TSTP Solution File: SEU519^1 by Satallax---3.5

View Problem - Process Solution

%------------------------------------------------------------------------------
% File     : Satallax---3.5
% Problem  : SEU519^1 : TPTP v8.1.0. Released v3.7.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s

% Computer : n022.cluster.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory   : 8042.1875MB
% OS       : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit  : 600s
% DateTime : Tue Jul 19 13:52:44 EDT 2022

% Result   : Theorem 26.05s 26.28s
% Output   : Proof 26.05s
% Verified : 
% SZS Type : Refutation
%            Derivation depth      :    3
%            Number of leaves      :  138
% Syntax   : Number of formulae    :  144 (  46 unt;  10 typ;  40 def)
%            Number of atoms       :  674 ( 126 equ;   0 cnn)
%            Maximal formula atoms :   89 (   5 avg)
%            Number of connectives : 1652 ( 271   ~;  45   |;   0   &; 873   @)
%                                         (  44 <=>; 419  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   41 (   5 avg)
%            Number of types       :    2 (   0 usr)
%            Number of type conns  :   53 (  53   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   95 (  93 usr;  88 con; 0-2 aty)
%            Number of variables   :  437 (  13   ^ 424   !;   0   ?; 437   :)

% Comments : 
%------------------------------------------------------------------------------
thf(ty_setunion,type,
    setunion: $i > $i ).

thf(ty_eigen__0,type,
    eigen__0: $i ).

thf(ty_emptyset,type,
    emptyset: $i ).

thf(ty_eigen__5,type,
    eigen__5: $i ).

thf(ty_descr,type,
    descr: ( $i > $o ) > $i ).

thf(ty_dsetconstr,type,
    dsetconstr: $i > ( $i > $o ) > $i ).

thf(ty_powerset,type,
    powerset: $i > $i ).

thf(ty_omega,type,
    omega: $i ).

thf(ty_in,type,
    in: $i > $i > $o ).

thf(ty_setadjoin,type,
    setadjoin: $i > $i > $i ).

thf(h0,assumption,
    ! [X1: $i > $o,X2: $i] :
      ( ( X1 @ X2 )
     => ( X1 @ ( eps__0 @ X1 ) ) ),
    introduced(assumption,[]) ).

thf(eigendef_eigen__0,definition,
    ( eigen__0
    = ( eps__0
      @ ^ [X1: $i] :
          ~ ( in @ emptyset @ ( powerset @ X1 ) ) ) ),
    introduced(definition,[new_symbols(definition,[eigen__0])]) ).

thf(eigendef_eigen__5,definition,
    ( eigen__5
    = ( eps__0
      @ ^ [X1: $i] :
          ~ ( ( in @ X1 @ emptyset )
           => ( in @ X1 @ eigen__0 ) ) ) ),
    introduced(definition,[new_symbols(definition,[eigen__5])]) ).

thf(sP1,plain,
    ( sP1
  <=> ( ! [X1: $i,X2: $i] :
          ( ( in @ X2 @ ( powerset @ X1 ) )
          = ( ! [X3: $i] :
                ( ( in @ X3 @ X2 )
               => ( in @ X3 @ X1 ) ) ) )
     => ( ! [X1: $i,X2: $i] :
            ( ( in @ X2 @ ( setunion @ X1 ) )
            = ( ~ ! [X3: $i] :
                    ( ( in @ X2 @ X3 )
                   => ~ ( in @ X3 @ X1 ) ) ) )
       => ( ( in @ emptyset @ omega )
         => ( ! [X1: $i] :
                ( ( in @ X1 @ omega )
               => ( in @ ( setadjoin @ X1 @ X1 ) @ omega ) )
           => ( ! [X1: $i] :
                  ( ~ ( ( in @ emptyset @ X1 )
                     => ~ ! [X2: $i] :
                            ( ~ ( ( in @ X2 @ omega )
                               => ~ ( in @ X2 @ X1 ) )
                           => ( in @ ( setadjoin @ X2 @ X2 ) @ X1 ) ) )
                 => ! [X2: $i] :
                      ( ( in @ X2 @ omega )
                     => ( in @ X2 @ X1 ) ) )
             => ( ! [X1: $i > $i > $o,X2: $i] :
                    ( ! [X3: $i] :
                        ( ( in @ X3 @ X2 )
                       => ~ ! [X4: $i] :
                              ( ( X1 @ X3 @ X4 )
                             => ~ ! [X5: $i] :
                                    ( ( X1 @ X3 @ X5 )
                                   => ( X4 = X5 ) ) ) )
                   => ~ ! [X3: $i] :
                          ~ ! [X4: $i] :
                              ( ( in @ X4 @ X3 )
                              = ( ~ ! [X5: $i] :
                                      ( ( in @ X5 @ X2 )
                                     => ~ ( X1 @ X5 @ X4 ) ) ) ) )
               => ( ! [X1: $i] :
                      ( ~ ! [X2: $i] :
                            ~ ( in @ X2 @ X1 )
                     => ~ ! [X2: $i] :
                            ( ( in @ X2 @ X1 )
                           => ~ ! [X3: $i] :
                                  ( ( in @ X3 @ X2 )
                                 => ~ ( in @ X3 @ X1 ) ) ) )
                 => ( ! [X1: $i] :
                        ~ ! [X2: $i] :
                            ( ~ ( ~ ( ! [X3: $i] :
                                        ( ( in @ X3 @ X2 )
                                       => ! [X4: $i] :
                                            ( ( in @ X4 @ X3 )
                                           => ( in @ X4 @ X1 ) ) )
                                   => ~ ! [X3: $i,X4: $i] :
                                          ( ~ ( ( in @ X3 @ X1 )
                                             => ~ ( in @ X4 @ X1 ) )
                                         => ( ! [X5: $i] :
                                                ( ( in @ X5 @ X2 )
                                               => ( ( in @ X3 @ X5 )
                                                  = ( in @ X4 @ X5 ) ) )
                                           => ( X3 = X4 ) ) ) )
                               => ~ ! [X3: $i,X4: $i] :
                                      ( ~ ( ( in @ X3 @ X2 )
                                         => ~ ( in @ X4 @ X2 ) )
                                     => ( ~ ! [X5: $i] :
                                              ( ( in @ X5 @ X3 )
                                             => ( in @ X5 @ X4 ) )
                                       => ! [X5: $i] :
                                            ( ( in @ X5 @ X4 )
                                           => ( in @ X5 @ X3 ) ) ) ) )
                           => ~ ! [X3: $i] :
                                  ( ~ ( ! [X4: $i] :
                                          ( ( in @ X4 @ X3 )
                                         => ( in @ X4 @ X1 ) )
                                     => ! [X4: $i] :
                                          ~ ( in @ X4 @ X3 ) )
                                 => ~ ! [X4: $i,X5: $i] :
                                        ( ~ ( ~ ( ( in @ X4 @ X2 )
                                               => ~ ( in @ X5 @ X3 ) )
                                           => ~ ! [X6: $i] :
                                                  ( ( in @ X6 @ X4 )
                                                 => ~ ( in @ X6 @ X3 ) ) )
                                       => ~ ! [X6: $i] :
                                              ( ( in @ X6 @ X2 )
                                             => ( ~ ! [X7: $i] :
                                                      ( ( in @ X7 @ X6 )
                                                     => ( in @ X7 @ X4 ) )
                                               => ( in @ X5 @ X6 ) ) ) ) ) )
                   => ( ! [X1: $i > $o] :
                          ( ~ ! [X2: $i] :
                                ( ( X1 @ X2 )
                               => ~ ! [X3: $i] :
                                      ( ( X1 @ X3 )
                                     => ( X2 = X3 ) ) )
                         => ( X1 @ ( descr @ X1 ) ) )
                     => ( ! [X1: $i,X2: $i > $o,X3: $i] :
                            ( ( in @ X3 @ X1 )
                           => ( ( X2 @ X3 )
                             => ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) ) ) )
                       => ( ! [X1: $i,X2: $i > $o,X3: $i] :
                              ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
                             => ( in @ X3 @ X1 ) )
                         => ( ! [X1: $i,X2: $i > $o,X3: $i] :
                                ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
                               => ( X2 @ X3 ) )
                           => ( ! [X1: $i > $o] :
                                  ( ~ ! [X2: $i] :
                                        ( ( X1 @ X2 )
                                       => ~ ! [X3: $i] :
                                              ( ( X1 @ X3 )
                                             => ( X2 = X3 ) ) )
                                 => ~ ! [X2: $i] :
                                        ( ( X1 @ X2 )
                                       => ~ ! [X3: $i] :
                                              ( ( X1 @ X3 )
                                             => ( X2 = X3 ) ) ) )
                             => ( ! [X1: $o,X2: $i] :
                                    ( ( in @ X2
                                      @ ( dsetconstr @ ( powerset @ emptyset )
                                        @ ^ [X3: $i] : X1 ) )
                                   => X1 )
                               => ( ! [X1: $i] :
                                      ( ( in @ X1 @ emptyset )
                                     => ! [X2: $o] : X2 )
                                 => ( ! [X1: $i] :
                                        ~ ( in @ X1 @ emptyset )
                                   => ( ! [X1: $i] :
                                          ~ ( in @ X1 @ emptyset )
                                     => ( ! [X1: $i > $o] :
                                            ( ~ ! [X2: $i] :
                                                  ( ( X1 @ X2 )
                                                 => ~ ! [X3: $i] :
                                                        ( ( X1 @ X3 )
                                                       => ( X2 = X3 ) ) )
                                           => ~ ! [X2: $i] :
                                                  ~ ( X1 @ X2 ) )
                                       => ( ! [X1: $i,X2: $i] :
                                              ( ! [X3: $i] :
                                                  ( ( in @ X3 @ X1 )
                                                 => ( in @ X3 @ X2 ) )
                                             => ( ! [X3: $i] :
                                                    ( ( in @ X3 @ X2 )
                                                   => ( in @ X3 @ X1 ) )
                                               => ( X1 = X2 ) ) )
                                         => ( ! [X1: $i] :
                                                ( ! [X2: $i] :
                                                    ~ ( in @ X2 @ X1 )
                                               => ( X1 = emptyset ) )
                                           => ( ! [X1: $i] :
                                                  ( ! [X2: $i] :
                                                      ~ ( in @ X2 @ X1 )
                                                 => ( X1 = emptyset ) )
                                             => ( ! [X1: $i,X2: $i > $o,X3: $i] :
                                                    ( ( in @ X3 @ X1 )
                                                   => ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
                                                      = ( X2 @ X3 ) ) )
                                               => ( ! [X1: $i] :
                                                      ( ( X1 != emptyset )
                                                     => ~ ! [X2: $i] :
                                                            ~ ( in @ X2 @ X1 ) )
                                                 => ( ! [X1: $i,X2: $i > $o,X3: $i] :
                                                        ( ( in @ X3 @ X1 )
                                                       => ( ( X2 @ X3 )
                                                         => ( ( dsetconstr @ X1 @ X2 )
                                                           != emptyset ) ) )
                                                   => ( ! [X1: $i] :
                                                          ( ~ ! [X2: $i] :
                                                                ~ ( in @ X2 @ X1 )
                                                         => ( X1 != emptyset ) )
                                                     => ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
                                                       => ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
                                                         => ( ! [X1: $i,X2: $i,X3: $i] :
                                                                ( ( in @ X3 @ X2 )
                                                               => ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
                                                           => ( ! [X1: $i,X2: $i,X3: $i] :
                                                                  ( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
                                                                 => ! [X4: $o] :
                                                                      ( ( ( X3 = X1 )
                                                                       => X4 )
                                                                     => ( ( ( in @ X3 @ X2 )
                                                                         => X4 )
                                                                       => X4 ) ) )
                                                             => ( ! [X1: $i,X2: $i,X3: $i] :
                                                                    ( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
                                                                   => ( ( X3 != X1 )
                                                                     => ( in @ X3 @ X2 ) ) )
                                                               => ( ! [X1: $i] :
                                                                      ( ( dsetconstr @ X1
                                                                        @ ^ [X2: $i] : ~ $false )
                                                                      = X1 )
                                                                 => ( ! [X1: $i,X2: $i] :
                                                                        ( ! [X3: $i] :
                                                                            ( ( in @ X3 @ X2 )
                                                                           => ( in @ X3 @ X1 ) )
                                                                       => ( in @ X2 @ ( powerset @ X1 ) ) )
                                                                   => ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
    introduced(definition,[new_symbols(definition,[sP1])]) ).

thf(sP2,plain,
    ( sP2
  <=> ( ! [X1: $i] :
          ( ( X1 != emptyset )
         => ~ ! [X2: $i] :
                ~ ( in @ X2 @ X1 ) )
     => ( ! [X1: $i,X2: $i > $o,X3: $i] :
            ( ( in @ X3 @ X1 )
           => ( ( X2 @ X3 )
             => ( ( dsetconstr @ X1 @ X2 )
               != emptyset ) ) )
       => ( ! [X1: $i] :
              ( ~ ! [X2: $i] :
                    ~ ( in @ X2 @ X1 )
             => ( X1 != emptyset ) )
         => ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
           => ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
             => ( ! [X1: $i,X2: $i,X3: $i] :
                    ( ( in @ X3 @ X2 )
                   => ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
               => ( ! [X1: $i,X2: $i,X3: $i] :
                      ( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
                     => ! [X4: $o] :
                          ( ( ( X3 = X1 )
                           => X4 )
                         => ( ( ( in @ X3 @ X2 )
                             => X4 )
                           => X4 ) ) )
                 => ( ! [X1: $i,X2: $i,X3: $i] :
                        ( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
                       => ( ( X3 != X1 )
                         => ( in @ X3 @ X2 ) ) )
                   => ( ! [X1: $i] :
                          ( ( dsetconstr @ X1
                            @ ^ [X2: $i] : ~ $false )
                          = X1 )
                     => ( ! [X1: $i,X2: $i] :
                            ( ! [X3: $i] :
                                ( ( in @ X3 @ X2 )
                               => ( in @ X3 @ X1 ) )
                           => ( in @ X2 @ ( powerset @ X1 ) ) )
                       => ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ),
    introduced(definition,[new_symbols(definition,[sP2])]) ).

thf(sP3,plain,
    ( sP3
  <=> ( ! [X1: $o,X2: $i] :
          ( ( in @ X2
            @ ( dsetconstr @ ( powerset @ emptyset )
              @ ^ [X3: $i] : X1 ) )
         => X1 )
     => ( ! [X1: $i] :
            ( ( in @ X1 @ emptyset )
           => ! [X2: $o] : X2 )
       => ( ! [X1: $i] :
              ~ ( in @ X1 @ emptyset )
         => ( ! [X1: $i] :
                ~ ( in @ X1 @ emptyset )
           => ( ! [X1: $i > $o] :
                  ( ~ ! [X2: $i] :
                        ( ( X1 @ X2 )
                       => ~ ! [X3: $i] :
                              ( ( X1 @ X3 )
                             => ( X2 = X3 ) ) )
                 => ~ ! [X2: $i] :
                        ~ ( X1 @ X2 ) )
             => ( ! [X1: $i,X2: $i] :
                    ( ! [X3: $i] :
                        ( ( in @ X3 @ X1 )
                       => ( in @ X3 @ X2 ) )
                   => ( ! [X3: $i] :
                          ( ( in @ X3 @ X2 )
                         => ( in @ X3 @ X1 ) )
                     => ( X1 = X2 ) ) )
               => ( ! [X1: $i] :
                      ( ! [X2: $i] :
                          ~ ( in @ X2 @ X1 )
                     => ( X1 = emptyset ) )
                 => ( ! [X1: $i] :
                        ( ! [X2: $i] :
                            ~ ( in @ X2 @ X1 )
                       => ( X1 = emptyset ) )
                   => ( ! [X1: $i,X2: $i > $o,X3: $i] :
                          ( ( in @ X3 @ X1 )
                         => ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
                            = ( X2 @ X3 ) ) )
                     => sP2 ) ) ) ) ) ) ) ) ) ),
    introduced(definition,[new_symbols(definition,[sP3])]) ).

thf(sP4,plain,
    ( sP4
  <=> ( ! [X1: $i,X2: $i,X3: $i] :
          ( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
         => ( ( X3 != X1 )
           => ( in @ X3 @ X2 ) ) )
     => ( ! [X1: $i] :
            ( ( dsetconstr @ X1
              @ ^ [X2: $i] : ~ $false )
            = X1 )
       => ( ! [X1: $i,X2: $i] :
              ( ! [X3: $i] :
                  ( ( in @ X3 @ X2 )
                 => ( in @ X3 @ X1 ) )
             => ( in @ X2 @ ( powerset @ X1 ) ) )
         => ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) ) ) ) ) ),
    introduced(definition,[new_symbols(definition,[sP4])]) ).

thf(sP5,plain,
    ( sP5
  <=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
          ( ( in @ X3 @ X1 )
         => ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
            = ( X2 @ X3 ) ) )
     => sP2 ) ),
    introduced(definition,[new_symbols(definition,[sP5])]) ).

thf(sP6,plain,
    ( sP6
  <=> ( ! [X1: $i,X2: $i,X3: $i] :
          ( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
          = ( ( X3 != X1 )
           => ( in @ X3 @ X2 ) ) )
     => sP1 ) ),
    introduced(definition,[new_symbols(definition,[sP6])]) ).

thf(sP7,plain,
    ( sP7
  <=> ( ! [X1: $i] :
          ( ( in @ X1 @ emptyset )
         => ! [X2: $o] : X2 )
     => ( ! [X1: $i] :
            ~ ( in @ X1 @ emptyset )
       => ( ! [X1: $i] :
              ~ ( in @ X1 @ emptyset )
         => ( ! [X1: $i > $o] :
                ( ~ ! [X2: $i] :
                      ( ( X1 @ X2 )
                     => ~ ! [X3: $i] :
                            ( ( X1 @ X3 )
                           => ( X2 = X3 ) ) )
               => ~ ! [X2: $i] :
                      ~ ( X1 @ X2 ) )
           => ( ! [X1: $i,X2: $i] :
                  ( ! [X3: $i] :
                      ( ( in @ X3 @ X1 )
                     => ( in @ X3 @ X2 ) )
                 => ( ! [X3: $i] :
                        ( ( in @ X3 @ X2 )
                       => ( in @ X3 @ X1 ) )
                   => ( X1 = X2 ) ) )
             => ( ! [X1: $i] :
                    ( ! [X2: $i] :
                        ~ ( in @ X2 @ X1 )
                   => ( X1 = emptyset ) )
               => ( ! [X1: $i] :
                      ( ! [X2: $i] :
                          ~ ( in @ X2 @ X1 )
                     => ( X1 = emptyset ) )
                 => sP5 ) ) ) ) ) ) ) ),
    introduced(definition,[new_symbols(definition,[sP7])]) ).

thf(sP8,plain,
    ( sP8
  <=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
     => ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
       => ( ! [X1: $i,X2: $i,X3: $i] :
              ( ( in @ X3 @ X2 )
             => ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
         => ( ! [X1: $i,X2: $i,X3: $i] :
                ( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
               => ! [X4: $o] :
                    ( ( ( X3 = X1 )
                     => X4 )
                   => ( ( ( in @ X3 @ X2 )
                       => X4 )
                     => X4 ) ) )
           => sP4 ) ) ) ) ),
    introduced(definition,[new_symbols(definition,[sP8])]) ).

thf(sP9,plain,
    ( sP9
  <=> ( ! [X1: $i,X2: $i] :
          ( ! [X3: $i] :
              ( ( in @ X3 @ X1 )
             => ( in @ X3 @ X2 ) )
         => ( ! [X3: $i] :
                ( ( in @ X3 @ X2 )
               => ( in @ X3 @ X1 ) )
           => ( X1 = X2 ) ) )
     => ( ! [X1: $i] :
            ( ! [X2: $i] :
                ~ ( in @ X2 @ X1 )
           => ( X1 = emptyset ) )
       => ( ! [X1: $i] :
              ( ! [X2: $i] :
                  ~ ( in @ X2 @ X1 )
             => ( X1 = emptyset ) )
         => sP5 ) ) ) ),
    introduced(definition,[new_symbols(definition,[sP9])]) ).

thf(sP10,plain,
    ( sP10
  <=> ( ! [X1: $i] :
          ( ~ ! [X2: $i] :
                ~ ( in @ X2 @ X1 )
         => ( X1 != emptyset ) )
     => sP8 ) ),
    introduced(definition,[new_symbols(definition,[sP10])]) ).

thf(sP11,plain,
    ( sP11
  <=> ( ! [X1: $i] :
          ~ ( in @ X1 @ emptyset )
     => ( ! [X1: $i] :
            ~ ( in @ X1 @ emptyset )
       => ( ! [X1: $i > $o] :
              ( ~ ! [X2: $i] :
                    ( ( X1 @ X2 )
                   => ~ ! [X3: $i] :
                          ( ( X1 @ X3 )
                         => ( X2 = X3 ) ) )
             => ~ ! [X2: $i] :
                    ~ ( X1 @ X2 ) )
         => sP9 ) ) ) ),
    introduced(definition,[new_symbols(definition,[sP11])]) ).

thf(sP12,plain,
    ( sP12
  <=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
          ( ( in @ X3 @ X1 )
         => ( ( X2 @ X3 )
           => ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) ) ) )
     => ( ! [X1: $i,X2: $i > $o,X3: $i] :
            ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
           => ( in @ X3 @ X1 ) )
       => ( ! [X1: $i,X2: $i > $o,X3: $i] :
              ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
             => ( X2 @ X3 ) )
         => ( ! [X1: $i > $o] :
                ( ~ ! [X2: $i] :
                      ( ( X1 @ X2 )
                     => ~ ! [X3: $i] :
                            ( ( X1 @ X3 )
                           => ( X2 = X3 ) ) )
               => ~ ! [X2: $i] :
                      ( ( X1 @ X2 )
                     => ~ ! [X3: $i] :
                            ( ( X1 @ X3 )
                           => ( X2 = X3 ) ) ) )
           => sP3 ) ) ) ) ),
    introduced(definition,[new_symbols(definition,[sP12])]) ).

thf(sP13,plain,
    ( sP13
  <=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
          ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
         => ( X2 @ X3 ) )
     => ( ! [X1: $i > $o] :
            ( ~ ! [X2: $i] :
                  ( ( X1 @ X2 )
                 => ~ ! [X3: $i] :
                        ( ( X1 @ X3 )
                       => ( X2 = X3 ) ) )
           => ~ ! [X2: $i] :
                  ( ( X1 @ X2 )
                 => ~ ! [X3: $i] :
                        ( ( X1 @ X3 )
                       => ( X2 = X3 ) ) ) )
       => sP3 ) ) ),
    introduced(definition,[new_symbols(definition,[sP13])]) ).

thf(sP14,plain,
    ( sP14
  <=> ( ! [X1: $i] :
          ( ! [X2: $i] :
              ~ ( in @ X2 @ X1 )
         => ( X1 = emptyset ) )
     => sP5 ) ),
    introduced(definition,[new_symbols(definition,[sP14])]) ).

thf(sP15,plain,
    ( sP15
  <=> ( ! [X1: $i] :
          ~ ! [X2: $i] :
              ( ~ ( ~ ( ! [X3: $i] :
                          ( ( in @ X3 @ X2 )
                         => ! [X4: $i] :
                              ( ( in @ X4 @ X3 )
                             => ( in @ X4 @ X1 ) ) )
                     => ~ ! [X3: $i,X4: $i] :
                            ( ~ ( ( in @ X3 @ X1 )
                               => ~ ( in @ X4 @ X1 ) )
                           => ( ! [X5: $i] :
                                  ( ( in @ X5 @ X2 )
                                 => ( ( in @ X3 @ X5 )
                                    = ( in @ X4 @ X5 ) ) )
                             => ( X3 = X4 ) ) ) )
                 => ~ ! [X3: $i,X4: $i] :
                        ( ~ ( ( in @ X3 @ X2 )
                           => ~ ( in @ X4 @ X2 ) )
                       => ( ~ ! [X5: $i] :
                                ( ( in @ X5 @ X3 )
                               => ( in @ X5 @ X4 ) )
                         => ! [X5: $i] :
                              ( ( in @ X5 @ X4 )
                             => ( in @ X5 @ X3 ) ) ) ) )
             => ~ ! [X3: $i] :
                    ( ~ ( ! [X4: $i] :
                            ( ( in @ X4 @ X3 )
                           => ( in @ X4 @ X1 ) )
                       => ! [X4: $i] :
                            ~ ( in @ X4 @ X3 ) )
                   => ~ ! [X4: $i,X5: $i] :
                          ( ~ ( ~ ( ( in @ X4 @ X2 )
                                 => ~ ( in @ X5 @ X3 ) )
                             => ~ ! [X6: $i] :
                                    ( ( in @ X6 @ X4 )
                                   => ~ ( in @ X6 @ X3 ) ) )
                         => ~ ! [X6: $i] :
                                ( ( in @ X6 @ X2 )
                               => ( ~ ! [X7: $i] :
                                        ( ( in @ X7 @ X6 )
                                       => ( in @ X7 @ X4 ) )
                                 => ( in @ X5 @ X6 ) ) ) ) ) )
     => ( ! [X1: $i > $o] :
            ( ~ ! [X2: $i] :
                  ( ( X1 @ X2 )
                 => ~ ! [X3: $i] :
                        ( ( X1 @ X3 )
                       => ( X2 = X3 ) ) )
           => ( X1 @ ( descr @ X1 ) ) )
       => sP12 ) ) ),
    introduced(definition,[new_symbols(definition,[sP15])]) ).

thf(sP16,plain,
    ( sP16
  <=> ( ! [X1: $i] :
          ( ! [X2: $i] :
              ~ ( in @ X2 @ X1 )
         => ( X1 = emptyset ) )
     => sP14 ) ),
    introduced(definition,[new_symbols(definition,[sP16])]) ).

thf(sP17,plain,
    ( sP17
  <=> ( ! [X1: $i] :
          ( ( in @ X1 @ emptyset )
         => ( in @ X1 @ eigen__0 ) )
     => ( in @ emptyset @ ( powerset @ eigen__0 ) ) ) ),
    introduced(definition,[new_symbols(definition,[sP17])]) ).

thf(sP18,plain,
    ( sP18
  <=> ( ! [X1: $i > $o] :
          ( ~ ! [X2: $i] :
                ( ( X1 @ X2 )
               => ~ ! [X3: $i] :
                      ( ( X1 @ X3 )
                     => ( X2 = X3 ) ) )
         => ~ ! [X2: $i] :
                ( ( X1 @ X2 )
               => ~ ! [X3: $i] :
                      ( ( X1 @ X3 )
                     => ( X2 = X3 ) ) ) )
     => sP3 ) ),
    introduced(definition,[new_symbols(definition,[sP18])]) ).

thf(sP19,plain,
    ( sP19
  <=> ( ! [X1: $i,X2: $i,X3: $i] :
          ( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
         => ! [X4: $o] :
              ( ( ( X3 = X1 )
               => X4 )
             => ( ( ( in @ X3 @ X2 )
                 => X4 )
               => X4 ) ) )
     => sP4 ) ),
    introduced(definition,[new_symbols(definition,[sP19])]) ).

thf(sP20,plain,
    ( sP20
  <=> ( ! [X1: $i > $i > $o,X2: $i] :
          ( ! [X3: $i] :
              ( ( in @ X3 @ X2 )
             => ~ ! [X4: $i] :
                    ( ( X1 @ X3 @ X4 )
                   => ~ ! [X5: $i] :
                          ( ( X1 @ X3 @ X5 )
                         => ( X4 = X5 ) ) ) )
         => ~ ! [X3: $i] :
                ~ ! [X4: $i] :
                    ( ( in @ X4 @ X3 )
                    = ( ~ ! [X5: $i] :
                            ( ( in @ X5 @ X2 )
                           => ~ ( X1 @ X5 @ X4 ) ) ) ) )
     => ( ! [X1: $i] :
            ( ~ ! [X2: $i] :
                  ~ ( in @ X2 @ X1 )
           => ~ ! [X2: $i] :
                  ( ( in @ X2 @ X1 )
                 => ~ ! [X3: $i] :
                        ( ( in @ X3 @ X2 )
                       => ~ ( in @ X3 @ X1 ) ) ) )
       => sP15 ) ) ),
    introduced(definition,[new_symbols(definition,[sP20])]) ).

thf(sP21,plain,
    ( sP21
  <=> ( ! [X1: $i > $o] :
          ( ~ ! [X2: $i] :
                ( ( X1 @ X2 )
               => ~ ! [X3: $i] :
                      ( ( X1 @ X3 )
                     => ( X2 = X3 ) ) )
         => ( X1 @ ( descr @ X1 ) ) )
     => sP12 ) ),
    introduced(definition,[new_symbols(definition,[sP21])]) ).

thf(sP22,plain,
    ( sP22
  <=> ( ! [X1: $i > $o] :
          ( ~ ! [X2: $i] :
                ( ( X1 @ X2 )
               => ~ ! [X3: $i] :
                      ( ( X1 @ X3 )
                     => ( X2 = X3 ) ) )
         => ~ ! [X2: $i] :
                ~ ( X1 @ X2 ) )
     => sP9 ) ),
    introduced(definition,[new_symbols(definition,[sP22])]) ).

thf(sP23,plain,
    ( sP23
  <=> ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) ) ),
    introduced(definition,[new_symbols(definition,[sP23])]) ).

thf(sP24,plain,
    ( sP24
  <=> ( ! [X1: $i] :
          ( ( in @ X1 @ omega )
         => ( in @ ( setadjoin @ X1 @ X1 ) @ omega ) )
     => ( ! [X1: $i] :
            ( ~ ( ( in @ emptyset @ X1 )
               => ~ ! [X2: $i] :
                      ( ~ ( ( in @ X2 @ omega )
                         => ~ ( in @ X2 @ X1 ) )
                     => ( in @ ( setadjoin @ X2 @ X2 ) @ X1 ) ) )
           => ! [X2: $i] :
                ( ( in @ X2 @ omega )
               => ( in @ X2 @ X1 ) ) )
       => sP20 ) ) ),
    introduced(definition,[new_symbols(definition,[sP24])]) ).

thf(sP25,plain,
    ( sP25
  <=> ( ! [X1: $i,X2: $i] :
          ( ( in @ X2 @ ( setunion @ X1 ) )
          = ( ~ ! [X3: $i] :
                  ( ( in @ X2 @ X3 )
                 => ~ ( in @ X3 @ X1 ) ) ) )
     => ( ( in @ emptyset @ omega )
       => sP24 ) ) ),
    introduced(definition,[new_symbols(definition,[sP25])]) ).

thf(sP26,plain,
    ( sP26
  <=> ( ( in @ emptyset @ omega )
     => sP24 ) ),
    introduced(definition,[new_symbols(definition,[sP26])]) ).

thf(sP27,plain,
    ( sP27
  <=> ( ! [X1: $i] :
          ( ( dsetconstr @ X1
            @ ^ [X2: $i] : ~ $false )
          = X1 )
     => ( ! [X1: $i,X2: $i] :
            ( ! [X3: $i] :
                ( ( in @ X3 @ X2 )
               => ( in @ X3 @ X1 ) )
           => ( in @ X2 @ ( powerset @ X1 ) ) )
       => sP23 ) ) ),
    introduced(definition,[new_symbols(definition,[sP27])]) ).

thf(sP28,plain,
    ( sP28
  <=> ( ! [X1: $i,X2: $i] :
          ( ! [X3: $i] :
              ( ( in @ X3 @ X2 )
             => ( in @ X3 @ X1 ) )
         => ( in @ X2 @ ( powerset @ X1 ) ) )
     => sP23 ) ),
    introduced(definition,[new_symbols(definition,[sP28])]) ).

thf(sP29,plain,
    ( sP29
  <=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
          ( ( in @ X3 @ X1 )
         => ( ( X2 @ X3 )
           => ( ( dsetconstr @ X1 @ X2 )
             != emptyset ) ) )
     => sP10 ) ),
    introduced(definition,[new_symbols(definition,[sP29])]) ).

thf(sP30,plain,
    ( sP30
  <=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
          ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
         => ( in @ X3 @ X1 ) )
     => sP13 ) ),
    introduced(definition,[new_symbols(definition,[sP30])]) ).

thf(sP31,plain,
    ( sP31
  <=> ! [X1: $i] :
        ( ( in @ X1 @ emptyset )
       => ( in @ X1 @ eigen__0 ) ) ),
    introduced(definition,[new_symbols(definition,[sP31])]) ).

thf(sP32,plain,
    ( sP32
  <=> ( ! [X1: $i] :
          ~ ( in @ X1 @ emptyset )
     => sP6 ) ),
    introduced(definition,[new_symbols(definition,[sP32])]) ).

thf(sP33,plain,
    ( sP33
  <=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
     => ( ! [X1: $i,X2: $i,X3: $i] :
            ( ( in @ X3 @ X2 )
           => ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
       => sP19 ) ) ),
    introduced(definition,[new_symbols(definition,[sP33])]) ).

thf(sP34,plain,
    ( sP34
  <=> ( ! [X1: $i,X2: $i] :
          ( ! [X3: $i] :
              ( ( in @ X3 @ X1 )
              = ( in @ X3 @ X2 ) )
         => ( X1 = X2 ) )
     => sP32 ) ),
    introduced(definition,[new_symbols(definition,[sP34])]) ).

thf(sP35,plain,
    ( sP35
  <=> ( ! [X1: $i,X2: $i,X3: $i] :
          ( ( in @ X3 @ X2 )
         => ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
     => sP19 ) ),
    introduced(definition,[new_symbols(definition,[sP35])]) ).

thf(sP36,plain,
    ( sP36
  <=> ( ! [X1: $i] :
          ( ~ ( ( in @ emptyset @ X1 )
             => ~ ! [X2: $i] :
                    ( ~ ( ( in @ X2 @ omega )
                       => ~ ( in @ X2 @ X1 ) )
                   => ( in @ ( setadjoin @ X2 @ X2 ) @ X1 ) ) )
         => ! [X2: $i] :
              ( ( in @ X2 @ omega )
             => ( in @ X2 @ X1 ) ) )
     => sP20 ) ),
    introduced(definition,[new_symbols(definition,[sP36])]) ).

thf(sP37,plain,
    ( sP37
  <=> ( ( in @ eigen__5 @ emptyset )
     => ( in @ eigen__5 @ eigen__0 ) ) ),
    introduced(definition,[new_symbols(definition,[sP37])]) ).

thf(sP38,plain,
    ( sP38
  <=> ( in @ eigen__5 @ emptyset ) ),
    introduced(definition,[new_symbols(definition,[sP38])]) ).

thf(sP39,plain,
    ( sP39
  <=> ! [X1: $i] :
        ( ! [X2: $i] :
            ( ( in @ X2 @ X1 )
           => ( in @ X2 @ eigen__0 ) )
       => ( in @ X1 @ ( powerset @ eigen__0 ) ) ) ),
    introduced(definition,[new_symbols(definition,[sP39])]) ).

thf(sP40,plain,
    ( sP40
  <=> ! [X1: $i,X2: $i] :
        ( ! [X3: $i] :
            ( ( in @ X3 @ X2 )
           => ( in @ X3 @ X1 ) )
       => ( in @ X2 @ ( powerset @ X1 ) ) ) ),
    introduced(definition,[new_symbols(definition,[sP40])]) ).

thf(sP41,plain,
    ( sP41
  <=> ! [X1: $i] :
        ~ ( in @ X1 @ emptyset ) ),
    introduced(definition,[new_symbols(definition,[sP41])]) ).

thf(sP42,plain,
    ( sP42
  <=> ( ! [X1: $i] :
          ( ~ ! [X2: $i] :
                ~ ( in @ X2 @ X1 )
         => ~ ! [X2: $i] :
                ( ( in @ X2 @ X1 )
               => ~ ! [X3: $i] :
                      ( ( in @ X3 @ X2 )
                     => ~ ( in @ X3 @ X1 ) ) ) )
     => sP15 ) ),
    introduced(definition,[new_symbols(definition,[sP42])]) ).

thf(sP43,plain,
    ( sP43
  <=> ( sP41
     => sP22 ) ),
    introduced(definition,[new_symbols(definition,[sP43])]) ).

thf(sP44,plain,
    ( sP44
  <=> ( in @ emptyset @ ( powerset @ eigen__0 ) ) ),
    introduced(definition,[new_symbols(definition,[sP44])]) ).

thf(def_exu,definition,
    ( exu
    = ( ^ [X1: $i > $o] :
          ~ ! [X2: $i] :
              ( ( X1 @ X2 )
             => ~ ! [X3: $i] :
                    ( ( X1 @ X3 )
                   => ( X2 = X3 ) ) ) ) ) ).

thf(def_setextAx,definition,
    ( setextAx
    = ( ! [X1: $i,X2: $i] :
          ( ! [X3: $i] :
              ( ( in @ X3 @ X1 )
              = ( in @ X3 @ X2 ) )
         => ( X1 = X2 ) ) ) ) ).

thf(def_emptysetAx,definition,
    emptysetAx = sP41 ).

thf(def_setadjoinAx,definition,
    ( setadjoinAx
    = ( ! [X1: $i,X2: $i,X3: $i] :
          ( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
          = ( ( X3 != X1 )
           => ( in @ X3 @ X2 ) ) ) ) ) ).

thf(def_powersetAx,definition,
    ( powersetAx
    = ( ! [X1: $i,X2: $i] :
          ( ( in @ X2 @ ( powerset @ X1 ) )
          = ( ! [X3: $i] :
                ( ( in @ X3 @ X2 )
               => ( in @ X3 @ X1 ) ) ) ) ) ) ).

thf(def_setunionAx,definition,
    ( setunionAx
    = ( ! [X1: $i,X2: $i] :
          ( ( in @ X2 @ ( setunion @ X1 ) )
          = ( ~ ! [X3: $i] :
                  ( ( in @ X2 @ X3 )
                 => ~ ( in @ X3 @ X1 ) ) ) ) ) ) ).

thf(def_omega0Ax,definition,
    ( omega0Ax
    = ( in @ emptyset @ omega ) ) ).

thf(def_omegaSAx,definition,
    ( omegaSAx
    = ( ! [X1: $i] :
          ( ( in @ X1 @ omega )
         => ( in @ ( setadjoin @ X1 @ X1 ) @ omega ) ) ) ) ).

thf(def_omegaIndAx,definition,
    ( omegaIndAx
    = ( ! [X1: $i] :
          ( ~ ( ( in @ emptyset @ X1 )
             => ~ ! [X2: $i] :
                    ( ~ ( ( in @ X2 @ omega )
                       => ~ ( in @ X2 @ X1 ) )
                   => ( in @ ( setadjoin @ X2 @ X2 ) @ X1 ) ) )
         => ! [X2: $i] :
              ( ( in @ X2 @ omega )
             => ( in @ X2 @ X1 ) ) ) ) ) ).

thf(def_replAx,definition,
    ( replAx
    = ( ! [X1: $i > $i > $o,X2: $i] :
          ( ! [X3: $i] :
              ( ( in @ X3 @ X2 )
             => ( exu @ ( X1 @ X3 ) ) )
         => ~ ! [X3: $i] :
                ~ ! [X4: $i] :
                    ( ( in @ X4 @ X3 )
                    = ( ~ ! [X5: $i] :
                            ( ( in @ X5 @ X2 )
                           => ~ ( X1 @ X5 @ X4 ) ) ) ) ) ) ) ).

thf(def_foundationAx,definition,
    ( foundationAx
    = ( ! [X1: $i] :
          ( ~ ! [X2: $i] :
                ~ ( in @ X2 @ X1 )
         => ~ ! [X2: $i] :
                ( ( in @ X2 @ X1 )
               => ~ ! [X3: $i] :
                      ( ( in @ X3 @ X2 )
                     => ~ ( in @ X3 @ X1 ) ) ) ) ) ) ).

thf(def_wellorderingAx,definition,
    ( wellorderingAx
    = ( ! [X1: $i] :
          ~ ! [X2: $i] :
              ( ~ ( ~ ( ! [X3: $i] :
                          ( ( in @ X3 @ X2 )
                         => ! [X4: $i] :
                              ( ( in @ X4 @ X3 )
                             => ( in @ X4 @ X1 ) ) )
                     => ~ ! [X3: $i,X4: $i] :
                            ( ~ ( ( in @ X3 @ X1 )
                               => ~ ( in @ X4 @ X1 ) )
                           => ( ! [X5: $i] :
                                  ( ( in @ X5 @ X2 )
                                 => ( ( in @ X3 @ X5 )
                                    = ( in @ X4 @ X5 ) ) )
                             => ( X3 = X4 ) ) ) )
                 => ~ ! [X3: $i,X4: $i] :
                        ( ~ ( ( in @ X3 @ X2 )
                           => ~ ( in @ X4 @ X2 ) )
                       => ( ~ ! [X5: $i] :
                                ( ( in @ X5 @ X3 )
                               => ( in @ X5 @ X4 ) )
                         => ! [X5: $i] :
                              ( ( in @ X5 @ X4 )
                             => ( in @ X5 @ X3 ) ) ) ) )
             => ~ ! [X3: $i] :
                    ( ~ ( ! [X4: $i] :
                            ( ( in @ X4 @ X3 )
                           => ( in @ X4 @ X1 ) )
                       => ! [X4: $i] :
                            ~ ( in @ X4 @ X3 ) )
                   => ~ ! [X4: $i,X5: $i] :
                          ( ~ ( ~ ( ( in @ X4 @ X2 )
                                 => ~ ( in @ X5 @ X3 ) )
                             => ~ ! [X6: $i] :
                                    ( ( in @ X6 @ X4 )
                                   => ~ ( in @ X6 @ X3 ) ) )
                         => ~ ! [X6: $i] :
                                ( ( in @ X6 @ X2 )
                               => ( ~ ! [X7: $i] :
                                        ( ( in @ X7 @ X6 )
                                       => ( in @ X7 @ X4 ) )
                                 => ( in @ X5 @ X6 ) ) ) ) ) ) ) ) ).

thf(def_descrp,definition,
    ( descrp
    = ( ! [X1: $i > $o] :
          ( ( exu @ X1 )
         => ( X1 @ ( descr @ X1 ) ) ) ) ) ).

thf(def_dsetconstrI,definition,
    ( dsetconstrI
    = ( ! [X1: $i,X2: $i > $o,X3: $i] :
          ( ( in @ X3 @ X1 )
         => ( ( X2 @ X3 )
           => ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) ) ) ) ) ) ).

thf(def_dsetconstrEL,definition,
    ( dsetconstrEL
    = ( ! [X1: $i,X2: $i > $o,X3: $i] :
          ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
         => ( in @ X3 @ X1 ) ) ) ) ).

thf(def_dsetconstrER,definition,
    ( dsetconstrER
    = ( ! [X1: $i,X2: $i > $o,X3: $i] :
          ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
         => ( X2 @ X3 ) ) ) ) ).

thf(def_exuE1,definition,
    ( exuE1
    = ( ! [X1: $i > $o] :
          ( ( exu @ X1 )
         => ~ ! [X2: $i] :
                ( ( X1 @ X2 )
               => ~ ! [X3: $i] :
                      ( ( X1 @ X3 )
                     => ( X2 = X3 ) ) ) ) ) ) ).

thf(def_prop2set,definition,
    ( prop2set
    = ( ^ [X1: $o] :
          ( dsetconstr @ ( powerset @ emptyset )
          @ ^ [X2: $i] : X1 ) ) ) ).

thf(def_prop2setE,definition,
    ( prop2setE
    = ( ! [X1: $o,X2: $i] :
          ( ( in @ X2 @ ( prop2set @ X1 ) )
         => X1 ) ) ) ).

thf(def_emptysetE,definition,
    ( emptysetE
    = ( ! [X1: $i] :
          ( ( in @ X1 @ emptyset )
         => ! [X2: $o] : X2 ) ) ) ).

thf(def_emptysetimpfalse,definition,
    emptysetimpfalse = sP41 ).

thf(def_notinemptyset,definition,
    notinemptyset = sP41 ).

thf(def_exuE3e,definition,
    ( exuE3e
    = ( ! [X1: $i > $o] :
          ( ( exu @ X1 )
         => ~ ! [X2: $i] :
                ~ ( X1 @ X2 ) ) ) ) ).

thf(def_setext,definition,
    ( setext
    = ( ! [X1: $i,X2: $i] :
          ( ! [X3: $i] :
              ( ( in @ X3 @ X1 )
             => ( in @ X3 @ X2 ) )
         => ( ! [X3: $i] :
                ( ( in @ X3 @ X2 )
               => ( in @ X3 @ X1 ) )
           => ( X1 = X2 ) ) ) ) ) ).

thf(def_emptyI,definition,
    ( emptyI
    = ( ! [X1: $i] :
          ( ! [X2: $i] :
              ~ ( in @ X2 @ X1 )
         => ( X1 = emptyset ) ) ) ) ).

thf(def_noeltsimpempty,definition,
    ( noeltsimpempty
    = ( ! [X1: $i] :
          ( ! [X2: $i] :
              ~ ( in @ X2 @ X1 )
         => ( X1 = emptyset ) ) ) ) ).

thf(def_setbeta,definition,
    ( setbeta
    = ( ! [X1: $i,X2: $i > $o,X3: $i] :
          ( ( in @ X3 @ X1 )
         => ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
            = ( X2 @ X3 ) ) ) ) ) ).

thf(def_nonempty,definition,
    ( nonempty
    = ( ^ [X1: $i] : ( X1 != emptyset ) ) ) ).

thf(def_nonemptyE1,definition,
    ( nonemptyE1
    = ( ! [X1: $i] :
          ( ( nonempty @ X1 )
         => ~ ! [X2: $i] :
                ~ ( in @ X2 @ X1 ) ) ) ) ).

thf(def_nonemptyI,definition,
    ( nonemptyI
    = ( ! [X1: $i,X2: $i > $o,X3: $i] :
          ( ( in @ X3 @ X1 )
         => ( ( X2 @ X3 )
           => ( nonempty @ ( dsetconstr @ X1 @ X2 ) ) ) ) ) ) ).

thf(def_nonemptyI1,definition,
    ( nonemptyI1
    = ( ! [X1: $i] :
          ( ~ ! [X2: $i] :
                ~ ( in @ X2 @ X1 )
         => ( nonempty @ X1 ) ) ) ) ).

thf(def_setadjoinIL,definition,
    ( setadjoinIL
    = ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) ) ) ) ).

thf(def_emptyinunitempty,definition,
    ( emptyinunitempty
    = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).

thf(def_setadjoinIR,definition,
    ( setadjoinIR
    = ( ! [X1: $i,X2: $i,X3: $i] :
          ( ( in @ X3 @ X2 )
         => ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) ) ) ) ).

thf(def_setadjoinE,definition,
    ( setadjoinE
    = ( ! [X1: $i,X2: $i,X3: $i] :
          ( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
         => ! [X4: $o] :
              ( ( ( X3 = X1 )
               => X4 )
             => ( ( ( in @ X3 @ X2 )
                 => X4 )
               => X4 ) ) ) ) ) ).

thf(def_setadjoinOr,definition,
    ( setadjoinOr
    = ( ! [X1: $i,X2: $i,X3: $i] :
          ( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
         => ( ( X3 != X1 )
           => ( in @ X3 @ X2 ) ) ) ) ) ).

thf(def_setoftrueEq,definition,
    ( setoftrueEq
    = ( ! [X1: $i] :
          ( ( dsetconstr @ X1
            @ ^ [X2: $i] : ~ $false )
          = X1 ) ) ) ).

thf(def_powersetI,definition,
    powersetI = sP40 ).

thf(emptyinPowerset,conjecture,
    sP34 ).

thf(h1,negated_conjecture,
    ~ sP34,
    inference(assume_negation,[status(cth)],[emptyinPowerset]) ).

thf(1,plain,
    ( ~ sP41
    | ~ sP38 ),
    inference(all_rule,[status(thm)],]) ).

thf(2,plain,
    ( sP37
    | sP38 ),
    inference(prop_rule,[status(thm)],]) ).

thf(3,plain,
    ( sP31
    | ~ sP37 ),
    inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__5]) ).

thf(4,plain,
    ( ~ sP40
    | sP39 ),
    inference(all_rule,[status(thm)],]) ).

thf(5,plain,
    ( ~ sP39
    | sP17 ),
    inference(all_rule,[status(thm)],]) ).

thf(6,plain,
    ( ~ sP17
    | ~ sP31
    | sP44 ),
    inference(prop_rule,[status(thm)],]) ).

thf(7,plain,
    ( sP23
    | ~ sP44 ),
    inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__0]) ).

thf(8,plain,
    ( sP28
    | ~ sP23 ),
    inference(prop_rule,[status(thm)],]) ).

thf(9,plain,
    ( sP28
    | sP40 ),
    inference(prop_rule,[status(thm)],]) ).

thf(10,plain,
    ( sP27
    | ~ sP28 ),
    inference(prop_rule,[status(thm)],]) ).

thf(11,plain,
    ( sP4
    | ~ sP27 ),
    inference(prop_rule,[status(thm)],]) ).

thf(12,plain,
    ( sP19
    | ~ sP4 ),
    inference(prop_rule,[status(thm)],]) ).

thf(13,plain,
    ( sP35
    | ~ sP19 ),
    inference(prop_rule,[status(thm)],]) ).

thf(14,plain,
    ( sP33
    | ~ sP35 ),
    inference(prop_rule,[status(thm)],]) ).

thf(15,plain,
    ( sP8
    | ~ sP33 ),
    inference(prop_rule,[status(thm)],]) ).

thf(16,plain,
    ( sP10
    | ~ sP8 ),
    inference(prop_rule,[status(thm)],]) ).

thf(17,plain,
    ( sP29
    | ~ sP10 ),
    inference(prop_rule,[status(thm)],]) ).

thf(18,plain,
    ( sP2
    | ~ sP29 ),
    inference(prop_rule,[status(thm)],]) ).

thf(19,plain,
    ( sP5
    | ~ sP2 ),
    inference(prop_rule,[status(thm)],]) ).

thf(20,plain,
    ( sP14
    | ~ sP5 ),
    inference(prop_rule,[status(thm)],]) ).

thf(21,plain,
    ( sP16
    | ~ sP14 ),
    inference(prop_rule,[status(thm)],]) ).

thf(22,plain,
    ( sP9
    | ~ sP16 ),
    inference(prop_rule,[status(thm)],]) ).

thf(23,plain,
    ( sP22
    | ~ sP9 ),
    inference(prop_rule,[status(thm)],]) ).

thf(24,plain,
    ( sP43
    | ~ sP22 ),
    inference(prop_rule,[status(thm)],]) ).

thf(25,plain,
    ( sP11
    | ~ sP43 ),
    inference(prop_rule,[status(thm)],]) ).

thf(26,plain,
    ( sP7
    | ~ sP11 ),
    inference(prop_rule,[status(thm)],]) ).

thf(27,plain,
    ( sP3
    | ~ sP7 ),
    inference(prop_rule,[status(thm)],]) ).

thf(28,plain,
    ( sP18
    | ~ sP3 ),
    inference(prop_rule,[status(thm)],]) ).

thf(29,plain,
    ( sP13
    | ~ sP18 ),
    inference(prop_rule,[status(thm)],]) ).

thf(30,plain,
    ( sP30
    | ~ sP13 ),
    inference(prop_rule,[status(thm)],]) ).

thf(31,plain,
    ( sP12
    | ~ sP30 ),
    inference(prop_rule,[status(thm)],]) ).

thf(32,plain,
    ( sP21
    | ~ sP12 ),
    inference(prop_rule,[status(thm)],]) ).

thf(33,plain,
    ( sP15
    | ~ sP21 ),
    inference(prop_rule,[status(thm)],]) ).

thf(34,plain,
    ( sP42
    | ~ sP15 ),
    inference(prop_rule,[status(thm)],]) ).

thf(35,plain,
    ( sP20
    | ~ sP42 ),
    inference(prop_rule,[status(thm)],]) ).

thf(36,plain,
    ( sP36
    | ~ sP20 ),
    inference(prop_rule,[status(thm)],]) ).

thf(37,plain,
    ( sP24
    | ~ sP36 ),
    inference(prop_rule,[status(thm)],]) ).

thf(38,plain,
    ( sP26
    | ~ sP24 ),
    inference(prop_rule,[status(thm)],]) ).

thf(39,plain,
    ( sP25
    | ~ sP26 ),
    inference(prop_rule,[status(thm)],]) ).

thf(40,plain,
    ( sP1
    | ~ sP25 ),
    inference(prop_rule,[status(thm)],]) ).

thf(41,plain,
    ( sP6
    | ~ sP1 ),
    inference(prop_rule,[status(thm)],]) ).

thf(42,plain,
    ( sP32
    | ~ sP6 ),
    inference(prop_rule,[status(thm)],]) ).

thf(43,plain,
    ( sP32
    | sP41 ),
    inference(prop_rule,[status(thm)],]) ).

thf(44,plain,
    ( sP34
    | ~ sP32 ),
    inference(prop_rule,[status(thm)],]) ).

thf(45,plain,
    $false,
    inference(prop_unsat,[status(thm),assumptions([h1,h0])],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,h1]) ).

thf(46,plain,
    $false,
    inference(eigenvar_choice,[status(thm),assumptions([h1]),eigenvar_choice(discharge,[h0])],[45,h0]) ).

thf(0,theorem,
    sP34,
    inference(contra,[status(thm),contra(discharge,[h1])],[45,h1]) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.12  % Problem  : SEU519^1 : TPTP v8.1.0. Released v3.7.0.
% 0.04/0.13  % Command  : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s
% 0.13/0.34  % Computer : n022.cluster.edu
% 0.13/0.34  % Model    : x86_64 x86_64
% 0.13/0.34  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34  % Memory   : 8042.1875MB
% 0.13/0.34  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34  % CPULimit : 300
% 0.13/0.34  % WCLimit  : 600
% 0.13/0.34  % DateTime : Mon Jun 20 02:37:01 EDT 2022
% 0.13/0.34  % CPUTime  : 
% 26.05/26.28  % SZS status Theorem
% 26.05/26.28  % Mode: mode454
% 26.05/26.28  % Inferences: 167
% 26.05/26.28  % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------