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
Axiom	ax-sep 5301*	Axiom scheme of separation. This is an axiom scheme of Zermelo and Zermelo-Fraenkel set theories. It was derived as axsep 5300 above and is therefore redundant in ZF set theory, which contains ax-rep 5284 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 3788. 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 5303, 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 5368 shows (showing the necessity of that condition in zfauscl 5303). Scheme Sep of [BellMachover] p. 463. (Contributed by NM, 11-Sep-2006.)
⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
Theorem	axsepg 5302*	A more general version of the axiom scheme of separation ax-sep 5301, 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 5301 with no additional set theory axioms. Note that it was also derived from ax-rep 5284 but without ax-sep 5301 as axsepgfromrep 5299. (Contributed by NM, 10-Dec-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) Remove dependency on ax-12 2174 and ax-13 2374 and shorten proof. (Revised by BJ, 6-Oct-2019.)
⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
Theorem	zfauscl 5303*	Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep 5301, we invoke the Axiom of Extensionality (indirectly via vtocl 3557), 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 5368 shows. (Contributed by NM, 21-Jun-1993.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))
Theorem	sepexlem 5304*	Lemma for sepex 5305. Use sepex 5305 instead. (Contributed by Matthew House, 19-Sep-2025.) (New usage is discouraged.)
⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 ∈ 𝑦) → ∃𝑧∀𝑥(𝑥 ∈ 𝑧 ↔ 𝜑))
Theorem	sepex 5305*	Convert implication to equivalence within an existence statement using the Separation Scheme (Aussonderung) ax-sep 5301. Similar to Theorem 1.3(ii) of [BellMachover] p. 463. (Contributed by Matthew House, 19-Sep-2025.)
⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 ∈ 𝑦) → ∃𝑧∀𝑥(𝑥 ∈ 𝑧 ↔ 𝜑))
Theorem	sepexi 5306*	Convert implication to equivalence within an existence statement using the Separation Scheme (Aussonderung) ax-sep 5301. Inference associated with sepex 5305. (Contributed by NM, 21-Jun-1993.) Generalize conclusion, extract closed form, and avoid ax-9 2115. (Revised by Matthew House, 19-Sep-2025.)
⊢ ∃𝑦∀𝑥(𝜑 → 𝑥 ∈ 𝑦)    ⇒   ⊢ ∃𝑧∀𝑥(𝑥 ∈ 𝑧 ↔ 𝜑)
Theorem	bm1.3iiOLD 5307*	Obsolete version of sepexi 5306 as of 18-Sep-2025. (Contributed by NM, 21-Jun-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∃𝑥∀𝑦(𝜑 → 𝑦 ∈ 𝑥)    ⇒   ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑)
Theorem	ax6vsep 5308*	Derive ax6v 1965 (a weakened version of ax-6 1964 where 𝑥 and 𝑦 must be distinct), from Separation ax-sep 5301 and Extensionality ax-ext 2705. See ax6 2386 for the derivation of ax-6 1964 from ax6v 1965. (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 5309*	Alternate proof of axnul 5310, proved from propositional calculus, ax-gen 1791, ax-4 1805, sp 2180, and ax-rep 5284. To check this, replace sp 2180 with the obsolete axiom ax-c5 38864 in the proof of axnulALT 5309 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 5310*	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 5301. 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 2710). This proof, suggested by Jeff Hoffman, uses only ax-4 1805 and ax-gen 1791 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 5301 implies the existence of at least one set. Note that Kunen's version of ax-sep 5301 (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 5309 for a proof directly from ax-rep 5284. This theorem should not be referenced by any proof. Instead, use ax-nul 5311 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 5311*	The Null Set Axiom of ZF set theory. It was derived as axnul 5310 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 5312	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 5311. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
⊢ ∅ ∈ V
Theorem	al0ssb 5313*	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 5314	Alternate proof of sseli 3990 illustrating the use of the weak deduction theorem to prove it from the inference sselii 3991. (Contributed by NM, 24-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐴 ⊆ 𝐵    ⇒   ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵)
Theorem	csbexg 5315	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 5316	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 5317	A version of unisn 4930 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 5318*	No set contains all sets. Theorem 41 of [Suppes] p. 30. (Contributed by NM, 23-Aug-1993.) Remove use of ax-12 2174 and ax-13 2374. (Revised by BJ, 31-May-2019.)
⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥
Theorem	vnex 5319	The universal class does not exist as a set. (Contributed by NM, 4-Jul-2005.)
⊢ ¬ ∃𝑥 𝑥 = V
Theorem	vprc 5320	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 5321	The universal class does not belong to any class. (Contributed by FL, 31-Dec-2006.)
⊢ ¬ V ∈ 𝐴
Theorem	inex1 5322	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 5323	Separation Scheme (Aussonderung) using class notation. (Contributed by NM, 27-Apr-1994.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐵 ∩ 𝐴) ∈ V
Theorem	inex1g 5324	Closed-form, generalized Separation Scheme. (Contributed by NM, 7-Apr-1995.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V)
Theorem	inex2g 5325	Sufficient condition for an intersection to be a set. Commuted form of inex1g 5324. (Contributed by Peter Mazsa, 19-Dec-2018.)
⊢ (𝐴 ∈ 𝑉 → (𝐵 ∩ 𝐴) ∈ V)
Theorem	ssex 5326	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 5301 (a.k.a. Subset Axiom). (Contributed by NM, 27-Apr-1994.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)
Theorem	ssexi 5327	The subset of a set is also a set. (Contributed by NM, 9-Sep-1993.)
⊢ 𝐵 ∈ V    &   ⊢ 𝐴 ⊆ 𝐵    ⇒   ⊢ 𝐴 ∈ V
Theorem	ssexg 5328	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 5329	A subclass of a set is a set. Deduction form of ssexg 5328. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐵 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ∈ V)
Theorem	abexd 5330*	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 5331*	Conditions for a class abstraction to be a set. Remark: This proof is shorter than a proof using abexd 5330. (Contributed by AV, 19-Apr-2025.)
⊢ (𝜑 → 𝑥 ∈ 𝐴)    &   ⊢ 𝐴 ∈ V    ⇒   ⊢ {𝑥 ∣ 𝜑} ∈ V
Theorem	prcssprc 5332	The superclass of a proper class is a proper class. (Contributed by AV, 27-Dec-2020.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ∉ V) → 𝐵 ∉ V)
Theorem	sselpwd 5333	Elementhood to a power set. (Contributed by Thierry Arnoux, 18-May-2020.)
⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐵)
Theorem	difexg 5334	Existence of a difference. (Contributed by NM, 26-May-1998.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ 𝐵) ∈ V)
Theorem	difexi 5335	Existence of a difference, inference version of difexg 5334. (Contributed by Glauco Siliprandi, 3-Mar-2021.) (Revised by AV, 26-Mar-2021.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∖ 𝐵) ∈ V
Theorem	difexd 5336	Existence of a difference. (Contributed by SN, 16-Jul-2024.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐴 ∖ 𝐵) ∈ V)
Theorem	zfausab 5337*	Separation Scheme (Aussonderung) in terms of a class abstraction. (Contributed by NM, 8-Jun-1994.)
⊢ 𝐴 ∈ V    ⇒   ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V
Theorem	elpw2g 5338	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 5339	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 5340	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 5341*	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 5342*	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 5343*	Obsolete version of rabexg 5342 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 5344*	Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 19-Jul-1996.)
⊢ 𝐴 ∈ V    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V
Theorem	rabexd 5345*	Separation Scheme in terms of a restricted class abstraction, deduction form of rabex2 5346. (Contributed by AV, 16-Jul-2019.)
⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜓}    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐵 ∈ V)
Theorem	rabex2 5346*	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 5347*	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 5348*	Membership in a class abstraction involving a subset. Unlike elabg 3676, 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 5349	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 5350	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 5351	The intersection of a nonempty class abstraction exists. (Contributed by NM, 21-Oct-2003.)
⊢ (∃𝑥𝜑 ↔ ∩ {𝑥 ∣ 𝜑} ∈ V)
Theorem	intexrab 5352	The intersection of a nonempty restricted class abstraction exists. (Contributed by NM, 21-Oct-2003.)
⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∩ {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V)
Theorem	iinexg 5353*	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 5354*	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 5355*	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 5356*	Two equivalent ways to express the Power Set Axiom. Note that ax-pow 5370 is not used by the proof. When ax-pow 5370 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 7783). (Contributed by NM, 22-Jun-2009.)
⊢ (𝒫 𝐴 ∈ V ↔ ∃𝑥∀𝑦(∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴) → 𝑦 ∈ 𝑥))
Theorem	pwnss 5357	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 5358	No set equals its power set. The sethood antecedent is necessary; compare pwv 4908. (Contributed by NM, 17-Nov-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ≠ 𝐴)
Theorem	difelpw 5359	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 5360*	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 3712) and to the empty set when 𝐴 is a proper class. (Contributed by NM, 16-Oct-2003.)
⊢ {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} ∈ V
Theorem	0elpw 5361	Every power class contains the empty set. (Contributed by NM, 25-Oct-2007.)
⊢ ∅ ∈ 𝒫 𝐴
Theorem	pwne0 5362	A power class is never empty. (Contributed by NM, 3-Sep-2018.)
⊢ 𝒫 𝐴 ≠ ∅
Theorem	0nep0 5363	The empty set and its power set are not equal. (Contributed by NM, 23-Dec-1993.)
⊢ ∅ ≠ {∅}
Theorem	0inp0 5364	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 5365	The removal of the empty set from a class does not affect its union. (Contributed by NM, 22-Mar-2004.)
⊢ ∪ (𝐴 ∖ {∅}) = ∪ 𝐴
Theorem	eqsnuniex 5366	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 5367*	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 5368*	In the Separation Scheme zfauscl 5303, 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 5369	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 5370*	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 5372 uses explicit subset notation. A version using class notation is pwex 5385. (Contributed by NM, 21-Jun-1993.)
⊢ ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)
Theorem	zfpow 5371*	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 5372*	A variant of the Axiom of Power Sets ax-pow 5370 using subset notation. Problem in [BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)
⊢ ∃𝑦∀𝑧(𝑧 ⊆ 𝑥 → 𝑧 ∈ 𝑦)
Theorem	axpow3 5373*	A variant of the Axiom of Power Sets ax-pow 5370. 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 5374*	Alternate proof of el 5447 using ax-9 2115 and ax-pow 5370 instead of ax-pr 5437. (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 5375*	Alternate proof of dtru 5446 using ax-pow 5370 instead of ax-pr 5437. See dtruALT 5393 for another proof using ax-pow 5370 instead of ax-pr 5437. (Contributed by NM, 7-Nov-2006.) Avoid ax-13 2374. (Revised by BJ, 31-May-2019.) Avoid ax-12 2174. (Revised by Rohan Ridenour, 9-Oct-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ¬ ∀𝑥 𝑥 = 𝑦
Theorem	dtrucor 5376*	Corollary of dtru 5446. 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 5377. (Contributed by NM, 27-Jun-2002.)
⊢ 𝑥 = 𝑦    ⇒   ⊢ 𝑥 ≠ 𝑦
Theorem	dtrucor2 5377	The theorem form of the deduction dtrucor 5376 leads to a contradiction, as mentioned in the "Wrong!" example at mmdeduction.html#bad 5376. Usage of this theorem is discouraged because it depends on ax-13 2374. (Contributed by NM, 20-Oct-2007.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → 𝑥 ≠ 𝑦)    ⇒   ⊢ (𝜑 ∧ ¬ 𝜑)
Theorem	dvdemo1 5378*	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 5379, 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 5379 for details on the "disjoint variable" mechanism. (The verb "bundle" to express this phenomenon was introduced by Raph Levien.) Note that dvdemo1 5378 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 2154 nor ax-13 2374. (Contributed by NM, 1-Dec-2006.) (Revised by BJ, 13-Jan-2024.)
⊢ ∃𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝑥)
Theorem	dvdemo2 5379*	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 5378, 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 5378 for details on the "disjoint variable" mechanism. Note that dvdemo2 5379 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 2154, ax-12 2174, ax-13 2374. (Contributed by NM, 1-Dec-2006.) Avoid ax-13 2374. (Revised by BJ, 13-Jan-2024.)
⊢ ∃𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝑥)
Theorem	nfnid 5380	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 5375 in its proof. (Contributed by Mario Carneiro, 8-Oct-2016.)
⊢ ¬ Ⅎ𝑥𝑥
Theorem	nfcvb 5381	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 2374. (New usage is discouraged.)
⊢ (Ⅎ𝑥𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
Theorem	vpwex 5382	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 5383 from vpwex 5382. (Revised by BJ, 10-Aug-2022.)
⊢ 𝒫 𝑥 ∈ V
Theorem	pwexg 5383	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 5384	Deduction version of the power set axiom. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝒫 𝐴 ∈ V)
Theorem	pwex 5385	Power set axiom expressed in class notation. (Contributed by NM, 21-Jun-1993.)
⊢ 𝐴 ∈ V    ⇒   ⊢ 𝒫 𝐴 ∈ V
Theorem	pwel 5386	Quantitative version of pwexg 5383: 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 5311 and ax-pr 5437 and shorten proof. (Revised by BJ, 13-Apr-2024.)
⊢ (𝐴 ∈ 𝐵 → 𝒫 𝐴 ∈ 𝒫 𝒫 ∪ 𝐵)
Theorem	abssexg 5387*	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)
Theorem	snexALT 5388	Alternate proof of snex 5441 using Power Set (ax-pow 5370) instead of Pairing (ax-pr 5437). Unlike in the proof of zfpair 5426, Replacement (ax-rep 5284) is not needed. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ {𝐴} ∈ V
Theorem	p0ex 5389	The power set of the empty set (the ordinal 1) is a set. See also p0exALT 5390. (Contributed by NM, 23-Dec-1993.)
⊢ {∅} ∈ V
Theorem	p0exALT 5390	Alternate proof of p0ex 5389 which is quite different and longer if snexALT 5388 is expanded. (Contributed by NM, 23-Dec-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ {∅} ∈ V
Theorem	pp0ex 5391	The power set of the power set of the empty set (the ordinal 2) is a set. (Contributed by NM, 24-Jun-1993.)
⊢ {∅, {∅}} ∈ V
Theorem	ord3ex 5392	The ordinal number 3 is a set, proved without the Axiom of Union ax-un 7753. (Contributed by NM, 2-May-2009.)
⊢ {∅, {∅}, {∅, {∅}}} ∈ V
Theorem	dtruALT 5393*	Alternate proof of dtru 5446 which requires more axioms but is shorter and may be easier to understand. Like dtruALT2 5375, it uses ax-pow 5370 rather than ax-pr 5437. Assuming that ZF set theory is consistent, we cannot prove this theorem unless we specify that 𝑥 and 𝑦 be distinct. Specifically, Theorem spcev 3605 requires that 𝑥 must not occur in the subexpression ¬ 𝑦 = {∅} in step 4 nor in the subexpression ¬ 𝑦 = ∅ in step 9. The proof verifier will require that 𝑥 and 𝑦 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.)
⊢ ¬ ∀𝑥 𝑥 = 𝑦
Theorem	axc16b 5394*	This theorem shows that Axiom ax-c16 38873 is redundant in the presence of Theorem dtruALT2 5375, which states simply that at least two things exist. This justifies the remark at mmzfcnd.html#twoness 5375 (which links to this theorem). (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by NM, 7-Nov-2006.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
Theorem	eunex 5395	Existential uniqueness implies there is a value for which the wff argument is false. (Contributed by NM, 24-Oct-2010.) (Proof shortened by BJ, 2-Jan-2023.)
⊢ (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑)
Theorem	eusv1 5396*	Two ways to express single-valuedness of a class expression 𝐴(𝑥). (Contributed by NM, 14-Oct-2010.)
⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 ↔ ∃𝑦∀𝑥 𝑦 = 𝐴)
Theorem	eusvnf 5397*	Even if 𝑥 is free in 𝐴, it is effectively bound when 𝐴(𝑥) is single-valued. (Contributed by NM, 14-Oct-2010.) (Revised by Mario Carneiro, 14-Oct-2016.)
⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → Ⅎ𝑥𝐴)
Theorem	eusvnfb 5398*	Two ways to say that 𝐴(𝑥) is a set expression that does not depend on 𝑥. (Contributed by Mario Carneiro, 18-Nov-2016.)
⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 ↔ (Ⅎ𝑥𝐴 ∧ 𝐴 ∈ V))
Theorem	eusv2i 5399*	Two ways to express single-valuedness of a class expression 𝐴(𝑥). (Contributed by NM, 14-Oct-2010.) (Revised by Mario Carneiro, 18-Nov-2016.)
⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → ∃!𝑦∃𝑥 𝑦 = 𝐴)
Theorem	eusv2nf 5400*	Two ways to express single-valuedness of a class expression 𝐴(𝑥). (Contributed by Mario Carneiro, 18-Nov-2016.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (∃!𝑦∃𝑥 𝑦 = 𝐴 ↔ Ⅎ𝑥𝐴)