P. 68 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 6701-6800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nndir 6701	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 6702	Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucelsucr 4612, but the forward direction, for all ordinals, implies excluded middle as seen as onsucelsucexmid 4634. (Contributed by Jim Kingdon, 25-Aug-2019.)
|- ( B e. om -> ( A e. B <-> suc A e. suc B ) )
Theorem	nnsucsssuc 6703	Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucsssucr 4613, but the forward direction, for all ordinals, implies excluded middle as seen as onsucsssucexmid 4631. (Contributed by Jim Kingdon, 25-Aug-2019.)
|- ( ( A e. om /\ B e. om ) -> ( A C_ B <-> suc A C_ suc B ) )
Theorem	nntri3or 6704	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 6705	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 6706	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 4614). The forward direction for all ordinals implies excluded middle (ordsucunielexmid 4635). (Contributed by Jim Kingdon, 13-Mar-2022.)
|- ( B e. om -> ( A e. U. B <-> suc A e. B ) )
Theorem	nntri1 6707	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 6708	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 6709	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 6710	Equality of natural numbers is decidable. Theorem 7.2.6 of [HoTT], p. (varies). For the specific case where B is zero, see nndceq0 4722. (Contributed by Jim Kingdon, 31-Aug-2019.)
|- ( ( A e. om /\ B e. om ) -> DECID A = B )
Theorem	nndcel 6711	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 6712	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 6713	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 6714	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 6715*	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 3824 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 6716*	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 6717*	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 6718	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 3824 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 6719	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 6720	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 6721	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 6722	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 6723	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 6724	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 6725	Weak ordering property of addition. (Contributed by NM, 9-Nov-2002.)
|- ( ( A e. om /\ B e. om ) -> A C_ ( B +o A ) )
Theorem	nnawordi 6726	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 6727	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 6728	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 6729	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 6730	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 6731	One is a natural number. (Contributed by NM, 29-Oct-1995.)
|- 1o e. om
Theorem	2onn 6732	The ordinal 2 is a natural number. (Contributed by NM, 28-Sep-2004.)
|- 2o e. om
Theorem	3onn 6733	The ordinal 3 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.)
|- 3o e. om
Theorem	4onn 6734	The ordinal 4 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.)
|- 4o e. om
Theorem	2ssom 6735	The ordinal 2 is included in the set of natural number ordinals. (Contributed by BJ, 5-Aug-2024.)
|- 2o C_ om
Theorem	nnm1 6736	Multiply an element of om by 1o. (Contributed by Mario Carneiro, 17-Nov-2014.)
|- ( A e. om -> ( A .o 1o ) = A )
Theorem	nnm2 6737	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 6738	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 6739*	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 6740*	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 6741	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 = (/) ) ) )
2.6.26  Equivalence relations and classes
Syntax	wer 6742	Extend the definition of a wff to include the equivalence predicate.
wff R Er A
Syntax	cec 6743	Extend the definition of a class to include equivalence class.
class [ A ] R
Syntax	cqs 6744	Extend the definition of a class to include quotient set.
class ( A /. R )
Definition	df-er 6745	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 6746 we derive a more typical definition. We show that an equivalence relation is reflexive, symmetric, and transitive in erref 6765, ersymb 6759, and ertr 6760. (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 6746*	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 6747	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 6746). 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 6748. (Contributed by NM, 23-Jul-1995.)
|- [ A ] R = ( R " { A } )
Theorem	dfec2 6748*	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 6749	An equivalence class modulo a set is a set. (Contributed by NM, 24-Jul-1995.)
|- ( R e. B -> [ A ] R e. _V )
Theorem	ecexr 6750	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 6751*	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 6752	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 6753	Equality theorem for equivalence predicate. (Contributed by Mario Carneiro, 12-Aug-2015.)
|- ( A = B -> ( R Er A <-> R Er B ) )
Theorem	errel 6754	An equivalence relation is a relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
|- ( R Er A -> Rel R )
Theorem	erdm 6755	The domain of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
|- ( R Er A -> dom R = A )
Theorem	ercl 6756	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 6757	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 6758	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 6759	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 6760	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 6761	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 6762	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 6763	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 6764	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 6765	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 6766	The converse of an equivalence relation is itself. (Contributed by Mario Carneiro, 12-Aug-2015.)
|- ( R Er A -> `' R = R )
Theorem	errn 6767	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 6768	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 6769	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 6770	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 6771*	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 6772	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 6773*	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 6774*	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 6775*	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 6776*	Lemma for eqer 6777. (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 6777*	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 6778	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 6779	The empty set is an equivalence relation on the empty set. (Contributed by Mario Carneiro, 5-Sep-2015.)
|- (/) Er (/)
Theorem	eceq1 6780	Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.)
|- ( A = B -> [ A ] C = [ B ] C )
Theorem	eceq1d 6781	Equality theorem for equivalence class (deduction form). (Contributed by Jim Kingdon, 31-Dec-2019.)
|- ( ph -> A = B )   =>    |- ( ph -> [ A ] C = [ B ] C )
Theorem	eceq2 6782	Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.)
|- ( A = B -> [ C ] A = [ C ] B )
Theorem	eceq2i 6783	Equality theorem for the A-coset and B-coset of C, inference version. (Contributed by Peter Mazsa, 11-May-2021.)
|- A = B   =>    |- [ C ] A = [ C ] B
Theorem	eceq2d 6784	Equality theorem for the A-coset and B-coset of C, deduction version. (Contributed by Peter Mazsa, 23-Apr-2021.)
|- ( ph -> A = B )   =>    |- ( ph -> [ C ] A = [ C ] B )
Theorem	elecg 6785	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 6786	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 6787	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 6788	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 6789*	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 6790	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 6791	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 6792	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 6793	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 6794	An equivalence class modulo the identity relation is a singleton. (Contributed by NM, 24-Oct-2004.)
|- [ A ] _I = { A }
Theorem	qseq1 6795	Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.)
|- ( A = B -> ( A /. C ) = ( B /. C ) )
Theorem	qseq2 6796	Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.)
|- ( A = B -> ( C /. A ) = ( C /. B ) )
Theorem	elqsg 6797*	Closed form of elqs 6798. (Contributed by Rodolfo Medina, 12-Oct-2010.)
|- ( B e. V -> ( B e. ( A /. R ) <-> E. x e. A B = [ x ] R ) )
Theorem	elqs 6798*	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 6799*	Membership in a quotient set. (Contributed by NM, 23-Jul-1995.)
|- ( B e. ( A /. R ) -> E. x e. A B = [ x ] R )
Theorem	ecelqsg 6800	Membership of an equivalence class in a quotient set. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 9-Jul-2014.)
|- ( ( R e. V /\ B e. A ) -> [ B ] R e. ( A /. R ) )