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Theorem List for Metamath Proof Explorer - 6501-6600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnnacom 6501 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.)

Theoremnnaordi 6502 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.)

Theoremnnaord 6503 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.)

Theoremnnaordr 6504 Ordering property of addition of natural numbers. (Contributed by NM, 9-Nov-2002.)

Theoremnnawordi 6505 Adding to both sides of an inequality in (Contributed by Scott Fenton, 16-Apr-2012.) (Revised by Mario Carneiro, 12-May-2012.)

Theoremnnaass 6506 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.)

Theoremnndi 6507 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.)

Theoremnnmass 6508 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.)

Theoremnnmsucr 6509 Multiplication with successor. Exercise 16 of [Enderton] p. 82. (Contributed by NM, 21-Sep-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremnnmcom 6510 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.)

Theoremnnaword 6511 Weak ordering property of addition. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremnnacan 6512 Cancellation law for addition of natural numbers. (Contributed by NM, 27-Oct-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremnnaword1 6513 Weak ordering property of addition. (Contributed by NM, 9-Nov-2002.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremnnaword2 6514 Weak ordering property of addition. (Contributed by NM, 9-Nov-2002.)

Theoremnnmordi 6515 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.)

Theoremnnmord 6516 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.)

Theoremnnmword 6517 Weak ordering property of ordinal multiplication. (Contributed by Mario Carneiro, 17-Nov-2014.)

Theoremnnmcan 6518 Cancellation law for multiplication of natural numbers. (Contributed by NM, 26-Oct-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremnnmwordi 6519 Weak ordering property of multiplication. (Contributed by Mario Carneiro, 17-Nov-2014.)

Theoremnnmwordri 6520 Weak ordering property of ordinal multiplication. Proposition 8.21 of [TakeutiZaring] p. 63, limited to natural numbers. (Contributed by Mario Carneiro, 17-Nov-2014.)

Theoremnnawordex 6521* Equivalence for weak ordering of natural numbers. (Contributed by NM, 8-Nov-2002.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremnnaordex 6522* Equivalence for ordering. Compare Exercise 23 of [Enderton] p. 88. (Contributed by NM, 5-Dec-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theorem1onn 6523 One is a natural number. (Contributed by NM, 29-Oct-1995.)

Theorem2onn 6524 The ordinal 2 is a natural number. (Contributed by NM, 28-Sep-2004.)

Theorem3onn 6525 The ordinal 3 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.)

Theorem4onn 6526 The ordinal 4 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.)

Theoremoaabslem 6527 Lemma for oaabs 6528. (Contributed by NM, 9-Dec-2004.)

Theoremoaabs 6528 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.)

Theoremoaabs2 6529 The absorption law oaabs 6528 is also a property of higher powers of . (Contributed by Mario Carneiro, 29-May-2015.)

Theoremomabslem 6530 Lemma for omabs 6531. (Contributed by Mario Carneiro, 30-May-2015.)

Theoremomabs 6531 Ordinal multiplication is also absorbed by powers of . (Contributed by Mario Carneiro, 30-May-2015.)

Theoremnnm1 6532 Multiply an element of by . (Contributed by Mario Carneiro, 17-Nov-2014.)

Theoremnnm2 6533 Multiply an element of by (Contributed by Scott Fenton, 18-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremnn2m 6534 Multiply an element of by (Contributed by Scott Fenton, 16-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremnnneo 6535 If an natural number is even, its successor is odd. (Contributed by Mario Carneiro, 16-Nov-2014.)

Theoremnneob 6536* A natural number is even iff its successor is odd. (Contributed by NM, 26-Jan-2006.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremomsmolem 6537* Lemma for omsmo 6538. (Contributed by NM, 30-Nov-2003.) (Revised by David Abernethy, 1-Jan-2014.)

Theoremomsmo 6538* 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.)

Theoremomopthlem1 6539 Lemma for omopthi 6541. (Contributed by Scott Fenton, 18-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremomopthlem2 6540 Lemma for omopthi 6541. (Contributed by Scott Fenton, 16-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremomopthi 6541 An ordered pair theorem for . Theorem 17.3 of [Quine] p. 124. This proof is adapted from nn0opthi 11163. (Contributed by Scott Fenton, 16-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremomopth 6542 An ordered pair theorem for finite integers. Analagous to nn0opthi 11163. (Contributed by Scott Fenton, 1-May-2012.) (Revised by Mario Carneiro, 12-May-2012.)

2.4.25  Equivalence relations and classes

Syntaxwer 6543 Extend the definition of a wff to include the equivalence predicate.

Syntaxcec 6544 Extend the definition of a class to include equivalence class.

Syntaxcqs 6545 Extend the definition of a class to include quotient set.

Definitiondf-er 6546 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 6547 we derive a more typical definition. We show that an equivalence relation is reflexive, symmetric, and transitive in erref 6566, ersymb 6560, and ertr 6561. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 2-Nov-2015.)

Theoremdfer2 6547* Alternate definition of equivalence predicate. (Contributed by NM, 3-Jan-1997.) (Revised by Mario Carneiro, 12-Aug-2015.)

Definitiondf-ec 6548 Define the -coset of . Exercise 35 of [Enderton] p. 61. This is called the equivalence class of modulo when is an equivalence relation (i.e. when ; see dfer2 6547). In this case, is a representative (member) of the equivalence class , which contains all sets that are equivalent to . Definition of [Enderton] p. 57 uses the notation (subscript) , although we simply follow the brackets by since we don't have subscripted expressions. For an alternate definition, see dfec2 6549. (Contributed by NM, 23-Jul-1995.)

Theoremdfec2 6549* Alternate definition of -coset of . Definition 34 of [Suppes] p. 81. (Contributed by NM, 3-Jan-1997.) (Proof shortened by Mario Carneiro, 9-Jul-2014.)

Theoremecexg 6550 An equivalence class modulo a set is a set. (Contributed by NM, 24-Jul-1995.)

Theoremecexr 6551 A nonempty equivalence class implies the representative is a set. (Contributed by Mario Carneiro, 9-Jul-2014.)

Definitiondf-qs 6552* Define quotient set. is usually an equivalence relation. Definition of [Enderton] p. 58. (Contributed by NM, 23-Jul-1995.)

Theoremereq1 6553 Equality theorem for equivalence predicate. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremereq2 6554 Equality theorem for equivalence predicate. (Contributed by Mario Carneiro, 12-Aug-2015.)

Theoremerrel 6555 An equivalence relation is a relation. (Contributed by Mario Carneiro, 12-Aug-2015.)

Theoremerdm 6556 The domain of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)

Theoremercl 6557 Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)

Theoremersym 6558 An equivalence relation is symmetric. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremercl2 6559 Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)

Theoremersymb 6560 An equivalence relation is symmetric. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremertr 6561 An equivalence relation is transitive. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremertrd 6562 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)

Theoremertr2d 6563 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)

Theoremertr3d 6564 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)

Theoremertr4d 6565 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)

Theoremerref 6566 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.)

Theoremercnv 6567 The converse of an equivalence relation is itself. (Contributed by Mario Carneiro, 12-Aug-2015.)

Theoremerrn 6568 The range and domain of an equivalence relation are equal. (Contributed by Rodolfo Medina, 11-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremerssxp 6569 An equivalence relation is a subset of the cartesian product of the field. (Contributed by Mario Carneiro, 12-Aug-2015.)

Theoremerex 6570 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.)

Theoremerexb 6571 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.)

Theoremiserd 6572* 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.)

Theorembrdifun 6573 Evaluate the incomparability relation. (Contributed by Mario Carneiro, 9-Jul-2014.)

Theoremswoer 6574* 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.)

Theoremswoord1 6575* The incomparability equivalence relation is compatible with the original order. (Contributed by Mario Carneiro, 31-Dec-2014.)

Theoremswoord2 6576* The incomparability equivalence relation is compatible with the original order. (Contributed by Mario Carneiro, 31-Dec-2014.)

Theoremswoso 6577* 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.)

Theoremeqerlem 6578* Lemma for eqer 6579. (Contributed by NM, 17-Mar-2008.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)

Theoremeqer 6579* Equivalence relation involving equality of dependent classes and . (Contributed by NM, 17-Mar-2008.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremider 6580 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.)

Theorem0er 6581 The empty set is an equivalence relation on the empty set. (Contributed by Mario Carneiro, 5-Sep-2015.)

Theoremeceq1 6582 Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.)

Theoremeceq2 6583 Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.)

Theoremelecg 6584 Membership in an equivalence class. Theorem 72 of [Suppes] p. 82. (Contributed by Mario Carneiro, 9-Jul-2014.)

Theoremelec 6585 Membership in an equivalence class. Theorem 72 of [Suppes] p. 82. (Contributed by NM, 23-Jul-1995.)

Theoremrelelec 6586 Membership in an equivalence class when is a relation. (Contributed by Mario Carneiro, 11-Sep-2015.)

Theoremecss 6587 An equivalence class is a subset of the domain. (Contributed by NM, 6-Aug-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremecdmn0 6588 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.)

Theoremereldm 6589 Equality of equivalence classes implies equivalence of domain membership. (Contributed by NM, 28-Jan-1996.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremerth 6590 Basic property of equivalence relations. Theorem 73 of [Suppes] p. 82. (Contributed by NM, 23-Jul-1995.) (Revised by Mario Carneiro, 6-Jul-2015.)

Theoremerth2 6591 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.)

Theoremerthi 6592 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.)

Theoremerdisj 6593 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.)

Theoremecidsn 6594 An equivalence class modulo the identity relation is a singleton. (Contributed by NM, 24-Oct-2004.)

Theoremqseq1 6595 Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.)

Theoremqseq2 6596 Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.)

Theoremelqsg 6597* Closed form of elqs 6598. (Contributed by Rodolfo Medina, 12-Oct-2010.)

Theoremelqs 6598* Membership in a quotient set. (Contributed by NM, 23-Jul-1995.)

Theoremelqsi 6599* Membership in a quotient set. (Contributed by NM, 23-Jul-1995.)

Theoremecelqsg 6600 Membership of an equivalence class in a quotient set. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 9-Jul-2014.)

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