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	mpteq12dva 5201*	An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 26-Jan-2017.) Remove dependency on ax-10 2182, ax-12 2219. (Revised by SN, 11-Nov-2024.)
⊢ (𝜑 → 𝐴 = 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))
Theorem	mpteq12dv 5202*	An equality inference for the maps-to notation. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 16-Dec-2013.) Remove dependency on ax-10 2182, ax-12 2219. (Revised by SN and GG, 1-Dec-2023.)
⊢ (𝜑 → 𝐴 = 𝐶)    &   ⊢ (𝜑 → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))
Theorem	mpteq12 5203*	An equality theorem for the maps-to notation. (Contributed by NM, 16-Dec-2013.)
⊢ ((𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))
Theorem	mpteq1 5204*	An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) (Proof shortened by SN, 11-Nov-2024.)
⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶))
Theorem	mpteq1d 5205*	An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 11-Jun-2016.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶))
Theorem	mpteq1i 5206	An equality theorem for the maps-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.) Remove all disjoint variable conditions. (Revised by SN, 11-Nov-2024.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶)
Theorem	mpteq2da 5207	Slightly more general equality inference for the maps-to notation. (Contributed by FL, 14-Sep-2013.) (Revised by Mario Carneiro, 16-Dec-2013.) (Proof shortened by SN, 11-Nov-2024.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶))
Theorem	mpteq2dva 5208*	Slightly more general equality inference for the maps-to notation. (Contributed by Scott Fenton, 25-Apr-2012.) Remove dependency on ax-10 2182. (Revised by SN, 11-Nov-2024.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶))
Theorem	mpteq2dv 5209*	An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 23-Aug-2014.)
⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶))
Theorem	mpteq2ia 5210	An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) (Proof shortened by SN, 11-Nov-2024.)
⊢ (𝑥 ∈ 𝐴 → 𝐵 = 𝐶)    ⇒   ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶)
Theorem	mpteq2i 5211	An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
⊢ 𝐵 = 𝐶    ⇒   ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶)
Theorem	mpteq12i 5212	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 5213*	Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ↦ 𝐵)
Theorem	nfmpt1 5214	Bound-variable hypothesis builder for the maps-to notation. (Contributed by FL, 17-Feb-2008.)
⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
Theorem	cbvmptf 5215*	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 2410. See cbvmptfg 5216 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 5216	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 2410. See cbvmptf 5215 for a version with more disjoint variable conditions, but not requiring ax-13 2410. (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 5217*	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 2410. See cbvmptg 5218 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 5218*	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 2410. See cbvmpt 5217 for a version with more disjoint variable conditions, but not requiring ax-13 2410. (Contributed by NM, 11-Sep-2011.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝐵    &   ⊢ Ⅎ𝑥𝐶    &   ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)    ⇒   ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶)
Theorem	cbvmptv 5219*	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 5220 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 5220*	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 2410. See cbvmptv 5219 for a version with more disjoint variable conditions, but not requiring ax-13 2410. (Contributed by Mario Carneiro, 19-Feb-2013.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)    ⇒   ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶)
Theorem	mptv 5221*	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 5222	Extend wff notation to include transitive classes. Notation from [TakeutiZaring] p. 35.
wff Tr 𝐴
Definition	df-tr 5223	Define the transitive class predicate. Not to be confused with a transitive relation (see cotr 6113). Definition of [Enderton] p. 71 extended to arbitrary classes. For alternate definitions, see dftr2 5224 (which is suggestive of the word "transitive"), dftr2c 5225, dftr3 5227, dftr4 5228, dftr5 5226, and (when 𝐴 is a set) unisuc 6443. 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 5224*	An alternate way of defining a transitive class. Exercise 7 of [TakeutiZaring] p. 40. Using dftr2c 5225 instead may avoid dependences on ax-11 2198. (Contributed by NM, 24-Apr-1994.)
⊢ (Tr 𝐴 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴))
Theorem	dftr2c 5225*	Variant of dftr2 5224 with commuted quantifiers, useful for shortening proofs and avoiding ax-11 2198. (Contributed by BTernaryTau, 28-Dec-2024.)
⊢ (Tr 𝐴 ↔ ∀𝑦∀𝑥((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴))
Theorem	dftr5 5226*	An alternate way of defining a transitive class. Definition 1.1 of [Schloeder] p. 1. (Contributed by NM, 20-Mar-2004.) Avoid ax-11 2198. (Revised by BTernaryTau, 28-Dec-2024.)
⊢ (Tr 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴)
Theorem	dftr3 5227*	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 5228	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 5229	Equality theorem for the transitive class predicate. (Contributed by NM, 17-Sep-1993.)
⊢ (𝐴 = 𝐵 → (Tr 𝐴 ↔ Tr 𝐵))
Theorem	trel 5230	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 5231	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 5232	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 5233	The union of transitive classes is transitive. (Contributed by Eric Schmidt, 19-Apr-2026.)
⊢ ((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴 ∪ 𝐵))
Theorem	trin 5234	The intersection of transitive classes is transitive. (Contributed by NM, 9-May-1994.)
⊢ ((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴 ∩ 𝐵))
Theorem	tr0 5235	The empty set is transitive. (Contributed by NM, 16-Sep-1993.)
⊢ Tr ∅
Theorem	trv 5236	The universe is transitive. (Contributed by NM, 14-Sep-2003.)
⊢ Tr V
Theorem	triun 5237	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 5238*	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 5239	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 5240*	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 5241	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 5242*	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 6624). 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 5250, where you can find some further remarks. A slightly more compact version is shown as axrep2 5245. A quite different variant is zfrep6 5254, which if used in place of ax-rep 5242 would also require that the Separation Scheme axsep 5260 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 9876 and the Boundedness Axiom bnd 9877. Many developments of set theory distinguish the uses of Replacement from uses of the weaker axioms of Separation axsep 5260, Null Set axnul 5270, and Pairing axpr 5399, 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 5261, ax-nul 5271, and ax-pr 5405 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 5243*	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 5242 → axrep1 5243 → axrep2 5245 → axrepnd 10578 → zfcndrep 10598 = ax-rep 5242. (Contributed by NM, 19-Nov-2005.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) Remove dependency on ax-13 2410. (Revised by BJ, 31-May-2019.)
⊢ ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ 𝜑)))
Theorem	axreplem 5244*	Lemma for axrep2 5245 and axrep3 5246. (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 5245*	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 2410. (Revised by BJ, 31-May-2019.)
⊢ ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑)))
Theorem	axrep3 5246*	Axiom of Replacement slightly strengthened from axrep2 5245; 𝑤 may occur free in 𝜑. (Contributed by NM, 2-Jan-1997.) Remove dependency on ax-13 2410. (Revised by BJ, 31-May-2019.)
⊢ ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑)))
Theorem	axrep4v 5247*	Version of axrep4 5248 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 5248*	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	axrep4OLD 5249*	Obsolete version of axrep4 5248 as of 18-Sep-2025. (Contributed by NM, 14-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Ⅎ𝑧𝜑    ⇒   ⊢ (∀𝑥∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧) → ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑)))
Theorem	axrep5 5250*	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 5251*	A condensed form of ax-rep 5242. (Contributed by SN, 18-Sep-2023.) (Proof shortened by Matthew House, 18-Sep-2025.)
⊢ (∀𝑤∃*𝑧𝜑 → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑥 𝜑))
Theorem	axrep6OLD 5252*	Obsolete version of axrep6 5251 as of 18-Sep-2025. (Contributed by SN, 18-Sep-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑤∃*𝑧𝜑 → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑥 𝜑))
Theorem	replem 5253*	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 5254*	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 5261 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 5242. (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 5255*	axrep6 5251 in class notation. It is equivalent to both ax-rep 5242 and abrexexg 7957, providing a direct link between the two. (Contributed by SN, 11-Dec-2024.)
⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥∃*𝑦𝜓) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜓} ∈ V)
Theorem	zfrepclf 5256*	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 5257*	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 5258*	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 5259*	A more general version axsepg 5262 of the axiom scheme of separation ax-sep 5261 derived from the axiom scheme of replacement ax-rep 5242 (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 2159 to ax-13 2410. (Revised by SN, 25-Sep-2023.) Use ax-sep 5261 instead (or axsepg 5262 if the extra generality is needed). (New usage is discouraged.)
⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
Theorem	axsep 5260*	Axiom scheme of separation ax-sep 5261 derived from the axiom scheme of replacement ax-rep 5242. The statement is identical to that of ax-sep 5261, and therefore shows that ax-sep 5261 is redundant when ax-rep 5242 is allowed. See ax-sep 5261 for more information. (Contributed by NM, 11-Sep-2006.) Use ax-sep 5261 instead. (New usage is discouraged.)
⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
Axiom	ax-sep 5261*	Axiom scheme of separation. This is an axiom scheme of Zermelo and Zermelo-Fraenkel set theories. It was derived as axsep 5260 above and is therefore redundant in ZF set theory, which contains ax-rep 5242 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 3752. 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 5263, 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 5335 shows (showing the necessity of that condition in zfauscl 5263). Scheme Sep of [BellMachover] p. 463. (Contributed by NM, 11-Sep-2006.)
⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
Theorem	axsepg 5262*	A more general version of the axiom scheme of separation ax-sep 5261, 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 5261 with no additional set theory axioms. Note that it was also derived from ax-rep 5242 but without ax-sep 5261 as axsepgfromrep 5259. (Contributed by NM, 10-Dec-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) Remove dependency on ax-12 2219 and ax-13 2410 and shorten proof. (Revised by BJ, 6-Oct-2019.)
⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
Theorem	zfauscl 5263*	Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep 5261, we invoke the Axiom of Extensionality (indirectly via vtocl 3534), 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 5335 shows. (Contributed by NM, 21-Jun-1993.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))
Theorem	sepexlem 5264*	Lemma for sepex 5265. Use sepex 5265 instead. (Contributed by Matthew House, 19-Sep-2025.) (New usage is discouraged.)
⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 ∈ 𝑦) → ∃𝑧∀𝑥(𝑥 ∈ 𝑧 ↔ 𝜑))
Theorem	sepex 5265*	Convert implication to equivalence within an existence statement using the Separation Scheme (Aussonderung) ax-sep 5261. 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 5266*	Convert implication to equivalence within an existence statement using the Separation Scheme (Aussonderung) ax-sep 5261. Inference associated with sepex 5265. (Contributed by NM, 21-Jun-1993.) Generalize conclusion, extract closed form, and avoid ax-9 2159. (Revised by Matthew House, 19-Sep-2025.)
⊢ ∃𝑦∀𝑥(𝜑 → 𝑥 ∈ 𝑦)    ⇒   ⊢ ∃𝑧∀𝑥(𝑥 ∈ 𝑧 ↔ 𝜑)
Theorem	bm1.3iiOLD 5267*	Obsolete version of sepexi 5266 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 5268*	Derive ax6v 1995 (a weakened version of ax-6 1994 where 𝑥 and 𝑦 must be distinct), from Separation ax-sep 5261 and Extensionality ax-ext 2741. See ax6 2422 for the derivation of ax-6 1994 from ax6v 1995. (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 5269*	Alternate proof of axnul 5270, proved from propositional calculus, ax-gen 1822, ax-4 1836, sp 2225, and ax-rep 5242. To check this, replace sp 2225 with the obsolete axiom ax-c5 39546 in the proof of axnulALT 5269 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 5270*	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 5261. 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 2746). This proof, suggested by Jeff Hoffman, uses only ax-4 1836 and ax-gen 1822 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 5261 implies the existence of at least one set. Note that Kunen's version of ax-sep 5261 (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 5269 for a proof directly from ax-rep 5242. This theorem should not be referenced by any proof. Instead, use ax-nul 5271 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 5271*	The Null Set Axiom of ZF set theory. It was derived as axnul 5270 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 5272	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 5271. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
⊢ ∅ ∈ V
Theorem	al0ssb 5273*	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 5274	Alternate proof of sseli 3941 illustrating the use of the weak deduction theorem to prove it from the inference sselii 3942. (Contributed by NM, 24-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐴 ⊆ 𝐵    ⇒   ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵)
Theorem	csbexg 5275	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 5276	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 5277	A version of unisn 4895 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 5278*	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 2219 and ax-13 2410. (Revised by BJ, 31-May-2019.) Extract from nalset 5279. (Revised by Matthew House, 12-Apr-2026.)
⊢ ∃𝑦 ¬ 𝑦 ∈ 𝑥
Theorem	nalset 5279*	No set contains all sets. Theorem 41 of [Suppes] p. 30. (Contributed by NM, 23-Aug-1993.) Extract exnelv 5278. (Revised by Matthew House, 12-Apr-2026.)
⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥
Theorem	nalsetOLD 5280*	Obsolete version of nalset 5279 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 5281	The universal class is not equal to any setvar. (Contributed by NM, 4-Jul-2005.) Extract from vnex 5282 and shorten proof. (Revised by BJ, 25-Apr-2026.)
⊢ ¬ 𝑥 = V
Theorem	vnex 5282	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 5283	Obsolete proof of vnex 5282 as of 25-Apr-2026. (Contributed by NM, 4-Jul-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ¬ ∃𝑥 𝑥 = V
Theorem	nvel 5284	The universal class does not belong to any class. (Contributed by FL, 31-Dec-2006.) Prove it without using vprc 5285, which is then proved as an instance of it. (Revised by BJ, 1-May-2026.)
⊢ ¬ V ∈ 𝐴
Theorem	vprc 5285	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 5286	Obsolete proof of vprc 5285, obsolete as of 1-May-2026. (Contributed by NM, 23-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ¬ V ∈ V
Theorem	nvelOLD 5287	Obsolete proof of nvel 5284, obsolete as of 1-May-2026. (Contributed by FL, 31-Dec-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ¬ V ∈ 𝐴
Theorem	inex1 5288	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 5289	Separation Scheme (Aussonderung) using class notation. (Contributed by NM, 27-Apr-1994.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐵 ∩ 𝐴) ∈ V
Theorem	inex1g 5290	Closed-form, generalized Separation Scheme. (Contributed by NM, 7-Apr-1995.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V)
Theorem	inex2g 5291	Sufficient condition for an intersection to be a set. Commuted form of inex1g 5290. (Contributed by Peter Mazsa, 19-Dec-2018.)
⊢ (𝐴 ∈ 𝑉 → (𝐵 ∩ 𝐴) ∈ V)
Theorem	ssex 5292	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 5261 (a.k.a. Subset Axiom). (Contributed by NM, 27-Apr-1994.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)
Theorem	ssexi 5293	The subset of a set is also a set. (Contributed by NM, 9-Sep-1993.)
⊢ 𝐵 ∈ V    &   ⊢ 𝐴 ⊆ 𝐵    ⇒   ⊢ 𝐴 ∈ V
Theorem	ssexg 5294	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 5295	A subclass of a set is a set. Deduction form of ssexg 5294. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐵 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ∈ V)
Theorem	abexd 5296*	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 5297*	Conditions for a class abstraction to be a set. Remark: This proof is shorter than a proof using abexd 5296. (Contributed by AV, 19-Apr-2025.)
⊢ (𝜑 → 𝑥 ∈ 𝐴)    &   ⊢ 𝐴 ∈ V    ⇒   ⊢ {𝑥 ∣ 𝜑} ∈ V
Theorem	prcssprc 5298	The superclass of a proper class is a proper class. (Contributed by AV, 27-Dec-2020.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ∉ V) → 𝐵 ∉ V)
Theorem	sselpwd 5299	Elementhood to a power set. (Contributed by Thierry Arnoux, 18-May-2020.)
⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐵)
Theorem	difexg 5300	Existence of a difference. (Contributed by NM, 26-May-1998.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ 𝐵) ∈ V)