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Theorem List for Intuitionistic Logic Explorer - 10501-10600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfacnn 10501 Value of the factorial function for positive integers. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)
 |-  ( N  e.  NN  ->  ( ! `  N )  =  (  seq 1 (  x.  ,  _I  ) `  N ) )
 
Theoremfac0 10502 The factorial of 0. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)
 |-  ( ! `  0
 )  =  1
 
Theoremfac1 10503 The factorial of 1. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)
 |-  ( ! `  1
 )  =  1
 
Theoremfacp1 10504 The factorial of a successor. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)
 |-  ( N  e.  NN0  ->  ( ! `  ( N  +  1 ) )  =  ( ( ! `
  N )  x.  ( N  +  1 ) ) )
 
Theoremfac2 10505 The factorial of 2. (Contributed by NM, 17-Mar-2005.)
 |-  ( ! `  2
 )  =  2
 
Theoremfac3 10506 The factorial of 3. (Contributed by NM, 17-Mar-2005.)
 |-  ( ! `  3
 )  =  6
 
Theoremfac4 10507 The factorial of 4. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  ( ! `  4
 )  = ; 2 4
 
Theoremfacnn2 10508 Value of the factorial function expressed recursively. (Contributed by NM, 2-Dec-2004.)
 |-  ( N  e.  NN  ->  ( ! `  N )  =  ( ( ! `  ( N  -  1 ) )  x.  N ) )
 
Theoremfaccl 10509 Closure of the factorial function. (Contributed by NM, 2-Dec-2004.)
 |-  ( N  e.  NN0  ->  ( ! `  N )  e.  NN )
 
Theoremfaccld 10510 Closure of the factorial function, deduction version of faccl 10509. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  ( ! `  N )  e. 
 NN )
 
Theoremfacne0 10511 The factorial function is nonzero. (Contributed by NM, 26-Apr-2005.)
 |-  ( N  e.  NN0  ->  ( ! `  N )  =/=  0 )
 
Theoremfacdiv 10512 A positive integer divides the factorial of an equal or larger number. (Contributed by NM, 2-May-2005.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN  /\  N  <_  M )  ->  ( ( ! `  M )  /  N )  e.  NN )
 
Theoremfacndiv 10513 No positive integer (greater than one) divides the factorial plus one of an equal or larger number. (Contributed by NM, 3-May-2005.)
 |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( 1  <  N  /\  N  <_  M ) )  ->  -.  ( ( ( ! `
  M )  +  1 )  /  N )  e.  ZZ )
 
Theoremfacwordi 10514 Ordering property of factorial. (Contributed by NM, 9-Dec-2005.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0  /\  M  <_  N )  ->  ( ! `  M )  <_  ( ! `  N ) )
 
Theoremfaclbnd 10515 A lower bound for the factorial function. (Contributed by NM, 17-Dec-2005.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( M ^ ( N  +  1 )
 )  <_  ( ( M ^ M )  x.  ( ! `  N ) ) )
 
Theoremfaclbnd2 10516 A lower bound for the factorial function. (Contributed by NM, 17-Dec-2005.)
 |-  ( N  e.  NN0  ->  ( ( 2 ^ N )  /  2
 )  <_  ( ! `  N ) )
 
Theoremfaclbnd3 10517 A lower bound for the factorial function. (Contributed by NM, 19-Dec-2005.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( M ^ N )  <_  ( ( M ^ M )  x.  ( ! `  N ) ) )
 
Theoremfaclbnd6 10518 Geometric lower bound for the factorial function, where N is usually held constant. (Contributed by Paul Chapman, 28-Dec-2007.)
 |-  ( ( N  e.  NN0  /\  M  e.  NN0 )  ->  ( ( ! `  N )  x.  (
 ( N  +  1 ) ^ M ) )  <_  ( ! `  ( N  +  M ) ) )
 
Theoremfacubnd 10519 An upper bound for the factorial function. (Contributed by Mario Carneiro, 15-Apr-2016.)
 |-  ( N  e.  NN0  ->  ( ! `  N ) 
 <_  ( N ^ N ) )
 
Theoremfacavg 10520 The product of two factorials is greater than or equal to the factorial of (the floor of) their average. (Contributed by NM, 9-Dec-2005.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( ! `  ( |_ `  ( ( M  +  N )  / 
 2 ) ) ) 
 <_  ( ( ! `  M )  x.  ( ! `  N ) ) )
 
4.6.9  The binomial coefficient operation
 
Syntaxcbc 10521 Extend class notation to include the binomial coefficient operation (combinatorial choose operation).
 class  _C
 
Definitiondf-bc 10522* Define the binomial coefficient operation. For example,  ( 5  _C  3 )  =  1 0 (ex-bc 13095).

In the literature, this function is often written as a column vector of the two arguments, or with the arguments as subscripts before and after the letter "C".  ( N  _C  K
) is read " N choose  K." Definition of binomial coefficient in [Gleason] p. 295. As suggested by Gleason, we define it to be 0 when  0  <_  k  <_  n does not hold. (Contributed by NM, 10-Jul-2005.)

 |- 
 _C  =  ( n  e.  NN0 ,  k  e. 
 ZZ  |->  if ( k  e.  ( 0 ... n ) ,  ( ( ! `  n )  /  ( ( ! `  ( n  -  k
 ) )  x.  ( ! `  k ) ) ) ,  0 ) )
 
Theorembcval 10523 Value of the binomial coefficient, 
N choose  K. Definition of binomial coefficient in [Gleason] p. 295. As suggested by Gleason, we define it to be 0 when  0  <_  K  <_  N does not hold. See bcval2 10524 for the value in the standard domain. (Contributed by NM, 10-Jul-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)
 |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  ( N  _C  K )  =  if ( K  e.  ( 0 ... N ) ,  (
 ( ! `  N )  /  ( ( ! `
  ( N  -  K ) )  x.  ( ! `  K ) ) ) ,  0 ) )
 
Theorembcval2 10524 Value of the binomial coefficient, 
N choose  K, in its standard domain. (Contributed by NM, 9-Jun-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)
 |-  ( K  e.  (
 0 ... N )  ->  ( N  _C  K )  =  ( ( ! `
  N )  /  ( ( ! `  ( N  -  K ) )  x.  ( ! `  K ) ) ) )
 
Theorembcval3 10525 Value of the binomial coefficient, 
N choose  K, outside of its standard domain. Remark in [Gleason] p. 295. (Contributed by NM, 14-Jul-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)
 |-  ( ( N  e.  NN0  /\  K  e.  ZZ  /\  -.  K  e.  ( 0
 ... N ) ) 
 ->  ( N  _C  K )  =  0 )
 
Theorembcval4 10526 Value of the binomial coefficient, 
N choose  K, outside of its standard domain. Remark in [Gleason] p. 295. (Contributed by NM, 14-Jul-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)
 |-  ( ( N  e.  NN0  /\  K  e.  ZZ  /\  ( K  <  0  \/  N  <  K ) )  ->  ( N  _C  K )  =  0 )
 
Theorembcrpcl 10527 Closure of the binomial coefficient in the positive reals. (This is mostly a lemma before we have bccl2 10542.) (Contributed by Mario Carneiro, 10-Mar-2014.)
 |-  ( K  e.  (
 0 ... N )  ->  ( N  _C  K )  e.  RR+ )
 
Theorembccmpl 10528 "Complementing" its second argument doesn't change a binary coefficient. (Contributed by NM, 21-Jun-2005.) (Revised by Mario Carneiro, 5-Mar-2014.)
 |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  ( N  _C  K )  =  ( N  _C  ( N  -  K ) ) )
 
Theorembcn0 10529  N choose 0 is 1. Remark in [Gleason] p. 296. (Contributed by NM, 17-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)
 |-  ( N  e.  NN0  ->  ( N  _C  0
 )  =  1 )
 
Theorembc0k 10530 The binomial coefficient " 0 choose  K " is 0 for a positive integer K. Note that  ( 0  _C  0 )  =  1 (see bcn0 10529). (Contributed by Alexander van der Vekens, 1-Jan-2018.)
 |-  ( K  e.  NN  ->  ( 0  _C  K )  =  0 )
 
Theorembcnn 10531  N choose  N is 1. Remark in [Gleason] p. 296. (Contributed by NM, 17-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)
 |-  ( N  e.  NN0  ->  ( N  _C  N )  =  1 )
 
Theorembcn1 10532 Binomial coefficient:  N choose  1. (Contributed by NM, 21-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)
 |-  ( N  e.  NN0  ->  ( N  _C  1
 )  =  N )
 
Theorembcnp1n 10533 Binomial coefficient:  N  +  1 choose  N. (Contributed by NM, 20-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)
 |-  ( N  e.  NN0  ->  ( ( N  +  1 )  _C  N )  =  ( N  +  1 ) )
 
Theorembcm1k 10534 The proportion of one binomial coefficient to another with  K decreased by 1. (Contributed by Mario Carneiro, 10-Mar-2014.)
 |-  ( K  e.  (
 1 ... N )  ->  ( N  _C  K )  =  ( ( N  _C  ( K  -  1 ) )  x.  ( ( N  -  ( K  -  1
 ) )  /  K ) ) )
 
Theorembcp1n 10535 The proportion of one binomial coefficient to another with  N increased by 1. (Contributed by Mario Carneiro, 10-Mar-2014.)
 |-  ( K  e.  (
 0 ... N )  ->  ( ( N  +  1 )  _C  K )  =  ( ( N  _C  K )  x.  ( ( N  +  1 )  /  (
 ( N  +  1 )  -  K ) ) ) )
 
Theorembcp1nk 10536 The proportion of one binomial coefficient to another with  N and  K increased by 1. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |-  ( K  e.  (
 0 ... N )  ->  ( ( N  +  1 )  _C  ( K  +  1 )
 )  =  ( ( N  _C  K )  x.  ( ( N  +  1 )  /  ( K  +  1
 ) ) ) )
 
Theorembcval5 10537 Write out the top and bottom parts of the binomial coefficient  ( N  _C  K )  =  ( N  x.  ( N  -  1 )  x. 
...  x.  ( ( N  -  K )  +  1 ) )  /  K ! explicitly. In this form, it is valid even for  N  <  K, although it is no longer valid for nonpositive  K. (Contributed by Mario Carneiro, 22-May-2014.) (Revised by Jim Kingdon, 23-Apr-2023.)
 |-  ( ( N  e.  NN0  /\  K  e.  NN )  ->  ( N  _C  K )  =  ( (  seq ( ( N  -  K )  +  1
 ) (  x.  ,  _I  ) `  N ) 
 /  ( ! `  K ) ) )
 
Theorembcn2 10538 Binomial coefficient:  N choose  2. (Contributed by Mario Carneiro, 22-May-2014.)
 |-  ( N  e.  NN0  ->  ( N  _C  2
 )  =  ( ( N  x.  ( N  -  1 ) ) 
 /  2 ) )
 
Theorembcp1m1 10539 Compute the binomial coefficient of 
( N  +  1 ) over  ( N  - 
1 ) (Contributed by Scott Fenton, 11-May-2014.) (Revised by Mario Carneiro, 22-May-2014.)
 |-  ( N  e.  NN0  ->  ( ( N  +  1 )  _C  ( N  -  1 ) )  =  ( ( ( N  +  1 )  x.  N )  / 
 2 ) )
 
Theorembcpasc 10540 Pascal's rule for the binomial coefficient, generalized to all integers  K. Equation 2 of [Gleason] p. 295. (Contributed by NM, 13-Jul-2005.) (Revised by Mario Carneiro, 10-Mar-2014.)
 |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  ( ( N  _C  K )  +  ( N  _C  ( K  -  1 ) ) )  =  ( ( N  +  1 )  _C  K ) )
 
Theorembccl 10541 A binomial coefficient, in its extended domain, is a nonnegative integer. (Contributed by NM, 10-Jul-2005.) (Revised by Mario Carneiro, 9-Nov-2013.)
 |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  ( N  _C  K )  e.  NN0 )
 
Theorembccl2 10542 A binomial coefficient, in its standard domain, is a positive integer. (Contributed by NM, 3-Jan-2006.) (Revised by Mario Carneiro, 10-Mar-2014.)
 |-  ( K  e.  (
 0 ... N )  ->  ( N  _C  K )  e.  NN )
 
Theorembcn2m1 10543 Compute the binomial coefficient " N choose 2 " from " ( N  -  1 ) choose 2 ": (N-1) + ( (N-1) 2 ) = ( N 2 ). (Contributed by Alexander van der Vekens, 7-Jan-2018.)
 |-  ( N  e.  NN  ->  ( ( N  -  1 )  +  (
 ( N  -  1
 )  _C  2 )
 )  =  ( N  _C  2 ) )
 
Theorembcn2p1 10544 Compute the binomial coefficient " ( N  +  1
) choose 2 " from " N choose 2 ": N + ( N 2 ) = ( (N+1) 2 ). (Contributed by Alexander van der Vekens, 8-Jan-2018.)
 |-  ( N  e.  NN0  ->  ( N  +  ( N  _C  2 ) )  =  ( ( N  +  1 )  _C  2 ) )
 
Theorempermnn 10545 The number of permutations of  N  -  R objects from a collection of  N objects is a positive integer. (Contributed by Jason Orendorff, 24-Jan-2007.)
 |-  ( R  e.  (
 0 ... N )  ->  ( ( ! `  N )  /  ( ! `  R ) )  e.  NN )
 
Theorembcnm1 10546 The binomial coefficent of  ( N  -  1 ) is  N. (Contributed by Scott Fenton, 16-May-2014.)
 |-  ( N  e.  NN0  ->  ( N  _C  ( N  -  1 ) )  =  N )
 
Theorem4bc3eq4 10547 The value of four choose three. (Contributed by Scott Fenton, 11-Jun-2016.)
 |-  ( 4  _C  3
 )  =  4
 
Theorem4bc2eq6 10548 The value of four choose two. (Contributed by Scott Fenton, 9-Jan-2017.)
 |-  ( 4  _C  2
 )  =  6
 
4.6.10  The ` # ` (set size) function
 
Syntaxchash 10549 Extend the definition of a class to include the set size function.
 class
 
Definitiondf-ihash 10550* Define the set size function ♯, which gives the cardinality of a finite set as a member of 
NN0, and assigns all infinite sets the value +oo. For example,  ( `  {
0 ,  1 ,  2 } )  =  3.

Note that we use the sharp sign (♯) for this function and we use the different character octothorpe (#) for the apartness relation (see df-ap 8364). We adopt the former notation from Corollary 8.2.4 of [AczelRathjen], p. 80 (although that work only defines it for finite sets).

This definition (in terms of  U. and 
~<_) is not taken directly from the literature, but for finite sets should be equivalent to the conventional definition that the size of a finite set is the unique natural number which is equinumerous to the given set. (Contributed by Jim Kingdon, 19-Feb-2022.)

 |- =  ( (frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 )  u.  { <. om , +oo >. } )  o.  ( x  e.  _V  |->  U.
 { y  e.  ( om  u.  { om }
 )  |  y  ~<_  x } ) )
 
Theoremhashinfuni 10551* The ordinal size of an infinite set is  om. (Contributed by Jim Kingdon, 20-Feb-2022.)
 |-  ( om  ~<_  A  ->  U.
 { y  e.  ( om  u.  { om }
 )  |  y  ~<_  A }  =  om )
 
Theoremhashinfom 10552 The value of the ♯ function on an infinite set. (Contributed by Jim Kingdon, 20-Feb-2022.)
 |-  ( om  ~<_  A  ->  ( `  A )  = +oo )
 
Theoremhashennnuni 10553* The ordinal size of a set equinumerous to an element of  om is that element of  om. (Contributed by Jim Kingdon, 20-Feb-2022.)
 |-  ( ( N  e.  om 
 /\  N  ~~  A )  ->  U. { y  e.  ( om  u.  { om } )  |  y  ~<_  A }  =  N )
 
Theoremhashennn 10554* The size of a set equinumerous to an element of  om. (Contributed by Jim Kingdon, 21-Feb-2022.)
 |-  ( ( N  e.  om 
 /\  N  ~~  A )  ->  ( `  A )  =  (frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 ) `  N ) )
 
Theoremhashcl 10555 Closure of the ♯ function. (Contributed by Paul Chapman, 26-Oct-2012.) (Revised by Mario Carneiro, 13-Jul-2014.)
 |-  ( A  e.  Fin  ->  ( `  A )  e. 
 NN0 )
 
Theoremhashfiv01gt1 10556 The size of a finite set is either 0 or 1 or greater than 1. (Contributed by Jim Kingdon, 21-Feb-2022.)
 |-  ( M  e.  Fin  ->  ( ( `  M )  =  0  \/  ( `  M )  =  1  \/  1  <  ( `  M ) ) )
 
Theoremhashfz1 10557 The set  ( 1 ... N ) has  N elements. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 15-Sep-2013.)
 |-  ( N  e.  NN0  ->  ( `  ( 1 ...
 N ) )  =  N )
 
Theoremhashen 10558 Two finite sets have the same number of elements iff they are equinumerous. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 15-Sep-2013.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( `  A )  =  ( `  B ) 
 <->  A  ~~  B ) )
 
Theoremhasheqf1o 10559* The size of two finite sets is equal if and only if there is a bijection mapping one of the sets onto the other. (Contributed by Alexander van der Vekens, 17-Dec-2017.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( `  A )  =  ( `  B ) 
 <-> 
 E. f  f : A -1-1-onto-> B ) )
 
Theoremfiinfnf1o 10560* There is no bijection between a finite set and an infinite set. By infnfi 6793 the theorem would also hold if "infinite" were expressed as  om  ~<_  B. (Contributed by Alexander van der Vekens, 25-Dec-2017.)
 |-  ( ( A  e.  Fin  /\  -.  B  e.  Fin )  ->  -.  E. f  f : A -1-1-onto-> B )
 
Theoremfocdmex 10561 The codomain of an onto function is a set if its domain is a set. (Contributed by AV, 4-May-2021.)
 |-  ( ( A  e.  V  /\  F : A -onto-> B )  ->  B  e.  _V )
 
Theoremfihasheqf1oi 10562 The size of two finite sets is equal if there is a bijection mapping one of the sets onto the other. (Contributed by Jim Kingdon, 21-Feb-2022.)
 |-  ( ( A  e.  Fin  /\  F : A -1-1-onto-> B )  ->  ( `  A )  =  ( `  B ) )
 
Theoremfihashf1rn 10563 The size of a finite set which is a one-to-one function is equal to the size of the function's range. (Contributed by Jim Kingdon, 21-Feb-2022.)
 |-  ( ( A  e.  Fin  /\  F : A -1-1-> B )  ->  ( `  F )  =  ( `  ran  F ) )
 
Theoremfihasheqf1od 10564 The size of two finite sets is equal if there is a bijection mapping one of the sets onto the other. (Contributed by Jim Kingdon, 21-Feb-2022.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  F : A -1-1-onto-> B )   =>    |-  ( ph  ->  ( `  A )  =  ( `  B ) )
 
Theoremfz1eqb 10565 Two possibly-empty 1-based finite sets of sequential integers are equal iff their endpoints are equal. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Mario Carneiro, 29-Mar-2014.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( ( 1 ...
 M )  =  ( 1 ... N )  <->  M  =  N )
 )
 
Theoremfiltinf 10566 The size of an infinite set is greater than the size of a finite set. (Contributed by Jim Kingdon, 21-Feb-2022.)
 |-  ( ( A  e.  Fin  /\  om  ~<_  B )  ->  ( `  A )  < 
 ( `  B ) )
 
Theoremisfinite4im 10567 A finite set is equinumerous to the range of integers from one up to the hash value of the set. (Contributed by Jim Kingdon, 22-Feb-2022.)
 |-  ( A  e.  Fin  ->  ( 1 ... ( `  A ) )  ~~  A )
 
Theoremfihasheq0 10568 Two ways of saying a finite set is empty. (Contributed by Paul Chapman, 26-Oct-2012.) (Revised by Mario Carneiro, 27-Jul-2014.) (Intuitionized by Jim Kingdon, 23-Feb-2022.)
 |-  ( A  e.  Fin  ->  ( ( `  A )  =  0  <->  A  =  (/) ) )
 
Theoremfihashneq0 10569 Two ways of saying a finite set is not empty. Also, "A is inhabited" would be equivalent by fin0 6783. (Contributed by Alexander van der Vekens, 23-Sep-2018.) (Intuitionized by Jim Kingdon, 23-Feb-2022.)
 |-  ( A  e.  Fin  ->  ( 0  <  ( `  A )  <->  A  =/=  (/) ) )
 
Theoremhashnncl 10570 Positive natural closure of the hash function. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |-  ( A  e.  Fin  ->  ( ( `  A )  e.  NN  <->  A  =/=  (/) ) )
 
Theoremhash0 10571 The empty set has size zero. (Contributed by Mario Carneiro, 8-Jul-2014.)
 |-  ( `  (/) )  =  0
 
Theoremhashsng 10572 The size of a singleton. (Contributed by Paul Chapman, 26-Oct-2012.) (Proof shortened by Mario Carneiro, 13-Feb-2013.)
 |-  ( A  e.  V  ->  ( `  { A }
 )  =  1 )
 
Theoremfihashen1 10573 A finite set has size 1 if and only if it is equinumerous to the ordinal 1. (Contributed by AV, 14-Apr-2019.) (Intuitionized by Jim Kingdon, 23-Feb-2022.)
 |-  ( A  e.  Fin  ->  ( ( `  A )  =  1  <->  A  ~~  1o )
 )
 
Theoremfihashfn 10574 A function on a finite set is equinumerous to its domain. (Contributed by Mario Carneiro, 12-Mar-2015.) (Intuitionized by Jim Kingdon, 24-Feb-2022.)
 |-  ( ( F  Fn  A  /\  A  e.  Fin )  ->  ( `  F )  =  ( `  A )
 )
 
Theoremfseq1hash 10575 The value of the size function on a finite 1-based sequence. (Contributed by Paul Chapman, 26-Oct-2012.) (Proof shortened by Mario Carneiro, 12-Mar-2015.)
 |-  ( ( N  e.  NN0  /\  F  Fn  ( 1
 ... N ) ) 
 ->  ( `  F )  =  N )
 
Theoremomgadd 10576 Mapping ordinal addition to integer addition. (Contributed by Jim Kingdon, 24-Feb-2022.)
 |-  G  = frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 )   =>    |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( G `  ( A  +o  B ) )  =  ( ( G `  A )  +  ( G `  B ) ) )
 
Theoremfihashdom 10577 Dominance relation for the size function. (Contributed by Jim Kingdon, 24-Feb-2022.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( `  A )  <_  ( `  B )  <->  A  ~<_  B ) )
 
Theoremhashunlem 10578 Lemma for hashun 10579. Ordinal size of the union. (Contributed by Jim Kingdon, 25-Feb-2022.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ph  ->  ( A  i^i  B )  =  (/) )   &    |-  ( ph  ->  N  e.  om )   &    |-  ( ph  ->  M  e.  om )   &    |-  ( ph  ->  A 
 ~~  N )   &    |-  ( ph  ->  B  ~~  M )   =>    |-  ( ph  ->  ( A  u.  B )  ~~  ( N  +o  M ) )
 
Theoremhashun 10579 The size of the union of disjoint finite sets is the sum of their sizes. (Contributed by Paul Chapman, 30-Nov-2012.) (Revised by Mario Carneiro, 15-Sep-2013.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  ->  ( `  ( A  u.  B ) )  =  (
 ( `  A )  +  ( `  B ) ) )
 
Theorem1elfz0hash 10580 1 is an element of the finite set of sequential nonnegative integers bounded by the size of a nonempty finite set. (Contributed by AV, 9-May-2020.)
 |-  ( ( A  e.  Fin  /\  A  =/=  (/) )  -> 
 1  e.  ( 0
 ... ( `  A )
 ) )
 
Theoremhashunsng 10581 The size of the union of a finite set with a disjoint singleton is one more than the size of the set. (Contributed by Paul Chapman, 30-Nov-2012.)
 |-  ( B  e.  V  ->  ( ( A  e.  Fin  /\  -.  B  e.  A )  ->  ( `  ( A  u.  { B } )
 )  =  ( ( `  A )  +  1 ) ) )
 
Theoremhashprg 10582 The size of an unordered pair. (Contributed by Mario Carneiro, 27-Sep-2013.) (Revised by Mario Carneiro, 5-May-2016.) (Revised by AV, 18-Sep-2021.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  =/=  B  <-> 
 ( `  { A ,  B } )  =  2 ) )
 
Theoremprhash2ex 10583 There is (at least) one set with two different elements: the unordered pair containing  0 and  1. In contrast to pr0hash2ex 10589, numbers are used instead of sets because their representation is shorter (and more comprehensive). (Contributed by AV, 29-Jan-2020.)
 |-  ( `  { 0 ,  1 } )  =  2
 
Theoremhashp1i 10584 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |-  A  e.  om   &    |-  B  =  suc  A   &    |-  ( `  A )  =  M   &    |-  ( M  +  1 )  =  N   =>    |-  ( `  B )  =  N
 
Theoremhash1 10585 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |-  ( `  1o )  =  1
 
Theoremhash2 10586 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |-  ( `  2o )  =  2
 
Theoremhash3 10587 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |-  ( `  3o )  =  3
 
Theoremhash4 10588 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |-  ( `  4o )  =  4
 
Theorempr0hash2ex 10589 There is (at least) one set with two different elements: the unordered pair containing the empty set and the singleton containing the empty set. (Contributed by AV, 29-Jan-2020.)
 |-  ( `  { (/) ,  { (/)
 } } )  =  2
 
Theoremfihashss 10590 The size of a subset is less than or equal to the size of its superset. (Contributed by Alexander van der Vekens, 14-Jul-2018.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  B  C_  A )  ->  ( `  B )  <_  ( `  A ) )
 
Theoremfiprsshashgt1 10591 The size of a superset of a proper unordered pair is greater than 1. (Contributed by AV, 6-Feb-2021.)
 |-  ( ( ( A  e.  V  /\  B  e.  W  /\  A  =/=  B )  /\  C  e.  Fin )  ->  ( { A ,  B }  C_  C  ->  2  <_  ( `  C ) ) )
 
Theoremfihashssdif 10592 The size of the difference of a finite set and a finite subset is the set's size minus the subset's. (Contributed by Jim Kingdon, 31-May-2022.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  B  C_  A )  ->  ( `  ( A  \  B ) )  =  ( ( `  A )  -  ( `  B ) ) )
 
Theoremhashdifsn 10593 The size of the difference of a finite set and a singleton subset is the set's size minus 1. (Contributed by Alexander van der Vekens, 6-Jan-2018.)
 |-  ( ( A  e.  Fin  /\  B  e.  A ) 
 ->  ( `  ( A  \  { B } )
 )  =  ( ( `  A )  -  1
 ) )
 
Theoremhashdifpr 10594 The size of the difference of a finite set and a proper ordered pair subset is the set's size minus 2. (Contributed by AV, 16-Dec-2020.)
 |-  ( ( A  e.  Fin  /\  ( B  e.  A  /\  C  e.  A  /\  B  =/=  C ) ) 
 ->  ( `  ( A  \  { B ,  C } ) )  =  ( ( `  A )  -  2 ) )
 
Theoremhashfz 10595 Value of the numeric cardinality of a nonempty integer range. (Contributed by Stefan O'Rear, 12-Sep-2014.) (Proof shortened by Mario Carneiro, 15-Apr-2015.)
 |-  ( B  e.  ( ZZ>=
 `  A )  ->  ( `  ( A ... B ) )  =  ( ( B  -  A )  +  1 )
 )
 
Theoremhashfzo 10596 Cardinality of a half-open set of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( B  e.  ( ZZ>=
 `  A )  ->  ( `  ( A..^ B ) )  =  ( B  -  A ) )
 
Theoremhashfzo0 10597 Cardinality of a half-open set of integers based at zero. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( B  e.  NN0  ->  ( `  ( 0..^ B ) )  =  B )
 
Theoremhashfzp1 10598 Value of the numeric cardinality of a (possibly empty) integer range. (Contributed by AV, 19-Jun-2021.)
 |-  ( B  e.  ( ZZ>=
 `  A )  ->  ( `  ( ( A  +  1 ) ... B ) )  =  ( B  -  A ) )
 
Theoremhashfz0 10599 Value of the numeric cardinality of a nonempty range of nonnegative integers. (Contributed by Alexander van der Vekens, 21-Jul-2018.)
 |-  ( B  e.  NN0  ->  ( `  ( 0 ...
 B ) )  =  ( B  +  1 ) )
 
Theoremhashxp 10600 The size of the Cartesian product of two finite sets is the product of their sizes. (Contributed by Paul Chapman, 30-Nov-2012.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( `  ( A  X.  B ) )  =  ( ( `  A )  x.  ( `  B ) ) )
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