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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | sepexi 5301* | Convert implication to equivalence within an existence statement using the Separation Scheme (Aussonderung) ax-sep 5296. Inference associated with sepex 5300. (Contributed by NM, 21-Jun-1993.) Generalize conclusion, extract closed form, and avoid ax-9 2118. (Revised by Matthew House, 19-Sep-2025.) |
| ⊢ ∃𝑦∀𝑥(𝜑 → 𝑥 ∈ 𝑦) ⇒ ⊢ ∃𝑧∀𝑥(𝑥 ∈ 𝑧 ↔ 𝜑) | ||
| Theorem | bm1.3iiOLD 5302* | Obsolete version of sepexi 5301 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 5303* | Derive ax6v 1968 (a weakened version of ax-6 1967 where 𝑥 and 𝑦 must be distinct), from Separation ax-sep 5296 and Extensionality ax-ext 2708. See ax6 2389 for the derivation of ax-6 1967 from ax6v 1968. (Contributed by NM, 12-Nov-2013.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦 | ||
| Theorem | axnulALT 5304* | Alternate proof of axnul 5305, proved from propositional calculus, ax-gen 1795, ax-4 1809, sp 2183, and ax-rep 5279. To check this, replace sp 2183 with the obsolete axiom ax-c5 38884 in the proof of axnulALT 5304 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 5305* |
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 5296. 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 2713).
This proof, suggested by Jeff Hoffman, uses only ax-4 1809 and ax-gen 1795 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 5296 implies the existence of at least one set. Note that Kunen's version of ax-sep 5296 (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 5304 for a proof directly from ax-rep 5279. This theorem should not be referenced by any proof. Instead, use ax-nul 5306 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 5306* | The Null Set Axiom of ZF set theory. It was derived as axnul 5305 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 5307 | 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 5306. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
| ⊢ ∅ ∈ V | ||
| Theorem | al0ssb 5308* | 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 5309 | Alternate proof of sseli 3979 illustrating the use of the weak deduction theorem to prove it from the inference sselii 3980. (Contributed by NM, 24-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ 𝐴 ⊆ 𝐵 ⇒ ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵) | ||
| Theorem | csbexg 5310 | 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 5311 | 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 5312 | A version of unisn 4926 without the 𝐴 ∈ V hypothesis. (Contributed by Stefan Allan, 14-Mar-2006.) |
| ⊢ ∪ {𝐴} ∈ {∅, 𝐴} | ||
| Theorem | nalset 5313* | No set contains all sets. Theorem 41 of [Suppes] p. 30. (Contributed by NM, 23-Aug-1993.) Remove use of ax-12 2177 and ax-13 2377. (Revised by BJ, 31-May-2019.) |
| ⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥 | ||
| Theorem | vnex 5314 | The universal class does not exist as a set. (Contributed by NM, 4-Jul-2005.) |
| ⊢ ¬ ∃𝑥 𝑥 = V | ||
| Theorem | vprc 5315 | 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 5316 | The universal class does not belong to any class. (Contributed by FL, 31-Dec-2006.) |
| ⊢ ¬ V ∈ 𝐴 | ||
| Theorem | inex1 5317 | 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 5318 | Separation Scheme (Aussonderung) using class notation. (Contributed by NM, 27-Apr-1994.) |
| ⊢ 𝐴 ∈ V ⇒ ⊢ (𝐵 ∩ 𝐴) ∈ V | ||
| Theorem | inex1g 5319 | Closed-form, generalized Separation Scheme. (Contributed by NM, 7-Apr-1995.) |
| ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V) | ||
| Theorem | inex2g 5320 | Sufficient condition for an intersection to be a set. Commuted form of inex1g 5319. (Contributed by Peter Mazsa, 19-Dec-2018.) |
| ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∩ 𝐴) ∈ V) | ||
| Theorem | ssex 5321 | 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 5296 (a.k.a. Subset Axiom). (Contributed by NM, 27-Apr-1994.) |
| ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V) | ||
| Theorem | ssexi 5322 | The subset of a set is also a set. (Contributed by NM, 9-Sep-1993.) |
| ⊢ 𝐵 ∈ V & ⊢ 𝐴 ⊆ 𝐵 ⇒ ⊢ 𝐴 ∈ V | ||
| Theorem | ssexg 5323 | 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 5324 | A subclass of a set is a set. Deduction form of ssexg 5323. (Contributed by David Moews, 1-May-2017.) |
| ⊢ (𝜑 → 𝐵 ∈ 𝐶) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ∈ V) | ||
| Theorem | abexd 5325* | 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 5326* | Conditions for a class abstraction to be a set. Remark: This proof is shorter than a proof using abexd 5325. (Contributed by AV, 19-Apr-2025.) |
| ⊢ (𝜑 → 𝑥 ∈ 𝐴) & ⊢ 𝐴 ∈ V ⇒ ⊢ {𝑥 ∣ 𝜑} ∈ V | ||
| Theorem | prcssprc 5327 | The superclass of a proper class is a proper class. (Contributed by AV, 27-Dec-2020.) |
| ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ∉ V) → 𝐵 ∉ V) | ||
| Theorem | sselpwd 5328 | Elementhood to a power set. (Contributed by Thierry Arnoux, 18-May-2020.) |
| ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐵) | ||
| Theorem | difexg 5329 | Existence of a difference. (Contributed by NM, 26-May-1998.) |
| ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ 𝐵) ∈ V) | ||
| Theorem | difexi 5330 | Existence of a difference, inference version of difexg 5329. (Contributed by Glauco Siliprandi, 3-Mar-2021.) (Revised by AV, 26-Mar-2021.) |
| ⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∖ 𝐵) ∈ V | ||
| Theorem | difexd 5331 | Existence of a difference. (Contributed by SN, 16-Jul-2024.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐴 ∖ 𝐵) ∈ V) | ||
| Theorem | zfausab 5332* | Separation Scheme (Aussonderung) in terms of a class abstraction. (Contributed by NM, 8-Jun-1994.) |
| ⊢ 𝐴 ∈ V ⇒ ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V | ||
| Theorem | elpw2g 5333 | 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 5334 | 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 5335 | 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 5336* | 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 5337* | Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 23-Oct-1999.) (Proof shortened by BJ, 24-Jul-2025.) |
| ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V) | ||
| Theorem | rabexgOLD 5338* | Obsolete version of rabexg 5337 as of 24-Jul-2025). (Contributed by NM, 23-Oct-1999.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V) | ||
| Theorem | rabex 5339* | Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 19-Jul-1996.) |
| ⊢ 𝐴 ∈ V ⇒ ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V | ||
| Theorem | rabexd 5340* | Separation Scheme in terms of a restricted class abstraction, deduction form of rabex2 5341. (Contributed by AV, 16-Jul-2019.) |
| ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜓} & ⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝐵 ∈ V) | ||
| Theorem | rabex2 5341* | 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 5342* | 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 5343* | Membership in a class abstraction involving a subset. Unlike elabg 3676, 𝐴 does not have to be a set. (Contributed by NM, 29-Aug-2006.) |
| ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ (𝑥 ⊆ 𝐵 ∧ 𝜑)} ↔ (𝐴 ⊆ 𝐵 ∧ 𝜓))) | ||
| Theorem | intex 5344 | 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 5345 | 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 5346 | The intersection of a nonempty class abstraction exists. (Contributed by NM, 21-Oct-2003.) |
| ⊢ (∃𝑥𝜑 ↔ ∩ {𝑥 ∣ 𝜑} ∈ V) | ||
| Theorem | intexrab 5347 | The intersection of a nonempty restricted class abstraction exists. (Contributed by NM, 21-Oct-2003.) |
| ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∩ {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V) | ||
| Theorem | iinexg 5348* | 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 5349* | 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 5350* | The intersection of a union ∪ 𝐴 with a class 𝐵 is equal to the union of the intersections of each element of 𝐴 with 𝐵. (Contributed by FL, 24-Mar-2007.) (Proof shortened by Wolf Lammen, 15-May-2025.) |
| ⊢ (∪ 𝐴 ∩ 𝐵) = ∪ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = (𝑦 ∩ 𝐵)} | ||
| Theorem | axpweq 5351* | Two equivalent ways to express the Power Set Axiom. Note that ax-pow 5365 is not used by the proof. When ax-pow 5365 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 7785). (Contributed by NM, 22-Jun-2009.) |
| ⊢ (𝒫 𝐴 ∈ V ↔ ∃𝑥∀𝑦(∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴) → 𝑦 ∈ 𝑥)) | ||
| Theorem | pwnss 5352 | The power set of a set is never a subset. (Contributed by Stefan O'Rear, 22-Feb-2015.) (Proof shortened by BJ, 24-Jul-2025.) |
| ⊢ (𝐴 ∈ 𝑉 → ¬ 𝒫 𝐴 ⊆ 𝐴) | ||
| Theorem | pwne 5353 | No set equals its power set. The sethood antecedent is necessary; compare pwv 4904. (Contributed by NM, 17-Nov-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.) |
| ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ≠ 𝐴) | ||
| Theorem | difelpw 5354 | 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 | class2set 5355* | The class of elements of 𝐴 "such that 𝐴 is a set" is a set. That class is equal to 𝐴 when 𝐴 is a set (see class2seteq 3710) and to the empty set when 𝐴 is a proper class. (Contributed by NM, 16-Oct-2003.) |
| ⊢ {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} ∈ V | ||
| Theorem | 0elpw 5356 | Every power class contains the empty set. (Contributed by NM, 25-Oct-2007.) |
| ⊢ ∅ ∈ 𝒫 𝐴 | ||
| Theorem | pwne0 5357 | A power class is never empty. (Contributed by NM, 3-Sep-2018.) |
| ⊢ 𝒫 𝐴 ≠ ∅ | ||
| Theorem | 0nep0 5358 | The empty set and its power set are not equal. (Contributed by NM, 23-Dec-1993.) |
| ⊢ ∅ ≠ {∅} | ||
| Theorem | 0inp0 5359 | 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 5360 | The removal of the empty set from a class does not affect its union. (Contributed by NM, 22-Mar-2004.) |
| ⊢ ∪ (𝐴 ∖ {∅}) = ∪ 𝐴 | ||
| Theorem | eqsnuniex 5361 | If a class is equal to the singleton of its union, then its union exists. (Contributed by BTernaryTau, 24-Sep-2024.) |
| ⊢ (𝐴 = {∪ 𝐴} → ∪ 𝐴 ∈ V) | ||
| Theorem | iin0 5362* | 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 5363* | In the Separation Scheme zfauscl 5298, we require that 𝑦 not occur in 𝜑 (which can be generalized to "not be free in"). Here we show special cases of 𝐴 and 𝜑 that result in a contradiction if that requirement is not met. (Contributed by NM, 8-Feb-2006.) (Proof shortened by BJ, 18-Nov-2023.) |
| ⊢ 𝐴 = {∅} & ⊢ (𝜑 ↔ ¬ 𝑥 ∈ 𝑦) ⇒ ⊢ ¬ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) | ||
| Theorem | intv 5364 | The intersection of the universal class is empty. (Contributed by NM, 11-Sep-2008.) |
| ⊢ ∩ V = ∅ | ||
| Axiom | ax-pow 5365* | 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 5367 uses explicit subset notation. A version using class notation is pwex 5380. (Contributed by NM, 21-Jun-1993.) |
| ⊢ ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦) | ||
| Theorem | zfpow 5366* | 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 5367* | A variant of the Axiom of Power Sets ax-pow 5365 using subset notation. Problem in [BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.) |
| ⊢ ∃𝑦∀𝑧(𝑧 ⊆ 𝑥 → 𝑧 ∈ 𝑦) | ||
| Theorem | axpow3 5368* | A variant of the Axiom of Power Sets ax-pow 5365. For any set 𝑥, there exists a set 𝑦 whose members are exactly the subsets of 𝑥 i.e. the power set of 𝑥. Axiom Pow of [BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.) |
| ⊢ ∃𝑦∀𝑧(𝑧 ⊆ 𝑥 ↔ 𝑧 ∈ 𝑦) | ||
| Theorem | elALT2 5369* | Alternate proof of el 5442 using ax-9 2118 and ax-pow 5365 instead of ax-pr 5432. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ ∃𝑦 𝑥 ∈ 𝑦 | ||
| Theorem | dtruALT2 5370* | Alternate proof of dtru 5441 using ax-pow 5365 instead of ax-pr 5432. See dtruALT 5388 for another proof using ax-pow 5365 instead of ax-pr 5432. (Contributed by NM, 7-Nov-2006.) Avoid ax-13 2377. (Revised by BJ, 31-May-2019.) Avoid ax-12 2177. (Revised by Rohan Ridenour, 9-Oct-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ ¬ ∀𝑥 𝑥 = 𝑦 | ||
| Theorem | dtrucor 5371* | Corollary of dtru 5441. 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 5372. (Contributed by NM, 27-Jun-2002.) |
| ⊢ 𝑥 = 𝑦 ⇒ ⊢ 𝑥 ≠ 𝑦 | ||
| Theorem | dtrucor2 5372 | The theorem form of the deduction dtrucor 5371 leads to a contradiction, as mentioned in the "Wrong!" example at mmdeduction.html#bad 5371. Usage of this theorem is discouraged because it depends on ax-13 2377. (Contributed by NM, 20-Oct-2007.) (New usage is discouraged.) |
| ⊢ (𝑥 = 𝑦 → 𝑥 ≠ 𝑦) ⇒ ⊢ (𝜑 ∧ ¬ 𝜑) | ||
| Theorem | dvdemo1 5373* |
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 5374, 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 5374 for details on the "disjoint variable" mechanism. (The verb "bundle" to express this phenomenon was introduced by Raph Levien.) Note that dvdemo1 5373 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 2157 nor ax-13 2377. (Contributed by NM, 1-Dec-2006.) (Revised by BJ, 13-Jan-2024.) |
| ⊢ ∃𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝑥) | ||
| Theorem | dvdemo2 5374* |
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 5373, 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 5373 for details on the "disjoint variable" mechanism. Note that dvdemo2 5374 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 2141, ax-11 2157, ax-12 2177, ax-13 2377. (Contributed by NM, 1-Dec-2006.) Avoid ax-13 2377. (Revised by BJ, 13-Jan-2024.) |
| ⊢ ∃𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝑥) | ||
| Theorem | nfnid 5375 | A setvar variable is not free from itself. This theorem is not true in a one-element domain, as illustrated by the use of dtruALT2 5370 in its proof. (Contributed by Mario Carneiro, 8-Oct-2016.) |
| ⊢ ¬ Ⅎ𝑥𝑥 | ||
| Theorem | nfcvb 5376 | The "distinctor" expression ¬ ∀𝑥𝑥 = 𝑦, stating that 𝑥 and 𝑦 are not the same variable, can be written in terms of Ⅎ in the obvious way. This theorem is not true in a one-element domain, because then Ⅎ𝑥𝑦 and ∀𝑥𝑥 = 𝑦 will both be true. (Contributed by Mario Carneiro, 8-Oct-2016.) Usage of this theorem is discouraged because it depends on ax-13 2377. (New usage is discouraged.) |
| ⊢ (Ⅎ𝑥𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦) | ||
| Theorem | vpwex 5377 | 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 5378 from vpwex 5377. (Revised by BJ, 10-Aug-2022.) |
| ⊢ 𝒫 𝑥 ∈ V | ||
| Theorem | pwexg 5378 | 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 5379 | Deduction version of the power set axiom. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝒫 𝐴 ∈ V) | ||
| Theorem | pwex 5380 | Power set axiom expressed in class notation. (Contributed by NM, 21-Jun-1993.) |
| ⊢ 𝐴 ∈ V ⇒ ⊢ 𝒫 𝐴 ∈ V | ||
| Theorem | pwel 5381 | Quantitative version of pwexg 5378: the powerset of an element of a class is an element of the double powerclass of the union of that class. Exercise 10 of [Enderton] p. 26. (Contributed by NM, 13-Jan-2007.) Remove use of ax-nul 5306 and ax-pr 5432 and shorten proof. (Revised by BJ, 13-Apr-2024.) |
| ⊢ (𝐴 ∈ 𝐵 → 𝒫 𝐴 ∈ 𝒫 𝒫 ∪ 𝐵) | ||
| Theorem | abssexg 5382* | 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 5383 | Alternate proof of snex 5436 using Power Set (ax-pow 5365) instead of Pairing (ax-pr 5432). Unlike in the proof of zfpair 5421, Replacement (ax-rep 5279) 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 5384 | The power set of the empty set (the ordinal 1) is a set. See also p0exALT 5385. (Contributed by NM, 23-Dec-1993.) |
| ⊢ {∅} ∈ V | ||
| Theorem | p0exALT 5385 | Alternate proof of p0ex 5384 which is quite different and longer if snexALT 5383 is expanded. (Contributed by NM, 23-Dec-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ {∅} ∈ V | ||
| Theorem | pp0ex 5386 | 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 5387 | The ordinal number 3 is a set, proved without the Axiom of Union ax-un 7755. (Contributed by NM, 2-May-2009.) |
| ⊢ {∅, {∅}, {∅, {∅}}} ∈ V | ||
| Theorem | dtruALT 5388* |
Alternate proof of dtru 5441 which requires more axioms but is shorter and
may be easier to understand. Like dtruALT2 5370, it uses ax-pow 5365 rather
than ax-pr 5432.
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 5389* | This theorem shows that Axiom ax-c16 38893 is redundant in the presence of Theorem dtruALT2 5370, which states simply that at least two things exist. This justifies the remark at mmzfcnd.html#twoness 5370 (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 5390 | 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 5391* | 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 5392* | 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 5393* | 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 5394* | 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 5395* | 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 5396* | 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 5397* | 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 5398* | Lemma for reusv2 5403. (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 5399* | Lemma for reusv2 5403. (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 5400* | Lemma for reusv2 5403. (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)) | ||
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