SET007 Axioms: SET007+51.ax


%------------------------------------------------------------------------------
% File     : SET007+51 : TPTP v7.5.0. 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 unit)
%            Number of atoms       :  520 (  10 equality)
%            Maximal formula depth :   10 (   6 average)
%            Number of connectives :  413 (  15 ~  ;   0  |; 188  &)
%                                         (  30 <=>; 180 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   21 (   0 propositional; 1-2 arity)
%            Number of functors    :   17 (   7 constant; 0-3 arity)
%            Number of variables   :  231 (   0 singleton; 220 !;  11 ?)
%            Maximal term depth    :    2 (   1 average)
% 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) ) ) ) )).
%------------------------------------------------------------------------------