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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | axnul 5201* |
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 5195. 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 2798).
This proof, suggested by Jeff Hoffman, uses only ax-4 1801 and ax-gen 1787 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 5195 implies the existence of at least one set. Note that Kunen's version of ax-sep 5195 (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 5200 for a proof directly from ax-rep 5182. This theorem should not be referenced by any proof. Instead, use ax-nul 5202 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 5202* | The Null Set Axiom of ZF set theory. It was derived as axnul 5201 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 5203 | 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 5202. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
⊢ ∅ ∈ V | ||
Theorem | al0ssb 5204* | 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 5205 | Alternate proof of sseli 3962 illustrating the use of the weak deduction theorem to prove it from the inference sselii 3963. (Contributed by NM, 24-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝐴 ⊆ 𝐵 ⇒ ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵) | ||
Theorem | csbexg 5206 | 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 5207 | 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 5208 | A version of unisn 4848 without the 𝐴 ∈ V hypothesis. (Contributed by Stefan Allan, 14-Mar-2006.) |
⊢ ∪ {𝐴} ∈ {∅, 𝐴} | ||
Theorem | nalset 5209* | No set contains all sets. Theorem 41 of [Suppes] p. 30. (Contributed by NM, 23-Aug-1993.) Remove use of ax-12 2167 and ax-13 2383. (Revised by BJ, 31-May-2019.) |
⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥 | ||
Theorem | vnex 5210 | The universal class does not exist as a set. (Contributed by NM, 4-Jul-2005.) |
⊢ ¬ ∃𝑥 𝑥 = V | ||
Theorem | vprc 5211 | 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 5212 | The universal class does not belong to any class. (Contributed by FL, 31-Dec-2006.) |
⊢ ¬ V ∈ 𝐴 | ||
Theorem | inex1 5213 | 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 5214 | Separation Scheme (Aussonderung) using class notation. (Contributed by NM, 27-Apr-1994.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐵 ∩ 𝐴) ∈ V | ||
Theorem | inex1g 5215 | Closed-form, generalized Separation Scheme. (Contributed by NM, 7-Apr-1995.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V) | ||
Theorem | inex2g 5216 | Sufficient condition for an intersection to be a set. Commuted form of inex1g 5215. (Contributed by Peter Mazsa, 19-Dec-2018.) |
⊢ (𝐴 ∈ 𝑉 → (𝐵 ∩ 𝐴) ∈ V) | ||
Theorem | ssex 5217 | 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 5195 (a.k.a. Subset Axiom). (Contributed by NM, 27-Apr-1994.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V) | ||
Theorem | ssexi 5218 | The subset of a set is also a set. (Contributed by NM, 9-Sep-1993.) |
⊢ 𝐵 ∈ V & ⊢ 𝐴 ⊆ 𝐵 ⇒ ⊢ 𝐴 ∈ V | ||
Theorem | ssexg 5219 | 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 5220 | A subclass of a set is a set. Deduction form of ssexg 5219. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐵 ∈ 𝐶) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ∈ V) | ||
Theorem | prcssprc 5221 | The superclass of a proper class is a proper class. (Contributed by AV, 27-Dec-2020.) |
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ∉ V) → 𝐵 ∉ V) | ||
Theorem | sselpwd 5222 | Elementhood to a power set. (Contributed by Thierry Arnoux, 18-May-2020.) |
⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐵) | ||
Theorem | difexg 5223 | Existence of a difference. (Contributed by NM, 26-May-1998.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ 𝐵) ∈ V) | ||
Theorem | difexi 5224 | Existence of a difference, inference version of difexg 5223. (Contributed by Glauco Siliprandi, 3-Mar-2021.) (Revised by AV, 26-Mar-2021.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∖ 𝐵) ∈ V | ||
Theorem | zfausab 5225* | Separation Scheme (Aussonderung) in terms of a class abstraction. (Contributed by NM, 8-Jun-1994.) |
⊢ 𝐴 ∈ V ⇒ ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V | ||
Theorem | rabexg 5226* | Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 23-Oct-1999.) |
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V) | ||
Theorem | rabex 5227* | Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 19-Jul-1996.) |
⊢ 𝐴 ∈ V ⇒ ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V | ||
Theorem | rabexd 5228* | Separation Scheme in terms of a restricted class abstraction, deduction form of rabex2 5229. (Contributed by AV, 16-Jul-2019.) |
⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜓} & ⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝐵 ∈ V) | ||
Theorem | rabex2 5229* | 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 5230* | 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 5231* | Membership in a class abstraction involving a subset. Unlike elabg 3665, 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 5232 | 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 5233 | 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 5234 | The intersection of a nonempty class abstraction exists. (Contributed by NM, 21-Oct-2003.) |
⊢ (∃𝑥𝜑 ↔ ∩ {𝑥 ∣ 𝜑} ∈ V) | ||
Theorem | intexrab 5235 | The intersection of a nonempty restricted class abstraction exists. (Contributed by NM, 21-Oct-2003.) |
⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∩ {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V) | ||
Theorem | iinexg 5236* | 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 5237* | 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 5238* | 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 5239 | 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 5240 | 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 5241 | Membership in a power class. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
⊢ 𝐵 ∈ 𝑉 & ⊢ 𝐴 ⊆ 𝐵 ⇒ ⊢ 𝐴 ∈ 𝒫 𝐵 | ||
Theorem | pwnss 5242 | The power set of a set is never a subset. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
⊢ (𝐴 ∈ 𝑉 → ¬ 𝒫 𝐴 ⊆ 𝐴) | ||
Theorem | pwne 5243 | No set equals its power set. The sethood antecedent is necessary; compare pwv 4829. (Contributed by NM, 17-Nov-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.) |
⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ≠ 𝐴) | ||
Theorem | difelpw 5244 | A difference is an element of the power set of its minuend. (Contributed by AV, 9-Oct-2023.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ 𝐵) ∈ 𝒫 𝐴) | ||
Theorem | rabelpw 5245* | 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 | class2set 5246* | Construct, from any class 𝐴, a set equal to it when the class exists and equal to the empty set when the class is proper. This theorem shows that the constructed set always exists. (Contributed by NM, 16-Oct-2003.) |
⊢ {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} ∈ V | ||
Theorem | class2seteq 5247* | Equality theorem based on class2set 5246. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Raph Levien, 30-Jun-2006.) |
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} = 𝐴) | ||
Theorem | 0elpw 5248 | Every power class contains the empty set. (Contributed by NM, 25-Oct-2007.) |
⊢ ∅ ∈ 𝒫 𝐴 | ||
Theorem | pwne0 5249 | A power class is never empty. (Contributed by NM, 3-Sep-2018.) |
⊢ 𝒫 𝐴 ≠ ∅ | ||
Theorem | 0nep0 5250 | The empty set and its power set are not equal. (Contributed by NM, 23-Dec-1993.) |
⊢ ∅ ≠ {∅} | ||
Theorem | 0inp0 5251 | 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 5252 | The removal of the empty set from a class does not affect its union. (Contributed by NM, 22-Mar-2004.) |
⊢ ∪ (𝐴 ∖ {∅}) = ∪ 𝐴 | ||
Theorem | iin0 5253* | 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 5254* | In the Separation Scheme zfauscl 5197, 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 | notzfausOLD 5255* | Obsolete proof of notzfaus 5254 as of 18-Nov-2023. (Contributed by NM, 8-Feb-2006.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝐴 = {∅} & ⊢ (𝜑 ↔ ¬ 𝑥 ∈ 𝑦) ⇒ ⊢ ¬ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) | ||
Theorem | intv 5256 | The intersection of the universal class is empty. (Contributed by NM, 11-Sep-2008.) |
⊢ ∩ V = ∅ | ||
Theorem | axpweq 5257* | Two equivalent ways to express the Power Set Axiom. Note that ax-pow 5258 is not used by the proof. When ax-pow 5258 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 7475). (Contributed by NM, 22-Jun-2009.) |
⊢ (𝒫 𝐴 ∈ V ↔ ∃𝑥∀𝑦(∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴) → 𝑦 ∈ 𝑥)) | ||
Axiom | ax-pow 5258* | 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 5260 uses explicit subset notation. A version using class notation is pwex 5273. (Contributed by NM, 21-Jun-1993.) |
⊢ ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦) | ||
Theorem | zfpow 5259* | 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 5260* | A variant of the Axiom of Power Sets ax-pow 5258 using subset notation. Problem in [BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.) |
⊢ ∃𝑦∀𝑧(𝑧 ⊆ 𝑥 → 𝑧 ∈ 𝑦) | ||
Theorem | axpow3 5261* | A variant of the Axiom of Power Sets ax-pow 5258. 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 | el 5262* | Every set is an element of some other set. See elALT 5326 for a shorter proof using more axioms. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
⊢ ∃𝑦 𝑥 ∈ 𝑦 | ||
Theorem | dtru 5263* |
At least two sets exist (or in terms of first-order logic, the universe
of discourse has two or more objects). Note that we may not substitute
the same variable for both 𝑥 and 𝑦 (as indicated by the
distinct
variable requirement), for otherwise we would contradict stdpc6 2026.
This theorem is proved directly from set theory axioms (no set theory definitions) and does not use ax-ext 2793 or ax-sep 5195. See dtruALT 5280 for a shorter proof using these axioms. The proof makes use of dummy variables 𝑧 and 𝑤 which do not appear in the final theorem. They must be distinct from each other and from 𝑥 and 𝑦. In other words, if we were to substitute 𝑥 for 𝑧 throughout the proof, the proof would fail. (Contributed by NM, 7-Nov-2006.) Avoid ax-13 2383. (Revised by Gino Giotto, 5-Sep-2023.) |
⊢ ¬ ∀𝑥 𝑥 = 𝑦 | ||
Theorem | dtrucor 5264* | Corollary of dtru 5263. 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 5265. (Contributed by NM, 27-Jun-2002.) |
⊢ 𝑥 = 𝑦 ⇒ ⊢ 𝑥 ≠ 𝑦 | ||
Theorem | dtrucor2 5265 | The theorem form of the deduction dtrucor 5264 leads to a contradiction, as mentioned in the "Wrong!" example at mmdeduction.html#bad 5264. (Contributed by NM, 20-Oct-2007.) |
⊢ (𝑥 = 𝑦 → 𝑥 ≠ 𝑦) ⇒ ⊢ (𝜑 ∧ ¬ 𝜑) | ||
Theorem | dvdemo1 5266* |
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 5267, 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 5267 for details on the "disjoint variable" mechanism. (The verb "bundle" to express this phenomenon was introduced by Raph Levien.) Note that dvdemo1 5266 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 2151 nor ax-13 2383. (Contributed by NM, 1-Dec-2006.) (Revised by BJ, 13-Jan-2024.) |
⊢ ∃𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝑥) | ||
Theorem | dvdemo2 5267* |
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 5266, 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 5266 for details on the "disjoint variable" mechanism. Note that dvdemo2 5267 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 2136, ax-11 2151, ax-12 2167, ax-13 2383. (Contributed by NM, 1-Dec-2006.) (Revised by BJ, 13-Jan-2024.) |
⊢ ∃𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝑥) | ||
Theorem | nfnid 5268 | A setvar variable is not free from itself. This theorem is not true in a one-element domain, as illustrated by the use of dtru 5263 in its proof. (Contributed by Mario Carneiro, 8-Oct-2016.) |
⊢ ¬ Ⅎ𝑥𝑥 | ||
Theorem | nfcvb 5269 | 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.) |
⊢ (Ⅎ𝑥𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦) | ||
Theorem | vpwex 5270 | 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 5271 from vpwex 5270. (Revised by BJ, 10-Aug-2022.) |
⊢ 𝒫 𝑥 ∈ V | ||
Theorem | pwexg 5271 | 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 5272 | Deduction version of the power set axiom. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝒫 𝐴 ∈ V) | ||
Theorem | pwex 5273 | Power set axiom expressed in class notation. (Contributed by NM, 21-Jun-1993.) |
⊢ 𝐴 ∈ V ⇒ ⊢ 𝒫 𝐴 ∈ V | ||
Theorem | abssexg 5274* | 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 5275 | Alternate proof of snex 5323 using Power Set (ax-pow 5258) instead of Pairing (ax-pr 5321). Unlike in the proof of zfpair 5313, Replacement (ax-rep 5182) 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 5276 | The power set of the empty set (the ordinal 1) is a set. See also p0exALT 5277. (Contributed by NM, 23-Dec-1993.) |
⊢ {∅} ∈ V | ||
Theorem | p0exALT 5277 | Alternate proof of p0ex 5276 which is quite different and longer if snexALT 5275 is expanded. (Contributed by NM, 23-Dec-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ {∅} ∈ V | ||
Theorem | pp0ex 5278 | 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 5279 | The ordinal number 3 is a set, proved without the Axiom of Union ax-un 7450. (Contributed by NM, 2-May-2009.) |
⊢ {∅, {∅}, {∅, {∅}}} ∈ V | ||
Theorem | dtruALT 5280* |
Alternate proof of dtru 5263 which requires more axioms but is shorter and
may be easier to understand.
Assuming that ZF set theory is consistent, we cannot prove this theorem unless we specify that 𝑥 and 𝑦 be distinct. Specifically, theorem spcev 3606 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 5281* | This theorem shows that axiom ax-c16 35910 is redundant in the presence of theorem dtru 5263, which states simply that at least two things exist. This justifies the remark at mmzfcnd.html#twoness 5263 (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 5282 | 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 5283* | 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 5284* | 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 5285* | 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 5286* | 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 5287* | 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) | ||
Theorem | eusv2 5288* | Two ways to express single-valuedness of a class expression 𝐴(𝑥). (Contributed by NM, 15-Oct-2010.) (Proof shortened by Mario Carneiro, 18-Nov-2016.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (∃!𝑦∃𝑥 𝑦 = 𝐴 ↔ ∃!𝑦∀𝑥 𝑦 = 𝐴) | ||
Theorem | reusv1 5289* | Two ways to express single-valuedness of a class expression 𝐶(𝑦). (Contributed by NM, 16-Dec-2012.) (Proof shortened by Mario Carneiro, 18-Nov-2016.) (Proof shortened by JJ, 7-Aug-2021.) |
⊢ (∃𝑦 ∈ 𝐵 𝜑 → (∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))) | ||
Theorem | reusv2lem1 5290* | Lemma for reusv2 5295. (Contributed by NM, 22-Oct-2010.) (Proof shortened by Mario Carneiro, 19-Nov-2016.) |
⊢ (𝐴 ≠ ∅ → (∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵)) | ||
Theorem | reusv2lem2 5291* | Lemma for reusv2 5295. (Contributed by NM, 27-Oct-2010.) (Proof shortened by Mario Carneiro, 19-Nov-2016.) (Proof shortened by JJ, 7-Aug-2021.) |
⊢ (∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵) | ||
Theorem | reusv2lem3 5292* | Lemma for reusv2 5295. (Contributed by NM, 14-Dec-2012.) (Proof shortened by Mario Carneiro, 19-Nov-2016.) |
⊢ (∀𝑦 ∈ 𝐴 𝐵 ∈ V → (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵)) | ||
Theorem | reusv2lem4 5293* | Lemma for reusv2 5295. (Contributed by NM, 13-Dec-2012.) |
⊢ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝑥 = 𝐶) ↔ ∃!𝑥∀𝑦 ∈ 𝐵 ((𝐶 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝐶)) | ||
Theorem | reusv2lem5 5294* | Lemma for reusv2 5295. (Contributed by NM, 4-Jan-2013.) (Proof shortened by Mario Carneiro, 19-Nov-2016.) |
⊢ ((∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐴 ∧ 𝐵 ≠ ∅) → (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 = 𝐶 ↔ ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 = 𝐶)) | ||
Theorem | reusv2 5295* | Two ways to express single-valuedness of a class expression 𝐶(𝑦) that is constant for those 𝑦 ∈ 𝐵 such that 𝜑. The first antecedent ensures that the constant value belongs to the existential uniqueness domain 𝐴, and the second ensures that 𝐶(𝑦) is evaluated for at least one 𝑦. (Contributed by NM, 4-Jan-2013.) (Proof shortened by Mario Carneiro, 19-Nov-2016.) |
⊢ ((∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐵 𝜑) → (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝑥 = 𝐶) ↔ ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))) | ||
Theorem | reusv3i 5296* | Two ways of expressing existential uniqueness via an indirect equality. (Contributed by NM, 23-Dec-2012.) |
⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝑧 → 𝐶 = 𝐷) ⇒ ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷)) | ||
Theorem | reusv3 5297* | Two ways to express single-valuedness of a class expression 𝐶(𝑦). See reusv1 5289 for the connection to uniqueness. (Contributed by NM, 27-Dec-2012.) |
⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝑧 → 𝐶 = 𝐷) ⇒ ⊢ (∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝐶 ∈ 𝐴) → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))) | ||
Theorem | eusv4 5298* | Two ways to express single-valuedness of a class expression 𝐵(𝑦). (Contributed by NM, 27-Oct-2010.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵) | ||
Theorem | alxfr 5299* | Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 18-Feb-2007.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((∀𝑦 𝐴 ∈ 𝐵 ∧ ∀𝑥∃𝑦 𝑥 = 𝐴) → (∀𝑥𝜑 ↔ ∀𝑦𝜓)) | ||
Theorem | ralxfrd 5300* | Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 15-Aug-2014.) (Proof shortened by Mario Carneiro, 19-Nov-2016.) (Proof shortened by JJ, 7-Aug-2021.) |
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑦 ∈ 𝐶 𝜒)) |
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