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Theorem List for Intuitionistic Logic Explorer - 7001-7100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfodjumkvlemres 7001* Lemma for fodjumkv 7002. The final result with  P expressed as a local definition. (Contributed by Jim Kingdon, 25-Mar-2023.)
 |-  ( ph  ->  M  e. Markov )   &    |-  ( ph  ->  F : M -onto-> ( A B ) )   &    |-  P  =  ( y  e.  M  |->  if ( E. z  e.  A  ( F `  y )  =  (inl `  z ) ,  (/) ,  1o ) )   =>    |-  ( ph  ->  ( A  =/=  (/)  ->  E. x  x  e.  A )
 )
 
Theoremfodjumkv 7002* A condition which ensures that a nonempty set is inhabited. (Contributed by Jim Kingdon, 25-Mar-2023.)
 |-  ( ph  ->  M  e. Markov )   &    |-  ( ph  ->  F : M -onto-> ( A B ) )   =>    |-  ( ph  ->  ( A  =/=  (/)  ->  E. x  x  e.  A )
 )
 
2.6.38  Cardinal numbers
 
Syntaxccrd 7003 Extend class definition to include the cardinal size function.
 class  card
 
Definitiondf-card 7004* Define the cardinal number function. The cardinal number of a set is the least ordinal number equinumerous to it. In other words, it is the "size" of the set. Definition of [Enderton] p. 197. Our notation is from Enderton. Other textbooks often use a double bar over the set to express this function. (Contributed by NM, 21-Oct-2003.)
 |- 
 card  =  ( x  e.  _V  |->  |^| { y  e. 
 On  |  y  ~~  x } )
 
Theoremcardcl 7005* The cardinality of a well-orderable set is an ordinal. (Contributed by Jim Kingdon, 30-Aug-2021.)
 |-  ( E. y  e. 
 On  y  ~~  A  ->  ( card `  A )  e.  On )
 
Theoremisnumi 7006 A set equinumerous to an ordinal is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  e.  On  /\  A  ~~  B )  ->  B  e.  dom  card
 )
 
Theoremfinnum 7007 Every finite set is numerable. (Contributed by Mario Carneiro, 4-Feb-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( A  e.  Fin  ->  A  e.  dom  card )
 
Theoremonenon 7008 Every ordinal number is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
 |-  ( A  e.  On  ->  A  e.  dom  card )
 
Theoremcardval3ex 7009* The value of  ( card `  A
). (Contributed by Jim Kingdon, 30-Aug-2021.)
 |-  ( E. x  e. 
 On  x  ~~  A  ->  ( card `  A )  =  |^| { y  e. 
 On  |  y  ~~  A } )
 
Theoremoncardval 7010* The value of the cardinal number function with an ordinal number as its argument. (Contributed by NM, 24-Nov-2003.) (Revised by Mario Carneiro, 13-Sep-2013.)
 |-  ( A  e.  On  ->  ( card `  A )  =  |^| { x  e. 
 On  |  x  ~~  A } )
 
Theoremcardonle 7011 The cardinal of an ordinal number is less than or equal to the ordinal number. Proposition 10.6(3) of [TakeutiZaring] p. 85. (Contributed by NM, 22-Oct-2003.)
 |-  ( A  e.  On  ->  ( card `  A )  C_  A )
 
Theoremcard0 7012 The cardinality of the empty set is the empty set. (Contributed by NM, 25-Oct-2003.)
 |-  ( card `  (/) )  =  (/)
 
Theoremcarden2bex 7013* If two numerable sets are equinumerous, then they have equal cardinalities. (Contributed by Jim Kingdon, 30-Aug-2021.)
 |-  ( ( A  ~~  B  /\  E. x  e. 
 On  x  ~~  A )  ->  ( card `  A )  =  ( card `  B ) )
 
Theorempm54.43 7014 Theorem *54.43 of [WhiteheadRussell] p. 360. (Contributed by NM, 4-Apr-2007.)
 |-  ( ( A  ~~  1o  /\  B  ~~  1o )  ->  ( ( A  i^i  B )  =  (/) 
 <->  ( A  u.  B )  ~~  2o ) )
 
Theorempr2nelem 7015 Lemma for pr2ne 7016. (Contributed by FL, 17-Aug-2008.)
 |-  ( ( A  e.  C  /\  B  e.  D  /\  A  =/=  B ) 
 ->  { A ,  B }  ~~  2o )
 
Theorempr2ne 7016 If an unordered pair has two elements they are different. (Contributed by FL, 14-Feb-2010.)
 |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( { A ,  B }  ~~  2o  <->  A  =/=  B ) )
 
Theoremexmidonfinlem 7017* Lemma for exmidonfin 7018. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.)
 |-  A  =  { { x  e.  { (/) }  |  ph
 } ,  { x  e.  { (/) }  |  -.  ph
 } }   =>    |-  ( om  =  ( On  i^i  Fin )  -> DECID  ph )
 
Theoremexmidonfin 7018 If a finite ordinal is a natural number, excluded middle follows. That excluded middle implies that a finite ordinal is a natural number is proved in the Metamath Proof Explorer. That a natural number is a finite ordinal is shown at nnfi 6734 and nnon 4493. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.)
 |-  ( om  =  ( On  i^i  Fin )  -> EXMID )
 
Theoremen2eleq 7019 Express a set of pair cardinality as the unordered pair of a given element and the other element. (Contributed by Stefan O'Rear, 22-Aug-2015.)
 |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  P  =  { X ,  U. ( P 
 \  { X }
 ) } )
 
Theoremen2other2 7020 Taking the other element twice in a pair gets back to the original element. (Contributed by Stefan O'Rear, 22-Aug-2015.)
 |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  U. ( P  \  { U. ( P  \  { X } ) }
 )  =  X )
 
Theoremdju1p1e2 7021 Disjoint union version of one plus one equals two. (Contributed by Jim Kingdon, 1-Jul-2022.)
 |-  ( 1o 1o )  ~~  2o
 
Theoreminfpwfidom 7022 The collection of finite subsets of a set dominates the set. (We use the weaker sethood assumption 
( ~P A  i^i  Fin )  e.  _V because this theorem also implies that  A is a set if  ~P A  i^i  Fin is.) (Contributed by Mario Carneiro, 17-May-2015.)
 |-  ( ( ~P A  i^i  Fin )  e.  _V  ->  A  ~<_  ( ~P A  i^i  Fin ) )
 
Theoremexmidfodomrlemeldju 7023 Lemma for exmidfodomr 7028. A variant of djur 6922. (Contributed by Jim Kingdon, 2-Jul-2022.)
 |-  ( ph  ->  A  C_ 
 1o )   &    |-  ( ph  ->  B  e.  ( A 1o )
 )   =>    |-  ( ph  ->  ( B  =  (inl `  (/) )  \/  B  =  (inr `  (/) ) ) )
 
Theoremexmidfodomrlemreseldju 7024 Lemma for exmidfodomrlemrALT 7027. A variant of eldju 6921. (Contributed by Jim Kingdon, 9-Jul-2022.)
 |-  ( ph  ->  A  C_ 
 1o )   &    |-  ( ph  ->  B  e.  ( A 1o )
 )   =>    |-  ( ph  ->  (
 ( (/)  e.  A  /\  B  =  ( (inl  |`  A ) `  (/) ) )  \/  B  =  ( (inr  |`  1o ) `  (/) ) ) )
 
Theoremexmidfodomrlemim 7025* Excluded middle implies the existence of a mapping from any set onto any inhabited set that it dominates. Proposition 1.1 of [PradicBrown2022], p. 2. (Contributed by Jim Kingdon, 1-Jul-2022.)
 |-  (EXMID 
 ->  A. x A. y
 ( ( E. z  z  e.  y  /\  y 
 ~<_  x )  ->  E. f  f : x -onto-> y ) )
 
Theoremexmidfodomrlemr 7026* The existence of a mapping from any set onto any inhabited set that it dominates implies excluded middle. Proposition 1.2 of [PradicBrown2022], p. 2. (Contributed by Jim Kingdon, 1-Jul-2022.)
 |-  ( A. x A. y ( ( E. z  z  e.  y  /\  y  ~<_  x )  ->  E. f  f : x -onto-> y )  -> EXMID )
 
TheoremexmidfodomrlemrALT 7027* The existence of a mapping from any set onto any inhabited set that it dominates implies excluded middle. Proposition 1.2 of [PradicBrown2022], p. 2. An alternative proof of exmidfodomrlemr 7026. In particular, this proof uses eldju 6921 instead of djur 6922 and avoids djulclb 6908. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Jim Kingdon, 9-Jul-2022.)
 |-  ( A. x A. y ( ( E. z  z  e.  y  /\  y  ~<_  x )  ->  E. f  f : x -onto-> y )  -> EXMID )
 
Theoremexmidfodomr 7028* Excluded middle is equivalent to the existence of a mapping from any set onto any inhabited set that it dominates. (Contributed by Jim Kingdon, 1-Jul-2022.)
 |-  (EXMID  <->  A. x A. y ( ( E. z  z  e.  y  /\  y  ~<_  x )  ->  E. f  f : x -onto-> y ) )
 
2.6.39  Axiom of Choice equivalents
 
Syntaxwac 7029 Formula for an abbreviation of the axiom of choice.
 wff CHOICE
 
Definitiondf-ac 7030* The expression CHOICE will be used as a readable shorthand for any form of the axiom of choice; all concrete forms are long, cryptic, have dummy variables, or all three, making it useful to have a short name. Similar to the Axiom of Choice (first form) of [Enderton] p. 49.

There are some decisions about how to write this definition especially around whether ax-setind 4422 is needed to show equivalence to other ways of stating choice, and about whether choice functions are available for nonempty sets or inhabited sets. (Contributed by Mario Carneiro, 22-Feb-2015.)

 |-  (CHOICE  <->  A. x E. f ( f  C_  x  /\  f  Fn  dom  x )
 )
 
Theoremacfun 7031* A convenient form of choice. The goal here is to state choice as the existence of a choice function on a set of inhabited sets, while making full use of our notation around functions and function values. (Contributed by Jim Kingdon, 20-Nov-2023.)
 |-  ( ph  -> CHOICE )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  A. x  e.  A  E. w  w  e.  x )   =>    |-  ( ph  ->  E. f
 ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  e.  x )
 )
 
Theoremexmidaclem 7032* Lemma for exmidac 7033. The result, with a few hypotheses to break out commonly used expressions. (Contributed by Jim Kingdon, 21-Nov-2023.)
 |-  A  =  { x  e.  { (/) ,  { (/) } }  |  ( x  =  (/)  \/  y  =  { (/) } ) }   &    |-  B  =  { x  e.  { (/) ,  { (/) } }  |  ( x  =  { (/)
 }  \/  y  =  { (/) } ) }   &    |-  C  =  { A ,  B }   =>    |-  (CHOICE 
 -> EXMID )
 
Theoremexmidac 7033 The axiom of choice implies excluded middle. See acexmid 5741 for more discussion of this theorem and a way of stating it without using CHOICE or EXMID. (Contributed by Jim Kingdon, 21-Nov-2023.)
 |-  (CHOICE 
 -> EXMID )
 
2.6.40  Cardinal number arithmetic
 
Theoremendjudisj 7034 Equinumerosity of a disjoint union and a union of two disjoint sets. (Contributed by Jim Kingdon, 30-Jul-2023.)
 |-  ( ( A  e.  V  /\  B  e.  W  /\  ( A  i^i  B )  =  (/) )  ->  ( A B )  ~~  ( A  u.  B ) )
 
Theoremdjuen 7035 Disjoint unions of equinumerous sets are equinumerous. (Contributed by Jim Kingdon, 30-Jul-2023.)
 |-  ( ( A  ~~  B  /\  C  ~~  D )  ->  ( A C ) 
 ~~  ( B D ) )
 
Theoremdjuenun 7036 Disjoint union is equinumerous to union for disjoint sets. (Contributed by Mario Carneiro, 29-Apr-2015.) (Revised by Jim Kingdon, 19-Aug-2023.)
 |-  ( ( A  ~~  B  /\  C  ~~  D  /\  ( B  i^i  D )  =  (/) )  ->  ( A C )  ~~  ( B  u.  D ) )
 
Theoremdju1en 7037 Cardinal addition with cardinal one (which is the same as ordinal one). Used in proof of Theorem 6J of [Enderton] p. 143. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  e.  V  /\  -.  A  e.  A )  ->  ( A 1o )  ~~  suc  A )
 
Theoremdju0en 7038 Cardinal addition with cardinal zero (the empty set). Part (a1) of proof of Theorem 6J of [Enderton] p. 143. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( A  e.  V  ->  ( A (/) )  ~~  A )
 
Theoremxp2dju 7039 Two times a cardinal number. Exercise 4.56(g) of [Mendelson] p. 258. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( 2o  X.  A )  =  ( A A )
 
Theoremdjucomen 7040 Commutative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258. (Contributed by NM, 24-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A B ) 
 ~~  ( B A ) )
 
Theoremdjuassen 7041 Associative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258. (Contributed by NM, 26-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X ) 
 ->  ( ( A B ) C )  ~~  ( A ( B C ) ) )
 
Theoremxpdjuen 7042 Cardinal multiplication distributes over cardinal addition. Theorem 6I(3) of [Enderton] p. 142. (Contributed by NM, 26-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X ) 
 ->  ( A  X.  ( B C ) )  ~~  ( ( A  X.  B ) ( A  X.  C ) ) )
 
Theoremdjudoml 7043 A set is dominated by its disjoint union with another. (Contributed by Jim Kingdon, 11-Jul-2023.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  A  ~<_  ( A B ) )
 
Theoremdjudomr 7044 A set is dominated by its disjoint union with another. (Contributed by Jim Kingdon, 11-Jul-2023.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  B  ~<_  ( A B ) )
 
PART 3  CHOICE PRINCIPLES

We have already introduced the full Axiom of Choice df-ac 7030 but since it implies excluded middle as shown at exmidac 7033, it is not especially relevant to us. In this section we define countable choice and dependent choice, which are not as strong as thus often considered in mathematics which seeks to avoid full excluded middle.

 
3.1  Countable Choice and Dependent Choice
 
3.1.1  Introduce Countable Choice
 
Syntaxwacc 7045 Formula for an abbreviation of countable choice.
 wff CCHOICE
 
Definitiondf-cc 7046* The expression CCHOICE will be used as a readable shorthand for any form of countable choice, analogous to df-ac 7030 for full choice. (Contributed by Jim Kingdon, 27-Nov-2023.)
 |-  (CCHOICE  <->  A. x ( dom  x  ~~ 
 om  ->  E. f ( f 
 C_  x  /\  f  Fn  dom  x ) ) )
 
Theoremccfunen 7047* Existence of a choice function for a countably infinite set. (Contributed by Jim Kingdon, 28-Nov-2023.)
 |-  ( ph  -> CCHOICE )   &    |-  ( ph  ->  A 
 ~~  om )   &    |-  ( ph  ->  A. x  e.  A  E. w  w  e.  x )   =>    |-  ( ph  ->  E. f
 ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  e.  x )
 )
 
PART 4  REAL AND COMPLEX NUMBERS

This section derives the basics of real and complex numbers.

To construct the real numbers constructively, we follow two main sources. The first is Metamath Proof Explorer, which has the advantage of being already formalized in metamath. Its disadvantage, for our purposes, is that it assumes the law of the excluded middle throughout. Since we have already developed natural numbers ( for example, nna0 6338 and similar theorems ), going from there to positive integers (df-ni 7080) and then positive rational numbers (df-nqqs 7124) does not involve a major change in approach compared with the Metamath Proof Explorer.

It is when we proceed to Dedekind cuts that we bring in more material from Section 11.2 of [HoTT], which focuses on the aspects of Dedekind cuts which are different without excluded middle. With excluded middle, it is natural to define the cut as the lower set only (as Metamath Proof Explorer does), but we define the cut as a pair of both the lower and upper sets, as [HoTT] does. There are also differences in how we handle order and replacing "not equal to zero" with "apart from zero".

 
4.1  Construction and axiomatization of real and complex numbers
 
4.1.1  Dedekind-cut construction of real and complex numbers
 
Syntaxcnpi 7048 The set of positive integers, which is the set of natural numbers  om with 0 removed.

Note: This is the start of the Dedekind-cut construction of real and complex numbers.

 class  N.
 
Syntaxcpli 7049 Positive integer addition.
 class  +N
 
Syntaxcmi 7050 Positive integer multiplication.
 class  .N
 
Syntaxclti 7051 Positive integer ordering relation.
 class  <N
 
Syntaxcplpq 7052 Positive pre-fraction addition.
 class  +pQ
 
Syntaxcmpq 7053 Positive pre-fraction multiplication.
 class  .pQ
 
Syntaxcltpq 7054 Positive pre-fraction ordering relation.
 class  <pQ
 
Syntaxceq 7055 Equivalence class used to construct positive fractions.
 class  ~Q
 
Syntaxcnq 7056 Set of positive fractions.
 class  Q.
 
Syntaxc1q 7057 The positive fraction constant 1.
 class  1Q
 
Syntaxcplq 7058 Positive fraction addition.
 class  +Q
 
Syntaxcmq 7059 Positive fraction multiplication.
 class  .Q
 
Syntaxcrq 7060 Positive fraction reciprocal operation.
 class  *Q
 
Syntaxcltq 7061 Positive fraction ordering relation.
 class  <Q
 
Syntaxceq0 7062 Equivalence class used to construct nonnegative fractions.
 class ~Q0
 
Syntaxcnq0 7063 Set of nonnegative fractions.
 class Q0
 
Syntaxc0q0 7064 The nonnegative fraction constant 0.
 class 0Q0
 
Syntaxcplq0 7065 Nonnegative fraction addition.
 class +Q0
 
Syntaxcmq0 7066 Nonnegative fraction multiplication.
 class ·Q0
 
Syntaxcnp 7067 Set of positive reals.
 class  P.
 
Syntaxc1p 7068 Positive real constant 1.
 class  1P
 
Syntaxcpp 7069 Positive real addition.
 class  +P.
 
Syntaxcmp 7070 Positive real multiplication.
 class  .P.
 
Syntaxcltp 7071 Positive real ordering relation.
 class  <P
 
Syntaxcer 7072 Equivalence class used to construct signed reals.
 class  ~R
 
Syntaxcnr 7073 Set of signed reals.
 class  R.
 
Syntaxc0r 7074 The signed real constant 0.
 class  0R
 
Syntaxc1r 7075 The signed real constant 1.
 class  1R
 
Syntaxcm1r 7076 The signed real constant -1.
 class  -1R
 
Syntaxcplr 7077 Signed real addition.
 class  +R
 
Syntaxcmr 7078 Signed real multiplication.
 class  .R
 
Syntaxcltr 7079 Signed real ordering relation.
 class  <R
 
Definitiondf-ni 7080 Define the class of positive integers. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by NM, 15-Aug-1995.)
 |- 
 N.  =  ( om  \  { (/) } )
 
Definitiondf-pli 7081 Define addition on positive integers. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by NM, 26-Aug-1995.)
 |- 
 +N  =  (  +o  |`  ( N.  X.  N. ) )
 
Definitiondf-mi 7082 Define multiplication on positive integers. This is a "temporary" set used in the construction of complex numbers and is intended to be used only by the construction. (Contributed by NM, 26-Aug-1995.)
 |- 
 .N  =  (  .o  |`  ( N.  X.  N. ) )
 
Definitiondf-lti 7083 Define 'less than' on positive integers. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by NM, 6-Feb-1996.)
 |- 
 <N  =  (  _E  i^i  ( N.  X.  N. ) )
 
Theoremelni 7084 Membership in the class of positive integers. (Contributed by NM, 15-Aug-1995.)
 |-  ( A  e.  N.  <->  ( A  e.  om  /\  A  =/= 
 (/) ) )
 
Theorempinn 7085 A positive integer is a natural number. (Contributed by NM, 15-Aug-1995.)
 |-  ( A  e.  N.  ->  A  e.  om )
 
Theorempion 7086 A positive integer is an ordinal number. (Contributed by NM, 23-Mar-1996.)
 |-  ( A  e.  N.  ->  A  e.  On )
 
Theorempiord 7087 A positive integer is ordinal. (Contributed by NM, 29-Jan-1996.)
 |-  ( A  e.  N.  ->  Ord  A )
 
Theoremniex 7088 The class of positive integers is a set. (Contributed by NM, 15-Aug-1995.)
 |- 
 N.  e.  _V
 
Theorem0npi 7089 The empty set is not a positive integer. (Contributed by NM, 26-Aug-1995.)
 |- 
 -.  (/)  e.  N.
 
Theoremelni2 7090 Membership in the class of positive integers. (Contributed by NM, 27-Nov-1995.)
 |-  ( A  e.  N.  <->  ( A  e.  om  /\  (/)  e.  A ) )
 
Theorem1pi 7091 Ordinal 'one' is a positive integer. (Contributed by NM, 29-Oct-1995.)
 |- 
 1o  e.  N.
 
Theoremaddpiord 7092 Positive integer addition in terms of ordinal addition. (Contributed by NM, 27-Aug-1995.)
 |-  ( ( A  e.  N. 
 /\  B  e.  N. )  ->  ( A  +N  B )  =  ( A  +o  B ) )
 
Theoremmulpiord 7093 Positive integer multiplication in terms of ordinal multiplication. (Contributed by NM, 27-Aug-1995.)
 |-  ( ( A  e.  N. 
 /\  B  e.  N. )  ->  ( A  .N  B )  =  ( A  .o  B ) )
 
Theoremmulidpi 7094 1 is an identity element for multiplication on positive integers. (Contributed by NM, 4-Mar-1996.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |-  ( A  e.  N.  ->  ( A  .N  1o )  =  A )
 
Theoremltpiord 7095 Positive integer 'less than' in terms of ordinal membership. (Contributed by NM, 6-Feb-1996.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( ( A  e.  N. 
 /\  B  e.  N. )  ->  ( A  <N  B  <->  A  e.  B )
 )
 
Theoremltsopi 7096 Positive integer 'less than' is a strict ordering. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Mario Carneiro, 10-Jul-2014.)
 |- 
 <N  Or  N.
 
Theorempitric 7097 Trichotomy for positive integers. (Contributed by Jim Kingdon, 21-Sep-2019.)
 |-  ( ( A  e.  N. 
 /\  B  e.  N. )  ->  ( A  <N  B  <->  -.  ( A  =  B  \/  B  <N  A )
 ) )
 
Theorempitri3or 7098 Trichotomy for positive integers. (Contributed by Jim Kingdon, 21-Sep-2019.)
 |-  ( ( A  e.  N. 
 /\  B  e.  N. )  ->  ( A  <N  B  \/  A  =  B  \/  B  <N  A )
 )
 
Theoremltdcpi 7099 Less-than for positive integers is decidable. (Contributed by Jim Kingdon, 12-Dec-2019.)
 |-  ( ( A  e.  N. 
 /\  B  e.  N. )  -> DECID  A  <N  B )
 
Theoremltrelpi 7100 Positive integer 'less than' is a relation on positive integers. (Contributed by NM, 8-Feb-1996.)
 |- 
 <N  C_  ( N.  X.  N. )
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