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Theorem List for Metamath Proof Explorer - 4301-4400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremordelss 4301 An element of an ordinal class is a subset of it. (Contributed by NM, 30-May-1994.)

Theoremtrssord 4302 A transitive subclass of an ordinal class is ordinal. (Contributed by NM, 29-May-1994.)

Theoremordirr 4303 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 4304 A member of an ordinal class is not equal to it. (Contributed by NM, 25-May-1998.)

Theoremordn2lp 4305 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 4306* 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 4307 An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. (Contributed by NM, 23-Apr-1994.)

Theoremtron 4308 The class of all ordinal numbers is transitive. (Contributed by NM, 4-May-2009.)

Theoremordelon 4309 An element of an ordinal class is an ordinal number. (Contributed by NM, 26-Oct-2003.)

Theoremonelon 4310 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 4311 Proposition 7.7 of [TakeutiZaring] p. 37. (Contributed by NM, 5-May-1994.)

Theoremordelssne 4312 Corollary 7.8 of [TakeutiZaring] p. 37. (Contributed by NM, 25-Nov-1995.)

Theoremordelpss 4313 Corollary 7.8 of [TakeutiZaring] p. 37. (Contributed by NM, 17-Jun-1998.)

Theoremordsseleq 4314 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 4315 The intersection of two ordinal classes is ordinal. Proposition 7.9 of [TakeutiZaring] p. 37. (Contributed by NM, 9-May-1994.)

Theoremonin 4316 The intersection of two ordinal numbers is an ordinal number. (Contributed by NM, 7-Apr-1995.)

Theoremordtri3or 4317 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 4318 A trichotomy law for ordinals. (Contributed by NM, 25-Mar-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremontri1 4319 A trichotomy law for ordinal numbers. (Contributed by NM, 6-Nov-2003.)

Theoremordtri2 4320 A trichotomy law for ordinals. (Contributed by NM, 25-Nov-1995.)

Theoremordtri3 4321 A trichotomy law for ordinals. (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremordtri4 4322 A trichotomy law for ordinals. (Contributed by NM, 1-Nov-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremorddisj 4323 An ordinal class and its singleton are disjoint. (Contributed by NM, 19-May-1998.)

Theoremonfr 4324 The ordinal class is well-founded. This lemma is needed for ordon 4465 in order to eliminate the need for the Axiom of Regularity. (Contributed by NM, 17-May-1994.)

Theoremonelpss 4325 Relationship between membership and proper subset of an ordinal number. (Contributed by NM, 15-Sep-1995.)

Theoremonsseleq 4326 Relationship between subset and membership of an ordinal number. (Contributed by NM, 15-Sep-1995.)

Theoremonelss 4327 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 4328 Transitive law for ordinal classes. (Contributed by NM, 12-Dec-2004.)

Theoremordtr2 4329 Transitive law for ordinal classes. (Contributed by NM, 12-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremordtr3 4330 Transitive law for ordinal classes. (Contributed by Mario Carneiro, 30-Dec-2014.)

Theoremontr1 4331 Transitive law for ordinal numbers. Theorem 7M(b) of [Enderton] p. 192. (Contributed by NM, 11-Aug-1994.)

Theoremontr2 4332 Transitive law for ordinal numbers. Exercise 3 of [TakeutiZaring] p. 40. (Contributed by NM, 6-Nov-2003.)

Theoremordunidif 4333 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 4334 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 4335* 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 4336* 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 4337 The empty set is an ordinal class. (Contributed by NM, 11-May-1994.)

Theorem0elon 4338 The empty set is an ordinal number. Corollary 7N(b) of [Enderton] p. 193. (Contributed by NM, 17-Sep-1993.)

Theoremord0eln0 4339 A non-empty ordinal contains the empty set. (Contributed by NM, 25-Nov-1995.)

Theoremon0eln0 4340 An ordinal number contains zero iff it is nonzero. (Contributed by NM, 6-Dec-2004.)

Theoremdflim2 4341 An alternate definition of a limit ordinal. (Contributed by NM, 4-Nov-2004.)

Theoreminton 4342 The intersection of the class of ordinal numbers is the empty set. (Contributed by NM, 20-Oct-2003.)

Theoremnlim0 4343 The empty set is not a limit ordinal. (Contributed by NM, 24-Mar-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremlimord 4344 A limit ordinal is ordinal. (Contributed by NM, 4-May-1995.)

Theoremlimuni 4345 A limit ordinal is its own supremum (union). (Contributed by NM, 4-May-1995.)

Theoremlimuni2 4346 The union of a limit ordinal is a limit ordinal. (Contributed by NM, 19-Sep-2006.)

Theorem0ellim 4347 A limit ordinal contains the empty set. (Contributed by NM, 15-May-1994.)

Theoremlimelon 4348 A limit ordinal class that is also a set is an ordinal number. (Contributed by NM, 26-Apr-2004.)

Theoremonn0 4349 The class of all ordinal numbers in not empty. (Contributed by NM, 17-Sep-1995.)

Theoremsuceq 4350 Equality of successors. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremelsuci 4351 Membership in a successor. This one-way implication does not require that either or be sets. (Contributed by NM, 6-Jun-1994.)

Theoremelsucg 4352 Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-1995.)

Theoremelsuc2g 4353 Variant of membership in a successor, requiring that rather than be a set. (Contributed by NM, 28-Oct-2003.)

Theoremelsuc 4354 Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-2003.)

Theoremelsuc2 4355 Membership in a successor. (Contributed by NM, 15-Sep-2003.)

Theoremnfsuc 4356 Bound-variable hypothesis builder for successor. (Contributed by NM, 15-Sep-2003.)

Theoremelelsuc 4357 Membership in a successor. (Contributed by NM, 20-Jun-1998.)

Theoremsucel 4358* Membership of a successor in another class. (Contributed by NM, 29-Jun-2004.)

Theoremsuc0 4359 The successor of the empty set. (Contributed by NM, 1-Feb-2005.)

Theoremsucprc 4360 A proper class is its own successor. (Contributed by NM, 3-Apr-1995.)

Theoremunisuc 4361 A transitive class is equal to the union of its successor. Combines Theorem 4E of [Enderton] p. 72 and Exercise 6 of [Enderton] p. 73. (Contributed by NM, 30-Aug-1993.)

Theoremsssucid 4362 A class is included in its own successor. Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized to arbitrary classes). (Contributed by NM, 31-May-1994.)

Theoremsucidg 4363 Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized). (Contributed by NM, 25-Mar-1995.) (Proof shortened by Scott Fenton, 20-Feb-2012.)

Theoremsucid 4364 A set belongs to its successor. (Contributed by NM, 22-Jun-1994.) (Proof shortened by Alan Sare, 18-Feb-2012.) (Proof shortened by Scott Fenton, 20-Feb-2012.)

Theoremnsuceq0 4365 No successor is empty. (Contributed by NM, 3-Apr-1995.)

Theoremeqelsuc 4366 A set belongs to the successor of an equal set. (Contributed by NM, 18-Aug-1994.)

Theoremiunsuc 4367* Inductive definition for the indexed union at a successor. (Contributed by Mario Carneiro, 4-Feb-2013.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)

Theoremsuctr 4368 The successor of a transtive class is transitive. (Contributed by Alan Sare, 11-Apr-2009.)

Theoremtrsuc 4369 A set whose successor belongs to a transitive class also belongs. (Contributed by NM, 5-Sep-2003.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)

Theoremtrsuc2OLD 4370 Obsolete proof of suctr 4368 as of 5-Apr-2016. The successor of a transitive set is transitive. (Contributed by Scott Fenton, 21-Feb-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremtrsucss 4371 A member of the successor of a transitive class is a subclass of it. (Contributed by NM, 4-Oct-2003.)

Theoremordsssuc 4372 A subset of an ordinal belongs to its successor. (Contributed by NM, 28-Nov-2003.)

Theoremonsssuc 4373 A subset of an ordinal number belongs to its successor. (Contributed by NM, 15-Sep-1995.)

Theoremordsssuc2 4374 An ordinal subset of an ordinal number belongs to its successor. (Contributed by NM, 1-Feb-2005.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)

Theoremonmindif 4375 When its successor is subtracted from a class of ordinal numbers, an ordinal number is less than the minimum of the resulting subclass. (Contributed by NM, 1-Dec-2003.)

Theoremordnbtwn 4376 There is no set between an ordinal class and its successor. Generalized Proposition 7.25 of [TakeutiZaring] p. 41. (Contributed by NM, 21-Jun-1998.)

Theoremonnbtwn 4377 There is no set between an ordinal number and its successor. Proposition 7.25 of [TakeutiZaring] p. 41. (Contributed by NM, 9-Jun-1994.)

Theoremsucssel 4378 A set whose successor is a subset of another class is a member of that class. (Contributed by NM, 16-Sep-1995.)

Theoremorddif 4379 Ordinal derived from its successor. (Contributed by NM, 20-May-1998.)

Theoremorduniss 4380 An ordinal class includes its union. (Contributed by NM, 13-Sep-2003.)

Theoremordtri2or 4381 A trichotomy law for ordinal classes. (Contributed by NM, 13-Sep-2003.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)

Theoremordtri2or2 4382 A trichotomy law for ordinal classes. (Contributed by NM, 2-Nov-2003.)

Theoremordssun 4383 Property of a subclass of the maximum (i.e. union) of two ordinals. (Contributed by NM, 28-Nov-2003.)

Theoremordequn 4384 The maximum (i.e. union) of two ordinals is either one or the other. Similar to Exercise 14 of [TakeutiZaring] p. 40. (Contributed by NM, 28-Nov-2003.)

Theoremordun 4385 The maximum (i.e. union) of two ordinals is ordinal. Exercise 12 of [TakeutiZaring] p. 40. (Contributed by NM, 28-Nov-2003.)

Theoremordunisssuc 4386 A subclass relationship for union and successor of ordinal classes. (Contributed by NM, 28-Nov-2003.)

Theoremsuc11 4387 The successor operation behaves like a one-to-one function. Compare Exercise 16 of [Enderton] p. 194. (Contributed by NM, 3-Sep-2003.)

Theoremonordi 4388 An ordinal number is an ordinal class. (Contributed by NM, 11-Jun-1994.)

Theoremontrci 4389 An ordinal number is a transitive class. (Contributed by NM, 11-Jun-1994.)

Theoremonirri 4390 An ordinal number is not a member of itself. Theorem 7M(c) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.)

Theoremoneli 4391 A member of an ordinal number is an ordinal number. Theorem 7M(a) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.)

Theoremonelssi 4392 A member of an ordinal number is a subset of it. (Contributed by NM, 11-Aug-1994.)

Theoremonssneli 4393 An ordering law for ordinal numbers. (Contributed by NM, 13-Jun-1994.)

Theoremonssnel2i 4394 An ordering law for ordinal numbers. (Contributed by NM, 13-Jun-1994.)

Theoremonelini 4395 An element of an ordinal number equals the intersection with it. (Contributed by NM, 11-Jun-1994.)

Theoremoneluni 4396 An ordinal number equals its union with any element. (Contributed by NM, 13-Jun-1994.)

Theoremonunisuci 4397 An ordinal number is equal to the union of its successor. (Contributed by NM, 12-Jun-1994.)

Theoremonsseli 4398 Subset is equivalent to membership or equality for ordinal numbers. (Contributed by NM, 15-Sep-1995.)

Theoremonun2i 4399 The union of two ordinal numbers is an ordinal number. (Contributed by NM, 13-Jun-1994.)

Theoremunizlim 4400 An ordinal equal to its own union is either zero or a limit ordinal. (Contributed by NM, 1-Oct-2003.)

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