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	el2oss1o 6501	Being an element of ordinal two implies being a subset of ordinal one. The converse is equivalent to excluded middle by ss1oel2o 15638. (Contributed by Jim Kingdon, 8-Aug-2022.)
|- ( A e. 2o -> A C_ 1o )
Theorem	oafnex 6502	The characteristic function for ordinal addition is defined everywhere. (Contributed by Jim Kingdon, 27-Jul-2019.)
|- ( x e. _V |-> suc x ) Fn _V
Theorem	sucinc 6503*	Successor is increasing. (Contributed by Jim Kingdon, 25-Jun-2019.)
|- F = ( z e. _V |-> suc z )   =>    |- A. x x C_ ( F ` x )
Theorem	sucinc2 6504*	Successor is increasing. (Contributed by Jim Kingdon, 14-Jul-2019.)
|- F = ( z e. _V |-> suc z )   =>    |- ( ( B e. On /\ A e. B ) -> ( F ` A ) C_ ( F ` B ) )
Theorem	fnoa 6505	Functionality and domain of ordinal addition. (Contributed by NM, 26-Aug-1995.) (Proof shortened by Mario Carneiro, 3-Jul-2019.)
|- +o Fn ( On X. On )
Theorem	oaexg 6506	Ordinal addition is a set. (Contributed by Mario Carneiro, 3-Jul-2019.)
|- ( ( A e. V /\ B e. W ) -> ( A +o B ) e. _V )
Theorem	omfnex 6507*	The characteristic function for ordinal multiplication is defined everywhere. (Contributed by Jim Kingdon, 23-Aug-2019.)
|- ( A e. V -> ( x e. _V |-> ( x +o A ) ) Fn _V )
Theorem	fnom 6508	Functionality and domain of ordinal multiplication. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 3-Jul-2019.)
|- .o Fn ( On X. On )
Theorem	omexg 6509	Ordinal multiplication is a set. (Contributed by Mario Carneiro, 3-Jul-2019.)
|- ( ( A e. V /\ B e. W ) -> ( A .o B ) e. _V )
Theorem	fnoei 6510	Functionality and domain of ordinal exponentiation. (Contributed by Mario Carneiro, 29-May-2015.) (Revised by Mario Carneiro, 3-Jul-2019.)
|- ↑o Fn ( On X. On )
Theorem	oeiexg 6511	Ordinal exponentiation is a set. (Contributed by Mario Carneiro, 3-Jul-2019.)
|- ( ( A e. V /\ B e. W ) -> ( A ↑o B ) e. _V )
Theorem	oav 6512*	Value of ordinal addition. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
|- ( ( A e. On /\ B e. On ) -> ( A +o B ) = ( rec ( ( x e. _V |-> suc x ) , A ) ` B ) )
Theorem	omv 6513*	Value of ordinal multiplication. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 23-Aug-2014.)
|- ( ( A e. On /\ B e. On ) -> ( A .o B ) = ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` B ) )
Theorem	oeiv 6514*	Value of ordinal exponentiation. (Contributed by Jim Kingdon, 9-Jul-2019.)
|- ( ( A e. On /\ B e. On ) -> ( A ↑o B ) = ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) )
Theorem	oa0 6515	Addition with zero. Proposition 8.3 of [TakeutiZaring] p. 57. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
|- ( A e. On -> ( A +o (/) ) = A )
Theorem	om0 6516	Ordinal multiplication with zero. Definition 8.15 of [TakeutiZaring] p. 62. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
|- ( A e. On -> ( A .o (/) ) = (/) )
Theorem	oei0 6517	Ordinal exponentiation with zero exponent. Definition 8.30 of [TakeutiZaring] p. 67. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
|- ( A e. On -> ( A ↑o (/) ) = 1o )
Theorem	oacl 6518	Closure law for ordinal addition. Proposition 8.2 of [TakeutiZaring] p. 57. (Contributed by NM, 5-May-1995.) (Constructive proof by Jim Kingdon, 26-Jul-2019.)
|- ( ( A e. On /\ B e. On ) -> ( A +o B ) e. On )
Theorem	omcl 6519	Closure law for ordinal multiplication. Proposition 8.16 of [TakeutiZaring] p. 57. (Contributed by NM, 3-Aug-2004.) (Constructive proof by Jim Kingdon, 26-Jul-2019.)
|- ( ( A e. On /\ B e. On ) -> ( A .o B ) e. On )
Theorem	oeicl 6520	Closure law for ordinal exponentiation. (Contributed by Jim Kingdon, 26-Jul-2019.)
|- ( ( A e. On /\ B e. On ) -> ( A ↑o B ) e. On )
Theorem	oav2 6521*	Value of ordinal addition. (Contributed by Mario Carneiro and Jim Kingdon, 12-Aug-2019.)
|- ( ( A e. On /\ B e. On ) -> ( A +o B ) = ( A u. U_ x e. B suc ( A +o x ) ) )
Theorem	oasuc 6522	Addition with successor. Definition 8.1 of [TakeutiZaring] p. 56. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
|- ( ( A e. On /\ B e. On ) -> ( A +o suc B ) = suc ( A +o B ) )
Theorem	omv2 6523*	Value of ordinal multiplication. (Contributed by Jim Kingdon, 23-Aug-2019.)
|- ( ( A e. On /\ B e. On ) -> ( A .o B ) = U_ x e. B ( ( A .o x ) +o A ) )
Theorem	onasuc 6524	Addition with successor. Theorem 4I(A2) of [Enderton] p. 79. (Contributed by Mario Carneiro, 16-Nov-2014.)
|- ( ( A e. On /\ B e. om ) -> ( A +o suc B ) = suc ( A +o B ) )
Theorem	oa1suc 6525	Addition with 1 is same as successor. Proposition 4.34(a) of [Mendelson] p. 266. (Contributed by NM, 29-Oct-1995.) (Revised by Mario Carneiro, 16-Nov-2014.)
|- ( A e. On -> ( A +o 1o ) = suc A )
Theorem	o1p1e2 6526	1 + 1 = 2 for ordinal numbers. (Contributed by NM, 18-Feb-2004.)
|- ( 1o +o 1o ) = 2o
Theorem	oawordi 6527	Weak ordering property of ordinal addition. (Contributed by Jim Kingdon, 27-Jul-2019.)
|- ( ( A e. On /\ B e. On /\ C e. On ) -> ( A C_ B -> ( C +o A ) C_ ( C +o B ) ) )
Theorem	oawordriexmid 6528*	A weak ordering property of ordinal addition which implies excluded middle. The property is proposition 8.7 of [TakeutiZaring] p. 59. Compare with oawordi 6527. (Contributed by Jim Kingdon, 15-May-2022.)
|- ( ( a e. On /\ b e. On /\ c e. On ) -> ( a C_ b -> ( a +o c ) C_ ( b +o c ) ) )   =>    |- ( ph \/ -. ph )
Theorem	oaword1 6529	An ordinal is less than or equal to its sum with another. Part of Exercise 5 of [TakeutiZaring] p. 62. (Contributed by NM, 6-Dec-2004.)
|- ( ( A e. On /\ B e. On ) -> A C_ ( A +o B ) )
Theorem	omsuc 6530	Multiplication with successor. Definition 8.15 of [TakeutiZaring] p. 62. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
|- ( ( A e. On /\ B e. On ) -> ( A .o suc B ) = ( ( A .o B ) +o A ) )
Theorem	onmsuc 6531	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. On /\ B e. om ) -> ( A .o suc B ) = ( ( A .o B ) +o A ) )
2.6.24  Natural number arithmetic
Theorem	nna0 6532	Addition with zero. Theorem 4I(A1) of [Enderton] p. 79. (Contributed by NM, 20-Sep-1995.)
|- ( A e. om -> ( A +o (/) ) = A )
Theorem	nnm0 6533	Multiplication with zero. Theorem 4J(A1) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.)
|- ( A e. om -> ( A .o (/) ) = (/) )
Theorem	nnasuc 6534	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 ) )
Theorem	nnmsuc 6535	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 ) )
Theorem	nna0r 6536	Addition to zero. Remark in proof of Theorem 4K(2) of [Enderton] p. 81. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
|- ( A e. om -> ( (/) +o A ) = A )
Theorem	nnm0r 6537	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 ) = (/) )
Theorem	nnacl 6538	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 )
Theorem	nnmcl 6539	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 6540	om is closed under addition. Inference form of nnacl 6538. (Contributed by Scott Fenton, 20-Apr-2012.)
|- A e. om   &    |- B e. om   =>    |- ( A +o B ) e. om
Theorem	nnmcli 6541	om is closed under multiplication. Inference form of nnmcl 6539. (Contributed by Scott Fenton, 20-Apr-2012.)
|- A e. om   &    |- B e. om   =>    |- ( A .o B ) e. om
Theorem	nnacom 6542	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 6543	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 6544	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 6545	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 6546	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 6547	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 6548	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 6549	Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucelsucr 4544, but the forward direction, for all ordinals, implies excluded middle as seen as onsucelsucexmid 4566. (Contributed by Jim Kingdon, 25-Aug-2019.)
|- ( B e. om -> ( A e. B <-> suc A e. suc B ) )
Theorem	nnsucsssuc 6550	Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucsssucr 4545, but the forward direction, for all ordinals, implies excluded middle as seen as onsucsssucexmid 4563. (Contributed by Jim Kingdon, 25-Aug-2019.)
|- ( ( A e. om /\ B e. om ) -> ( A C_ B <-> suc A C_ suc B ) )
Theorem	nntri3or 6551	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 6552	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 6553	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 4546). The forward direction for all ordinals implies excluded middle (ordsucunielexmid 4567). (Contributed by Jim Kingdon, 13-Mar-2022.)
|- ( B e. om -> ( A e. U. B <-> suc A e. B ) )
Theorem	nntri1 6554	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 6555	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 6556	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 6557	Equality of natural numbers is decidable. Theorem 7.2.6 of [HoTT], p. (varies). For the specific case where B is zero, see nndceq0 4654. (Contributed by Jim Kingdon, 31-Aug-2019.)
|- ( ( A e. om /\ B e. om ) -> DECID A = B )
Theorem	nndcel 6558	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 6559	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 6560	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 6561	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 6562*	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 3768 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 6563*	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 6564*	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 6565	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 3768 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 6566	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 6567	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 6568	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 6569	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 6570	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 6571	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 6572	Weak ordering property of addition. (Contributed by NM, 9-Nov-2002.)
|- ( ( A e. om /\ B e. om ) -> A C_ ( B +o A ) )
Theorem	nnawordi 6573	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 6574	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 6575	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 6576	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 6577	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 6578	One is a natural number. (Contributed by NM, 29-Oct-1995.)
|- 1o e. om
Theorem	2onn 6579	The ordinal 2 is a natural number. (Contributed by NM, 28-Sep-2004.)
|- 2o e. om
Theorem	3onn 6580	The ordinal 3 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.)
|- 3o e. om
Theorem	4onn 6581	The ordinal 4 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.)
|- 4o e. om
Theorem	2ssom 6582	The ordinal 2 is included in the set of natural number ordinals. (Contributed by BJ, 5-Aug-2024.)
|- 2o C_ om
Theorem	nnm1 6583	Multiply an element of om by 1o. (Contributed by Mario Carneiro, 17-Nov-2014.)
|- ( A e. om -> ( A .o 1o ) = A )
Theorem	nnm2 6584	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 6585	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 6586*	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 6587*	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 6588	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 6589	Extend the definition of a wff to include the equivalence predicate.
wff R Er A
Syntax	cec 6590	Extend the definition of a class to include equivalence class.
class [ A ] R
Syntax	cqs 6591	Extend the definition of a class to include quotient set.
class ( A /. R )
Definition	df-er 6592	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 6593 we derive a more typical definition. We show that an equivalence relation is reflexive, symmetric, and transitive in erref 6612, ersymb 6606, and ertr 6607. (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 6593*	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 6594	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 6593). 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 6595. (Contributed by NM, 23-Jul-1995.)
|- [ A ] R = ( R " { A } )
Theorem	dfec2 6595*	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 6596	An equivalence class modulo a set is a set. (Contributed by NM, 24-Jul-1995.)
|- ( R e. B -> [ A ] R e. _V )
Theorem	ecexr 6597	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 6598*	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 6599	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 6600	Equality theorem for equivalence predicate. (Contributed by Mario Carneiro, 12-Aug-2015.)
|- ( A = B -> ( R Er A <-> R Er B ) )