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	nnmcl 6501	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 )
Theorem	nnacli 6502	om is closed under addition. Inference form of nnacl 6500. (Contributed by Scott Fenton, 20-Apr-2012.)
|- A e. om   &    |- B e. om   =>    |- ( A +o B ) e. om
Theorem	nnmcli 6503	om is closed under multiplication. Inference form of nnmcl 6501. (Contributed by Scott Fenton, 20-Apr-2012.)
|- A e. om   &    |- B e. om   =>    |- ( A .o B ) e. om
Theorem	nnacom 6504	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 ) )
Theorem	nnaass 6505	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 ) ) )
Theorem	nndi 6506	Distributive law for natural numbers (left-distributivity). 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 ) ) )
Theorem	nnmass 6507	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 6508	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 6509	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 6510	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 6511	Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucelsucr 4522, but the forward direction, for all ordinals, implies excluded middle as seen as onsucelsucexmid 4544. (Contributed by Jim Kingdon, 25-Aug-2019.)
|- ( B e. om -> ( A e. B <-> suc A e. suc B ) )
Theorem	nnsucsssuc 6512	Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucsssucr 4523, but the forward direction, for all ordinals, implies excluded middle as seen as onsucsssucexmid 4541. (Contributed by Jim Kingdon, 25-Aug-2019.)
|- ( ( A e. om /\ B e. om ) -> ( A C_ B <-> suc A C_ suc B ) )
Theorem	nntri3or 6513	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 6514	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 6515	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 4524). The forward direction for all ordinals implies excluded middle (ordsucunielexmid 4545). (Contributed by Jim Kingdon, 13-Mar-2022.)
|- ( B e. om -> ( A e. U. B <-> suc A e. B ) )
Theorem	nntri1 6516	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 6517	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 6518	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 6519	Equality of natural numbers is decidable. Theorem 7.2.6 of [HoTT], p. (varies). For the specific case where B is zero, see nndceq0 4632. (Contributed by Jim Kingdon, 31-Aug-2019.)
|- ( ( A e. om /\ B e. om ) -> DECID A = B )
Theorem	nndcel 6520	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 6521	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 6522	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 6523	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 6524*	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 3753 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 6525*	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 6526*	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 6527	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 3753 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 6528	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 6529	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 6530	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 6531	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 6532	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 6533	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 6534	Weak ordering property of addition. (Contributed by NM, 9-Nov-2002.)
|- ( ( A e. om /\ B e. om ) -> A C_ ( B +o A ) )
Theorem	nnawordi 6535	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 6536	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 6537	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 6538	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 6539	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 6540	One is a natural number. (Contributed by NM, 29-Oct-1995.)
|- 1o e. om
Theorem	2onn 6541	The ordinal 2 is a natural number. (Contributed by NM, 28-Sep-2004.)
|- 2o e. om
Theorem	3onn 6542	The ordinal 3 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.)
|- 3o e. om
Theorem	4onn 6543	The ordinal 4 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.)
|- 4o e. om
Theorem	2ssom 6544	The ordinal 2 is included in the set of natural number ordinals. (Contributed by BJ, 5-Aug-2024.)
|- 2o C_ om
Theorem	nnm1 6545	Multiply an element of om by 1o. (Contributed by Mario Carneiro, 17-Nov-2014.)
|- ( A e. om -> ( A .o 1o ) = A )
Theorem	nnm2 6546	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 6547	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 6548*	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 6549*	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 6550	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 6551	Extend the definition of a wff to include the equivalence predicate.
wff R Er A
Syntax	cec 6552	Extend the definition of a class to include equivalence class.
class [ A ] R
Syntax	cqs 6553	Extend the definition of a class to include quotient set.
class ( A /. R )
Definition	df-er 6554	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 6555 we derive a more typical definition. We show that an equivalence relation is reflexive, symmetric, and transitive in erref 6574, ersymb 6568, and ertr 6569. (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 6555*	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 6556	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 6555). 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 6557. (Contributed by NM, 23-Jul-1995.)
|- [ A ] R = ( R " { A } )
Theorem	dfec2 6557*	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 6558	An equivalence class modulo a set is a set. (Contributed by NM, 24-Jul-1995.)
|- ( R e. B -> [ A ] R e. _V )
Theorem	ecexr 6559	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 6560*	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 6561	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 6562	Equality theorem for equivalence predicate. (Contributed by Mario Carneiro, 12-Aug-2015.)
|- ( A = B -> ( R Er A <-> R Er B ) )
Theorem	errel 6563	An equivalence relation is a relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
|- ( R Er A -> Rel R )
Theorem	erdm 6564	The domain of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
|- ( R Er A -> dom R = A )
Theorem	ercl 6565	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 6566	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 6567	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 6568	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 6569	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 6570	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 6571	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 6572	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 6573	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 6574	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 6575	The converse of an equivalence relation is itself. (Contributed by Mario Carneiro, 12-Aug-2015.)
|- ( R Er A -> `' R = R )
Theorem	errn 6576	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 6577	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 6578	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 6579	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 6580*	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 6581	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 6582*	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 6583*	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 6584*	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 6585*	Lemma for eqer 6586. (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 6586*	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 6587	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 6588	The empty set is an equivalence relation on the empty set. (Contributed by Mario Carneiro, 5-Sep-2015.)
|- (/) Er (/)
Theorem	eceq1 6589	Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.)
|- ( A = B -> [ A ] C = [ B ] C )
Theorem	eceq1d 6590	Equality theorem for equivalence class (deduction form). (Contributed by Jim Kingdon, 31-Dec-2019.)
|- ( ph -> A = B )   =>    |- ( ph -> [ A ] C = [ B ] C )
Theorem	eceq2 6591	Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.)
|- ( A = B -> [ C ] A = [ C ] B )
Theorem	elecg 6592	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 6593	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 6594	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 6595	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 6596*	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 6597	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 6598	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 6599	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 6600	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 )