B,C,D) = k8_zf_lang(A,k14_zf_lang(B,C,D))
& k17_zf_lang(A,B,C,D) = k13_zf_lang(A,k15_zf_lang(B,C,D)) ) ) ) ) ) ).
fof(t25_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ~ ( ~ v2_zf_lang(A)
& ~ v3_zf_lang(A)
& ~ v4_zf_lang(A)
& ~ v5_zf_lang(A)
& ~ v6_zf_lang(A) ) ) ).
fof(t26_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ~ ( ~ v7_zf_lang(A)
& ~ v4_zf_lang(A)
& ~ v5_zf_lang(A)
& ~ v6_zf_lang(A) ) ) ).
fof(t27_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( v7_zf_lang(A)
=> k3_finseq_1(A) = np__3 ) ) ).
fof(t28_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ~ ( ~ v7_zf_lang(A)
& ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ~ r1_xreal_0(k1_nat_1(k3_finseq_1(B),np__1),k3_finseq_1(A)) ) ) ) ).
fof(t29_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> r1_xreal_0(np__3,k3_finseq_1(A)) ) ).
fof(t30_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( k3_finseq_1(A) = np__3
=> v7_zf_lang(A) ) ) ).
fof(t31_zf_lang,axiom,
! [A] :
( m2_subset_1(A,k5_numbers,k1_zf_lang)
=> ! [B] :
( m2_subset_1(B,k5_numbers,k1_zf_lang)
=> ( k1_funct_1(k4_zf_lang(A,B),np__1) = np__0
& k1_funct_1(k5_zf_lang(A,B),np__1) = np__1 ) ) ) ).
fof(t32_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> k1_funct_1(k6_zf_lang(A),np__1) = np__2 ) ).
fof(t33_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> k1_funct_1(k7_zf_lang(A,B),np__1) = np__3 ) ) ).
fof(t34_zf_lang,axiom,
! [A] :
( m2_subset_1(A,k5_numbers,k1_zf_lang)
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> k1_funct_1(k8_zf_lang(A,B),np__1) = np__4 ) ) ).
fof(t35_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( v2_zf_lang(A)
=> k1_funct_1(A,np__1) = np__0 ) ) ).
fof(t36_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( v3_zf_lang(A)
=> k1_funct_1(A,np__1) = np__1 ) ) ).
fof(t37_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( v4_zf_lang(A)
=> k1_funct_1(A,np__1) = np__2 ) ) ).
fof(t38_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( v5_zf_lang(A)
=> k1_funct_1(A,np__1) = np__3 ) ) ).
fof(t39_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( v6_zf_lang(A)
=> k1_funct_1(A,np__1) = np__4 ) ) ).
fof(t40_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ~ ( ~ ( v2_zf_lang(A)
& k1_funct_1(A,np__1) = np__0 )
& ~ ( v3_zf_lang(A)
& k1_funct_1(A,np__1) = np__1 )
& ~ ( v4_zf_lang(A)
& k1_funct_1(A,np__1) = np__2 )
& ~ ( v5_zf_lang(A)
& k1_funct_1(A,np__1) = np__3 )
& ~ ( v6_zf_lang(A)
& k1_funct_1(A,np__1) = np__4 ) ) ) ).
fof(t41_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( k1_funct_1(A,np__1) = np__0
=> v2_zf_lang(A) ) ) ).
fof(t42_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( k1_funct_1(A,np__1) = np__1
=> v3_zf_lang(A) ) ) ).
fof(t43_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( k1_funct_1(A,np__1) = np__2
=> v4_zf_lang(A) ) ) ).
fof(t44_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( k1_funct_1(A,np__1) = np__3
=> v5_zf_lang(A) ) ) ).
fof(t45_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( k1_funct_1(A,np__1) = np__4
=> v6_zf_lang(A) ) ) ).
fof(t46_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ( A = k7_finseq_1(B,C)
=> A = B ) ) ) ) ).
fof(t47_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> ! [D] :
( ( v1_zf_lang(D)
& m2_finseq_1(D,k5_numbers) )
=> ( k7_zf_lang(A,B) = k7_zf_lang(C,D)
=> ( A = C
& B = D ) ) ) ) ) ) ).
fof(t48_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> ! [D] :
( ( v1_zf_lang(D)
& m2_finseq_1(D,k5_numbers) )
=> ( k10_zf_lang(A,B) = k10_zf_lang(C,D)
=> ( A = C
& B = D ) ) ) ) ) ) ).
fof(t49_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> ! [D] :
( ( v1_zf_lang(D)
& m2_finseq_1(D,k5_numbers) )
=> ( k11_zf_lang(A,B) = k11_zf_lang(C,D)
=> ( A = C
& B = D ) ) ) ) ) ) ).
fof(t50_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> ! [D] :
( ( v1_zf_lang(D)
& m2_finseq_1(D,k5_numbers) )
=> ( k12_zf_lang(A,B) = k12_zf_lang(C,D)
=> ( A = C
& B = D ) ) ) ) ) ) ).
fof(t51_zf_lang,axiom,
! [A] :
( m2_subset_1(A,k5_numbers,k1_zf_lang)
=> ! [B] :
( m2_subset_1(B,k5_numbers,k1_zf_lang)
=> ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> ! [D] :
( ( v1_zf_lang(D)
& m2_finseq_1(D,k5_numbers) )
=> ( k13_zf_lang(A,C) = k13_zf_lang(B,D)
=> ( A = B
& C = D ) ) ) ) ) ) ).
fof(d28_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( v7_zf_lang(A)
=> k18_zf_lang(A) = k1_funct_1(A,np__2) ) ) ).
fof(d29_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( v7_zf_lang(A)
=> k19_zf_lang(A) = k1_funct_1(A,np__3) ) ) ).
fof(t52_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( v7_zf_lang(A)
=> ( k18_zf_lang(A) = k1_funct_1(A,np__2)
& k19_zf_lang(A) = k1_funct_1(A,np__3) ) ) ) ).
fof(t53_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( v2_zf_lang(A)
=> A = k4_zf_lang(k18_zf_lang(A),k19_zf_lang(A)) ) ) ).
fof(t54_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( v3_zf_lang(A)
=> A = k5_zf_lang(k18_zf_lang(A),k19_zf_lang(A)) ) ) ).
fof(d30_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( v4_zf_lang(A)
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ( B = k20_zf_lang(A)
<=> k6_zf_lang(B) = A ) ) ) ) ).
fof(d31_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( ( v5_zf_lang(A)
| v8_zf_lang(A) )
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ( ( v5_zf_lang(A)
=> ( B = k21_zf_lang(A)
<=> ? [C] :
( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers)
& k7_zf_lang(B,C) = A ) ) )
& ( ~ v5_zf_lang(A)
=> ( B = k21_zf_lang(A)
<=> ? [C] :
( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers)
& k10_zf_lang(B,C) = A ) ) ) ) ) ) ) ).
fof(d32_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( ( v5_zf_lang(A)
| v8_zf_lang(A) )
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ( ( v5_zf_lang(A)
=> ( B = k22_zf_lang(A)
<=> ? [C] :
( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers)
& k7_zf_lang(C,B) = A ) ) )
& ( ~ v5_zf_lang(A)
=> ( B = k22_zf_lang(A)
<=> ? [C] :
( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers)
& k10_zf_lang(C,B) = A ) ) ) ) ) ) ) ).
fof(t55_zf_lang,axiom,
$true ).
fof(t56_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ( v5_zf_lang(A)
=> ( ~ ( B = k21_zf_lang(A)
& ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> k7_zf_lang(B,C) != A ) )
& ( ? [C] :
( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers)
& k7_zf_lang(B,C) = A )
=> B = k21_zf_lang(A) )
& ~ ( B = k22_zf_lang(A)
& ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> k7_zf_lang(C,B) != A ) )
& ( ? [C] :
( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers)
& k7_zf_lang(C,B) = A )
=> B = k22_zf_lang(A) ) ) ) ) ) ).
fof(t57_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ( v8_zf_lang(A)
=> ( ~ ( B = k21_zf_lang(A)
& ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> k10_zf_lang(B,C) != A ) )
& ( ? [C] :
( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers)
& k10_zf_lang(B,C) = A )
=> B = k21_zf_lang(A) )
& ~ ( B = k22_zf_lang(A)
& ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> k10_zf_lang(C,B) != A ) )
& ( ? [C] :
( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers)
& k10_zf_lang(C,B) = A )
=> B = k22_zf_lang(A) ) ) ) ) ) ).
fof(t58_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( v5_zf_lang(A)
=> A = k7_zf_lang(k21_zf_lang(A),k22_zf_lang(A)) ) ) ).
fof(t59_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( v8_zf_lang(A)
=> A = k10_zf_lang(k21_zf_lang(A),k22_zf_lang(A)) ) ) ).
fof(d33_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( ( v6_zf_lang(A)
| v11_zf_lang(A) )
=> ! [B] :
( m2_subset_1(B,k5_numbers,k1_zf_lang)
=> ( ( v6_zf_lang(A)
=> ( B = k23_zf_lang(A)
<=> ? [C] :
( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers)
& k8_zf_lang(B,C) = A ) ) )
& ( ~ v6_zf_lang(A)
=> ( B = k23_zf_lang(A)
<=> ? [C] :
( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers)
& k13_zf_lang(B,C) = A ) ) ) ) ) ) ) ).
fof(d34_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( ( v6_zf_lang(A)
| v11_zf_lang(A) )
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ( ( v6_zf_lang(A)
=> ( B = k24_zf_lang(A)
<=> ? [C] :
( m2_subset_1(C,k5_numbers,k1_zf_lang)
& k8_zf_lang(C,B) = A ) ) )
& ( ~ v6_zf_lang(A)
=> ( B = k24_zf_lang(A)
<=> ? [C] :
( m2_subset_1(C,k5_numbers,k1_zf_lang)
& k13_zf_lang(C,B) = A ) ) ) ) ) ) ) ).
fof(t60_zf_lang,axiom,
! [A] :
( m2_subset_1(A,k5_numbers,k1_zf_lang)
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> ( v6_zf_lang(B)
=> ( ~ ( A = k23_zf_lang(B)
& ! [D] :
( ( v1_zf_lang(D)
& m2_finseq_1(D,k5_numbers) )
=> k8_zf_lang(A,D) != B ) )
& ( ? [D] :
( v1_zf_lang(D)
& m2_finseq_1(D,k5_numbers)
& k8_zf_lang(A,D) = B )
=> A = k23_zf_lang(B) )
& ~ ( C = k24_zf_lang(B)
& ! [D] :
( m2_subset_1(D,k5_numbers,k1_zf_lang)
=> k8_zf_lang(D,C) != B ) )
& ( ? [D] :
( m2_subset_1(D,k5_numbers,k1_zf_lang)
& k8_zf_lang(D,C) = B )
=> C = k24_zf_lang(B) ) ) ) ) ) ) ).
fof(t61_zf_lang,axiom,
! [A] :
( m2_subset_1(A,k5_numbers,k1_zf_lang)
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> ( v11_zf_lang(B)
=> ( ~ ( A = k23_zf_lang(B)
& ! [D] :
( ( v1_zf_lang(D)
& m2_finseq_1(D,k5_numbers) )
=> k13_zf_lang(A,D) != B ) )
& ( ? [D] :
( v1_zf_lang(D)
& m2_finseq_1(D,k5_numbers)
& k13_zf_lang(A,D) = B )
=> A = k23_zf_lang(B) )
& ~ ( C = k24_zf_lang(B)
& ! [D] :
( m2_subset_1(D,k5_numbers,k1_zf_lang)
=> k13_zf_lang(D,C) != B ) )
& ( ? [D] :
( m2_subset_1(D,k5_numbers,k1_zf_lang)
& k13_zf_lang(D,C) = B )
=> C = k24_zf_lang(B) ) ) ) ) ) ) ).
fof(t62_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( v6_zf_lang(A)
=> A = k8_zf_lang(k23_zf_lang(A),k24_zf_lang(A)) ) ) ).
fof(t63_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( v11_zf_lang(A)
=> A = k13_zf_lang(k23_zf_lang(A),k24_zf_lang(A)) ) ) ).
fof(d35_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( v9_zf_lang(A)
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ( B = k25_zf_lang(A)
<=> ? [C] :
( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers)
& A = k11_zf_lang(B,C) ) ) ) ) ) ).
fof(d36_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( v9_zf_lang(A)
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ( B = k26_zf_lang(A)
<=> ? [C] :
( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers)
& A = k11_zf_lang(C,B) ) ) ) ) ) ).
fof(t64_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ( v9_zf_lang(A)
=> ( ~ ( B = k25_zf_lang(A)
& ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> A != k11_zf_lang(B,C) ) )
& ( ? [C] :
( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers)
& A = k11_zf_lang(B,C) )
=> B = k25_zf_lang(A) )
& ~ ( B = k26_zf_lang(A)
& ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> A != k11_zf_lang(C,B) ) )
& ( ? [C] :
( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers)
& A = k11_zf_lang(C,B) )
=> B = k26_zf_lang(A) ) ) ) ) ) ).
fof(t65_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( v9_zf_lang(A)
=> A = k11_zf_lang(k25_zf_lang(A),k26_zf_lang(A)) ) ) ).
fof(d37_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( v10_zf_lang(A)
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ( B = k27_zf_lang(A)
<=> ? [C] :
( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers)
& A = k12_zf_lang(B,C) ) ) ) ) ) ).
fof(d38_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( v10_zf_lang(A)
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ( B = k28_zf_lang(A)
<=> ? [C] :
( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers)
& A = k12_zf_lang(C,B) ) ) ) ) ) ).
fof(t66_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ( v10_zf_lang(A)
=> ( ~ ( B = k27_zf_lang(A)
& ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> A != k12_zf_lang(B,C) ) )
& ( ? [C] :
( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers)
& A = k12_zf_lang(B,C) )
=> B = k27_zf_lang(A) )
& ~ ( B = k28_zf_lang(A)
& ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> A != k12_zf_lang(C,B) ) )
& ( ? [C] :
( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers)
& A = k12_zf_lang(C,B) )
=> B = k28_zf_lang(A) ) ) ) ) ) ).
fof(t67_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( v10_zf_lang(A)
=> A = k12_zf_lang(k27_zf_lang(A),k28_zf_lang(A)) ) ) ).
fof(d39_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ( r1_zf_lang(A,B)
<=> ~ ( B != k6_zf_lang(A)
& ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> ( B != k7_zf_lang(A,C)
& B != k7_zf_lang(C,A) ) )
& ! [C] :
( m2_subset_1(C,k5_numbers,k1_zf_lang)
=> B != k8_zf_lang(C,A) ) ) ) ) ) ).
fof(t68_zf_lang,axiom,
$true ).
fof(t69_zf_lang,axiom,
! [A] :
( m2_subset_1(A,k5_numbers,k1_zf_lang)
=> ! [B] :
( m2_subset_1(B,k5_numbers,k1_zf_lang)
=> ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> ~ r1_zf_lang(C,k4_zf_lang(A,B)) ) ) ) ).
fof(t70_zf_lang,axiom,
! [A] :
( m2_subset_1(A,k5_numbers,k1_zf_lang)
=> ! [B] :
( m2_subset_1(B,k5_numbers,k1_zf_lang)
=> ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> ~ r1_zf_lang(C,k5_zf_lang(A,B)) ) ) ) ).
fof(t71_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ( r1_zf_lang(A,k6_zf_lang(B))
<=> A = B ) ) ) ).
fof(t72_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> ( r1_zf_lang(A,k7_zf_lang(B,C))
<=> ( A = B
| A = C ) ) ) ) ) ).
fof(t73_zf_lang,axiom,
! [A] :
( m2_subset_1(A,k5_numbers,k1_zf_lang)
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> ( r1_zf_lang(B,k8_zf_lang(A,C))
<=> B = C ) ) ) ) ).
fof(t74_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ~ ( v7_zf_lang(A)
& r1_zf_lang(B,A) ) ) ) ).
fof(t75_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ( v4_zf_lang(A)
=> ( r1_zf_lang(B,A)
<=> B = k20_zf_lang(A) ) ) ) ) ).
fof(t76_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ( v5_zf_lang(A)
=> ( r1_zf_lang(B,A)
<=> ( B = k21_zf_lang(A)
| B = k22_zf_lang(A) ) ) ) ) ) ).
fof(t77_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ( v6_zf_lang(A)
=> ( r1_zf_lang(B,A)
<=> B = k24_zf_lang(A) ) ) ) ) ).
fof(d40_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ( r2_zf_lang(A,B)
<=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& ? [D] :
( v1_relat_1(D)
& v1_funct_1(D)
& v1_finseq_1(D)
& r1_xreal_0(np__1,C)
& k3_finseq_1(D) = C
& k1_funct_1(D,np__1) = A
& k1_funct_1(D,C) = B
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__1,E)
& ~ r1_xreal_0(C,E)
& ! [F] :
( ( v1_zf_lang(F)
& m2_finseq_1(F,k5_numbers) )
=> ! [G] :
( ( v1_zf_lang(G)
& m2_finseq_1(G,k5_numbers) )
=> ~ ( k1_funct_1(D,E) = F
& k1_funct_1(D,k1_nat_1(E,np__1)) = G
& r1_zf_lang(F,G) ) ) ) ) ) ) ) ) ) ) ).
fof(t78_zf_lang,axiom,
$true ).
fof(t79_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> r2_zf_lang(A,A) ) ).
fof(d41_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ( r3_zf_lang(A,B)
<=> ( r2_zf_lang(A,B)
& A != B ) ) ) ) ).
fof(t80_zf_lang,axiom,
$true ).
fof(t81_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ~ ( r1_zf_lang(A,B)
& r1_xreal_0(k3_finseq_1(B),k3_finseq_1(A)) ) ) ) ).
fof(t82_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ( r1_zf_lang(A,B)
=> r3_zf_lang(A,B) ) ) ) ).
fof(t83_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ~ ( r3_zf_lang(A,B)
& r1_xreal_0(k3_finseq_1(B),k3_finseq_1(A)) ) ) ) ).
fof(t84_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ~ ( r3_zf_lang(A,B)
& ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> ~ r1_zf_lang(C,B) ) ) ) ) ).
fof(t85_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> ( ( r3_zf_lang(A,B)
& r3_zf_lang(B,C) )
=> r3_zf_lang(A,C) ) ) ) ) ).
fof(t86_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> ( ( r2_zf_lang(A,B)
& r2_zf_lang(B,C) )
=> r2_zf_lang(A,C) ) ) ) ) ).
fof(t87_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ( ( r2_zf_lang(A,B)
& r2_zf_lang(B,A) )
=> A = B ) ) ) ).
fof(t88_zf_lang,axiom,
! [A] :
( m2_subset_1(A,k5_numbers,k1_zf_lang)
=> ! [B] :
( m2_subset_1(B,k5_numbers,k1_zf_lang)
=> ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> ~ r3_zf_lang(C,k4_zf_lang(A,B)) ) ) ) ).
fof(t89_zf_lang,axiom,
! [A] :
( m2_subset_1(A,k5_numbers,k1_zf_lang)
=> ! [B] :
( m2_subset_1(B,k5_numbers,k1_zf_lang)
=> ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> ~ r3_zf_lang(C,k5_zf_lang(A,B)) ) ) ) ).
fof(t90_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ( r3_zf_lang(A,k6_zf_lang(B))
=> r2_zf_lang(A,B) ) ) ) ).
fof(t91_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> ~ ( r3_zf_lang(A,k7_zf_lang(B,C))
& ~ r2_zf_lang(A,B)
& ~ r2_zf_lang(A,C) ) ) ) ) ).
fof(t92_zf_lang,axiom,
! [A] :
( m2_subset_1(A,k5_numbers,k1_zf_lang)
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> ( r3_zf_lang(B,k8_zf_lang(A,C))
=> r2_zf_lang(B,C) ) ) ) ) ).
fof(t93_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ~ ( v7_zf_lang(A)
& r3_zf_lang(B,A) ) ) ) ).
fof(t94_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( v4_zf_lang(A)
=> r3_zf_lang(k20_zf_lang(A),A) ) ) ).
fof(t95_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( v5_zf_lang(A)
=> ( r3_zf_lang(k21_zf_lang(A),A)
& r3_zf_lang(k22_zf_lang(A),A) ) ) ) ).
fof(t96_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( v6_zf_lang(A)
=> r3_zf_lang(k24_zf_lang(A),A) ) ) ).
fof(t97_zf_lang,axiom,
! [A] :
( m2_subset_1(A,k5_numbers,k1_zf_lang)
=> ! [B] :
( m2_subset_1(B,k5_numbers,k1_zf_lang)
=> ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> ( r2_zf_lang(C,k4_zf_lang(A,B))
<=> C = k4_zf_lang(A,B) ) ) ) ) ).
fof(t98_zf_lang,axiom,
! [A] :
( m2_subset_1(A,k5_numbers,k1_zf_lang)
=> ! [B] :
( m2_subset_1(B,k5_numbers,k1_zf_lang)
=> ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> ( r2_zf_lang(C,k5_zf_lang(A,B))
<=> C = k5_zf_lang(A,B) ) ) ) ) ).
fof(d42_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( B = k29_zf_lang(A)
<=> ! [C] :
( r2_hidden(C,B)
<=> ? [D] :
( v1_zf_lang(D)
& m2_finseq_1(D,k5_numbers)
& D = C
& r2_zf_lang(D,A) ) ) ) ) ).
fof(t99_zf_lang,axiom,
$true ).
fof(t100_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ( r2_hidden(A,k29_zf_lang(B))
=> r2_zf_lang(A,B) ) ) ) ).
fof(t101_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ( r2_zf_lang(A,B)
=> r1_tarski(k29_zf_lang(A),k29_zf_lang(B)) ) ) ) ).
fof(t102_zf_lang,axiom,
! [A] :
( m2_subset_1(A,k5_numbers,k1_zf_lang)
=> ! [B] :
( m2_subset_1(B,k5_numbers,k1_zf_lang)
=> k29_zf_lang(k4_zf_lang(A,B)) = k1_tarski(k4_zf_lang(A,B)) ) ) ).
fof(t103_zf_lang,axiom,
! [A] :
( m2_subset_1(A,k5_numbers,k1_zf_lang)
=> ! [B] :
( m2_subset_1(B,k5_numbers,k1_zf_lang)
=> k29_zf_lang(k5_zf_lang(A,B)) = k1_tarski(k5_zf_lang(A,B)) ) ) ).
fof(t104_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> k29_zf_lang(k6_zf_lang(A)) = k2_xboole_0(k29_zf_lang(A),k1_tarski(k6_zf_lang(A))) ) ).
fof(t105_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> k29_zf_lang(k7_zf_lang(A,B)) = k2_xboole_0(k2_xboole_0(k29_zf_lang(A),k29_zf_lang(B)),k1_tarski(k7_zf_lang(A,B))) ) ) ).
fof(t106_zf_lang,axiom,
! [A] :
( m2_subset_1(A,k5_numbers,k1_zf_lang)
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> k29_zf_lang(k8_zf_lang(A,B)) = k2_xboole_0(k29_zf_lang(B),k1_tarski(k8_zf_lang(A,B))) ) ) ).
fof(t107_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( v7_zf_lang(A)
<=> k29_zf_lang(A) = k1_tarski(A) ) ) ).
fof(t108_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( v4_zf_lang(A)
=> k29_zf_lang(A) = k2_xboole_0(k29_zf_lang(k20_zf_lang(A)),k1_tarski(A)) ) ) ).
fof(t109_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( v5_zf_lang(A)
=> k29_zf_lang(A) = k2_xboole_0(k2_xboole_0(k29_zf_lang(k21_zf_lang(A)),k29_zf_lang(k22_zf_lang(A))),k1_tarski(A)) ) ) ).
fof(t110_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( v6_zf_lang(A)
=> k29_zf_lang(A) = k2_xboole_0(k29_zf_lang(k24_zf_lang(A)),k1_tarski(A)) ) ) ).
fof(t111_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> ( r2_hidden(B,k29_zf_lang(C))
=> ( ( ~ r1_zf_lang(A,B)
& ~ r3_zf_lang(A,B)
& ~ r2_zf_lang(A,B) )
| r2_hidden(A,k29_zf_lang(C)) ) ) ) ) ) ).
fof(s1_zf_lang,axiom,
( ( ! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( v7_zf_lang(A)
=> p1_s1_zf_lang(A) ) )
& ! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( ( v4_zf_lang(A)
& p1_s1_zf_lang(k20_zf_lang(A)) )
=> p1_s1_zf_lang(A) ) )
& ! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( ( v5_zf_lang(A)
& p1_s1_zf_lang(k21_zf_lang(A))
& p1_s1_zf_lang(k22_zf_lang(A)) )
=> p1_s1_zf_lang(A) ) )
& ! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( ( v6_zf_lang(A)
& p1_s1_zf_lang(k24_zf_lang(A)) )
=> p1_s1_zf_lang(A) ) ) )
=> ! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> p1_s1_zf_lang(A) ) ) ).
fof(s2_zf_lang,axiom,
( ! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ( ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ( r3_zf_lang(B,A)
=> p1_s2_zf_lang(B) ) )
=> p1_s2_zf_lang(A) ) )
=> ! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> p1_s2_zf_lang(A) ) ) ).
fof(dt_k1_zf_lang,axiom,
m1_subset_1(k1_zf_lang,k1_zfmisc_1(k5_numbers)) ).
fof(dt_k2_zf_lang,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> m2_subset_1(k2_zf_lang(A),k5_numbers,k1_zf_lang) ) ).
fof(dt_k3_zf_lang,axiom,
! [A] :
( m1_subset_1(A,k1_zf_lang)
=> m2_finseq_1(k3_zf_lang(A),k5_numbers) ) ).
fof(redefinition_k3_zf_lang,axiom,
! [A] :
( m1_subset_1(A,k1_zf_lang)
=> k3_zf_lang(A) = k5_finseq_1(A) ) ).
fof(dt_k4_zf_lang,axiom,
! [A,B] :
( ( m1_subset_1(A,k1_zf_lang)
& m1_subset_1(B,k1_zf_lang) )
=> m2_finseq_1(k4_zf_lang(A,B),k5_numbers) ) ).
fof(dt_k5_zf_lang,axiom,
! [A,B] :
( ( m1_subset_1(A,k1_zf_lang)
& m1_subset_1(B,k1_zf_lang) )
=> m2_finseq_1(k5_zf_lang(A,B),k5_numbers) ) ).
fof(dt_k6_zf_lang,axiom,
! [A] :
( m1_finseq_1(A,k5_numbers)
=> m2_finseq_1(k6_zf_lang(A),k5_numbers) ) ).
fof(dt_k7_zf_lang,axiom,
! [A,B] :
( ( m1_finseq_1(A,k5_numbers)
& m1_finseq_1(B,k5_numbers) )
=> m2_finseq_1(k7_zf_lang(A,B),k5_numbers) ) ).
fof(dt_k8_zf_lang,axiom,
! [A,B] :
( ( m1_subset_1(A,k1_zf_lang)
& m1_finseq_1(B,k5_numbers) )
=> m2_finseq_1(k8_zf_lang(A,B),k5_numbers) ) ).
fof(dt_k9_zf_lang,axiom,
~ v1_xboole_0(k9_zf_lang) ).
fof(dt_k10_zf_lang,axiom,
! [A,B] :
( ( v1_zf_lang(A)
& m1_finseq_1(A,k5_numbers)
& v1_zf_lang(B)
& m1_finseq_1(B,k5_numbers) )
=> ( v1_zf_lang(k10_zf_lang(A,B))
& m2_finseq_1(k10_zf_lang(A,B),k5_numbers) ) ) ).
fof(dt_k11_zf_lang,axiom,
! [A,B] :
( ( v1_zf_lang(A)
& m1_finseq_1(A,k5_numbers)
& v1_zf_lang(B)
& m1_finseq_1(B,k5_numbers) )
=> ( v1_zf_lang(k11_zf_lang(A,B))
& m2_finseq_1(k11_zf_lang(A,B),k5_numbers) ) ) ).
fof(dt_k12_zf_lang,axiom,
! [A,B] :
( ( v1_zf_lang(A)
& m1_finseq_1(A,k5_numbers)
& v1_zf_lang(B)
& m1_finseq_1(B,k5_numbers) )
=> ( v1_zf_lang(k12_zf_lang(A,B))
& m2_finseq_1(k12_zf_lang(A,B),k5_numbers) ) ) ).
fof(dt_k13_zf_lang,axiom,
! [A,B] :
( ( m1_subset_1(A,k1_zf_lang)
& v1_zf_lang(B)
& m1_finseq_1(B,k5_numbers) )
=> ( v1_zf_lang(k13_zf_lang(A,B))
& m2_finseq_1(k13_zf_lang(A,B),k5_numbers) ) ) ).
fof(dt_k14_zf_lang,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k1_zf_lang)
& m1_subset_1(B,k1_zf_lang)
& v1_zf_lang(C)
& m1_finseq_1(C,k5_numbers) )
=> ( v1_zf_lang(k14_zf_lang(A,B,C))
& m2_finseq_1(k14_zf_lang(A,B,C),k5_numbers) ) ) ).
fof(dt_k15_zf_lang,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k1_zf_lang)
& m1_subset_1(B,k1_zf_lang)
& v1_zf_lang(C)
& m1_finseq_1(C,k5_numbers) )
=> ( v1_zf_lang(k15_zf_lang(A,B,C))
& m2_finseq_1(k15_zf_lang(A,B,C),k5_numbers) ) ) ).
fof(dt_k16_zf_lang,axiom,
! [A,B,C,D] :
( ( m1_subset_1(A,k1_zf_lang)
& m1_subset_1(B,k1_zf_lang)
& m1_subset_1(C,k1_zf_lang)
& v1_zf_lang(D)
& m1_finseq_1(D,k5_numbers) )
=> ( v1_zf_lang(k16_zf_lang(A,B,C,D))
& m2_finseq_1(k16_zf_lang(A,B,C,D),k5_numbers) ) ) ).
fof(dt_k17_zf_lang,axiom,
! [A,B,C,D] :
( ( m1_subset_1(A,k1_zf_lang)
& m1_subset_1(B,k1_zf_lang)
& m1_subset_1(C,k1_zf_lang)
& v1_zf_lang(D)
& m1_finseq_1(D,k5_numbers) )
=> ( v1_zf_lang(k17_zf_lang(A,B,C,D))
& m2_finseq_1(k17_zf_lang(A,B,C,D),k5_numbers) ) ) ).
fof(dt_k18_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m1_finseq_1(A,k5_numbers) )
=> m2_subset_1(k18_zf_lang(A),k5_numbers,k1_zf_lang) ) ).
fof(dt_k19_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m1_finseq_1(A,k5_numbers) )
=> m2_subset_1(k19_zf_lang(A),k5_numbers,k1_zf_lang) ) ).
fof(dt_k20_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m1_finseq_1(A,k5_numbers) )
=> ( v1_zf_lang(k20_zf_lang(A))
& m2_finseq_1(k20_zf_lang(A),k5_numbers) ) ) ).
fof(dt_k21_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m1_finseq_1(A,k5_numbers) )
=> ( v1_zf_lang(k21_zf_lang(A))
& m2_finseq_1(k21_zf_lang(A),k5_numbers) ) ) ).
fof(dt_k22_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m1_finseq_1(A,k5_numbers) )
=> ( v1_zf_lang(k22_zf_lang(A))
& m2_finseq_1(k22_zf_lang(A),k5_numbers) ) ) ).
fof(dt_k23_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m1_finseq_1(A,k5_numbers) )
=> m2_subset_1(k23_zf_lang(A),k5_numbers,k1_zf_lang) ) ).
fof(dt_k24_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m1_finseq_1(A,k5_numbers) )
=> ( v1_zf_lang(k24_zf_lang(A))
& m2_finseq_1(k24_zf_lang(A),k5_numbers) ) ) ).
fof(dt_k25_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m1_finseq_1(A,k5_numbers) )
=> ( v1_zf_lang(k25_zf_lang(A))
& m2_finseq_1(k25_zf_lang(A),k5_numbers) ) ) ).
fof(dt_k26_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m1_finseq_1(A,k5_numbers) )
=> ( v1_zf_lang(k26_zf_lang(A))
& m2_finseq_1(k26_zf_lang(A),k5_numbers) ) ) ).
fof(dt_k27_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m1_finseq_1(A,k5_numbers) )
=> ( v1_zf_lang(k27_zf_lang(A))
& m2_finseq_1(k27_zf_lang(A),k5_numbers) ) ) ).
fof(dt_k28_zf_lang,axiom,
! [A] :
( ( v1_zf_lang(A)
& m1_finseq_1(A,k5_numbers) )
=> ( v1_zf_lang(k28_zf_lang(A))
& m2_finseq_1(k28_zf_lang(A),k5_numbers) ) ) ).
fof(dt_k29_zf_lang,axiom,
$true ).
fof(d1_zf_lang,axiom,
k1_zf_lang = a_0_0_zf_lang ).
fof(fraenkel_a_0_0_zf_lang,axiom,
! [A] :
( r2_hidden(A,a_0_0_zf_lang)
<=> ? [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
& A = B
& r1_xreal_0(np__5,B) ) ) ).
%------------------------------------------------------------------------------