SET007 Axioms: SET007+51.ax
%------------------------------------------------------------------------------
% File : SET007+51 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : On the Sets Inhabited by Numbers
% 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 : membered [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 122 ( 0 unt; 0 def)
% Number of atoms : 520 ( 10 equ)
% Maximal formula atoms : 8 ( 4 avg)
% Number of connectives : 413 ( 15 ~; 0 |; 188 &)
% ( 30 <=>; 180 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 6 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of predicates : 21 ( 20 usr; 0 prp; 1-2 aty)
% Number of functors : 17 ( 17 usr; 7 con; 0-3 aty)
% Number of variables : 231 ( 220 !; 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(cc1_membered,axiom,
! [A] :
( v5_membered(A)
=> v4_membered(A) ) ).
fof(cc2_membered,axiom,
! [A] :
( v4_membered(A)
=> v3_membered(A) ) ).
fof(cc3_membered,axiom,
! [A] :
( v3_membered(A)
=> v2_membered(A) ) ).
fof(cc4_membered,axiom,
! [A] :
( v2_membered(A)
=> v1_membered(A) ) ).
fof(rc1_membered,axiom,
? [A] :
( ~ v1_xboole_0(A)
& v1_membered(A)
& v2_membered(A)
& v3_membered(A)
& v4_membered(A)
& v5_membered(A) ) ).
fof(cc5_membered,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k2_numbers))
=> v1_membered(A) ) ).
fof(cc6_membered,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> ( v1_membered(A)
& v2_membered(A) ) ) ).
fof(cc7_membered,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k3_numbers))
=> ( v1_membered(A)
& v2_membered(A)
& v3_membered(A) ) ) ).
fof(cc8_membered,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k4_numbers))
=> ( v1_membered(A)
& v2_membered(A)
& v3_membered(A)
& v4_membered(A) ) ) ).
fof(cc9_membered,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k5_numbers))
=> ( v1_membered(A)
& v2_membered(A)
& v3_membered(A)
& v4_membered(A)
& v5_membered(A) ) ) ).
fof(fc1_membered,axiom,
( ~ v1_xboole_0(k2_numbers)
& v1_membered(k2_numbers) ) ).
fof(fc2_membered,axiom,
( ~ v1_xboole_0(k1_numbers)
& v1_membered(k1_numbers)
& v2_membered(k1_numbers) ) ).
fof(fc3_membered,axiom,
( ~ v1_xboole_0(k3_numbers)
& v1_membered(k3_numbers)
& v2_membered(k3_numbers)
& v3_membered(k3_numbers) ) ).
fof(fc4_membered,axiom,
( ~ v1_xboole_0(k4_numbers)
& v1_membered(k4_numbers)
& v2_membered(k4_numbers)
& v3_membered(k4_numbers)
& v4_membered(k4_numbers) ) ).
fof(fc5_membered,axiom,
( v1_membered(k5_ordinal2)
& v2_membered(k5_ordinal2)
& v3_membered(k5_ordinal2)
& v4_membered(k5_ordinal2)
& v5_membered(k5_ordinal2) ) ).
fof(cc10_membered,axiom,
! [A] :
( v1_membered(A)
=> ! [B] :
( m1_subset_1(B,A)
=> v1_xcmplx_0(B) ) ) ).
fof(cc11_membered,axiom,
! [A] :
( v2_membered(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ( v1_xcmplx_0(B)
& v1_xreal_0(B) ) ) ) ).
fof(cc12_membered,axiom,
! [A] :
( v3_membered(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ( v1_xcmplx_0(B)
& v1_xreal_0(B)
& v1_rat_1(B) ) ) ) ).
fof(cc13_membered,axiom,
! [A] :
( v4_membered(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ( v1_xcmplx_0(B)
& v1_xreal_0(B)
& v1_int_1(B)
& v1_rat_1(B) ) ) ) ).
fof(cc14_membered,axiom,
! [A] :
( v5_membered(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ( v1_xcmplx_0(B)
& v4_ordinal2(B)
& v1_xreal_0(B)
& v1_int_1(B)
& v1_rat_1(B) ) ) ) ).
fof(fc6_membered,axiom,
( v1_xboole_0(k1_xboole_0)
& v1_membered(k1_xboole_0)
& v2_membered(k1_xboole_0)
& v3_membered(k1_xboole_0)
& v4_membered(k1_xboole_0)
& v5_membered(k1_xboole_0) ) ).
fof(cc15_membered,axiom,
! [A] :
( v1_xboole_0(A)
=> ( v1_membered(A)
& v2_membered(A)
& v3_membered(A)
& v4_membered(A)
& v5_membered(A) ) ) ).
fof(fc7_membered,axiom,
! [A] :
( v1_xcmplx_0(A)
=> v1_membered(k1_tarski(A)) ) ).
fof(fc8_membered,axiom,
! [A] :
( v1_xreal_0(A)
=> ( v1_membered(k1_tarski(A))
& v2_membered(k1_tarski(A)) ) ) ).
fof(fc9_membered,axiom,
! [A] :
( v1_rat_1(A)
=> ( v1_membered(k1_tarski(A))
& v2_membered(k1_tarski(A))
& v3_membered(k1_tarski(A)) ) ) ).
fof(fc10_membered,axiom,
! [A] :
( v1_int_1(A)
=> ( v1_membered(k1_tarski(A))
& v2_membered(k1_tarski(A))
& v3_membered(k1_tarski(A))
& v4_membered(k1_tarski(A)) ) ) ).
fof(fc11_membered,axiom,
! [A] :
( v4_ordinal2(A)
=> ( v1_membered(k1_tarski(A))
& v2_membered(k1_tarski(A))
& v3_membered(k1_tarski(A))
& v4_membered(k1_tarski(A))
& v5_membered(k1_tarski(A)) ) ) ).
fof(fc12_membered,axiom,
! [A,B] :
( ( v1_xcmplx_0(A)
& v1_xcmplx_0(B) )
=> v1_membered(k2_tarski(A,B)) ) ).
fof(fc13_membered,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& v1_xreal_0(B) )
=> ( v1_membered(k2_tarski(A,B))
& v2_membered(k2_tarski(A,B)) ) ) ).
fof(fc14_membered,axiom,
! [A,B] :
( ( v1_rat_1(A)
& v1_rat_1(B) )
=> ( v1_membered(k2_tarski(A,B))
& v2_membered(k2_tarski(A,B))
& v3_membered(k2_tarski(A,B)) ) ) ).
fof(fc15_membered,axiom,
! [A,B] :
( ( v1_int_1(A)
& v1_int_1(B) )
=> ( v1_membered(k2_tarski(A,B))
& v2_membered(k2_tarski(A,B))
& v3_membered(k2_tarski(A,B))
& v4_membered(k2_tarski(A,B)) ) ) ).
fof(fc16_membered,axiom,
! [A,B] :
( ( v4_ordinal2(A)
& v4_ordinal2(B) )
=> ( v1_membered(k2_tarski(A,B))
& v2_membered(k2_tarski(A,B))
& v3_membered(k2_tarski(A,B))
& v4_membered(k2_tarski(A,B))
& v5_membered(k2_tarski(A,B)) ) ) ).
fof(fc17_membered,axiom,
! [A,B,C] :
( ( v1_xcmplx_0(A)
& v1_xcmplx_0(B)
& v1_xcmplx_0(C) )
=> v1_membered(k1_enumset1(A,B,C)) ) ).
fof(fc18_membered,axiom,
! [A,B,C] :
( ( v1_xreal_0(A)
& v1_xreal_0(B)
& v1_xreal_0(C) )
=> ( v1_membered(k1_enumset1(A,B,C))
& v2_membered(k1_enumset1(A,B,C)) ) ) ).
fof(fc19_membered,axiom,
! [A,B,C] :
( ( v1_rat_1(A)
& v1_rat_1(B)
& v1_rat_1(C) )
=> ( v1_membered(k1_enumset1(A,B,C))
& v2_membered(k1_enumset1(A,B,C))
& v3_membered(k1_enumset1(A,B,C)) ) ) ).
fof(fc20_membered,axiom,
! [A,B,C] :
( ( v1_int_1(A)
& v1_int_1(B)
& v1_int_1(C) )
=> ( v1_membered(k1_enumset1(A,B,C))
& v2_membered(k1_enumset1(A,B,C))
& v3_membered(k1_enumset1(A,B,C))
& v4_membered(k1_enumset1(A,B,C)) ) ) ).
fof(fc21_membered,axiom,
! [A,B,C] :
( ( v4_ordinal2(A)
& v4_ordinal2(B)
& v4_ordinal2(C) )
=> ( v1_membered(k1_enumset1(A,B,C))
& v2_membered(k1_enumset1(A,B,C))
& v3_membered(k1_enumset1(A,B,C))
& v4_membered(k1_enumset1(A,B,C))
& v5_membered(k1_enumset1(A,B,C)) ) ) ).
fof(cc16_membered,axiom,
! [A] :
( v1_membered(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> v1_membered(B) ) ) ).
fof(cc17_membered,axiom,
! [A] :
( v2_membered(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ( v1_membered(B)
& v2_membered(B) ) ) ) ).
fof(cc18_membered,axiom,
! [A] :
( v3_membered(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ( v1_membered(B)
& v2_membered(B)
& v3_membered(B) ) ) ) ).
fof(cc19_membered,axiom,
! [A] :
( v4_membered(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ( v1_membered(B)
& v2_membered(B)
& v3_membered(B)
& v4_membered(B) ) ) ) ).
fof(cc20_membered,axiom,
! [A] :
( v5_membered(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ( v1_membered(B)
& v2_membered(B)
& v3_membered(B)
& v4_membered(B)
& v5_membered(B) ) ) ) ).
fof(fc22_membered,axiom,
! [A,B] :
( ( v1_membered(A)
& v1_membered(B) )
=> v1_membered(k2_xboole_0(A,B)) ) ).
fof(fc23_membered,axiom,
! [A,B] :
( ( v2_membered(A)
& v2_membered(B) )
=> ( v1_membered(k2_xboole_0(A,B))
& v2_membered(k2_xboole_0(A,B)) ) ) ).
fof(fc24_membered,axiom,
! [A,B] :
( ( v3_membered(A)
& v3_membered(B) )
=> ( v1_membered(k2_xboole_0(A,B))
& v2_membered(k2_xboole_0(A,B))
& v3_membered(k2_xboole_0(A,B)) ) ) ).
fof(fc25_membered,axiom,
! [A,B] :
( ( v4_membered(A)
& v4_membered(B) )
=> ( v1_membered(k2_xboole_0(A,B))
& v2_membered(k2_xboole_0(A,B))
& v3_membered(k2_xboole_0(A,B))
& v4_membered(k2_xboole_0(A,B)) ) ) ).
fof(fc26_membered,axiom,
! [A,B] :
( ( v5_membered(A)
& v5_membered(B) )
=> ( v1_membered(k2_xboole_0(A,B))
& v2_membered(k2_xboole_0(A,B))
& v3_membered(k2_xboole_0(A,B))
& v4_membered(k2_xboole_0(A,B))
& v5_membered(k2_xboole_0(A,B)) ) ) ).
fof(fc27_membered,axiom,
! [A,B] :
( v1_membered(A)
=> v1_membered(k3_xboole_0(A,B)) ) ).
fof(fc28_membered,axiom,
! [A,B] :
( v1_membered(A)
=> v1_membered(k3_xboole_0(B,A)) ) ).
fof(fc29_membered,axiom,
! [A,B] :
( v2_membered(A)
=> ( v1_membered(k3_xboole_0(A,B))
& v2_membered(k3_xboole_0(A,B)) ) ) ).
fof(fc30_membered,axiom,
! [A,B] :
( v2_membered(A)
=> ( v1_membered(k3_xboole_0(B,A))
& v2_membered(k3_xboole_0(B,A)) ) ) ).
fof(fc31_membered,axiom,
! [A,B] :
( v3_membered(A)
=> ( v1_membered(k3_xboole_0(A,B))
& v2_membered(k3_xboole_0(A,B))
& v3_membered(k3_xboole_0(A,B)) ) ) ).
fof(fc32_membered,axiom,
! [A,B] :
( v3_membered(A)
=> ( v1_membered(k3_xboole_0(B,A))
& v2_membered(k3_xboole_0(B,A))
& v3_membered(k3_xboole_0(B,A)) ) ) ).
fof(fc33_membered,axiom,
! [A,B] :
( v4_membered(A)
=> ( v1_membered(k3_xboole_0(A,B))
& v2_membered(k3_xboole_0(A,B))
& v3_membered(k3_xboole_0(A,B))
& v4_membered(k3_xboole_0(A,B)) ) ) ).
fof(fc34_membered,axiom,
! [A,B] :
( v4_membered(A)
=> ( v1_membered(k3_xboole_0(B,A))
& v2_membered(k3_xboole_0(B,A))
& v3_membered(k3_xboole_0(B,A))
& v4_membered(k3_xboole_0(B,A)) ) ) ).
fof(fc35_membered,axiom,
! [A,B] :
( v5_membered(A)
=> ( v1_membered(k3_xboole_0(A,B))
& v2_membered(k3_xboole_0(A,B))
& v3_membered(k3_xboole_0(A,B))
& v4_membered(k3_xboole_0(A,B))
& v5_membered(k3_xboole_0(A,B)) ) ) ).
fof(fc36_membered,axiom,
! [A,B] :
( v5_membered(A)
=> ( v1_membered(k3_xboole_0(B,A))
& v2_membered(k3_xboole_0(B,A))
& v3_membered(k3_xboole_0(B,A))
& v4_membered(k3_xboole_0(B,A))
& v5_membered(k3_xboole_0(B,A)) ) ) ).
fof(fc37_membered,axiom,
! [A,B] :
( v1_membered(A)
=> v1_membered(k4_xboole_0(A,B)) ) ).
fof(fc38_membered,axiom,
! [A,B] :
( v2_membered(A)
=> ( v1_membered(k4_xboole_0(A,B))
& v2_membered(k4_xboole_0(A,B)) ) ) ).
fof(fc39_membered,axiom,
! [A,B] :
( v3_membered(A)
=> ( v1_membered(k4_xboole_0(A,B))
& v2_membered(k4_xboole_0(A,B))
& v3_membered(k4_xboole_0(A,B)) ) ) ).
fof(fc40_membered,axiom,
! [A,B] :
( v4_membered(A)
=> ( v1_membered(k4_xboole_0(A,B))
& v2_membered(k4_xboole_0(A,B))
& v3_membered(k4_xboole_0(A,B))
& v4_membered(k4_xboole_0(A,B)) ) ) ).
fof(fc41_membered,axiom,
! [A,B] :
( v5_membered(A)
=> ( v1_membered(k4_xboole_0(A,B))
& v2_membered(k4_xboole_0(A,B))
& v3_membered(k4_xboole_0(A,B))
& v4_membered(k4_xboole_0(A,B))
& v5_membered(k4_xboole_0(A,B)) ) ) ).
fof(fc42_membered,axiom,
! [A,B] :
( ( v1_membered(A)
& v1_membered(B) )
=> v1_membered(k5_xboole_0(A,B)) ) ).
fof(fc43_membered,axiom,
! [A,B] :
( ( v2_membered(A)
& v2_membered(B) )
=> ( v1_membered(k5_xboole_0(A,B))
& v2_membered(k5_xboole_0(A,B)) ) ) ).
fof(fc44_membered,axiom,
! [A,B] :
( ( v3_membered(A)
& v3_membered(B) )
=> ( v1_membered(k5_xboole_0(A,B))
& v2_membered(k5_xboole_0(A,B))
& v3_membered(k5_xboole_0(A,B)) ) ) ).
fof(fc45_membered,axiom,
! [A,B] :
( ( v4_membered(A)
& v4_membered(B) )
=> ( v1_membered(k5_xboole_0(A,B))
& v2_membered(k5_xboole_0(A,B))
& v3_membered(k5_xboole_0(A,B))
& v4_membered(k5_xboole_0(A,B)) ) ) ).
fof(fc46_membered,axiom,
! [A,B] :
( ( v5_membered(A)
& v5_membered(B) )
=> ( v1_membered(k5_xboole_0(A,B))
& v2_membered(k5_xboole_0(A,B))
& v3_membered(k5_xboole_0(A,B))
& v4_membered(k5_xboole_0(A,B))
& v5_membered(k5_xboole_0(A,B)) ) ) ).
fof(d1_membered,axiom,
! [A] :
( v1_membered(A)
<=> ! [B] :
( r2_hidden(B,A)
=> v1_xcmplx_0(B) ) ) ).
fof(d2_membered,axiom,
! [A] :
( v2_membered(A)
<=> ! [B] :
( r2_hidden(B,A)
=> v1_xreal_0(B) ) ) ).
fof(d3_membered,axiom,
! [A] :
( v3_membered(A)
<=> ! [B] :
( r2_hidden(B,A)
=> v1_rat_1(B) ) ) ).
fof(d4_membered,axiom,
! [A] :
( v4_membered(A)
<=> ! [B] :
( r2_hidden(B,A)
=> v1_int_1(B) ) ) ).
fof(d5_membered,axiom,
! [A] :
( v5_membered(A)
<=> ! [B] :
( r2_hidden(B,A)
=> v4_ordinal2(B) ) ) ).
fof(t1_membered,axiom,
! [A] :
( v1_membered(A)
=> r1_tarski(A,k2_numbers) ) ).
fof(t2_membered,axiom,
! [A] :
( v2_membered(A)
=> r1_tarski(A,k1_numbers) ) ).
fof(t3_membered,axiom,
! [A] :
( v3_membered(A)
=> r1_tarski(A,k3_numbers) ) ).
fof(t4_membered,axiom,
! [A] :
( v4_membered(A)
=> r1_tarski(A,k4_numbers) ) ).
fof(t5_membered,axiom,
! [A] :
( v5_membered(A)
=> r1_tarski(A,k5_numbers) ) ).
fof(t6_membered,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_membered(A) )
=> ? [B] :
( v1_xcmplx_0(B)
& r2_hidden(B,A) ) ) ).
fof(t7_membered,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v2_membered(A) )
=> ? [B] :
( v1_xreal_0(B)
& r2_hidden(B,A) ) ) ).
fof(t8_membered,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v3_membered(A) )
=> ? [B] :
( v1_rat_1(B)
& r2_hidden(B,A) ) ) ).
fof(t9_membered,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_membered(A) )
=> ? [B] :
( v1_int_1(B)
& r2_hidden(B,A) ) ) ).
fof(t10_membered,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v5_membered(A) )
=> ? [B] :
( v4_ordinal2(B)
& r2_hidden(B,A) ) ) ).
fof(t11_membered,axiom,
! [A] :
( v1_membered(A)
=> ( ! [B] :
( v1_xcmplx_0(B)
=> r2_hidden(B,A) )
=> A = k2_numbers ) ) ).
fof(t12_membered,axiom,
! [A] :
( v2_membered(A)
=> ( ! [B] :
( v1_xreal_0(B)
=> r2_hidden(B,A) )
=> A = k1_numbers ) ) ).
fof(t13_membered,axiom,
! [A] :
( v3_membered(A)
=> ( ! [B] :
( v1_rat_1(B)
=> r2_hidden(B,A) )
=> A = k3_numbers ) ) ).
fof(t14_membered,axiom,
! [A] :
( v4_membered(A)
=> ( ! [B] :
( v1_int_1(B)
=> r2_hidden(B,A) )
=> A = k4_numbers ) ) ).
fof(t15_membered,axiom,
! [A] :
( v5_membered(A)
=> ( ! [B] :
( v4_ordinal2(B)
=> r2_hidden(B,A) )
=> A = k5_numbers ) ) ).
fof(t16_membered,axiom,
! [A,B] :
( v1_membered(B)
=> ( r1_tarski(A,B)
=> v1_membered(A) ) ) ).
fof(t17_membered,axiom,
! [A,B] :
( v2_membered(B)
=> ( r1_tarski(A,B)
=> v2_membered(A) ) ) ).
fof(t18_membered,axiom,
! [A,B] :
( v3_membered(B)
=> ( r1_tarski(A,B)
=> v3_membered(A) ) ) ).
fof(t19_membered,axiom,
! [A,B] :
( v4_membered(B)
=> ( r1_tarski(A,B)
=> v4_membered(A) ) ) ).
fof(t20_membered,axiom,
! [A,B] :
( v5_membered(B)
=> ( r1_tarski(A,B)
=> v5_membered(A) ) ) ).
fof(d6_membered,axiom,
! [A] :
( v1_membered(A)
=> ! [B] :
( v1_membered(B)
=> ( r1_tarski(A,B)
<=> ! [C] :
( v1_xcmplx_0(C)
=> ( r2_hidden(C,A)
=> r2_hidden(C,B) ) ) ) ) ) ).
fof(d7_membered,axiom,
! [A] :
( v2_membered(A)
=> ! [B] :
( v2_membered(B)
=> ( r1_tarski(A,B)
<=> ! [C] :
( v1_xreal_0(C)
=> ( r2_hidden(C,A)
=> r2_hidden(C,B) ) ) ) ) ) ).
fof(d8_membered,axiom,
! [A] :
( v3_membered(A)
=> ! [B] :
( v3_membered(B)
=> ( r1_tarski(A,B)
<=> ! [C] :
( v1_rat_1(C)
=> ( r2_hidden(C,A)
=> r2_hidden(C,B) ) ) ) ) ) ).
fof(d9_membered,axiom,
! [A] :
( v4_membered(A)
=> ! [B] :
( v4_membered(B)
=> ( r1_tarski(A,B)
<=> ! [C] :
( v1_int_1(C)
=> ( r2_hidden(C,A)
=> r2_hidden(C,B) ) ) ) ) ) ).
fof(d10_membered,axiom,
! [A] :
( v5_membered(A)
=> ! [B] :
( v5_membered(B)
=> ( r1_tarski(A,B)
<=> ! [C] :
( v4_ordinal2(C)
=> ( r2_hidden(C,A)
=> r2_hidden(C,B) ) ) ) ) ) ).
fof(d11_membered,axiom,
! [A] :
( v1_membered(A)
=> ! [B] :
( v1_membered(B)
=> ( A = B
<=> ! [C] :
( v1_xcmplx_0(C)
=> ( r2_hidden(C,A)
<=> r2_hidden(C,B) ) ) ) ) ) ).
fof(d12_membered,axiom,
! [A] :
( v2_membered(A)
=> ! [B] :
( v2_membered(B)
=> ( A = B
<=> ! [C] :
( v1_xreal_0(C)
=> ( r2_hidden(C,A)
<=> r2_hidden(C,B) ) ) ) ) ) ).
fof(d13_membered,axiom,
! [A] :
( v3_membered(A)
=> ! [B] :
( v3_membered(B)
=> ( A = B
<=> ! [C] :
( v1_rat_1(C)
=> ( r2_hidden(C,A)
<=> r2_hidden(C,B) ) ) ) ) ) ).
fof(d14_membered,axiom,
! [A] :
( v4_membered(A)
=> ! [B] :
( v4_membered(B)
=> ( A = B
<=> ! [C] :
( v1_int_1(C)
=> ( r2_hidden(C,A)
<=> r2_hidden(C,B) ) ) ) ) ) ).
fof(d15_membered,axiom,
! [A] :
( v5_membered(A)
=> ! [B] :
( v5_membered(B)
=> ( A = B
<=> ! [C] :
( v4_ordinal2(C)
=> ( r2_hidden(C,A)
<=> r2_hidden(C,B) ) ) ) ) ) ).
fof(d16_membered,axiom,
! [A] :
( v1_membered(A)
=> ! [B] :
( v1_membered(B)
=> ( r1_xboole_0(A,B)
<=> ! [C] :
( v1_xcmplx_0(C)
=> ~ ( r2_hidden(C,A)
& r2_hidden(C,B) ) ) ) ) ) ).
fof(d17_membered,axiom,
! [A] :
( v2_membered(A)
=> ! [B] :
( v2_membered(B)
=> ( r1_xboole_0(A,B)
<=> ! [C] :
( v1_xreal_0(C)
=> ~ ( r2_hidden(C,A)
& r2_hidden(C,B) ) ) ) ) ) ).
fof(d18_membered,axiom,
! [A] :
( v3_membered(A)
=> ! [B] :
( v3_membered(B)
=> ( r1_xboole_0(A,B)
<=> ! [C] :
( v1_rat_1(C)
=> ~ ( r2_hidden(C,A)
& r2_hidden(C,B) ) ) ) ) ) ).
fof(d19_membered,axiom,
! [A] :
( v4_membered(A)
=> ! [B] :
( v4_membered(B)
=> ( r1_xboole_0(A,B)
<=> ! [C] :
( v1_int_1(C)
=> ~ ( r2_hidden(C,A)
& r2_hidden(C,B) ) ) ) ) ) ).
fof(d20_membered,axiom,
! [A] :
( v5_membered(A)
=> ! [B] :
( v5_membered(B)
=> ( r1_xboole_0(A,B)
<=> ! [C] :
( v4_ordinal2(C)
=> ~ ( r2_hidden(C,A)
& r2_hidden(C,B) ) ) ) ) ) ).
fof(t21_membered,axiom,
! [A] :
( ! [B] :
( r2_hidden(B,A)
=> v1_membered(B) )
=> v1_membered(k3_tarski(A)) ) ).
fof(t22_membered,axiom,
! [A] :
( ! [B] :
( r2_hidden(B,A)
=> v2_membered(B) )
=> v2_membered(k3_tarski(A)) ) ).
fof(t23_membered,axiom,
! [A] :
( ! [B] :
( r2_hidden(B,A)
=> v3_membered(B) )
=> v3_membered(k3_tarski(A)) ) ).
fof(t24_membered,axiom,
! [A] :
( ! [B] :
( r2_hidden(B,A)
=> v4_membered(B) )
=> v4_membered(k3_tarski(A)) ) ).
fof(t25_membered,axiom,
! [A] :
( ! [B] :
( r2_hidden(B,A)
=> v5_membered(B) )
=> v5_membered(k3_tarski(A)) ) ).
fof(t26_membered,axiom,
! [A,B] :
( ( r2_hidden(B,A)
& v1_membered(B) )
=> v1_membered(k1_setfam_1(A)) ) ).
fof(t27_membered,axiom,
! [A,B] :
( ( r2_hidden(B,A)
& v2_membered(B) )
=> v2_membered(k1_setfam_1(A)) ) ).
fof(t28_membered,axiom,
! [A,B] :
( ( r2_hidden(B,A)
& v3_membered(B) )
=> v3_membered(k1_setfam_1(A)) ) ).
fof(t29_membered,axiom,
! [A,B] :
( ( r2_hidden(B,A)
& v4_membered(B) )
=> v4_membered(k1_setfam_1(A)) ) ).
fof(t30_membered,axiom,
! [A,B] :
( ( r2_hidden(B,A)
& v5_membered(B) )
=> v5_membered(k1_setfam_1(A)) ) ).
fof(s1_membered,axiom,
? [A] :
( v1_membered(A)
& ! [B] :
( v1_xcmplx_0(B)
=> ( r2_hidden(B,A)
<=> p1_s1_membered(B) ) ) ) ).
fof(s2_membered,axiom,
? [A] :
( v2_membered(A)
& ! [B] :
( v1_xreal_0(B)
=> ( r2_hidden(B,A)
<=> p1_s2_membered(B) ) ) ) ).
fof(s3_membered,axiom,
? [A] :
( v3_membered(A)
& ! [B] :
( v1_rat_1(B)
=> ( r2_hidden(B,A)
<=> p1_s3_membered(B) ) ) ) ).
fof(s4_membered,axiom,
? [A] :
( v4_membered(A)
& ! [B] :
( v1_int_1(B)
=> ( r2_hidden(B,A)
<=> p1_s4_membered(B) ) ) ) ).
fof(s5_membered,axiom,
? [A] :
( v5_membered(A)
& ! [B] :
( v4_ordinal2(B)
=> ( r2_hidden(B,A)
<=> p1_s5_membered(B) ) ) ) ).
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