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Theorem List for Metamath Proof Explorer - 6601-6700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremoeordi 6601 Ordering law for ordinal exponentiation. Proposition 8.33 of [TakeutiZaring] p. 67. (Contributed by NM, 5-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.)
 |-  ( ( B  e.  On  /\  C  e.  ( On  \  2o ) ) 
 ->  ( A  e.  B  ->  ( C  ^o  A )  e.  ( C  ^o  B ) ) )
 
Theoremoeord 6602 Ordering property of ordinal exponentiation. Corollary 8.34 of [TakeutiZaring] p. 68 and its converse. (Contributed by NM, 6-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.)
 |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  ( A  e.  B  <->  ( C  ^o  A )  e.  ( C  ^o  B ) ) )
 
Theoremoecan 6603 Left cancellation law for ordinal exponentiation. (Contributed by NM, 6-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.)
 |-  ( ( A  e.  ( On  \  2o )  /\  B  e.  On  /\  C  e.  On )  ->  ( ( A  ^o  B )  =  ( A  ^o  C )  <->  B  =  C ) )
 
Theoremoeword 6604 Weak ordering property of ordinal exponentiation. (Contributed by NM, 6-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.)
 |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  ( A  C_  B  <->  ( C  ^o  A )  C_  ( C 
 ^o  B ) ) )
 
Theoremoewordi 6605 Weak ordering property of ordinal exponentiation. (Contributed by NM, 6-Jan-2005.)
 |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  C )  ->  ( A  C_  B  ->  ( C  ^o  A )  C_  ( C 
 ^o  B ) ) )
 
Theoremoewordri 6606 Weak ordering property of ordinal exponentiation. Proposition 8.35 of [TakeutiZaring] p. 68. (Contributed by NM, 6-Jan-2005.)
 |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( A  e.  B  ->  ( A  ^o  C )  C_  ( B 
 ^o  C ) ) )
 
Theoremoeworde 6607 Ordinal exponentiation compared to its exponent. Proposition 8.37 of [TakeutiZaring] p. 68. (Contributed by NM, 7-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.)
 |-  ( ( A  e.  ( On  \  2o )  /\  B  e.  On )  ->  B  C_  ( A  ^o  B ) )
 
Theoremoeordsuc 6608 Ordering property of ordinal exponentiation with a successor exponent. Corollary 8.36 of [TakeutiZaring] p. 68. (Contributed by NM, 7-Jan-2005.)
 |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( A  e.  B  ->  ( A  ^o  suc 
 C )  e.  ( B  ^o  suc  C )
 ) )
 
Theoremoelim2 6609* Ordinal exponentiation with a limit exponent. Part of Exercise 4.36 of [Mendelson] p. 250. (Contributed by NM, 6-Jan-2005.)
 |-  ( ( A  e.  On  /\  ( B  e.  C  /\  Lim  B )
 )  ->  ( A  ^o  B )  =  U_ x  e.  ( B  \  1o ) ( A 
 ^o  x ) )
 
Theoremoeoalem 6610 Lemma for oeoa 6611. (Contributed by Eric Schmidt, 26-May-2009.)
 |-  A  e.  On   &    |-  (/)  e.  A   &    |-  B  e.  On   =>    |-  ( C  e.  On  ->  ( A  ^o  ( B  +o  C ) )  =  ( ( A 
 ^o  B )  .o  ( A  ^o  C ) ) )
 
Theoremoeoa 6611 Sum of exponents law for ordinal exponentiation. Theorem 8R of [Enderton] p. 238. Also Proposition 8.41 of [TakeutiZaring] p. 69. (Contributed by Eric Schmidt, 26-May-2009.)
 |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( A  ^o  ( B  +o  C ) )  =  ( ( A 
 ^o  B )  .o  ( A  ^o  C ) ) )
 
Theoremoeoelem 6612 Lemma for oeoe 6613. (Contributed by Eric Schmidt, 26-May-2009.)
 |-  A  e.  On   &    |-  (/)  e.  A   =>    |-  (
 ( B  e.  On  /\  C  e.  On )  ->  ( ( A  ^o  B )  ^o  C )  =  ( A  ^o  ( B  .o  C ) ) )
 
Theoremoeoe 6613 Product of exponents law for ordinal exponentiation. Theorem 8S of [Enderton] p. 238. Also Proposition 8.42 of [TakeutiZaring] p. 70. (Contributed by Eric Schmidt, 26-May-2009.)
 |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( ( A  ^o  B )  ^o  C )  =  ( A  ^o  ( B  .o  C ) ) )
 
Theoremoelimcl 6614 The ordinal exponential with a limit ordinal is a limit ordinal. (Contributed by Mario Carneiro, 29-May-2015.)
 |-  ( ( A  e.  ( On  \  2o )  /\  ( B  e.  C  /\  Lim  B ) ) 
 ->  Lim  ( A  ^o  B ) )
 
Theoremoeeulem 6615* Lemma for oeeu 6617. (Contributed by Mario Carneiro, 28-Feb-2013.)
 |-  X  =  U. |^| { x  e.  On  |  B  e.  ( A  ^o  x ) }   =>    |-  ( ( A  e.  ( On  \  2o )  /\  B  e.  ( On  \  1o )
 )  ->  ( X  e.  On  /\  ( A 
 ^o  X )  C_  B  /\  B  e.  ( A  ^o  suc  X )
 ) )
 
Theoremoeeui 6616* The division algorithm for ordinal exponentiation. (This version of oeeu 6617 gives an explicit expression for the unique solution of the equation, in terms of the solution  P to omeu 6599.) (Contributed by Mario Carneiro, 25-May-2015.)
 |-  X  =  U. |^| { x  e.  On  |  B  e.  ( A  ^o  x ) }   &    |-  P  =  ( iota w E. y  e.  On  E. z  e.  ( A  ^o  X ) ( w  = 
 <. y ,  z >.  /\  ( ( ( A 
 ^o  X )  .o  y )  +o  z
 )  =  B ) )   &    |-  Y  =  ( 1st `  P )   &    |-  Z  =  ( 2nd `  P )   =>    |-  ( ( A  e.  ( On  \  2o )  /\  B  e.  ( On  \  1o ) )  ->  ( ( ( C  e.  On  /\  D  e.  ( A  \  1o )  /\  E  e.  ( A  ^o  C ) ) 
 /\  ( ( ( A  ^o  C )  .o  D )  +o  E )  =  B ) 
 <->  ( C  =  X  /\  D  =  Y  /\  E  =  Z )
 ) )
 
Theoremoeeu 6617* 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 6618 Addition with zero. Theorem 4I(A1) of [Enderton] p. 79. (Contributed by NM, 20-Sep-1995.)
 |-  ( A  e.  om  ->  ( A  +o  (/) )  =  A )
 
Theoremnnm0 6619 Multiplication with zero. Theorem 4J(A1) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.)
 |-  ( A  e.  om  ->  ( A  .o  (/) )  =  (/) )
 
Theoremnnasuc 6620 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 6621 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 6622 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 6623 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 6553) so that we can avoid ax-rep 4147, 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 6624 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 6625 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 6626 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 6627 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 6628  om is closed under addition. Inference form of nnacl 6625. (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 6629  om is closed under multiplication. Inference form of nnmcl 6626. (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 6630 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 6631 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 6632 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 6633 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 6634 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 6635 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 6636 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 6637 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 6638 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 6639 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 6640 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 6641 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 6642 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 6643 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 6644 Weak ordering property of addition. (Contributed by NM, 9-Nov-2002.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  A  C_  ( B  +o  A ) )
 
Theoremnnmordi 6645 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 6646 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 6647 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 6648 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 6649 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 6650 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 6651* 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 6652* 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 6653 One is a natural number. (Contributed by NM, 29-Oct-1995.)
 |- 
 1o  e.  om
 
Theorem2onn 6654 The ordinal 2 is a natural number. (Contributed by NM, 28-Sep-2004.)
 |- 
 2o  e.  om
 
Theorem3onn 6655 The ordinal 3 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |- 
 3o  e.  om
 
Theorem4onn 6656 The ordinal 4 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |- 
 4o  e.  om
 
Theoremoaabslem 6657 Lemma for oaabs 6658. (Contributed by NM, 9-Dec-2004.)
 |-  ( ( om  e.  On  /\  A  e.  om )  ->  ( A  +o  om )  =  om )
 
Theoremoaabs 6658 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 6659 The absorption law oaabs 6658 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 6660 Lemma for omabs 6661. (Contributed by Mario Carneiro, 30-May-2015.)
 |-  ( ( om  e.  On  /\  A  e.  om  /\  (/)  e.  A )  ->  ( A  .o  om )  =  om )
 
Theoremomabs 6661 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 6662 Multiply an element of  om by  1o. (Contributed by Mario Carneiro, 17-Nov-2014.)
 |-  ( A  e.  om  ->  ( A  .o  1o )  =  A )
 
Theoremnnm2 6663 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 6664 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 6665 If a 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 6666* 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 6667* Lemma for omsmo 6668. (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 6668* 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 6669 Lemma for omopthi 6671. (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 6670 Lemma for omopthi 6671. (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 6671 An ordered pair theorem for  om. Theorem 17.3 of [Quine] p. 124. This proof is adapted from nn0opthi 11301. (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 6672 An ordered pair theorem for finite integers. Analagous to nn0opthi 11301. (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 6673 Extend the definition of a wff to include the equivalence predicate.
 wff  R  Er  A
 
Syntaxcec 6674 Extend the definition of a class to include equivalence class.
 class  [ A ] R
 
Syntaxcqs 6675 Extend the definition of a class to include quotient set.
 class  ( A /. R )
 
Definitiondf-er 6676 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 6677 we derive a more typical definition. We show that an equivalence relation is reflexive, symmetric, and transitive in erref 6696, ersymb 6690, and ertr 6691. (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 6677* 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 6678 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 6677). 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 6679. (Contributed by NM, 23-Jul-1995.)
 |- 
 [ A ] R  =  ( R " { A } )
 
Theoremdfec2 6679* 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 6680 An equivalence class modulo a set is a set. (Contributed by NM, 24-Jul-1995.)
 |-  ( R  e.  B  ->  [ A ] R  e.  _V )
 
Theoremecexr 6681 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 6682* 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 6683 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 6684 Equality theorem for equivalence predicate. (Contributed by Mario Carneiro, 12-Aug-2015.)
 |-  ( A  =  B  ->  ( R  Er  A  <->  R  Er  B ) )
 
Theoremerrel 6685 An equivalence relation is a relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
 |-  ( R  Er  A  ->  Rel  R )
 
Theoremerdm 6686 The domain of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
 |-  ( R  Er  A  ->  dom  R  =  A )
 
Theoremercl 6687 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 6688 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 6689 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 6690 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 6691 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 6692 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 6693 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 6694 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 6695 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 6696 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 6697 The converse of an equivalence relation is itself. (Contributed by Mario Carneiro, 12-Aug-2015.)
 |-  ( R  Er  A  ->  `' R  =  R )
 
Theoremerrn 6698 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 6699 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 6700 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 )
 )
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