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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | mpteq1iOLD 5201* | An equality theorem for the maps-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶) | ||
Theorem | mpteq2da 5202 | 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 | mpteq2daOLD 5203 | Obsolete version of mpteq2da 5202 as of 11-Nov-2024. (Contributed by FL, 14-Sep-2013.) (Revised by Mario Carneiro, 16-Dec-2013.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶)) | ||
Theorem | mpteq2dva 5204* | Slightly more general equality inference for the maps-to notation. (Contributed by Scott Fenton, 25-Apr-2012.) Remove dependency on ax-10 2138. (Revised by SN, 11-Nov-2024.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶)) | ||
Theorem | mpteq2dvaOLD 5205* | Obsolete version of mpteq2dva 5204 as of 11-Nov-2024. (Contributed by Scott Fenton, 25-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶)) | ||
Theorem | mpteq2dv 5206* | 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 5207 | 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 | mpteq2iaOLD 5208 | Obsolete version of mpteq2ia 5207 as of 11-Nov-2024. (Contributed by Mario Carneiro, 16-Dec-2013.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝑥 ∈ 𝐴 → 𝐵 = 𝐶) ⇒ ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) | ||
Theorem | mpteq2i 5209 | 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 5210 | 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 5211* | Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ↦ 𝐵) | ||
Theorem | nfmpt1 5212 | Bound-variable hypothesis builder for the maps-to notation. (Contributed by FL, 17-Feb-2008.) |
⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵) | ||
Theorem | cbvmptf 5213* | 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 2372. See cbvmptfg 5214 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 17-Jan-2024.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑦𝐵 & ⊢ Ⅎ𝑥𝐶 & ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) ⇒ ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶) | ||
Theorem | cbvmptfg 5214 | 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 2372. See cbvmptf 5213 for a version with more disjoint variable conditions, but not requiring ax-13 2372. (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 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.) Add disjoint variable condition to avoid ax-13 2372. See cbvmptg 5216 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 17-Jan-2024.) |
⊢ Ⅎ𝑦𝐵 & ⊢ Ⅎ𝑥𝐶 & ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) ⇒ ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶) | ||
Theorem | cbvmptg 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 2372. See cbvmpt 5215 for a version with more disjoint variable conditions, but not requiring ax-13 2372. (Contributed by NM, 11-Sep-2011.) (New usage is discouraged.) |
⊢ Ⅎ𝑦𝐵 & ⊢ Ⅎ𝑥𝐶 & ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) ⇒ ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶) | ||
Theorem | cbvmptv 5217* | 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 5219 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 17-Nov-2024.) |
⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) ⇒ ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶) | ||
Theorem | cbvmptvOLD 5218* | Obsolete version of cbvmptv 5217 as of 17-Nov-2024. (Contributed by Mario Carneiro, 19-Feb-2013.) Add disjoint variable condition to avoid ax-13 2372. See cbvmptvg 5219 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 17-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) ⇒ ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶) | ||
Theorem | cbvmptvg 5219* | 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 2372. See cbvmptv 5217 for a version with more disjoint variable conditions, but not requiring ax-13 2372. (Contributed by Mario Carneiro, 19-Feb-2013.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) ⇒ ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶) | ||
Theorem | mptv 5220* | Function with universal domain in maps-to notation. (Contributed by NM, 16-Aug-2013.) |
⊢ (𝑥 ∈ V ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐵} | ||
Syntax | wtr 5221 | Extend wff notation to include transitive classes. Notation from [TakeutiZaring] p. 35. |
wff Tr 𝐴 | ||
Definition | df-tr 5222 | Define the transitive class predicate. Not to be confused with a transitive relation (see cotr 6061). Definition of [Enderton] p. 71 extended to arbitrary classes. For alternate definitions, see dftr2 5223 (which is suggestive of the word "transitive"), dftr2c 5224, dftr3 5227, dftr4 5228, dftr5 5225, and (when 𝐴 is a set) unisuc 6393. 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 5223* | An alternate way of defining a transitive class. Exercise 7 of [TakeutiZaring] p. 40. Using dftr2c 5224 instead may avoid dependences on ax-11 2155. (Contributed by NM, 24-Apr-1994.) |
⊢ (Tr 𝐴 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴)) | ||
Theorem | dftr2c 5224* | Variant of dftr2 5223 with commuted quantifiers, useful for shortening proofs and avoiding ax-11 2155. (Contributed by BTernaryTau, 28-Dec-2024.) |
⊢ (Tr 𝐴 ↔ ∀𝑦∀𝑥((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴)) | ||
Theorem | dftr5 5225* | An alternate way of defining a transitive class. (Contributed by NM, 20-Mar-2004.) Avoid ax-11 2155. (Revised by BTernaryTau, 28-Dec-2024.) |
⊢ (Tr 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴) | ||
Theorem | dftr5OLD 5226* | Obsolete version of dftr5 5225 as of 28-Dec-2024. (Contributed by NM, 20-Mar-2004.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (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 | trin 5233 | The intersection of transitive classes is transitive. (Contributed by NM, 9-May-1994.) |
⊢ ((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴 ∩ 𝐵)) | ||
Theorem | tr0 5234 | The empty set is transitive. (Contributed by NM, 16-Sep-1993.) |
⊢ Tr ∅ | ||
Theorem | trv 5235 | The universe is transitive. (Contributed by NM, 14-Sep-2003.) |
⊢ Tr V | ||
Theorem | triun 5236 | 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 5237* | 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 5238 | 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 5239* | 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 5240 | 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) | ||
Axiom | ax-rep 5241* |
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 6585). 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 5247, where you can
find some further remarks. A slightly more compact version is shown as
axrep2 5244. A quite different variant is zfrep6 7878, which if used in
place of ax-rep 5241 would also require that the Separation Scheme
axsep 5254
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 9761 and the Boundedness Axiom bnd 9762. Many developments of set theory distinguish the uses of Replacement from uses of the weaker axioms of Separation axsep 5254, Null Set axnul 5261, and Pairing axpr 5382, 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 5255, ax-nul 5262, and ax-pr 5383 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 5242* | 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 5241 → axrep1 5242 → axrep2 5244 → axrepnd 10464 → zfcndrep 10484 = ax-rep 5241. (Contributed by NM, 19-Nov-2005.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) Remove dependency on ax-13 2372. (Revised by BJ, 31-May-2019.) |
⊢ ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ 𝜑))) | ||
Theorem | axreplem 5243* | Lemma for axrep2 5244 and axrep3 5245. (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 5244* | 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 2372. (Revised by BJ, 31-May-2019.) |
⊢ ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑))) | ||
Theorem | axrep3 5245* | Axiom of Replacement slightly strengthened from axrep2 5244; 𝑤 may occur free in 𝜑. (Contributed by NM, 2-Jan-1997.) Remove dependency on ax-13 2372. (Revised by BJ, 31-May-2019.) |
⊢ ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑))) | ||
Theorem | axrep4 5246* | 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 5247* | 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 5248* | A condensed form of ax-rep 5241. (Contributed by SN, 18-Sep-2023.) |
⊢ (∀𝑤∃*𝑧𝜑 → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑥 𝜑)) | ||
Theorem | axrep6g 5249* | axrep6 5248 in class notation. It is equivalent to both ax-rep 5241 and abrexexg 7884, providing a direct link between the two. (Contributed by SN, 11-Dec-2024.) |
⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥∃*𝑦𝜓) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜓} ∈ V) | ||
Theorem | zfrepclf 5250* | 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 5251* | 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 5252* | 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 | ||
Theorem | axsepgfromrep 5253* | A more general version axsepg 5256 of the axiom scheme of separation ax-sep 5255 derived from the axiom scheme of replacement ax-rep 5241 (and first-order logic). The extra generality consists in the absence of a disjoint variable condition on 𝑧, 𝜑 (that is, variable 𝑧 may occur in formula 𝜑). See linked statements for more information. (Contributed by NM, 11-Sep-2006.) Remove dependencies on ax-9 2117 to ax-13 2372. (Revised by SN, 25-Sep-2023.) Use ax-sep 5255 instead (or axsepg 5256 if the extra generality is needed). (New usage is discouraged.) |
⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) | ||
Theorem | axsep 5254* | Axiom scheme of separation ax-sep 5255 derived from the axiom scheme of replacement ax-rep 5241. The statement is identical to that of ax-sep 5255, and therefore shows that ax-sep 5255 is redundant when ax-rep 5241 is allowed. See ax-sep 5255 for more information. (Contributed by NM, 11-Sep-2006.) Use ax-sep 5255 instead. (New usage is discouraged.) |
⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) | ||
Axiom | ax-sep 5255* |
Axiom scheme of separation. This is an axiom scheme of Zermelo and
Zermelo-Fraenkel set theories.
It was derived as axsep 5254 above and is therefore redundant in ZF set theory, which contains ax-rep 5241 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 zfauscl 5257, 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 5317 shows (showing the necessity of that condition in zfauscl 5257). Scheme Sep of [BellMachover] p. 463. (Contributed by NM, 11-Sep-2006.) |
⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) | ||
Theorem | axsepg 5256* | A more general version of the axiom scheme of separation ax-sep 5255, 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 5255 with no additional set theory axioms. Note that it was also derived from ax-rep 5241 but without ax-sep 5255 as axsepgfromrep 5253. (Contributed by NM, 10-Dec-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) Remove dependency on ax-12 2172 and ax-13 2372 and shorten proof. (Revised by BJ, 6-Oct-2019.) |
⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) | ||
Theorem | zfauscl 5257* |
Separation Scheme (Aussonderung) using a class variable. To derive this
from ax-sep 5255, we invoke the Axiom of Extensionality
(indirectly via
vtocl 3517), 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 5317 shows. (Contributed by NM, 21-Jun-1993.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) | ||
Theorem | bm1.3ii 5258* | Convert implication to equivalence using the Separation Scheme (Aussonderung) ax-sep 5255. 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 5259* | Derive ax6v 1973 (a weakened version of ax-6 1972 where 𝑥 and 𝑦 must be distinct), from Separation ax-sep 5255 and Extensionality ax-ext 2709. See ax6 2384 for the derivation of ax-6 1972 from ax6v 1973. (Contributed by NM, 12-Nov-2013.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦 | ||
Theorem | axnulALT 5260* | Alternate proof of axnul 5261, proved from propositional calculus, ax-gen 1798, ax-4 1812, sp 2177, and ax-rep 5241. To check this, replace sp 2177 with the obsolete axiom ax-c5 37231 in the proof of axnulALT 5260 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 5261* |
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 5255. 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 2714).
This proof, suggested by Jeff Hoffman, uses only ax-4 1812 and ax-gen 1798 from predicate calculus, which are valid in "free logic" i.e. logic holding in an empty domain (see Axiom A5 and Rule R2 of [LeBlanc] p. 277). Thus, our ax-sep 5255 implies the existence of at least one set. Note that Kunen's version of ax-sep 5255 (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 5260 for a proof directly from ax-rep 5241. This theorem should not be referenced by any proof. Instead, use ax-nul 5262 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 5262* | The Null Set Axiom of ZF set theory. It was derived as axnul 5261 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 5263 | 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 5262. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
⊢ ∅ ∈ V | ||
Theorem | al0ssb 5264* | 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 5265 | Alternate proof of sseli 3939 illustrating the use of the weak deduction theorem to prove it from the inference sselii 3940. (Contributed by NM, 24-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝐴 ⊆ 𝐵 ⇒ ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵) | ||
Theorem | csbexg 5266 | 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 5267 | 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 5268 | A version of unisn 4886 without the 𝐴 ∈ V hypothesis. (Contributed by Stefan Allan, 14-Mar-2006.) |
⊢ ∪ {𝐴} ∈ {∅, 𝐴} | ||
Theorem | nalset 5269* | No set contains all sets. Theorem 41 of [Suppes] p. 30. (Contributed by NM, 23-Aug-1993.) Remove use of ax-12 2172 and ax-13 2372. (Revised by BJ, 31-May-2019.) |
⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥 | ||
Theorem | vnex 5270 | The universal class does not exist as a set. (Contributed by NM, 4-Jul-2005.) |
⊢ ¬ ∃𝑥 𝑥 = V | ||
Theorem | vprc 5271 | 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 5272 | The universal class does not belong to any class. (Contributed by FL, 31-Dec-2006.) |
⊢ ¬ V ∈ 𝐴 | ||
Theorem | inex1 5273 | 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 5274 | Separation Scheme (Aussonderung) using class notation. (Contributed by NM, 27-Apr-1994.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐵 ∩ 𝐴) ∈ V | ||
Theorem | inex1g 5275 | Closed-form, generalized Separation Scheme. (Contributed by NM, 7-Apr-1995.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V) | ||
Theorem | inex2g 5276 | Sufficient condition for an intersection to be a set. Commuted form of inex1g 5275. (Contributed by Peter Mazsa, 19-Dec-2018.) |
⊢ (𝐴 ∈ 𝑉 → (𝐵 ∩ 𝐴) ∈ V) | ||
Theorem | ssex 5277 | 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 5255 (a.k.a. Subset Axiom). (Contributed by NM, 27-Apr-1994.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V) | ||
Theorem | ssexi 5278 | The subset of a set is also a set. (Contributed by NM, 9-Sep-1993.) |
⊢ 𝐵 ∈ V & ⊢ 𝐴 ⊆ 𝐵 ⇒ ⊢ 𝐴 ∈ V | ||
Theorem | ssexg 5279 | 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 5280 | A subclass of a set is a set. Deduction form of ssexg 5279. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐵 ∈ 𝐶) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ∈ V) | ||
Theorem | prcssprc 5281 | The superclass of a proper class is a proper class. (Contributed by AV, 27-Dec-2020.) |
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ∉ V) → 𝐵 ∉ V) | ||
Theorem | sselpwd 5282 | Elementhood to a power set. (Contributed by Thierry Arnoux, 18-May-2020.) |
⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐵) | ||
Theorem | difexg 5283 | Existence of a difference. (Contributed by NM, 26-May-1998.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ 𝐵) ∈ V) | ||
Theorem | difexi 5284 | Existence of a difference, inference version of difexg 5283. (Contributed by Glauco Siliprandi, 3-Mar-2021.) (Revised by AV, 26-Mar-2021.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∖ 𝐵) ∈ V | ||
Theorem | difexd 5285 | Existence of a difference. (Contributed by SN, 16-Jul-2024.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐴 ∖ 𝐵) ∈ V) | ||
Theorem | zfausab 5286* | Separation Scheme (Aussonderung) in terms of a class abstraction. (Contributed by NM, 8-Jun-1994.) |
⊢ 𝐴 ∈ V ⇒ ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V | ||
Theorem | rabexg 5287* | Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 23-Oct-1999.) |
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V) | ||
Theorem | rabex 5288* | Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 19-Jul-1996.) |
⊢ 𝐴 ∈ V ⇒ ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V | ||
Theorem | rabexd 5289* | Separation Scheme in terms of a restricted class abstraction, deduction form of rabex2 5290. (Contributed by AV, 16-Jul-2019.) |
⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜓} & ⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝐵 ∈ V) | ||
Theorem | rabex2 5290* | 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 5291* | 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 5292* | Membership in a class abstraction involving a subset. Unlike elabg 3627, 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 5293 | 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 5294 | 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 5295 | The intersection of a nonempty class abstraction exists. (Contributed by NM, 21-Oct-2003.) |
⊢ (∃𝑥𝜑 ↔ ∩ {𝑥 ∣ 𝜑} ∈ V) | ||
Theorem | intexrab 5296 | The intersection of a nonempty restricted class abstraction exists. (Contributed by NM, 21-Oct-2003.) |
⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∩ {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V) | ||
Theorem | iinexg 5297* | 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 5298* | 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 5299* | 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.) |
⊢ (∪ 𝐴 ∩ 𝐵) = ∪ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = (𝑦 ∩ 𝐵)} | ||
Theorem | elpw2g 5300 | Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 7-Aug-2000.) |
⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)) |
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