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Theorem List for Metamath Proof Explorer - 23101-23200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremballotlemfrc 23101* Express the value of  ( F `  ( R `  C )
) in terms of the newly defined  .^. (Contributed by Thierry Arnoux, 21-Apr-2017.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   &    |-  R  =  ( c  e.  ( O 
 \  E )  |->  ( ( S `  c
 ) " c ) )   &    |-  .^  =  ( u  e. 
 Fin ,  v  e.  Fin  |->  ( ( # `  (
 v  i^i  u )
 )  -  ( # `  ( v  \  u ) ) ) )   =>    |-  ( ( C  e.  ( O  \  E ) 
 /\  J  e.  (
 1 ... ( I `  C ) ) ) 
 ->  ( ( F `  ( R `  C ) ) `  J )  =  ( C  .^  ( ( ( S `
  C ) `  J ) ... ( I `  C ) ) ) )
 
Theoremballotlemfrci 23102* Reverse counting preserves a tie at the first tie. (Contributed by Thierry Arnoux, 21-Apr-2017.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   &    |-  R  =  ( c  e.  ( O 
 \  E )  |->  ( ( S `  c
 ) " c ) )   &    |-  .^  =  ( u  e. 
 Fin ,  v  e.  Fin  |->  ( ( # `  (
 v  i^i  u )
 )  -  ( # `  ( v  \  u ) ) ) )   =>    |-  ( C  e.  ( O  \  E )  ->  ( ( F `  ( R `  C ) ) `  ( I `
  C ) )  =  0 )
 
Theoremballotlemfrceq 23103* Value of  F for a reverse counting  ( R `  C ). (Contributed by Thierry Arnoux, 27-Apr-2017.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   &    |-  R  =  ( c  e.  ( O 
 \  E )  |->  ( ( S `  c
 ) " c ) )   &    |-  .^  =  ( u  e. 
 Fin ,  v  e.  Fin  |->  ( ( # `  (
 v  i^i  u )
 )  -  ( # `  ( v  \  u ) ) ) )   =>    |-  ( ( C  e.  ( O  \  E ) 
 /\  J  e.  (
 1 ... ( I `  C ) ) ) 
 ->  ( ( F `  C ) `  (
 ( ( S `  C ) `  J )  -  1 ) )  =  -u ( ( F `
  ( R `  C ) ) `  J ) )
 
Theoremballotlemfrcn0 23104* Value of  F for a reversed counting  ( R `  C ), before the first tie, cannot be zero . (Contributed by Thierry Arnoux, 25-Apr-2017.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   &    |-  R  =  ( c  e.  ( O 
 \  E )  |->  ( ( S `  c
 ) " c ) )   =>    |-  ( ( C  e.  ( O  \  E ) 
 /\  J  e.  (
 1 ... ( M  +  N ) )  /\  J  <  ( I `  C ) )  ->  ( ( F `  ( R `  C ) ) `  J )  =/=  0 )
 
Theoremballotlemrc 23105* Range of  R. (Contributed by Thierry Arnoux, 19-Apr-2017.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   &    |-  R  =  ( c  e.  ( O 
 \  E )  |->  ( ( S `  c
 ) " c ) )   =>    |-  ( C  e.  ( O  \  E )  ->  ( R `  C )  e.  ( O  \  E ) )
 
Theoremballotlemirc 23106* Applying  R does not change first ties. (Contributed by Thierry Arnoux, 19-Apr-2017.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   &    |-  R  =  ( c  e.  ( O 
 \  E )  |->  ( ( S `  c
 ) " c ) )   =>    |-  ( C  e.  ( O  \  E )  ->  ( I `  ( R `
  C ) )  =  ( I `  C ) )
 
Theoremballotlemrinv0 23107* Lemma for ballotlemrinv 23108. (Contributed by Thierry Arnoux, 18-Apr-2017.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   &    |-  R  =  ( c  e.  ( O 
 \  E )  |->  ( ( S `  c
 ) " c ) )   =>    |-  ( ( C  e.  ( O  \  E ) 
 /\  D  =  ( ( S `  C ) " C ) ) 
 ->  ( D  e.  ( O  \  E )  /\  C  =  ( ( S `  D ) " D ) ) )
 
Theoremballotlemrinv 23108*  R is its own inverse : it is an involution. (Contributed by Thierry Arnoux, 10-Apr-2017.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   &    |-  R  =  ( c  e.  ( O 
 \  E )  |->  ( ( S `  c
 ) " c ) )   =>    |-  `' R  =  R
 
Theoremballotlem1ri 23109* When the vote on the first tie is for A, the first vote is also for A on the reverse counting. (Contributed by Thierry Arnoux, 18-Apr-2017.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   &    |-  R  =  ( c  e.  ( O 
 \  E )  |->  ( ( S `  c
 ) " c ) )   =>    |-  ( C  e.  ( O  \  E )  ->  ( 1  e.  ( R `  C )  <->  ( I `  C )  e.  C ) )
 
Theoremballotlem7 23110*  R is a bijection between two subsets of  ( O  \  E
): one where a vote for A is picked first, and one where a vote for B is picked first (Contributed by Thierry Arnoux, 12-Dec-2016.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   &    |-  R  =  ( c  e.  ( O 
 \  E )  |->  ( ( S `  c
 ) " c ) )   =>    |-  ( R  |`  { c  e.  ( O  \  E )  |  1  e.  c } ) : {
 c  e.  ( O 
 \  E )  |  1  e.  c } -1-1-onto-> {
 c  e.  ( O 
 \  E )  |  -.  1  e.  c }
 
Theoremballotlem8 23111* There are as many countings with ties starting with a ballot for A as there are starting with a ballot for B. (Contributed by Thierry Arnoux, 7-Dec-2016.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   &    |-  R  =  ( c  e.  ( O 
 \  E )  |->  ( ( S `  c
 ) " c ) )   =>    |-  ( # `  { c  e.  ( O  \  E )  |  1  e.  c } )  =  ( # `  { c  e.  ( O  \  E )  |  -.  1  e.  c } )
 
Theoremballotth 23112* Bertrand's ballot problem : the probability that A is ahead throughout the counting. (Contributed by Thierry Arnoux, 7-Dec-2016.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   &    |-  R  =  ( c  e.  ( O 
 \  E )  |->  ( ( S `  c
 ) " c ) )   =>    |-  ( P `  E )  =  ( ( M  -  N )  /  ( M  +  N ) )
 
Theorem3o1cs 23113 Deduction eliminating disjunct. (Contributed by Thierry Arnoux, 19-Dec-2016.)
 |-  (
 ( ph  \/  ps  \/  ch )  ->  th )   =>    |-  ( ph  ->  th )
 
Theorem3o2cs 23114 Deduction eliminating disjunct. (Contributed by Thierry Arnoux, 19-Dec-2016.)
 |-  (
 ( ph  \/  ps  \/  ch )  ->  th )   =>    |-  ( ps  ->  th )
 
Theorem3o3cs 23115 Deduction eliminating disjunct. (Contributed by Thierry Arnoux, 19-Dec-2016.)
 |-  (
 ( ph  \/  ps  \/  ch )  ->  th )   =>    |-  ( ch  ->  th )
 
18.3.2  Division in the extended real number system
 
Syntaxcxdiv 23116 Extend class notation to include division of extended reals.
 class /𝑒
 
Definitiondf-xdiv 23117* Define division over extended real numbers. (Contributed by Thierry Arnoux, 17-Dec-2016.)
 |- /𝑒  =  ( x  e.  RR* ,  y  e.  ( RR  \  {
 0 } )  |->  (
 iota_ z  e.  RR* (
 y x e z )  =  x ) )
 
Theoremxdivval 23118* Value of division: the (unique) element  x such that  ( B  x.  x )  =  A. This is meaningful only when  B is nonzero. (Contributed by Thierry Arnoux, 17-Dec-2016.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( A /𝑒  B )  =  (
 iota_ x  e.  RR* ( B x e x )  =  A ) )
 
Theoremxrecex 23119* Existence of reciprocal of nonzero real number. (Contributed by Thierry Arnoux, 17-Dec-2016.)
 |-  (
 ( A  e.  RR  /\  A  =/=  0 ) 
 ->  E. x  e.  RR  ( A x e x )  =  1 )
 
Theoremxmulcand 23120 Cancellation law for extended multiplication. (Contributed by Thierry Arnoux, 17-Dec-2016.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  B  e.  RR* )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  C  =/=  0
 )   =>    |-  ( ph  ->  (
 ( C x e A )  =  ( C x e B )  <->  A  =  B )
 )
 
Theoremxreceu 23121* Existential uniqueness of reciprocals. Theorem I.8 of [Apostol] p. 18. (Contributed by Thierry Arnoux, 17-Dec-2016.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  E! x  e.  RR*  ( B x e x )  =  A )
 
Theoremxdivcld 23122 Closure law for the extended division. (Contributed by Thierry Arnoux, 15-Mar-2017.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  B  =/=  0 )   =>    |-  ( ph  ->  ( A /𝑒 
 B )  e.  RR* )
 
Theoremxdivcl 23123 Closure law for the extended division. (Contributed by Thierry Arnoux, 15-Mar-2017.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( A /𝑒  B )  e.  RR* )
 
Theoremxdivmul 23124 Relationship between division and multiplication. (Contributed by Thierry Arnoux, 24-Dec-2016.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR*  /\  ( C  e.  RR  /\  C  =/=  0 ) )  ->  ( ( A /𝑒  C )  =  B  <->  ( C x e B )  =  A ) )
 
Theoremrexdiv 23125 The extended real division operation when both arguments are real. (Contributed by Thierry Arnoux, 18-Dec-2016.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( A /𝑒  B )  =  ( A  /  B ) )
 
Theoremxdivrec 23126 Relationship between division and reciprocal. (Contributed by Thierry Arnoux, 5-Jul-2017.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( A /𝑒  B )  =  ( A x e ( 1 /𝑒 
 B ) ) )
 
Theoremxdivid 23127 A number divided by itself is one. (Contributed by Thierry Arnoux, 18-Dec-2016.)
 |-  (
 ( A  e.  RR  /\  A  =/=  0 ) 
 ->  ( A /𝑒  A )  =  1 )
 
Theoremxdiv0 23128 Division into zero is zero. (Contributed by Thierry Arnoux, 18-Dec-2016.)
 |-  (
 ( A  e.  RR  /\  A  =/=  0 ) 
 ->  ( 0 /𝑒  A )  =  0 )
 
Theoremxdiv0rp 23129 Division into zero is zero. (Contributed by Thierry Arnoux, 18-Dec-2016.)
 |-  ( A  e.  RR+  ->  (
 0 /𝑒  A )  =  0
 )
 
Theoremxgtpnf 23130 An extended real which is greater than plus infinity is plus infinity. (Contributed by Thierry Arnoux, 18-Dec-2016.)
 |-  ( A  e.  RR*  ->  (  +oo  <_  A  <->  A  =  +oo ) )
 
Theoremeliccioo 23131 Membership in a closed interval of extended reals vs. the same open interval (Contributed by Thierry Arnoux, 18-Dec-2016.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  ( C  e.  ( A [,] B )  <->  ( C  =  A  \/  C  e.  ( A (,) B )  \/  C  =  B ) ) )
 
Theoremelxrge02 23132 Elementhood in the set of nonnegative extended reals. (Contributed by Thierry Arnoux, 18-Dec-2016.)
 |-  ( A  e.  ( 0 [,]  +oo )  <->  ( A  =  0  \/  A  e.  RR+  \/  A  =  +oo )
 )
 
Theoremxdivpnfrp 23133 Plus infinity divided by a positive real number is plus infinity. (Contributed by Thierry Arnoux, 18-Dec-2016.)
 |-  ( A  e.  RR+  ->  (  +oo /𝑒  A )  =  +oo )
 
Theoremrpxdivcld 23134 Closure law for extended division of positive reals. (Contributed by Thierry Arnoux, 18-Dec-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  ( A /𝑒  B )  e.  RR+ )
 
Theoremxrpxdivcld 23135 Closure law for extended division of positive extended reals. (Contributed by Thierry Arnoux, 18-Dec-2016.)
 |-  ( ph  ->  A  e.  (
 0 [,]  +oo ) )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  ( A /𝑒  B )  e.  ( 0 [,]  +oo ) )
 
18.3.3  Propositional Calculus - misc additions
 
Theorembisimpd 23136 Removing of a condition for a biconditionnal connective. (Contributed by Thierry Arnoux, 23-Oct-2016.)
 |-  ( ph  ->  ( ps  ->  th ) )   &    |-  ( ph  ->  ( ch  ->  th )
 )   &    |-  ( ph  ->  ( th  ->  ( ps  <->  ch ) ) )   =>    |-  ( ph  ->  ( ps  <->  ch ) )
 
Theorembisimp 23137 Removing of a condition for a biconditionnal connective. (Contributed by Thierry Arnoux, 23-Oct-2016.)
 |-  ( ps  ->  th )   &    |-  ( ch  ->  th )   &    |-  ( th  ->  ( ps  <->  ch ) )   =>    |-  ( ps  <->  ch )
 
Theorembian1d 23138 Adding a superfluous conjunct in a biconditionnal. (Contributed by Thierry Arnoux, 26-Feb-2017.)
 |-  ( ph  ->  ( ps  <->  ( ch  /\  th ) ) )   =>    |-  ( ph  ->  ( ( ch  /\  ps ) 
 <->  ( ch  /\  th ) ) )
 
Theoremor3di 23139 Distributive law for disjunction. (Contributed by Thierry Arnoux, 3-Jul-2017.)
 |-  (
 ( ph  \/  ( ps  /\  ch  /\  ta ) )  <->  ( ( ph  \/  ps )  /\  ( ph  \/  ch )  /\  ( ph  \/  ta )
 ) )
 
Theoremor3dir 23140 Distributive law for disjunction. (Contributed by Thierry Arnoux, 3-Jul-2017.)
 |-  (
 ( ( ph  /\  ps  /\ 
 ch )  \/  ta ) 
 <->  ( ( ph  \/  ta )  /\  ( ps 
 \/  ta )  /\  ( ch  \/  ta ) ) )
 
18.3.4  Subclass relations - misc additions
 
Theoremssrd 23141 Deduction rule based on subclass definition. (Contributed by Thierry Arnoux, 8-Mar-2017.)
 |-  F/ x ph   &    |-  F/_ x A   &    |-  F/_ x B   &    |-  ( ph  ->  ( x  e.  A  ->  x  e.  B ) )   =>    |-  ( ph  ->  A  C_  B )
 
Theoremeqrd 23142 Deduce equality of classes from equivalence of membership. (Contributed by Thierry Arnoux, 21-Mar-2017.)
 |-  F/ x ph   &    |-  F/_ x A   &    |-  F/_ x B   &    |-  ( ph  ->  ( x  e.  A  <->  x  e.  B ) )   =>    |-  ( ph  ->  A  =  B )
 
Theoremdifneqnul 23143 If the difference of two sets is not empty, then the sets are not equal. (Contributed by Thierry Arnoux, 28-Feb-2017.)
 |-  (
 ( A  \  B )  =/=  (/)  ->  A  =/=  B )
 
Theoremdifeq 23144 Rewriting an equation with set difference, without using quantifiers. (Contributed by Thierry Arnoux, 24-Sep-2017.)
 |-  (
 ( A  \  B )  =  C  <->  ( ( C  i^i  B )  =  (/)  /\  ( C  u.  B )  =  ( A  u.  B ) ) )
 
18.3.5  Restricted Quantification - misc additions
 
Theoremabeq2f 23145 Equality of a class variable and a class abstraction. In this version, the fact that  x is a non-free variable in  A is explicitely stated as a hypothesis. (Contributed by Thierry Arnoux, 11-May-2017.)
 |-  F/_ x A   =>    |-  ( A  =  { x  |  ph }  <->  A. x ( x  e.  A  <->  ph ) )
 
Theoremeqvincg 23146* A variable introduction law for class equality, deduction version. (Contributed by Thierry Arnoux, 2-Mar-2017.)
 |-  ( A  e.  V  ->  ( A  =  B  <->  E. x ( x  =  A  /\  x  =  B ) ) )
 
Theoremraleqbid 23147 Equality deduction for restricted universal quantifier. (Contributed by Thierry Arnoux, 8-Mar-2017.)
 |-  F/ x ph   &    |-  F/_ x A   &    |-  F/_ x B   &    |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  (
 A. x  e.  A  ps 
 <-> 
 A. x  e.  B  ch ) )
 
Theoremrexeqbid 23148 Equality deduction for restricted universal quantifier. (Contributed by Thierry Arnoux, 8-Mar-2017.)
 |-  F/ x ph   &    |-  F/_ x A   &    |-  F/_ x B   &    |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( E. x  e.  A  ps 
 <-> 
 E. x  e.  B  ch ) )
 
Theoremralcom4f 23149* Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Revised by Thierry Arnoux, 8-Mar-2017.)
 |-  F/_ y A   =>    |-  ( A. x  e.  A  A. y ph  <->  A. y A. x  e.  A  ph )
 
Theoremrexcom4f 23150* Commutation of restricted and unrestricted existential quantifiers. (Contributed by NM, 12-Apr-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Revised by Thierry Arnoux, 8-Mar-2017.)
 |-  F/_ y A   =>    |-  ( E. x  e.  A  E. y ph  <->  E. y E. x  e.  A  ph )
 
Theoremrabid2f 23151 An "identity" law for restricted class abstraction. (Contributed by NM, 9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) (Revised by Thierry Arnoux, 13-Mar-2017.)
 |-  F/_ x A   =>    |-  ( A  =  { x  e.  A  |  ph
 } 
 <-> 
 A. x  e.  A  ph )
 
Theoremrabss3d 23152* Subclass law for restricted abstraction. (Contributed by Thierry Arnoux, 25-Sep-2017.)
 |-  (
 ( ph  /\  ( x  e.  A  /\  ps ) )  ->  x  e.  B )   =>    |-  ( ph  ->  { x  e.  A  |  ps }  C_ 
 { x  e.  B  |  ps } )
 
Theorem2reuswap2 23153* A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Thierry Arnoux, 7-Apr-2017.)
 |-  ( A. x  e.  A  E* y ( y  e.  B  /\  ph )  ->  ( E! x  e.  A  E. y  e.  B  ph  ->  E! y  e.  B  E. x  e.  A  ph ) )
 
Theoremreuxfr3d 23154* Transfer existential uniqueness from a variable  x to another variable  y contained in expression  A. Cf. reuxfr2d 4573 (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)
 |-  (
 ( ph  /\  y  e.  C )  ->  A  e.  B )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  E* y  e.  C x  =  A )   =>    |-  ( ph  ->  ( E! x  e.  B  E. y  e.  C  ( x  =  A  /\  ps )  <->  E! y  e.  C  ps ) )
 
Theoremreuxfr4d 23155* Transfer existential uniqueness from a variable  x to another variable  y contained in expression  A. Cf. reuxfrd 4575 (Contributed by Thierry Arnoux, 7-Apr-2017.)
 |-  (
 ( ph  /\  y  e.  C )  ->  A  e.  B )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  E! y  e.  C  x  =  A )   &    |-  (
 ( ph  /\  x  =  A )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E! x  e.  B  ps 
 <->  E! y  e.  C  ch ) )
 
Theoremrexunirn 23156* Restricted existential quantification over the union of the range of a function. Cf. rexrn 5683 and eluni2 3847. (Contributed by Thierry Arnoux, 19-Sep-2017.)
 |-  F  =  ( x  e.  A  |->  B )   &    |-  ( x  e.  A  ->  B  e.  V )   =>    |-  ( E. x  e.  A  E. y  e.  B  ph  ->  E. y  e.  U. ran  F ph )
 
18.3.6  Substitution (without distinct variables) - misc additions
 
Theoremsbcss12g 23157* Set substitution into the both argument of a subset relation. (Contributed by Thierry Arnoux, 25-Jan-2017.)
 |-  ( A  e.  V  ->  (
 [. A  /  x ]. B  C_  C  <->  [_ A  /  x ]_ B  C_  [_ A  /  x ]_ C ) )
 
Theoremclelsb3f 23158 Substitution applied to an atomic wff (class version of elsb3 2055). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (Revised by Thierry Arnoux, 13-Mar-2017.)
 |-  F/_ y A   =>    |-  ( [ x  /  y ] y  e.  A  <->  x  e.  A )
 
18.3.7  Existential Uniqueness - misc additions
 
Theoremmo5f 23159* Alternate definition of "at most one." (Contributed by Thierry Arnoux, 1-Mar-2017.)
 |-  F/ i ph   &    |-  F/ j ph   =>    |-  ( E* x ph  <->  A. i A. j
 ( ( [ i  /  x ] ph  /\  [
 j  /  x ] ph )  ->  i  =  j ) )
 
Theoremnmo 23160* Negation of "at most one". (Contributed by Thierry Arnoux, 26-Feb-2017.)
 |-  F/ y ph   =>    |-  ( -.  E* x ph  <->  A. y E. x (
 ph  /\  x  =/=  y ) )
 
Theoremmoimd 23161* "At most one" is preserved through implication (notice wff reversal). (Contributed by Thierry Arnoux, 25-Feb-2017.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( E* x ch  ->  E* x ps ) )
 
TheoremrmoxfrdOLD 23162* Transfer "at most one" restricted quantification from a variable  x to another variable  y contained in expression 
A. (Contributed by Thierry Arnoux, 7-Apr-2017.)
 |-  (
 ( ph  /\  y  e.  C )  ->  A  e.  B )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  E! y  e.  C  x  =  A )   &    |-  (
 ( ph  /\  x  =  A )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E* x ( x  e.  B  /\  ps ) 
 <->  E* y ( y  e.  C  /\  ch ) ) )
 
Theoremrmoxfrd 23163* Transfer "at most one" restricted quantification from a variable  x to another variable  y contained in expression 
A. (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)
 |-  (
 ( ph  /\  y  e.  C )  ->  A  e.  B )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  E! y  e.  C  x  =  A )   &    |-  (
 ( ph  /\  x  =  A )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E* x  e.  B ps 
 <->  E* y  e.  C ch ) )
 
Theoremssrmo 23164 "At most one" existential quantification restricted to a subclass. (Contributed by Thierry Arnoux, 8-Oct-2017.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  ( A  C_  B  ->  ( E* x  e.  B ph  ->  E* x  e.  A ph ) )
 
18.3.8  Conditional operator - misc additions
 
Theoremifbieq12d2 23165 Equivalence deduction for conditional operators. (Contributed by Thierry Arnoux, 14-Feb-2017.)
 |-  ( ph  ->  ( ps  <->  ch ) )   &    |-  (
 ( ph  /\  ps )  ->  A  =  C )   &    |-  ( ( ph  /\  -.  ps )  ->  B  =  D )   =>    |-  ( ph  ->  if ( ps ,  A ,  B )  =  if ( ch ,  C ,  D ) )
 
Theoremovif 23166 Move a conditional outside of an operation (Contributed by Thierry Arnoux, 25-Jan-2017.)
 |-  ( if ( ph ,  A ,  B ) F C )  =  if ( ph ,  ( A F C ) ,  ( B F C ) )
 
Theoremelimifd 23167 Elimination of a conditional operator contained in a wff  ch. (Contributed by Thierry Arnoux, 25-Jan-2017.)
 |-  ( ph  ->  ( if ( ps ,  A ,  B )  =  A  ->  ( ch  <->  th ) ) )   &    |-  ( ph  ->  ( if ( ps ,  A ,  B )  =  B  ->  ( ch  <->  ta ) ) )   =>    |-  ( ph  ->  ( ch  <->  (
 ( ps  /\  th )  \/  ( -.  ps  /\ 
 ta ) ) ) )
 
Theoremelim2if 23168 Elimination of two conditional operators contained in a wff  ch. (Contributed by Thierry Arnoux, 25-Jan-2017.)
 |-  ( if ( ph ,  A ,  if ( ps ,  B ,  C )
 )  =  A  ->  ( ch  <->  th ) )   &    |-  ( if ( ph ,  A ,  if ( ps ,  B ,  C )
 )  =  B  ->  ( ch  <->  ta ) )   &    |-  ( if ( ph ,  A ,  if ( ps ,  B ,  C )
 )  =  C  ->  ( ch  <->  et ) )   =>    |-  ( ch  <->  ( ( ph  /\ 
 th )  \/  ( -.  ph  /\  ( ( ps  /\  ta )  \/  ( -.  ps  /\  et ) ) ) ) )
 
Theoremelim2ifim 23169 Elimination of two conditional operators for an implication. (Contributed by Thierry Arnoux, 25-Jan-2017.)
 |-  ( if ( ph ,  A ,  if ( ps ,  B ,  C )
 )  =  A  ->  ( ch  <->  th ) )   &    |-  ( if ( ph ,  A ,  if ( ps ,  B ,  C )
 )  =  B  ->  ( ch  <->  ta ) )   &    |-  ( if ( ph ,  A ,  if ( ps ,  B ,  C )
 )  =  C  ->  ( ch  <->  et ) )   &    |-  ( ph  ->  th )   &    |-  ( ( -.  ph  /\  ps )  ->  ta )   &    |-  ( ( -.  ph  /\  -.  ps )  ->  et )   =>    |- 
 ch
 
18.3.9  Indexed union - misc additions
 
Theoremiuneq12daf 23170 Equality deduction for indexed union, deduction version. (Contributed by Thierry Arnoux, 13-Mar-2017.)
 |-  F/ x ph   &    |-  F/_ x A   &    |-  F/_ x B   &    |-  ( ph  ->  A  =  B )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  =  D )   =>    |-  ( ph  ->  U_ x  e.  A  C  =  U_ x  e.  B  D )
 
Theoremiuneq12df 23171 Equality deduction for indexed union, deduction version. (Contributed by Thierry Arnoux, 31-Dec-2016.)
 |-  F/ x ph   &    |-  F/_ x A   &    |-  F/_ x B   &    |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  U_ x  e.  A  C  =  U_ x  e.  B  D )
 
Theoremssiun3 23172* Subset equivalence for an indexed union. (Contributed by Thierry Arnoux, 17-Oct-2016.)
 |-  ( A. y  e.  C  E. x  e.  A  y  e.  B  <->  C  C_  U_ x  e.  A  B )
 
Theoremssiun2sf 23173 Subset relationship for an indexed union. (Contributed by Thierry Arnoux, 31-Dec-2016.)
 |-  F/_ x A   &    |-  F/_ x C   &    |-  F/_ x D   &    |-  ( x  =  C  ->  B  =  D )   =>    |-  ( C  e.  A  ->  D  C_  U_ x  e.  A  B )
 
Theoremiuninc 23174* The union of an increasing collection of sets is its last element. (Contributed by Thierry Arnoux, 22-Jan-2017.)
 |-  ( ph  ->  F  Fn  NN )   &    |-  ( ( ph  /\  n  e.  NN )  ->  ( F `  n )  C_  ( F `  ( n  +  1 ) ) )   =>    |-  ( ( ph  /\  i  e.  NN )  ->  U_ n  e.  ( 1 ... i
 ) ( F `  n )  =  ( F `  i ) )
 
Theoremiundifdifd 23175* The intersection of a set is the complement of the union of the complements. (Contributed by Thierry Arnoux, 19-Dec-2016.)
 |-  ( A  C_  ~P O  ->  ( A  =/=  (/)  ->  |^| A  =  ( O  \  U_ x  e.  A  ( O  \  x ) ) ) )
 
Theoremiundifdif 23176* The intersection of a set is the complement of the union of the complements. TODO shorten using iundifdifd 23175 (Contributed by Thierry Arnoux, 4-Sep-2016.)
 |-  O  e.  _V   &    |-  A  C_  ~P O   =>    |-  ( A  =/=  (/)  ->  |^| A  =  ( O  \  U_ x  e.  A  ( O  \  x ) ) )
 
Theoremiunrdx 23177* Re-index an indexed union. (Contributed by Thierry Arnoux, 6-Apr-2017.)
 |-  ( ph  ->  F : A -onto-> C )   &    |-  ( ( ph  /\  y  =  ( F `
  x ) ) 
 ->  D  =  B )   =>    |-  ( ph  ->  U_ x  e.  A  B  =  U_ y  e.  C  D )
 
18.3.10  Miscellaneous
 
Theoremceqsexv2d 23178* Elimination of an existential quantifier, using implicit substitution. (Contributed by Thierry Arnoux, 10-Sep-2016.)
 |-  A  e.  _V   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ps   =>    |- 
 E. x ph
 
Theoremr19.41vv 23179* Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90. Version with two quantifiers (Contributed by Thierry Arnoux, 25-Jan-2017.)
 |-  ( E. x  e.  A  E. y  e.  B  ( ph  /\  ps )  <->  ( E. x  e.  A  E. y  e.  B  ph 
 /\  ps ) )
 
Theoremrabbidva2 23180* Equivalent wff's yield equal restricted class abstractions. (Contributed by Thierry Arnoux, 4-Feb-2017.)
 |-  ( ph  ->  ( ( x  e.  A  /\  ps ) 
 <->  ( x  e.  B  /\  ch ) ) )   =>    |-  ( ph  ->  { x  e.  A  |  ps }  =  { x  e.  B  |  ch } )
 
Theoremabrexdomjm 23181* An indexed set is dominated by the indexing set. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  (
 y  e.  A  ->  E* x ph )   =>    |-  ( A  e.  V  ->  { x  |  E. y  e.  A  ph
 }  ~<_  A )
 
Theoremabrexdom2jm 23182* An indexed set is dominated by the indexing set. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( A  e.  V  ->  { x  |  E. y  e.  A  x  =  B } 
 ~<_  A )
 
Theoreminin 23183 Intersection with an intersection (Contributed by Thierry Arnoux, 27-Dec-2016.)
 |-  ( A  i^i  ( A  i^i  B ) )  =  ( A  i^i  B )
 
Theoremsumpr 23184* A sum over a pair is the sum of the elements. (Contributed by Thierry Arnoux, 12-Dec-2016.)
 |-  (
 k  =  A  ->  C  =  D )   &    |-  (
 k  =  B  ->  C  =  E )   &    |-  ( ph  ->  ( D  e.  CC  /\  E  e.  CC ) )   &    |-  ( ph  ->  ( A  e.  V  /\  B  e.  W )
 )   &    |-  ( ph  ->  A  =/=  B )   =>    |-  ( ph  ->  sum_ k  e.  { A ,  B } C  =  ( D  +  E )
 )
 
Theoreminfi1 23185 The interection of two sets is finite if one of them is. (Contributed by Thierry Arnoux, 14-Feb-2017.)
 |-  ( A  e.  Fin  ->  ( A  i^i  B )  e. 
 Fin )
 
Theoremrabfi 23186* A restricted class built from a finite set is finite. (Contributed by Thierry Arnoux, 14-Feb-2017.)
 |-  ( A  e.  Fin  ->  { x  e.  A  |  ph }  e.  Fin )
 
TheoremrabexgfGS 23187 Separation Scheme in terms of a restricted class abstraction. To be removed in profit of Glauco's equivalent version. (Contributed by Thierry Arnoux, 11-May-2017.)
 |-  F/_ x A   =>    |-  ( A  e.  V  ->  { x  e.  A  |  ph }  e.  _V )
 
Theoremfneq12 23188 Equality theorem for function predicate with domain. (Contributed by Thierry Arnoux, 31-Jan-2017.)
 |-  (
 ( F  =  G  /\  A  =  B ) 
 ->  ( F  Fn  A  <->  G  Fn  B ) )
 
Theoremhashresfn 23189 Restriction of the domain of the size function. (Contributed by Thierry Arnoux, 31-Jan-2017.)
 |-  ( #  |`  A )  Fn  A
 
Theoremdmhashres 23190 Restriction of the domain of the size function. (Contributed by Thierry Arnoux, 12-Jan-2017.)
 |-  dom  ( #  |`  A )  =  A
 
Theoremcntnevol 23191 Counting and Lebesgue measure are different. (Contributed by Thierry Arnoux, 27-Jan-2017.)
 |-  ( #  |`  ~P O )  =/= 
 vol
 
Theoremdisjex 23192* Two ways to say that two classes are disjoint (or equal). (Contributed by Thierry Arnoux, 4-Oct-2016.)
 |-  (
 ( E. z ( z  e.  A  /\  z  e.  B )  ->  A  =  B )  <-> 
 ( A  =  B  \/  ( A  i^i  B )  =  (/) ) )
 
Theoremdisjexc 23193* A variant of disjex 23192, applicable for more generic families. (Contributed by Thierry Arnoux, 4-Oct-2016.)
 |-  ( x  =  y  ->  A  =  B )   =>    |-  ( ( E. z ( z  e.  A  /\  z  e.  B )  ->  x  =  y )  ->  ( A  =  B  \/  ( A  i^i  B )  =  (/) ) )
 
Theoremrmo3f 23194* Restricted "at most one" using explicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)
 |-  F/_ x A   &    |-  F/_ y A   &    |-  F/ y ph   =>    |-  ( E* x  e.  A ph  <->  A. x  e.  A  A. y  e.  A  (
 ( ph  /\  [ y  /  x ] ph )  ->  x  =  y ) )
 
Theoremrmo4fOLD 23195* Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by Thierry Arnoux, 11-Oct-2016.) (Revised by Thierry Arnoux, 8-Mar-2017.)
 |-  F/_ x A   &    |-  F/_ y A   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E* x ( x  e.  A  /\  ph )  <->  A. x  e.  A  A. y  e.  A  ( ( ph  /\  ps )  ->  x  =  y ) )
 
Theoremrmo4f 23196* Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by Thierry Arnoux, 11-Oct-2016.) (Revised by Thierry Arnoux, 8-Mar-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)
 |-  F/_ x A   &    |-  F/_ y A   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E* x  e.  A ph  <->  A. x  e.  A  A. y  e.  A  ( ( ph  /\  ps )  ->  x  =  y ) )
 
Theoremr19.29d2r 23197 Theorem 19.29 of [Margaris] p. 90 with two restricted quantifiers, deduction version (Contributed by Thierry Arnoux, 30-Jan-2017.)
 |-  ( ph  ->  A. x  e.  A  A. y  e.  B  ps )   &    |-  ( ph  ->  E. x  e.  A  E. y  e.  B  ch )   =>    |-  ( ph  ->  E. x  e.  A  E. y  e.  B  ( ps  /\  ch ) )
 
Theorem19.9d2rf 23198 A deduction version of one direction of 19.9 1795 with two variables (Contributed by Thierry Arnoux, 20-Mar-2017.)
 |-  F/ y ph   &    |-  ( ph  ->  F/ x ps )   &    |-  ( ph  ->  F/ y ps )   &    |-  ( ph  ->  E. x  e.  A  E. y  e.  B  ps )   =>    |-  ( ph  ->  ps )
 
Theorem19.9d2r 23199* A deduction version of one direction of 19.9 1795 with two variables (Contributed by Thierry Arnoux, 30-Jan-2017.)
 |-  ( ph  ->  F/ x ps )   &    |-  ( ph  ->  F/ y ps )   &    |-  ( ph  ->  E. x  e.  A  E. y  e.  B  ps )   =>    |-  ( ph  ->  ps )
 
Theoremprsspwg 23200 An unordered pair belongs to the power class of a class iff each member belongs to the class. (Contributed by Thierry Arnoux, 3-Oct-2016.)
 |-  (
 ( A  e.  _V  /\  B  e.  _V )  ->  ( { A ,  B }  C_  ~P C  <->  ( A  C_  C  /\  B  C_  C ) ) )
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