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Theorem List for Intuitionistic Logic Explorer - 4101-4200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremposs 4101 Subset theorem for the partial ordering predicate. (Contributed by NM, 27-Mar-1997.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
 |-  ( A  C_  B  ->  ( R  Po  B  ->  R  Po  A ) )
 
Theorempoeq1 4102 Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.)
 |-  ( R  =  S  ->  ( R  Po  A  <->  S  Po  A ) )
 
Theorempoeq2 4103 Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.)
 |-  ( A  =  B  ->  ( R  Po  A  <->  R  Po  B ) )
 
Theoremnfpo 4104 Bound-variable hypothesis builder for partial orders. (Contributed by Stefan O'Rear, 20-Jan-2015.)
 |-  F/_ x R   &    |-  F/_ x A   =>    |-  F/ x  R  Po  A
 
Theoremnfso 4105 Bound-variable hypothesis builder for total orders. (Contributed by Stefan O'Rear, 20-Jan-2015.)
 |-  F/_ x R   &    |-  F/_ x A   =>    |-  F/ x  R  Or  A
 
Theorempocl 4106 Properties of partial order relation in class notation. (Contributed by NM, 27-Mar-1997.)
 |-  ( R  Po  A  ->  ( ( B  e.  A  /\  C  e.  A  /\  D  e.  A ) 
 ->  ( -.  B R B  /\  ( ( B R C  /\  C R D )  ->  B R D ) ) ) )
 
Theoremispod 4107* Sufficient conditions for a partial order. (Contributed by NM, 9-Jul-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  -.  x R x )   &    |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A ) )  ->  ( ( x R y  /\  y R z )  ->  x R z ) )   =>    |-  ( ph  ->  R  Po  A )
 
Theoremswopolem 4108* Perform the substitutions into the strict weak ordering law. (Contributed by Mario Carneiro, 31-Dec-2014.)
 |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A )
 )  ->  ( x R y  ->  ( x R z  \/  z R y ) ) )   =>    |-  ( ( ph  /\  ( X  e.  A  /\  Y  e.  A  /\  Z  e.  A )
 )  ->  ( X R Y  ->  ( X R Z  \/  Z R Y ) ) )
 
Theoremswopo 4109* A strict weak order is a partial order. (Contributed by Mario Carneiro, 9-Jul-2014.)
 |-  ( ( ph  /\  (
 y  e.  A  /\  z  e.  A )
 )  ->  ( y R z  ->  -.  z R y ) )   &    |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A )
 )  ->  ( x R y  ->  ( x R z  \/  z R y ) ) )   =>    |-  ( ph  ->  R  Po  A )
 
Theorempoirr 4110 A partial order relation is irreflexive. (Contributed by NM, 27-Mar-1997.)
 |-  ( ( R  Po  A  /\  B  e.  A )  ->  -.  B R B )
 
Theorempotr 4111 A partial order relation is a transitive relation. (Contributed by NM, 27-Mar-1997.)
 |-  ( ( R  Po  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A ) )  ->  ( ( B R C  /\  C R D )  ->  B R D ) )
 
Theorempo2nr 4112 A partial order relation has no 2-cycle loops. (Contributed by NM, 27-Mar-1997.)
 |-  ( ( R  Po  A  /\  ( B  e.  A  /\  C  e.  A ) )  ->  -.  ( B R C  /\  C R B ) )
 
Theorempo3nr 4113 A partial order relation has no 3-cycle loops. (Contributed by NM, 27-Mar-1997.)
 |-  ( ( R  Po  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A ) )  ->  -.  ( B R C  /\  C R D  /\  D R B ) )
 
Theorempo0 4114 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 4115* A function preserves a partial order relation. (Contributed by Jeff Madsen, 18-Jun-2011.)
 |-  S  =  { <. x ,  y >.  |  X R Y }   &    |-  ( x  =  y  ->  X  =  Y )   =>    |-  ( ( R  Po  B  /\  A. x  e.  A  X  e.  B )  ->  S  Po  A )
 
Theoremsopo 4116 A strict linear order is a strict partial order. (Contributed by NM, 28-Mar-1997.)
 |-  ( R  Or  A  ->  R  Po  A )
 
Theoremsoss 4117 Subset theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 |-  ( A  C_  B  ->  ( R  Or  B  ->  R  Or  A ) )
 
Theoremsoeq1 4118 Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.)
 |-  ( R  =  S  ->  ( R  Or  A  <->  S  Or  A ) )
 
Theoremsoeq2 4119 Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.)
 |-  ( A  =  B  ->  ( R  Or  A  <->  R  Or  B ) )
 
Theoremsonr 4120 A strict order relation is irreflexive. (Contributed by NM, 24-Nov-1995.)
 |-  ( ( R  Or  A  /\  B  e.  A )  ->  -.  B R B )
 
Theoremsotr 4121 A strict order relation is a transitive relation. (Contributed by NM, 21-Jan-1996.)
 |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A ) )  ->  ( ( B R C  /\  C R D )  ->  B R D ) )
 
Theoremissod 4122* An irreflexive, transitive, trichotomous relation is a linear ordering (in the sense of df-iso 4100). (Contributed by NM, 21-Jan-1996.) (Revised by Mario Carneiro, 9-Jul-2014.)
 |-  ( ph  ->  R  Po  A )   &    |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  A ) )  ->  ( x R y  \/  x  =  y  \/  y R x ) )   =>    |-  ( ph  ->  R  Or  A )
 
Theoremsowlin 4123 A strict order relation satisfies weak linearity. (Contributed by Jim Kingdon, 6-Oct-2018.)
 |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A ) )  ->  ( B R C  ->  ( B R D  \/  D R C ) ) )
 
Theoremso2nr 4124 A strict order relation has no 2-cycle loops. (Contributed by NM, 21-Jan-1996.)
 |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A ) )  ->  -.  ( B R C  /\  C R B ) )
 
Theoremso3nr 4125 A strict order relation has no 3-cycle loops. (Contributed by NM, 21-Jan-1996.)
 |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A ) )  ->  -.  ( B R C  /\  C R D  /\  D R B ) )
 
Theoremsotricim 4126 One direction of sotritric 4127 holds for all weakly linear orders. (Contributed by Jim Kingdon, 28-Sep-2019.)
 |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A ) )  ->  ( B R C  ->  -.  ( B  =  C  \/  C R B ) ) )
 
Theoremsotritric 4127 A trichotomy relationship, given a trichotomous order. (Contributed by Jim Kingdon, 28-Sep-2019.)
 |-  R  Or  A   &    |-  (
 ( B  e.  A  /\  C  e.  A ) 
 ->  ( B R C  \/  B  =  C  \/  C R B ) )   =>    |-  ( ( B  e.  A  /\  C  e.  A )  ->  ( B R C 
 <->  -.  ( B  =  C  \/  C R B ) ) )
 
Theoremsotritrieq 4128 A trichotomy relationship, given a trichotomous order. (Contributed by Jim Kingdon, 13-Dec-2019.)
 |-  R  Or  A   &    |-  (
 ( B  e.  A  /\  C  e.  A ) 
 ->  ( B R C  \/  B  =  C  \/  C R B ) )   =>    |-  ( ( B  e.  A  /\  C  e.  A )  ->  ( B  =  C 
 <->  -.  ( B R C  \/  C R B ) ) )
 
Theoremso0 4129 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  (/)
 
2.3.9  Founded and set-like relations
 
Syntaxwfrfor 4130 Extend wff notation to include the well-founded predicate.
 wff FrFor  R A S
 
Syntaxwfr 4131 Extend wff notation to include the well-founded predicate. Read: '  R is a well-founded relation on 
A.'
 wff  R  Fr  A
 
Syntaxwse 4132 Extend wff notation to include the set-like predicate. Read: '  R is set-like on  A.'
 wff  R Se  A
 
Syntaxwwe 4133 Extend wff notation to include the well-ordering predicate. Read: '  R well-orders  A.'
 wff  R  We  A
 
Definitiondf-frfor 4134* Define the well-founded relation predicate where  A might be a proper class. By passing in  S we allow it potentially to be a proper class rather than a set. (Contributed by Jim Kingdon and Mario Carneiro, 22-Sep-2021.)
 |-  (FrFor  R A S  <->  (
 A. x  e.  A  ( A. y  e.  A  ( y R x 
 ->  y  e.  S )  ->  x  e.  S )  ->  A  C_  S ) )
 
Definitiondf-frind 4135* Define the well-founded relation predicate. In the presence of excluded middle, there are a variety of equivalent ways to define this. In our case, this definition, in terms of an inductive principle, works better than one along the lines of "there is an element which is minimal when A is ordered by R". Because  s is constrained to be a set (not a proper class) here, sometimes it may be necessary to use FrFor directly rather than via  Fr. (Contributed by Jim Kingdon and Mario Carneiro, 21-Sep-2021.)
 |-  ( R  Fr  A  <->  A. sFrFor  R A s )
 
Definitiondf-se 4136* Define the set-like predicate. (Contributed by Mario Carneiro, 19-Nov-2014.)
 |-  ( R Se  A  <->  A. x  e.  A  { y  e.  A  |  y R x }  e.  _V )
 
Definitiondf-wetr 4137* Define the well-ordering predicate. It is unusual to define "well-ordering" in the absence of excluded middle, but we mean an ordering which is like the ordering which we have for ordinals (for example, it does not entail trichotomy because ordinals don't have that as seen at ordtriexmid 4313). Given excluded middle, well-ordering is usually defined to require trichotomy (and the defintion of  Fr is typically also different). (Contributed by Mario Carneiro and Jim Kingdon, 23-Sep-2021.)
 |-  ( R  We  A  <->  ( R  Fr  A  /\  A. x  e.  A  A. y  e.  A  A. z  e.  A  ( ( x R y  /\  y R z )  ->  x R z ) ) )
 
Theoremseex 4138* 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  A  /\  B  e.  A ) 
 ->  { x  e.  A  |  x R B }  e.  _V )
 
Theoremexse 4139 Any relation on a set is set-like on it. (Contributed by Mario Carneiro, 22-Jun-2015.)
 |-  ( A  e.  V  ->  R Se  A )
 
Theoremsess1 4140 Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
 |-  ( R  C_  S  ->  ( S Se  A  ->  R Se 
 A ) )
 
Theoremsess2 4141 Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
 |-  ( A  C_  B  ->  ( R Se  B  ->  R Se 
 A ) )
 
Theoremseeq1 4142 Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
 |-  ( R  =  S  ->  ( R Se  A  <->  S Se  A )
 )
 
Theoremseeq2 4143 Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
 |-  ( A  =  B  ->  ( R Se  A  <->  R Se  B )
 )
 
Theoremnfse 4144 Bound-variable hypothesis builder for set-like relations. (Contributed by Mario Carneiro, 24-Jun-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)
 |-  F/_ x R   &    |-  F/_ x A   =>    |-  F/ x  R Se  A
 
Theoremepse 4145 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  A
 
Theoremfrforeq1 4146 Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.)
 |-  ( R  =  S  ->  (FrFor  R A T  <-> FrFor  S A T ) )
 
Theoremfreq1 4147 Equality theorem for the well-founded predicate. (Contributed by NM, 9-Mar-1997.)
 |-  ( R  =  S  ->  ( R  Fr  A  <->  S  Fr  A ) )
 
Theoremfrforeq2 4148 Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.)
 |-  ( A  =  B  ->  (FrFor  R A T  <-> FrFor  R B T ) )
 
Theoremfreq2 4149 Equality theorem for the well-founded predicate. (Contributed by NM, 3-Apr-1994.)
 |-  ( A  =  B  ->  ( R  Fr  A  <->  R  Fr  B ) )
 
Theoremfrforeq3 4150 Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.)
 |-  ( S  =  T  ->  (FrFor  R A S  <-> FrFor  R A T ) )
 
Theoremnffrfor 4151 Bound-variable hypothesis builder for well-founded relations. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)
 |-  F/_ x R   &    |-  F/_ x A   &    |-  F/_ x S   =>    |- 
 F/ xFrFor  R A S
 
Theoremnffr 4152 Bound-variable hypothesis builder for well-founded relations. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)
 |-  F/_ x R   &    |-  F/_ x A   =>    |-  F/ x  R  Fr  A
 
Theoremfrirrg 4153 A well-founded relation is irreflexive. This is the case where  A exists. (Contributed by Jim Kingdon, 21-Sep-2021.)
 |-  ( ( R  Fr  A  /\  A  e.  V  /\  B  e.  A ) 
 ->  -.  B R B )
 
Theoremfr0 4154 Any relation is well-founded on the empty set. (Contributed by NM, 17-Sep-1993.)
 |-  R  Fr  (/)
 
Theoremfrind 4155* Induction over a well-founded set. (Contributed by Jim Kingdon, 28-Sep-2021.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   &    |-  (
 ( ch  /\  x  e.  A )  ->  ( A. y  e.  A  ( y R x 
 ->  ps )  ->  ph )
 )   &    |-  ( ch  ->  R  Fr  A )   &    |-  ( ch  ->  A  e.  V )   =>    |-  ( ( ch 
 /\  x  e.  A )  ->  ph )
 
Theoremefrirr 4156 Irreflexivity of the epsilon relation: a class founded by epsilon is not a member of itself. (Contributed by NM, 18-Apr-1994.) (Revised by Mario Carneiro, 22-Jun-2015.)
 |-  (  _E  Fr  A  ->  -.  A  e.  A )
 
Theoremtz7.2 4157 Similar to Theorem 7.2 of [TakeutiZaring] p. 35, of except that the Axiom of Regularity is not required due to antecedent  _E  Fr  A. (Contributed by NM, 4-May-1994.)
 |-  ( ( Tr  A  /\  _E  Fr  A  /\  B  e.  A )  ->  ( B  C_  A  /\  B  =/=  A ) )
 
Theoremnfwe 4158 Bound-variable hypothesis builder for well-orderings. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)
 |-  F/_ x R   &    |-  F/_ x A   =>    |-  F/ x  R  We  A
 
Theoremweeq1 4159 Equality theorem for the well-ordering predicate. (Contributed by NM, 9-Mar-1997.)
 |-  ( R  =  S  ->  ( R  We  A  <->  S  We  A ) )
 
Theoremweeq2 4160 Equality theorem for the well-ordering predicate. (Contributed by NM, 3-Apr-1994.)
 |-  ( A  =  B  ->  ( R  We  A  <->  R  We  B ) )
 
Theoremwefr 4161 A well-ordering is well-founded. (Contributed by NM, 22-Apr-1994.)
 |-  ( R  We  A  ->  R  Fr  A )
 
Theoremwepo 4162 A well-ordering is a partial ordering. (Contributed by Jim Kingdon, 23-Sep-2021.)
 |-  ( ( R  We  A  /\  A  e.  V )  ->  R  Po  A )
 
Theoremwetrep 4163* An epsilon well-ordering is a transitive relation. (Contributed by NM, 22-Apr-1994.)
 |-  ( (  _E  We  A  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A ) )  ->  ( ( x  e.  y  /\  y  e.  z )  ->  x  e.  z ) )
 
Theoremwe0 4164 Any relation is a well-ordering of the empty set. (Contributed by NM, 16-Mar-1997.)
 |-  R  We  (/)
 
2.3.10  Ordinals
 
Syntaxword 4165 Extend the definition of a wff to include the ordinal predicate.
 wff  Ord  A
 
Syntaxcon0 4166 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.)
 class  On
 
Syntaxwlim 4167 Extend the definition of a wff to include the limit ordinal predicate.
 wff  Lim  A
 
Syntaxcsuc 4168 Extend class notation to include the successor function.
 class  suc  A
 
Definitiondf-iord 4169* 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 4170 instead for naming consistency with set.mm. (New usage is discouraged.)
 |-  ( Ord  A  <->  ( Tr  A  /\  A. x  e.  A  Tr  x ) )
 
Theoremdford3 4170* Alias for df-iord 4169. Use it instead of df-iord 4169 for naming consistency with set.mm. (Contributed by Jim Kingdon, 10-Oct-2018.)
 |-  ( Ord  A  <->  ( Tr  A  /\  A. x  e.  A  Tr  x ) )
 
Definitiondf-on 4171 Define the class of all ordinal numbers. Definition 7.11 of [TakeutiZaring] p. 38. (Contributed by NM, 5-Jun-1994.)
 |- 
 On  =  { x  |  Ord  x }
 
Definitiondf-ilim 4172 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  A  =/=  (/) to  (/)  e.  A (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 4173 instead for naming consistency with set.mm. (New usage is discouraged.)
 |-  ( Lim  A  <->  ( Ord  A  /\  (/)  e.  A  /\  A  =  U. A ) )
 
Theoremdflim2 4173 Alias for df-ilim 4172. Use it instead of df-ilim 4172 for naming consistency with set.mm. (Contributed by NM, 4-Nov-2004.)
 |-  ( Lim  A  <->  ( Ord  A  /\  (/)  e.  A  /\  A  =  U. A ) )
 
Definitiondf-suc 4174 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 4215). 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  A  =  ( A  u.  { A }
 )
 
Theoremordeq 4175 Equality theorem for the ordinal predicate. (Contributed by NM, 17-Sep-1993.)
 |-  ( A  =  B  ->  ( Ord  A  <->  Ord  B ) )
 
Theoremelong 4176 An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)
 |-  ( A  e.  V  ->  ( A  e.  On  <->  Ord  A ) )
 
Theoremelon 4177 An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)
 |-  A  e.  _V   =>    |-  ( A  e.  On 
 <-> 
 Ord  A )
 
Theoremeloni 4178 An ordinal number has the ordinal property. (Contributed by NM, 5-Jun-1994.)
 |-  ( A  e.  On  ->  Ord  A )
 
Theoremelon2 4179 An ordinal number is an ordinal set. (Contributed by NM, 8-Feb-2004.)
 |-  ( A  e.  On  <->  ( Ord  A  /\  A  e.  _V ) )
 
Theoremlimeq 4180 Equality theorem for the limit predicate. (Contributed by NM, 22-Apr-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 |-  ( A  =  B  ->  ( Lim  A  <->  Lim  B ) )
 
Theoremordtr 4181 An ordinal class is transitive. (Contributed by NM, 3-Apr-1994.)
 |-  ( Ord  A  ->  Tr  A )
 
Theoremordelss 4182 An element of an ordinal class is a subset of it. (Contributed by NM, 30-May-1994.)
 |-  ( ( Ord  A  /\  B  e.  A ) 
 ->  B  C_  A )
 
Theoremtrssord 4183 A transitive subclass of an ordinal class is ordinal. (Contributed by NM, 29-May-1994.)
 |-  ( ( Tr  A  /\  A  C_  B  /\  Ord 
 B )  ->  Ord  A )
 
Theoremordelord 4184 An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. (Contributed by NM, 23-Apr-1994.)
 |-  ( ( Ord  A  /\  B  e.  A ) 
 ->  Ord  B )
 
Theoremtron 4185 The class of all ordinal numbers is transitive. (Contributed by NM, 4-May-2009.)
 |- 
 Tr  On
 
Theoremordelon 4186 An element of an ordinal class is an ordinal number. (Contributed by NM, 26-Oct-2003.)
 |-  ( ( Ord  A  /\  B  e.  A ) 
 ->  B  e.  On )
 
Theoremonelon 4187 An element of an ordinal number is an ordinal number. Theorem 2.2(iii) of [BellMachover] p. 469. (Contributed by NM, 26-Oct-2003.)
 |-  ( ( A  e.  On  /\  B  e.  A )  ->  B  e.  On )
 
Theoremordin 4188 The intersection of two ordinal classes is ordinal. Proposition 7.9 of [TakeutiZaring] p. 37. (Contributed by NM, 9-May-1994.)
 |-  ( ( Ord  A  /\  Ord  B )  ->  Ord  ( A  i^i  B ) )
 
Theoremonin 4189 The intersection of two ordinal numbers is an ordinal number. (Contributed by NM, 7-Apr-1995.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  i^i  B )  e.  On )
 
Theoremonelss 4190 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.)
 |-  ( A  e.  On  ->  ( B  e.  A  ->  B  C_  A )
 )
 
Theoremordtr1 4191 Transitive law for ordinal classes. (Contributed by NM, 12-Dec-2004.)
 |-  ( Ord  C  ->  ( ( A  e.  B  /\  B  e.  C ) 
 ->  A  e.  C ) )
 
Theoremontr1 4192 Transitive law for ordinal numbers. Theorem 7M(b) of [Enderton] p. 192. (Contributed by NM, 11-Aug-1994.)
 |-  ( C  e.  On  ->  ( ( A  e.  B  /\  B  e.  C )  ->  A  e.  C ) )
 
Theoremonintss 4193* 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.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( A  e.  On  ->  ( ps  ->  |^|
 { x  e.  On  |  ph }  C_  A ) )
 
Theoremord0 4194 The empty set is an ordinal class. (Contributed by NM, 11-May-1994.)
 |- 
 Ord  (/)
 
Theorem0elon 4195 The empty set is an ordinal number. Corollary 7N(b) of [Enderton] p. 193. (Contributed by NM, 17-Sep-1993.)
 |-  (/)  e.  On
 
Theoreminton 4196 The intersection of the class of ordinal numbers is the empty set. (Contributed by NM, 20-Oct-2003.)
 |- 
 |^| On  =  (/)
 
Theoremnlim0 4197 The empty set is not a limit ordinal. (Contributed by NM, 24-Mar-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 |- 
 -.  Lim  (/)
 
Theoremlimord 4198 A limit ordinal is ordinal. (Contributed by NM, 4-May-1995.)
 |-  ( Lim  A  ->  Ord 
 A )
 
Theoremlimuni 4199 A limit ordinal is its own supremum (union). (Contributed by NM, 4-May-1995.)
 |-  ( Lim  A  ->  A  =  U. A )
 
Theoremlimuni2 4200 The union of a limit ordinal is a limit ordinal. (Contributed by NM, 19-Sep-2006.)
 |-  ( Lim  A  ->  Lim  U. A )
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