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Theorem List for Metamath Proof Explorer - 5201-5300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremtrss 5201 An element of a transitive class is a subset of the class. (Contributed by NM, 7-Aug-1994.) (Proof shortened by JJ, 26-Jul-2021.)
(Tr 𝐴 → (𝐵𝐴𝐵𝐴))
 
Theoremtrin 5202 The intersection of transitive classes is transitive. (Contributed by NM, 9-May-1994.)
((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴𝐵))
 
Theoremtr0 5203 The empty set is transitive. (Contributed by NM, 16-Sep-1993.)
Tr ∅
 
Theoremtrv 5204 The universe is transitive. (Contributed by NM, 14-Sep-2003.)
Tr V
 
Theoremtriun 5205 An indexed union of a class of transitive sets is transitive. (Contributed by Mario Carneiro, 16-Nov-2014.)
(∀𝑥𝐴 Tr 𝐵 → Tr 𝑥𝐴 𝐵)
 
Theoremtruni 5206* 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.)
(∀𝑥𝐴 Tr 𝑥 → Tr 𝐴)
 
Theoremtriin 5207 An indexed intersection of a class of transitive sets is transitive. (Contributed by BJ, 3-Oct-2022.)
(∀𝑥𝐴 Tr 𝐵 → Tr 𝑥𝐴 𝐵)
 
Theoremtrint 5208* The intersection of a class of transitive sets is transitive. Exercise 5(b) of [Enderton] p. 73. (Contributed by Scott Fenton, 25-Feb-2011.) (Proof shortened by BJ, 3-Oct-2022.)
(∀𝑥𝐴 Tr 𝑥 → Tr 𝐴)
 
Theoremtrintss 5209 Any nonempty transitive class includes its intersection. Exercise 3 in [TakeutiZaring] p. 44 (which mistakenly does not include the nonemptiness hypothesis). (Contributed by Scott Fenton, 3-Mar-2011.) (Proof shortened by Andrew Salmon, 14-Nov-2011.)
((Tr 𝐴𝐴 ≠ ∅) → 𝐴𝐴)
 
2.2  ZF Set Theory - add the Axiom of Replacement
 
2.2.1  Introduce the Axiom of Replacement
 
Axiomax-rep 5210* 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 6529). Although 𝜑 may be any wff whatsoever, this axiom is useful (i.e. its antecedent is satisfied) when we are given some function and 𝜑 encodes the predicate "the value of the function at 𝑤 is 𝑧". Thus, 𝜑 will ordinarily have free variables 𝑤 and 𝑧- think of it informally as 𝜑(𝑤, 𝑧). We prefix 𝜑 with the quantifier 𝑦 in order to "protect" the axiom from any 𝜑 containing 𝑦, thus allowing us to eliminate any restrictions on 𝜑. Another common variant is derived as axrep5 5216, where you can find some further remarks. A slightly more compact version is shown as axrep2 5213. A quite different variant is zfrep6 7806, which if used in place of ax-rep 5210 would also require that the Separation Scheme axsep 5223 be stated as a separate axiom.

There is a very strong generalization of Replacement that doesn't demand function-like behavior of 𝜑. Two versions of this generalization are called the Collection Principle cp 9658 and the Boundedness Axiom bnd 9659.

Many developments of set theory distinguish the uses of Replacement from uses of the weaker axioms of Separation axsep 5223, Null Set axnul 5230, and Pairing axpr 5352, 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 5224, ax-nul 5231, and ax-pr 5353 below the theorems that prove them. (Contributed by NM, 23-Dec-1993.)

(∀𝑤𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦) → ∃𝑦𝑧(𝑧𝑦 ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝜑)))
 
Theoremaxrep1 5211* The version of the Axiom of Replacement used in the Metamath Solitaire applet https://us.metamath.org/mmsolitaire/mms.html. Equivalence is shown via the path ax-rep 5210 axrep1 5211 axrep2 5213 axrepnd 10359 zfcndrep 10379 = ax-rep 5210. (Contributed by NM, 19-Nov-2005.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) Remove dependency on ax-13 2373. (Revised by BJ, 31-May-2019.)
𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦𝜑)))
 
Theoremaxreplem 5212* Lemma for axrep2 5213 and axrep3 5214. (Contributed by BJ, 6-Aug-2022.)
(𝑥 = 𝑦 → (∃𝑢(𝜑 → ∀𝑣(𝜓 ↔ ∃𝑤(𝑧𝑥𝜒))) ↔ ∃𝑢(𝜑 → ∀𝑣(𝜓 ↔ ∃𝑤(𝑧𝑦𝜒)))))
 
Theoremaxrep2 5213* Axiom of Replacement expressed with the fewest number of different variables and without any restrictions on 𝜑. (Contributed by NM, 15-Aug-2003.) Remove dependency on ax-13 2373. (Revised by BJ, 31-May-2019.)
𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)))
 
Theoremaxrep3 5214* Axiom of Replacement slightly strengthened from axrep2 5213; 𝑤 may occur free in 𝜑. (Contributed by NM, 2-Jan-1997.) Remove dependency on ax-13 2373. (Revised by BJ, 31-May-2019.)
𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)))
 
Theoremaxrep4 5215* A more traditional version of the Axiom of Replacement. (Contributed by NM, 14-Aug-1994.)
𝑧𝜑       (∀𝑥𝑧𝑦(𝜑𝑦 = 𝑧) → ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑)))
 
Theoremaxrep5 5216* Axiom of Replacement (similar to Axiom Rep of [BellMachover] p. 463). The antecedent tells us 𝜑 is analogous to a "function" from 𝑥 to 𝑦 (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 𝑧 that corresponds to the "image" of 𝜑 restricted to some other set 𝑤. The hypothesis says 𝑧 must not be free in 𝜑. (Contributed by NM, 26-Nov-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)
𝑧𝜑       (∀𝑥(𝑥𝑤 → ∃𝑧𝑦(𝜑𝑦 = 𝑧)) → ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑)))
 
Theoremaxrep6 5217* A condensed form of ax-rep 5210. (Contributed by SN, 18-Sep-2023.)
(∀𝑤∃*𝑧𝜑 → ∃𝑦𝑧(𝑧𝑦 ↔ ∃𝑤𝑥 𝜑))
 
Theoremaxrep6g 5218* axrep6 5217 in class notation. It is equivalent to both ax-rep 5210 and abrexexg 7812, providing a direct link between the two. (Contributed by SN, 11-Dec-2024.)
((𝐴𝑉 ∧ ∀𝑥∃*𝑦𝜓) → {𝑦 ∣ ∃𝑥𝐴 𝜓} ∈ V)
 
Theoremzfrepclf 5219* An inference based on the Axiom of Replacement. Typically, 𝜑 defines a function from 𝑥 to 𝑦. (Contributed by NM, 26-Nov-1995.)
𝑥𝐴    &   𝐴 ∈ V    &   (𝑥𝐴 → ∃𝑧𝑦(𝜑𝑦 = 𝑧))       𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝐴𝜑))
 
Theoremzfrep3cl 5220* An inference based on the Axiom of Replacement. Typically, 𝜑 defines a function from 𝑥 to 𝑦. (Contributed by NM, 26-Nov-1995.)
𝐴 ∈ V    &   (𝑥𝐴 → ∃𝑧𝑦(𝜑𝑦 = 𝑧))       𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝐴𝜑))
 
Theoremzfrep4 5221* A version of Replacement using class abstractions. (Contributed by NM, 26-Nov-1995.)
{𝑥𝜑} ∈ V    &   (𝜑 → ∃𝑧𝑦(𝜓𝑦 = 𝑧))       {𝑦 ∣ ∃𝑥(𝜑𝜓)} ∈ V
 
2.2.2  Derive the Axiom of Separation
 
Theoremaxsepgfromrep 5222* A more general version axsepg 5225 of the axiom scheme of separation ax-sep 5224 derived from the axiom scheme of replacement ax-rep 5210 (and first-order logic). The extra generality consists in the absence of a disjoint variable condition on 𝑧, 𝜑 (that is, variable 𝑧 may occur in formula 𝜑). See linked statements for more information. (Contributed by NM, 11-Sep-2006.) Remove dependencies on ax-9 2117 to ax-13 2373. (Revised by SN, 25-Sep-2023.) Use ax-sep 5224 instead (or axsepg 5225 if the extra generality is needed). (New usage is discouraged.)
𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))
 
Theoremaxsep 5223* Axiom scheme of separation ax-sep 5224 derived from the axiom scheme of replacement ax-rep 5210. The statement is identical to that of ax-sep 5224, and therefore shows that ax-sep 5224 is redundant when ax-rep 5210 is allowed. See ax-sep 5224 for more information. (Contributed by NM, 11-Sep-2006.) Use ax-sep 5224 instead. (New usage is discouraged.)
𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))
 
Axiomax-sep 5224* Axiom scheme of separation. This is an axiom scheme of Zermelo and Zermelo-Fraenkel set theories.

It was derived as axsep 5223 above and is therefore redundant in ZF set theory, which contains ax-rep 5210 as an axiom (contrary to Zermelo set theory). We state it as a separate axiom here so that some of its uses can be identified more easily. Some textbooks present the axiom scheme of separation as a separate axiom scheme in order to show that much of set theory can be derived without the stronger axiom scheme of replacement (which is not part of Zermelo set theory).

The axiom scheme of separation is a weak form of Frege's axiom scheme of (unrestricted) comprehension, in that it conditions it with the condition 𝑥𝑧, so that it asserts the existence of a collection only if it is smaller than some other collection 𝑧 that already exists. This prevents Russell's paradox ru 3716. In some texts, this scheme is called "Aussonderung" (German for "separation") or "Subset Axiom".

The variable 𝑥 can occur in the formula 𝜑, which in textbooks is often written 𝜑(𝑥). To specify this in the Metamath language, we omit the distinct variable condition ($d) that 𝑥 not occur in 𝜑.

For a version using a class variable, see zfauscl 5226, which requires the axiom of extensionality as well as the axiom scheme of separation for its derivation.

If we omit the requirement that 𝑦 not occur in 𝜑, we can derive a contradiction, as notzfaus 5286 shows (showing the necessity of that condition in zfauscl 5226).

Scheme Sep of [BellMachover] p. 463. (Contributed by NM, 11-Sep-2006.)

𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))
 
Theoremaxsepg 5225* A more general version of the axiom scheme of separation ax-sep 5224, where variable 𝑧 can also occur (in addition to 𝑥) in formula 𝜑, which can therefore be thought of as 𝜑(𝑥, 𝑧). This version is derived from the more restrictive ax-sep 5224 with no additional set theory axioms. Note that it was also derived from ax-rep 5210 but without ax-sep 5224 as axsepgfromrep 5222. (Contributed by NM, 10-Dec-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) Remove dependency on ax-12 2172 and ax-13 2373 and shorten proof. (Revised by BJ, 6-Oct-2019.)
𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))
 
Theoremzfauscl 5226* Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep 5224, we invoke the Axiom of Extensionality (indirectly via vtocl 3499), which is needed for the justification of class variable notation.

If we omit the requirement that 𝑦 not occur in 𝜑, we can derive a contradiction, as notzfaus 5286 shows. (Contributed by NM, 21-Jun-1993.)

𝐴 ∈ V       𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))
 
Theorembm1.3ii 5227* Convert implication to equivalence using the Separation Scheme (Aussonderung) ax-sep 5224. Similar to Theorem 1.3(ii) of [BellMachover] p. 463. (Contributed by NM, 21-Jun-1993.)
𝑥𝑦(𝜑𝑦𝑥)       𝑥𝑦(𝑦𝑥𝜑)
 
Theoremax6vsep 5228* Derive ax6v 1973 (a weakened version of ax-6 1972 where 𝑥 and 𝑦 must be distinct), from Separation ax-sep 5224 and Extensionality ax-ext 2710. See ax6 2385 for the derivation of ax-6 1972 from ax6v 1973. (Contributed by NM, 12-Nov-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
¬ ∀𝑥 ¬ 𝑥 = 𝑦
 
2.2.3  Derive the Null Set Axiom
 
TheoremaxnulALT 5229* Alternate proof of axnul 5230, proved from propositional calculus, ax-gen 1798, ax-4 1812, sp 2177, and ax-rep 5210. To check this, replace sp 2177 with the obsolete axiom ax-c5 36904 in the proof of axnulALT 5229 and type the Metamath program "MM> SHOW TRACE_BACK axnulALT / AXIOMS" command. (Contributed by Jeff Hoffman, 3-Feb-2008.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝑦 ¬ 𝑦𝑥
 
Theoremaxnul 5230* 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 5224. 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 nulmo 2715).

This proof, suggested by Jeff Hoffman, uses only ax-4 1812 and ax-gen 1798 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 5224 implies the existence of at least one set. Note that Kunen's version of ax-sep 5224 (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 𝑥𝑥 = 𝑥 (Axiom 0 of [Kunen] p. 10).

See axnulALT 5229 for a proof directly from ax-rep 5210.

This theorem should not be referenced by any proof. Instead, use ax-nul 5231 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.)

𝑥𝑦 ¬ 𝑦𝑥
 
Axiomax-nul 5231* The Null Set Axiom of ZF set theory. It was derived as axnul 5230 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.)
𝑥𝑦 ¬ 𝑦𝑥
 
Theorem0ex 5232 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 5231. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
∅ ∈ V
 
Theoremal0ssb 5233* The empty set is the unique class which is a subclass of any set. (Contributed by AV, 24-Aug-2022.)
(∀𝑦 𝑋𝑦𝑋 = ∅)
 
TheoremsseliALT 5234 Alternate proof of sseli 3918 illustrating the use of the weak deduction theorem to prove it from the inference sselii 3919. (Contributed by NM, 24-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐴𝐵       (𝐶𝐴𝐶𝐵)
 
Theoremcsbexg 5235 The existence of proper substitution into a class. (Contributed by NM, 10-Nov-2005.) (Revised by NM, 17-Aug-2018.)
(∀𝑥 𝐵𝑊𝐴 / 𝑥𝐵 ∈ V)
 
Theoremcsbex 5236 The existence of proper substitution into a class. (Contributed by NM, 7-Aug-2007.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Revised by NM, 17-Aug-2018.)
𝐵 ∈ V       𝐴 / 𝑥𝐵 ∈ V
 
Theoremunisn2 5237 A version of unisn 4862 without the 𝐴 ∈ V hypothesis. (Contributed by Stefan Allan, 14-Mar-2006.)
{𝐴} ∈ {∅, 𝐴}
 
2.2.4  Theorems requiring subset and intersection existence
 
Theoremnalset 5238* No set contains all sets. Theorem 41 of [Suppes] p. 30. (Contributed by NM, 23-Aug-1993.) Remove use of ax-12 2172 and ax-13 2373. (Revised by BJ, 31-May-2019.)
¬ ∃𝑥𝑦 𝑦𝑥
 
Theoremvnex 5239 The universal class does not exist as a set. (Contributed by NM, 4-Jul-2005.)
¬ ∃𝑥 𝑥 = V
 
Theoremvprc 5240 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.)
¬ V ∈ V
 
Theoremnvel 5241 The universal class does not belong to any class. (Contributed by FL, 31-Dec-2006.)
¬ V ∈ 𝐴
 
Theoreminex1 5242 Separation Scheme (Aussonderung) using class notation. Compare Exercise 4 of [TakeutiZaring] p. 22. (Contributed by NM, 21-Jun-1993.)
𝐴 ∈ V       (𝐴𝐵) ∈ V
 
Theoreminex2 5243 Separation Scheme (Aussonderung) using class notation. (Contributed by NM, 27-Apr-1994.)
𝐴 ∈ V       (𝐵𝐴) ∈ V
 
Theoreminex1g 5244 Closed-form, generalized Separation Scheme. (Contributed by NM, 7-Apr-1995.)
(𝐴𝑉 → (𝐴𝐵) ∈ V)
 
Theoreminex2g 5245 Sufficient condition for an intersection to be a set. Commuted form of inex1g 5244. (Contributed by Peter Mazsa, 19-Dec-2018.)
(𝐴𝑉 → (𝐵𝐴) ∈ V)
 
Theoremssex 5246 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 5224 (a.k.a. Subset Axiom). (Contributed by NM, 27-Apr-1994.)
𝐵 ∈ V       (𝐴𝐵𝐴 ∈ V)
 
Theoremssexi 5247 The subset of a set is also a set. (Contributed by NM, 9-Sep-1993.)
𝐵 ∈ V    &   𝐴𝐵       𝐴 ∈ V
 
Theoremssexg 5248 The subset of a set is also a set. Exercise 3 of [TakeutiZaring] p. 22 (generalized). (Contributed by NM, 14-Aug-1994.)
((𝐴𝐵𝐵𝐶) → 𝐴 ∈ V)
 
Theoremssexd 5249 A subclass of a set is a set. Deduction form of ssexg 5248. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐵𝐶)    &   (𝜑𝐴𝐵)       (𝜑𝐴 ∈ V)
 
Theoremprcssprc 5250 The superclass of a proper class is a proper class. (Contributed by AV, 27-Dec-2020.)
((𝐴𝐵𝐴 ∉ V) → 𝐵 ∉ V)
 
Theoremsselpwd 5251 Elementhood to a power set. (Contributed by Thierry Arnoux, 18-May-2020.)
(𝜑𝐵𝑉)    &   (𝜑𝐴𝐵)       (𝜑𝐴 ∈ 𝒫 𝐵)
 
Theoremdifexg 5252 Existence of a difference. (Contributed by NM, 26-May-1998.)
(𝐴𝑉 → (𝐴𝐵) ∈ V)
 
Theoremdifexi 5253 Existence of a difference, inference version of difexg 5252. (Contributed by Glauco Siliprandi, 3-Mar-2021.) (Revised by AV, 26-Mar-2021.)
𝐴 ∈ V       (𝐴𝐵) ∈ V
 
Theoremdifexd 5254 Existence of a difference. (Contributed by SN, 16-Jul-2024.)
(𝜑𝐴𝑉)       (𝜑 → (𝐴𝐵) ∈ V)
 
Theoremzfausab 5255* Separation Scheme (Aussonderung) in terms of a class abstraction. (Contributed by NM, 8-Jun-1994.)
𝐴 ∈ V       {𝑥 ∣ (𝑥𝐴𝜑)} ∈ V
 
Theoremrabexg 5256* Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 23-Oct-1999.)
(𝐴𝑉 → {𝑥𝐴𝜑} ∈ V)
 
Theoremrabex 5257* Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 19-Jul-1996.)
𝐴 ∈ V       {𝑥𝐴𝜑} ∈ V
 
Theoremrabexd 5258* Separation Scheme in terms of a restricted class abstraction, deduction form of rabex2 5259. (Contributed by AV, 16-Jul-2019.)
𝐵 = {𝑥𝐴𝜓}    &   (𝜑𝐴𝑉)       (𝜑𝐵 ∈ V)
 
Theoremrabex2 5259* Separation Scheme in terms of a restricted class abstraction. (Contributed by AV, 16-Jul-2019.) (Revised by AV, 26-Mar-2021.)
𝐵 = {𝑥𝐴𝜓}    &   𝐴 ∈ V       𝐵 ∈ V
 
Theoremrab2ex 5260* A class abstraction based on a class abstraction based on a set is a set. (Contributed by AV, 16-Jul-2019.) (Revised by AV, 26-Mar-2021.)
𝐵 = {𝑦𝐴𝜓}    &   𝐴 ∈ V       {𝑥𝐵𝜑} ∈ V
 
Theoremelssabg 5261* Membership in a class abstraction involving a subset. Unlike elabg 3608, 𝐴 does not have to be a set. (Contributed by NM, 29-Aug-2006.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐵𝑉 → (𝐴 ∈ {𝑥 ∣ (𝑥𝐵𝜑)} ↔ (𝐴𝐵𝜓)))
 
Theoremintex 5262 The intersection of a nonempty class exists. Exercise 5 of [TakeutiZaring] p. 44 and its converse. (Contributed by NM, 13-Aug-2002.)
(𝐴 ≠ ∅ ↔ 𝐴 ∈ V)
 
Theoremintnex 5263 If a class intersection is not a set, it must be the universe. (Contributed by NM, 3-Jul-2005.)
𝐴 ∈ V ↔ 𝐴 = V)
 
Theoremintexab 5264 The intersection of a nonempty class abstraction exists. (Contributed by NM, 21-Oct-2003.)
(∃𝑥𝜑 {𝑥𝜑} ∈ V)
 
Theoremintexrab 5265 The intersection of a nonempty restricted class abstraction exists. (Contributed by NM, 21-Oct-2003.)
(∃𝑥𝐴 𝜑 {𝑥𝐴𝜑} ∈ V)
 
Theoremiinexg 5266* The existence of a class intersection. 𝑥 is normally a free-variable parameter in 𝐵, which should be read 𝐵(𝑥). (Contributed by FL, 19-Sep-2011.)
((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵𝐶) → 𝑥𝐴 𝐵 ∈ V)
 
Theoremintabs 5267* Absorption of a redundant conjunct in the intersection of a class abstraction. (Contributed by NM, 3-Jul-2005.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   (𝑥 = {𝑦𝜓} → (𝜑𝜒))    &   ( {𝑦𝜓} ⊆ 𝐴𝜒)        {𝑥 ∣ (𝑥𝐴𝜑)} = {𝑥𝜑}
 
Theoreminuni 5268* The intersection of a union 𝐴 with a class 𝐵 is equal to the union of the intersections of each element of 𝐴 with 𝐵. (Contributed by FL, 24-Mar-2007.)
( 𝐴𝐵) = {𝑥 ∣ ∃𝑦𝐴 𝑥 = (𝑦𝐵)}
 
Theoremelpw2g 5269 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 7-Aug-2000.)
(𝐵𝑉 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
 
Theoremelpw2 5270 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 11-Oct-2007.)
𝐵 ∈ V       (𝐴 ∈ 𝒫 𝐵𝐴𝐵)
 
Theoremelpwi2 5271 Membership in a power class. (Contributed by Glauco Siliprandi, 3-Mar-2021.) (Proof shortened by Wolf Lammen, 26-May-2024.)
𝐵𝑉    &   𝐴𝐵       𝐴 ∈ 𝒫 𝐵
 
Theoremelpwi2OLD 5272 Obsolete version of elpwi2 5271 as of 26-May-2024. (Contributed by Glauco Siliprandi, 3-Mar-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐵𝑉    &   𝐴𝐵       𝐴 ∈ 𝒫 𝐵
 
Theorempwnss 5273 The power set of a set is never a subset. (Contributed by Stefan O'Rear, 22-Feb-2015.)
(𝐴𝑉 → ¬ 𝒫 𝐴𝐴)
 
Theorempwne 5274 No set equals its power set. The sethood antecedent is necessary; compare pwv 4837. (Contributed by NM, 17-Nov-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
(𝐴𝑉 → 𝒫 𝐴𝐴)
 
Theoremdifelpw 5275 A difference is an element of the power set of its minuend. (Contributed by AV, 9-Oct-2023.)
(𝐴𝑉 → (𝐴𝐵) ∈ 𝒫 𝐴)
 
Theoremrabelpw 5276* A restricted class abstraction is an element of the power set of its restricting set. (Contributed by AV, 9-Oct-2023.)
(𝐴𝑉 → {𝑥𝐴𝜑} ∈ 𝒫 𝐴)
 
2.2.5  Theorems requiring empty set existence
 
Theoremclass2set 5277* Construct, from any class 𝐴, 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.)
{𝑥𝐴𝐴 ∈ V} ∈ V
 
Theoremclass2seteq 5278* Equality theorem based on class2set 5277. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Raph Levien, 30-Jun-2006.)
(𝐴𝑉 → {𝑥𝐴𝐴 ∈ V} = 𝐴)
 
Theorem0elpw 5279 Every power class contains the empty set. (Contributed by NM, 25-Oct-2007.)
∅ ∈ 𝒫 𝐴
 
Theorempwne0 5280 A power class is never empty. (Contributed by NM, 3-Sep-2018.)
𝒫 𝐴 ≠ ∅
 
Theorem0nep0 5281 The empty set and its power set are not equal. (Contributed by NM, 23-Dec-1993.)
∅ ≠ {∅}
 
Theorem0inp0 5282 Something cannot be equal to both the null set and the power set of the null set. (Contributed by NM, 21-Jun-1993.)
(𝐴 = ∅ → ¬ 𝐴 = {∅})
 
Theoremunidif0 5283 The removal of the empty set from a class does not affect its union. (Contributed by NM, 22-Mar-2004.)
(𝐴 ∖ {∅}) = 𝐴
 
Theoremeqsnuniex 5284 If a class is equal to the singleton of its union, then its union exists. (Contributed by BTernaryTau, 24-Sep-2024.)
(𝐴 = { 𝐴} → 𝐴 ∈ V)
 
Theoremiin0 5285* An indexed intersection of the empty set, with a nonempty index set, is empty. (Contributed by NM, 20-Oct-2005.)
(𝐴 ≠ ∅ ↔ 𝑥𝐴 ∅ = ∅)
 
Theoremnotzfaus 5286* In the Separation Scheme zfauscl 5226, we require that 𝑦 not occur in 𝜑 (which can be generalized to "not be free in"). Here we show special cases of 𝐴 and 𝜑 that result in a contradiction if that requirement is not met. (Contributed by NM, 8-Feb-2006.) (Proof shortened by BJ, 18-Nov-2023.)
𝐴 = {∅}    &   (𝜑 ↔ ¬ 𝑥𝑦)        ¬ ∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))
 
Theoremintv 5287 The intersection of the universal class is empty. (Contributed by NM, 11-Sep-2008.)
V = ∅
 
Theoremaxpweq 5288* Two equivalent ways to express the Power Set Axiom. Note that ax-pow 5289 is not used by the proof. When ax-pow 5289 is assumed and 𝐴 is a set, both sides of the biconditional hold. In ZF, both sides hold if and only if 𝐴 is a set (see pwexr 7624). (Contributed by NM, 22-Jun-2009.)
(𝒫 𝐴 ∈ V ↔ ∃𝑥𝑦(∀𝑧(𝑧𝑦𝑧𝐴) → 𝑦𝑥))
 
2.3  ZF Set Theory - add the Axiom of Power Sets
 
2.3.1  Introduce the Axiom of Power Sets
 
Axiomax-pow 5289* Axiom of Power Sets. An axiom of Zermelo-Fraenkel set theory. It states that a set 𝑦 exists that includes the power set of a given set 𝑥 i.e. contains every subset of 𝑥. The variant axpow2 5291 uses explicit subset notation. A version using class notation is pwex 5304. (Contributed by NM, 21-Jun-1993.)
𝑦𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦)
 
Theoremzfpow 5290* Axiom of Power Sets expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.)
𝑥𝑦(∀𝑥(𝑥𝑦𝑥𝑧) → 𝑦𝑥)
 
Theoremaxpow2 5291* A variant of the Axiom of Power Sets ax-pow 5289 using subset notation. Problem in [BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)
𝑦𝑧(𝑧𝑥𝑧𝑦)
 
Theoremaxpow3 5292* A variant of the Axiom of Power Sets ax-pow 5289. For any set 𝑥, there exists a set 𝑦 whose members are exactly the subsets of 𝑥 i.e. the power set of 𝑥. Axiom Pow of [BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)
𝑦𝑧(𝑧𝑥𝑧𝑦)
 
TheoremelALT2 5293* Alternate proof of el 5358 using ax-9 2117 and ax-pow 5289 instead of ax-pr 5353. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑦 𝑥𝑦
 
TheoremdtruALT2 5294* Alternate proof of dtru 5360 using ax-pow 5289 instead of ax-pr 5353. (Contributed by NM, 7-Nov-2006.) Avoid ax-13 2373. (Revised by Gino Giotto, 5-Sep-2023.) Avoid ax-12 2172. (Revised by Rohan Ridenour, 9-Oct-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
¬ ∀𝑥 𝑥 = 𝑦
 
Theoremdtrucor 5295* Corollary of dtru 5360. 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 5296. (Contributed by NM, 27-Jun-2002.)
𝑥 = 𝑦       𝑥𝑦
 
Theoremdtrucor2 5296 The theorem form of the deduction dtrucor 5295 leads to a contradiction, as mentioned in the "Wrong!" example at mmdeduction.html#bad 5295. Usage of this theorem is discouraged because it depends on ax-13 2373. (Contributed by NM, 20-Oct-2007.) (New usage is discouraged.)
(𝑥 = 𝑦𝑥𝑦)       (𝜑 ∧ ¬ 𝜑)
 
Theoremdvdemo1 5297* Demonstration of a theorem that requires the setvar variables 𝑥 and 𝑦 to be disjoint (but without any other disjointness conditions, and in particular, none on 𝑧).

That theorem bundles the theorems (𝑥(𝑥 = 𝑦𝑧𝑥) with 𝑥, 𝑦, 𝑧 disjoint), often called its "principal instance", and the two "degenerate instances" (𝑥(𝑥 = 𝑦𝑥𝑥) with 𝑥, 𝑦 disjoint) and (𝑥(𝑥 = 𝑦𝑦𝑥) with 𝑥, 𝑦 disjoint).

Compare with dvdemo2 5298, which has the same principal instance and one common degenerate instance but crucially differs in the other degenerate instance.

See https://us.metamath.org/mpeuni/mmset.html#distinct 5298 for details on the "disjoint variable" mechanism. (The verb "bundle" to express this phenomenon was introduced by Raph Levien.)

Note that dvdemo1 5297 is partially bundled, in that the pairs of setvar variables 𝑥, 𝑧 and 𝑦, 𝑧 need not be disjoint, and in spite of that, its proof does not require ax-11 2155 nor ax-13 2373. (Contributed by NM, 1-Dec-2006.) (Revised by BJ, 13-Jan-2024.)

𝑥(𝑥 = 𝑦𝑧𝑥)
 
Theoremdvdemo2 5298* Demonstration of a theorem that requires the setvar variables 𝑥 and 𝑧 to be disjoint (but without any other disjointness conditions, and in particular, none on 𝑦).

That theorem bundles the theorems (𝑥(𝑥 = 𝑦𝑧𝑥) with 𝑥, 𝑦, 𝑧 disjoint), often called its "principal instance", and the two "degenerate instances" (𝑥(𝑥 = 𝑥𝑧𝑥) with 𝑥, 𝑧 disjoint) and (𝑥(𝑥 = 𝑧𝑧𝑥) with 𝑥, 𝑧 disjoint).

Compare with dvdemo1 5297, which has the same principal instance and one common degenerate instance but crucially differs in the other degenerate instance.

See https://us.metamath.org/mpeuni/mmset.html#distinct 5297 for details on the "disjoint variable" mechanism.

Note that dvdemo2 5298 is partially bundled, in that the pairs of setvar variables 𝑥, 𝑦 and 𝑦, 𝑧 need not be disjoint, and in spite of that, its proof does not require any of the auxiliary axioms ax-10 2138, ax-11 2155, ax-12 2172, ax-13 2373. (Contributed by NM, 1-Dec-2006.) (Revised by BJ, 13-Jan-2024.)

𝑥(𝑥 = 𝑦𝑧𝑥)
 
Theoremnfnid 5299 A setvar variable is not free from itself. This theorem is not true in a one-element domain, as illustrated by the use of dtruALT2 5294 in its proof. (Contributed by Mario Carneiro, 8-Oct-2016.)
¬ 𝑥𝑥
 
Theoremnfcvb 5300 The "distinctor" expression ¬ ∀𝑥𝑥 = 𝑦, stating that 𝑥 and 𝑦 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 𝑥𝑦 and 𝑥𝑥 = 𝑦 will both be true. (Contributed by Mario Carneiro, 8-Oct-2016.) Usage of this theorem is discouraged because it depends on ax-13 2373. (New usage is discouraged.)
(𝑥𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
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