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Theorem List for Metamath Proof Explorer - 4201-4300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremepel 4201 The epsilon relation and the membership relation are the same. (Contributed by NM, 13-Aug-1995.)

Definitiondf-id 4202* Define the identity relation. Definition 9.15 of [Quine] p. 64. For example, and (ex-id 20634). (Contributed by NM, 13-Aug-1995.)

Theoremdfid3 4203 A stronger version of df-id 4202 that doesn't require and to be distinct. Ordinarily, we wouldn't use this as a definition, since non-distinct dummy variables would make soundness verification more difficult (as the proof here shows). The proof can be instructive in showing how distinct variable requirements may be eliminated, a task that is not necessarily obvious. (Contributed by NM, 5-Feb-2008.) (Revised by Mario Carneiro, 18-Nov-2016.)

Theoremdfid2 4204 Alternate definition of the identity relation. (Contributed by NM, 15-Mar-2007.)

2.3.7  Partial and complete ordering

Syntaxwpo 4205 Extend wff notation to include the strict partial ordering predicate. Read: ' is a partial order on .'

Syntaxwor 4206 Extend wff notation to include the strict complete ordering predicate. Read: ' orders .'

Definitiondf-po 4207* Define the strict partial order predicate. Definition of [Enderton] p. 168. The expression means is a partial order on . For example, is true, while is false (ex-po 20635). (Contributed by NM, 16-Mar-1997.)

Definitiondf-so 4208* Define the strict complete (linear) order predicate. The expression is true if relationship orders . For example, is true (ltso 8783). (Contributed by NM, 21-Jan-1996.)

Theoremposs 4209 Subset theorem for the partial ordering predicate. (Contributed by NM, 27-Mar-1997.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)

Theorempoeq1 4210 Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.)

Theorempoeq2 4211 Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.)

Theoremnfpo 4212 Bound-variable hypothesis builder for partial orders. (Contributed by Stefan O'Rear, 20-Jan-2015.)

Theoremnfso 4213 Bound-variable hypothesis builder for total orders. (Contributed by Stefan O'Rear, 20-Jan-2015.)

Theorempocl 4214 Properties of partial order relation in class notation. (Contributed by NM, 27-Mar-1997.)

Theoremispod 4215* Sufficient conditions for a partial order. (Contributed by NM, 9-Jul-2014.)

Theoremswopolem 4216* Perform the substitutions into the strict weak ordering law. (Contributed by Mario Carneiro, 31-Dec-2014.)

Theoremswopo 4217* A strict weak order is a partial order. (Contributed by Mario Carneiro, 9-Jul-2014.)

Theorempoirr 4218 A partial order relation is irreflexive. (Contributed by NM, 27-Mar-1997.)

Theorempotr 4219 A partial order relation is a transitive relation. (Contributed by NM, 27-Mar-1997.)

Theorempo2nr 4220 A partial order relation has no 2-cycle loops. (Contributed by NM, 27-Mar-1997.)

Theorempo3nr 4221 A partial order relation has no 3-cycle loops. (Contributed by NM, 27-Mar-1997.)

Theorempo0 4222 Any relation is a partial ordering of the empty set. (Contributed by NM, 28-Mar-1997.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theorempofun 4223* A function preserves a partial order relation. (Contributed by Jeff Madsen, 18-Jun-2011.)

Theoremsopo 4224 A strict linear order is a strict partial order. (Contributed by NM, 28-Mar-1997.)

Theoremsoss 4225 Subset theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremsoeq1 4226 Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.)

Theoremsoeq2 4227 Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.)

Theoremsonr 4228 A strict order relation is irreflexive. (Contributed by NM, 24-Nov-1995.)

Theoremsotr 4229 A strict order relation is a transitive relation. (Contributed by NM, 21-Jan-1996.)

Theoremsolin 4230 A strict order relation is linear (satisfies trichotomy). (Contributed by NM, 21-Jan-1996.)

Theoremso2nr 4231 A strict order relation has no 2-cycle loops. (Contributed by NM, 21-Jan-1996.)

Theoremso3nr 4232 A strict order relation has no 3-cycle loops. (Contributed by NM, 21-Jan-1996.)

Theoremsotric 4233 A strict order relation satisfies strict trichotomy. (Contributed by NM, 19-Feb-1996.)

Theoremsotrieq 4234 Trichotomy law for strict order relation. (Contributed by NM, 9-Apr-1996.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremsotrieq2 4235 Trichotomy law for strict order relation. (Contributed by NM, 5-May-1999.)

Theoremsotr2 4236 A transitivity relation. (Read and implies .) (Contributed by Mario Carneiro, 10-May-2013.)

Theoremissod 4237* A irreflexive, transitive, linear relation is a strict ordering. (Contributed by NM, 21-Jan-1996.) (Revised by Mario Carneiro, 9-Jul-2014.)

Theoremissoi 4238* A irreflexive, transitive, linear relation is a strict ordering. (Contributed by NM, 21-Jan-1996.) (Revised by Mario Carneiro, 9-Jul-2014.)

Theoremisso2i 4239* Deduce strict ordering from its properties. (Contributed by NM, 29-Jan-1996.) (Revised by Mario Carneiro, 9-Jul-2014.)

Theoremso0 4240 Any relation is a strict ordering of the empty set. (Contributed by NM, 16-Mar-1997.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremsomo 4241* A totally ordered set has at most one minimal element. (Contributed by Mario Carneiro, 24-Jun-2015.)

2.3.8  Founded and well-ordering relations

Syntaxwfr 4242 Extend wff notation to include the well-founded predicate. Read: ' is a well-founded relation on .'

Syntaxwse 4243 Extend wff notation to include the set-like predicate. Read: ' is set-like on .'
Se

Syntaxwwe 4244 Extend wff notation to include the well-ordering predicate. Read: ' well-orders .'

Definitiondf-fr 4245* Define the well-founded relation predicate. Definition 6.24(1) of [TakeutiZaring] p. 30. For alternate definitions, see dffr2 4251 and dffr3 4952. (Contributed by NM, 3-Apr-1994.)

Definitiondf-se 4246* Define the set-like predicate. (Contributed by Mario Carneiro, 19-Nov-2014.)
Se

Definitiondf-we 4247 Define the well-ordering predicate. For an alternate definition, see dfwe2 4464. (Contributed by NM, 3-Apr-1994.)

Theoremfri 4248* Property of well-founded relation (one direction of definition). (Contributed by NM, 18-Mar-1997.)

Theoremseex 4249* The -preimage of an element of the base set in a set-like relation is a set. (Contributed by Mario Carneiro, 19-Nov-2014.)
Se

Theoremexse 4250 Any relation on a set is set-like on it. (Contributed by Mario Carneiro, 22-Jun-2015.)
Se

Theoremdffr2 4251* Alternate definition of well-founded relation. Similar to Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by NM, 17-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Proof shortened by Mario Carneiro, 23-Jun-2015.)

Theoremfrc 4252* Property of well-founded relation (one direction of definition using class variables). (Contributed by NM, 17-Feb-2004.) (Revised by Mario Carneiro, 19-Nov-2014.)

Theoremfrss 4253 Subset theorem for the well-founded predicate. Exercise 1 of [TakeutiZaring] p. 31. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremsess1 4254 Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
Se Se

Theoremsess2 4255 Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
Se Se

Theoremfreq1 4256 Equality theorem for the well-founded predicate. (Contributed by NM, 9-Mar-1997.)

Theoremfreq2 4257 Equality theorem for the well-founded predicate. (Contributed by NM, 3-Apr-1994.)

Theoremseeq1 4258 Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
Se Se

Theoremseeq2 4259 Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
Se Se

Theoremnffr 4260 Bound-variable hypothesis builder for well-founded relations. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)

Theoremnfse 4261 Bound-variable hypothesis builder for set-like relations. (Contributed by Mario Carneiro, 24-Jun-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)
Se

Theoremnfwe 4262 Bound-variable hypothesis builder for well-orderings. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)

Theoremfrirr 4263 A well-founded relation is irreflexive. Special case of Proposition 6.23 of [TakeutiZaring] p. 30. (Contributed by NM, 2-Jan-1994.) (Revised by Mario Carneiro, 22-Jun-2015.)

Theoremfr2nr 4264 A well-founded relation has no 2-cycle loops. Special case of Proposition 6.23 of [TakeutiZaring] p. 30. (Contributed by NM, 30-May-1994.) (Revised by Mario Carneiro, 22-Jun-2015.)

Theoremfr0 4265 Any relation is well-founded on the empty set. (Contributed by NM, 17-Sep-1993.)

Theoremfrminex 4266* If an element of a well-founded set satisfies a property , then there is a minimal element that satisfies . (Contributed by Jeff Madsen, 18-Jun-2010.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)

Theoremefrirr 4267 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.)

Theoremefrn2lp 4268 A set founded by epsilon contains no 2-cycle loops. (Contributed by NM, 19-Apr-1994.)

Theoremepse 4269 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.)
Se

Theoremtz7.2 4270 Similar to Theorem 7.2 of [TakeutiZaring] p. 35, of except that the Axiom of Regularity is not required due to antecedent . (Contributed by NM, 4-May-1994.)

Theoremdfepfr 4271* An alternate way of saying that the epsilon relation is well-founded. (Contributed by NM, 17-Feb-2004.) (Revised by Mario Carneiro, 23-Jun-2015.)

Theoremepfrc 4272* A subset of an epsilon-founded class has a minimal element. (Contributed by NM, 17-Feb-2004.) (Revised by David Abernethy, 22-Feb-2011.)

Theoremwess 4273 Subset theorem for the well-ordering predicate. Exercise 4 of [TakeutiZaring] p. 31. (Contributed by NM, 19-Apr-1994.)

Theoremweeq1 4274 Equality theorem for the well-ordering predicate. (Contributed by NM, 9-Mar-1997.)

Theoremweeq2 4275 Equality theorem for the well-ordering predicate. (Contributed by NM, 3-Apr-1994.)

Theoremwefr 4276 A well-ordering is well-founded. (Contributed by NM, 22-Apr-1994.)

Theoremweso 4277 A well-ordering is a strict ordering. (Contributed by NM, 16-Mar-1997.)

Theoremwecmpep 4278 The elements of an epsilon well-ordering are comparable. (Contributed by NM, 17-May-1994.)

Theoremwetrep 4279 An epsilon well-ordering is a transitive relation. (Contributed by NM, 22-Apr-1994.)

Theoremwefrc 4280* A non-empty (possibly proper) subclass of a class well-ordered by has a minimal element. Special case of Proposition 6.26 of [TakeutiZaring] p. 31. (Contributed by NM, 17-Feb-2004.)

Theoremwe0 4281 Any relation is a well-ordering of the empty set. (Contributed by NM, 16-Mar-1997.)

Theoremwereu 4282* A subset of a well-ordered set has a unique minimal element. (Contributed by NM, 18-Mar-1997.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremwereu2 4283* All nonempty (possibly proper) subclasses of , which has a well-founded relation , have -minimal elements. Proposition 6.26 of [TakeutiZaring] p. 31. (Contributed by Scott Fenton, 29-Jan-2011.) (Revised by Mario Carneiro, 24-Jun-2015.)
Se

2.3.9  Ordinals

Syntaxword 4284 Extend the definition of a wff to include the ordinal predicate.

Syntaxcon0 4285 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.)

Syntaxwlim 4286 Extend the definition of a wff to include the limit ordinal predicate.

Syntaxcsuc 4287 Extend class notation to include the successor function.

Definitiondf-ord 4288 Define the ordinal predicate, which is true for a class that is transitive and is well-ordered by the epsilon relation. Variant of definition of [BellMachover] p. 468. (Contributed by NM, 17-Sep-1993.)

Definitiondf-on 4289 Define the class of all ordinal numbers. Definition 7.11 of [TakeutiZaring] p. 38. (Contributed by NM, 5-Jun-1994.)

Definitiondf-lim 4290 Define the limit ordinal predicate, which is true for a non-empty ordinal that 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. See dflim2 4341, dflim3 4529, and dflim4 for alternate definitions. (Contributed by NM, 22-Apr-1994.)

Definitiondf-suc 4291 Define the successor of a class. When applied to an ordinal number, the successor means the same thing as "plus 1" (see oa1suc 6416). 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 affect on a proper class (sucprc 4360), so that the successor of any ordinal class is still an ordinal class (ordsuc 4496), simplifying certain proofs. 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.)

Theoremordeq 4292 Equality theorem for the ordinal predicate. (Contributed by NM, 17-Sep-1993.)

Theoremelong 4293 An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)

Theoremelon 4294 An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)

Theoremeloni 4295 An ordinal number has the ordinal property. (Contributed by NM, 5-Jun-1994.)

Theoremelon2 4296 An ordinal number is an ordinal set. (Contributed by NM, 8-Feb-2004.)

Theoremlimeq 4297 Equality theorem for the limit predicate. (Contributed by NM, 22-Apr-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremordwe 4298 Epsilon well orders every ordinal. Proposition 7.4 of [TakeutiZaring] p. 36. (Contributed by NM, 3-Apr-1994.)

Theoremordtr 4299 An ordinal class is transitive. (Contributed by NM, 3-Apr-1994.)

Theoremordfr 4300 Epsilon is well-founded on an ordinal class. (Contributed by NM, 22-Apr-1994.)

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