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Theorem List for Intuitionistic Logic Explorer - 4201-4300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremssopab2 4201 Equivalence of ordered pair abstraction subclass and implication. (Contributed by NM, 27-Dec-1996.) (Revised by Mario Carneiro, 19-May-2013.)
 |-  ( A. x A. y ( ph  ->  ps )  ->  { <. x ,  y >.  |  ph }  C_  {
 <. x ,  y >.  |  ps } )
 
Theoremssopab2b 4202 Equivalence of ordered pair abstraction subclass and implication. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
 |-  ( { <. x ,  y >.  |  ph }  C_  {
 <. x ,  y >.  |  ps }  <->  A. x A. y
 ( ph  ->  ps )
 )
 
Theoremssopab2i 4203 Inference of ordered pair abstraction subclass from implication. (Contributed by NM, 5-Apr-1995.)
 |-  ( ph  ->  ps )   =>    |-  { <. x ,  y >.  |  ph } 
 C_  { <. x ,  y >.  |  ps }
 
Theoremssopab2dv 4204* Inference of ordered pair abstraction subclass from implication. (Contributed by NM, 19-Jan-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  {
 <. x ,  y >.  |  ps }  C_  { <. x ,  y >.  |  ch } )
 
Theoremeqopab2b 4205 Equivalence of ordered pair abstraction equality and biconditional. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( { <. x ,  y >.  |  ph }  =  { <. x ,  y >.  |  ps }  <->  A. x A. y
 ( ph  <->  ps ) )
 
Theoremopabm 4206* Inhabited ordered pair class abstraction. (Contributed by Jim Kingdon, 29-Sep-2018.)
 |-  ( E. z  z  e.  { <. x ,  y >.  |  ph }  <->  E. x E. y ph )
 
Theoremiunopab 4207* Move indexed union inside an ordered-pair abstraction. (Contributed by Stefan O'Rear, 20-Feb-2015.)
 |-  U_ z  e.  A  { <. x ,  y >.  |  ph }  =  { <. x ,  y >.  |  E. z  e.  A  ph }
 
2.3.6  Power class of union and intersection
 
Theorempwin 4208 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 ( A  i^i  B )  =  ( ~P A  i^i  ~P B )
 
Theorempwunss 4209 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 A  u.  ~P B )  C_  ~P ( A  u.  B )
 
Theorempwssunim 4210 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.)
 |-  ( ( A  C_  B  \/  B  C_  A )  ->  ~P ( A  u.  B )  C_  ( ~P A  u.  ~P B ) )
 
Theorempwundifss 4211 Break up the power class of a union into a union of smaller classes. (Contributed by Jim Kingdon, 30-Sep-2018.)
 |-  ( ( ~P ( A  u.  B )  \  ~P A )  u.  ~P A )  C_  ~P ( A  u.  B )
 
Theorempwunim 4212 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.)
 |-  ( ( A  C_  B  \/  B  C_  A )  ->  ~P ( A  u.  B )  =  ( ~P A  u.  ~P B ) )
 
2.3.7  Epsilon and identity relations
 
Syntaxcep 4213 Extend class notation to include the epsilon relation.
 class  _E
 
Syntaxcid 4214 Extend the definition of a class to include identity relation.
 class  _I
 
Definitiondf-eprel 4215* Define the epsilon relation. Similar to Definition 6.22 of [TakeutiZaring] p. 30. The epsilon relation and set membership are the same, that is,  ( A  _E  B  <->  A  e.  B ) when  B is a set by epelg 4216. Thus, 5  _E { 1 , 5 }. (Contributed by NM, 13-Aug-1995.)
 |- 
 _E  =  { <. x ,  y >.  |  x  e.  y }
 
Theoremepelg 4216 The epsilon relation and membership are the same. General version of epel 4218. (Contributed by Scott Fenton, 27-Mar-2011.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( B  e.  V  ->  ( A  _E  B  <->  A  e.  B ) )
 
Theoremepelc 4217 The epsilon relationship and the membership relation are the same. (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  B  e.  _V   =>    |-  ( A  _E  B 
 <->  A  e.  B )
 
Theoremepel 4218 The epsilon relation and the membership relation are the same. (Contributed by NM, 13-Aug-1995.)
 |-  ( x  _E  y  <->  x  e.  y )
 
Definitiondf-id 4219* 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  =  { <. x ,  y >.  |  x  =  y }
 
2.3.8  Partial and complete ordering
 
Syntaxwpo 4220 Extend wff notation to include the strict partial ordering predicate. Read: '  R is a partial order on  A.'
 wff  R  Po  A
 
Syntaxwor 4221 Extend wff notation to include the strict linear ordering predicate. Read: '  R orders  A.'
 wff  R  Or  A
 
Definitiondf-po 4222* Define the strict partial order predicate. Definition of [Enderton] p. 168. The expression  R  Po  A means  R is a partial order on  A. (Contributed by NM, 16-Mar-1997.)
 |-  ( R  Po  A  <->  A. x  e.  A  A. y  e.  A  A. z  e.  A  ( -.  x R x  /\  ( ( x R y  /\  y R z )  ->  x R z ) ) )
 
Definitiondf-iso 4223* Define the strict linear order predicate. The expression  R  Or  A is true if relationship  R orders  A. The property  x R y  ->  ( x R z  \/  z R y ) is called weak linearity by Proposition 11.2.3 of [HoTT], p. (varies). If we assumed excluded middle, it would be equivalent to trichotomy, 
x R y  \/  x  =  y  \/  y R x. (Contributed by NM, 21-Jan-1996.) (Revised by Jim Kingdon, 4-Oct-2018.)
 |-  ( R  Or  A  <->  ( R  Po  A  /\  A. x  e.  A  A. y  e.  A  A. z  e.  A  ( x R y  ->  ( x R z  \/  z R y ) ) ) )
 
Theoremposs 4224 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 4225 Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.)
 |-  ( R  =  S  ->  ( R  Po  A  <->  S  Po  A ) )
 
Theorempoeq2 4226 Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.)
 |-  ( A  =  B  ->  ( R  Po  A  <->  R  Po  B ) )
 
Theoremnfpo 4227 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 4228 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 4229 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 4230* 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 4231* 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 4232* 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 4233 A partial order relation is irreflexive. (Contributed by NM, 27-Mar-1997.)
 |-  ( ( R  Po  A  /\  B  e.  A )  ->  -.  B R B )
 
Theorempotr 4234 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 4235 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 4236 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 4237 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 4238* 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 4239 A strict linear order is a strict partial order. (Contributed by NM, 28-Mar-1997.)
 |-  ( R  Or  A  ->  R  Po  A )
 
Theoremsoss 4240 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 4241 Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.)
 |-  ( R  =  S  ->  ( R  Or  A  <->  S  Or  A ) )
 
Theoremsoeq2 4242 Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.)
 |-  ( A  =  B  ->  ( R  Or  A  <->  R  Or  B ) )
 
Theoremsonr 4243 A strict order relation is irreflexive. (Contributed by NM, 24-Nov-1995.)
 |-  ( ( R  Or  A  /\  B  e.  A )  ->  -.  B R B )
 
Theoremsotr 4244 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 4245* An irreflexive, transitive, trichotomous relation is a linear ordering (in the sense of df-iso 4223). (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 4246 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 4247 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 4248 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 4249 One direction of sotritric 4250 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 4250 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 4251 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 4252 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 4253 Extend wff notation to include the well-founded predicate.
 wff FrFor  R A S
 
Syntaxwfr 4254 Extend wff notation to include the well-founded predicate. Read: '  R is a well-founded relation on 
A.'
 wff  R  Fr  A
 
Syntaxwse 4255 Extend wff notation to include the set-like predicate. Read: '  R is set-like on  A.'
 wff  R Se  A
 
Syntaxwwe 4256 Extend wff notation to include the well-ordering predicate. Read: '  R well-orders  A.'
 wff  R  We  A
 
Definitiondf-frfor 4257* 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 4258* 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 4259* 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 4260* 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 do not have that as seen at ordtriexmid 4441). Given excluded middle, well-ordering is usually defined to require trichotomy (and the definition 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 4261* 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 4262 Any relation on a set is set-like on it. (Contributed by Mario Carneiro, 22-Jun-2015.)
 |-  ( A  e.  V  ->  R Se  A )
 
Theoremsess1 4263 Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
 |-  ( R  C_  S  ->  ( S Se  A  ->  R Se 
 A ) )
 
Theoremsess2 4264 Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
 |-  ( A  C_  B  ->  ( R Se  B  ->  R Se 
 A ) )
 
Theoremseeq1 4265 Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
 |-  ( R  =  S  ->  ( R Se  A  <->  S Se  A )
 )
 
Theoremseeq2 4266 Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
 |-  ( A  =  B  ->  ( R Se  A  <->  R Se  B )
 )
 
Theoremnfse 4267 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 4268 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 4269 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 4270 Equality theorem for the well-founded predicate. (Contributed by NM, 9-Mar-1997.)
 |-  ( R  =  S  ->  ( R  Fr  A  <->  S  Fr  A ) )
 
Theoremfrforeq2 4271 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 4272 Equality theorem for the well-founded predicate. (Contributed by NM, 3-Apr-1994.)
 |-  ( A  =  B  ->  ( R  Fr  A  <->  R  Fr  B ) )
 
Theoremfrforeq3 4273 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 4274 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 4275 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 4276 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 4277 Any relation is well-founded on the empty set. (Contributed by NM, 17-Sep-1993.)
 |-  R  Fr  (/)
 
Theoremfrind 4278* 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 4279 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 4280 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 4281 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 4282 Equality theorem for the well-ordering predicate. (Contributed by NM, 9-Mar-1997.)
 |-  ( R  =  S  ->  ( R  We  A  <->  S  We  A ) )
 
Theoremweeq2 4283 Equality theorem for the well-ordering predicate. (Contributed by NM, 3-Apr-1994.)
 |-  ( A  =  B  ->  ( R  We  A  <->  R  We  B ) )
 
Theoremwefr 4284 A well-ordering is well-founded. (Contributed by NM, 22-Apr-1994.)
 |-  ( R  We  A  ->  R  Fr  A )
 
Theoremwepo 4285 A well-ordering is a partial ordering. (Contributed by Jim Kingdon, 23-Sep-2021.)
 |-  ( ( R  We  A  /\  A  e.  V )  ->  R  Po  A )
 
Theoremwetrep 4286* 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 4287 Any relation is a well-ordering of the empty set. (Contributed by NM, 16-Mar-1997.)
 |-  R  We  (/)
 
2.3.10  Ordinals
 
Syntaxword 4288 Extend the definition of a wff to include the ordinal predicate.
 wff  Ord  A
 
Syntaxcon0 4289 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 4290 Extend the definition of a wff to include the limit ordinal predicate.
 wff  Lim  A
 
Syntaxcsuc 4291 Extend class notation to include the successor function.
 class  suc  A
 
Definitiondf-iord 4292* 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 4293 instead for naming consistency with set.mm. (New usage is discouraged.)
 |-  ( Ord  A  <->  ( Tr  A  /\  A. x  e.  A  Tr  x ) )
 
Theoremdford3 4293* Alias for df-iord 4292. Use it instead of df-iord 4292 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 4294 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 4295 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 4296 instead for naming consistency with set.mm. (New usage is discouraged.)
 |-  ( Lim  A  <->  ( Ord  A  /\  (/)  e.  A  /\  A  =  U. A ) )
 
Theoremdflim2 4296 Alias for df-ilim 4295. Use it instead of df-ilim 4295 for naming consistency with set.mm. (Contributed by NM, 4-Nov-2004.)
 |-  ( Lim  A  <->  ( Ord  A  /\  (/)  e.  A  /\  A  =  U. A ) )
 
Definitiondf-suc 4297 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 4338). 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 4298 Equality theorem for the ordinal predicate. (Contributed by NM, 17-Sep-1993.)
 |-  ( A  =  B  ->  ( Ord  A  <->  Ord  B ) )
 
Theoremelong 4299 An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)
 |-  ( A  e.  V  ->  ( A  e.  On  <->  Ord  A ) )
 
Theoremelon 4300 An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)
 |-  A  e.  _V   =>    |-  ( A  e.  On 
 <-> 
 Ord  A )
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