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Theorem List for Metamath Proof Explorer - 6601-6700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremoeeu 6601* The division algorithm for ordinal exponentiation. (Contributed by Mario Carneiro, 25-May-2015.)
 |-  ( ( A  e.  ( On  \  2o )  /\  B  e.  ( On  \  1o ) )  ->  E! w E. x  e. 
 On  E. y  e.  ( A  \  1o ) E. z  e.  ( A  ^o  x ) ( w  =  <. x ,  y ,  z >.  /\  ( ( ( A  ^o  x )  .o  y )  +o  z )  =  B ) )
 
2.4.26  Natural number arithmetic
 
Theoremnna0 6602 Addition with zero. Theorem 4I(A1) of [Enderton] p. 79. (Contributed by NM, 20-Sep-1995.)
 |-  ( A  e.  om  ->  ( A  +o  (/) )  =  A )
 
Theoremnnm0 6603 Multiplication with zero. Theorem 4J(A1) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.)
 |-  ( A  e.  om  ->  ( A  .o  (/) )  =  (/) )
 
Theoremnnasuc 6604 Addition with successor. Theorem 4I(A2) of [Enderton] p. 79. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( A  +o  suc 
 B )  =  suc  ( A  +o  B ) )
 
Theoremnnmsuc 6605 Multiplication with successor. Theorem 4J(A2) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( A  .o  suc 
 B )  =  ( ( A  .o  B )  +o  A ) )
 
Theoremnnesuc 6606 Exponentiation with a successor exponent. Definition 8.30 of [TakeutiZaring] p. 67. (Contributed by Mario Carneiro, 14-Nov-2014.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( A  ^o  suc 
 B )  =  ( ( A  ^o  B )  .o  A ) )
 
Theoremnna0r 6607 Addition to zero. Remark in proof of Theorem 4K(2) of [Enderton] p. 81. Note: In this and later theorems, we deliberately avoid the more general ordinal versions of these theorems (in this case oa0r 6537) so that we can avoid ax-rep 4131, which is not needed for finite recursive definitions. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
 |-  ( A  e.  om  ->  ( (/)  +o  A )  =  A )
 
Theoremnnm0r 6608 Multiplication with zero. Exercise 16 of [Enderton] p. 82. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  ( A  e.  om  ->  ( (/)  .o  A )  =  (/) )
 
Theoremnnacl 6609 Closure of addition of natural numbers. Proposition 8.9 of [TakeutiZaring] p. 59. (Contributed by NM, 20-Sep-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( A  +o  B )  e.  om )
 
Theoremnnmcl 6610 Closure of multiplication of natural numbers. Proposition 8.17 of [TakeutiZaring] p. 63. (Contributed by NM, 20-Sep-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( A  .o  B )  e.  om )
 
Theoremnnecl 6611 Closure of exponentiation of natural numbers. Proposition 8.17 of [TakeutiZaring] p. 63. (Contributed by NM, 24-Mar-2007.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( A  ^o  B )  e.  om )
 
Theoremnnacli 6612  om is closed under addition. Inference form of nnacl 6609. (Contributed by Scott Fenton, 20-Apr-2012.) (Revised by Mario Carneiro, 12-May-2012.)
 |-  A  e.  om   &    |-  B  e.  om   =>    |-  ( A  +o  B )  e.  om
 
Theoremnnmcli 6613  om is closed under multiplication. Inference form of nnmcl 6610. (Contributed by Scott Fenton, 20-Apr-2012.) (Revised by Mario Carneiro, 12-May-2012.)
 |-  A  e.  om   &    |-  B  e.  om   =>    |-  ( A  .o  B )  e.  om
 
Theoremnnarcl 6614 Reverse closure law for addition of natural numbers. Exercise 1 of [TakeutiZaring] p. 62 and its converse. (Contributed by NM, 12-Dec-2004.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  +o  B )  e. 
 om 
 <->  ( A  e.  om  /\  B  e.  om )
 ) )
 
Theoremnnacom 6615 Addition of natural numbers is commutative. Theorem 4K(2) of [Enderton] p. 81. (Contributed by NM, 6-May-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( A  +o  B )  =  ( B  +o  A ) )
 
Theoremnnaordi 6616 Ordering property of addition. Proposition 8.4 of [TakeutiZaring] p. 58, limited to natural numbers. (Contributed by NM, 3-Feb-1996.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  ( ( B  e.  om 
 /\  C  e.  om )  ->  ( A  e.  B  ->  ( C  +o  A )  e.  ( C  +o  B ) ) )
 
Theoremnnaord 6617 Ordering property of addition. Proposition 8.4 of [TakeutiZaring] p. 58, limited to natural numbers, and its converse. (Contributed by NM, 7-Mar-1996.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  ( ( A  e.  om 
 /\  B  e.  om  /\  C  e.  om )  ->  ( A  e.  B  <->  ( C  +o  A )  e.  ( C  +o  B ) ) )
 
Theoremnnaordr 6618 Ordering property of addition of natural numbers. (Contributed by NM, 9-Nov-2002.)
 |-  ( ( A  e.  om 
 /\  B  e.  om  /\  C  e.  om )  ->  ( A  e.  B  <->  ( A  +o  C )  e.  ( B  +o  C ) ) )
 
Theoremnnawordi 6619 Adding to both sides of an inequality in  om (Contributed by Scott Fenton, 16-Apr-2012.) (Revised by Mario Carneiro, 12-May-2012.)
 |-  ( ( A  e.  om 
 /\  B  e.  om  /\  C  e.  om )  ->  ( A  C_  B  ->  ( A  +o  C )  C_  ( B  +o  C ) ) )
 
Theoremnnaass 6620 Addition of natural numbers is associative. Theorem 4K(1) of [Enderton] p. 81. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  ( ( A  e.  om 
 /\  B  e.  om  /\  C  e.  om )  ->  ( ( A  +o  B )  +o  C )  =  ( A  +o  ( B  +o  C ) ) )
 
Theoremnndi 6621 Distributive law for natural numbers. Theorem 4K(3) of [Enderton] p. 81. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  ( ( A  e.  om 
 /\  B  e.  om  /\  C  e.  om )  ->  ( A  .o  ( B  +o  C ) )  =  ( ( A  .o  B )  +o  ( A  .o  C ) ) )
 
Theoremnnmass 6622 Multiplication of natural numbers is associative. Theorem 4K(4) of [Enderton] p. 81. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  ( ( A  e.  om 
 /\  B  e.  om  /\  C  e.  om )  ->  ( ( A  .o  B )  .o  C )  =  ( A  .o  ( B  .o  C ) ) )
 
Theoremnnmsucr 6623 Multiplication with successor. Exercise 16 of [Enderton] p. 82. (Contributed by NM, 21-Sep-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( suc  A  .o  B )  =  ( ( A  .o  B )  +o  B ) )
 
Theoremnnmcom 6624 Multiplication of natural numbers is commutative. Theorem 4K(5) of [Enderton] p. 81. (Contributed by NM, 21-Sep-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( A  .o  B )  =  ( B  .o  A ) )
 
Theoremnnaword 6625 Weak ordering property of addition. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  ( ( A  e.  om 
 /\  B  e.  om  /\  C  e.  om )  ->  ( A  C_  B  <->  ( C  +o  A ) 
 C_  ( C  +o  B ) ) )
 
Theoremnnacan 6626 Cancellation law for addition of natural numbers. (Contributed by NM, 27-Oct-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  ( ( A  e.  om 
 /\  B  e.  om  /\  C  e.  om )  ->  ( ( A  +o  B )  =  ( A  +o  C )  <->  B  =  C ) )
 
Theoremnnaword1 6627 Weak ordering property of addition. (Contributed by NM, 9-Nov-2002.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  A  C_  ( A  +o  B ) )
 
Theoremnnaword2 6628 Weak ordering property of addition. (Contributed by NM, 9-Nov-2002.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  A  C_  ( B  +o  A ) )
 
Theoremnnmordi 6629 Ordering property of multiplication. Half of Proposition 8.19 of [TakeutiZaring] p. 63, limited to natural numbers. (Contributed by NM, 18-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  ( ( ( B  e.  om  /\  C  e.  om )  /\  (/)  e.  C )  ->  ( A  e.  B  ->  ( C  .o  A )  e.  ( C  .o  B ) ) )
 
Theoremnnmord 6630 Ordering property of multiplication. Proposition 8.19 of [TakeutiZaring] p. 63, limited to natural numbers. (Contributed by NM, 22-Jan-1996.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  ( ( A  e.  om 
 /\  B  e.  om  /\  C  e.  om )  ->  ( ( A  e.  B  /\  (/)  e.  C )  <-> 
 ( C  .o  A )  e.  ( C  .o  B ) ) )
 
Theoremnnmword 6631 Weak ordering property of ordinal multiplication. (Contributed by Mario Carneiro, 17-Nov-2014.)
 |-  ( ( ( A  e.  om  /\  B  e.  om  /\  C  e.  om )  /\  (/)  e.  C )  ->  ( A  C_  B 
 <->  ( C  .o  A )  C_  ( C  .o  B ) ) )
 
Theoremnnmcan 6632 Cancellation law for multiplication of natural numbers. (Contributed by NM, 26-Oct-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  ( ( ( A  e.  om  /\  B  e.  om  /\  C  e.  om )  /\  (/)  e.  A )  ->  ( ( A  .o  B )  =  ( A  .o  C ) 
 <->  B  =  C ) )
 
Theoremnnmwordi 6633 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 6634 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 6635* 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 6636* 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 6637 One is a natural number. (Contributed by NM, 29-Oct-1995.)
 |- 
 1o  e.  om
 
Theorem2onn 6638 The ordinal 2 is a natural number. (Contributed by NM, 28-Sep-2004.)
 |- 
 2o  e.  om
 
Theorem3onn 6639 The ordinal 3 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |- 
 3o  e.  om
 
Theorem4onn 6640 The ordinal 4 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |- 
 4o  e.  om
 
Theoremoaabslem 6641 Lemma for oaabs 6642. (Contributed by NM, 9-Dec-2004.)
 |-  ( ( om  e.  On  /\  A  e.  om )  ->  ( A  +o  om )  =  om )
 
Theoremoaabs 6642 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 6643 The absorption law oaabs 6642 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 6644 Lemma for omabs 6645. (Contributed by Mario Carneiro, 30-May-2015.)
 |-  ( ( om  e.  On  /\  A  e.  om  /\  (/)  e.  A )  ->  ( A  .o  om )  =  om )
 
Theoremomabs 6645 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 6646 Multiply an element of  om by  1o. (Contributed by Mario Carneiro, 17-Nov-2014.)
 |-  ( A  e.  om  ->  ( A  .o  1o )  =  A )
 
Theoremnnm2 6647 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 6648 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 6649 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 6650* 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 6651* Lemma for omsmo 6652. (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 6652* 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 6653 Lemma for omopthi 6655. (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 6654 Lemma for omopthi 6655. (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 6655 An ordered pair theorem for  om. Theorem 17.3 of [Quine] p. 124. This proof is adapted from nn0opthi 11285. (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 6656 An ordered pair theorem for finite integers. Analagous to nn0opthi 11285. (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 ) ) )
 
2.4.27  Equivalence relations and classes
 
Syntaxwer 6657 Extend the definition of a wff to include the equivalence predicate.
 wff  R  Er  A
 
Syntaxcec 6658 Extend the definition of a class to include equivalence class.
 class  [ A ] R
 
Syntaxcqs 6659 Extend the definition of a class to include quotient set.
 class  ( A /. R )
 
Definitiondf-er 6660 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 6661 we derive a more typical definition. We show that an equivalence relation is reflexive, symmetric, and transitive in erref 6680, ersymb 6674, and ertr 6675. (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 6661* 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 6662 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 6661). 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 6663. (Contributed by NM, 23-Jul-1995.)
 |- 
 [ A ] R  =  ( R " { A } )
 
Theoremdfec2 6663* 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 6664 An equivalence class modulo a set is a set. (Contributed by NM, 24-Jul-1995.)
 |-  ( R  e.  B  ->  [ A ] R  e.  _V )
 
Theoremecexr 6665 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 6666* 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 6667 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 6668 Equality theorem for equivalence predicate. (Contributed by Mario Carneiro, 12-Aug-2015.)
 |-  ( A  =  B  ->  ( R  Er  A  <->  R  Er  B ) )
 
Theoremerrel 6669 An equivalence relation is a relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
 |-  ( R  Er  A  ->  Rel  R )
 
Theoremerdm 6670 The domain of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
 |-  ( R  Er  A  ->  dom  R  =  A )
 
Theoremercl 6671 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 6672 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 6673 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 6674 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 6675 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 6676 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 6677 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 6678 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 6679 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 6680 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 6681 The converse of an equivalence relation is itself. (Contributed by Mario Carneiro, 12-Aug-2015.)
 |-  ( R  Er  A  ->  `' R  =  R )
 
Theoremerrn 6682 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 6683 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 6684 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 6685 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 6686* 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 6687 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 6688* 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 6689* 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 6690* 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 6691* 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 6692* Lemma for eqer 6693. (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 6693* 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 6694 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 6695 The empty set is an equivalence relation on the empty set. (Contributed by Mario Carneiro, 5-Sep-2015.)
 |-  (/)  Er  (/)
 
Theoremeceq1 6696 Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.)
 |-  ( A  =  B  ->  [ A ] C  =  [ B ] C )
 
Theoremeceq2 6697 Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.)
 |-  ( A  =  B  ->  [ C ] A  =  [ C ] B )
 
Theoremelecg 6698 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 6699 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 6700 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 ) )
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