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Theorem List for Intuitionistic Logic Explorer - 4001-4100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremopelopabaf 4001* The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 3999 uses bound-variable hypotheses in place of distinct variable conditions." (Contributed by Mario Carneiro, 19-Dec-2013.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)

 F/   &     F/   &     _V   &     _V   &       =>     <. ,  >.  { <. ,  >.  |  }
 
Theoremopelopabf 4002* The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 3999 uses bound-variable hypotheses in place of distinct variable conditions." (Contributed by NM, 19-Dec-2008.)

 F/   &     F/   &     _V   &     _V   &       &       =>     <. ,  >.  { <. ,  >.  |  }
 
Theoremssopab2 4003 Equivalence of ordered pair abstraction subclass and implication. (Contributed by NM, 27-Dec-1996.) (Revised by Mario Carneiro, 19-May-2013.)
 { <. ,  >.  |  }  C_  {
 <. ,  >.  |  }
 
Theoremssopab2b 4004 Equivalence of ordered pair abstraction subclass and implication. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
 { <. ,  >.  |  }  C_  {
 <. ,  >.  |  }
 
Theoremssopab2i 4005 Inference of ordered pair abstraction subclass from implication. (Contributed by NM, 5-Apr-1995.)
   =>    
 { <. ,  >.  |  }  C_  {
 <. ,  >.  |  }
 
Theoremssopab2dv 4006* Inference of ordered pair abstraction subclass from implication. (Contributed by NM, 19-Jan-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
   =>     { <. ,  >.  |  } 
 C_  { <. ,  >.  |  }
 
Theoremeqopab2b 4007 Equivalence of ordered pair abstraction equality and biconditional. (Contributed by Mario Carneiro, 4-Jan-2017.)
 { <. ,  >.  |  }  {
 <. ,  >.  |  }
 
Theoremopabm 4008* Inhabited ordered pair class abstraction. (Contributed by Jim Kingdon, 29-Sep-2018.)
 { <. ,  >.  |  }
 
Theoremiunopab 4009* Move indexed union inside an ordered-pair abstraction. (Contributed by Stefan O'Rear, 20-Feb-2015.)
 U_  {
 <. ,  >.  |  }  { <. ,  >.  |  }
 
2.3.5  Power class of union and intersection
 
Theorempwin 4010 The power class of the intersection of two classes is the intersection of their power classes. Exercise 4.12(j) of [Mendelson] p. 235. (Contributed by NM, 23-Nov-2003.)

 ~P  i^i  ~P  i^i  ~P
 
Theorempwunss 4011 The power class of the union of two classes includes the union of their power classes. Exercise 4.12(k) of [Mendelson] p. 235. (Contributed by NM, 23-Nov-2003.)
 ~P  u.  ~P  C_  ~P  u.
 
Theorempwssunim 4012 The power class of the union of two classes is a subset of the union of their power classes, if one class is a subclass of the other. One direction of Exercise 4.12(l) of [Mendelson] p. 235. (Contributed by Jim Kingdon, 30-Sep-2018.)
 C_  C_  ~P  u.  C_  ~P  u.  ~P
 
Theorempwundifss 4013 Break up the power class of a union into a union of smaller classes. (Contributed by Jim Kingdon, 30-Sep-2018.)
 ~P  u.  \  ~P  u.  ~P  C_  ~P  u.
 
Theorempwunim 4014 The power class of the union of two classes equals the union of their power classes, iff one class is a subclass of the other. Part of Exercise 7(b) of [Enderton] p. 28. (Contributed by Jim Kingdon, 30-Sep-2018.)
 C_  C_  ~P  u.  ~P  u.  ~P
 
2.3.6  Epsilon and identity relations
 
Syntaxcep 4015 Extend class notation to include the epsilon relation.

 _E
 
Syntaxcid 4016 Extend the definition of a class to include identity relation.

 _I
 
Definitiondf-eprel 4017* Define the epsilon relation. Similar to Definition 6.22 of [TakeutiZaring] p. 30. The epsilon relation and set membership are the same, that is,  _E when is a set by epelg 4018. Thus, 5  _E { 1 , 5 }. (Contributed by NM, 13-Aug-1995.)

 _E  { <. ,  >.  |  }
 
Theoremepelg 4018 The epsilon relation and membership are the same. General version of epel 4020. (Contributed by Scott Fenton, 27-Mar-2011.) (Revised by Mario Carneiro, 28-Apr-2015.)
 V  _E
 
Theoremepelc 4019 The epsilon relationship and the membership relation are the same. (Contributed by Scott Fenton, 11-Apr-2012.)
 _V   =>     _E
 
Theoremepel 4020 The epsilon relation and the membership relation are the same. (Contributed by NM, 13-Aug-1995.)
 _E
 
Definitiondf-id 4021* Define the identity relation. Definition 9.15 of [Quine] p. 64. For example, 5  _I 5 and 4  _I 5. (Contributed by NM, 13-Aug-1995.)

 _I  { <. ,  >.  |  }
 
2.3.7  Partial and complete ordering
 
Syntaxwpo 4022 Extend wff notation to include the strict partial ordering predicate. Read: '  R is a partial order on .'
 R  Po
 
Syntaxwor 4023 Extend wff notation to include the strict linear ordering predicate. Read: '  R orders .'
 R  Or
 
Definitiondf-po 4024* Define the strict partial order predicate. Definition of [Enderton] p. 168. The expression  R  Po means  R is a partial order on . (Contributed by NM, 16-Mar-1997.)
 R  Po  R  R  R  R
 
Definitiondf-iso 4025* Define the strict linear order predicate. The expression  R  Or is true if relationship  R orders . The property  R  R  R is called weak linearity by Proposition 11.2.3 of [HoTT], p. (varies). If we assumed excluded middle, it would be equivalent to trichotomy,  R  R. (Contributed by NM, 21-Jan-1996.) (Revised by Jim Kingdon, 4-Oct-2018.)
 R  Or  R  Po  R  R  R
 
Theoremposs 4026 Subset theorem for the partial ordering predicate. (Contributed by NM, 27-Mar-1997.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
 C_  R  Po  R  Po
 
Theorempoeq1 4027 Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.)
 R  S  R  Po  S  Po
 
Theorempoeq2 4028 Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.)
 R  Po  R  Po
 
Theoremnfpo 4029 Bound-variable hypothesis builder for partial orders. (Contributed by Stefan O'Rear, 20-Jan-2015.)
 F/_ R   &     F/_   =>     F/  R  Po
 
Theoremnfso 4030 Bound-variable hypothesis builder for total orders. (Contributed by Stefan O'Rear, 20-Jan-2015.)
 F/_ R   &     F/_   =>     F/  R  Or
 
Theorempocl 4031 Properties of partial order relation in class notation. (Contributed by NM, 27-Mar-1997.)
 R  Po  C  D  R  R C  C R D  R D
 
Theoremispod 4032* Sufficient conditions for a partial order. (Contributed by NM, 9-Jul-2014.)
 R   &     R  R  R   =>     R  Po
 
Theoremswopolem 4033* Perform the substitutions into the strict weak ordering law. (Contributed by Mario Carneiro, 31-Dec-2014.)
 R  R  R   =>     X  Y  Z  X R Y  X R Z  Z R Y
 
Theoremswopo 4034* A strict weak order is a partial order. (Contributed by Mario Carneiro, 9-Jul-2014.)
 R  R   &     R  R  R   =>     R  Po
 
Theorempoirr 4035 A partial order relation is irreflexive. (Contributed by NM, 27-Mar-1997.)
 R  Po  R
 
Theorempotr 4036 A partial order relation is a transitive relation. (Contributed by NM, 27-Mar-1997.)
 R  Po  C  D  R C  C R D  R D
 
Theorempo2nr 4037 A partial order relation has no 2-cycle loops. (Contributed by NM, 27-Mar-1997.)
 R  Po  C  R C  C R
 
Theorempo3nr 4038 A partial order relation has no 3-cycle loops. (Contributed by NM, 27-Mar-1997.)
 R  Po  C  D  R C  C R D  D R
 
Theorempo0 4039 Any relation is a partial ordering of the empty set. (Contributed by NM, 28-Mar-1997.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 R  Po  (/)
 
Theorempofun 4040* A function preserves a partial order relation. (Contributed by Jeff Madsen, 18-Jun-2011.)
 S  { <. ,  >.  |  X R Y }   &     X  Y   =>     R  Po  X  S  Po
 
Theoremsopo 4041 A strict linear order is a strict partial order. (Contributed by NM, 28-Mar-1997.)
 R  Or  R  Po
 
Theoremsoss 4042 Subset theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 C_  R  Or  R  Or
 
Theoremsoeq1 4043 Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.)
 R  S  R  Or  S  Or
 
Theoremsoeq2 4044 Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.)
 R  Or  R  Or
 
Theoremsonr 4045 A strict order relation is irreflexive. (Contributed by NM, 24-Nov-1995.)
 R  Or  R
 
Theoremsotr 4046 A strict order relation is a transitive relation. (Contributed by NM, 21-Jan-1996.)
 R  Or  C  D  R C  C R D  R D
 
Theoremissod 4047* An irreflexive, transitive, trichotomous relation is a linear ordering (in the sense of df-iso 4025). (Contributed by NM, 21-Jan-1996.) (Revised by Mario Carneiro, 9-Jul-2014.)
 R  Po    &     R  R   =>     R  Or
 
Theoremsowlin 4048 A strict order relation satisfies weak linearity. (Contributed by Jim Kingdon, 6-Oct-2018.)
 R  Or  C  D  R C  R D  D R C
 
Theoremso2nr 4049 A strict order relation has no 2-cycle loops. (Contributed by NM, 21-Jan-1996.)
 R  Or  C  R C  C R
 
Theoremso3nr 4050 A strict order relation has no 3-cycle loops. (Contributed by NM, 21-Jan-1996.)
 R  Or  C  D  R C  C R D  D R
 
Theoremsotricim 4051 One direction of sotritric 4052 holds for all weakly linear orders. (Contributed by Jim Kingdon, 28-Sep-2019.)
 R  Or  C  R C  C  C R
 
Theoremsotritric 4052 A trichotomy relationship, given a trichotomous order. (Contributed by Jim Kingdon, 28-Sep-2019.)
 R  Or    &     C  R C  C  C R   =>     C  R C  C  C R
 
Theoremsotritrieq 4053 A trichotomy relationship, given a trichotomous order. (Contributed by Jim Kingdon, 13-Dec-2019.)
 R  Or    &     C  R C  C  C R   =>     C  C  R C  C R
 
Theoremso0 4054 Any relation is a strict ordering of the empty set. (Contributed by NM, 16-Mar-1997.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 R  Or  (/)
 
2.3.8  Set-like relations
 
Syntaxwse 4055 Extend wff notation to include the set-like predicate. Read: '  R is set-like on .'
 R Se
 
Definitiondf-se 4056* Define the set-like predicate. (Contributed by Mario Carneiro, 19-Nov-2014.)
 R Se  {  |  R }  _V
 
Theoremseex 4057* The  R-preimage of an element of the base set in a set-like relation is a set. (Contributed by Mario Carneiro, 19-Nov-2014.)
 R Se  {  |  R }  _V
 
Theoremexse 4058 Any relation on a set is set-like on it. (Contributed by Mario Carneiro, 22-Jun-2015.)
 V  R Se
 
Theoremsess1 4059 Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
 R  C_  S  S Se  R Se
 
Theoremsess2 4060 Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
 C_  R Se  R Se
 
Theoremseeq1 4061 Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
 R  S  R Se  S Se
 
Theoremseeq2 4062 Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
 R Se  R Se
 
Theoremnfse 4063 Bound-variable hypothesis builder for set-like relations. (Contributed by Mario Carneiro, 24-Jun-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)
 F/_ R   &     F/_   =>     F/  R Se
 
Theoremepse 4064 The epsilon relation is set-like on any class. (This is the origin of the term "set-like": a set-like relation "acts like" the epsilon relation of sets and their elements.) (Contributed by Mario Carneiro, 22-Jun-2015.)

 _E Se
 
2.3.9  Ordinals
 
Syntaxword 4065 Extend the definition of a wff to include the ordinal predicate.
 Ord
 
Syntaxcon0 4066 Extend the definition of a class to include the class of all ordinal numbers. (The 0 in the name prevents creating a file called con.html, which causes problems in Windows.)

 On
 
Syntaxwlim 4067 Extend the definition of a wff to include the limit ordinal predicate.
 Lim
 
Syntaxcsuc 4068 Extend class notation to include the successor function.
 suc
 
Definitiondf-iord 4069* Define the ordinal predicate, which is true for a class that is transitive and whose elements are transitive. Definition of ordinal in [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic". (Contributed by Jim Kingdon, 10-Oct-2018.) Use its alias dford3 4070 instead for naming consistency with set.mm. (New usage is discouraged.)
 Ord  Tr  Tr
 
Theoremdford3 4070* Alias for df-iord 4069. Use it instead of df-iord 4069 for naming consistency with set.mm. (Contributed by Jim Kingdon, 10-Oct-2018.)
 Ord  Tr  Tr
 
Definitiondf-on 4071 Define the class of all ordinal numbers. Definition 7.11 of [TakeutiZaring] p. 38. (Contributed by NM, 5-Jun-1994.)

 On  {  |  Ord  }
 
Definitiondf-ilim 4072 Define the limit ordinal predicate, which is true for an ordinal that has the empty set as an element and is not a successor (i.e. that is the union of itself). Our definition combines the definition of Lim of [BellMachover] p. 471 and Exercise 1 of [TakeutiZaring] p. 42, and then changes  =/=  (/) to  (/) (which would be equivalent given the law of the excluded middle, but which is not for us). (Contributed by Jim Kingdon, 11-Nov-2018.) Use its alias dflim2 4073 instead for naming consistency with set.mm. (New usage is discouraged.)
 Lim  Ord  (/)  U.
 
Theoremdflim2 4073 Alias for df-ilim 4072. Use it instead of df-ilim 4072 for naming consistency with set.mm. (Contributed by NM, 4-Nov-2004.)
 Lim  Ord  (/)  U.
 
Definitiondf-suc 4074 Define the successor of a class. When applied to an ordinal number, the successor means the same thing as "plus 1". Definition 7.22 of [TakeutiZaring] p. 41, who use "+ 1" to denote this function. Our definition is a generalization to classes. Although it is not conventional to use it with proper classes, it has no effect on a proper class (sucprc 4115). Some authors denote the successor operation with a prime (apostrophe-like) symbol, such as Definition 6 of [Suppes] p. 134 and the definition of successor in [Mendelson] p. 246 (who uses the symbol "Suc" as a predicate to mean "is a successor ordinal"). The definition of successor of [Enderton] p. 68 denotes the operation with a plus-sign superscript. (Contributed by NM, 30-Aug-1993.)

 suc  u.  { }
 
Theoremordeq 4075 Equality theorem for the ordinal predicate. (Contributed by NM, 17-Sep-1993.)
 Ord  Ord
 
Theoremelong 4076 An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)
 V  On  Ord
 
Theoremelon 4077 An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)
 _V   =>     On  Ord
 
Theoremeloni 4078 An ordinal number has the ordinal property. (Contributed by NM, 5-Jun-1994.)
 On  Ord
 
Theoremelon2 4079 An ordinal number is an ordinal set. (Contributed by NM, 8-Feb-2004.)
 On 
 Ord  _V
 
Theoremlimeq 4080 Equality theorem for the limit predicate. (Contributed by NM, 22-Apr-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 Lim  Lim
 
Theoremordtr 4081 An ordinal class is transitive. (Contributed by NM, 3-Apr-1994.)
 Ord  Tr
 
Theoremordelss 4082 An element of an ordinal class is a subset of it. (Contributed by NM, 30-May-1994.)
 Ord  C_
 
Theoremtrssord 4083 A transitive subclass of an ordinal class is ordinal. (Contributed by NM, 29-May-1994.)
 Tr  C_  Ord  Ord
 
Theoremordelord 4084 An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. (Contributed by NM, 23-Apr-1994.)
 Ord  Ord
 
Theoremtron 4085 The class of all ordinal numbers is transitive. (Contributed by NM, 4-May-2009.)

 Tr  On
 
Theoremordelon 4086 An element of an ordinal class is an ordinal number. (Contributed by NM, 26-Oct-2003.)
 Ord  On
 
Theoremonelon 4087 An element of an ordinal number is an ordinal number. Theorem 2.2(iii) of [BellMachover] p. 469. (Contributed by NM, 26-Oct-2003.)
 On  On
 
Theoremordin 4088 The intersection of two ordinal classes is ordinal. Proposition 7.9 of [TakeutiZaring] p. 37. (Contributed by NM, 9-May-1994.)
 Ord 
 Ord  Ord  i^i
 
Theoremonin 4089 The intersection of two ordinal numbers is an ordinal number. (Contributed by NM, 7-Apr-1995.)
 On  On  i^i  On
 
Theoremonelss 4090 An element of an ordinal number is a subset of the number. (Contributed by NM, 5-Jun-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 On  C_
 
Theoremordtr1 4091 Transitive law for ordinal classes. (Contributed by NM, 12-Dec-2004.)
 Ord  C  C  C
 
Theoremontr1 4092 Transitive law for ordinal numbers. Theorem 7M(b) of [Enderton] p. 192. (Contributed by NM, 11-Aug-1994.)
 C  On  C  C
 
Theoremonintss 4093* If a property is true for an ordinal number, then the minimum ordinal number for which it is true is smaller or equal. Theorem Schema 61 of [Suppes] p. 228. (Contributed by NM, 3-Oct-2003.)
   =>     On  |^| {  On  | 
 }  C_
 
Theoremord0 4094 The empty set is an ordinal class. (Contributed by NM, 11-May-1994.)

 Ord  (/)
 
Theorem0elon 4095 The empty set is an ordinal number. Corollary 7N(b) of [Enderton] p. 193. (Contributed by NM, 17-Sep-1993.)
 (/)  On
 
Theoreminton 4096 The intersection of the class of ordinal numbers is the empty set. (Contributed by NM, 20-Oct-2003.)

 |^| On  (/)
 
Theoremnlim0 4097 The empty set is not a limit ordinal. (Contributed by NM, 24-Mar-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 Lim  (/)
 
Theoremlimord 4098 A limit ordinal is ordinal. (Contributed by NM, 4-May-1995.)
 Lim  Ord
 
Theoremlimuni 4099 A limit ordinal is its own supremum (union). (Contributed by NM, 4-May-1995.)
 Lim  U.
 
Theoremlimuni2 4100 The union of a limit ordinal is a limit ordinal. (Contributed by NM, 19-Sep-2006.)
 Lim  Lim  U.
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