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Theorem List for Metamath Proof Explorer - 6601-6700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnnmwordi 6601 Weak ordering property of multiplication. (Contributed by Mario Carneiro, 17-Nov-2014.)
 |-  ( ( A  e.  om 
 /\  B  e.  om  /\  C  e.  om )  ->  ( A  C_  B  ->  ( C  .o  A )  C_  ( C  .o  B ) ) )
 
Theoremnnmwordri 6602 Weak ordering property of ordinal multiplication. Proposition 8.21 of [TakeutiZaring] p. 63, limited to natural numbers. (Contributed by Mario Carneiro, 17-Nov-2014.)
 |-  ( ( A  e.  om 
 /\  B  e.  om  /\  C  e.  om )  ->  ( A  C_  B  ->  ( A  .o  C )  C_  ( B  .o  C ) ) )
 
Theoremnnawordex 6603* Equivalence for weak ordering of natural numbers. (Contributed by NM, 8-Nov-2002.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( A  C_  B 
 <-> 
 E. x  e.  om  ( A  +o  x )  =  B )
 )
 
Theoremnnaordex 6604* Equivalence for ordering. Compare Exercise 23 of [Enderton] p. 88. (Contributed by NM, 5-Dec-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( A  e.  B 
 <-> 
 E. x  e.  om  ( (/)  e.  x  /\  ( A  +o  x )  =  B )
 ) )
 
Theorem1onn 6605 One is a natural number. (Contributed by NM, 29-Oct-1995.)
 |- 
 1o  e.  om
 
Theorem2onn 6606 The ordinal 2 is a natural number. (Contributed by NM, 28-Sep-2004.)
 |- 
 2o  e.  om
 
Theorem3onn 6607 The ordinal 3 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |- 
 3o  e.  om
 
Theorem4onn 6608 The ordinal 4 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |- 
 4o  e.  om
 
Theoremoaabslem 6609 Lemma for oaabs 6610. (Contributed by NM, 9-Dec-2004.)
 |-  ( ( om  e.  On  /\  A  e.  om )  ->  ( A  +o  om )  =  om )
 
Theoremoaabs 6610 Ordinal addition absorbs a natural number added to the left of a transfinite number. Proposition 8.10 of [TakeutiZaring] p. 59. (Contributed by NM, 9-Dec-2004.) (Proof shortened by Mario Carneiro, 29-May-2015.)
 |-  ( ( ( A  e.  om  /\  B  e.  On )  /\  om  C_  B )  ->  ( A  +o  B )  =  B )
 
Theoremoaabs2 6611 The absorption law oaabs 6610 is also a property of higher powers of  om. (Contributed by Mario Carneiro, 29-May-2015.)
 |-  ( ( ( A  e.  ( om  ^o  C )  /\  B  e.  On )  /\  ( om  ^o  C )  C_  B )  ->  ( A  +o  B )  =  B )
 
Theoremomabslem 6612 Lemma for omabs 6613. (Contributed by Mario Carneiro, 30-May-2015.)
 |-  ( ( om  e.  On  /\  A  e.  om  /\  (/)  e.  A )  ->  ( A  .o  om )  =  om )
 
Theoremomabs 6613 Ordinal multiplication is also absorbed by powers of  om. (Contributed by Mario Carneiro, 30-May-2015.)
 |-  ( ( ( A  e.  om  /\  (/)  e.  A )  /\  ( B  e.  On  /\  (/)  e.  B ) )  ->  ( A  .o  ( om  ^o  B ) )  =  ( om  ^o  B ) )
 
Theoremnnm1 6614 Multiply an element of  om by  1o. (Contributed by Mario Carneiro, 17-Nov-2014.)
 |-  ( A  e.  om  ->  ( A  .o  1o )  =  A )
 
Theoremnnm2 6615 Multiply an element of  om by  2o (Contributed by Scott Fenton, 18-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |-  ( A  e.  om  ->  ( A  .o  2o )  =  ( A  +o  A ) )
 
Theoremnn2m 6616 Multiply an element of  om by  2o (Contributed by Scott Fenton, 16-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |-  ( A  e.  om  ->  ( 2o  .o  A )  =  ( A  +o  A ) )
 
Theoremnnneo 6617 If an natural number is even, its successor is odd. (Contributed by Mario Carneiro, 16-Nov-2014.)
 |-  ( ( A  e.  om 
 /\  B  e.  om  /\  C  =  ( 2o 
 .o  A ) ) 
 ->  -.  suc  C  =  ( 2o  .o  B ) )
 
Theoremnneob 6618* A natural number is even iff its successor is odd. (Contributed by NM, 26-Jan-2006.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  ( A  e.  om  ->  ( E. x  e. 
 om  A  =  ( 2o  .o  x )  <->  -.  E. x  e.  om  suc 
 A  =  ( 2o 
 .o  x ) ) )
 
Theoremomsmolem 6619* Lemma for omsmo 6620. (Contributed by NM, 30-Nov-2003.) (Revised by David Abernethy, 1-Jan-2014.)
 |-  ( y  e.  om  ->  ( ( ( A 
 C_  On  /\  F : om
 --> A )  /\  A. x  e.  om  ( F `
  x )  e.  ( F `  suc  x ) )  ->  (
 z  e.  y  ->  ( F `  z )  e.  ( F `  y ) ) ) )
 
Theoremomsmo 6620* A strictly monotonic ordinal function on the set of natural numbers is one-to-one. (Contributed by NM, 30-Nov-2003.) (Revised by David Abernethy, 1-Jan-2014.)
 |-  ( ( ( A 
 C_  On  /\  F : om
 --> A )  /\  A. x  e.  om  ( F `
  x )  e.  ( F `  suc  x ) )  ->  F : om -1-1-> A )
 
Theoremomopthlem1 6621 Lemma for omopthi 6623. (Contributed by Scott Fenton, 18-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |-  A  e.  om   &    |-  C  e.  om   =>    |-  ( A  e.  C  ->  ( ( A  .o  A )  +o  ( A  .o  2o ) )  e.  ( C  .o  C ) )
 
Theoremomopthlem2 6622 Lemma for omopthi 6623. (Contributed by Scott Fenton, 16-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |-  A  e.  om   &    |-  B  e.  om   &    |-  C  e.  om   &    |-  D  e.  om   =>    |-  ( ( A  +o  B )  e.  C  ->  -.  ( ( C  .o  C )  +o  D )  =  (
 ( ( A  +o  B )  .o  ( A  +o  B ) )  +o  B ) )
 
Theoremomopthi 6623 An ordered pair theorem for  om. Theorem 17.3 of [Quine] p. 124. This proof is adapted from nn0opthi 11251. (Contributed by Scott Fenton, 16-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |-  A  e.  om   &    |-  B  e.  om   &    |-  C  e.  om   &    |-  D  e.  om   =>    |-  ( ( ( ( A  +o  B )  .o  ( A  +o  B ) )  +o  B )  =  (
 ( ( C  +o  D )  .o  ( C  +o  D ) )  +o  D )  <->  ( A  =  C  /\  B  =  D ) )
 
Theoremomopth 6624 An ordered pair theorem for finite integers. Analagous to nn0opthi 11251. (Contributed by Scott Fenton, 1-May-2012.) (Revised by Mario Carneiro, 12-May-2012.)
 |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( C  e.  om  /\  D  e.  om ) )  ->  ( ( ( ( A  +o  B )  .o  ( A  +o  B ) )  +o  B )  =  (
 ( ( C  +o  D )  .o  ( C  +o  D ) )  +o  D )  <->  ( A  =  C  /\  B  =  D ) ) )
 
2.4.25  Equivalence relations and classes
 
Syntaxwer 6625 Extend the definition of a wff to include the equivalence predicate.
 wff  R  Er  A
 
Syntaxcec 6626 Extend the definition of a class to include equivalence class.
 class  [ A ] R
 
Syntaxcqs 6627 Extend the definition of a class to include quotient set.
 class  ( A /. R )
 
Definitiondf-er 6628 Define the equivalence relation predicate. Our notation is not standard. A formal notation doesn't seem to exist in the literature; instead only informal English tends to be used. The present definition, although somewhat cryptic, nicely avoids dummy variables. In dfer2 6629 we derive a more typical definition. We show that an equivalence relation is reflexive, symmetric, and transitive in erref 6648, ersymb 6642, and ertr 6643. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 2-Nov-2015.)
 |-  ( R  Er  A  <->  ( Rel  R  /\  dom  R  =  A  /\  ( `' R  u.  ( R  o.  R ) ) 
 C_  R ) )
 
Theoremdfer2 6629* Alternate definition of equivalence predicate. (Contributed by NM, 3-Jan-1997.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( R  Er  A  <->  ( Rel  R  /\  dom  R  =  A  /\  A. x A. y A. z
 ( ( x R y  ->  y R x )  /\  ( ( x R y  /\  y R z )  ->  x R z ) ) ) )
 
Definitiondf-ec 6630 Define the  R-coset of  A. Exercise 35 of [Enderton] p. 61. This is called the equivalence class of  A modulo  R when  R is an equivalence relation (i.e. when  Er  R; see dfer2 6629). In this case,  A is a representative (member) of the equivalence class  [ A ] R, which contains all sets that are equivalent to  A. Definition of [Enderton] p. 57 uses the notation  [ A ] (subscript)  R, although we simply follow the brackets by  R since we don't have subscripted expressions. For an alternate definition, see dfec2 6631. (Contributed by NM, 23-Jul-1995.)
 |- 
 [ A ] R  =  ( R " { A } )
 
Theoremdfec2 6631* Alternate definition of  R-coset of  A. Definition 34 of [Suppes] p. 81. (Contributed by NM, 3-Jan-1997.) (Proof shortened by Mario Carneiro, 9-Jul-2014.)
 |-  ( A  e.  V  ->  [ A ] R  =  { y  |  A R y } )
 
Theoremecexg 6632 An equivalence class modulo a set is a set. (Contributed by NM, 24-Jul-1995.)
 |-  ( R  e.  B  ->  [ A ] R  e.  _V )
 
Theoremecexr 6633 A nonempty equivalence class implies the representative is a set. (Contributed by Mario Carneiro, 9-Jul-2014.)
 |-  ( A  e.  [ B ] R  ->  B  e.  _V )
 
Definitiondf-qs 6634* Define quotient set.  R is usually an equivalence relation. Definition of [Enderton] p. 58. (Contributed by NM, 23-Jul-1995.)
 |-  ( A /. R )  =  { y  |  E. x  e.  A  y  =  [ x ] R }
 
Theoremereq1 6635 Equality theorem for equivalence predicate. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( R  =  S  ->  ( R  Er  A  <->  S  Er  A ) )
 
Theoremereq2 6636 Equality theorem for equivalence predicate. (Contributed by Mario Carneiro, 12-Aug-2015.)
 |-  ( A  =  B  ->  ( R  Er  A  <->  R  Er  B ) )
 
Theoremerrel 6637 An equivalence relation is a relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
 |-  ( R  Er  A  ->  Rel  R )
 
Theoremerdm 6638 The domain of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
 |-  ( R  Er  A  ->  dom  R  =  A )
 
Theoremercl 6639 Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
 |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  A R B )   =>    |-  ( ph  ->  A  e.  X )
 
Theoremersym 6640 An equivalence relation is symmetric. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  A R B )   =>    |-  ( ph  ->  B R A )
 
Theoremercl2 6641 Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
 |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  A R B )   =>    |-  ( ph  ->  B  e.  X )
 
Theoremersymb 6642 An equivalence relation is symmetric. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( ph  ->  R  Er  X )   =>    |-  ( ph  ->  ( A R B  <->  B R A ) )
 
Theoremertr 6643 An equivalence relation is transitive. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( ph  ->  R  Er  X )   =>    |-  ( ph  ->  (
 ( A R B  /\  B R C ) 
 ->  A R C ) )
 
Theoremertrd 6644 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)
 |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  A R B )   &    |-  ( ph  ->  B R C )   =>    |-  ( ph  ->  A R C )
 
Theoremertr2d 6645 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)
 |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  A R B )   &    |-  ( ph  ->  B R C )   =>    |-  ( ph  ->  C R A )
 
Theoremertr3d 6646 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)
 |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  B R A )   &    |-  ( ph  ->  B R C )   =>    |-  ( ph  ->  A R C )
 
Theoremertr4d 6647 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)
 |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  A R B )   &    |-  ( ph  ->  C R B )   =>    |-  ( ph  ->  A R C )
 
Theoremerref 6648 An equivalence relation is reflexive on its field. Compare Theorem 3M of [Enderton] p. 56. (Contributed by Mario Carneiro, 6-May-2013.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  A  e.  X )   =>    |-  ( ph  ->  A R A )
 
Theoremercnv 6649 The converse of an equivalence relation is itself. (Contributed by Mario Carneiro, 12-Aug-2015.)
 |-  ( R  Er  A  ->  `' R  =  R )
 
Theoremerrn 6650 The range and domain of an equivalence relation are equal. (Contributed by Rodolfo Medina, 11-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( R  Er  A  ->  ran  R  =  A )
 
Theoremerssxp 6651 An equivalence relation is a subset of the cartesian product of the field. (Contributed by Mario Carneiro, 12-Aug-2015.)
 |-  ( R  Er  A  ->  R  C_  ( A  X.  A ) )
 
Theoremerex 6652 An equivalence relation is a set if its domain is a set. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Proof shortened by Mario Carneiro, 12-Aug-2015.)
 |-  ( R  Er  A  ->  ( A  e.  V  ->  R  e.  _V )
 )
 
Theoremerexb 6653 An equivalence relation is a set if and only if its domain is a set. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( R  Er  A  ->  ( R  e.  _V  <->  A  e.  _V ) )
 
Theoremiserd 6654* A reflexive, symmetric, transitive relation is an equivalence relation on its domain. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ( ph  /\  x R y )  ->  y R x )   &    |-  (
 ( ph  /\  ( x R y  /\  y R z ) ) 
 ->  x R z )   &    |-  ( ph  ->  ( x  e.  A  <->  x R x ) )   =>    |-  ( ph  ->  R  Er  A )
 
Theorembrdifun 6655 Evaluate the incomparability relation. (Contributed by Mario Carneiro, 9-Jul-2014.)
 |-  R  =  ( ( X  X.  X ) 
 \  (  .<  u.  `'  .<  ) )   =>    |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A R B 
 <->  -.  ( A  .<  B  \/  B  .<  A ) ) )
 
Theoremswoer 6656* Incomparability under a strict weak partial order is an equivalence relation. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  R  =  ( ( X  X.  X ) 
 \  (  .<  u.  `'  .<  ) )   &    |-  ( ( ph  /\  ( y  e.  X  /\  z  e.  X ) )  ->  ( y 
 .<  z  ->  -.  z  .<  y ) )   &    |-  (
 ( ph  /\  ( x  e.  X  /\  y  e.  X  /\  z  e.  X ) )  ->  ( x  .<  y  ->  ( x  .<  z  \/  z  .<  y )
 ) )   =>    |-  ( ph  ->  R  Er  X )
 
Theoremswoord1 6657* The incomparability equivalence relation is compatible with the original order. (Contributed by Mario Carneiro, 31-Dec-2014.)
 |-  R  =  ( ( X  X.  X ) 
 \  (  .<  u.  `'  .<  ) )   &    |-  ( ( ph  /\  ( y  e.  X  /\  z  e.  X ) )  ->  ( y 
 .<  z  ->  -.  z  .<  y ) )   &    |-  (
 ( ph  /\  ( x  e.  X  /\  y  e.  X  /\  z  e.  X ) )  ->  ( x  .<  y  ->  ( x  .<  z  \/  z  .<  y )
 ) )   &    |-  ( ph  ->  B  e.  X )   &    |-  ( ph  ->  C  e.  X )   &    |-  ( ph  ->  A R B )   =>    |-  ( ph  ->  ( A  .<  C  <->  B  .<  C ) )
 
Theoremswoord2 6658* The incomparability equivalence relation is compatible with the original order. (Contributed by Mario Carneiro, 31-Dec-2014.)
 |-  R  =  ( ( X  X.  X ) 
 \  (  .<  u.  `'  .<  ) )   &    |-  ( ( ph  /\  ( y  e.  X  /\  z  e.  X ) )  ->  ( y 
 .<  z  ->  -.  z  .<  y ) )   &    |-  (
 ( ph  /\  ( x  e.  X  /\  y  e.  X  /\  z  e.  X ) )  ->  ( x  .<  y  ->  ( x  .<  z  \/  z  .<  y )
 ) )   &    |-  ( ph  ->  B  e.  X )   &    |-  ( ph  ->  C  e.  X )   &    |-  ( ph  ->  A R B )   =>    |-  ( ph  ->  ( C  .<  A  <->  C  .<  B ) )
 
Theoremswoso 6659* If the incomparability relation is equivalent to equality in a subset, then the partial order strictly orders the subset. (Contributed by Mario Carneiro, 30-Dec-2014.)
 |-  R  =  ( ( X  X.  X ) 
 \  (  .<  u.  `'  .<  ) )   &    |-  ( ( ph  /\  ( y  e.  X  /\  z  e.  X ) )  ->  ( y 
 .<  z  ->  -.  z  .<  y ) )   &    |-  (
 ( ph  /\  ( x  e.  X  /\  y  e.  X  /\  z  e.  X ) )  ->  ( x  .<  y  ->  ( x  .<  z  \/  z  .<  y )
 ) )   &    |-  ( ph  ->  Y 
 C_  X )   &    |-  (
 ( ph  /\  ( x  e.  Y  /\  y  e.  Y  /\  x R y ) )  ->  x  =  y )   =>    |-  ( ph  ->  .<  Or  Y )
 
Theoremeqerlem 6660* Lemma for eqer 6661. (Contributed by NM, 17-Mar-2008.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)
 |-  ( x  =  y 
 ->  A  =  B )   &    |-  R  =  { <. x ,  y >.  |  A  =  B }   =>    |-  ( z R w  <->  [_ z  /  x ]_ A  =  [_ w  /  x ]_ A )
 
Theoremeqer 6661* Equivalence relation involving equality of dependent classes  A
( x ) and  B ( y ). (Contributed by NM, 17-Mar-2008.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( x  =  y 
 ->  A  =  B )   &    |-  R  =  { <. x ,  y >.  |  A  =  B }   =>    |-  R  Er  _V
 
Theoremider 6662 The identity relation is an equivalence relation. (Contributed by NM, 10-May-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Proof shortened by Mario Carneiro, 9-Jul-2014.)
 |- 
 _I  Er  _V
 
Theorem0er 6663 The empty set is an equivalence relation on the empty set. (Contributed by Mario Carneiro, 5-Sep-2015.)
 |-  (/)  Er  (/)
 
Theoremeceq1 6664 Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.)
 |-  ( A  =  B  ->  [ A ] C  =  [ B ] C )
 
Theoremeceq2 6665 Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.)
 |-  ( A  =  B  ->  [ C ] A  =  [ C ] B )
 
Theoremelecg 6666 Membership in an equivalence class. Theorem 72 of [Suppes] p. 82. (Contributed by Mario Carneiro, 9-Jul-2014.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  e.  [ B ] R  <->  B R A ) )
 
Theoremelec 6667 Membership in an equivalence class. Theorem 72 of [Suppes] p. 82. (Contributed by NM, 23-Jul-1995.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  e.  [ B ] R  <->  B R A )
 
Theoremrelelec 6668 Membership in an equivalence class when  R is a relation. (Contributed by Mario Carneiro, 11-Sep-2015.)
 |-  ( Rel  R  ->  ( A  e.  [ B ] R  <->  B R A ) )
 
Theoremecss 6669 An equivalence class is a subset of the domain. (Contributed by NM, 6-Aug-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( ph  ->  R  Er  X )   =>    |-  ( ph  ->  [ A ] R  C_  X )
 
Theoremecdmn0 6670 A representative of a nonempty equivalence class belongs to the domain of the equivalence relation. (Contributed by NM, 15-Feb-1996.) (Revised by Mario Carneiro, 9-Jul-2014.)
 |-  ( A  e.  dom  R  <->  [ A ] R  =/=  (/) )
 
Theoremereldm 6671 Equality of equivalence classes implies equivalence of domain membership. (Contributed by NM, 28-Jan-1996.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  [ A ] R  =  [ B ] R )   =>    |-  ( ph  ->  ( A  e.  X  <->  B  e.  X ) )
 
Theoremerth 6672 Basic property of equivalence relations. Theorem 73 of [Suppes] p. 82. (Contributed by NM, 23-Jul-1995.) (Revised by Mario Carneiro, 6-Jul-2015.)
 |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  A  e.  X )   =>    |-  ( ph  ->  ( A R B  <->  [ A ] R  =  [ B ] R ) )
 
Theoremerth2 6673 Basic property of equivalence relations. Compare Theorem 73 of [Suppes] p. 82. Assumes membership of the second argument in the domain. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 6-Jul-2015.)
 |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  B  e.  X )   =>    |-  ( ph  ->  ( A R B  <->  [ A ] R  =  [ B ] R ) )
 
Theoremerthi 6674 Basic property of equivalence relations. Part of Lemma 3N of [Enderton] p. 57. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
 |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  A R B )   =>    |-  ( ph  ->  [ A ] R  =  [ B ] R )
 
Theoremerdisj 6675 Equivalence classes do not overlap. In other words, two equivalence classes are either equal or disjoint. Theorem 74 of [Suppes] p. 83. (Contributed by NM, 15-Jun-2004.) (Revised by Mario Carneiro, 9-Jul-2014.)
 |-  ( R  Er  X  ->  ( [ A ] R  =  [ B ] R  \/  ( [ A ] R  i^i  [ B ] R )  =  (/) ) )
 
Theoremecidsn 6676 An equivalence class modulo the identity relation is a singleton. (Contributed by NM, 24-Oct-2004.)
 |- 
 [ A ]  _I  =  { A }
 
Theoremqseq1 6677 Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.)
 |-  ( A  =  B  ->  ( A /. C )  =  ( B /. C ) )
 
Theoremqseq2 6678 Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.)
 |-  ( A  =  B  ->  ( C /. A )  =  ( C /. B ) )
 
Theoremelqsg 6679* Closed form of elqs 6680. (Contributed by Rodolfo Medina, 12-Oct-2010.)
 |-  ( B  e.  V  ->  ( B  e.  ( A /. R )  <->  E. x  e.  A  B  =  [ x ] R ) )
 
Theoremelqs 6680* Membership in a quotient set. (Contributed by NM, 23-Jul-1995.)
 |-  B  e.  _V   =>    |-  ( B  e.  ( A /. R )  <->  E. x  e.  A  B  =  [ x ] R )
 
Theoremelqsi 6681* Membership in a quotient set. (Contributed by NM, 23-Jul-1995.)
 |-  ( B  e.  ( A /. R )  ->  E. x  e.  A  B  =  [ x ] R )
 
Theoremecelqsg 6682 Membership of an equivalence class in a quotient set. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 9-Jul-2014.)
 |-  ( ( R  e.  V  /\  B  e.  A )  ->  [ B ] R  e.  ( A /. R ) )
 
Theoremecelqsi 6683 Membership of an equivalence class in a quotient set. (Contributed by NM, 25-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
 |-  R  e.  _V   =>    |-  ( B  e.  A  ->  [ B ] R  e.  ( A /. R ) )
 
Theoremecopqsi 6684 "Closure" law for equivalence class of ordered pairs. (Contributed by NM, 25-Mar-1996.)
 |-  R  e.  _V   &    |-  S  =  ( ( A  X.  A ) /. R )   =>    |-  ( ( B  e.  A  /\  C  e.  A )  ->  [ <. B ,  C >. ] R  e.  S )
 
Theoremqsexg 6685 A quotient set exists. (Contributed by FL, 19-May-2007.) (Revised by Mario Carneiro, 9-Jul-2014.)
 |-  ( A  e.  V  ->  ( A /. R )  e.  _V )
 
Theoremqsex 6686 A quotient set exists. (Contributed by NM, 14-Aug-1995.)
 |-  A  e.  _V   =>    |-  ( A /. R )  e.  _V
 
Theoremuniqs 6687 The union of a quotient set. (Contributed by NM, 9-Dec-2008.)
 |-  ( R  e.  V  ->  U. ( A /. R )  =  ( R " A ) )
 
Theoremqsss 6688 A quotient set is a set of subsets of the base set. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( ph  ->  R  Er  A )   =>    |-  ( ph  ->  ( A /. R )  C_  ~P A )
 
Theoremuniqs2 6689 The union of a quotient set. (Contributed by Mario Carneiro, 11-Jul-2014.)
 |-  ( ph  ->  R  Er  A )   &    |-  ( ph  ->  R  e.  V )   =>    |-  ( ph  ->  U. ( A /. R )  =  A )
 
Theoremsnec 6690 The singleton of an equivalence class. (Contributed by NM, 29-Jan-1999.) (Revised by Mario Carneiro, 9-Jul-2014.)
 |-  A  e.  _V   =>    |-  { [ A ] R }  =  ( { A } /. R )
 
Theoremecqs 6691 Equivalence class in terms of quotient set. (Contributed by NM, 29-Jan-1999.)
 |-  R  e.  _V   =>    |-  [ A ] R  =  U. ( { A } /. R )
 
Theoremecid 6692 A set is equal to its converse epsilon coset. (Note: converse epsilon is not an equivalence relation.) (Contributed by NM, 13-Aug-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
 |-  A  e.  _V   =>    |-  [ A ] `'  _E  =  A
 
Theoremqsid 6693 A set is equal to its quotient set mod converse epsilon. (Note: converse epsilon is not an equivalence relation.) (Contributed by NM, 13-Aug-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
 |-  ( A /. `'  _E  )  =  A
 
Theoremectocld 6694* Implicit substitution of class for equivalence class. (Contributed by Mario Carneiro, 9-Jul-2014.)
 |-  S  =  ( B
 /. R )   &    |-  ( [ x ] R  =  A  ->  ( ph  <->  ps ) )   &    |-  (
 ( ch  /\  x  e.  B )  ->  ph )   =>    |-  (
 ( ch  /\  A  e.  S )  ->  ps )
 
Theoremectocl 6695* Implicit substitution of class for equivalence class. (Contributed by NM, 23-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
 |-  S  =  ( B
 /. R )   &    |-  ( [ x ] R  =  A  ->  ( ph  <->  ps ) )   &    |-  ( x  e.  B  ->  ph )   =>    |-  ( A  e.  S  ->  ps )
 
Theoremelqsn0 6696 A quotient set doesn't contain the empty set. (Contributed by NM, 24-Aug-1995.)
 |-  ( ( dom  R  =  A  /\  B  e.  ( A /. R ) )  ->  B  =/=  (/) )
 
Theoremecelqsdm 6697 Membership of an equivalence class in a quotient set. (Contributed by NM, 30-Jul-1995.)
 |-  ( ( dom  R  =  A  /\  [ B ] R  e.  ( A /. R ) ) 
 ->  B  e.  A )
 
Theoremxpider 6698 A square cross product is an equivalence relation (in general it's not a poset). (Contributed by FL, 31-Jul-2009.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( A  X.  A )  Er  A
 
Theoremiiner 6699* The intersection of a nonempty family of equivalence relations is an equivalence relation. (Contributed by Mario Carneiro, 27-Sep-2015.)
 |-  ( ( A  =/=  (/)  /\  A. x  e.  A  R  Er  B )  ->  |^|_
 x  e.  A  R  Er  B )
 
Theoremriiner 6700* The relative intersection of a family of equivalence relations is an equivalence relation. (Contributed by Mario Carneiro, 27-Sep-2015.)
 |-  ( A. x  e.  A  R  Er  B  ->  ( ( B  X.  B )  i^i  |^|_ x  e.  A  R )  Er  B )
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