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Theorem List for Intuitionistic Logic Explorer - 8601-8700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremxp1d2m1eqxm1d2 8601 A complex number increased by 1, then divided by 2, then decreased by 1 equals the complex number decreased by 1 and then divided by 2. (Contributed by AV, 24-May-2020.)
 |-  ( X  e.  CC  ->  ( ( ( X  +  1 )  / 
 2 )  -  1
 )  =  ( ( X  -  1 ) 
 /  2 ) )
 
Theoremdiv4p1lem1div2 8602 An integer greater than 5, divided by 4 and increased by 1, is less than or equal to the half of the integer minus 1. (Contributed by AV, 8-Jul-2021.)
 |-  ( ( N  e.  RR  /\  6  <_  N )  ->  ( ( N 
 /  4 )  +  1 )  <_  ( ( N  -  1 ) 
 /  2 ) )
 
3.4.6  The Archimedean property
 
Theoremarch 8603* Archimedean property of real numbers. For any real number, there is an integer greater than it. Theorem I.29 of [Apostol] p. 26. (Contributed by NM, 21-Jan-1997.)
 |-  ( A  e.  RR  ->  E. n  e.  NN  A  <  n )
 
Theoremnnrecl 8604* There exists a positive integer whose reciprocal is less than a given positive real. Exercise 3 of [Apostol] p. 28. (Contributed by NM, 8-Nov-2004.)
 |-  ( ( A  e.  RR  /\  0  <  A )  ->  E. n  e.  NN  ( 1  /  n )  <  A )
 
Theorembndndx 8605* A bounded real sequence  A ( k ) is less than or equal to at least one of its indices. (Contributed by NM, 18-Jan-2008.)
 |-  ( E. x  e. 
 RR  A. k  e.  NN  ( A  e.  RR  /\  A  <_  x )  ->  E. k  e.  NN  A  <_  k )
 
3.4.7  Nonnegative integers (as a subset of complex numbers)
 
Syntaxcn0 8606 Extend class notation to include the class of nonnegative integers.
 class  NN0
 
Definitiondf-n0 8607 Define the set of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
 |- 
 NN0  =  ( NN  u.  { 0 } )
 
Theoremelnn0 8608 Nonnegative integers expressed in terms of naturals and zero. (Contributed by Raph Levien, 10-Dec-2002.)
 |-  ( A  e.  NN0  <->  ( A  e.  NN  \/  A  =  0 )
 )
 
Theoremnnssnn0 8609 Positive naturals are a subset of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
 |- 
 NN  C_  NN0
 
Theoremnn0ssre 8610 Nonnegative integers are a subset of the reals. (Contributed by Raph Levien, 10-Dec-2002.)
 |- 
 NN0  C_  RR
 
Theoremnn0sscn 8611 Nonnegative integers are a subset of the complex numbers.) (Contributed by NM, 9-May-2004.)
 |- 
 NN0  C_  CC
 
Theoremnn0ex 8612 The set of nonnegative integers exists. (Contributed by NM, 18-Jul-2004.)
 |- 
 NN0  e.  _V
 
Theoremnnnn0 8613 A positive integer is a nonnegative integer. (Contributed by NM, 9-May-2004.)
 |-  ( A  e.  NN  ->  A  e.  NN0 )
 
Theoremnnnn0i 8614 A positive integer is a nonnegative integer. (Contributed by NM, 20-Jun-2005.)
 |-  N  e.  NN   =>    |-  N  e.  NN0
 
Theoremnn0re 8615 A nonnegative integer is a real number. (Contributed by NM, 9-May-2004.)
 |-  ( A  e.  NN0  ->  A  e.  RR )
 
Theoremnn0cn 8616 A nonnegative integer is a complex number. (Contributed by NM, 9-May-2004.)
 |-  ( A  e.  NN0  ->  A  e.  CC )
 
Theoremnn0rei 8617 A nonnegative integer is a real number. (Contributed by NM, 14-May-2003.)
 |-  A  e.  NN0   =>    |-  A  e.  RR
 
Theoremnn0cni 8618 A nonnegative integer is a complex number. (Contributed by NM, 14-May-2003.)
 |-  A  e.  NN0   =>    |-  A  e.  CC
 
Theoremdfn2 8619 The set of positive integers defined in terms of nonnegative integers. (Contributed by NM, 23-Sep-2007.) (Proof shortened by Mario Carneiro, 13-Feb-2013.)
 |- 
 NN  =  ( NN0  \  { 0 } )
 
Theoremelnnne0 8620 The positive integer property expressed in terms of difference from zero. (Contributed by Stefan O'Rear, 12-Sep-2015.)
 |-  ( N  e.  NN  <->  ( N  e.  NN0  /\  N  =/=  0 ) )
 
Theorem0nn0 8621 0 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
 |-  0  e.  NN0
 
Theorem1nn0 8622 1 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
 |-  1  e.  NN0
 
Theorem2nn0 8623 2 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
 |-  2  e.  NN0
 
Theorem3nn0 8624 3 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  3  e.  NN0
 
Theorem4nn0 8625 4 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  4  e.  NN0
 
Theorem5nn0 8626 5 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  5  e.  NN0
 
Theorem6nn0 8627 6 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  6  e.  NN0
 
Theorem7nn0 8628 7 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  7  e.  NN0
 
Theorem8nn0 8629 8 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  8  e.  NN0
 
Theorem9nn0 8630 9 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  9  e.  NN0
 
Theoremnn0ge0 8631 A nonnegative integer is greater than or equal to zero. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 16-May-2014.)
 |-  ( N  e.  NN0  -> 
 0  <_  N )
 
Theoremnn0nlt0 8632 A nonnegative integer is not less than zero. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( A  e.  NN0  ->  -.  A  <  0 )
 
Theoremnn0ge0i 8633 Nonnegative integers are nonnegative. (Contributed by Raph Levien, 10-Dec-2002.)
 |-  N  e.  NN0   =>    |-  0  <_  N
 
Theoremnn0le0eq0 8634 A nonnegative integer is less than or equal to zero iff it is equal to zero. (Contributed by NM, 9-Dec-2005.)
 |-  ( N  e.  NN0  ->  ( N  <_  0  <->  N  =  0
 ) )
 
Theoremnn0p1gt0 8635 A nonnegative integer increased by 1 is greater than 0. (Contributed by Alexander van der Vekens, 3-Oct-2018.)
 |-  ( N  e.  NN0  -> 
 0  <  ( N  +  1 ) )
 
Theoremnnnn0addcl 8636 A positive integer plus a nonnegative integer is a positive integer. (Contributed by NM, 20-Apr-2005.) (Proof shortened by Mario Carneiro, 16-May-2014.)
 |-  ( ( M  e.  NN  /\  N  e.  NN0 )  ->  ( M  +  N )  e.  NN )
 
Theoremnn0nnaddcl 8637 A nonnegative integer plus a positive integer is a positive integer. (Contributed by NM, 22-Dec-2005.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN )  ->  ( M  +  N )  e.  NN )
 
Theorem0mnnnnn0 8638 The result of subtracting a positive integer from 0 is not a nonnegative integer. (Contributed by Alexander van der Vekens, 19-Mar-2018.)
 |-  ( N  e.  NN  ->  ( 0  -  N )  e/  NN0 )
 
Theoremun0addcl 8639 If  S is closed under addition, then so is  S  u.  { 0 }. (Contributed by Mario Carneiro, 17-Jul-2014.)
 |-  ( ph  ->  S  C_ 
 CC )   &    |-  T  =  ( S  u.  { 0 } )   &    |-  ( ( ph  /\  ( M  e.  S  /\  N  e.  S ) )  ->  ( M  +  N )  e.  S )   =>    |-  ( ( ph  /\  ( M  e.  T  /\  N  e.  T )
 )  ->  ( M  +  N )  e.  T )
 
Theoremun0mulcl 8640 If  S is closed under multiplication, then so is  S  u.  { 0 }. (Contributed by Mario Carneiro, 17-Jul-2014.)
 |-  ( ph  ->  S  C_ 
 CC )   &    |-  T  =  ( S  u.  { 0 } )   &    |-  ( ( ph  /\  ( M  e.  S  /\  N  e.  S ) )  ->  ( M  x.  N )  e.  S )   =>    |-  ( ( ph  /\  ( M  e.  T  /\  N  e.  T )
 )  ->  ( M  x.  N )  e.  T )
 
Theoremnn0addcl 8641 Closure of addition of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 17-Jul-2014.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( M  +  N )  e.  NN0 )
 
Theoremnn0mulcl 8642 Closure of multiplication of nonnegative integers. (Contributed by NM, 22-Jul-2004.) (Proof shortened by Mario Carneiro, 17-Jul-2014.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( M  x.  N )  e.  NN0 )
 
Theoremnn0addcli 8643 Closure of addition of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)
 |-  M  e.  NN0   &    |-  N  e.  NN0   =>    |-  ( M  +  N )  e.  NN0
 
Theoremnn0mulcli 8644 Closure of multiplication of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)
 |-  M  e.  NN0   &    |-  N  e.  NN0   =>    |-  ( M  x.  N )  e.  NN0
 
Theoremnn0p1nn 8645 A nonnegative integer plus 1 is a positive integer. (Contributed by Raph Levien, 30-Jun-2006.) (Revised by Mario Carneiro, 16-May-2014.)
 |-  ( N  e.  NN0  ->  ( N  +  1
 )  e.  NN )
 
Theorempeano2nn0 8646 Second Peano postulate for nonnegative integers. (Contributed by NM, 9-May-2004.)
 |-  ( N  e.  NN0  ->  ( N  +  1
 )  e.  NN0 )
 
Theoremnnm1nn0 8647 A positive integer minus 1 is a nonnegative integer. (Contributed by Jason Orendorff, 24-Jan-2007.) (Revised by Mario Carneiro, 16-May-2014.)
 |-  ( N  e.  NN  ->  ( N  -  1
 )  e.  NN0 )
 
Theoremelnn0nn 8648 The nonnegative integer property expressed in terms of positive integers. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
 |-  ( N  e.  NN0  <->  ( N  e.  CC  /\  ( N  +  1 )  e.  NN ) )
 
Theoremelnnnn0 8649 The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-May-2004.)
 |-  ( N  e.  NN  <->  ( N  e.  CC  /\  ( N  -  1 )  e. 
 NN0 ) )
 
Theoremelnnnn0b 8650 The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 1-Sep-2005.)
 |-  ( N  e.  NN  <->  ( N  e.  NN0  /\  0  <  N ) )
 
Theoremelnnnn0c 8651 The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-Jan-2006.)
 |-  ( N  e.  NN  <->  ( N  e.  NN0  /\  1  <_  N ) )
 
Theoremnn0addge1 8652 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)
 |-  ( ( A  e.  RR  /\  N  e.  NN0 )  ->  A  <_  ( A  +  N )
 )
 
Theoremnn0addge2 8653 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)
 |-  ( ( A  e.  RR  /\  N  e.  NN0 )  ->  A  <_  ( N  +  A )
 )
 
Theoremnn0addge1i 8654 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)
 |-  A  e.  RR   &    |-  N  e.  NN0   =>    |-  A  <_  ( A  +  N )
 
Theoremnn0addge2i 8655 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)
 |-  A  e.  RR   &    |-  N  e.  NN0   =>    |-  A  <_  ( N  +  A )
 
Theoremnn0le2xi 8656 A nonnegative integer is less than or equal to twice itself. (Contributed by Raph Levien, 10-Dec-2002.)
 |-  N  e.  NN0   =>    |-  N  <_  ( 2  x.  N )
 
Theoremnn0lele2xi 8657 'Less than or equal to' implies 'less than or equal to twice' for nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
 |-  M  e.  NN0   &    |-  N  e.  NN0   =>    |-  ( N  <_  M  ->  N  <_  ( 2  x.  M ) )
 
Theoremnn0supp 8658 Two ways to write the support of a function on  NN0. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  ( F : I --> NN0  ->  ( `' F " ( _V  \  {
 0 } ) )  =  ( `' F " NN ) )
 
Theoremnnnn0d 8659 A positive integer is a nonnegative integer. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  A  e.  NN0 )
 
Theoremnn0red 8660 A nonnegative integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN0 )   =>    |-  ( ph  ->  A  e.  RR )
 
Theoremnn0cnd 8661 A nonnegative integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN0 )   =>    |-  ( ph  ->  A  e.  CC )
 
Theoremnn0ge0d 8662 A nonnegative integer is greater than or equal to zero. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN0 )   =>    |-  ( ph  ->  0  <_  A )
 
Theoremnn0addcld 8663 Closure of addition of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN0 )   &    |-  ( ph  ->  B  e.  NN0 )   =>    |-  ( ph  ->  ( A  +  B )  e.  NN0 )
 
Theoremnn0mulcld 8664 Closure of multiplication of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN0 )   &    |-  ( ph  ->  B  e.  NN0 )   =>    |-  ( ph  ->  ( A  x.  B )  e. 
 NN0 )
 
Theoremnn0readdcl 8665 Closure law for addition of reals, restricted to nonnegative integers. (Contributed by Alexander van der Vekens, 6-Apr-2018.)
 |-  ( ( A  e.  NN0  /\  B  e.  NN0 )  ->  ( A  +  B )  e.  RR )
 
Theoremnn0ge2m1nn 8666 If a nonnegative integer is greater than or equal to two, the integer decreased by 1 is a positive integer. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (Revised by AV, 4-Jan-2020.)
 |-  ( ( N  e.  NN0  /\  2  <_  N ) 
 ->  ( N  -  1
 )  e.  NN )
 
Theoremnn0ge2m1nn0 8667 If a nonnegative integer is greater than or equal to two, the integer decreased by 1 is also a nonnegative integer. (Contributed by Alexander van der Vekens, 1-Aug-2018.)
 |-  ( ( N  e.  NN0  /\  2  <_  N ) 
 ->  ( N  -  1
 )  e.  NN0 )
 
Theoremnn0nndivcl 8668 Closure law for dividing of a nonnegative integer by a positive integer. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
 |-  ( ( K  e.  NN0  /\  L  e.  NN )  ->  ( K  /  L )  e.  RR )
 
3.4.8  Extended nonnegative integers

The function values of the hash (set size) function are either nonnegative integers or positive infinity. To avoid the need to distinguish between finite and infinite sets (and therefore if the set size is a nonnegative integer or positive infinity), it is useful to provide a definition of the set of nonnegative integers extended by positive infinity, analogously to the extension of the real numbers  RR*, see df-xr 7470.

 
Syntaxcxnn0 8669 The set of extended nonnegative integers.
 class NN0*
 
Definitiondf-xnn0 8670 Define the set of extended nonnegative integers that includes positive infinity. Analogue of the extension of the real numbers  RR*, see df-xr 7470. If we assumed excluded middle, this would be essentially the same as ℕ as defined at df-nninf 6735 but in its absence the relationship between the two is more complicated. (Contributed by AV, 10-Dec-2020.)
 |- NN0*  =  ( NN0  u.  { +oo } )
 
Theoremelxnn0 8671 An extended nonnegative integer is either a standard nonnegative integer or positive infinity. (Contributed by AV, 10-Dec-2020.)
 |-  ( A  e. NN0*  <->  ( A  e.  NN0 
 \/  A  = +oo ) )
 
Theoremnn0ssxnn0 8672 The standard nonnegative integers are a subset of the extended nonnegative integers. (Contributed by AV, 10-Dec-2020.)
 |- 
 NN0  C_ NN0*
 
Theoremnn0xnn0 8673 A standard nonnegative integer is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
 |-  ( A  e.  NN0  ->  A  e. NN0* )
 
Theoremxnn0xr 8674 An extended nonnegative integer is an extended real. (Contributed by AV, 10-Dec-2020.)
 |-  ( A  e. NN0*  ->  A  e.  RR* )
 
Theorem0xnn0 8675 Zero is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
 |-  0  e. NN0*
 
Theorempnf0xnn0 8676 Positive infinity is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
 |- +oo  e. NN0*
 
Theoremnn0nepnf 8677 No standard nonnegative integer equals positive infinity. (Contributed by AV, 10-Dec-2020.)
 |-  ( A  e.  NN0  ->  A  =/= +oo )
 
Theoremnn0xnn0d 8678 A standard nonnegative integer is an extended nonnegative integer, deduction form. (Contributed by AV, 10-Dec-2020.)
 |-  ( ph  ->  A  e.  NN0 )   =>    |-  ( ph  ->  A  e. NN0* )
 
Theoremnn0nepnfd 8679 No standard nonnegative integer equals positive infinity, deduction form. (Contributed by AV, 10-Dec-2020.)
 |-  ( ph  ->  A  e.  NN0 )   =>    |-  ( ph  ->  A  =/= +oo )
 
Theoremxnn0nemnf 8680 No extended nonnegative integer equals negative infinity. (Contributed by AV, 10-Dec-2020.)
 |-  ( A  e. NN0*  ->  A  =/= -oo )
 
Theoremxnn0xrnemnf 8681 The extended nonnegative integers are extended reals without negative infinity. (Contributed by AV, 10-Dec-2020.)
 |-  ( A  e. NN0*  ->  ( A  e.  RR*  /\  A  =/= -oo ) )
 
Theoremxnn0nnn0pnf 8682 An extended nonnegative integer which is not a standard nonnegative integer is positive infinity. (Contributed by AV, 10-Dec-2020.)
 |-  ( ( N  e. NN0*  /\ 
 -.  N  e.  NN0 )  ->  N  = +oo )
 
3.4.9  Integers (as a subset of complex numbers)
 
Syntaxcz 8683 Extend class notation to include the class of integers.
 class  ZZ
 
Definitiondf-z 8684 Define the set of integers, which are the positive and negative integers together with zero. Definition of integers in [Apostol] p. 22. The letter Z abbreviates the German word Zahlen meaning "numbers." (Contributed by NM, 8-Jan-2002.)
 |- 
 ZZ  =  { n  e.  RR  |  ( n  =  0  \/  n  e.  NN  \/  -u n  e.  NN ) }
 
Theoremelz 8685 Membership in the set of integers. (Contributed by NM, 8-Jan-2002.)
 |-  ( N  e.  ZZ  <->  ( N  e.  RR  /\  ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN )
 ) )
 
Theoremnnnegz 8686 The negative of a positive integer is an integer. (Contributed by NM, 12-Jan-2002.)
 |-  ( N  e.  NN  -> 
 -u N  e.  ZZ )
 
Theoremzre 8687 An integer is a real. (Contributed by NM, 8-Jan-2002.)
 |-  ( N  e.  ZZ  ->  N  e.  RR )
 
Theoremzcn 8688 An integer is a complex number. (Contributed by NM, 9-May-2004.)
 |-  ( N  e.  ZZ  ->  N  e.  CC )
 
Theoremzrei 8689 An integer is a real number. (Contributed by NM, 14-Jul-2005.)
 |-  A  e.  ZZ   =>    |-  A  e.  RR
 
Theoremzssre 8690 The integers are a subset of the reals. (Contributed by NM, 2-Aug-2004.)
 |- 
 ZZ  C_  RR
 
Theoremzsscn 8691 The integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
 |- 
 ZZ  C_  CC
 
Theoremzex 8692 The set of integers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |- 
 ZZ  e.  _V
 
Theoremelnnz 8693 Positive integer property expressed in terms of integers. (Contributed by NM, 8-Jan-2002.)
 |-  ( N  e.  NN  <->  ( N  e.  ZZ  /\  0  <  N ) )
 
Theorem0z 8694 Zero is an integer. (Contributed by NM, 12-Jan-2002.)
 |-  0  e.  ZZ
 
Theorem0zd 8695 Zero is an integer, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( ph  ->  0  e.  ZZ )
 
Theoremelnn0z 8696 Nonnegative integer property expressed in terms of integers. (Contributed by NM, 9-May-2004.)
 |-  ( N  e.  NN0  <->  ( N  e.  ZZ  /\  0  <_  N ) )
 
Theoremelznn0nn 8697 Integer property expressed in terms nonnegative integers and positive integers. (Contributed by NM, 10-May-2004.)
 |-  ( N  e.  ZZ  <->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
 
Theoremelznn0 8698 Integer property expressed in terms of nonnegative integers. (Contributed by NM, 9-May-2004.)
 |-  ( N  e.  ZZ  <->  ( N  e.  RR  /\  ( N  e.  NN0  \/  -u N  e.  NN0 ) ) )
 
Theoremelznn 8699 Integer property expressed in terms of positive integers and nonnegative integers. (Contributed by NM, 12-Jul-2005.)
 |-  ( N  e.  ZZ  <->  ( N  e.  RR  /\  ( N  e.  NN  \/  -u N  e.  NN0 )
 ) )
 
Theoremnnssz 8700 Positive integers are a subset of integers. (Contributed by NM, 9-Jan-2002.)
 |- 
 NN  C_  ZZ
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