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Theorem List for Metamath Proof Explorer - 4301-4400   *Has distinct variable group(s)
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Theoremmptv 4301* Function with universal domain in maps-to notation. (Contributed by NM, 16-Aug-2013.)

2.1.24  Transitive classes

Syntaxwtr 4302 Extend wff notation to include transitive classes. Notation from [TakeutiZaring] p. 35.

Definitiondf-tr 4303 Define the transitive class predicate. Not to be confused with a transitive relation (see cotr 5246). Definition of [Enderton] p. 71 extended to arbitrary classes. For alternate definitions, see dftr2 4304 (which is suggestive of the word "transitive"), dftr3 4306, dftr4 4307, dftr5 4305, and (when is a set) unisuc 4657. The term "complete" is used instead of "transitive" in Definition 3 of [Suppes] p. 130. (Contributed by NM, 29-Aug-1993.)

Theoremdftr2 4304* An alternate way of defining a transitive class. Exercise 7 of [TakeutiZaring] p. 40. (Contributed by NM, 24-Apr-1994.)

Theoremdftr5 4305* An alternate way of defining a transitive class. (Contributed by NM, 20-Mar-2004.)

Theoremdftr3 4306* An alternate way of defining a transitive class. Definition 7.1 of [TakeutiZaring] p. 35. (Contributed by NM, 29-Aug-1993.)

Theoremdftr4 4307 An alternate way of defining a transitive class. Definition of [Enderton] p. 71. (Contributed by NM, 29-Aug-1993.)

Theoremtreq 4308 Equality theorem for the transitive class predicate. (Contributed by NM, 17-Sep-1993.)

Theoremtrel 4309 In a transitive class, the membership relation is transitive. (Contributed by NM, 19-Apr-1994.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremtrel3 4310 In a transitive class, the membership relation is transitive. (Contributed by NM, 19-Apr-1994.)

Theoremtrss 4311 An element of a transitive class is a subset of the class. (Contributed by NM, 7-Aug-1994.)

Theoremtrin 4312 The intersection of transitive classes is transitive. (Contributed by NM, 9-May-1994.)

Theoremtr0 4313 The empty set is transitive. (Contributed by NM, 16-Sep-1993.)

Theoremtrv 4314 The universe is transitive. (Contributed by NM, 14-Sep-2003.)

Theoremtriun 4315* The indexed union of a class of transitive sets is transitive. (Contributed by Mario Carneiro, 16-Nov-2014.)

Theoremtruni 4316* 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 4317* 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 4318 If is transitive and non-null, then is a subset of . (Contributed by Scott Fenton, 3-Mar-2011.)

Theoremtrint0 4319 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 4320* 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 5531). 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 1626. Another common variant is derived as axrep5 4325, where you can find some further remarks. A slightly more compact version is shown as axrep2 4322. A quite different variant is zfrep6 5968, which if used in place of ax-rep 4320 would also require that the Separation Scheme axsep 4329 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 7815 and the Boundedness Axiom bnd 7816.

Many developments of set theory distinguish the uses of Replacement from uses the weaker axioms of Separation axsep 4329, Null Set axnul 4337, and Pairing axpr 4402, 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 4330, ax-nul 4338, and ax-pr 4403 below the theorems that prove them. (Contributed by NM, 23-Dec-1993.)

Theoremaxrep1 4321* 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 4320 axrep1 4321 axrep2 4322 axrepnd 8469 zfcndrep 8489 = ax-rep 4320. (Contributed by NM, 19-Nov-2005.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)

Theoremaxrep2 4322* Axiom of Replacement expressed with the fewest number of different variables and without any restrictions on . (Contributed by NM, 15-Aug-2003.)

Theoremaxrep3 4323* Axiom of Replacement slightly strengthened from axrep2 4322; may occur free in . (Contributed by NM, 2-Jan-1997.)

Theoremaxrep4 4324* A more traditional version of the Axiom of Replacement. (Contributed by NM, 14-Aug-1994.)

Theoremaxrep5 4325* 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 4326* An inference rule based on the Axiom of Replacement. Typically, defines a function from to . (Contributed by NM, 26-Nov-1995.)

Theoremzfrep3cl 4327* An inference rule based on the Axiom of Replacement. Typically, defines a function from to . (Contributed by NM, 26-Nov-1995.)

Theoremzfrep4 4328* A version of Replacement using class abstractions. (Contributed by NM, 26-Nov-1995.)

2.2.2  Derive the Axiom of Separation

Theoremaxsep 4329* 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 4320. 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 3160. 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 4332, 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 4374 shows (contradicting zfauscl 4332). However, as axsep2 4331 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 4330 from ax-rep 4320.

This theorem should not be referenced by any proof. Instead, use ax-sep 4330 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 4330* The Axiom of Separation of ZF set theory. See axsep 4329 for more information. It was derived as axsep 4329 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 4331* A less restrictive version of the Separation Scheme axsep 4329, 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 4330 with no additional set theory axioms. (Contributed by NM, 10-Dec-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)

Theoremzfauscl 4332* Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep 4330, we invoke the Axiom of Extensionality (indirectly via vtocl 3006), 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 4374 shows. (Contributed by NM, 5-Aug-1993.)

Theorembm1.3ii 4333* Convert implication to equivalence using the Separation Scheme (Aussonderung) ax-sep 4330. Similar to Theorem 1.3ii of [BellMachover] p. 463. (Contributed by NM, 5-Aug-1993.)

Theoremax9vsep 4334* Derive a weakened version of ax-9 1666 ( i.e. ax9v 1667), where and must be distinct, from Separation ax-sep 4330 and Extensionality ax-ext 2417. See ax9 1953 for the derivation of ax-9 1666 from ax9v 1667. (Contributed by NM, 12-Nov-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

2.2.3  Derive the Null Set Axiom

Theoremzfnuleu 4335* Show the uniqueness of the empty set (using the Axiom of Extensionality via bm1.1 2421 to strengthen the hypothesis in the form of axnul 4337). (Contributed by NM, 22-Dec-2007.)

TheoremaxnulALT 4336* Prove axnul 4337 directly from ax-rep 4320 using none of the equality axioms ax-8 1687 through ax-15 2220 provided we accept sp 1763 as an axiom. Replace sp 1763 with the obsolete ax-4 2212 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 4337* 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 4330. 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 4335).

This proof, suggested by Jeff Hoffman, uses only ax-5 1566 and ax-gen 1555 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 4330 implies the existence of at least one set. Note that Kunen's version of ax-sep 4330 (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 4336 for a proof directly from ax-rep 4320.

This theorem should not be referenced by any proof. Instead, use ax-nul 4338 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 4338* The Null Set Axiom of ZF set theory. It was derived as axnul 4337 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 4339 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 4338. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

2.2.4  Theorems requiring subset and intersection existence

Theoremnalset 4340* No set contains all sets. Theorem 41 of [Suppes] p. 30. (Contributed by NM, 23-Aug-1993.)

Theoremvprc 4341 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 4342 The universal class doesn't belong to any class. (Contributed by FL, 31-Dec-2006.)

Theoremvnex 4343 The universal class does not exist. (Contributed by NM, 4-Jul-2005.)

Theoreminex1 4344 Separation Scheme (Aussonderung) using class notation. Compare Exercise 4 of [TakeutiZaring] p. 22. (Contributed by NM, 5-Aug-1993.)

Theoreminex2 4345 Separation Scheme (Aussonderung) using class notation. (Contributed by NM, 27-Apr-1994.)

Theoreminex1g 4346 Closed-form, generalized Separation Scheme. (Contributed by NM, 7-Apr-1995.)

Theoremssex 4347 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 4330 (a.k.a. Subset Axiom). (Contributed by NM, 27-Apr-1994.)

Theoremssexi 4348 The subset of a set is also a set. (Contributed by NM, 9-Sep-1993.)

Theoremssexg 4349 The subset of a set is also a set. Exercise 3 of [TakeutiZaring] p. 22 (generalized). (Contributed by NM, 14-Aug-1994.)

Theoremssexd 4350 A subclass of a set is a set. Deduction form of ssexg 4349. (Contributed by David Moews, 1-May-2017.)

Theoremdifexg 4351 Existence of a difference. (Contributed by NM, 26-May-1998.)

Theoremzfausab 4352* Separation Scheme (Aussonderung) in terms of a class abstraction. (Contributed by NM, 8-Jun-1994.)

Theoremrabexg 4353* Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 23-Oct-1999.)

Theoremrabex 4354* Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 19-Jul-1996.)

Theoremelssabg 4355* Membership in a class abstraction involving a subset. Unlike elabg 3083, does not have to be a set. (Contributed by NM, 29-Aug-2006.)

Theoremintex 4356 The intersection of a non-empty class exists. Exercise 5 of [TakeutiZaring] p. 44 and its converse. (Contributed by NM, 13-Aug-2002.)

Theoremintnex 4357 If a class intersection is not a set, it must be the universe. (Contributed by NM, 3-Jul-2005.)

Theoremintexab 4358 The intersection of a non-empty class abstraction exists. (Contributed by NM, 21-Oct-2003.)

Theoremintexrab 4359 The intersection of a non-empty restricted class abstraction exists. (Contributed by NM, 21-Oct-2003.)

Theoremiinexg 4360* The existence of an indexed union. is normally a free-variable parameter in , which should be read . (Contributed by FL, 19-Sep-2011.)

Theoremintabs 4361* Absorption of a redundant conjunct in the intersection of a class abstraction. (Contributed by NM, 3-Jul-2005.)

Theoreminuni 4362* 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 4363 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 7-Aug-2000.)

Theoremelpw2 4364 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 11-Oct-2007.)

Theorempwnss 4365 The power set of a set is never a subset. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theorempwne 4366 No set equals its power set. The sethood antecedent is necessary; compare pwv 4014. (Contributed by NM, 17-Nov-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)

2.2.5  Theorems requiring empty set existence

Theoremclass2set 4367* 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 4368* Equality theorem based on class2set 4367. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Raph Levien, 30-Jun-2006.)

Theorem0elpw 4369 Every power class contains the empty set. (Contributed by NM, 25-Oct-2007.)

Theorem0nep0 4370 The empty set and its power set are not equal. (Contributed by NM, 23-Dec-1993.)

Theorem0inp0 4371 Something cannot be equal to both the null set and the power set of the null set. (Contributed by NM, 30-Sep-2003.)

Theoremunidif0 4372 The removal of the empty set from a class does not affect its union. (Contributed by NM, 22-Mar-2004.)

Theoremiin0 4373* An indexed intersection of the empty set, with a non-empty index set, is empty. (Contributed by NM, 20-Oct-2005.)

Theoremnotzfaus 4374* In the Separation Scheme zfauscl 4332, 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 4375 The intersection of the universal class is empty. (Contributed by NM, 11-Sep-2008.)

Theoremaxpweq 4376* Two equivalent ways to express the Power Set Axiom. Note that ax-pow 4377 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 4377* 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 4379 uses explicit subset notation. A version using class notation is pwex 4382. (Contributed by NM, 5-Aug-1993.)

Theoremzfpow 4378* Axiom of Power Sets expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.)

Theoremaxpow2 4379* A variant of the Axiom of Power Sets ax-pow 4377 using subset notation. Problem in {BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)

Theoremaxpow3 4380* A variant of the Axiom of Power Sets ax-pow 4377. 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 4381* Every set is an element of some other set. See elALT 4407 for a shorter proof using more axioms. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theorempwex 4382 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 4383 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 4384* Existence of a class of subsets. (Contributed by NM, 15-Jul-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

TheoremsnexALT 4385 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 4401, Replacement is not needed. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) See also snex 4405. (Proof modification is discouraged.) (New usage is discouraged.)

Theoremp0ex 4386 The power set of the empty set (the ordinal 1) is a set. See also p0exALT 4387. (Contributed by NM, 23-Dec-1993.)

Theoremp0exALT 4387 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 4386 if snexALT 4385 is expanded. (Contributed by NM, 23-Dec-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorempp0ex 4388 The power set of the power set of the empty set (the ordinal 2) is a set. (Contributed by NM, 5-Aug-1993.)

Theoremord3ex 4389 The ordinal number 3 is a set, proved without the Axiom of Union ax-un 4701. (Contributed by NM, 2-May-2009.)

Theoremdtru 4390* At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Note that we may not substitute the same variable for both and (as indicated by the distinct variable requirement), for otherwise we would contradict stdpc6 1699.

This theorem is proved directly from set theory axioms (no set theory definitions) and does not use ax-ext 2417 or ax-sep 4330. See dtruALT 4396 for a shorter proof using these axioms.

The proof makes use of dummy variables and which do not appear in the final theorem. They must be distinct from each other and from and . In other words, if we were to substitute for throughout the proof, the proof would fail. (Contributed by NM, 7-Nov-2006.)

Theoremax16b 4391* This theorem shows that axiom ax-16 2221 is redundant in the presence of theorem dtru 4390, which states simply that at least two things exist. This justifies the remark at http://us.metamath.org/mpeuni/mmzfcnd.html#twoness (which links to this theorem). (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by NM, 7-Nov-2006.)

Theoremeunex 4392 Existential uniqueness implies there is a value for which the wff argument is false. (Contributed by NM, 24-Oct-2010.)

Theoremnfnid 4393 A set variable is not free from itself. The proof relies on dtru 4390, that is, it is not true in a one-element domain. (Contributed by Mario Carneiro, 8-Oct-2016.)

Theoremnfcvb 4394* The "distinctor" expression , stating that and are not the same variable, can be written in terms of in the obvious way. This theorem is not true in a one-element domain, because then and will both be true. (Contributed by Mario Carneiro, 8-Oct-2016.)

Theorempwuni 4395 A class is a subclass of the power class of its union. Exercise 6(b) of [Enderton] p. 38. (Contributed by NM, 14-Oct-1996.)

TheoremdtruALT 4396* A version of dtru 4390 ("two things exist") with a shorter proof that uses more axioms but may be easier to understand.

Assuming that ZF set theory is consistent, we cannot prove this theorem unless we specify that and be distinct. Specifically, theorem spcev 3043 requires that must not occur in the subexpression in step 4 nor in the subexpression in step 9. The proof verifier will require that and be in a distinct variable group to ensure this. You can check this by deleting the \$d statement in set.mm and rerunning the verifier, which will print a detailed explanation of the distinct variable violation. (Contributed by NM, 15-Jul-1994.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremdtrucor 4397* Corollary of dtru 4390. This example illustrates the danger of blindly trusting the standard Deduction Theorem without accounting for free variables: the theorem form of this deduction is not valid, as shown by dtrucor2 4398. (Contributed by NM, 27-Jun-2002.)

Theoremdtrucor2 4398 The theorem form of the deduction dtrucor 4397 leads to a contradiction, as mentioned in the "Wrong!" example at http://us.metamath.org/mpeuni/mmdeduction.html#bad. (Contributed by NM, 20-Oct-2007.)

Theoremdvdemo1 4399* Demonstration of a theorem (scheme) that requires (meta)variables and to be distinct, but no others. It bundles the theorem schemes and . Compare dvdemo2 4400. ("Bundles" is a term introduced by Raph Levien.) (Contributed by NM, 1-Dec-2006.)

Theoremdvdemo2 4400* Demonstration of a theorem (scheme) that requires (meta)variables and to be distinct, but no others. It bundles the theorem schemes and . Compare dvdemo1 4399. (Contributed by NM, 1-Dec-2006.)

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