P. 54 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 5301-5400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	trint 5301*	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.)
⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∩ 𝐴)
Theorem	trintss 5302	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 𝐴 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ⊆ 𝐴)
2.2  ZF Set Theory - add the Axiom of Replacement
2.2.1  Introduce the Axiom of Replacement
Axiom	ax-rep 5303*	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 6666). 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 5309, where you can find some further remarks. A slightly more compact version is shown as axrep2 5306. A quite different variant is zfrep6 7995, which if used in place of ax-rep 5303 would also require that the Separation Scheme axsep 5316 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 9960 and the Boundedness Axiom bnd 9961. Many developments of set theory distinguish the uses of Replacement from uses of the weaker axioms of Separation axsep 5316, Null Set axnul 5323, and Pairing axpr 5446, 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 5317, ax-nul 5324, and ax-pr 5447 below the theorems that prove them. (Contributed by NM, 23-Dec-1993.)
⊢ (∀𝑤∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
Theorem	axrep1 5304*	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 5303 → axrep1 5304 → axrep2 5306 → axrepnd 10663 → zfcndrep 10683 = ax-rep 5303. (Contributed by NM, 19-Nov-2005.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) Remove dependency on ax-13 2380. (Revised by BJ, 31-May-2019.)
⊢ ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ 𝜑)))
Theorem	axreplem 5305*	Lemma for axrep2 5306 and axrep3 5307. (Contributed by BJ, 6-Aug-2022.)
⊢ (𝑥 = 𝑦 → (∃𝑢(𝜑 → ∀𝑣(𝜓 ↔ ∃𝑤(𝑧 ∈ 𝑥 ∧ 𝜒))) ↔ ∃𝑢(𝜑 → ∀𝑣(𝜓 ↔ ∃𝑤(𝑧 ∈ 𝑦 ∧ 𝜒)))))
Theorem	axrep2 5306*	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 2380. (Revised by BJ, 31-May-2019.)
⊢ ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑)))
Theorem	axrep3 5307*	Axiom of Replacement slightly strengthened from axrep2 5306; 𝑤 may occur free in 𝜑. (Contributed by NM, 2-Jan-1997.) Remove dependency on ax-13 2380. (Revised by BJ, 31-May-2019.)
⊢ ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑)))
Theorem	axrep4 5308*	A more traditional version of the Axiom of Replacement. (Contributed by NM, 14-Aug-1994.)
⊢ Ⅎ𝑧𝜑    ⇒   ⊢ (∀𝑥∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧) → ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑)))
Theorem	axrep5 5309*	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.)
⊢ Ⅎ𝑧𝜑    ⇒   ⊢ (∀𝑥(𝑥 ∈ 𝑤 → ∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧)) → ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑)))
Theorem	axrep6 5310*	A condensed form of ax-rep 5303. (Contributed by SN, 18-Sep-2023.)
⊢ (∀𝑤∃*𝑧𝜑 → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑥 𝜑))
Theorem	axrep6g 5311*	axrep6 5310 in class notation. It is equivalent to both ax-rep 5303 and abrexexg 8001, providing a direct link between the two. (Contributed by SN, 11-Dec-2024.)
⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥∃*𝑦𝜓) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜓} ∈ V)
Theorem	zfrepclf 5312*	An inference based on the Axiom of Replacement. Typically, 𝜑 defines a function from 𝑥 to 𝑦. (Contributed by NM, 26-Nov-1995.)
⊢ Ⅎ𝑥𝐴    &   ⊢ 𝐴 ∈ V    &   ⊢ (𝑥 ∈ 𝐴 → ∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧))    ⇒   ⊢ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
Theorem	zfrep3cl 5313*	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    &   ⊢ (𝑥 ∈ 𝐴 → ∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧))    ⇒   ⊢ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
Theorem	zfrep4 5314*	A version of Replacement using class abstractions. (Contributed by NM, 26-Nov-1995.)
⊢ {𝑥 ∣ 𝜑} ∈ V    &   ⊢ (𝜑 → ∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧))    ⇒   ⊢ {𝑦 ∣ ∃𝑥(𝜑 ∧ 𝜓)} ∈ V
2.2.2  Derive the Axiom of Separation
Theorem	axsepgfromrep 5315*	A more general version axsepg 5318 of the axiom scheme of separation ax-sep 5317 derived from the axiom scheme of replacement ax-rep 5303 (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 2118 to ax-13 2380. (Revised by SN, 25-Sep-2023.) Use ax-sep 5317 instead (or axsepg 5318 if the extra generality is needed). (New usage is discouraged.)
⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
Theorem	axsep 5316*	Axiom scheme of separation ax-sep 5317 derived from the axiom scheme of replacement ax-rep 5303. The statement is identical to that of ax-sep 5317, and therefore shows that ax-sep 5317 is redundant when ax-rep 5303 is allowed. See ax-sep 5317 for more information. (Contributed by NM, 11-Sep-2006.) Use ax-sep 5317 instead. (New usage is discouraged.)
⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
Axiom	ax-sep 5317*	Axiom scheme of separation. This is an axiom scheme of Zermelo and Zermelo-Fraenkel set theories. It was derived as axsep 5316 above and is therefore redundant in ZF set theory, which contains ax-rep 5303 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 𝑥 ∈ 𝑧, 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 3802. 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 5319, 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 5381 shows (showing the necessity of that condition in zfauscl 5319). Scheme Sep of [BellMachover] p. 463. (Contributed by NM, 11-Sep-2006.)
⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
Theorem	axsepg 5318*	A more general version of the axiom scheme of separation ax-sep 5317, 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 5317 with no additional set theory axioms. Note that it was also derived from ax-rep 5303 but without ax-sep 5317 as axsepgfromrep 5315. (Contributed by NM, 10-Dec-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) Remove dependency on ax-12 2178 and ax-13 2380 and shorten proof. (Revised by BJ, 6-Oct-2019.)
⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
Theorem	zfauscl 5319*	Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep 5317, we invoke the Axiom of Extensionality (indirectly via vtocl 3570), 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 5381 shows. (Contributed by NM, 21-Jun-1993.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))
Theorem	bm1.3ii 5320*	Convert implication to equivalence using the Separation Scheme (Aussonderung) ax-sep 5317. Similar to Theorem 1.3(ii) of [BellMachover] p. 463. (Contributed by NM, 21-Jun-1993.)
⊢ ∃𝑥∀𝑦(𝜑 → 𝑦 ∈ 𝑥)    ⇒   ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑)
Theorem	ax6vsep 5321*	Derive ax6v 1968 (a weakened version of ax-6 1967 where 𝑥 and 𝑦 must be distinct), from Separation ax-sep 5317 and Extensionality ax-ext 2711. See ax6 2392 for the derivation of ax-6 1967 from ax6v 1968. (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
Theorem	axnulALT 5322*	Alternate proof of axnul 5323, proved from propositional calculus, ax-gen 1793, ax-4 1807, sp 2184, and ax-rep 5303. To check this, replace sp 2184 with the obsolete axiom ax-c5 38839 in the proof of axnulALT 5322 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.)
⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥
Theorem	axnul 5323*	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 5317. 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 2716). This proof, suggested by Jeff Hoffman, uses only ax-4 1807 and ax-gen 1793 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 5317 implies the existence of at least one set. Note that Kunen's version of ax-sep 5317 (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 5322 for a proof directly from ax-rep 5303. This theorem should not be referenced by any proof. Instead, use ax-nul 5324 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.)
⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥
Axiom	ax-nul 5324*	The Null Set Axiom of ZF set theory. It was derived as axnul 5323 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.)
⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥
Theorem	0ex 5325	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 5324. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
⊢ ∅ ∈ V
Theorem	al0ssb 5326*	The empty set is the unique class which is a subclass of any set. (Contributed by AV, 24-Aug-2022.)
⊢ (∀𝑦 𝑋 ⊆ 𝑦 ↔ 𝑋 = ∅)
Theorem	sseliALT 5327	Alternate proof of sseli 4004 illustrating the use of the weak deduction theorem to prove it from the inference sselii 4005. (Contributed by NM, 24-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐴 ⊆ 𝐵    ⇒   ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵)
Theorem	csbexg 5328	The existence of proper substitution into a class. (Contributed by NM, 10-Nov-2005.) (Revised by NM, 17-Aug-2018.)
⊢ (∀𝑥 𝐵 ∈ 𝑊 → ⦋𝐴 / 𝑥⦌𝐵 ∈ V)
Theorem	csbex 5329	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 / 𝑥⦌𝐵 ∈ V
Theorem	unisn2 5330	A version of unisn 4950 without the 𝐴 ∈ V hypothesis. (Contributed by Stefan Allan, 14-Mar-2006.)
⊢ ∪ {𝐴} ∈ {∅, 𝐴}
2.2.4  Theorems requiring subset and intersection existence
Theorem	nalset 5331*	No set contains all sets. Theorem 41 of [Suppes] p. 30. (Contributed by NM, 23-Aug-1993.) Remove use of ax-12 2178 and ax-13 2380. (Revised by BJ, 31-May-2019.)
⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥
Theorem	vnex 5332	The universal class does not exist as a set. (Contributed by NM, 4-Jul-2005.)
⊢ ¬ ∃𝑥 𝑥 = V
Theorem	vprc 5333	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
Theorem	nvel 5334	The universal class does not belong to any class. (Contributed by FL, 31-Dec-2006.)
⊢ ¬ V ∈ 𝐴
Theorem	inex1 5335	Separation Scheme (Aussonderung) using class notation. Compare Exercise 4 of [TakeutiZaring] p. 22. (Contributed by NM, 21-Jun-1993.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∩ 𝐵) ∈ V
Theorem	inex2 5336	Separation Scheme (Aussonderung) using class notation. (Contributed by NM, 27-Apr-1994.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐵 ∩ 𝐴) ∈ V
Theorem	inex1g 5337	Closed-form, generalized Separation Scheme. (Contributed by NM, 7-Apr-1995.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V)
Theorem	inex2g 5338	Sufficient condition for an intersection to be a set. Commuted form of inex1g 5337. (Contributed by Peter Mazsa, 19-Dec-2018.)
⊢ (𝐴 ∈ 𝑉 → (𝐵 ∩ 𝐴) ∈ V)
Theorem	ssex 5339	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 5317 (a.k.a. Subset Axiom). (Contributed by NM, 27-Apr-1994.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)
Theorem	ssexi 5340	The subset of a set is also a set. (Contributed by NM, 9-Sep-1993.)
⊢ 𝐵 ∈ V    &   ⊢ 𝐴 ⊆ 𝐵    ⇒   ⊢ 𝐴 ∈ V
Theorem	ssexg 5341	The subset of a set is also a set. Exercise 3 of [TakeutiZaring] p. 22 (generalized). (Contributed by NM, 14-Aug-1994.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V)
Theorem	ssexd 5342	A subclass of a set is a set. Deduction form of ssexg 5341. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐵 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ∈ V)
Theorem	abexd 5343*	Conditions for a class abstraction to be a set, deduction form. (Contributed by AV, 19-Apr-2025.)
⊢ ((𝜑 ∧ 𝜓) → 𝑥 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → {𝑥 ∣ 𝜓} ∈ V)
Theorem	abex 5344*	Conditions for a class abstraction to be a set. Remark: This proof is shorter than a proof using abexd 5343. (Contributed by AV, 19-Apr-2025.)
⊢ (𝜑 → 𝑥 ∈ 𝐴)    &   ⊢ 𝐴 ∈ V    ⇒   ⊢ {𝑥 ∣ 𝜑} ∈ V
Theorem	prcssprc 5345	The superclass of a proper class is a proper class. (Contributed by AV, 27-Dec-2020.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ∉ V) → 𝐵 ∉ V)
Theorem	sselpwd 5346	Elementhood to a power set. (Contributed by Thierry Arnoux, 18-May-2020.)
⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐵)
Theorem	difexg 5347	Existence of a difference. (Contributed by NM, 26-May-1998.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ 𝐵) ∈ V)
Theorem	difexi 5348	Existence of a difference, inference version of difexg 5347. (Contributed by Glauco Siliprandi, 3-Mar-2021.) (Revised by AV, 26-Mar-2021.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∖ 𝐵) ∈ V
Theorem	difexd 5349	Existence of a difference. (Contributed by SN, 16-Jul-2024.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐴 ∖ 𝐵) ∈ V)
Theorem	zfausab 5350*	Separation Scheme (Aussonderung) in terms of a class abstraction. (Contributed by NM, 8-Jun-1994.)
⊢ 𝐴 ∈ V    ⇒   ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V
Theorem	elpw2g 5351	Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 7-Aug-2000.)
⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))
Theorem	elpw2 5352	Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 11-Oct-2007.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)
Theorem	elpwi2 5353	Membership in a power class. (Contributed by Glauco Siliprandi, 3-Mar-2021.) (Proof shortened by Wolf Lammen, 26-May-2024.)
⊢ 𝐵 ∈ 𝑉    &   ⊢ 𝐴 ⊆ 𝐵    ⇒   ⊢ 𝐴 ∈ 𝒫 𝐵
Theorem	rabelpw 5354*	A restricted class abstraction is an element of the power set of its restricting set. (Contributed by AV, 9-Oct-2023.)
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ 𝒫 𝐴)
Theorem	rabexg 5355*	Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 23-Oct-1999.) (Proof shortened by BJ, 24-Jul-2025.)
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V)
Theorem	rabexgOLD 5356*	Obsolete proof of rabexg 5355 as of 24-Jul-2025). (Contributed by NM, 23-Oct-1999.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V)
Theorem	rabex 5357*	Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 19-Jul-1996.)
⊢ 𝐴 ∈ V    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V
Theorem	rabexd 5358*	Separation Scheme in terms of a restricted class abstraction, deduction form of rabex2 5359. (Contributed by AV, 16-Jul-2019.)
⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜓}    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐵 ∈ V)
Theorem	rabex2 5359*	Separation Scheme in terms of a restricted class abstraction. (Contributed by AV, 16-Jul-2019.) (Revised by AV, 26-Mar-2021.)
⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜓}    &   ⊢ 𝐴 ∈ V    ⇒   ⊢ 𝐵 ∈ V
Theorem	rab2ex 5360*	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 = {𝑦 ∈ 𝐴 ∣ 𝜓}    &   ⊢ 𝐴 ∈ V    ⇒   ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} ∈ V
Theorem	elssabg 5361*	Membership in a class abstraction involving a subset. Unlike elabg 3690, 𝐴 does not have to be a set. (Contributed by NM, 29-Aug-2006.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ (𝑥 ⊆ 𝐵 ∧ 𝜑)} ↔ (𝐴 ⊆ 𝐵 ∧ 𝜓)))
Theorem	intex 5362	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)
Theorem	intnex 5363	If a class intersection is not a set, it must be the universe. (Contributed by NM, 3-Jul-2005.)
⊢ (¬ ∩ 𝐴 ∈ V ↔ ∩ 𝐴 = V)
Theorem	intexab 5364	The intersection of a nonempty class abstraction exists. (Contributed by NM, 21-Oct-2003.)
⊢ (∃𝑥𝜑 ↔ ∩ {𝑥 ∣ 𝜑} ∈ V)
Theorem	intexrab 5365	The intersection of a nonempty restricted class abstraction exists. (Contributed by NM, 21-Oct-2003.)
⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∩ {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V)
Theorem	iinexg 5366*	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 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ V)
Theorem	intabs 5367*	Absorption of a redundant conjunct in the intersection of a class abstraction. (Contributed by NM, 3-Jul-2005.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = ∩ {𝑦 ∣ 𝜓} → (𝜑 ↔ 𝜒))    &   ⊢ (∩ {𝑦 ∣ 𝜓} ⊆ 𝐴 ∧ 𝜒)    ⇒   ⊢ ∩ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} = ∩ {𝑥 ∣ 𝜑}
Theorem	inuni 5368*	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.) (Proof shortened by Wolf Lammen, 15-May-2025.)
⊢ (∪ 𝐴 ∩ 𝐵) = ∪ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = (𝑦 ∩ 𝐵)}
Theorem	axpweq 5369*	Two equivalent ways to express the Power Set Axiom. Note that ax-pow 5383 is not used by the proof. When ax-pow 5383 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 7800). (Contributed by NM, 22-Jun-2009.)
⊢ (𝒫 𝐴 ∈ V ↔ ∃𝑥∀𝑦(∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴) → 𝑦 ∈ 𝑥))
Theorem	pwnss 5370	The power set of a set is never a subset. (Contributed by Stefan O'Rear, 22-Feb-2015.) (Proof shortened by BJ, 24-Jul-2025.)
⊢ (𝐴 ∈ 𝑉 → ¬ 𝒫 𝐴 ⊆ 𝐴)
Theorem	pwne 5371	No set equals its power set. The sethood antecedent is necessary; compare pwv 4928. (Contributed by NM, 17-Nov-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ≠ 𝐴)
Theorem	difelpw 5372	A difference is an element of the power set of its minuend. (Contributed by AV, 9-Oct-2023.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ 𝐵) ∈ 𝒫 𝐴)
2.2.5  Theorems requiring empty set existence
Theorem	class2set 5373*	The class of elements of 𝐴 "such that 𝐴 is a set" is a set. That class is equal to 𝐴 when 𝐴 is a set (see class2seteq 3726) and to the empty set when 𝐴 is a proper class. (Contributed by NM, 16-Oct-2003.)
⊢ {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} ∈ V
Theorem	0elpw 5374	Every power class contains the empty set. (Contributed by NM, 25-Oct-2007.)
⊢ ∅ ∈ 𝒫 𝐴
Theorem	pwne0 5375	A power class is never empty. (Contributed by NM, 3-Sep-2018.)
⊢ 𝒫 𝐴 ≠ ∅
Theorem	0nep0 5376	The empty set and its power set are not equal. (Contributed by NM, 23-Dec-1993.)
⊢ ∅ ≠ {∅}
Theorem	0inp0 5377	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 = {∅})
Theorem	unidif0 5378	The removal of the empty set from a class does not affect its union. (Contributed by NM, 22-Mar-2004.)
⊢ ∪ (𝐴 ∖ {∅}) = ∪ 𝐴
Theorem	eqsnuniex 5379	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)
Theorem	iin0 5380*	An indexed intersection of the empty set, with a nonempty index set, is empty. (Contributed by NM, 20-Oct-2005.)
⊢ (𝐴 ≠ ∅ ↔ ∩ 𝑥 ∈ 𝐴 ∅ = ∅)
Theorem	notzfaus 5381*	In the Separation Scheme zfauscl 5319, 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 ↔ ¬ 𝑥 ∈ 𝑦)    ⇒   ⊢ ¬ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))
Theorem	intv 5382	The intersection of the universal class is empty. (Contributed by NM, 11-Sep-2008.)
⊢ ∩ V = ∅
2.3  ZF Set Theory - add the Axiom of Power Sets
2.3.1  Introduce the Axiom of Power Sets
Axiom	ax-pow 5383*	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 5385 uses explicit subset notation. A version using class notation is pwex 5398. (Contributed by NM, 21-Jun-1993.)
⊢ ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)
Theorem	zfpow 5384*	Axiom of Power Sets expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.)
⊢ ∃𝑥∀𝑦(∀𝑥(𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)
Theorem	axpow2 5385*	A variant of the Axiom of Power Sets ax-pow 5383 using subset notation. Problem in [BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)
⊢ ∃𝑦∀𝑧(𝑧 ⊆ 𝑥 → 𝑧 ∈ 𝑦)
Theorem	axpow3 5386*	A variant of the Axiom of Power Sets ax-pow 5383. 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.)
⊢ ∃𝑦∀𝑧(𝑧 ⊆ 𝑥 ↔ 𝑧 ∈ 𝑦)
Theorem	elALT2 5387*	Alternate proof of el 5457 using ax-9 2118 and ax-pow 5383 instead of ax-pr 5447. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∃𝑦 𝑥 ∈ 𝑦
Theorem	dtruALT2 5388*	Alternate proof of dtru 5456 using ax-pow 5383 instead of ax-pr 5447. See dtruALT 5406 for another proof using ax-pow 5383 instead of ax-pr 5447. (Contributed by NM, 7-Nov-2006.) Avoid ax-13 2380. (Revised by BJ, 31-May-2019.) Avoid ax-12 2178. (Revised by Rohan Ridenour, 9-Oct-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ¬ ∀𝑥 𝑥 = 𝑦
Theorem	dtrucor 5389*	Corollary of dtru 5456. 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 5390. (Contributed by NM, 27-Jun-2002.)
⊢ 𝑥 = 𝑦    ⇒   ⊢ 𝑥 ≠ 𝑦
Theorem	dtrucor2 5390	The theorem form of the deduction dtrucor 5389 leads to a contradiction, as mentioned in the "Wrong!" example at mmdeduction.html#bad 5389. Usage of this theorem is discouraged because it depends on ax-13 2380. (Contributed by NM, 20-Oct-2007.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → 𝑥 ≠ 𝑦)    ⇒   ⊢ (𝜑 ∧ ¬ 𝜑)
Theorem	dvdemo1 5391*	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 = 𝑦 → 𝑧 ∈ 𝑥) with 𝑥, 𝑦, 𝑧 disjoint), often called its "principal instance", and the two "degenerate instances" (⊢ ∃𝑥(𝑥 = 𝑦 → 𝑥 ∈ 𝑥) with 𝑥, 𝑦 disjoint) and (⊢ ∃𝑥(𝑥 = 𝑦 → 𝑦 ∈ 𝑥) with 𝑥, 𝑦 disjoint). Compare with dvdemo2 5392, 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 5392 for details on the "disjoint variable" mechanism. (The verb "bundle" to express this phenomenon was introduced by Raph Levien.) Note that dvdemo1 5391 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 2158 nor ax-13 2380. (Contributed by NM, 1-Dec-2006.) (Revised by BJ, 13-Jan-2024.)
⊢ ∃𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝑥)
Theorem	dvdemo2 5392*	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 = 𝑦 → 𝑧 ∈ 𝑥) with 𝑥, 𝑦, 𝑧 disjoint), often called its "principal instance", and the two "degenerate instances" (⊢ ∃𝑥(𝑥 = 𝑥 → 𝑧 ∈ 𝑥) with 𝑥, 𝑧 disjoint) and (⊢ ∃𝑥(𝑥 = 𝑧 → 𝑧 ∈ 𝑥) with 𝑥, 𝑧 disjoint). Compare with dvdemo1 5391, 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 5391 for details on the "disjoint variable" mechanism. Note that dvdemo2 5392 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 2141, ax-11 2158, ax-12 2178, ax-13 2380. (Contributed by NM, 1-Dec-2006.) Avoid ax-13 2380. (Revised by BJ, 13-Jan-2024.)
⊢ ∃𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝑥)
Theorem	nfnid 5393	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 5388 in its proof. (Contributed by Mario Carneiro, 8-Oct-2016.)
⊢ ¬ Ⅎ𝑥𝑥
Theorem	nfcvb 5394	The "distinctor" expression ¬ ∀𝑥𝑥 = 𝑦, stating that 𝑥 and 𝑦 are not the same variable, can be written in terms of Ⅎ in the obvious way. This theorem is not true in a one-element domain, because then Ⅎ𝑥𝑦 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 2380. (New usage is discouraged.)
⊢ (Ⅎ𝑥𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
Theorem	vpwex 5395	Power set axiom: the powerclass of a set is a set. Axiom 4 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) Revised to prove pwexg 5396 from vpwex 5395. (Revised by BJ, 10-Aug-2022.)
⊢ 𝒫 𝑥 ∈ V
Theorem	pwexg 5396	Power set axiom expressed in class notation, with the sethood requirement as an antecedent. (Contributed by NM, 30-Oct-2003.)
⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
Theorem	pwexd 5397	Deduction version of the power set axiom. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝒫 𝐴 ∈ V)
Theorem	pwex 5398	Power set axiom expressed in class notation. (Contributed by NM, 21-Jun-1993.)
⊢ 𝐴 ∈ V    ⇒   ⊢ 𝒫 𝐴 ∈ V
Theorem	pwel 5399	Quantitative version of pwexg 5396: the powerset of an element of a class is an element of the double powerclass of the union of that class. Exercise 10 of [Enderton] p. 26. (Contributed by NM, 13-Jan-2007.) Remove use of ax-nul 5324 and ax-pr 5447 and shorten proof. (Revised by BJ, 13-Apr-2024.)
⊢ (𝐴 ∈ 𝐵 → 𝒫 𝐴 ∈ 𝒫 𝒫 ∪ 𝐵)
Theorem	abssexg 5400*	Existence of a class of subsets. (Contributed by NM, 15-Jul-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ∈ V)