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Theorem List for Metamath Proof Explorer - 4101-4200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremalxfr 4101* Transfer universal quantification from a variable to another variable contained in expression .
   =>   
 
Theoremralxfrd 4102* Transfer universal quantification from a variable to another variable contained in expression .
   &       &       =>   
 
Theoremrexxfrd 4103* Transfer universal quantification from a variable to another variable contained in expression . (Contributed by FL, 10-Apr-2007.)
   &       &       =>   
 
Theoremralxfr2d 4104* Transfer universal quantification from a variable to another variable contained in expression . (Contributed by Mario Carneiro, 20-Aug-2014.)
   &       &       =>   
 
Theoremrexxfr2d 4105* Transfer universal quantification from a variable to another variable contained in expression . (Contributed by Mario Carneiro, 20-Aug-2014.)
   &       &       =>   
 
TheoremralxfrdOLD 4106* Obsolete version of ralxfrd 4102 as of 14-Aug-2014.
   &       &       =>   
 
TheoremrexxfrdOLD 4107* Obsolete version of rexxfrd 4103 as of 14-Aug-2014.
   &       &       =>   
 
Theoremralxfr 4108* Transfer universal quantification from a variable to another variable contained in expression .
   &       &       =>   
 
TheoremralxfrALT 4109* Transfer universal quantification from a variable to another variable contained in expression .
   &       &       =>   
 
Theoremrexxfr 4110* Transfer existence from a variable to another variable contained in expression .
   &       &       =>   
 
TheoremralxfrOLD 4111* Obsolete version of ralxfr 4108 as of 14-Aug-2014.
   &       &       =>   
 
TheoremralxfrALTOLD 4112* Obsolete version of ralxfrALT 4109 as of 14-Aug-2014.
   &       &       =>   
 
TheoremrexxfrOLD 4113* Obsolete version of rexxfr 4110 as of 14-Aug-2014.
   &       &       =>   
 
Theoremrabxfrd 4114* Class builder membership after substituting an expression (containing ) for in the class expression .
   &       &       &       &       =>   
 
Theoremrabxfr 4115* Class builder membership after substituting an expression (containing ) for in the class expression .
   &       &       &       &       =>   
 
Theoremreuxfr2d 4116* Transfer existential uniqueness from a variable to another variable contained in expression .
   &       =>   
 
Theoremreuxfr2 4117* Transfer existential uniqueness from a variable to another variable contained in expression .
   &       =>   
 
Theoremreuxfrd 4118* Transfer existential uniqueness from a variable to another variable contained in expression . Use reuhypd 4120 to eliminate the second hypothesis.
   &       &       =>   
 
Theoremreuxfr 4119* Transfer existential uniqueness from a variable to another variable contained in expression . Use reuhyp 4121 to eliminate the second hypothesis.
   &       &       =>   
 
Theoremreuhypd 4120* A theorem useful for eliminating the restricted existential uniqueness hypotheses in riotaxfrd 6444.
   &       =>   
 
Theoremreuhyp 4121* A theorem useful for eliminating the restricted existential uniqueness hypotheses in reuxfr 4119.
   &       =>   
 
Theoremreuunixfr 4122* Change the variable in the expression for "the unique such that " to another variable contained in expression . Use reuhyp 4121 to eliminate the last hypothesis.
   &       &       &       &       &       =>   
 
Theoremuniexb 4123 The Axiom of Union and its converse. A class is a set iff its union is a set.
 
Theorempwexb 4124 The Axiom of Power Sets and its converse. A class is a set iff its power class is a set.
 
Theoremuniv 4125 The union of the universe is the universe. Exercise 4.12(c) of [Mendelson] p. 235.
 
Theoremeldifpw 4126 Membership in a power class difference.
   =>   
 
Theoremelpwun 4127 Membership in the power class of a union.
   =>   
 
Theoremelpwunsn 4128 Membership in an extension of a power class.
 
Theoremop1stb 4129 Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (See op2ndb 4689 to extract the second member, op1sta 4686 for an alternate version, and op1st 5619 for the preferred version.)
   &       =>   
 
Theoremiunpw 4130* An indexed union of a power class in terms of the power class of the union of its index. Part of Exercise 24(b) of [Enderton] p. 33.
   =>   
 
Theoremfr3nr 4131 A well-founded relation has no 3-cycle loops. Special case of Proposition 6.23 of [TakeutiZaring] p. 30.
 
Theoremepne3 4132 A set well-founded by epsilon contains no 3-cycle loops.
 
Theoremdfwe2 4133* Alternate definition of well-ordering. Definition 6.24(2) of [TakeutiZaring] p. 30. (The proof was shortened by Andrew Salmon, 12-Aug-2011.)
 
2.4.2  Ordinals (continued)
 
Theoremordon 4134 The class of all ordinal numbers is ordinal. Proposition 7.12 of [TakeutiZaring] p. 38, but without using the Axiom of Regularity.
 
Theoremepweon 4135 The epsilon relation well-orders the class of ordinal numbers. Proposition 4.8(g) of [Mendelson] p. 244.
 
Theoremonprc 4136 No set contains all ordinal numbers. Proposition 7.13 of [TakeutiZaring] p. 38, but without using the Axiom of Regularity. This is also known as the Burali-Forti paradox (remark in [Enderton] p. 194). In 1897, Cesare Burali-Forti noticed that since the "set" of all ordinal numbers is an ordinal class (ordon 4134), it must be both an element of the set of all ordinal numbers yet greater than every such element. ZF set theory resolves this paradox by not allowing the class of all ordinal numbers to be a set (so instead it is a proper class). Here we prove the denial of its existence.
 
Theoremssorduni 4137 The union of a class of ordinal numbers is ordinal. Proposition 7.19 of [TakeutiZaring] p. 40. (The proof was shortened by Andrew Salmon, 12-Aug-2011.)
 
Theoremssonuni 4138 The union of a set of ordinal numbers is an ordinal number. Theorem 9 of [Suppes] p. 132.
 
Theoremssonunii 4139 The union of a set of ordinal numbers is an ordinal number. Corollary 7N(d) of [Enderton] p. 193.
   =>   
 
Theoremordeleqon 4140 A way to express the ordinal property of a class in terms of the class of ordinal numbers. Corollary 7.14 of [TakeutiZaring] p. 38 and its converse.
 
Theoremordsson 4141 Any ordinal class is a subclass of the class of ordinal numbers. Corollary 7.15 of [TakeutiZaring] p. 38. (The proof was shortened by Andrew Salmon, 12-Aug-2011.)
 
Theoremonss 4142 An ordinal number is a subset of the class of ordinal numbers.
 
Theoremssonprc 4143 Two ways of saying a class of ordinals is unbounded. (Contributed by Mario Carneiro, 8-Jun-2013.)
 
Theoremonuni 4144 The union of an ordinal number is an ordinal number.
 
Theoremorduni 4145 The union of an ordinal class is ordinal.
 
Theoremonint 4146 The intersection (infimum) of a non-empty class of ordinal numbers belongs to the class. Compare Exercise 4 of [TakeutiZaring] p. 45.
 
Theoremonint0 4147 The intersection of a class of ordinal numbers is zero iff the class contains zero.
 
Theoremonssmin 4148* A non-empty class of ordinal numbers has a smallest member. Exercise 9 of [TakeutiZaring] p. 40.
 
Theoremonminsb 4149 If a property is true for some ordinal number, it is true for a minimal ordinal number. This version uses implicit substitution. Theorem Schema 62 of [Suppes] p. 228.
   &       =>   
 
Theoremonminesb 4150 If a property is true for some ordinal number, it is true for a minimal ordinal number. This version uses explicit substitution. Theorem Schema 62 of [Suppes] p. 228.
 
Theoremoninton 4151 The intersection of a non-empty collection of ordinal numbers is an ordinal number. Compare Exercise 6 of [TakeutiZaring] p. 44.
 
Theoremonintrab 4152 The intersection of a class of ordinal numbers exists iff it is an ordinal number.
 
Theoremonintrab2 4153 An existence condition equivalent to an intersection's being an ordinal number.
 
Theoremonnmin 4154 No member of a set of ordinal numbers belongs to its minimum.
 
Theoremonnminsb 4155* An ordinal number smaller than the minimum of a set of ordinal numbers does not have the property determining that set. is the wff resulting from the substitution of for in wff .
   =>   
 
Theoremoneqmin 4156* A way to show that an ordinal number equals the minimum of a non-empty collection of ordinal numbers: it must be in the collection, and it must not be larger than any member of the collection.
 
Theorembm2.5ii 4157* Problem 2.5(ii) of [BellMachover] p. 471.
   =>   
 
Theoremonminex 4158* If a wff is true for an ordinal number, there is a smallest ordinal number for which it is true.
   =>   
 
Theoremsucon 4159 The class of all ordinal numbers is its own successor.
 
Theoremsucexb 4160 A successor exists iff its class argument exists.
 
Theoremsucexg 4161 The successor of a set is a set (generalization).
 
Theoremsucex 4162 The successor of a set is a set.
   =>   
 
Theoremonmindif2 4163 The minimum of a class of ordinal numbers is less than the minimum of that class with its minimum removed.
 
Theoremsuceloni 4164 The successor of an ordinal number is an ordinal number. Proposition 7.24 of [TakeutiZaring] p. 41.
 
Theoremordsuc 4165 The successor of an ordinal class is ordinal.
 
Theoremordpwsuc 4166 The collection of ordinals in the power class of an ordinal is its successor.
 
Theoremonpwsuc 4167 The collection of ordinal numbers in the power set of an ordinal number is its successor.
 
Theoremsucelon 4168 The successor of an ordinal number is an ordinal number.
 
Theoremordsucss 4169 The successor of an element of an ordinal class is a subset of it.
 
Theoremonpsssuc 4170 An ordinal number is a proper subset of its successor. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 
Theoremordelsuc 4171 A set belongs to an ordinal iff its successor is a subset of the ordinal. Exercise 8 of [TakeutiZaring] p. 42 and its converse.
 
Theoremonsucmin 4172* The successor of an ordinal number is the smallest larger ordinal number.
 
Theoremordsucelsuc 4173 Membership is inherited by successors. Generalization of Exercise 9 of [TakeutiZaring] p. 42. (The proof was shortened by Andrew Salmon, 12-Aug-2011.)
 
Theoremordsucsssuc 4174 The subclass relationship between two ordinal classes is inherited by their successors.
 
Theoremordsucuniel 4175 Given an element of the union of an ordinal , is an element of itself. (Contributed by Scott Fenton, 28-Mar-2012.) (The proof was shortened by Mario Carneiro, 29-May-2015.)
 
Theoremordsucun 4176 The successor of the maximum (i.e. union) of two ordinals is the maximum of their successors.
 
Theoremordunpr 4177 The maximum of two ordinals is equal to one of them.
 
Theoremordunel 4178 The maximum of two ordinals belongs to a third if each of them do.
 
Theoremonsucuni 4179 A class of ordinal numbers is a subclass of the successor of its union. Similar to Proposition 7.26 of [TakeutiZaring] p. 41.
 
Theoremordsucuni 4180 An ordinal class is a subclass of the successor of its union.
 
Theoremorduniorsuc 4181 An ordinal class is either its union or the successor of its union. If we adopt the view that zero is a limit ordinal, this means every ordinal class is either a limit or a successor.
 
Theoremunon 4182 The class of all ordinal numbers is its own union. Exercise 11 of [TakeutiZaring] p. 40.
 
Theoremordunisuc 4183 An ordinal class is equal to the union of its successor. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
 
Theoremorduniss2 4184* The union of the ordinal subsets of an ordinal number is that number.
 
Theoremonsucuni2 4185 A successor ordinal is the successor of its union. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
 
Theorem0elsuc 4186 The successor of an ordinal class contains the empty set.
 
Theoremlimon 4187 The class of ordinal numbers is a limit ordinal.
 
Theoremonssi 4188 An ordinal number is a subset of .
   =>   
 
Theoremonsuci 4189 The successor of an ordinal number is an ordinal number. Corollary 7N(c) of [Enderton] p. 193.
   =>   
 
Theoremonuniorsuci 4190 An ordinal number is either its own union (if zero or a limit ordinal) or the successor of its union.
   =>   
 
Theoremonuninsuci 4191* A limit ordinal is not a successor ordinal.
   =>   
 
Theoremonsucssi 4192 A set belongs to an ordinal number iff its successor is a subset of the ordinal number. Exercise 8 of [TakeutiZaring] p. 42 and its converse.
   &       =>   
 
Theoremnlimsucg 4193 A successor is not a limit ordinal. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
 
Theoremorduninsuc 4194* An ordinal equal to its union is not a successor.
 
Theoremordunisuc2 4195* An ordinal equal to its union contains the successor of each of its members.
 
Theoremordzsl 4196* An ordinal is zero, a successor ordinal, or a limit ordinal.
 
Theoremonzsl 4197* An ordinal number is zero, a successor ordinal, or a limit ordinal number. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
 
Theoremdflim3 4198* An alternate definition of a limit ordinal, which is any ordinal that is neither zero nor a successor. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
 
Theoremdflim4 4199* An alternate definition of a limit ordinal.
 
Theoremlimsuc 4200 The successor of a member of a limit ordinal is also a member.
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