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Theorem List for Metamath Proof Explorer - 4101-4200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmpteq1d 4101* An equality theorem for the maps to notation. (Contributed by Mario Carneiro, 11-Jun-2016.)

Theoremmpteq2ia 4102 An equality inference for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)

Theoremmpteq2i 4103 An equality inference for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)

Theoremmpteq12i 4104 An equality inference for the maps to notation. (Contributed by Scott Fenton, 27-Oct-2010.) (Revised by Mario Carneiro, 16-Dec-2013.)

Theoremmpteq2da 4105 Slightly more general equality inference for the maps to notation. (Contributed by FL, 14-Sep-2013.) (Revised by Mario Carneiro, 16-Dec-2013.)

Theoremmpteq2dva 4106* Slightly more general equality inference for the maps to notation. (Contributed by Scott Fenton, 25-Apr-2012.)

Theoremmpteq2dv 4107* An equality inference for the maps to notation. (Contributed by Mario Carneiro, 23-Aug-2014.)

Theoremnfmpt 4108* Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.)

Theoremnfmpt1 4109 Bound-variable hypothesis builder for the maps-to notation. (Contributed by FL, 17-Feb-2008.)

Theoremcbvmpt 4110* Rule to change the bound variable in a maps-to function, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by NM, 11-Sep-2011.)

Theoremcbvmptv 4111* Rule to change the bound variable in a maps-to function, using implicit substitution. (Contributed by Mario Carneiro, 19-Feb-2013.)

Theoremmptv 4112* Function with universal domain in maps-to notation. (Contributed by NM, 16-Aug-2013.)

2.1.24  Transitive classes

Syntaxwtr 4113 Extend wff notation to include transitive classes. Notation from [TakeutiZaring] p. 35.

Definitiondf-tr 4114 Define the transitive class predicate. Not to be confused with a transitive relation (see cotr 5055). Definition of [Enderton] p. 71 extended to arbitrary classes. For alternate definitions, see dftr2 4115 (which is suggestive of the word "transitive"), dftr3 4117, dftr4 4118, dftr5 4116, and (when is a set) unisuc 4468. The term "complete" is used instead of "transitive" in Definition 3 of [Suppes] p. 130. (Contributed by NM, 29-Aug-1993.)

Theoremdftr2 4115* An alternate way of defining a transitive class. Exercise 7 of [TakeutiZaring] p. 40. (Contributed by NM, 24-Apr-1994.)

Theoremdftr5 4116* An alternate way of defining a transitive class. (Contributed by NM, 20-Mar-2004.)

Theoremdftr3 4117* An alternate way of defining a transitive class. Definition 7.1 of [TakeutiZaring] p. 35. (Contributed by NM, 29-Aug-1993.)

Theoremdftr4 4118 An alternate way of defining a transitive class. Definition of [Enderton] p. 71. (Contributed by NM, 29-Aug-1993.)

Theoremtreq 4119 Equality theorem for the transitive class predicate. (Contributed by NM, 17-Sep-1993.)

Theoremtrel 4120 In a transitive class, the membership relation is transitive. (Contributed by NM, 19-Apr-1994.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremtrel3 4121 In a transitive class, the membership relation is transitive. (Contributed by NM, 19-Apr-1994.)

Theoremtrss 4122 An element of a transitive class is a subset of the class. (Contributed by NM, 7-Aug-1994.)

Theoremtrin 4123 The intersection of transitive classes is transitive. (Contributed by NM, 9-May-1994.)

Theoremtr0 4124 The empty set is transitive. (Contributed by NM, 16-Sep-1993.)

Theoremtrv 4125 The universe is transitive. (Contributed by NM, 14-Sep-2003.)

Theoremtriun 4126* The indexed union of a class of transitive sets is transitive. (Contributed by Mario Carneiro, 16-Nov-2014.)

Theoremtruni 4127* The union of a class of transitive sets is transitive. Exercise 5(a) of [Enderton] p. 73. (Contributed by Scott Fenton, 21-Feb-2011.) (Proof shortened by Mario Carneiro, 26-Apr-2014.)

Theoremtrint 4128* The intersection of a class of transitive sets is transitive. Exercise 5(b) of [Enderton] p. 73. (Contributed by Scott Fenton, 25-Feb-2011.)

Theoremtrintss 4129 If is transitive and non-null, then is a subset of . (Contributed by Scott Fenton, 3-Mar-2011.)

Theoremtrint0 4130 Any non-empty transitive class includes its intersection. Exercise 2 in [TakeutiZaring] p. 44. (Contributed by Andrew Salmon, 14-Nov-2011.)

2.2  ZF Set Theory - add the Axiom of Replacement

2.2.1  Introduce the Axiom of Replacement

Axiomax-rep 4131* Axiom of Replacement. An axiom scheme of Zermelo-Fraenkel set theory. Axiom 5 of [TakeutiZaring] p. 19. It tells us that the image of any set under a function is also a set (see the variant funimaex 5330). Although may be any wff whatsoever, this axiom is useful (i.e. its antecedent is satisfied) when we are given some function and encodes the predicate "the value of the function at is ." Thus will ordinarily have free variables and - think of it informally as . We prefix with the quantifier in order to "protect" the axiom from any containing , thus allowing us to eliminate any restrictions on . This makes the axiom usable in a formalization that omits the logically redundant axiom ax-17 1603. Another common variant is derived as axrep5 4136, where you can find some further remarks. A slightly more compact version is shown as axrep2 4133. A quite different variant is zfrep6 5748, which if used in place of ax-rep 4131 would also require that the Separation Scheme axsep 4140 be stated as a separate axiom.

There is very a strong generalization of Replacement that doesn't demand function-like behavior of . Two versions of this generalization are called the Collection Principle cp 7561 and the Boundedness Axiom bnd 7562.

Many developments of set theory distinguish the uses of Replacement from uses the weaker axioms of Separation axsep 4140, Null Set axnul 4148, and Pairing axpr 4213, all of which we derive from Replacement. In order to make it easier to identify the uses of those redundant axioms, we restate them as axioms ax-sep 4141, ax-nul 4149, and ax-pr 4214 below the theorems that prove them. (Contributed by NM, 23-Dec-1993.)

Theoremaxrep1 4132* The version of the Axiom of Replacement used in the Metamath Solitaire applet http://us.metamath.org/mmsolitaire/mms.html. Equivalence is shown via the path ax-rep 4131 axrep1 4132 axrep2 4133 axrepnd 8216 zfcndrep 8236 = ax-rep 4131. (Contributed by NM, 19-Nov-2005.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)

Theoremaxrep2 4133* Axiom of Replacement expressed with the fewest number of different variables and without any restrictions on . (Contributed by NM, 15-Aug-2003.)

Theoremaxrep3 4134* Axiom of Replacement slightly strengthened from axrep2 4133; may occur free in . (Contributed by NM, 2-Jan-1997.)

Theoremaxrep4 4135* A more traditional version of the Axiom of Replacement. (Contributed by NM, 14-Aug-1994.)

Theoremaxrep5 4136* Axiom of Replacement (similar to Axiom Rep of [BellMachover] p. 463). The antecedent tells us is analogous to a "function" from to (although it is not really a function since it is a wff and not a class). In the consequent we postulate the existence of a set that corresponds to the "image" of restricted to some other set . The hypothesis says must not be free in . (Contributed by NM, 26-Nov-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)

Theoremzfrepclf 4137* An inference rule based on the Axiom of Replacement. Typically, defines a function from to . (Contributed by NM, 26-Nov-1995.)

Theoremzfrep3cl 4138* An inference rule based on the Axiom of Replacement. Typically, defines a function from to . (Contributed by NM, 26-Nov-1995.)

Theoremzfrep4 4139* A version of Replacement using class abstractions. (Contributed by NM, 26-Nov-1995.)

2.2.2  Derive the Axiom of Separation

Theoremaxsep 4140* Separation Scheme, which is an axiom scheme of Zermelo's original theory. Scheme Sep of [BellMachover] p. 463. As we show here, it is redundant if we assume Replacement in the form of ax-rep 4131. Some textbooks present Separation as a separate axiom scheme in order to show that much of set theory can be derived without the stronger Replacement. The Separation Scheme is a weak form of Frege's Axiom of Comprehension, conditioning it (with ) so that it asserts the existence of a collection only if it is smaller than some other collection that already exists. This prevents Russell's paradox ru 2990. In some texts, this scheme is called "Aussonderung" or the Subset Axiom.

The variable can appear free in the wff , which in textbooks is often written . To specify this in the Metamath language, we omit the distinct variable requirement (\$d) that not appear in .

For a version using a class variable, see zfauscl 4143, which requires the Axiom of Extensionality as well as Separation for its derivation.

If we omit the requirement that not occur in , we can derive a contradiction, as notzfaus 4185 shows (contradicting zfauscl 4143). However, as axsep2 4142 shows, we can eliminate the restriction that not occur in .

Note: the distinct variable restriction that not occur in is actually redundant in this particular proof, but we keep it since its purpose is to demonstrate the derivation of the exact ax-sep 4141 from ax-rep 4131.

This theorem should not be referenced by any proof. Instead, use ax-sep 4141 below so that the uses of the Axiom of Separation can be more easily identified. (Contributed by NM, 11-Sep-2006.) (New usage is discouraged.)

Axiomax-sep 4141* The Axiom of Separation of ZF set theory. See axsep 4140 for more information. It was derived as axsep 4140 above and is therefore redundant, but we state it as a separate axiom here so that its uses can be identified more easily. (Contributed by NM, 11-Sep-2006.)

Theoremaxsep2 4142* A less restrictive version of the Separation Scheme axsep 4140, where variables and can both appear free in the wff , which can therefore be thought of as . This version was derived from the more restrictive ax-sep 4141 with no additional set theory axioms. (Contributed by NM, 10-Dec-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)

Theoremzfauscl 4143* Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep 4141, we invoke the Axiom of Extensionality (indirectly via vtocl 2838), which is needed for the justification of class variable notation.

If we omit the requirement that not occur in , we can derive a contradiction, as notzfaus 4185 shows. (Contributed by NM, 5-Aug-1993.)

Theorembm1.3ii 4144* Convert implication to equivalence using the Separation Scheme (Aussonderung) ax-sep 4141. Similar to Theorem 1.3ii of [BellMachover] p. 463. (Contributed by NM, 5-Aug-1993.)

Theoremax9vsep 4145* Derive a weakened version of ax9 1889 ( i.e. ax9v 1636), where and must be distinct, from Separation ax-sep 4141 and Extensionality ax-ext 2264. See ax9 1889 for the derivation of ax9 1889 from ax9v 1636. (Contributed by NM, 12-Nov-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

2.2.3  Derive the Null Set Axiom

Theoremzfnuleu 4146* Show the uniqueness of the empty set (using the Axiom of Extensionality via bm1.1 2268 to strengthen the hypothesis in the form of axnul 4148). (Contributed by NM, 22-Dec-2007.)

TheoremaxnulALT 4147* Prove axnul 4148 directly from ax-rep 4131 using none of the equality axioms ax-8 1643 through ax-15 2082 provided we accept sp 1716 as an axiom. Replace sp 1716 with the obsolete ax-4 2074 to see this in 'show traceback'. (Contributed by Jeff Hoffman, 3-Feb-2008.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremaxnul 4148* The Null Set Axiom of ZF set theory: there exists a set with no elements. Axiom of Empty Set of [Enderton] p. 18. In some textbooks, this is presented as a separate axiom; here we show it can be derived from Separation ax-sep 4141. This version of the Null Set Axiom tells us that at least one empty set exists, but does not tells us that it is unique - we need the Axiom of Extensionality to do that (see zfnuleu 4146).

This proof, suggested by Jeff Hoffman, uses only ax-5 1544 and ax-gen 1533 from predicate calculus, which are valid in "free logic" i.e. logic holding in an empty domain (see Axiom A5 and Rule R2 of [LeBlanc] p. 277). Thus our ax-sep 4141 implies the existence of at least one set. Note that Kunen's version of ax-sep 4141 (Axiom 3 of [Kunen] p. 11) does not imply the existence of a set because his is universally closed i.e. prefixed with universal quantifiers to eliminate all free variables. His existence is provided by a separate axiom stating (Axiom 0 of [Kunen] p. 10).

See axnulALT 4147 for a proof directly from ax-rep 4131.

This theorem should not be referenced by any proof. Instead, use ax-nul 4149 below so that the uses of the Null Set Axiom can be more easily identified. (Contributed by Jeff Hoffman, 3-Feb-2008.) (Revised by NM, 4-Feb-2008.) (New usage is discouraged.) (Proof modification is discouraged.)

Axiomax-nul 4149* The Null Set Axiom of ZF set theory. It was derived as axnul 4148 above and is therefore redundant, but we state it as a separate axiom here so that its uses can be identified more easily. (Contributed by NM, 7-Aug-2003.)

Theorem0ex 4150 The Null Set Axiom of ZF set theory: the empty set exists. Corollary 5.16 of [TakeutiZaring] p. 20. For the unabbreviated version, see ax-nul 4149. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

2.2.4  Theorems requiring subset and intersection existence

Theoremnalset 4151* No set contains all sets. Theorem 41 of [Suppes] p. 30. (Contributed by NM, 23-Aug-1993.)

Theoremvprc 4152 The universal class is not a member of itself (and thus is not a set). Proposition 5.21 of [TakeutiZaring] p. 21; our proof, however, does not depend on the Axiom of Regularity. (Contributed by NM, 23-Aug-1993.)

Theoremnvel 4153 The universal class doesn't belong to any class. (Contributed by FL, 31-Dec-2006.)

Theoremvnex 4154 The universal class does not exist. (Contributed by NM, 4-Jul-2005.)

Theoreminex1 4155 Separation Scheme (Aussonderung) using class notation. Compare Exercise 4 of [TakeutiZaring] p. 22. (Contributed by NM, 5-Aug-1993.)

Theoreminex2 4156 Separation Scheme (Aussonderung) using class notation. (Contributed by NM, 27-Apr-1994.)

Theoreminex1g 4157 Closed-form, generalized Separation Scheme. (Contributed by NM, 7-Apr-1995.)

Theoremssex 4158 The subset of a set is also a set. Exercise 3 of [TakeutiZaring] p. 22. This is one way to express the Axiom of Separation ax-sep 4141 (a.k.a. Subset Axiom). (Contributed by NM, 27-Apr-1994.)

Theoremssexi 4159 The subset of a set is also a set. (Contributed by NM, 9-Sep-1993.)

Theoremssexg 4160 The subset of a set is also a set. Exercise 3 of [TakeutiZaring] p. 22 (generalized). (Contributed by NM, 14-Aug-1994.)

Theoremssexd 4161 A subclass of a set is a set. Deduction form of ssexg 4160. (Contributed by David Moews, 1-May-2017.)

Theoremdifexg 4162 Existence of a difference. (Contributed by NM, 26-May-1998.)

Theoremzfausab 4163* Separation Scheme (Aussonderung) in terms of a class abstraction. (Contributed by NM, 8-Jun-1994.)

Theoremrabexg 4164* Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 23-Oct-1999.)

Theoremrabex 4165* Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 19-Jul-1996.)

Theoremelssabg 4166* Membership in a class abstraction involving a subset. Unlike elabg 2915, does not have to be a set. (Contributed by NM, 29-Aug-2006.)

Theoremintex 4167 The intersection of a non-empty class exists. Exercise 5 of [TakeutiZaring] p. 44 and its converse. (Contributed by NM, 13-Aug-2002.)

Theoremintnex 4168 If a class intersection is not a set, it must be the universe. (Contributed by NM, 3-Jul-2005.)

Theoremintexab 4169 The intersection of a non-empty class abstraction exists. (Contributed by NM, 21-Oct-2003.)

Theoremintexrab 4170 The intersection of a non-empty restricted class abstraction exists. (Contributed by NM, 21-Oct-2003.)

Theoremiinexg 4171* The existence of an indexed union. is normally a free-variable parameter in , which should be read . (Contributed by FL, 19-Sep-2011.)

Theoremintabs 4172* Absorption of a redundant conjunct in the intersection of a class abstraction. (Contributed by NM, 3-Jul-2005.)

Theoreminuni 4173* The intersection of a union with a class is equal to the union of the intersections of each element of with . (Contributed by FL, 24-Mar-2007.)

Theoremelpw2g 4174 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 7-Aug-2000.)

Theoremelpw2 4175 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 11-Oct-2007.)

Theorempwnss 4176 The power set of a set is never a subset. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theorempwne 4177 No set equals its power set. The sethood antecedent is necessary; compare pwv 3826. (Contributed by NM, 17-Nov-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)

2.2.5  Theorems requiring empty set existence

Theoremclass2set 4178* Construct, from any class , a set equal to it when the class exists and equal to the empty set when the class is proper. This theorem shows that the constructed set always exists. (Contributed by NM, 16-Oct-2003.)

Theoremclass2seteq 4179* Equality theorem based on class2set 4178. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Raph Levien, 30-Jun-2006.)

Theorem0elpw 4180 Every power class contains the empty set. (Contributed by NM, 25-Oct-2007.)

Theorem0nep0 4181 The empty set and its power set are not equal. (Contributed by NM, 23-Dec-1993.)

Theorem0inp0 4182 Something cannot be equal to both the null set and the power set of the null set. (Contributed by NM, 30-Sep-2003.)

Theoremunidif0 4183 The removal of the empty set from a class does not affect its union. (Contributed by NM, 22-Mar-2004.)

Theoremiin0 4184* An indexed intersection of the empty set, with a non-empty index set, is empty. (Contributed by NM, 20-Oct-2005.)

Theoremnotzfaus 4185* In the Separation Scheme zfauscl 4143, we require that not occur in (which can be generalized to "not be free in"). Here we show special cases of and that result in a contradiction by violating this requirement. (Contributed by NM, 8-Feb-2006.)

Theoremintv 4186 The intersection of the universal class is empty. (Contributed by NM, 11-Sep-2008.)

Theoremaxpweq 4187* Two equivalent ways to express the Power Set Axiom. Note that ax-pow 4188 is not used by the proof. (Contributed by NM, 22-Jun-2009.)

2.3  ZF Set Theory - add the Axiom of Power Sets

2.3.1  Introduce the Axiom of Power Sets

Axiomax-pow 4188* Axiom of Power Sets. An axiom of Zermelo-Fraenkel set theory. It states that a set exists that includes the power set of a given set i.e. contains every subset of . The variant axpow2 4190 uses explicit subset notation. A version using class notation is pwex 4193. (Contributed by NM, 5-Aug-1993.)

Theoremzfpow 4189* Axiom of Power Sets expressed with fewest number of different variables. (Contributed by NM, 14-Aug-2003.)

Theoremaxpow2 4190* A variant of the Axiom of Power Sets ax-pow 4188 using subset notation. Problem in {BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)

Theoremaxpow3 4191* A variant of the Axiom of Power Sets ax-pow 4188. For any set , there exists a set whose members are exactly the subsets of i.e. the power set of . Axiom Pow of [BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)

Theoremel 4192* Every set is an element of some other set. See elALT 4218 for a shorter proof using more axioms. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theorempwex 4193 Power set axiom expressed in class notation. Axiom 4 of [TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theorempwexg 4194 Power set axiom expressed in class notation, with the sethood requirement as an antecedent. Axiom 4 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Oct-2003.)

Theoremabssexg 4195* Existence of a class of subsets. (Contributed by NM, 15-Jul-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

TheoremsnexALT 4196 A singleton is a set. Theorem 7.13 of [Quine] p. 51, but proved using only Extensionality, Power Set, and Separation. Unlike the proof of zfpair 4212, Replacement is not needed. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) See also snex 4216. (Proof modification is discouraged.) (New usage is discouraged.)

Theoremp0ex 4197 The power set of the empty set (the ordinal 1) is a set. See also p0exALT 4198. (Contributed by NM, 23-Dec-1993.)

Theoremp0exALT 4198 The power set of the empty set (the ordinal 1) is a set. Alternate proof which is longer and quite different from the proof of p0ex 4197 if snexALT 4196 is expanded. (Contributed by NM, 23-Dec-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorempp0ex 4199 The power set of the power set of the empty set (the ordinal 2) is a set. (Contributed by NM, 5-Aug-1993.)

Theoremord3ex 4200 The ordinal number 3 is a set, proved without the Axiom of Union ax-un 4512. (Contributed by NM, 2-May-2009.)

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