SET007 Axioms: SET007+74.ax


%------------------------------------------------------------------------------
% File     : SET007+74 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : The Reflection Theorem
% 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_refle [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   52 (  14 unit)
%            Number of atoms       :  278 (  26 equality)
%            Maximal formula depth :   20 (   6 average)
%            Number of connectives :  255 (  29 ~  ;   4  |; 126  &)
%                                         (   4 <=>;  92 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   33 (   1 propositional; 0-3 arity)
%            Number of functors    :   46 (  18 constant; 0-4 arity)
%            Number of variables   :  106 (   0 singleton;  94 !;  12 ?)
%            Maximal term depth    :    5 (   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(rc1_zf_refle,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ? [B] :
          ( m1_subset_1(B,A)
          & ~ v1_xboole_0(B) ) ) )).

fof(rc2_zf_refle,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ? [B] :
          ( m1_ordinal1(B,A)
          & v1_relat_1(B)
          & v2_relat_1(B)
          & v1_funct_1(B)
          & v5_ordinal1(B)
          & v1_zf_refle(B,A) ) ) )).

fof(t1_zf_refle,axiom,(
    $true )).

fof(t2_zf_refle,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => r2_zf_model(A,k7_zf_model) ) )).

fof(t3_zf_refle,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => r2_zf_model(A,k8_zf_model) ) )).

fof(t4_zf_refle,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ( r2_hidden(k5_ordinal2,A)
       => r2_zf_model(A,k9_zf_model) ) ) )).

fof(t5_zf_refle,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => r2_zf_model(A,k10_zf_model) ) )).

fof(t6_zf_refle,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ! [B] :
          ( ( v1_zf_lang(B)
            & m2_finseq_1(B,k5_numbers) )
         => ( r1_xboole_0(k1_enumset1(k2_zf_lang(np__0),k2_zf_lang(np__1),k2_zf_lang(np__2)),k2_zf_model(B))
           => r2_zf_model(A,k11_zf_model(B)) ) ) ) )).

fof(t7_zf_refle,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ( r2_hidden(k5_ordinal2,A)
       => v1_zf_model(A) ) ) )).

fof(d1_zf_refle,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ( r1_tarski(A,B)
          <=> ! [C] :
                ( v3_ordinal1(C)
               => ( r2_hidden(C,A)
                 => r2_hidden(C,B) ) ) ) ) ) )).

fof(t8_zf_refle,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ! [B] :
          ( m1_ordinal4(B,A)
        <=> r2_hidden(B,k2_ordinal2(A)) ) ) )).

fof(d2_zf_refle,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_classes2(B) )
         => ! [C] :
              ( m1_ordinal4(C,B)
             => k1_zf_refle(A,B,C) = k3_card_3(k8_relat_1(B,k7_relat_1(A,k4_classes1(C)))) ) ) ) )).

fof(t9_zf_refle,axiom,(
    $true )).

fof(t10_zf_refle,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A) )
     => ! [B] :
          ( v3_ordinal1(B)
         => ( v1_relat_1(k7_relat_1(A,k4_classes1(B)))
            & v1_funct_1(k7_relat_1(A,k4_classes1(B)))
            & v5_ordinal1(k7_relat_1(A,k4_classes1(B))) ) ) ) )).

fof(t11_zf_refle,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_ordinal2(A) )
     => ! [B] :
          ( v3_ordinal1(B)
         => ( v1_relat_1(k7_relat_1(A,k4_classes1(B)))
            & v1_funct_1(k7_relat_1(A,k4_classes1(B)))
            & v5_ordinal1(k7_relat_1(A,k4_classes1(B)))
            & v1_ordinal2(k7_relat_1(A,k4_classes1(B))) ) ) ) )).

fof(t12_zf_refle,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_ordinal2(A) )
     => v3_ordinal1(k3_card_3(A)) ) )).

fof(t13_zf_refle,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v5_ordinal1(B)
        & v1_ordinal2(B) )
     => v3_ordinal1(k3_card_3(k8_relat_1(A,B))) ) )).

fof(t14_zf_refle,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => k2_ordinal2(k4_classes1(A)) = A ) )).

fof(t15_zf_refle,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v5_ordinal1(B)
            & v1_ordinal2(B) )
         => k7_relat_1(B,k4_classes1(A)) = k2_ordinal1(B,A) ) ) )).

fof(t16_zf_refle,axiom,(
    $true )).

fof(t17_zf_refle,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ! [B] :
          ( m1_ordinal4(B,A)
         => ! [C] :
              ( m2_ordinal4(C,A)
             => ( k2_zf_refle(C,A,B) = k3_card_3(k2_ordinal1(C,B))
                & k2_zf_refle(k2_ordinal1(C,B),A,B) = k3_card_3(k2_ordinal1(C,B)) ) ) ) ) )).

fof(d3_zf_refle,axiom,(
    $true )).

fof(d4_zf_refle,axiom,(
    $true )).

fof(d5_zf_refle,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ! [B] :
          ( m1_ordinal1(B,A)
         => ( v1_zf_refle(B,A)
          <=> k1_relat_1(B) = k2_ordinal2(A) ) ) ) )).

fof(t18_zf_refle,axiom,(
    $true )).

fof(t19_zf_refle,axiom,(
    $true )).

fof(t20_zf_refle,axiom,(
    $true )).

fof(t21_zf_refle,axiom,(
    $true )).

fof(t22_zf_refle,axiom,(
    $true )).

fof(t23_zf_refle,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ! [B] :
          ( m1_ordinal4(B,A)
         => ! [C] :
              ( ( v2_relat_1(C)
                & v1_zf_refle(C,A)
                & m1_ordinal1(C,A) )
             => r2_hidden(B,k1_relat_1(C)) ) ) ) )).

fof(t24_zf_refle,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ! [B] :
          ( m1_ordinal4(B,A)
         => ! [C] :
              ( ( v2_relat_1(C)
                & v1_zf_refle(C,A)
                & m1_ordinal1(C,A) )
             => r1_tarski(k5_zf_refle(A,C,B),k4_zf_refle(A,C)) ) ) ) )).

fof(t25_zf_refle,axiom,(
    r2_tarski(k5_numbers,k1_zf_lang) )).

fof(t26_zf_refle,axiom,(
    $true )).

fof(t27_zf_refle,axiom,(
    ! [A] : r1_tarski(k7_ordinal2(A),k1_ordinal1(k3_tarski(k2_ordinal2(A)))) )).

fof(t28_zf_refle,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ! [B] :
          ( r2_hidden(B,A)
         => r2_hidden(k7_ordinal2(B),A) ) ) )).

fof(t29_zf_refle,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ! [B] :
          ( ( v2_relat_1(B)
            & v1_zf_refle(B,A)
            & m1_ordinal1(B,A) )
         => ( ( r2_hidden(k5_ordinal2,A)
              & ! [C] :
                  ( m1_ordinal4(C,A)
                 => ! [D] :
                      ( m1_ordinal4(D,A)
                     => ( r2_hidden(C,D)
                       => r1_tarski(k5_zf_refle(A,B,C),k5_zf_refle(A,B,D)) ) ) )
              & ! [C] :
                  ( m1_ordinal4(C,A)
                 => ( v4_ordinal1(C)
                   => ( C = k1_xboole_0
                      | k5_zf_refle(A,B,C) = k3_card_3(k2_ordinal1(B,C)) ) ) ) )
           => ! [C] :
                ( ( v1_zf_lang(C)
                  & m2_finseq_1(C,k5_numbers) )
               => ? [D] :
                    ( m2_ordinal4(D,A)
                    & v2_ordinal2(D)
                    & v3_ordinal2(D)
                    & ! [E] :
                        ( m1_ordinal4(E,A)
                       => ( k4_ordinal4(A,D,E) = E
                         => ( k1_xboole_0 = E
                            | ! [F] :
                                ( ( v1_funct_1(F)
                                  & v1_funct_2(F,k1_zf_lang,k5_zf_refle(A,B,E))
                                  & m2_relset_1(F,k1_zf_lang,k5_zf_refle(A,B,E)) )
                               => ( r1_zf_model(k4_zf_refle(A,B),k2_zf_lang1(k1_zf_lang,k5_zf_refle(A,B,E),k4_zf_refle(A,B),F),C)
                                <=> r1_zf_model(k5_zf_refle(A,B,E),F,C) ) ) ) ) ) ) ) ) ) ) )).

fof(s1_zf_refle,axiom,
    ( ! [A] :
        ( m1_subset_1(A,f1_s1_zf_refle)
       => ? [B] :
            ( v3_ordinal1(B)
            & p1_s1_zf_refle(A,B) ) )
   => ? [A] :
        ( v1_relat_1(A)
        & v1_funct_1(A)
        & k1_relat_1(A) = f1_s1_zf_refle
        & ! [B] :
            ( m1_subset_1(B,f1_s1_zf_refle)
           => ? [C] :
                ( v3_ordinal1(C)
                & C = k1_funct_1(A,B)
                & p1_s1_zf_refle(B,C)
                & ! [D] :
                    ( v3_ordinal1(D)
                   => ( p1_s1_zf_refle(B,D)
                     => r1_tarski(C,D) ) ) ) ) ) )).

fof(s2_zf_refle,axiom,
    ( ! [A] :
        ( m1_subset_1(A,f2_s2_zf_refle)
       => ? [B] :
            ( m1_ordinal4(B,f1_s2_zf_refle)
            & p1_s2_zf_refle(A,B) ) )
   => ? [A] :
        ( v1_relat_1(A)
        & v1_funct_1(A)
        & k1_relat_1(A) = f2_s2_zf_refle
        & ! [B] :
            ( m1_subset_1(B,f2_s2_zf_refle)
           => ? [C] :
                ( m1_ordinal4(C,f1_s2_zf_refle)
                & C = k1_funct_1(A,B)
                & p1_s2_zf_refle(B,C)
                & ! [D] :
                    ( m1_ordinal4(D,f1_s2_zf_refle)
                   => ( p1_s2_zf_refle(B,D)
                     => r1_tarski(C,D) ) ) ) ) ) )).

fof(s3_zf_refle,axiom,
    ( ( ! [A] :
          ( m1_ordinal4(A,f1_s3_zf_refle)
         => ! [B] :
              ( m1_ordinal4(B,f1_s3_zf_refle)
             => ! [C] :
                  ( m1_ordinal4(C,f1_s3_zf_refle)
                 => ( ( p1_s3_zf_refle(A,B)
                      & p1_s3_zf_refle(A,C) )
                   => B = C ) ) ) )
      & ! [A] :
          ( m1_ordinal4(A,f1_s3_zf_refle)
         => ? [B] :
              ( m1_ordinal4(B,f1_s3_zf_refle)
              & p1_s3_zf_refle(A,B) ) ) )
   => ? [A] :
        ( m2_ordinal4(A,f1_s3_zf_refle)
        & ! [B] :
            ( m1_ordinal4(B,f1_s3_zf_refle)
           => p1_s3_zf_refle(B,k4_ordinal4(f1_s3_zf_refle,A,B)) ) ) )).

fof(s4_zf_refle,axiom,(
    ? [A] :
      ( m2_ordinal4(A,f1_s4_zf_refle)
      & k4_ordinal4(f1_s4_zf_refle,A,k2_ordinal4(f1_s4_zf_refle)) = f2_s4_zf_refle
      & ! [B] :
          ( m1_ordinal4(B,f1_s4_zf_refle)
         => k4_ordinal4(f1_s4_zf_refle,A,k6_ordinal4(f1_s4_zf_refle,B)) = f3_s4_zf_refle(B,k4_ordinal4(f1_s4_zf_refle,A,B)) )
      & ! [B] :
          ( m1_ordinal4(B,f1_s4_zf_refle)
         => ( v4_ordinal1(B)
           => ( B = k2_ordinal4(f1_s4_zf_refle)
              | k4_ordinal4(f1_s4_zf_refle,A,B) = f4_s4_zf_refle(B,k2_ordinal1(A,B)) ) ) ) ) )).

fof(s5_zf_refle,axiom,
    ( ( p1_s5_zf_refle(k2_ordinal4(f1_s5_zf_refle))
      & ! [A] :
          ( m1_ordinal4(A,f1_s5_zf_refle)
         => ( p1_s5_zf_refle(A)
           => p1_s5_zf_refle(k6_ordinal4(f1_s5_zf_refle,A)) ) )
      & ! [A] :
          ( m1_ordinal4(A,f1_s5_zf_refle)
         => ( ( v4_ordinal1(A)
              & ! [B] :
                  ( m1_ordinal4(B,f1_s5_zf_refle)
                 => ( r2_hidden(B,A)
                   => p1_s5_zf_refle(B) ) ) )
           => ( A = k2_ordinal4(f1_s5_zf_refle)
              | p1_s5_zf_refle(A) ) ) ) )
   => ! [A] :
        ( m1_ordinal4(A,f1_s5_zf_refle)
       => p1_s5_zf_refle(A) ) )).

fof(dt_k1_zf_refle,axiom,(
    $true )).

fof(dt_k2_zf_refle,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_ordinal2(A)
        & ~ v1_xboole_0(B)
        & v1_classes2(B)
        & m1_ordinal4(C,B) )
     => m1_ordinal4(k2_zf_refle(A,B,C),B) ) )).

fof(redefinition_k2_zf_refle,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_ordinal2(A)
        & ~ v1_xboole_0(B)
        & v1_classes2(B)
        & m1_ordinal4(C,B) )
     => k2_zf_refle(A,B,C) = k1_zf_refle(A,B,C) ) )).

fof(dt_k3_zf_refle,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & m1_ordinal4(B,A)
        & m1_ordinal4(C,A) )
     => m1_ordinal4(k3_zf_refle(A,B,C),A) ) )).

fof(commutativity_k3_zf_refle,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & m1_ordinal4(B,A)
        & m1_ordinal4(C,A) )
     => k3_zf_refle(A,B,C) = k3_zf_refle(A,C,B) ) )).

fof(idempotence_k3_zf_refle,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & m1_ordinal4(B,A)
        & m1_ordinal4(C,A) )
     => k3_zf_refle(A,B,B) = B ) )).

fof(redefinition_k3_zf_refle,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & m1_ordinal4(B,A)
        & m1_ordinal4(C,A) )
     => k3_zf_refle(A,B,C) = k2_xboole_0(B,C) ) )).

fof(dt_k4_zf_refle,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & v2_relat_1(B)
        & v1_zf_refle(B,A)
        & m1_ordinal1(B,A) )
     => ( ~ v1_xboole_0(k4_zf_refle(A,B))
        & m1_subset_1(k4_zf_refle(A,B),k1_zfmisc_1(A)) ) ) )).

fof(redefinition_k4_zf_refle,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & v2_relat_1(B)
        & v1_zf_refle(B,A)
        & m1_ordinal1(B,A) )
     => k4_zf_refle(A,B) = k3_card_3(B) ) )).

fof(dt_k5_zf_refle,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & v2_relat_1(B)
        & v1_zf_refle(B,A)
        & m1_ordinal1(B,A)
        & m1_ordinal4(C,A) )
     => ( ~ v1_xboole_0(k5_zf_refle(A,B,C))
        & m1_subset_1(k5_zf_refle(A,B,C),A) ) ) )).

fof(redefinition_k5_zf_refle,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & v2_relat_1(B)
        & v1_zf_refle(B,A)
        & m1_ordinal1(B,A)
        & m1_ordinal4(C,A) )
     => k5_zf_refle(A,B,C) = k1_funct_1(B,C) ) )).
%------------------------------------------------------------------------------