SET007 Axioms: SET007+123.ax


%------------------------------------------------------------------------------
% File     : SET007+123 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Monotonic and Continuous Real Function
% 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    : fcont_3 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   40 (   4 unit)
%            Number of atoms       :  222 (  15 equality)
%            Maximal formula depth :   19 (   7 average)
%            Number of connectives :  214 (  32 ~  ;  16  |;  79  &)
%                                         (   2 <=>;  85 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   27 (   0 propositional; 1-3 arity)
%            Number of functors    :   27 (   4 constant; 0-4 arity)
%            Number of variables   :   75 (   0 singleton;  73 !;   2 ?)
%            Maximal term depth    :    4 (   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(fc1_fcont_3,axiom,
    ( v1_membered(k2_subset_1(k1_numbers))
    & v2_membered(k2_subset_1(k1_numbers))
    & v2_rcomp_1(k2_subset_1(k1_numbers))
    & v3_rcomp_1(k2_subset_1(k1_numbers)) )).

fof(fc2_fcont_3,axiom,
    ( v1_xboole_0(k1_subset_1(k1_numbers))
    & v1_membered(k1_subset_1(k1_numbers))
    & v2_membered(k1_subset_1(k1_numbers))
    & v3_membered(k1_subset_1(k1_numbers))
    & v4_membered(k1_subset_1(k1_numbers))
    & v5_membered(k1_subset_1(k1_numbers))
    & v2_rcomp_1(k1_subset_1(k1_numbers))
    & v3_rcomp_1(k1_subset_1(k1_numbers)) )).

fof(fc3_fcont_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( v1_membered(k4_limfunc1(A))
        & v2_membered(k4_limfunc1(A))
        & v3_rcomp_1(k4_limfunc1(A)) ) ) )).

fof(fc4_fcont_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( v1_membered(k12_prob_1(A))
        & v2_membered(k12_prob_1(A))
        & v3_rcomp_1(k12_prob_1(A)) ) ) )).

fof(fc5_fcont_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( v1_membered(k3_limfunc1(A))
        & v2_membered(k3_limfunc1(A))
        & v2_rcomp_1(k3_limfunc1(A)) ) ) )).

fof(fc6_fcont_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( v1_membered(k2_limfunc1(A))
        & v2_membered(k2_limfunc1(A))
        & v2_rcomp_1(k2_limfunc1(A)) ) ) )).

fof(t1_fcont_3,axiom,(
    v2_rcomp_1(k2_subset_1(k1_numbers)) )).

fof(t2_fcont_3,axiom,(
    v3_rcomp_1(k1_subset_1(k1_numbers)) )).

fof(t3_fcont_3,axiom,(
    v2_rcomp_1(k1_subset_1(k1_numbers)) )).

fof(t4_fcont_3,axiom,(
    v3_rcomp_1(k2_subset_1(k1_numbers)) )).

fof(t5_fcont_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => v2_rcomp_1(k3_limfunc1(A)) ) )).

fof(t6_fcont_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => v2_rcomp_1(k2_limfunc1(A)) ) )).

fof(t7_fcont_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => v3_rcomp_1(k4_limfunc1(A)) ) )).

fof(t8_fcont_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => v3_rcomp_1(k12_prob_1(A)) ) )).

fof(t9_fcont_3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( v1_xreal_0(C)
             => ( ( ~ r1_xreal_0(C,np__0)
                  & r2_hidden(A,k2_rcomp_1(k6_xcmplx_0(B,C),k2_xcmplx_0(B,C))) )
              <=> ? [D] :
                    ( m1_subset_1(D,k1_numbers)
                    & A = k3_real_1(B,D)
                    & ~ r1_xreal_0(C,k18_complex1(D)) ) ) ) ) ) )).

fof(t10_fcont_3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( v1_xreal_0(C)
             => ( ( ~ r1_xreal_0(C,np__0)
                  & r2_hidden(A,k2_rcomp_1(k6_xcmplx_0(B,C),k2_xcmplx_0(B,C))) )
              <=> r2_hidden(k5_real_1(A,B),k2_rcomp_1(k4_xcmplx_0(C),C)) ) ) ) ) )).

fof(t11_fcont_3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => k2_limfunc1(A) = k2_xboole_0(k1_tarski(A),k12_prob_1(A)) ) )).

fof(t12_fcont_3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => k3_limfunc1(A) = k2_xboole_0(k1_tarski(A),k4_limfunc1(A)) ) )).

fof(t13_fcont_3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( v1_xreal_0(C)
             => ( ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => k2_seq_1(k5_numbers,k1_numbers,B,D) = k6_xcmplx_0(C,k6_real_1(A,k3_real_1(D,np__1))) )
               => ( v4_seq_2(B)
                  & k2_seq_2(B) = C ) ) ) ) ) )).

fof(t14_fcont_3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( v1_xreal_0(C)
             => ( ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => k2_seq_1(k5_numbers,k1_numbers,B,D) = k2_xcmplx_0(C,k6_real_1(A,k3_real_1(D,np__1))) )
               => ( v4_seq_2(B)
                  & k2_seq_2(B) = C ) ) ) ) ) )).

fof(t15_fcont_3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ~ ( r1_fcont_1(C,A)
                  & k2_seq_1(k1_numbers,k1_numbers,C,A) != B
                  & ? [D] :
                      ( m1_rcomp_1(D,A)
                      & r1_tarski(D,k4_relset_1(k1_numbers,k1_numbers,C)) )
                  & ! [D] :
                      ( m1_rcomp_1(D,A)
                     => ~ ( r1_tarski(D,k4_relset_1(k1_numbers,k1_numbers,C))
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ~ ( r2_hidden(E,D)
                                  & k2_seq_1(k1_numbers,k1_numbers,C,E) = B ) ) ) ) ) ) ) ) )).

fof(t16_fcont_3,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & m2_relset_1(B,k1_numbers,k1_numbers) )
     => ( ( r1_rfunct_2(B,A)
          | r2_rfunct_2(B,A) )
       => v2_funct_1(k2_partfun1(k1_numbers,k1_numbers,B,A)) ) ) )).

fof(t17_fcont_3,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v2_funct_1(B)
        & m2_relset_1(B,k1_numbers,k1_numbers) )
     => ( r1_rfunct_2(B,A)
       => r1_rfunct_2(k2_partfun2(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,B,A)),k10_relset_1(k1_numbers,k1_numbers,B,A)) ) ) )).

fof(t18_fcont_3,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v2_funct_1(B)
        & m2_relset_1(B,k1_numbers,k1_numbers) )
     => ( r2_rfunct_2(B,A)
       => r2_rfunct_2(k2_partfun2(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,B,A)),k10_relset_1(k1_numbers,k1_numbers,B,A)) ) ) )).

fof(t19_fcont_3,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & m2_relset_1(B,k1_numbers,k1_numbers) )
     => ( ( r1_tarski(A,k4_relset_1(k1_numbers,k1_numbers,B))
          & r5_rfunct_2(B,A) )
       => ( ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => k10_relset_1(k1_numbers,k1_numbers,B,A) != k12_prob_1(C) )
          | r2_fcont_1(B,A) ) ) ) )).

fof(t20_fcont_3,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & m2_relset_1(B,k1_numbers,k1_numbers) )
     => ( ( r1_tarski(A,k4_relset_1(k1_numbers,k1_numbers,B))
          & r5_rfunct_2(B,A) )
       => ( ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => k10_relset_1(k1_numbers,k1_numbers,B,A) != k4_limfunc1(C) )
          | r2_fcont_1(B,A) ) ) ) )).

fof(t21_fcont_3,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & m2_relset_1(B,k1_numbers,k1_numbers) )
     => ( ( r1_tarski(A,k4_relset_1(k1_numbers,k1_numbers,B))
          & r5_rfunct_2(B,A) )
       => ( ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => k10_relset_1(k1_numbers,k1_numbers,B,A) != k2_limfunc1(C) )
          | r2_fcont_1(B,A) ) ) ) )).

fof(t22_fcont_3,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & m2_relset_1(B,k1_numbers,k1_numbers) )
     => ( ( r1_tarski(A,k4_relset_1(k1_numbers,k1_numbers,B))
          & r5_rfunct_2(B,A) )
       => ( ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => k10_relset_1(k1_numbers,k1_numbers,B,A) != k3_limfunc1(C) )
          | r2_fcont_1(B,A) ) ) ) )).

fof(t23_fcont_3,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & m2_relset_1(B,k1_numbers,k1_numbers) )
     => ( ( r1_tarski(A,k4_relset_1(k1_numbers,k1_numbers,B))
          & r5_rfunct_2(B,A) )
       => ( ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => k10_relset_1(k1_numbers,k1_numbers,B,A) != k2_rcomp_1(C,D) ) )
          | r2_fcont_1(B,A) ) ) ) )).

fof(t24_fcont_3,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & m2_relset_1(B,k1_numbers,k1_numbers) )
     => ( ( r1_tarski(A,k4_relset_1(k1_numbers,k1_numbers,B))
          & r5_rfunct_2(B,A)
          & k10_relset_1(k1_numbers,k1_numbers,B,A) = k1_numbers )
       => r2_fcont_1(B,A) ) ) )).

fof(t25_fcont_3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v2_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( r1_tarski(k2_rcomp_1(A,B),k4_relset_1(k1_numbers,k1_numbers,C))
               => ( ( ~ r1_rfunct_2(C,k2_rcomp_1(A,B))
                    & ~ r2_rfunct_2(C,k2_rcomp_1(A,B)) )
                  | r2_fcont_1(k2_partfun2(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,C,k2_rcomp_1(A,B))),k10_relset_1(k1_numbers,k1_numbers,C,k2_rcomp_1(A,B))) ) ) ) ) ) )).

fof(t26_fcont_3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v2_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( r1_tarski(k12_prob_1(A),k4_relset_1(k1_numbers,k1_numbers,B))
           => ( ( ~ r1_rfunct_2(B,k12_prob_1(A))
                & ~ r2_rfunct_2(B,k12_prob_1(A)) )
              | r2_fcont_1(k2_partfun2(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,B,k12_prob_1(A))),k10_relset_1(k1_numbers,k1_numbers,B,k12_prob_1(A))) ) ) ) ) )).

fof(t27_fcont_3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v2_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( r1_tarski(k4_limfunc1(A),k4_relset_1(k1_numbers,k1_numbers,B))
           => ( ( ~ r1_rfunct_2(B,k4_limfunc1(A))
                & ~ r2_rfunct_2(B,k4_limfunc1(A)) )
              | r2_fcont_1(k2_partfun2(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,B,k4_limfunc1(A))),k10_relset_1(k1_numbers,k1_numbers,B,k4_limfunc1(A))) ) ) ) ) )).

fof(t28_fcont_3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v2_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( r1_tarski(k2_limfunc1(A),k4_relset_1(k1_numbers,k1_numbers,B))
           => ( ( ~ r1_rfunct_2(B,k2_limfunc1(A))
                & ~ r2_rfunct_2(B,k2_limfunc1(A)) )
              | r2_fcont_1(k2_partfun2(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,B,k2_limfunc1(A))),k10_relset_1(k1_numbers,k1_numbers,B,k2_limfunc1(A))) ) ) ) ) )).

fof(t29_fcont_3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v2_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( r1_tarski(k3_limfunc1(A),k4_relset_1(k1_numbers,k1_numbers,B))
           => ( ( ~ r1_rfunct_2(B,k3_limfunc1(A))
                & ~ r2_rfunct_2(B,k3_limfunc1(A)) )
              | r2_fcont_1(k2_partfun2(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,B,k3_limfunc1(A))),k10_relset_1(k1_numbers,k1_numbers,B,k3_limfunc1(A))) ) ) ) ) )).

fof(t30_fcont_3,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v2_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ( v1_partfun1(A,k1_numbers,k1_numbers)
       => ( ( ~ r1_rfunct_2(A,k2_subset_1(k1_numbers))
            & ~ r2_rfunct_2(A,k2_subset_1(k1_numbers)) )
          | r2_fcont_1(k2_partfun2(k1_numbers,k1_numbers,A),k5_relset_1(k1_numbers,k1_numbers,A)) ) ) ) )).

fof(t31_fcont_3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( r2_fcont_1(C,k2_rcomp_1(A,B))
               => ( ( ~ r1_rfunct_2(C,k2_rcomp_1(A,B))
                    & ~ r2_rfunct_2(C,k2_rcomp_1(A,B)) )
                  | v3_rcomp_1(k5_relset_1(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,C,k2_rcomp_1(A,B)))) ) ) ) ) ) )).

fof(t32_fcont_3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( r2_fcont_1(B,k12_prob_1(A))
           => ( ( ~ r1_rfunct_2(B,k12_prob_1(A))
                & ~ r2_rfunct_2(B,k12_prob_1(A)) )
              | v3_rcomp_1(k5_relset_1(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,B,k12_prob_1(A)))) ) ) ) ) )).

fof(t33_fcont_3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( r2_fcont_1(B,k4_limfunc1(A))
           => ( ( ~ r1_rfunct_2(B,k4_limfunc1(A))
                & ~ r2_rfunct_2(B,k4_limfunc1(A)) )
              | v3_rcomp_1(k5_relset_1(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,B,k4_limfunc1(A)))) ) ) ) ) )).

fof(t34_fcont_3,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ( r2_fcont_1(A,k2_subset_1(k1_numbers))
       => ( ( ~ r1_rfunct_2(A,k2_subset_1(k1_numbers))
            & ~ r2_rfunct_2(A,k2_subset_1(k1_numbers)) )
          | v3_rcomp_1(k5_relset_1(k1_numbers,k1_numbers,A)) ) ) ) )).
%------------------------------------------------------------------------------