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	df1o2 6701	Expanded value of the ordinal number 1. (Contributed by NM, 4-Nov-2002.)
|- 1o = { (/) }
Theorem	df2o3 6702	Expanded value of the ordinal number 2. (Contributed by Mario Carneiro, 14-Aug-2015.)
|- 2o = { (/) , 1o }
Theorem	df2o2 6703	Expanded value of the ordinal number 2. (Contributed by NM, 29-Jan-2004.)
|- 2o = { (/) , { (/) } }
Theorem	2oex 6704	2o is a set. (Contributed by BJ, 6-Apr-2019.) (Proof shortened by Zhi Wang, 19-Sep-2024.)
|- 2o e. _V
Theorem	1n0 6705	Ordinal one is not equal to ordinal zero. (Contributed by NM, 26-Dec-2004.)
|- 1o =/= (/)
Theorem	xp01disj 6706	Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by NM, 2-Jun-2007.)
|- ( ( A X. { (/) } ) i^i ( C X. { 1o } ) ) = (/)
Theorem	xp01disjl 6707	Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by Jim Kingdon, 11-Jul-2023.)
|- ( ( { (/) } X. A ) i^i ( { 1o } X. C ) ) = (/)
Theorem	ordgt0ge1 6708	Two ways to express that an ordinal class is positive. (Contributed by NM, 21-Dec-2004.)
|- ( Ord A -> ( (/) e. A <-> 1o C_ A ) )
Theorem	ordge1n0im 6709	An ordinal greater than or equal to 1 is nonzero. (Contributed by Jim Kingdon, 26-Jun-2019.)
|- ( Ord A -> ( 1o C_ A -> A =/= (/) ) )
Theorem	el1o 6710	Membership in ordinal one. (Contributed by NM, 5-Jan-2005.)
|- ( A e. 1o <-> A = (/) )
Theorem	dif1o 6711	Two ways to say that A is a nonzero number of the set B. (Contributed by Mario Carneiro, 21-May-2015.)
|- ( A e. ( B \ 1o ) <-> ( A e. B /\ A =/= (/) ) )
Theorem	2oconcl 6712	Closure of the pair swapping function on 2o. (Contributed by Mario Carneiro, 27-Sep-2015.)
|- ( A e. 2o -> ( 1o \ A ) e. 2o )
Theorem	0lt1o 6713	Ordinal zero is less than ordinal one. (Contributed by NM, 5-Jan-2005.)
|- (/) e. 1o
Theorem	0lt2o 6714	Ordinal zero is less than ordinal two. (Contributed by Jim Kingdon, 31-Jul-2022.)
|- (/) e. 2o
Theorem	1lt2o 6715	Ordinal one is less than ordinal two. (Contributed by Jim Kingdon, 31-Jul-2022.)
|- 1o e. 2o
Theorem	el2oss1o 6716	Being an element of ordinal two implies being a subset of ordinal one. The converse is equivalent to excluded middle by ss1oel2o 17017. (Contributed by Jim Kingdon, 8-Aug-2022.)
|- ( A e. 2o -> A C_ 1o )
Theorem	oafnex 6717	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 6718*	Successor is increasing. (Contributed by Jim Kingdon, 25-Jun-2019.)
|- F = ( z e. _V |-> suc z )   =>    |- A. x x C_ ( F ` x )
Theorem	sucinc2 6719*	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 6720	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 6721	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 6722*	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 6723	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 6724	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 6725	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 6726	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 6727*	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 6728*	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 6729*	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 6730	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 6731	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 6732	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 6733	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 6734	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 6735	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 6736*	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 6737	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 6738*	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 6739	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 6740	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 6741	1 + 1 = 2 for ordinal numbers. (Contributed by NM, 18-Feb-2004.)
|- ( 1o +o 1o ) = 2o
Theorem	oawordi 6742	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 6743*	A weak ordering property of ordinal addition which implies excluded middle. The property is proposition 8.7 of [TakeutiZaring] p. 59. Compare with oawordi 6742. (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 6744	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 6745	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 6746	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 ) )
2.6.25  Natural number arithmetic
Theorem	nna0 6747	Addition with zero. Theorem 4I(A1) of [Enderton] p. 79. (Contributed by NM, 20-Sep-1995.)
|- ( A e. om -> ( A +o (/) ) = A )
Theorem	nnm0 6748	Multiplication with zero. Theorem 4J(A1) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.)
|- ( A e. om -> ( A .o (/) ) = (/) )
Theorem	nnasuc 6749	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 6750	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 6751	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 6752	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 6753	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 6754	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 6755	om is closed under addition. Inference form of nnacl 6753. (Contributed by Scott Fenton, 20-Apr-2012.)
|- A e. om   &    |- B e. om   =>    |- ( A +o B ) e. om
Theorem	nnmcli 6756	om is closed under multiplication. Inference form of nnmcl 6754. (Contributed by Scott Fenton, 20-Apr-2012.)
|- A e. om   &    |- B e. om   =>    |- ( A .o B ) e. om
Theorem	nnacom 6757	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 6758	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 6759	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 6760	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 6761	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 6762	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 6763	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 6764	Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucelsucr 4655, but the forward direction, for all ordinals, implies excluded middle as seen as onsucelsucexmid 4677. (Contributed by Jim Kingdon, 25-Aug-2019.)
|- ( B e. om -> ( A e. B <-> suc A e. suc B ) )
Theorem	nnsucsssuc 6765	Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucsssucr 4656, but the forward direction, for all ordinals, implies excluded middle as seen as onsucsssucexmid 4674. (Contributed by Jim Kingdon, 25-Aug-2019.)
|- ( ( A e. om /\ B e. om ) -> ( A C_ B <-> suc A C_ suc B ) )
Theorem	nntri3or 6766	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 6767	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 6768	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 4657). The forward direction for all ordinals implies excluded middle (ordsucunielexmid 4678). (Contributed by Jim Kingdon, 13-Mar-2022.)
|- ( B e. om -> ( A e. U. B <-> suc A e. B ) )
Theorem	nntri1 6769	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 6770	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 6771	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 6772	Equality of natural numbers is decidable. Theorem 7.2.6 of [HoTT], p. (varies). For the specific case where B is zero, see nndceq0 4765. (Contributed by Jim Kingdon, 31-Aug-2019.)
|- ( ( A e. om /\ B e. om ) -> DECID A = B )
Theorem	nndcel 6773	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 6774	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 6775	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 6776	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 6777*	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 3861 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 6778*	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 6779*	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 6780	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 3861 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 6781	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 6782	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 6783	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 6784	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 6785	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 6786	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 6787	Weak ordering property of addition. (Contributed by NM, 9-Nov-2002.)
|- ( ( A e. om /\ B e. om ) -> A C_ ( B +o A ) )
Theorem	nnawordi 6788	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 6789	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 6790	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 6791	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 6792	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 6793	One is a natural number. (Contributed by NM, 29-Oct-1995.)
|- 1o e. om
Theorem	2onn 6794	The ordinal 2 is a natural number. (Contributed by NM, 28-Sep-2004.)
|- 2o e. om
Theorem	3onn 6795	The ordinal 3 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.)
|- 3o e. om
Theorem	4onn 6796	The ordinal 4 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.)
|- 4o e. om
Theorem	2ssom 6797	The ordinal 2 is included in the set of natural number ordinals. (Contributed by BJ, 5-Aug-2024.)
|- 2o C_ om
Theorem	nnm1 6798	Multiply an element of om by 1o. (Contributed by Mario Carneiro, 17-Nov-2014.)
|- ( A e. om -> ( A .o 1o ) = A )
Theorem	nnm2 6799	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 6800	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 ) )