P. 66 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 6501-6600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nnmass 6501	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 ) ) )
Theorem	nnmsucr 6502	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 ) )
Theorem	nnmcom 6503	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 ) )
Theorem	nndir 6504	Distributive law for natural numbers (right-distributivity). (Contributed by Jim Kingdon, 3-Dec-2019.)
|- ( ( A e. om /\ B e. om /\ C e. om ) -> ( ( A +o B ) .o C ) = ( ( A .o C ) +o ( B .o C ) ) )
Theorem	nnsucelsuc 6505	Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucelsucr 4519, but the forward direction, for all ordinals, implies excluded middle as seen as onsucelsucexmid 4541. (Contributed by Jim Kingdon, 25-Aug-2019.)
|- ( B e. om -> ( A e. B <-> suc A e. suc B ) )
Theorem	nnsucsssuc 6506	Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucsssucr 4520, but the forward direction, for all ordinals, implies excluded middle as seen as onsucsssucexmid 4538. (Contributed by Jim Kingdon, 25-Aug-2019.)
|- ( ( A e. om /\ B e. om ) -> ( A C_ B <-> suc A C_ suc B ) )
Theorem	nntri3or 6507	Trichotomy for natural numbers. (Contributed by Jim Kingdon, 25-Aug-2019.)
|- ( ( A e. om /\ B e. om ) -> ( A e. B \/ A = B \/ B e. A ) )
Theorem	nntri2 6508	A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 28-Aug-2019.)
|- ( ( A e. om /\ B e. om ) -> ( A e. B <-> -. ( A = B \/ B e. A ) ) )
Theorem	nnsucuniel 6509	Given an element A of the union of a natural number B, suc A is an element of B itself. The reverse direction holds for all ordinals (sucunielr 4521). The forward direction for all ordinals implies excluded middle (ordsucunielexmid 4542). (Contributed by Jim Kingdon, 13-Mar-2022.)
|- ( B e. om -> ( A e. U. B <-> suc A e. B ) )
Theorem	nntri1 6510	A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 28-Aug-2019.)
|- ( ( A e. om /\ B e. om ) -> ( A C_ B <-> -. B e. A ) )
Theorem	nntri3 6511	A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 15-May-2020.)
|- ( ( A e. om /\ B e. om ) -> ( A = B <-> ( -. A e. B /\ -. B e. A ) ) )
Theorem	nntri2or2 6512	A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 15-Sep-2021.)
|- ( ( A e. om /\ B e. om ) -> ( A C_ B \/ B C_ A ) )
Theorem	nndceq 6513	Equality of natural numbers is decidable. Theorem 7.2.6 of [HoTT], p. (varies). For the specific case where B is zero, see nndceq0 4629. (Contributed by Jim Kingdon, 31-Aug-2019.)
|- ( ( A e. om /\ B e. om ) -> DECID A = B )
Theorem	nndcel 6514	Set membership between two natural numbers is decidable. (Contributed by Jim Kingdon, 6-Sep-2019.)
|- ( ( A e. om /\ B e. om ) -> DECID A e. B )
Theorem	nnsseleq 6515	For natural numbers, inclusion is equivalent to membership or equality. (Contributed by Jim Kingdon, 16-Sep-2021.)
|- ( ( A e. om /\ B e. om ) -> ( A C_ B <-> ( A e. B \/ A = B ) ) )
Theorem	nnsssuc 6516	A natural number is a subset of another natural number if and only if it belongs to its successor. (Contributed by Jim Kingdon, 22-Jul-2023.)
|- ( ( A e. om /\ B e. om ) -> ( A C_ B <-> A e. suc B ) )
Theorem	nntr2 6517	Transitive law for natural numbers. (Contributed by Jim Kingdon, 22-Jul-2023.)
|- ( ( A e. om /\ C e. om ) -> ( ( A C_ B /\ B e. C ) -> A e. C ) )
Theorem	dcdifsnid 6518*	If we remove a single element from a set with decidable equality then put it back in, we end up with the original set. This strengthens difsnss 3750 from subset to equality but the proof relies on equality being decidable. (Contributed by Jim Kingdon, 17-Jun-2022.)
|- ( ( A. x e. A A. y e. A DECID x = y /\ B e. A ) -> ( ( A \ { B } ) u. { B } ) = A )
Theorem	fnsnsplitdc 6519*	Split a function into a single point and all the rest. (Contributed by Stefan O'Rear, 27-Feb-2015.) (Revised by Jim Kingdon, 29-Jan-2023.)
|- ( ( A. x e. A A. y e. A DECID x = y /\ F Fn A /\ X e. A ) -> F = ( ( F |` ( A \ { X } ) ) u. { <. X , ( F ` X ) >. } ) )
Theorem	funresdfunsndc 6520*	Restricting a function to a domain without one element of the domain of the function, and adding a pair of this element and the function value of the element results in the function itself, where equality is decidable. (Contributed by AV, 2-Dec-2018.) (Revised by Jim Kingdon, 30-Jan-2023.)
|- ( ( A. x e. dom F A. y e. dom FDECID x = y /\ Fun F /\ X e. dom F ) -> ( ( F |` ( _V \ { X } ) ) u. { <. X , ( F ` X ) >. } ) = F )
Theorem	nndifsnid 6521	If we remove a single element from a natural number then put it back in, we end up with the original natural number. This strengthens difsnss 3750 from subset to equality but the proof relies on equality being decidable. (Contributed by Jim Kingdon, 31-Aug-2021.)
|- ( ( A e. om /\ B e. A ) -> ( ( A \ { B } ) u. { B } ) = A )
Theorem	nnaordi 6522	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 ) ) )
Theorem	nnaord 6523	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 ) ) )
Theorem	nnaordr 6524	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 ) ) )
Theorem	nnaword 6525	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 ) ) )
Theorem	nnacan 6526	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 ) )
Theorem	nnaword1 6527	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 ) )
Theorem	nnaword2 6528	Weak ordering property of addition. (Contributed by NM, 9-Nov-2002.)
|- ( ( A e. om /\ B e. om ) -> A C_ ( B +o A ) )
Theorem	nnawordi 6529	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 ) ) )
Theorem	nnmordi 6530	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 ) ) )
Theorem	nnmord 6531	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 ) ) )
Theorem	nnmword 6532	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 ) ) )
Theorem	nnmcan 6533	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 ) )
Theorem	1onn 6534	One is a natural number. (Contributed by NM, 29-Oct-1995.)
|- 1o e. om
Theorem	2onn 6535	The ordinal 2 is a natural number. (Contributed by NM, 28-Sep-2004.)
|- 2o e. om
Theorem	3onn 6536	The ordinal 3 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.)
|- 3o e. om
Theorem	4onn 6537	The ordinal 4 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.)
|- 4o e. om
Theorem	2ssom 6538	The ordinal 2 is included in the set of natural number ordinals. (Contributed by BJ, 5-Aug-2024.)
|- 2o C_ om
Theorem	nnm1 6539	Multiply an element of om by 1o. (Contributed by Mario Carneiro, 17-Nov-2014.)
|- ( A e. om -> ( A .o 1o ) = A )
Theorem	nnm2 6540	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 ) )
Theorem	nn2m 6541	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 ) )
Theorem	nnaordex 6542*	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 ) ) )
Theorem	nnawordex 6543*	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 ) )
Theorem	nnm00 6544	The product of two natural numbers is zero iff at least one of them is zero. (Contributed by Jim Kingdon, 11-Nov-2004.)
|- ( ( A e. om /\ B e. om ) -> ( ( A .o B ) = (/) <-> ( A = (/) \/ B = (/) ) ) )
2.6.25  Equivalence relations and classes
Syntax	wer 6545	Extend the definition of a wff to include the equivalence predicate.
wff R Er A
Syntax	cec 6546	Extend the definition of a class to include equivalence class.
class [ A ] R
Syntax	cqs 6547	Extend the definition of a class to include quotient set.
class ( A /. R )
Definition	df-er 6548	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 6549 we derive a more typical definition. We show that an equivalence relation is reflexive, symmetric, and transitive in erref 6568, ersymb 6562, and ertr 6563. (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 ) )
Theorem	dfer2 6549*	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 ) ) ) )
Definition	df-ec 6550	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 6549). 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 6551. (Contributed by NM, 23-Jul-1995.)
|- [ A ] R = ( R " { A } )
Theorem	dfec2 6551*	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 } )
Theorem	ecexg 6552	An equivalence class modulo a set is a set. (Contributed by NM, 24-Jul-1995.)
|- ( R e. B -> [ A ] R e. _V )
Theorem	ecexr 6553	An inhabited equivalence class implies the representative is a set. (Contributed by Mario Carneiro, 9-Jul-2014.)
|- ( A e. [ B ] R -> B e. _V )
Definition	df-qs 6554*	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 }
Theorem	ereq1 6555	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 ) )
Theorem	ereq2 6556	Equality theorem for equivalence predicate. (Contributed by Mario Carneiro, 12-Aug-2015.)
|- ( A = B -> ( R Er A <-> R Er B ) )
Theorem	errel 6557	An equivalence relation is a relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
|- ( R Er A -> Rel R )
Theorem	erdm 6558	The domain of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
|- ( R Er A -> dom R = A )
Theorem	ercl 6559	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 )
Theorem	ersym 6560	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 )
Theorem	ercl2 6561	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 )
Theorem	ersymb 6562	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 ) )
Theorem	ertr 6563	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 ) )
Theorem	ertrd 6564	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 )
Theorem	ertr2d 6565	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 )
Theorem	ertr3d 6566	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 )
Theorem	ertr4d 6567	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 )
Theorem	erref 6568	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 )
Theorem	ercnv 6569	The converse of an equivalence relation is itself. (Contributed by Mario Carneiro, 12-Aug-2015.)
|- ( R Er A -> `' R = R )
Theorem	errn 6570	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 )
Theorem	erssxp 6571	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 ) )
Theorem	erex 6572	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 ) )
Theorem	erexb 6573	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 ) )
Theorem	iserd 6574*	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 )
Theorem	brdifun 6575	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 ) ) )
Theorem	swoer 6576*	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 )
Theorem	swoord1 6577*	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 ) )
Theorem	swoord2 6578*	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 ) )
Theorem	eqerlem 6579*	Lemma for eqer 6580. (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 )
Theorem	eqer 6580*	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
Theorem	ider 6581	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
Theorem	0er 6582	The empty set is an equivalence relation on the empty set. (Contributed by Mario Carneiro, 5-Sep-2015.)
|- (/) Er (/)
Theorem	eceq1 6583	Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.)
|- ( A = B -> [ A ] C = [ B ] C )
Theorem	eceq1d 6584	Equality theorem for equivalence class (deduction form). (Contributed by Jim Kingdon, 31-Dec-2019.)
|- ( ph -> A = B )   =>    |- ( ph -> [ A ] C = [ B ] C )
Theorem	eceq2 6585	Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.)
|- ( A = B -> [ C ] A = [ C ] B )
Theorem	elecg 6586	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 ) )
Theorem	elec 6587	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 )
Theorem	relelec 6588	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 ) )
Theorem	ecss 6589	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 )
Theorem	ecdmn0m 6590*	A representative of an inhabited equivalence class belongs to the domain of the equivalence relation. (Contributed by Jim Kingdon, 21-Aug-2019.)
|- ( A e. dom R <-> E. x x e. [ A ] R )
Theorem	ereldm 6591	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 ) )
Theorem	erth 6592	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 ) )
Theorem	erth2 6593	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 ) )
Theorem	erthi 6594	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 )
Theorem	ecidsn 6595	An equivalence class modulo the identity relation is a singleton. (Contributed by NM, 24-Oct-2004.)
|- [ A ] _I = { A }
Theorem	qseq1 6596	Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.)
|- ( A = B -> ( A /. C ) = ( B /. C ) )
Theorem	qseq2 6597	Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.)
|- ( A = B -> ( C /. A ) = ( C /. B ) )
Theorem	elqsg 6598*	Closed form of elqs 6599. (Contributed by Rodolfo Medina, 12-Oct-2010.)
|- ( B e. V -> ( B e. ( A /. R ) <-> E. x e. A B = [ x ] R ) )
Theorem	elqs 6599*	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 )
Theorem	elqsi 6600*	Membership in a quotient set. (Contributed by NM, 23-Jul-1995.)
|- ( B e. ( A /. R ) -> E. x e. A B = [ x ] R )