## TPTP Problem File: SEV420^1.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SEV420^1 : TPTP v7.5.0. Released v4.1.0.
% Domain   : Set Theory
% Problem  : Size of disjoint sets' union
% Version  : Especial.
% English  : If |A|=|A'| & |B|=|B'| & |A' ^ B'|=0, then |A U B| =< |A' U B'|

% Refs     : [Kun10] Kuncak (2009), Email to Geoff Sutcliffe
% Source   : [Kun10]
% Names    :

% Status   : Theorem
% Rating   : 0.45 v7.5.0, 0.14 v7.4.0, 0.11 v7.3.0, 0.22 v7.2.0, 0.25 v7.1.0, 0.00 v6.2.0, 0.29 v6.1.0, 0.14 v5.5.0, 0.17 v5.4.0, 0.40 v5.2.0, 0.20 v4.1.0, 1.00 v4.0.0
% Syntax   : Number of formulae    :   41 (   0 unit;  20 type;  20 defn)
%            Number of atoms       :  160 (  28 equality;  96 variable)
%            Maximal formula depth :   13 (   6 average)
%            Number of connectives :   88 (   5   ~;   3   |;  12   &;  61   @)
%                                         (   0 <=>;   7  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :  124 ( 124   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   23 (  20   :;   0   =)
%            Number of variables   :   62 (   1 sgn;   9   !;   5   ?;  48   ^)
%                                         (  62   :;   0  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      : TH0_THM_EQU_NAR

%------------------------------------------------------------------------------
include('Axioms/SET008^0.ax').
%------------------------------------------------------------------------------
thf(is_function_type,type,(
is_function: ( \$i > \$o ) > ( \$i > \$i ) > ( \$i > \$o ) > \$o )).

thf(is_function,definition,
( is_function
= ( ^ [X: \$i > \$o,F: \$i > \$i,Y: \$i > \$o] :
! [E: \$i] :
( ( X @ E )
=> ( Y @ ( F @ E ) ) ) ) )).

thf(injection_type,type,(
injection: ( \$i > \$o ) > ( \$i > \$i ) > ( \$i > \$o ) > \$o )).

thf(injection,definition,
( injection
= ( ^ [X: \$i > \$o,F: \$i > \$i,Y: \$i > \$o] :
( ( is_function @ X @ F @ Y )
=> ! [E1: \$i,E2: \$i] :
( ( ( X @ E1 )
& ( X @ E2 )
& ( ( F @ E1 )
= ( F @ E2 ) ) )
=> ( E1 = E2 ) ) ) ) )).

thf(surjection_type,type,(
surjection: ( \$i > \$o ) > ( \$i > \$i ) > ( \$i > \$o ) > \$o )).

thf(surjection,definition,
( surjection
= ( ^ [X: \$i > \$o,F: \$i > \$i,Y: \$i > \$o] :
( ( is_function @ X @ F @ Y )
=> ! [E1: \$i] :
( ( Y @ E1 )
=> ? [E2: \$i] :
( ( X @ E2 )
& ( ( F @ E2 )
= E1 ) ) ) ) ) )).

thf(bijection_type,type,(
bijection: ( \$i > \$o ) > ( \$i > \$i ) > ( \$i > \$o ) > \$o )).

thf(bijection,definition,
( bijection
= ( ^ [X: \$i > \$o,F: \$i > \$i,Y: \$i > \$o] :
( ( injection @ X @ F @ Y )
& ( surjection @ X @ F @ Y ) ) ) )).

thf(equinumerous_type,type,(
equinumerous: ( \$i > \$o ) > ( \$i > \$o ) > \$o )).

thf(equinumerous,definition,
( equinumerous
= ( ^ [X: \$i > \$o,Y: \$i > \$o] :
? [F: \$i > \$i] :
( bijection @ X @ F @ Y ) ) )).

thf(embedding_type,type,(
embedding: ( \$i > \$o ) > ( \$i > \$o ) > \$o )).

thf(embedding,definition,
( embedding
= ( ^ [X: \$i > \$o,Y: \$i > \$o] :
? [F: \$i > \$i] :
( injection @ X @ F @ Y ) ) )).

thf(prove,conjecture,(
! [A: \$i > \$o,Ap: \$i > \$o,B: \$i > \$o,Bp: \$i > \$o] :
( ( ( equinumerous @ A @ Ap )
& ( equinumerous @ B @ Bp )
& ( ( intersection @ Ap @ Bp )
= emptyset ) )
=> ( embedding @ ( union @ A @ B ) @ ( union @ Ap @ Bp ) ) ) )).

%------------------------------------------------------------------------------

```