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

Theorem3brtr3d 4201 Substitution of equality into both sides of a binary relation. (Contributed by NM, 18-Oct-1999.)

Theorem3brtr4d 4202 Substitution of equality into both sides of a binary relation. (Contributed by NM, 21-Feb-2005.)

Theorem3brtr3g 4203 Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)

Theorem3brtr4g 4204 Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)

Theoremsyl5eqbr 4205 B chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)

Theoremsyl5eqbrr 4206 B chained equality inference for a binary relation. (Contributed by NM, 17-Sep-2004.)

Theoremsyl5breq 4207 B chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)

Theoremsyl5breqr 4208 B chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)

Theoremsyl6eqbr 4209 A chained equality inference for a binary relation. (Contributed by NM, 12-Oct-1999.)

Theoremsyl6eqbrr 4210 A chained equality inference for a binary relation. (Contributed by NM, 4-Jan-2006.)

Theoremsyl6breq 4211 A chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)

Theoremsyl6breqr 4212 A chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)

Theoremssbrd 4213 Deduction from a subclass relationship of binary relations. (Contributed by NM, 30-Apr-2004.)

Theoremssbri 4214 Inference from a subclass relationship of binary relations. (Contributed by NM, 28-Mar-2007.) (Revised by Mario Carneiro, 8-Feb-2015.)

Theoremnfbrd 4215 Deduction version of bound-variable hypothesis builder nfbr 4216. (Contributed by NM, 13-Dec-2005.) (Revised by Mario Carneiro, 14-Oct-2016.)

Theoremnfbr 4216 Bound-variable hypothesis builder for binary relation. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 14-Oct-2016.)

Theorembrab1 4217* Relationship between a binary relation and a class abstraction. (Contributed by Andrew Salmon, 8-Jul-2011.)

Theorembrun 4218 The union of two binary relations. (Contributed by NM, 21-Dec-2008.)

Theorembrin 4219 The intersection of two relations. (Contributed by FL, 7-Oct-2008.)

Theorembrdif 4220 The difference of two binary relations. (Contributed by Scott Fenton, 11-Apr-2011.)

Theoremsbcbrg 4221 Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremsbcbr12g 4222* Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)

Theoremsbcbr1g 4223* Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)

Theoremsbcbr2g 4224* Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)

2.1.23  Ordered-pair class abstractions (class builders)

Syntaxcopab 4225 Extend class notation to include ordered-pair class abstraction (class builder).

Syntaxcmpt 4226 Extend the definition of a class to include maps-to notation for defining a function via a rule.

Definitiondf-opab 4227* Define the class abstraction of a collection of ordered pairs. Definition 3.3 of [Monk1] p. 34. Usually and are distinct, although the definition doesn't strictly require it (see dfid2 4460 for a case where they are not distinct). The brace notation is called "class abstraction" by Quine; it is also (more commonly) called a "class builder" in the literature. An alternate definition using no existential quantifiers is shown by dfopab2 6360. For example, (ex-opab 21693). (Contributed by NM, 4-Jul-1994.)

Definitiondf-mpt 4228* Define maps-to notation for defining a function via a rule. Read as "the function defined by the map from (in ) to ." The class expression is the value of the function at and normally contains the variable . An example is the square function for complex numbers, . Similar to the definition of mapping in [ChoquetDD] p. 2. (Contributed by NM, 17-Feb-2008.)

Theoremopabss 4229* The collection of ordered pairs in a class is a subclass of it. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremopabbid 4230 Equivalent wff's yield equal ordered-pair class abstractions (deduction rule). (Contributed by NM, 21-Feb-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremopabbidv 4231* Equivalent wff's yield equal ordered-pair class abstractions (deduction rule). (Contributed by NM, 15-May-1995.)

Theoremopabbii 4232 Equivalent wff's yield equal class abstractions. (Contributed by NM, 15-May-1995.)

Theoremnfopab 4233* Bound-variable hypothesis builder for class abstraction. (Contributed by NM, 1-Sep-1999.) (Unnecessary distinct variable restrictions were removed by Andrew Salmon, 11-Jul-2011.)

Theoremnfopab1 4234 The first abstraction variable in an ordered-pair class abstraction (class builder) is effectively not free. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)

Theoremnfopab2 4235 The second abstraction variable in an ordered-pair class abstraction (class builder) is effectively not free. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)

Theoremcbvopab 4236* Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 14-Sep-2003.)

Theoremcbvopabv 4237* Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 15-Oct-1996.)

Theoremcbvopab1 4238* Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 6-Oct-2004.) (Revised by Mario Carneiro, 14-Oct-2016.)

Theoremcbvopab2 4239* Change second bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 22-Aug-2013.)

Theoremcbvopab1s 4240* Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 31-Jul-2003.)

Theoremcbvopab1v 4241* Rule used to change the first bound variable in an ordered pair abstraction, using implicit substitution. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)

Theoremcbvopab2v 4242* Rule used to change the second bound variable in an ordered pair abstraction, using implicit substitution. (Contributed by NM, 2-Sep-1999.)

Theoremcsbopabg 4243* Move substitution into a class abstraction. (Contributed by NM, 6-Aug-2007.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)

Theoremunopab 4244 Union of two ordered pair class abstractions. (Contributed by NM, 30-Sep-2002.)

Theoremmpteq12f 4245 An equality theorem for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)

Theoremmpteq12dva 4246* An equality inference for the maps to notation. (Contributed by Mario Carneiro, 26-Jan-2017.)

Theoremmpteq12dv 4247* An equality inference for the maps to notation. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 16-Dec-2013.)

Theoremmpteq12 4248* An equality theorem for the maps to notation. (Contributed by NM, 16-Dec-2013.)

Theoremmpteq1 4249* An equality theorem for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)

Theoremmpteq1d 4250* An equality theorem for the maps to notation. (Contributed by Mario Carneiro, 11-Jun-2016.)

Theoremmpteq2ia 4251 An equality inference for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)

Theoremmpteq2i 4252 An equality inference for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)

Theoremmpteq12i 4253 An equality inference for the maps to notation. (Contributed by Scott Fenton, 27-Oct-2010.) (Revised by Mario Carneiro, 16-Dec-2013.)

Theoremmpteq2da 4254 Slightly more general equality inference for the maps to notation. (Contributed by FL, 14-Sep-2013.) (Revised by Mario Carneiro, 16-Dec-2013.)

Theoremmpteq2dva 4255* Slightly more general equality inference for the maps to notation. (Contributed by Scott Fenton, 25-Apr-2012.)

Theoremmpteq2dv 4256* An equality inference for the maps to notation. (Contributed by Mario Carneiro, 23-Aug-2014.)

Theoremnfmpt 4257* Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.)

Theoremnfmpt1 4258 Bound-variable hypothesis builder for the maps-to notation. (Contributed by FL, 17-Feb-2008.)

Theoremcbvmpt 4259* 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 4260* Rule to change the bound variable in a maps-to function, using implicit substitution. (Contributed by Mario Carneiro, 19-Feb-2013.)

Theoremmptv 4261* Function with universal domain in maps-to notation. (Contributed by NM, 16-Aug-2013.)

2.1.24  Transitive classes

Syntaxwtr 4262 Extend wff notation to include transitive classes. Notation from [TakeutiZaring] p. 35.

Definitiondf-tr 4263 Define the transitive class predicate. Not to be confused with a transitive relation (see cotr 5205). Definition of [Enderton] p. 71 extended to arbitrary classes. For alternate definitions, see dftr2 4264 (which is suggestive of the word "transitive"), dftr3 4266, dftr4 4267, dftr5 4265, and (when is a set) unisuc 4617. The term "complete" is used instead of "transitive" in Definition 3 of [Suppes] p. 130. (Contributed by NM, 29-Aug-1993.)

Theoremdftr2 4264* An alternate way of defining a transitive class. Exercise 7 of [TakeutiZaring] p. 40. (Contributed by NM, 24-Apr-1994.)

Theoremdftr5 4265* An alternate way of defining a transitive class. (Contributed by NM, 20-Mar-2004.)

Theoremdftr3 4266* An alternate way of defining a transitive class. Definition 7.1 of [TakeutiZaring] p. 35. (Contributed by NM, 29-Aug-1993.)

Theoremdftr4 4267 An alternate way of defining a transitive class. Definition of [Enderton] p. 71. (Contributed by NM, 29-Aug-1993.)

Theoremtreq 4268 Equality theorem for the transitive class predicate. (Contributed by NM, 17-Sep-1993.)

Theoremtrel 4269 In a transitive class, the membership relation is transitive. (Contributed by NM, 19-Apr-1994.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremtrel3 4270 In a transitive class, the membership relation is transitive. (Contributed by NM, 19-Apr-1994.)

Theoremtrss 4271 An element of a transitive class is a subset of the class. (Contributed by NM, 7-Aug-1994.)

Theoremtrin 4272 The intersection of transitive classes is transitive. (Contributed by NM, 9-May-1994.)

Theoremtr0 4273 The empty set is transitive. (Contributed by NM, 16-Sep-1993.)

Theoremtrv 4274 The universe is transitive. (Contributed by NM, 14-Sep-2003.)

Theoremtriun 4275* The indexed union of a class of transitive sets is transitive. (Contributed by Mario Carneiro, 16-Nov-2014.)

Theoremtruni 4276* 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 4277* 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 4278 If is transitive and non-null, then is a subset of . (Contributed by Scott Fenton, 3-Mar-2011.)

Theoremtrint0 4279 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 4280* 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 5490). 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 1623. Another common variant is derived as axrep5 4285, where you can find some further remarks. A slightly more compact version is shown as axrep2 4282. A quite different variant is zfrep6 5927, which if used in place of ax-rep 4280 would also require that the Separation Scheme axsep 4289 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 7771 and the Boundedness Axiom bnd 7772.

Many developments of set theory distinguish the uses of Replacement from uses the weaker axioms of Separation axsep 4289, Null Set axnul 4297, and Pairing axpr 4362, 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 4290, ax-nul 4298, and ax-pr 4363 below the theorems that prove them. (Contributed by NM, 23-Dec-1993.)

Theoremaxrep1 4281* 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 4280 axrep1 4281 axrep2 4282 axrepnd 8425 zfcndrep 8445 = ax-rep 4280. (Contributed by NM, 19-Nov-2005.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)

Theoremaxrep2 4282* Axiom of Replacement expressed with the fewest number of different variables and without any restrictions on . (Contributed by NM, 15-Aug-2003.)

Theoremaxrep3 4283* Axiom of Replacement slightly strengthened from axrep2 4282; may occur free in . (Contributed by NM, 2-Jan-1997.)

Theoremaxrep4 4284* A more traditional version of the Axiom of Replacement. (Contributed by NM, 14-Aug-1994.)

Theoremaxrep5 4285* 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 4286* An inference rule based on the Axiom of Replacement. Typically, defines a function from to . (Contributed by NM, 26-Nov-1995.)

Theoremzfrep3cl 4287* An inference rule based on the Axiom of Replacement. Typically, defines a function from to . (Contributed by NM, 26-Nov-1995.)

Theoremzfrep4 4288* A version of Replacement using class abstractions. (Contributed by NM, 26-Nov-1995.)

2.2.2  Derive the Axiom of Separation

Theoremaxsep 4289* 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 4280. 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 3120. 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 4292, 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 4334 shows (contradicting zfauscl 4292). However, as axsep2 4291 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 4290 from ax-rep 4280.

This theorem should not be referenced by any proof. Instead, use ax-sep 4290 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 4290* The Axiom of Separation of ZF set theory. See axsep 4289 for more information. It was derived as axsep 4289 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 4291* A less restrictive version of the Separation Scheme axsep 4289, 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 4290 with no additional set theory axioms. (Contributed by NM, 10-Dec-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)

Theoremzfauscl 4292* Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep 4290, we invoke the Axiom of Extensionality (indirectly via vtocl 2966), 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 4334 shows. (Contributed by NM, 5-Aug-1993.)

Theorembm1.3ii 4293* Convert implication to equivalence using the Separation Scheme (Aussonderung) ax-sep 4290. Similar to Theorem 1.3ii of [BellMachover] p. 463. (Contributed by NM, 5-Aug-1993.)

Theoremax9vsep 4294* Derive a weakened version of ax-9 1662 ( i.e. ax9v 1663), where and must be distinct, from Separation ax-sep 4290 and Extensionality ax-ext 2385. See ax9 1949 for the derivation of ax-9 1662 from ax9v 1663. (Contributed by NM, 12-Nov-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

2.2.3  Derive the Null Set Axiom

Theoremzfnuleu 4295* Show the uniqueness of the empty set (using the Axiom of Extensionality via bm1.1 2389 to strengthen the hypothesis in the form of axnul 4297). (Contributed by NM, 22-Dec-2007.)

TheoremaxnulALT 4296* Prove axnul 4297 directly from ax-rep 4280 using none of the equality axioms ax-8 1683 through ax-15 2193 provided we accept sp 1759 as an axiom. Replace sp 1759 with the obsolete ax-4 2185 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 4297* 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 4290. This version of the Null Set Axiom tells us that at least one empty set exists, but does not tell us that it is unique - we need the Axiom of Extensionality to do that (see zfnuleu 4295).

This proof, suggested by Jeff Hoffman, uses only ax-5 1563 and ax-gen 1552 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 4290 implies the existence of at least one set. Note that Kunen's version of ax-sep 4290 (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 4296 for a proof directly from ax-rep 4280.

This theorem should not be referenced by any proof. Instead, use ax-nul 4298 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 4298* The Null Set Axiom of ZF set theory. It was derived as axnul 4297 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 4299 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 4298. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

2.2.4  Theorems requiring subset and intersection existence

Theoremnalset 4300* No set contains all sets. Theorem 41 of [Suppes] p. 30. (Contributed by NM, 23-Aug-1993.)

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