P. 53 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 5201-5300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	mpteq12i 5201	An equality inference for the maps-to notation. (Contributed by Scott Fenton, 27-Oct-2010.) (Revised by Mario Carneiro, 16-Dec-2013.)
⊢ 𝐴 = 𝐶    &   ⊢ 𝐵 = 𝐷    ⇒   ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)
Theorem	nfmpt 5202*	Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ↦ 𝐵)
Theorem	nfmpt1 5203	Bound-variable hypothesis builder for the maps-to notation. (Contributed by FL, 17-Feb-2008.)
⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
Theorem	cbvmptf 5204*	Rule to change the bound variable in a maps-to function, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by NM, 11-Sep-2011.) (Revised by Thierry Arnoux, 9-Mar-2017.) Add disjoint variable condition to avoid ax-13 2401. See cbvmptfg 5205 for a less restrictive version requiring more axioms. (Revised by GG, 17-Jan-2024.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑦𝐵    &   ⊢ Ⅎ𝑥𝐶    &   ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)    ⇒   ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶)
Theorem	cbvmptfg 5205	Rule to change the bound variable in a maps-to function, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2401. See cbvmptf 5204 for a version with more disjoint variable conditions, but not requiring ax-13 2401. (Contributed by NM, 11-Sep-2011.) (Revised by Thierry Arnoux, 9-Mar-2017.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑦𝐵    &   ⊢ Ⅎ𝑥𝐶    &   ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)    ⇒   ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶)
Theorem	cbvmpt 5206*	Rule to change the bound variable in a maps-to function, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by NM, 11-Sep-2011.) Add disjoint variable condition to avoid ax-13 2401. See cbvmptg 5207 for a less restrictive version requiring more axioms. (Revised by GG, 17-Jan-2024.)
⊢ Ⅎ𝑦𝐵    &   ⊢ Ⅎ𝑥𝐶    &   ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)    ⇒   ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶)
Theorem	cbvmptg 5207*	Rule to change the bound variable in a maps-to function, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2401. See cbvmpt 5206 for a version with more disjoint variable conditions, but not requiring ax-13 2401. (Contributed by NM, 11-Sep-2011.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝐵    &   ⊢ Ⅎ𝑥𝐶    &   ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)    ⇒   ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶)
Theorem	cbvmptv 5208*	Rule to change the bound variable in a maps-to function, using implicit substitution. (Contributed by Mario Carneiro, 19-Feb-2013.) Add disjoint variable condition to avoid auxiliary axioms . See cbvmptvg 5209 for a less restrictive version requiring more axioms. (Revised by GG, 17-Nov-2024.)
⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)    ⇒   ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶)
Theorem	cbvmptvg 5209*	Rule to change the bound variable in a maps-to function, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2401. See cbvmptv 5208 for a version with more disjoint variable conditions, but not requiring ax-13 2401. (Contributed by Mario Carneiro, 19-Feb-2013.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)    ⇒   ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶)
Theorem	mptv 5210*	Function with universal domain in maps-to notation. (Contributed by NM, 16-Aug-2013.)
⊢ (𝑥 ∈ V ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐵}
2.1.26  Transitive classes
Syntax	wtr 5211	Extend wff notation to include transitive classes. Notation from [TakeutiZaring] p. 35.
wff Tr 𝐴
Definition	df-tr 5212	Define the transitive class predicate. Not to be confused with a transitive relation (see cotr 6105). Definition of [Enderton] p. 71 extended to arbitrary classes. For alternate definitions, see dftr2 5213 (which is suggestive of the word "transitive"), dftr2c 5214, dftr3 5216, dftr4 5217, dftr5 5215, and (when 𝐴 is a set) unisuc 6444. The term "complete" is used instead of "transitive" in Definition 3 of [Suppes] p. 130. (Contributed by NM, 29-Aug-1993.)
⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴)
Theorem	dftr2 5213*	An alternate way of defining a transitive class. Exercise 7 of [TakeutiZaring] p. 40. Using dftr2c 5214 instead may avoid dependences on ax-11 2193. (Contributed by NM, 24-Apr-1994.)
⊢ (Tr 𝐴 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴))
Theorem	dftr2c 5214*	Variant of dftr2 5213 with commuted quantifiers, useful for shortening proofs and avoiding ax-11 2193. (Contributed by BTernaryTau, 28-Dec-2024.)
⊢ (Tr 𝐴 ↔ ∀𝑦∀𝑥((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴))
Theorem	dftr5 5215*	An alternate way of defining a transitive class. Definition 1.1 of [Schloeder] p. 1. (Contributed by NM, 20-Mar-2004.) Avoid ax-11 2193. (Revised by BTernaryTau, 28-Dec-2024.)
⊢ (Tr 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴)
Theorem	dftr3 5216*	An alternate way of defining a transitive class. Definition 7.1 of [TakeutiZaring] p. 35. (Contributed by NM, 29-Aug-1993.)
⊢ (Tr 𝐴 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐴)
Theorem	dftr4 5217	An alternate way of defining a transitive class. Definition of [Enderton] p. 71. (Contributed by NM, 29-Aug-1993.)
⊢ (Tr 𝐴 ↔ 𝐴 ⊆ 𝒫 𝐴)
Theorem	treq 5218	Equality theorem for the transitive class predicate. (Contributed by NM, 17-Sep-1993.)
⊢ (𝐴 = 𝐵 → (Tr 𝐴 ↔ Tr 𝐵))
Theorem	trel 5219	In a transitive class, the membership relation is transitive. (Contributed by NM, 19-Apr-1994.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
⊢ (Tr 𝐴 → ((𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → 𝐵 ∈ 𝐴))
Theorem	trel3 5220	In a transitive class, the membership relation is transitive. (Contributed by NM, 19-Apr-1994.)
⊢ (Tr 𝐴 → ((𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐷 ∧ 𝐷 ∈ 𝐴) → 𝐵 ∈ 𝐴))
Theorem	trss 5221	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 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))
Theorem	trun 5222	The union of transitive classes is transitive. (Contributed by Eric Schmidt, 19-Apr-2026.)
⊢ ((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴 ∪ 𝐵))
Theorem	trin 5223	The intersection of transitive classes is transitive. (Contributed by NM, 9-May-1994.)
⊢ ((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴 ∩ 𝐵))
Theorem	tr0 5224	The empty set is transitive. (Contributed by NM, 16-Sep-1993.)
⊢ Tr ∅
Theorem	trv 5225	The universe is transitive. (Contributed by NM, 14-Sep-2003.)
⊢ Tr V
Theorem	triun 5226	An indexed union of a class of transitive sets is transitive. (Contributed by Mario Carneiro, 16-Nov-2014.)
⊢ (∀𝑥 ∈ 𝐴 Tr 𝐵 → Tr ∪ 𝑥 ∈ 𝐴 𝐵)
Theorem	truni 5227*	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.)
⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∪ 𝐴)
Theorem	triin 5228	An indexed intersection of a class of transitive sets is transitive. (Contributed by BJ, 3-Oct-2022.)
⊢ (∀𝑥 ∈ 𝐴 Tr 𝐵 → Tr ∩ 𝑥 ∈ 𝐴 𝐵)
Theorem	trint 5229*	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 5230	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 5231*	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 6627). 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 5238, where you can find some further remarks. A slightly more compact version is shown as axrep2 5234. A quite different variant is zfrep6 5241, which if used in place of ax-rep 5231 would also require that the Separation Scheme axsep 5247 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 9957 and the Boundedness Axiom bnd 9958. Many developments of set theory distinguish the uses of Replacement from uses of the weaker axioms of Separation axsep 5247, Null Set axnul 5258, and Pairing axpr 5388, 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 5248, ax-nul 5259, and ax-pr 5390 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 5232*	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 5231 → axrep1 5232 → axrep2 5234 → axrepnd 10682 → zfcndrep 10702 = ax-rep 5231. (Contributed by NM, 19-Nov-2005.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) Remove dependency on ax-13 2401. (Revised by BJ, 31-May-2019.)
⊢ ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ 𝜑)))
Theorem	axreplem 5233*	Lemma for axrep2 5234 and axrep3 5235. (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 5234*	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 2401. (Revised by BJ, 31-May-2019.)
⊢ ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑)))
Theorem	axrep3 5235*	Axiom of Replacement slightly strengthened from axrep2 5234; 𝑤 may occur free in 𝜑. (Contributed by NM, 2-Jan-1997.) Remove dependency on ax-13 2401. (Revised by BJ, 31-May-2019.)
⊢ ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑)))
Theorem	axrep4v 5236*	Version of axrep4 5237 with a disjoint variable condition, requiring fewer axioms. (Contributed by Matthew House, 18-Sep-2025.)
⊢ (∀𝑥∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧) → ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑)))
Theorem	axrep4 5237*	A more traditional version of the Axiom of Replacement. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Matthew House, 18-Sep-2025.)
⊢ Ⅎ𝑧𝜑    ⇒   ⊢ (∀𝑥∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧) → ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑)))
Theorem	axrep5 5238*	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 5239*	A condensed form of ax-rep 5231. (Contributed by SN, 18-Sep-2023.) (Proof shortened by Matthew House, 18-Sep-2025.)
⊢ (∀𝑤∃*𝑧𝜑 → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑥 𝜑))
Theorem	replem 5240*	A lemma for variants of the axiom of replacement: if we can form the set of images of the functional relation, then we can also form a set containing all its images. The converse requires the axiom of separation. (Contributed by BJ, 5-Apr-2026.)
⊢ ((∀𝑥 ∈ 𝑧 ∃𝑦𝜑 ∧ ∃𝑤∀𝑦(𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 𝜑)) → ∃𝑤∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑤 𝜑)
Theorem	zfrep6 5241*	A version of the Axiom of Replacement. Normally 𝜑 would have free variables 𝑥 and 𝑦. Axiom 6 of [Kunen] p. 12. The Separation Scheme ax-sep 5248 cannot be derived from this version and must be stated as a separate axiom in an axiom system (such as Kunen's) that uses this version in place of our ax-rep 5231. (Contributed by NM, 10-Oct-2003.) Shorten proof and reduce axiom dependencies. (Revised by BJ, 5-Apr-2026.)
⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → ∃𝑤∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑤 𝜑)
Theorem	axrep6g 5242*	axrep6 5239 in class notation. It is equivalent to both ax-rep 5231 and abrexexg 7973, providing a direct link between the two. (Contributed by SN, 11-Dec-2024.)
⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥∃*𝑦𝜓) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜓} ∈ V)
Theorem	zfrepclf 5243*	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 5244*	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 5245*	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 5246*	A more general version axsepg 5249 of the axiom scheme of separation ax-sep 5248 derived from the axiom scheme of replacement ax-rep 5231 (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 2154 to ax-13 2401. (Revised by SN, 25-Sep-2023.) Use ax-sep 5248 instead (or axsepg 5249 if the extra generality is needed). (New usage is discouraged.)
⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
Theorem	axsep 5247*	Axiom scheme of separation ax-sep 5248 derived from the axiom scheme of replacement ax-rep 5231. The statement is identical to that of ax-sep 5248, and therefore shows that ax-sep 5248 is redundant when ax-rep 5231 is allowed. See ax-sep 5248 for more information. (Contributed by NM, 11-Sep-2006.) Use ax-sep 5248 instead. (New usage is discouraged.)
⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
Axiom	ax-sep 5248*	Axiom scheme of separation. This is an axiom scheme of Zermelo and Zermelo-Fraenkel set theories. It was derived as axsep 5247 above and is therefore redundant in ZF set theory, which contains ax-rep 5231 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 3737. 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 sepg 5250, 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 notsep 5324 shows (showing the necessity of that condition in sepgi 5251, hence in sepg 5250 and ax-sep 5248). Scheme Sep of [BellMachover] p. 463. (Contributed by NM, 11-Sep-2006.)
⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
Theorem	axsepg 5249*	A more general version of the axiom scheme of separation ax-sep 5248, 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 5248 with no additional set theory axioms. Note that it was also derived from ax-rep 5231 but without ax-sep 5248 as axsepgfromrep 5246. (Contributed by NM, 10-Dec-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) Remove dependency on ax-12 2212 and ax-13 2401 and shorten proof. (Revised by BJ, 6-Oct-2019.)
⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
Theorem	sepg 5250*	Version of the axiom of separation where the "containing set" is a class variable and the sethood assumption is in the antecedent. (Contributed by NM, 21-Jun-1993.) Put sepgi 5251 in closed form. (Revised by BJ, 2-Jul-2022.)
⊢ (𝐴 ∈ 𝑉 → ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))
Theorem	sepgi 5251*	Inference associated with sepg 5250. The requirement that 𝑦 not occur in 𝜑 is necessary, as notsep 5324 shows. (Contributed by NM, 21-Jun-1993.) (Revised by BJ, 14-Jul-2026.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))
Theorem	zfausclOLD 5252*	Obsolete version of sepgi 5251 as of 14-Jul-2026. (Contributed by NM, 21-Jun-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))
Theorem	sepexlem 5253*	Lemma for sepex 5254. Use sepex 5254 instead. (Contributed by Matthew House, 19-Sep-2025.) (New usage is discouraged.)
⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 ∈ 𝑦) → ∃𝑧∀𝑥(𝑥 ∈ 𝑧 ↔ 𝜑))
Theorem	sepex 5254*	Convert implication to equivalence within an existence statement using the Separation Scheme (Aussonderung) ax-sep 5248. 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 5255*	Convert implication to equivalence within an existence statement using the Separation Scheme (Aussonderung) ax-sep 5248. Inference associated with sepex 5254. (Contributed by NM, 21-Jun-1993.) Generalize conclusion, extract closed form, and avoid ax-9 2154. (Revised by Matthew House, 19-Sep-2025.)
⊢ ∃𝑦∀𝑥(𝜑 → 𝑥 ∈ 𝑦)    ⇒   ⊢ ∃𝑧∀𝑥(𝑥 ∈ 𝑧 ↔ 𝜑)
Theorem	ax6vsep 5256*	Derive ax6v 2000 (a weakened version of ax-6 1999 where 𝑥 and 𝑦 must be distinct), from Separation ax-sep 5248 and Extensionality ax-ext 2732. See ax6 2413 for the derivation of ax-6 1999 from ax6v 2000. (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 5257*	Alternate proof of axnul 5258, proved from propositional calculus, ax-gen 1827, ax-4 1841, sp 2219, and ax-rep 5231. To check this, replace sp 2219 with the obsolete axiom ax-c5 40033 in the proof of axnulALT 5257 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 5258*	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 5248. 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 2737). This proof, suggested by Jeff Hoffman, uses only ax-4 1841 and ax-gen 1827 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 5248 implies the existence of at least one set. Note that Kunen's version of ax-sep 5248 (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 5257 for a proof directly from ax-rep 5231. This theorem should not be referenced by any proof. Instead, use ax-nul 5259 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 5259*	The Null Set Axiom of ZF set theory. It was derived as axnul 5258 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 5260	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 5259. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
⊢ ∅ ∈ V
Theorem	al0ssb 5261*	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 5262	Alternate proof of sseli 3904 illustrating the use of the weak deduction theorem to prove it from the inference sselii 3905. (Contributed by NM, 24-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐴 ⊆ 𝐵    ⇒   ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵)
Theorem	csbexg 5263	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 5264	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 5265	A version of unisn 4885 without the 𝐴 ∈ V hypothesis. (Contributed by Stefan Allan, 14-Mar-2006.)
⊢ ∪ {𝐴} ∈ {∅, 𝐴}
2.2.4  Theorems requiring subset and intersection existence
Theorem	exnelv 5266*	For any set 𝑥, there is a set not contained in 𝑥. The proof is based on Russell's paradox. (Contributed by NM, 23-Aug-1993.) Remove use of ax-12 2212 and ax-13 2401. (Revised by BJ, 31-May-2019.) Extract from nalset 5267. (Revised by Matthew House, 12-Apr-2026.)
⊢ ∃𝑦 ¬ 𝑦 ∈ 𝑥
Theorem	nalset 5267*	No set contains all sets. Theorem 41 of [Suppes] p. 30. (Contributed by NM, 23-Aug-1993.) Extract exnelv 5266. (Revised by Matthew House, 12-Apr-2026.)
⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥
Theorem	nalsetOLD 5268*	Obsolete version of nalset 5267 as of 12-Apr-2026. (Contributed by NM, 23-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥
Theorem	vneqv 5269	The universal class is not equal to any setvar. (Contributed by NM, 4-Jul-2005.) Extract from vnex 5270 and shorten proof. (Revised by BJ, 25-Apr-2026.)
⊢ ¬ 𝑥 = V
Theorem	vnex 5270	The universal class does not exist as a set. (Contributed by NM, 4-Jul-2005.) (Proof shortened by BJ, 25-Apr-2026.)
⊢ ¬ ∃𝑥 𝑥 = V
Theorem	vnexOLD 5271	Obsolete proof of vnex 5270 as of 25-Apr-2026. (Contributed by NM, 4-Jul-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ¬ ∃𝑥 𝑥 = V
Theorem	nvel 5272	The universal class does not belong to any class. (Contributed by FL, 31-Dec-2006.) Prove it without using vprc 5273, which is then proved as an instance of it. (Revised by BJ, 1-May-2026.)
⊢ ¬ V ∈ 𝐴
Theorem	vprc 5273	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.) (Proof shortened by BJ, 1-May-2026.)
⊢ ¬ V ∈ V
Theorem	vprcOLD 5274	Obsolete proof of vprc 5273, obsolete as of 1-May-2026. (Contributed by NM, 23-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ¬ V ∈ V
Theorem	nvelOLD 5275	Obsolete proof of nvel 5272, obsolete as of 1-May-2026. (Contributed by FL, 31-Dec-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ¬ V ∈ 𝐴
Theorem	inex1 5276	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 5277	Separation Scheme (Aussonderung) using class notation. (Contributed by NM, 27-Apr-1994.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐵 ∩ 𝐴) ∈ V
Theorem	inex1g 5278	Closed-form, generalized Separation Scheme. (Contributed by NM, 7-Apr-1995.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V)
Theorem	inex2g 5279	Sufficient condition for an intersection to be a set. Commuted form of inex1g 5278. (Contributed by Peter Mazsa, 19-Dec-2018.)
⊢ (𝐴 ∈ 𝑉 → (𝐵 ∩ 𝐴) ∈ V)
Theorem	ssexg 5280	A subclass of a set is a set. Exercise 3 of [TakeutiZaring] p. 22 (generalized). (Contributed by NM, 14-Aug-1994.) (Proof shortened by BJ, 18-Jul-2026.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V)
Theorem	ssex 5281	A subclass of a set is a set. Exercise 3 of [TakeutiZaring] p. 22. This is one way to express the Axiom of Separation ax-sep 5248 (a.k.a. Subset Axiom). (Contributed by NM, 27-Apr-1994.) (Proof shortened by BJ, 18-Jul-2026.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)
Theorem	ssexOLD 5282	Obsolete version of ssex 5281 as of 18-Jul-2026. (Contributed by NM, 27-Apr-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)
Theorem	ssexi 5283	A subclass of a set is a set. (Contributed by NM, 9-Sep-1993.)
⊢ 𝐵 ∈ V    &   ⊢ 𝐴 ⊆ 𝐵    ⇒   ⊢ 𝐴 ∈ V
Theorem	ssexgOLD 5284	Obsolete version of ssexg 5280 as of 18-Jul-2026. (Contributed by NM, 14-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V)
Theorem	ssexd 5285	A subclass of a set is a set. Deduction form of ssexg 5280. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐵 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ∈ V)
Theorem	abexd 5286*	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 5287*	Conditions for a class abstraction to be a set. Remark: This proof is shorter than a proof using abexd 5286. (Contributed by AV, 19-Apr-2025.)
⊢ (𝜑 → 𝑥 ∈ 𝐴)    &   ⊢ 𝐴 ∈ V    ⇒   ⊢ {𝑥 ∣ 𝜑} ∈ V
Theorem	prcssprc 5288	The superclass of a proper class is a proper class. (Contributed by AV, 27-Dec-2020.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ∉ V) → 𝐵 ∉ V)
Theorem	sselpwd 5289	Membership in a power set. (Contributed by Thierry Arnoux, 18-May-2020.)
⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐵)
Theorem	difexg 5290	Existence of a difference. (Contributed by NM, 26-May-1998.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ 𝐵) ∈ V)
Theorem	difexi 5291	Existence of a difference, inference version of difexg 5290. (Contributed by Glauco Siliprandi, 3-Mar-2021.) (Revised by AV, 26-Mar-2021.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∖ 𝐵) ∈ V
Theorem	difexd 5292	Existence of a difference. (Contributed by SN, 16-Jul-2024.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐴 ∖ 𝐵) ∈ V)
Theorem	sepab 5293*	Separation Scheme (Aussonderung) in terms of a class abstraction. Prefer using the more natural statement rabexg 5298. (Contributed by NM, 8-Jun-1994.) Put in closed form. (Revised by BJ, 18-Jul-2026.)
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V)
Theorem	elpw2g 5294	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 5295	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 5296	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 5297*	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 5298*	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	rabex 5299*	Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 19-Jul-1996.)
⊢ 𝐴 ∈ V    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V
Theorem	rabexd 5300*	Separation Scheme in terms of a restricted class abstraction, deduction form of rabex2 5301. (Contributed by AV, 16-Jul-2019.)
⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜓}    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐵 ∈ V)