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Theorem List for Intuitionistic Logic Explorer - 6301-6400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfrecfcllem 6301* Lemma for frecfcl 6302. Just giving a name to a common expression to simplify the proof. (Contributed by Jim Kingdon, 30-Mar-2022.)
 |-  G  = recs ( ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `  m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) }
 ) )   =>    |-  ( ( A. z  e.  S  ( F `  z )  e.  S  /\  A  e.  S ) 
 -> frec ( F ,  A ) : om --> S )
 
Theoremfrecfcl 6302* Finite recursion yields a function on the natural numbers. (Contributed by Jim Kingdon, 30-Mar-2022.)
 |-  ( ( A. z  e.  S  ( F `  z )  e.  S  /\  A  e.  S ) 
 -> frec ( F ,  A ) : om --> S )
 
Theoremfrecsuclem 6303* Lemma for frecsuc 6304. Just giving a name to a common expression to simplify the proof. (Contributed by Jim Kingdon, 29-Mar-2022.)
 |-  G  =  ( g  e.  _V  |->  { x  |  ( E. m  e. 
 om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
 g `  m )
 ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) }
 )   =>    |-  ( ( A. z  e.  S  ( F `  z )  e.  S  /\  A  e.  S  /\  B  e.  om )  ->  (frec ( F ,  A ) `  suc  B )  =  ( F `
  (frec ( F ,  A ) `  B ) ) )
 
Theoremfrecsuc 6304* The successor value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 31-Mar-2022.)
 |-  ( ( A. z  e.  S  ( F `  z )  e.  S  /\  A  e.  S  /\  B  e.  om )  ->  (frec ( F ,  A ) `  suc  B )  =  ( F `
  (frec ( F ,  A ) `  B ) ) )
 
Theoremfrecrdg 6305* Transfinite recursion restricted to omega.

Given a suitable characteristic function, df-frec 6288 produces the same results as df-irdg 6267 restricted to  om.

Presumably the theorem would also hold if  F  Fn  _V were changed to  A. z ( F `  z )  e.  _V. (Contributed by Jim Kingdon, 29-Aug-2019.)

 |-  ( ph  ->  F  Fn  _V )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  A. x  x  C_  ( F `  x ) )   =>    |-  ( ph  -> frec ( F ,  A )  =  ( rec ( F ,  A )  |`  om ) )
 
2.6.22  Ordinal arithmetic
 
Syntaxc1o 6306 Extend the definition of a class to include the ordinal number 1.
 class  1o
 
Syntaxc2o 6307 Extend the definition of a class to include the ordinal number 2.
 class  2o
 
Syntaxc3o 6308 Extend the definition of a class to include the ordinal number 3.
 class  3o
 
Syntaxc4o 6309 Extend the definition of a class to include the ordinal number 4.
 class  4o
 
Syntaxcoa 6310 Extend the definition of a class to include the ordinal addition operation.
 class  +o
 
Syntaxcomu 6311 Extend the definition of a class to include the ordinal multiplication operation.
 class  .o
 
Syntaxcoei 6312 Extend the definition of a class to include the ordinal exponentiation operation.
 classo
 
Definitiondf-1o 6313 Define the ordinal number 1. (Contributed by NM, 29-Oct-1995.)
 |- 
 1o  =  suc  (/)
 
Definitiondf-2o 6314 Define the ordinal number 2. (Contributed by NM, 18-Feb-2004.)
 |- 
 2o  =  suc  1o
 
Definitiondf-3o 6315 Define the ordinal number 3. (Contributed by Mario Carneiro, 14-Jul-2013.)
 |- 
 3o  =  suc  2o
 
Definitiondf-4o 6316 Define the ordinal number 4. (Contributed by Mario Carneiro, 14-Jul-2013.)
 |- 
 4o  =  suc  3o
 
Definitiondf-oadd 6317* Define the ordinal addition operation. (Contributed by NM, 3-May-1995.)
 |- 
 +o  =  ( x  e.  On ,  y  e.  On  |->  ( rec (
 ( z  e.  _V  |->  suc  z ) ,  x ) `  y ) )
 
Definitiondf-omul 6318* Define the ordinal multiplication operation. (Contributed by NM, 26-Aug-1995.)
 |- 
 .o  =  ( x  e.  On ,  y  e.  On  |->  ( rec (
 ( z  e.  _V  |->  ( z  +o  x ) ) ,  (/) ) `  y ) )
 
Definitiondf-oexpi 6319* Define the ordinal exponentiation operation.

This definition is similar to a conventional definition of exponentiation except that it defines  (/)o  A to be  1o for all  A  e.  On, in order to avoid having different cases for whether the base is  (/) or not. (Contributed by Mario Carneiro, 4-Jul-2019.)

 |-o  =  ( x  e.  On ,  y  e.  On  |->  ( rec ( ( z  e.  _V  |->  ( z  .o  x ) ) ,  1o ) `  y ) )
 
Theorem1on 6320 Ordinal 1 is an ordinal number. (Contributed by NM, 29-Oct-1995.)
 |- 
 1o  e.  On
 
Theorem1oex 6321 Ordinal 1 is a set. (Contributed by BJ, 4-Jul-2022.)
 |- 
 1o  e.  _V
 
Theorem2on 6322 Ordinal 2 is an ordinal number. (Contributed by NM, 18-Feb-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
 |- 
 2o  e.  On
 
Theorem2on0 6323 Ordinal two is not zero. (Contributed by Scott Fenton, 17-Jun-2011.)
 |- 
 2o  =/=  (/)
 
Theorem3on 6324 Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |- 
 3o  e.  On
 
Theorem4on 6325 Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |- 
 4o  e.  On
 
Theoremdf1o2 6326 Expanded value of the ordinal number 1. (Contributed by NM, 4-Nov-2002.)
 |- 
 1o  =  { (/) }
 
Theoremdf2o3 6327 Expanded value of the ordinal number 2. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |- 
 2o  =  { (/) ,  1o }
 
Theoremdf2o2 6328 Expanded value of the ordinal number 2. (Contributed by NM, 29-Jan-2004.)
 |- 
 2o  =  { (/) ,  { (/)
 } }
 
Theorem1n0 6329 Ordinal one is not equal to ordinal zero. (Contributed by NM, 26-Dec-2004.)
 |- 
 1o  =/=  (/)
 
Theoremxp01disj 6330 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 }
 ) )  =  (/)
 
Theoremxp01disjl 6331 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 ) )  =  (/)
 
Theoremordgt0ge1 6332 Two ways to express that an ordinal class is positive. (Contributed by NM, 21-Dec-2004.)
 |-  ( Ord  A  ->  ( (/)  e.  A  <->  1o  C_  A ) )
 
Theoremordge1n0im 6333 An ordinal greater than or equal to 1 is nonzero. (Contributed by Jim Kingdon, 26-Jun-2019.)
 |-  ( Ord  A  ->  ( 1o  C_  A  ->  A  =/=  (/) ) )
 
Theoremel1o 6334 Membership in ordinal one. (Contributed by NM, 5-Jan-2005.)
 |-  ( A  e.  1o  <->  A  =  (/) )
 
Theoremdif1o 6335 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  =/=  (/) ) )
 
Theorem2oconcl 6336 Closure of the pair swapping function on  2o. (Contributed by Mario Carneiro, 27-Sep-2015.)
 |-  ( A  e.  2o  ->  ( 1o  \  A )  e.  2o )
 
Theorem0lt1o 6337 Ordinal zero is less than ordinal one. (Contributed by NM, 5-Jan-2005.)
 |-  (/)  e.  1o
 
Theorem0lt2o 6338 Ordinal zero is less than ordinal two. (Contributed by Jim Kingdon, 31-Jul-2022.)
 |-  (/)  e.  2o
 
Theorem1lt2o 6339 Ordinal one is less than ordinal two. (Contributed by Jim Kingdon, 31-Jul-2022.)
 |- 
 1o  e.  2o
 
Theoremoafnex 6340 The characteristic function for ordinal addition is defined everywhere. (Contributed by Jim Kingdon, 27-Jul-2019.)
 |-  ( x  e.  _V  |->  suc  x )  Fn  _V
 
Theoremsucinc 6341* Successor is increasing. (Contributed by Jim Kingdon, 25-Jun-2019.)
 |-  F  =  ( z  e.  _V  |->  suc  z
 )   =>    |- 
 A. x  x  C_  ( F `  x )
 
Theoremsucinc2 6342* 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 ) )
 
Theoremfnoa 6343 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 )
 
Theoremoaexg 6344 Ordinal addition is a set. (Contributed by Mario Carneiro, 3-Jul-2019.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  +o  B )  e.  _V )
 
Theoremomfnex 6345* 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 )
 
Theoremfnom 6346 Functionality and domain of ordinal multiplication. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 3-Jul-2019.)
 |- 
 .o  Fn  ( On  X. 
 On )
 
Theoremomexg 6347 Ordinal multiplication is a set. (Contributed by Mario Carneiro, 3-Jul-2019.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  .o  B )  e.  _V )
 
Theoremfnoei 6348 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 )
 
Theoremoeiexg 6349 Ordinal exponentiation is a set. (Contributed by Mario Carneiro, 3-Jul-2019.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( Ao  B )  e.  _V )
 
Theoremoav 6350* 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 ) )
 
Theoremomv 6351* 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 ) )
 
Theoremoeiv 6352* Value of ordinal exponentiation. (Contributed by Jim Kingdon, 9-Jul-2019.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( Ao  B )  =  ( rec (
 ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B ) )
 
Theoremoa0 6353 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 )
 
Theoremom0 6354 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  (/) )  =  (/) )
 
Theoremoei0 6355 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  ->  ( Ao  (/) )  =  1o )
 
Theoremoacl 6356 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 )
 
Theoremomcl 6357 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 )
 
Theoremoeicl 6358 Closure law for ordinal exponentiation. (Contributed by Jim Kingdon, 26-Jul-2019.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( Ao  B )  e.  On )
 
Theoremoav2 6359* 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 ) ) )
 
Theoremoasuc 6360 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 ) )
 
Theoremomv2 6361* 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 ) )
 
Theoremonasuc 6362 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 ) )
 
Theoremoa1suc 6363 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 )
 
Theoremo1p1e2 6364 1 + 1 = 2 for ordinal numbers. (Contributed by NM, 18-Feb-2004.)
 |-  ( 1o  +o  1o )  =  2o
 
Theoremoawordi 6365 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 ) ) )
 
Theoremoawordriexmid 6366* A weak ordering property of ordinal addition which implies excluded middle. The property is proposition 8.7 of [TakeutiZaring] p. 59. Compare with oawordi 6365. (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 )
 
Theoremoaword1 6367 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 ) )
 
Theoremomsuc 6368 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 ) )
 
Theoremonmsuc 6369 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 ) )
 
2.6.23  Natural number arithmetic
 
Theoremnna0 6370 Addition with zero. Theorem 4I(A1) of [Enderton] p. 79. (Contributed by NM, 20-Sep-1995.)
 |-  ( A  e.  om  ->  ( A  +o  (/) )  =  A )
 
Theoremnnm0 6371 Multiplication with zero. Theorem 4J(A1) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.)
 |-  ( A  e.  om  ->  ( A  .o  (/) )  =  (/) )
 
Theoremnnasuc 6372 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 ) )
 
Theoremnnmsuc 6373 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 ) )
 
Theoremnna0r 6374 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 )
 
Theoremnnm0r 6375 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 )  =  (/) )
 
Theoremnnacl 6376 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 )
 
Theoremnnmcl 6377 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 )
 
Theoremnnacli 6378  om is closed under addition. Inference form of nnacl 6376. (Contributed by Scott Fenton, 20-Apr-2012.)
 |-  A  e.  om   &    |-  B  e.  om   =>    |-  ( A  +o  B )  e.  om
 
Theoremnnmcli 6379  om is closed under multiplication. Inference form of nnmcl 6377. (Contributed by Scott Fenton, 20-Apr-2012.)
 |-  A  e.  om   &    |-  B  e.  om   =>    |-  ( A  .o  B )  e.  om
 
Theoremnnacom 6380 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 ) )
 
Theoremnnaass 6381 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 ) ) )
 
Theoremnndi 6382 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 ) ) )
 
Theoremnnmass 6383 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 ) ) )
 
Theoremnnmsucr 6384 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 ) )
 
Theoremnnmcom 6385 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 ) )
 
Theoremnndir 6386 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 ) ) )
 
Theoremnnsucelsuc 6387 Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucelsucr 4424, but the forward direction, for all ordinals, implies excluded middle as seen as onsucelsucexmid 4445. (Contributed by Jim Kingdon, 25-Aug-2019.)
 |-  ( B  e.  om  ->  ( A  e.  B  <->  suc 
 A  e.  suc  B ) )
 
Theoremnnsucsssuc 6388 Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucsssucr 4425, but the forward direction, for all ordinals, implies excluded middle as seen as onsucsssucexmid 4442. (Contributed by Jim Kingdon, 25-Aug-2019.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( A  C_  B 
 <-> 
 suc  A  C_  suc  B ) )
 
Theoremnntri3or 6389 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 ) )
 
Theoremnntri2 6390 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 ) ) )
 
Theoremnnsucuniel 6391 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 4426). The forward direction for all ordinals implies excluded middle (ordsucunielexmid 4446). (Contributed by Jim Kingdon, 13-Mar-2022.)
 |-  ( B  e.  om  ->  ( A  e.  U. B 
 <-> 
 suc  A  e.  B ) )
 
Theoremnntri1 6392 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 ) )
 
Theoremnntri3 6393 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 ) ) )
 
Theoremnntri2or2 6394 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 ) )
 
Theoremnndceq 6395 Equality of natural numbers is decidable. Theorem 7.2.6 of [HoTT], p. (varies). For the specific case where  B is zero, see nndceq0 4531. (Contributed by Jim Kingdon, 31-Aug-2019.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  -> DECID  A  =  B )
 
Theoremnndcel 6396 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 )
 
Theoremnnsseleq 6397 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 ) ) )
 
Theoremnnsssuc 6398 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 ) )
 
Theoremnntr2 6399 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 ) )
 
Theoremdcdifsnid 6400* 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 3666 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 )
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