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Theorem List for Metamath Proof Explorer - 4501-4600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-fr 4501* Define the well-founded relation predicate. Definition 6.24(1) of [TakeutiZaring] p. 30. For alternate definitions, see dffr2 4507 and dffr3 5195. (Contributed by NM, 3-Apr-1994.)

Definitiondf-se 4502* Define the set-like predicate. (Contributed by Mario Carneiro, 19-Nov-2014.)
Se

Definitiondf-we 4503 Define the well-ordering predicate. For an alternate definition, see dfwe2 4721. (Contributed by NM, 3-Apr-1994.)

Theoremfri 4504* Property of well-founded relation (one direction of definition). (Contributed by NM, 18-Mar-1997.)

Theoremseex 4505* 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 4506 Any relation on a set is set-like on it. (Contributed by Mario Carneiro, 22-Jun-2015.)
Se

Theoremdffr2 4507* 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 4508* 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 4509 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 4510 Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
Se Se

Theoremsess2 4511 Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
Se Se

Theoremfreq1 4512 Equality theorem for the well-founded predicate. (Contributed by NM, 9-Mar-1997.)

Theoremfreq2 4513 Equality theorem for the well-founded predicate. (Contributed by NM, 3-Apr-1994.)

Theoremseeq1 4514 Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
Se Se

Theoremseeq2 4515 Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
Se Se

Theoremnffr 4516 Bound-variable hypothesis builder for well-founded relations. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)

Theoremnfse 4517 Bound-variable hypothesis builder for set-like relations. (Contributed by Mario Carneiro, 24-Jun-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)
Se

Theoremnfwe 4518 Bound-variable hypothesis builder for well-orderings. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)

Theoremfrirr 4519 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 4520 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 4521 Any relation is well-founded on the empty set. (Contributed by NM, 17-Sep-1993.)

Theoremfrminex 4522* 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 4523 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 4524 A set founded by epsilon contains no 2-cycle loops. (Contributed by NM, 19-Apr-1994.)

Theoremepse 4525 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 4526 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 4527* 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 4528* 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 4529 Subset theorem for the well-ordering predicate. Exercise 4 of [TakeutiZaring] p. 31. (Contributed by NM, 19-Apr-1994.)

Theoremweeq1 4530 Equality theorem for the well-ordering predicate. (Contributed by NM, 9-Mar-1997.)

Theoremweeq2 4531 Equality theorem for the well-ordering predicate. (Contributed by NM, 3-Apr-1994.)

Theoremwefr 4532 A well-ordering is well-founded. (Contributed by NM, 22-Apr-1994.)

Theoremweso 4533 A well-ordering is a strict ordering. (Contributed by NM, 16-Mar-1997.)

Theoremwecmpep 4534 The elements of an epsilon well-ordering are comparable. (Contributed by NM, 17-May-1994.)

Theoremwetrep 4535 An epsilon well-ordering is a transitive relation. (Contributed by NM, 22-Apr-1994.)

Theoremwefrc 4536* 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 4537 Any relation is a well-ordering of the empty set. (Contributed by NM, 16-Mar-1997.)

Theoremwereu 4538* 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 4539* 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 4540 Extend the definition of a wff to include the ordinal predicate.

Syntaxcon0 4541 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 4542 Extend the definition of a wff to include the limit ordinal predicate.

Syntaxcsuc 4543 Extend class notation to include the successor function.

Definitiondf-ord 4544 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 4545 Define the class of all ordinal numbers. Definition 7.11 of [TakeutiZaring] p. 38. (Contributed by NM, 5-Jun-1994.)

Definitiondf-lim 4546 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 4597, dflim3 4786, and dflim4 for alternate definitions. (Contributed by NM, 22-Apr-1994.)

Definitiondf-suc 4547 Define the successor of a class. When applied to an ordinal number, the successor means the same thing as "plus 1" (see oa1suc 6734). 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 4616), so that the successor of any ordinal class is still an ordinal class (ordsuc 4753), 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 4548 Equality theorem for the ordinal predicate. (Contributed by NM, 17-Sep-1993.)

Theoremelong 4549 An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)

Theoremelon 4550 An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)

Theoremeloni 4551 An ordinal number has the ordinal property. (Contributed by NM, 5-Jun-1994.)

Theoremelon2 4552 An ordinal number is an ordinal set. (Contributed by NM, 8-Feb-2004.)

Theoremlimeq 4553 Equality theorem for the limit predicate. (Contributed by NM, 22-Apr-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremordwe 4554 Epsilon well-orders every ordinal. Proposition 7.4 of [TakeutiZaring] p. 36. (Contributed by NM, 3-Apr-1994.)

Theoremordtr 4555 An ordinal class is transitive. (Contributed by NM, 3-Apr-1994.)

Theoremordfr 4556 Epsilon is well-founded on an ordinal class. (Contributed by NM, 22-Apr-1994.)

Theoremordelss 4557 An element of an ordinal class is a subset of it. (Contributed by NM, 30-May-1994.)

Theoremtrssord 4558 A transitive subclass of an ordinal class is ordinal. (Contributed by NM, 29-May-1994.)

Theoremordirr 4559 Epsilon irreflexivity of ordinals: no ordinal class is a member of itself. Theorem 2.2(i) of [BellMachover] p. 469, generalized to classes. We prove this without invoking the Axiom of Regularity. (Contributed by NM, 2-Jan-1994.)

Theoremnordeq 4560 A member of an ordinal class is not equal to it. (Contributed by NM, 25-May-1998.)

Theoremordn2lp 4561 An ordinal class cannot an element of one of its members. Variant of first part of Theorem 2.2(vii) of [BellMachover] p. 469. (Contributed by NM, 3-Apr-1994.)

Theoremtz7.5 4562* A subclass (possibly proper) of an ordinal class has a minimal element. Proposition 7.5 of [TakeutiZaring] p. 36. (Contributed by NM, 18-Feb-2004.) (Revised by David Abernethy, 16-Mar-2011.)

Theoremordelord 4563 An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. (Contributed by NM, 23-Apr-1994.)

Theoremtron 4564 The class of all ordinal numbers is transitive. (Contributed by NM, 4-May-2009.)

Theoremordelon 4565 An element of an ordinal class is an ordinal number. (Contributed by NM, 26-Oct-2003.)

Theoremonelon 4566 An element of an ordinal number is an ordinal number. Theorem 2.2(iii) of [BellMachover] p. 469. (Contributed by NM, 26-Oct-2003.)

Theoremtz7.7 4567 Proposition 7.7 of [TakeutiZaring] p. 37. (Contributed by NM, 5-May-1994.)

Theoremordelssne 4568 Corollary 7.8 of [TakeutiZaring] p. 37. (Contributed by NM, 25-Nov-1995.)

Theoremordelpss 4569 Corollary 7.8 of [TakeutiZaring] p. 37. (Contributed by NM, 17-Jun-1998.)

Theoremordsseleq 4570 For ordinal classes, subclass is equivalent to membership or equality. (Contributed by NM, 25-Nov-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremordin 4571 The intersection of two ordinal classes is ordinal. Proposition 7.9 of [TakeutiZaring] p. 37. (Contributed by NM, 9-May-1994.)

Theoremonin 4572 The intersection of two ordinal numbers is an ordinal number. (Contributed by NM, 7-Apr-1995.)

Theoremordtri3or 4573 A trichotomy law for ordinals. Proposition 7.10 of [TakeutiZaring] p. 38. (Contributed by NM, 10-May-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremordtri1 4574 A trichotomy law for ordinals. (Contributed by NM, 25-Mar-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremontri1 4575 A trichotomy law for ordinal numbers. (Contributed by NM, 6-Nov-2003.)

Theoremordtri2 4576 A trichotomy law for ordinals. (Contributed by NM, 25-Nov-1995.)

Theoremordtri3 4577 A trichotomy law for ordinals. (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremordtri4 4578 A trichotomy law for ordinals. (Contributed by NM, 1-Nov-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremorddisj 4579 An ordinal class and its singleton are disjoint. (Contributed by NM, 19-May-1998.)

Theoremonfr 4580 The ordinal class is well-founded. This lemma is needed for ordon 4722 in order to eliminate the need for the Axiom of Regularity. (Contributed by NM, 17-May-1994.)

Theoremonelpss 4581 Relationship between membership and proper subset of an ordinal number. (Contributed by NM, 15-Sep-1995.)

Theoremonsseleq 4582 Relationship between subset and membership of an ordinal number. (Contributed by NM, 15-Sep-1995.)

Theoremonelss 4583 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.)

Theoremordtr1 4584 Transitive law for ordinal classes. (Contributed by NM, 12-Dec-2004.)

Theoremordtr2 4585 Transitive law for ordinal classes. (Contributed by NM, 12-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremordtr3 4586 Transitive law for ordinal classes. (Contributed by Mario Carneiro, 30-Dec-2014.)

Theoremontr1 4587 Transitive law for ordinal numbers. Theorem 7M(b) of [Enderton] p. 192. (Contributed by NM, 11-Aug-1994.)

Theoremontr2 4588 Transitive law for ordinal numbers. Exercise 3 of [TakeutiZaring] p. 40. (Contributed by NM, 6-Nov-2003.)

Theoremordunidif 4589 The union of an ordinal stays the same if a subset equal to one of its elements is removed. (Contributed by NM, 10-Dec-2004.)

Theoremordintdif 4590 If is smaller than , then it equals the intersection of the difference. Exercise 11 in [TakeutiZaring] p. 44. (Contributed by Andrew Salmon, 14-Nov-2011.)

Theoremonintss 4591* 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.)

Theoremoneqmini 4592* A way to show that an ordinal number equals the minimum of a collection of ordinal numbers: it must be in the collection, and it must not be larger than any member of the collection. (Contributed by NM, 14-Nov-2003.)

Theoremord0 4593 The empty set is an ordinal class. (Contributed by NM, 11-May-1994.)

Theorem0elon 4594 The empty set is an ordinal number. Corollary 7N(b) of [Enderton] p. 193. (Contributed by NM, 17-Sep-1993.)

Theoremord0eln0 4595 A non-empty ordinal contains the empty set. (Contributed by NM, 25-Nov-1995.)

Theoremon0eln0 4596 An ordinal number contains zero iff it is nonzero. (Contributed by NM, 6-Dec-2004.)

Theoremdflim2 4597 An alternate definition of a limit ordinal. (Contributed by NM, 4-Nov-2004.)

Theoreminton 4598 The intersection of the class of ordinal numbers is the empty set. (Contributed by NM, 20-Oct-2003.)

Theoremnlim0 4599 The empty set is not a limit ordinal. (Contributed by NM, 24-Mar-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremlimord 4600 A limit ordinal is ordinal. (Contributed by NM, 4-May-1995.)

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