ITP001 Axioms: ITP104^5.ax


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% File     : ITP104^5 : TPTP v9.0.0. Bugfixed v7.5.0.
% Domain   : Interactive Theorem Proving
% Axioms   : HOL4 set theory export, chainy mode
% Version  : [BG+19] axioms.
% English  :

% Refs     : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
%          : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source   : [BG+19]
% Names    : ucord^2.ax [Gau20]
%          : HL4104^5.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :    9 (   1 unt;   1 typ;   0 def)
%            Number of atoms       :  213 (   1 equ;   0 cnn)
%            Maximal formula atoms :   62 (  23 avg)
%            Number of connectives :  250 (   3   ~;   0   |;   0   &; 245   @)
%                                         (   1 <=>;   1  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   19 (  11 avg; 245 nst)
%            Number of types       :    1 (   0 usr)
%            Number of type conns  :    1 (   1   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   20 (  19 usr;  18 con; 0-2 aty)
%            Number of variables   :   11 (   2   ^   9   !;   0   ?;  11   :)
% SPC      : TH0_SAT_EQU_NAR

% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
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thf(tp_c_2Eucord_2Eomega1,type,
    c_2Eucord_2Eomega1: del > $i ).

thf(mem_c_2Eucord_2Eomega1,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eucord_2Eomega1 @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) ) ).

thf(conj_thm_2Eucord_2Eucinf__uncountable,axiom,
    ! [A_27a: del] :
      ~ ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) ) ) ) ).

thf(conj_thm_2Eucord_2EUnum__cardlt__ucinf,axiom,
    ! [A_27a: del] :
      ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) @ ty_2Enum_2Enum ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ).

thf(conj_thm_2Eucord_2EUnum__cardle__ucinf,axiom,
    ! [A_27a: del] : ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ty_2Enum_2Enum @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) ) ) ) ).

thf(conj_thm_2Eucord_2Eucord__sup__exists__lemma,axiom,
    ! [A_27a: del] :
      ( p
      @ ( ap
        @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Eordinal_2Eordinal @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) )
          @ ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Eordinal_2Eordinal @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) @ ( ty_2Eordinal_2Eordinal @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) )
            @ ( lam @ ( ty_2Eordinal_2Eordinal @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) )
              @ ^ [V0a: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) @ bool ) @ V0a ) @ ( ap @ ( c_2Epred__set_2Ecountable @ ( ty_2Eordinal_2Eordinal @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) ) @ ( ap @ ( c_2Eordinal_2Epreds @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) @ V0a ) ) ) ) ) )
        @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) ) ) ) ).

thf(ax_thm_2Eucord_2Eomega1__def,axiom,
    ! [A_27a: del] :
      ( ( c_2Eucord_2Eomega1 @ A_27a )
      = ( ap @ ( c_2Eordinal_2Esup @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) )
        @ ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Eordinal_2Eordinal @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) @ ( ty_2Eordinal_2Eordinal @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) )
          @ ( lam @ ( ty_2Eordinal_2Eordinal @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) )
            @ ^ [V0a: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) @ bool ) @ V0a ) @ ( ap @ ( c_2Epred__set_2Ecountable @ ( ty_2Eordinal_2Eordinal @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) ) @ ( ap @ ( c_2Eordinal_2Epreds @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) @ V0a ) ) ) ) ) ) ) ).

thf(conj_thm_2Eucord_2Ex__lt__omega1__countable,axiom,
    ! [A_27a: del,V0x: $i] :
      ( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) )
     => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) @ V0x ) @ ( c_2Eucord_2Eomega1 @ A_27a ) ) )
      <=> ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ( ty_2Eordinal_2Eordinal @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) ) @ ( ap @ ( c_2Eordinal_2Epreds @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) @ V0x ) ) ) ) ) ).

thf(conj_thm_2Eucord_2Eomega1__not__countable,axiom,
    ! [A_27a: del] :
      ~ ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ( ty_2Eordinal_2Eordinal @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) ) @ ( ap @ ( c_2Eordinal_2Epreds @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) @ ( c_2Eucord_2Eomega1 @ A_27a ) ) ) ) ).

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