SET007 Axioms: SET007+114.ax
%------------------------------------------------------------------------------
% File : SET007+114 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Schemes of Existence of Some Types of Functions
% 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 : scheme1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 26 ( 0 unt; 0 def)
% Number of atoms : 472 ( 81 equ)
% Maximal formula atoms : 42 ( 18 avg)
% Number of connectives : 555 ( 109 ~; 5 |; 267 &)
% ( 13 <=>; 161 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 12 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 56 ( 55 usr; 0 prp; 1-3 aty)
% Number of functors : 117 ( 117 usr; 48 con; 0-4 aty)
% Number of variables : 124 ( 101 !; 23 ?)
% 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_scheme1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ~ ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( A != k2_nat_1(np__2,B)
& A != k1_nat_1(k2_nat_1(np__2,B),np__1) ) ) ) ).
fof(t2_scheme1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ~ ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( A != k2_nat_1(np__3,B)
& A != k1_nat_1(k2_nat_1(np__3,B),np__1)
& A != k1_nat_1(k2_nat_1(np__3,B),np__2) ) ) ) ).
fof(t3_scheme1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ~ ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( A != k2_nat_1(np__4,B)
& A != k1_nat_1(k2_nat_1(np__4,B),np__1)
& A != k1_nat_1(k2_nat_1(np__4,B),np__2)
& A != k1_nat_1(k2_nat_1(np__4,B),np__3) ) ) ) ).
fof(t4_scheme1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ~ ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( A != k2_nat_1(np__5,B)
& A != k1_nat_1(k2_nat_1(np__5,B),np__1)
& A != k1_nat_1(k2_nat_1(np__5,B),np__2)
& A != k1_nat_1(k2_nat_1(np__5,B),np__3)
& A != k1_nat_1(k2_nat_1(np__5,B),np__4) ) ) ) ).
fof(s1_scheme1,axiom,
( ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ? [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
& r1_xreal_0(A,B)
& p1_s1_scheme1(k2_seq_1(k5_numbers,k1_numbers,f1_s1_scheme1,B)) ) )
=> ? [A] :
( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers)
& m1_seqm_3(A,f1_s1_scheme1)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> p1_s1_scheme1(k2_seq_1(k5_numbers,k1_numbers,A,B)) )
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( C = k2_seq_1(k5_numbers,k1_numbers,f1_s1_scheme1,B)
=> p1_s1_scheme1(C) ) )
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,f1_s1_scheme1,B) != k2_seq_1(k5_numbers,k1_numbers,A,C) ) ) ) ) ) ).
fof(s2_scheme1,axiom,
? [A] :
( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( k2_seq_1(k5_numbers,k1_numbers,A,k2_nat_1(np__2,B)) = f1_s2_scheme1(B)
& k2_seq_1(k5_numbers,k1_numbers,A,k1_nat_1(k2_nat_1(np__2,B),np__1)) = f2_s2_scheme1(B) ) ) ) ).
fof(s3_scheme1,axiom,
? [A] :
( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( k2_seq_1(k5_numbers,k1_numbers,A,k2_nat_1(np__3,B)) = f1_s3_scheme1(B)
& k2_seq_1(k5_numbers,k1_numbers,A,k1_nat_1(k2_nat_1(np__3,B),np__1)) = f2_s3_scheme1(B)
& k2_seq_1(k5_numbers,k1_numbers,A,k1_nat_1(k2_nat_1(np__3,B),np__2)) = f3_s3_scheme1(B) ) ) ) ).
fof(s4_scheme1,axiom,
? [A] :
( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( k2_seq_1(k5_numbers,k1_numbers,A,k2_nat_1(np__4,B)) = f1_s4_scheme1(B)
& k2_seq_1(k5_numbers,k1_numbers,A,k1_nat_1(k2_nat_1(np__4,B),np__1)) = f2_s4_scheme1(B)
& k2_seq_1(k5_numbers,k1_numbers,A,k1_nat_1(k2_nat_1(np__4,B),np__2)) = f3_s4_scheme1(B)
& k2_seq_1(k5_numbers,k1_numbers,A,k1_nat_1(k2_nat_1(np__4,B),np__3)) = f4_s4_scheme1(B) ) ) ) ).
fof(s5_scheme1,axiom,
? [A] :
( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( k2_seq_1(k5_numbers,k1_numbers,A,k2_nat_1(np__5,B)) = f1_s5_scheme1(B)
& k2_seq_1(k5_numbers,k1_numbers,A,k1_nat_1(k2_nat_1(np__5,B),np__1)) = f2_s5_scheme1(B)
& k2_seq_1(k5_numbers,k1_numbers,A,k1_nat_1(k2_nat_1(np__5,B),np__2)) = f3_s5_scheme1(B)
& k2_seq_1(k5_numbers,k1_numbers,A,k1_nat_1(k2_nat_1(np__5,B),np__3)) = f4_s5_scheme1(B)
& k2_seq_1(k5_numbers,k1_numbers,A,k1_nat_1(k2_nat_1(np__5,B),np__4)) = f5_s5_scheme1(B) ) ) ) ).
fof(s6_scheme1,axiom,
( ! [A] :
( m1_subset_1(A,f1_s6_scheme1)
=> ~ ( p1_s6_scheme1(A)
& p2_s6_scheme1(A) ) )
=> ? [A] :
( v1_funct_1(A)
& m2_relset_1(A,f1_s6_scheme1,f2_s6_scheme1)
& ! [B] :
( m1_subset_1(B,f1_s6_scheme1)
=> ( r2_hidden(B,k4_relset_1(f1_s6_scheme1,f2_s6_scheme1,A))
<=> ( p1_s6_scheme1(B)
| p2_s6_scheme1(B) ) ) )
& ! [B] :
( m1_subset_1(B,f1_s6_scheme1)
=> ( r2_hidden(B,k4_relset_1(f1_s6_scheme1,f2_s6_scheme1,A))
=> ( ( p1_s6_scheme1(B)
=> k1_funct_1(A,B) = f3_s6_scheme1(B) )
& ( p2_s6_scheme1(B)
=> k1_funct_1(A,B) = f4_s6_scheme1(B) ) ) ) ) ) ) ).
fof(s7_scheme1,axiom,
( ! [A] :
( m1_subset_1(A,f1_s7_scheme1)
=> ( ( p1_s7_scheme1(A)
& p2_s7_scheme1(A) )
=> f3_s7_scheme1(A) = f4_s7_scheme1(A) ) )
=> ? [A] :
( v1_funct_1(A)
& m2_relset_1(A,f1_s7_scheme1,f2_s7_scheme1)
& ! [B] :
( m1_subset_1(B,f1_s7_scheme1)
=> ( r2_hidden(B,k4_relset_1(f1_s7_scheme1,f2_s7_scheme1,A))
<=> ( p1_s7_scheme1(B)
| p2_s7_scheme1(B) ) ) )
& ! [B] :
( m1_subset_1(B,f1_s7_scheme1)
=> ( r2_hidden(B,k4_relset_1(f1_s7_scheme1,f2_s7_scheme1,A))
=> ( ( p1_s7_scheme1(B)
=> k1_funct_1(A,B) = f3_s7_scheme1(B) )
& ( p2_s7_scheme1(B)
=> k1_funct_1(A,B) = f4_s7_scheme1(B) ) ) ) ) ) ) ).
fof(s8_scheme1,axiom,
? [A] :
( v1_funct_1(A)
& m2_relset_1(A,f1_s8_scheme1,f2_s8_scheme1)
& v1_partfun1(A,f1_s8_scheme1,f2_s8_scheme1)
& ! [B] :
( m1_subset_1(B,f1_s8_scheme1)
=> ( r2_hidden(B,k4_relset_1(f1_s8_scheme1,f2_s8_scheme1,A))
=> ( ( p1_s8_scheme1(B)
=> k1_funct_1(A,B) = f3_s8_scheme1(B) )
& ( ~ p1_s8_scheme1(B)
=> k1_funct_1(A,B) = f4_s8_scheme1(B) ) ) ) ) ) ).
fof(s9_scheme1,axiom,
( ! [A] :
( m1_subset_1(A,f1_s9_scheme1)
=> ( ~ ( p1_s9_scheme1(A)
& p2_s9_scheme1(A) )
& ~ ( p1_s9_scheme1(A)
& p3_s9_scheme1(A) )
& ~ ( p2_s9_scheme1(A)
& p3_s9_scheme1(A) ) ) )
=> ? [A] :
( v1_funct_1(A)
& m2_relset_1(A,f1_s9_scheme1,f2_s9_scheme1)
& ! [B] :
( m1_subset_1(B,f1_s9_scheme1)
=> ( r2_hidden(B,k4_relset_1(f1_s9_scheme1,f2_s9_scheme1,A))
<=> ~ ( ~ p1_s9_scheme1(B)
& ~ p2_s9_scheme1(B)
& ~ p3_s9_scheme1(B) ) ) )
& ! [B] :
( m1_subset_1(B,f1_s9_scheme1)
=> ( r2_hidden(B,k4_relset_1(f1_s9_scheme1,f2_s9_scheme1,A))
=> ( ( p1_s9_scheme1(B)
=> k1_funct_1(A,B) = f3_s9_scheme1(B) )
& ( p2_s9_scheme1(B)
=> k1_funct_1(A,B) = f4_s9_scheme1(B) )
& ( p3_s9_scheme1(B)
=> k1_funct_1(A,B) = f5_s9_scheme1(B) ) ) ) ) ) ) ).
fof(s10_scheme1,axiom,
( ! [A] :
( m1_subset_1(A,f1_s10_scheme1)
=> ( ( ( p1_s10_scheme1(A)
& p2_s10_scheme1(A) )
=> f3_s10_scheme1(A) = f4_s10_scheme1(A) )
& ( ( p1_s10_scheme1(A)
& p3_s10_scheme1(A) )
=> f3_s10_scheme1(A) = f5_s10_scheme1(A) )
& ( ( p2_s10_scheme1(A)
& p3_s10_scheme1(A) )
=> f4_s10_scheme1(A) = f5_s10_scheme1(A) ) ) )
=> ? [A] :
( v1_funct_1(A)
& m2_relset_1(A,f1_s10_scheme1,f2_s10_scheme1)
& ! [B] :
( m1_subset_1(B,f1_s10_scheme1)
=> ( r2_hidden(B,k4_relset_1(f1_s10_scheme1,f2_s10_scheme1,A))
<=> ~ ( ~ p1_s10_scheme1(B)
& ~ p2_s10_scheme1(B)
& ~ p3_s10_scheme1(B) ) ) )
& ! [B] :
( m1_subset_1(B,f1_s10_scheme1)
=> ( r2_hidden(B,k4_relset_1(f1_s10_scheme1,f2_s10_scheme1,A))
=> ( ( p1_s10_scheme1(B)
=> k1_funct_1(A,B) = f3_s10_scheme1(B) )
& ( p2_s10_scheme1(B)
=> k1_funct_1(A,B) = f4_s10_scheme1(B) )
& ( p3_s10_scheme1(B)
=> k1_funct_1(A,B) = f5_s10_scheme1(B) ) ) ) ) ) ) ).
fof(s11_scheme1,axiom,
( ! [A] :
( m1_subset_1(A,f1_s11_scheme1)
=> ( ~ ( p1_s11_scheme1(A)
& p2_s11_scheme1(A) )
& ~ ( p1_s11_scheme1(A)
& p3_s11_scheme1(A) )
& ~ ( p1_s11_scheme1(A)
& p4_s11_scheme1(A) )
& ~ ( p2_s11_scheme1(A)
& p3_s11_scheme1(A) )
& ~ ( p2_s11_scheme1(A)
& p4_s11_scheme1(A) )
& ~ ( p3_s11_scheme1(A)
& p4_s11_scheme1(A) ) ) )
=> ? [A] :
( v1_funct_1(A)
& m2_relset_1(A,f1_s11_scheme1,f2_s11_scheme1)
& ! [B] :
( m1_subset_1(B,f1_s11_scheme1)
=> ( r2_hidden(B,k4_relset_1(f1_s11_scheme1,f2_s11_scheme1,A))
<=> ~ ( ~ p1_s11_scheme1(B)
& ~ p2_s11_scheme1(B)
& ~ p3_s11_scheme1(B)
& ~ p4_s11_scheme1(B) ) ) )
& ! [B] :
( m1_subset_1(B,f1_s11_scheme1)
=> ( r2_hidden(B,k4_relset_1(f1_s11_scheme1,f2_s11_scheme1,A))
=> ( ( p1_s11_scheme1(B)
=> k1_funct_1(A,B) = f3_s11_scheme1(B) )
& ( p2_s11_scheme1(B)
=> k1_funct_1(A,B) = f4_s11_scheme1(B) )
& ( p3_s11_scheme1(B)
=> k1_funct_1(A,B) = f5_s11_scheme1(B) )
& ( p4_s11_scheme1(B)
=> k1_funct_1(A,B) = f6_s11_scheme1(B) ) ) ) ) ) ) ).
fof(s12_scheme1,axiom,
( ( ! [A] :
~ ( r2_hidden(A,f1_s12_scheme1)
& p1_s12_scheme1(A)
& p2_s12_scheme1(A) )
& ! [A] :
( ( r2_hidden(A,f1_s12_scheme1)
& p1_s12_scheme1(A) )
=> r2_hidden(f3_s12_scheme1(A),f2_s12_scheme1) )
& ! [A] :
( ( r2_hidden(A,f1_s12_scheme1)
& p2_s12_scheme1(A) )
=> r2_hidden(f4_s12_scheme1(A),f2_s12_scheme1) ) )
=> ? [A] :
( v1_funct_1(A)
& m2_relset_1(A,f1_s12_scheme1,f2_s12_scheme1)
& ! [B] :
( r2_hidden(B,k4_relset_1(f1_s12_scheme1,f2_s12_scheme1,A))
<=> ( r2_hidden(B,f1_s12_scheme1)
& ( p1_s12_scheme1(B)
| p2_s12_scheme1(B) ) ) )
& ! [B] :
( r2_hidden(B,k4_relset_1(f1_s12_scheme1,f2_s12_scheme1,A))
=> ( ( p1_s12_scheme1(B)
=> k1_funct_1(A,B) = f3_s12_scheme1(B) )
& ( p2_s12_scheme1(B)
=> k1_funct_1(A,B) = f4_s12_scheme1(B) ) ) ) ) ) ).
fof(s13_scheme1,axiom,
( ( ! [A] :
( r2_hidden(A,f1_s13_scheme1)
=> ( ~ ( p1_s13_scheme1(A)
& p2_s13_scheme1(A) )
& ~ ( p1_s13_scheme1(A)
& p3_s13_scheme1(A) )
& ~ ( p2_s13_scheme1(A)
& p3_s13_scheme1(A) ) ) )
& ! [A] :
( ( r2_hidden(A,f1_s13_scheme1)
& p1_s13_scheme1(A) )
=> r2_hidden(f3_s13_scheme1(A),f2_s13_scheme1) )
& ! [A] :
( ( r2_hidden(A,f1_s13_scheme1)
& p2_s13_scheme1(A) )
=> r2_hidden(f4_s13_scheme1(A),f2_s13_scheme1) )
& ! [A] :
( ( r2_hidden(A,f1_s13_scheme1)
& p3_s13_scheme1(A) )
=> r2_hidden(f5_s13_scheme1(A),f2_s13_scheme1) ) )
=> ? [A] :
( v1_funct_1(A)
& m2_relset_1(A,f1_s13_scheme1,f2_s13_scheme1)
& ! [B] :
( r2_hidden(B,k4_relset_1(f1_s13_scheme1,f2_s13_scheme1,A))
<=> ( r2_hidden(B,f1_s13_scheme1)
& ~ ( ~ p1_s13_scheme1(B)
& ~ p2_s13_scheme1(B)
& ~ p3_s13_scheme1(B) ) ) )
& ! [B] :
( r2_hidden(B,k4_relset_1(f1_s13_scheme1,f2_s13_scheme1,A))
=> ( ( p1_s13_scheme1(B)
=> k1_funct_1(A,B) = f3_s13_scheme1(B) )
& ( p2_s13_scheme1(B)
=> k1_funct_1(A,B) = f4_s13_scheme1(B) )
& ( p3_s13_scheme1(B)
=> k1_funct_1(A,B) = f5_s13_scheme1(B) ) ) ) ) ) ).
fof(s14_scheme1,axiom,
( ( ! [A] :
( r2_hidden(A,f1_s14_scheme1)
=> ( ~ ( p1_s14_scheme1(A)
& p2_s14_scheme1(A) )
& ~ ( p1_s14_scheme1(A)
& p3_s14_scheme1(A) )
& ~ ( p1_s14_scheme1(A)
& p4_s14_scheme1(A) )
& ~ ( p2_s14_scheme1(A)
& p3_s14_scheme1(A) )
& ~ ( p2_s14_scheme1(A)
& p4_s14_scheme1(A) )
& ~ ( p3_s14_scheme1(A)
& p4_s14_scheme1(A) ) ) )
& ! [A] :
( ( r2_hidden(A,f1_s14_scheme1)
& p1_s14_scheme1(A) )
=> r2_hidden(f3_s14_scheme1(A),f2_s14_scheme1) )
& ! [A] :
( ( r2_hidden(A,f1_s14_scheme1)
& p2_s14_scheme1(A) )
=> r2_hidden(f4_s14_scheme1(A),f2_s14_scheme1) )
& ! [A] :
( ( r2_hidden(A,f1_s14_scheme1)
& p3_s14_scheme1(A) )
=> r2_hidden(f5_s14_scheme1(A),f2_s14_scheme1) )
& ! [A] :
( ( r2_hidden(A,f1_s14_scheme1)
& p4_s14_scheme1(A) )
=> r2_hidden(f6_s14_scheme1(A),f2_s14_scheme1) ) )
=> ? [A] :
( v1_funct_1(A)
& m2_relset_1(A,f1_s14_scheme1,f2_s14_scheme1)
& ! [B] :
( r2_hidden(B,k4_relset_1(f1_s14_scheme1,f2_s14_scheme1,A))
<=> ( r2_hidden(B,f1_s14_scheme1)
& ~ ( ~ p1_s14_scheme1(B)
& ~ p2_s14_scheme1(B)
& ~ p3_s14_scheme1(B)
& ~ p4_s14_scheme1(B) ) ) )
& ! [B] :
( r2_hidden(B,k4_relset_1(f1_s14_scheme1,f2_s14_scheme1,A))
=> ( ( p1_s14_scheme1(B)
=> k1_funct_1(A,B) = f3_s14_scheme1(B) )
& ( p2_s14_scheme1(B)
=> k1_funct_1(A,B) = f4_s14_scheme1(B) )
& ( p3_s14_scheme1(B)
=> k1_funct_1(A,B) = f5_s14_scheme1(B) )
& ( p4_s14_scheme1(B)
=> k1_funct_1(A,B) = f6_s14_scheme1(B) ) ) ) ) ) ).
fof(s15_scheme1,axiom,
( ! [A] :
( m1_subset_1(A,f1_s15_scheme1)
=> ! [B] :
( m1_subset_1(B,f2_s15_scheme1)
=> ~ ( p1_s15_scheme1(A,B)
& p2_s15_scheme1(A,B) ) ) )
=> ? [A] :
( v1_funct_1(A)
& m2_relset_1(A,k2_zfmisc_1(f1_s15_scheme1,f2_s15_scheme1),f3_s15_scheme1)
& ! [B] :
( m1_subset_1(B,f1_s15_scheme1)
=> ! [C] :
( m1_subset_1(C,f2_s15_scheme1)
=> ( r2_hidden(k4_tarski(B,C),k4_relset_1(k2_zfmisc_1(f1_s15_scheme1,f2_s15_scheme1),f3_s15_scheme1,A))
<=> ( p1_s15_scheme1(B,C)
| p2_s15_scheme1(B,C) ) ) ) )
& ! [B] :
( m1_subset_1(B,f1_s15_scheme1)
=> ! [C] :
( m1_subset_1(C,f2_s15_scheme1)
=> ( r2_hidden(k4_tarski(B,C),k4_relset_1(k2_zfmisc_1(f1_s15_scheme1,f2_s15_scheme1),f3_s15_scheme1,A))
=> ( ( p1_s15_scheme1(B,C)
=> k1_funct_1(A,k4_tarski(B,C)) = f4_s15_scheme1(B,C) )
& ( p2_s15_scheme1(B,C)
=> k1_funct_1(A,k4_tarski(B,C)) = f5_s15_scheme1(B,C) ) ) ) ) ) ) ) ).
fof(s16_scheme1,axiom,
( ! [A] :
( m1_subset_1(A,f1_s16_scheme1)
=> ! [B] :
( m1_subset_1(B,f2_s16_scheme1)
=> ( ~ ( p1_s16_scheme1(A,B)
& p2_s16_scheme1(A,B) )
& ~ ( p1_s16_scheme1(A,B)
& p3_s16_scheme1(A,B) )
& ~ ( p2_s16_scheme1(A,B)
& p3_s16_scheme1(A,B) ) ) ) )
=> ? [A] :
( v1_funct_1(A)
& m2_relset_1(A,k2_zfmisc_1(f1_s16_scheme1,f2_s16_scheme1),f3_s16_scheme1)
& ! [B] :
( m1_subset_1(B,f1_s16_scheme1)
=> ! [C] :
( m1_subset_1(C,f2_s16_scheme1)
=> ( r2_hidden(k4_tarski(B,C),k4_relset_1(k2_zfmisc_1(f1_s16_scheme1,f2_s16_scheme1),f3_s16_scheme1,A))
<=> ~ ( ~ p1_s16_scheme1(B,C)
& ~ p2_s16_scheme1(B,C)
& ~ p3_s16_scheme1(B,C) ) ) ) )
& ! [B] :
( m1_subset_1(B,f1_s16_scheme1)
=> ! [C] :
( m1_subset_1(C,f2_s16_scheme1)
=> ( r2_hidden(k4_tarski(B,C),k4_relset_1(k2_zfmisc_1(f1_s16_scheme1,f2_s16_scheme1),f3_s16_scheme1,A))
=> ( ( p1_s16_scheme1(B,C)
=> k1_funct_1(A,k4_tarski(B,C)) = f4_s16_scheme1(B,C) )
& ( p2_s16_scheme1(B,C)
=> k1_funct_1(A,k4_tarski(B,C)) = f5_s16_scheme1(B,C) )
& ( p3_s16_scheme1(B,C)
=> k1_funct_1(A,k4_tarski(B,C)) = f6_s16_scheme1(B,C) ) ) ) ) ) ) ) ).
fof(s17_scheme1,axiom,
( ( ! [A,B] :
~ ( r2_hidden(A,f1_s17_scheme1)
& r2_hidden(B,f2_s17_scheme1)
& p1_s17_scheme1(A,B)
& p2_s17_scheme1(A,B) )
& ! [A,B] :
( ( r2_hidden(A,f1_s17_scheme1)
& r2_hidden(B,f2_s17_scheme1)
& p1_s17_scheme1(A,B) )
=> r2_hidden(f4_s17_scheme1(A,B),f3_s17_scheme1) )
& ! [A,B] :
( ( r2_hidden(A,f1_s17_scheme1)
& r2_hidden(B,f2_s17_scheme1)
& p2_s17_scheme1(A,B) )
=> r2_hidden(f5_s17_scheme1(A,B),f3_s17_scheme1) ) )
=> ? [A] :
( v1_funct_1(A)
& m2_relset_1(A,k2_zfmisc_1(f1_s17_scheme1,f2_s17_scheme1),f3_s17_scheme1)
& ! [B,C] :
( r2_hidden(k4_tarski(B,C),k4_relset_1(k2_zfmisc_1(f1_s17_scheme1,f2_s17_scheme1),f3_s17_scheme1,A))
<=> ( r2_hidden(B,f1_s17_scheme1)
& r2_hidden(C,f2_s17_scheme1)
& ( p1_s17_scheme1(B,C)
| p2_s17_scheme1(B,C) ) ) )
& ! [B,C] :
( r2_hidden(k4_tarski(B,C),k4_relset_1(k2_zfmisc_1(f1_s17_scheme1,f2_s17_scheme1),f3_s17_scheme1,A))
=> ( ( p1_s17_scheme1(B,C)
=> k1_funct_1(A,k4_tarski(B,C)) = f4_s17_scheme1(B,C) )
& ( p2_s17_scheme1(B,C)
=> k1_funct_1(A,k4_tarski(B,C)) = f5_s17_scheme1(B,C) ) ) ) ) ) ).
fof(s18_scheme1,axiom,
( ( ! [A,B] :
( ( r2_hidden(A,f1_s18_scheme1)
& r2_hidden(B,f2_s18_scheme1) )
=> ( ~ ( p1_s18_scheme1(A,B)
& p2_s18_scheme1(A,B) )
& ~ ( p1_s18_scheme1(A,B)
& p3_s18_scheme1(A,B) )
& ~ ( p2_s18_scheme1(A,B)
& p3_s18_scheme1(A,B) ) ) )
& ! [A,B] :
( ( r2_hidden(A,f1_s18_scheme1)
& r2_hidden(B,f2_s18_scheme1)
& p1_s18_scheme1(A,B) )
=> r2_hidden(f4_s18_scheme1(A,B),f3_s18_scheme1) )
& ! [A,B] :
( ( r2_hidden(A,f1_s18_scheme1)
& r2_hidden(B,f2_s18_scheme1)
& p2_s18_scheme1(A,B) )
=> r2_hidden(f5_s18_scheme1(A,B),f3_s18_scheme1) )
& ! [A,B] :
( ( r2_hidden(A,f1_s18_scheme1)
& r2_hidden(B,f2_s18_scheme1)
& p3_s18_scheme1(A,B) )
=> r2_hidden(f6_s18_scheme1(A,B),f3_s18_scheme1) ) )
=> ? [A] :
( v1_funct_1(A)
& m2_relset_1(A,k2_zfmisc_1(f1_s18_scheme1,f2_s18_scheme1),f3_s18_scheme1)
& ! [B,C] :
( r2_hidden(k4_tarski(B,C),k4_relset_1(k2_zfmisc_1(f1_s18_scheme1,f2_s18_scheme1),f3_s18_scheme1,A))
<=> ( r2_hidden(B,f1_s18_scheme1)
& r2_hidden(C,f2_s18_scheme1)
& ~ ( ~ p1_s18_scheme1(B,C)
& ~ p2_s18_scheme1(B,C)
& ~ p3_s18_scheme1(B,C) ) ) )
& ! [B,C] :
( r2_hidden(k4_tarski(B,C),k4_relset_1(k2_zfmisc_1(f1_s18_scheme1,f2_s18_scheme1),f3_s18_scheme1,A))
=> ( ( p1_s18_scheme1(B,C)
=> k1_funct_1(A,k4_tarski(B,C)) = f4_s18_scheme1(B,C) )
& ( p2_s18_scheme1(B,C)
=> k1_funct_1(A,k4_tarski(B,C)) = f5_s18_scheme1(B,C) )
& ( p3_s18_scheme1(B,C)
=> k1_funct_1(A,k4_tarski(B,C)) = f6_s18_scheme1(B,C) ) ) ) ) ) ).
fof(s19_scheme1,axiom,
( ( ! [A] :
( m1_subset_1(A,f1_s19_scheme1)
=> ( ~ ( p1_s19_scheme1(A)
& p2_s19_scheme1(A) )
& ~ ( p1_s19_scheme1(A)
& p3_s19_scheme1(A) )
& ~ ( p2_s19_scheme1(A)
& p3_s19_scheme1(A) ) ) )
& ! [A] :
( m1_subset_1(A,f1_s19_scheme1)
=> ~ ( ~ p1_s19_scheme1(A)
& ~ p2_s19_scheme1(A)
& ~ p3_s19_scheme1(A) ) ) )
=> ? [A] :
( v1_funct_1(A)
& v1_funct_2(A,f1_s19_scheme1,f2_s19_scheme1)
& m2_relset_1(A,f1_s19_scheme1,f2_s19_scheme1)
& ! [B] :
( m1_subset_1(B,f1_s19_scheme1)
=> ( ( p1_s19_scheme1(B)
=> k8_funct_2(f1_s19_scheme1,f2_s19_scheme1,A,B) = f3_s19_scheme1(B) )
& ( p2_s19_scheme1(B)
=> k8_funct_2(f1_s19_scheme1,f2_s19_scheme1,A,B) = f4_s19_scheme1(B) )
& ( p3_s19_scheme1(B)
=> k8_funct_2(f1_s19_scheme1,f2_s19_scheme1,A,B) = f5_s19_scheme1(B) ) ) ) ) ) ).
fof(s20_scheme1,axiom,
( ( ! [A] :
( m1_subset_1(A,f1_s20_scheme1)
=> ( ~ ( p1_s20_scheme1(A)
& p2_s20_scheme1(A) )
& ~ ( p1_s20_scheme1(A)
& p3_s20_scheme1(A) )
& ~ ( p1_s20_scheme1(A)
& p4_s20_scheme1(A) )
& ~ ( p2_s20_scheme1(A)
& p3_s20_scheme1(A) )
& ~ ( p2_s20_scheme1(A)
& p4_s20_scheme1(A) )
& ~ ( p3_s20_scheme1(A)
& p4_s20_scheme1(A) ) ) )
& ! [A] :
( m1_subset_1(A,f1_s20_scheme1)
=> ~ ( ~ p1_s20_scheme1(A)
& ~ p2_s20_scheme1(A)
& ~ p3_s20_scheme1(A)
& ~ p4_s20_scheme1(A) ) ) )
=> ? [A] :
( v1_funct_1(A)
& v1_funct_2(A,f1_s20_scheme1,f2_s20_scheme1)
& m2_relset_1(A,f1_s20_scheme1,f2_s20_scheme1)
& ! [B] :
( m1_subset_1(B,f1_s20_scheme1)
=> ( ( p1_s20_scheme1(B)
=> k8_funct_2(f1_s20_scheme1,f2_s20_scheme1,A,B) = f3_s20_scheme1(B) )
& ( p2_s20_scheme1(B)
=> k8_funct_2(f1_s20_scheme1,f2_s20_scheme1,A,B) = f4_s20_scheme1(B) )
& ( p3_s20_scheme1(B)
=> k8_funct_2(f1_s20_scheme1,f2_s20_scheme1,A,B) = f5_s20_scheme1(B) )
& ( p4_s20_scheme1(B)
=> k8_funct_2(f1_s20_scheme1,f2_s20_scheme1,A,B) = f6_s20_scheme1(B) ) ) ) ) ) ).
fof(s21_scheme1,axiom,
? [A] :
( v1_funct_1(A)
& v1_funct_2(A,k2_zfmisc_1(f1_s21_scheme1,f2_s21_scheme1),f3_s21_scheme1)
& m2_relset_1(A,k2_zfmisc_1(f1_s21_scheme1,f2_s21_scheme1),f3_s21_scheme1)
& ! [B] :
( m1_subset_1(B,f1_s21_scheme1)
=> ! [C] :
( m1_subset_1(C,f2_s21_scheme1)
=> ( r2_hidden(k4_tarski(B,C),k4_relset_1(k2_zfmisc_1(f1_s21_scheme1,f2_s21_scheme1),f3_s21_scheme1,A))
=> ( ( p1_s21_scheme1(B,C)
=> k1_funct_1(A,k4_tarski(B,C)) = f4_s21_scheme1(B,C) )
& ( ~ p1_s21_scheme1(B,C)
=> k1_funct_1(A,k4_tarski(B,C)) = f5_s21_scheme1(B,C) ) ) ) ) ) ) ).
fof(s22_scheme1,axiom,
( ( ! [A] :
( m1_subset_1(A,f1_s22_scheme1)
=> ! [B] :
( m1_subset_1(B,f2_s22_scheme1)
=> ( ~ ( p1_s22_scheme1(A,B)
& p2_s22_scheme1(A,B) )
& ~ ( p1_s22_scheme1(A,B)
& p3_s22_scheme1(A,B) )
& ~ ( p2_s22_scheme1(A,B)
& p3_s22_scheme1(A,B) ) ) ) )
& ! [A] :
( m1_subset_1(A,f1_s22_scheme1)
=> ! [B] :
( m1_subset_1(B,f2_s22_scheme1)
=> ~ ( ~ p1_s22_scheme1(A,B)
& ~ p2_s22_scheme1(A,B)
& ~ p3_s22_scheme1(A,B) ) ) ) )
=> ? [A] :
( v1_funct_1(A)
& v1_funct_2(A,k2_zfmisc_1(f1_s22_scheme1,f2_s22_scheme1),f3_s22_scheme1)
& m2_relset_1(A,k2_zfmisc_1(f1_s22_scheme1,f2_s22_scheme1),f3_s22_scheme1)
& ! [B] :
( m1_subset_1(B,f1_s22_scheme1)
=> ! [C] :
( m1_subset_1(C,f2_s22_scheme1)
=> ( r2_hidden(k4_tarski(B,C),k4_relset_1(k2_zfmisc_1(f1_s22_scheme1,f2_s22_scheme1),f3_s22_scheme1,A))
<=> ~ ( ~ p1_s22_scheme1(B,C)
& ~ p2_s22_scheme1(B,C)
& ~ p3_s22_scheme1(B,C) ) ) ) )
& ! [B] :
( m1_subset_1(B,f1_s22_scheme1)
=> ! [C] :
( m1_subset_1(C,f2_s22_scheme1)
=> ( r2_hidden(k4_tarski(B,C),k4_relset_1(k2_zfmisc_1(f1_s22_scheme1,f2_s22_scheme1),f3_s22_scheme1,A))
=> ( ( p1_s22_scheme1(B,C)
=> k1_funct_1(A,k4_tarski(B,C)) = f4_s22_scheme1(B,C) )
& ( p2_s22_scheme1(B,C)
=> k1_funct_1(A,k4_tarski(B,C)) = f5_s22_scheme1(B,C) )
& ( p3_s22_scheme1(B,C)
=> k1_funct_1(A,k4_tarski(B,C)) = f6_s22_scheme1(B,C) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------