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

Theoreminex1g 4301 Closed-form, generalized Separation Scheme. (Contributed by NM, 7-Apr-1995.)

Theoremssex 4302 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 4285 (a.k.a. Subset Axiom). (Contributed by NM, 27-Apr-1994.)

Theoremssexi 4303 The subset of a set is also a set. (Contributed by NM, 9-Sep-1993.)

Theoremssexg 4304 The subset of a set is also a set. Exercise 3 of [TakeutiZaring] p. 22 (generalized). (Contributed by NM, 14-Aug-1994.)

Theoremssexd 4305 A subclass of a set is a set. Deduction form of ssexg 4304. (Contributed by David Moews, 1-May-2017.)

Theoremdifexg 4306 Existence of a difference. (Contributed by NM, 26-May-1998.)

Theoremzfausab 4307* Separation Scheme (Aussonderung) in terms of a class abstraction. (Contributed by NM, 8-Jun-1994.)

Theoremrabexg 4308* Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 23-Oct-1999.)

Theoremrabex 4309* Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 19-Jul-1996.)

Theoremelssabg 4310* Membership in a class abstraction involving a subset. Unlike elabg 3040, does not have to be a set. (Contributed by NM, 29-Aug-2006.)

Theoremintex 4311 The intersection of a non-empty class exists. Exercise 5 of [TakeutiZaring] p. 44 and its converse. (Contributed by NM, 13-Aug-2002.)

Theoremintnex 4312 If a class intersection is not a set, it must be the universe. (Contributed by NM, 3-Jul-2005.)

Theoremintexab 4313 The intersection of a non-empty class abstraction exists. (Contributed by NM, 21-Oct-2003.)

Theoremintexrab 4314 The intersection of a non-empty restricted class abstraction exists. (Contributed by NM, 21-Oct-2003.)

Theoremiinexg 4315* The existence of an indexed union. is normally a free-variable parameter in , which should be read . (Contributed by FL, 19-Sep-2011.)

Theoremintabs 4316* Absorption of a redundant conjunct in the intersection of a class abstraction. (Contributed by NM, 3-Jul-2005.)

Theoreminuni 4317* 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 4318 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 7-Aug-2000.)

Theoremelpw2 4319 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 11-Oct-2007.)

Theorempwnss 4320 The power set of a set is never a subset. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theorempwne 4321 No set equals its power set. The sethood antecedent is necessary; compare pwv 3970. (Contributed by NM, 17-Nov-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)

2.2.5  Theorems requiring empty set existence

Theoremclass2set 4322* 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 4323* Equality theorem based on class2set 4322. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Raph Levien, 30-Jun-2006.)

Theorem0elpw 4324 Every power class contains the empty set. (Contributed by NM, 25-Oct-2007.)

Theorem0nep0 4325 The empty set and its power set are not equal. (Contributed by NM, 23-Dec-1993.)

Theorem0inp0 4326 Something cannot be equal to both the null set and the power set of the null set. (Contributed by NM, 30-Sep-2003.)

Theoremunidif0 4327 The removal of the empty set from a class does not affect its union. (Contributed by NM, 22-Mar-2004.)

Theoremiin0 4328* An indexed intersection of the empty set, with a non-empty index set, is empty. (Contributed by NM, 20-Oct-2005.)

Theoremnotzfaus 4329* In the Separation Scheme zfauscl 4287, 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 4330 The intersection of the universal class is empty. (Contributed by NM, 11-Sep-2008.)

Theoremaxpweq 4331* Two equivalent ways to express the Power Set Axiom. Note that ax-pow 4332 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 4332* 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 4334 uses explicit subset notation. A version using class notation is pwex 4337. (Contributed by NM, 5-Aug-1993.)

Theoremzfpow 4333* Axiom of Power Sets expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.)

Theoremaxpow2 4334* A variant of the Axiom of Power Sets ax-pow 4332 using subset notation. Problem in {BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)

Theoremaxpow3 4335* A variant of the Axiom of Power Sets ax-pow 4332. 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 4336* Every set is an element of some other set. See elALT 4362 for a shorter proof using more axioms. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theorempwex 4337 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 4338 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 4339* Existence of a class of subsets. (Contributed by NM, 15-Jul-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

TheoremsnexALT 4340 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 4356, Replacement is not needed. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) See also snex 4360. (Proof modification is discouraged.) (New usage is discouraged.)

Theoremp0ex 4341 The power set of the empty set (the ordinal 1) is a set. See also p0exALT 4342. (Contributed by NM, 23-Dec-1993.)

Theoremp0exALT 4342 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 4341 if snexALT 4340 is expanded. (Contributed by NM, 23-Dec-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorempp0ex 4343 The power set of the power set of the empty set (the ordinal 2) is a set. (Contributed by NM, 5-Aug-1993.)

Theoremord3ex 4344 The ordinal number 3 is a set, proved without the Axiom of Union ax-un 4655. (Contributed by NM, 2-May-2009.)

Theoremdtru 4345* 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 1695.

This theorem is proved directly from set theory axioms (no set theory definitions) and does not use ax-ext 2382 or ax-sep 4285. See dtruALT 4351 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 4346* This theorem shows that axiom ax-16 2192 is redundant in the presence of theorem dtru 4345, 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 4347 Existential uniqueness implies there is a value for which the wff argument is false. (Contributed by NM, 24-Oct-2010.)

Theoremnfnid 4348 A set variable is not free from itself. The proof relies on dtru 4345, that is, it is not true in a one-element domain. (Contributed by Mario Carneiro, 8-Oct-2016.)

Theoremnfcvb 4349* 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 4350 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 4351* A version of dtru 4345 ("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 3000 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 4352* Corollary of dtru 4345. 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 4353. (Contributed by NM, 27-Jun-2002.)

Theoremdtrucor2 4353 The theorem form of the deduction dtrucor 4352 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 4354* Demonstration of a theorem (scheme) that requires (meta)variables and to be distinct, but no others. It bundles the theorem schemes and . Compare dvdemo2 4355. ("Bundles" is a term introduced by Raph Levien.) (Contributed by NM, 1-Dec-2006.)

Theoremdvdemo2 4355* Demonstration of a theorem (scheme) that requires (meta)variables and to be distinct, but no others. It bundles the theorem schemes and . Compare dvdemo1 4354. (Contributed by NM, 1-Dec-2006.)

2.3.2  Derive the Axiom of Pairing

Theoremzfpair 4356 The Axiom of Pairing of Zermelo-Fraenkel set theory. Axiom 2 of [TakeutiZaring] p. 15. In some textbooks this is stated as a separate axiom; here we show it is redundant since it can be derived from the other axioms.

This theorem should not be referenced by any proof other than axpr 4357. Instead, use zfpair2 4359 below so that the uses of the Axiom of Pairing can be more easily identified. (Contributed by NM, 18-Oct-1995.) (New usage is discouraged.)

Theoremaxpr 4357* Unabbreviated version of the Axiom of Pairing of ZF set theory, derived as a theorem from the other axioms.

This theorem should not be referenced by any proof. Instead, use ax-pr 4358 below so that the uses of the Axiom of Pairing can be more easily identified. (Contributed by NM, 14-Nov-2006.) (New usage is discouraged.)

Axiomax-pr 4358* The Axiom of Pairing of ZF set theory. It was derived as theorem axpr 4357 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, 14-Nov-2006.)

Theoremzfpair2 4359 Derive the abbreviated version of the Axiom of Pairing from ax-pr 4358. See zfpair 4356 for its derivation from the other axioms. (Contributed by NM, 14-Nov-2006.)

Theoremsnex 4360 A singleton is a set. Theorem 7.13 of [Quine] p. 51, proved using Extensionality, Separation, and Pairing. See also snexALT 4340. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 19-May-2013.) (Proof modification is discouraged.)

Theoremprex 4361 The Axiom of Pairing using class variables. Theorem 7.13 of [Quine] p. 51. By virtue of its definition, an unordered pair remains a set (even though no longer a pair) even when its components are proper classes (see prprc 3873), so we can dispense with hypotheses requiring them to be sets. (Contributed by NM, 5-Aug-1993.)

TheoremelALT 4362* Every set is an element of some other set. This has a shorter proof than el 4336 but uses more axioms. (Contributed by NM, 4-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

TheoremdtruALT2 4363* An alternative proof of dtru 4345 ("two things exist") using ax-pr 4358 instead of ax-pow 4332. (Contributed by Mario Carneiro, 31-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremsnelpwi 4364 A singleton of a set belongs to the power class of a class containing the set. (Contributed by Alan Sare, 25-Aug-2011.)

Theoremsnelpw 4365 A singleton of a set belongs to the power class of a class containing the set. (Contributed by NM, 1-Apr-1998.)

Theoremprelpwi 4366 A pair of two sets belongs to the power class of a class containing those two sets. (Contributed by Thierry Arnoux, 10-Mar-2017.)

Theoremrext 4367* A theorem similar to extensionality, requiring the existence of a singleton. Exercise 8 of [TakeutiZaring] p. 16. (Contributed by NM, 10-Aug-1993.)

Theoremsspwb 4368 Classes are subclasses if and only if their power classes are subclasses. Exercise 18 of [TakeutiZaring] p. 18. (Contributed by NM, 13-Oct-1996.)

Theoremunipw 4369 A class equals the union of its power class. Exercise 6(a) of [Enderton] p. 38. (Contributed by NM, 14-Oct-1996.) (Proof shortened by Alan Sare, 28-Dec-2008.)

Theorempwel 4370 Membership of a power class. Exercise 10 of [Enderton] p. 26. (Contributed by NM, 13-Jan-2007.)

Theorempwtr 4371 A class is transitive iff its power class is transitive. (Contributed by Alan Sare, 25-Aug-2011.) (Revised by Mario Carneiro, 15-Jun-2014.)

Theoremssextss 4372* An extensionality-like principle defining subclass in terms of subsets. (Contributed by NM, 30-Jun-2004.)

Theoremssext 4373* An extensionality-like principle that uses the subset instead of the membership relation: two classes are equal iff they have the same subsets. (Contributed by NM, 30-Jun-2004.)

Theoremnssss 4374* Negation of subclass relationship. Compare nss 3363. (Contributed by NM, 30-Jun-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theorempweqb 4375 Classes are equal if and only if their power classes are equal. Exercise 19 of [TakeutiZaring] p. 18. (Contributed by NM, 13-Oct-1996.)

Theoremintid 4376* The intersection of all sets to which a set belongs is the singleton of that set. (Contributed by NM, 5-Jun-2009.)

Theoremmoabex 4377 "At most one" existence implies a class abstraction exists. (Contributed by NM, 30-Dec-1996.)

Theoremrmorabex 4378 Restricted "at most one" existence implies a restricted class abstraction exists. (Contributed by NM, 17-Jun-2017.)

Theoremeuabex 4379 The abstraction of a wff with existential uniqueness exists. (Contributed by NM, 25-Nov-1994.)

Theoremnnullss 4380* A non-empty class (even if proper) has a non-empty subset. (Contributed by NM, 23-Aug-2003.)

Theoremexss 4381* Restricted existence in a class (even if proper) implies restricted existence in a subset. (Contributed by NM, 23-Aug-2003.)

Theoremopex 4382 An ordered pair of classes is a set. Exercise 7 of [TakeutiZaring] p. 16. (Contributed by NM, 18-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremotex 4383 An ordered triple of classes is a set. (Contributed by NM, 3-Apr-2015.)

Theoremelop 4384 An ordered pair has two elements. Exercise 3 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremopi1 4385 One of the two elements in an ordered pair. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremopi2 4386 One of the two elements of an ordered pair. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.)

2.3.3  Ordered pair theorem

Theoremopnz 4387 An ordered pair is nonempty iff the arguments are sets. (Contributed by NM, 24-Jan-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremopnzi 4388 An ordered pair is nonempty if the arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremopth1 4389 Equality of the first members of equal ordered pairs. (Contributed by NM, 28-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremopth 4390 The ordered pair theorem. If two ordered pairs are equal, their first elements are equal and their second elements are equal. Exercise 6 of [TakeutiZaring] p. 16. Note that and are not required to be sets due our specific ordered pair definition. (Contributed by NM, 28-May-1995.)

Theoremopthg 4391 Ordered pair theorem. and are not required to be sets under our specific ordered pair definition. (Contributed by NM, 14-Oct-2005.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremopthg2 4392 Ordered pair theorem. (Contributed by NM, 14-Oct-2005.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremopth2 4393 Ordered pair theorem. (Contributed by NM, 21-Sep-2014.)

Theoremotth2 4394 Ordered triple theorem, with triple express with ordered pairs. (Contributed by NM, 1-May-1995.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremotth 4395 Ordered triple theorem. (Contributed by NM, 25-Sep-2014.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremeqvinop 4396* A variable introduction law for ordered pairs. Analog of Lemma 15 of [Monk2] p. 109. (Contributed by NM, 28-May-1995.)

Theoremcopsexg 4397* Substitution of class for ordered pair . (Contributed by NM, 27-Dec-1996.) (Revised by Andrew Salmon, 11-Jul-2011.)

Theoremcopsex2t 4398* Closed theorem form of copsex2g 4399. (Contributed by NM, 17-Feb-2013.)

Theoremcopsex2g 4399* Implicit substitution inference for ordered pairs. (Contributed by NM, 28-May-1995.)

Theoremcopsex4g 4400* An implicit substitution inference for 2 ordered pairs. (Contributed by NM, 5-Aug-1995.)

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