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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | zfauscl 5301* |
Separation Scheme (Aussonderung) using a class variable. To derive this
from ax-sep 5299, we invoke the Axiom of Extensionality
(indirectly via
vtocl 3545), 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 5361 shows. (Contributed by NM, 21-Jun-1993.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) | ||
Theorem | bm1.3ii 5302* | Convert implication to equivalence using the Separation Scheme (Aussonderung) ax-sep 5299. 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 5303* | Derive ax6v 1971 (a weakened version of ax-6 1970 where 𝑥 and 𝑦 must be distinct), from Separation ax-sep 5299 and Extensionality ax-ext 2702. See ax6 2382 for the derivation of ax-6 1970 from ax6v 1971. (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 1796, ax-4 1810, sp 2175, and ax-rep 5285. To check this, replace sp 2175 with the obsolete axiom ax-c5 38057 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 5299. 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 2707).
This proof, suggested by Jeff Hoffman, uses only ax-4 1810 and ax-gen 1796 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 5299 implies the existence of at least one set. Note that Kunen's version of ax-sep 5299 (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 5285. 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 3978 illustrating the use of the weak deduction theorem to prove it from the inference sselii 3979. (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 4930 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 2170 and ax-13 2370. (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 5299 (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 | prcssprc 5325 | The superclass of a proper class is a proper class. (Contributed by AV, 27-Dec-2020.) |
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ∉ V) → 𝐵 ∉ V) | ||
Theorem | sselpwd 5326 | Elementhood to a power set. (Contributed by Thierry Arnoux, 18-May-2020.) |
⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐵) | ||
Theorem | difexg 5327 | Existence of a difference. (Contributed by NM, 26-May-1998.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ 𝐵) ∈ V) | ||
Theorem | difexi 5328 | Existence of a difference, inference version of difexg 5327. (Contributed by Glauco Siliprandi, 3-Mar-2021.) (Revised by AV, 26-Mar-2021.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∖ 𝐵) ∈ V | ||
Theorem | difexd 5329 | Existence of a difference. (Contributed by SN, 16-Jul-2024.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐴 ∖ 𝐵) ∈ V) | ||
Theorem | zfausab 5330* | Separation Scheme (Aussonderung) in terms of a class abstraction. (Contributed by NM, 8-Jun-1994.) |
⊢ 𝐴 ∈ V ⇒ ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V | ||
Theorem | rabexg 5331* | Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 23-Oct-1999.) |
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V) | ||
Theorem | rabex 5332* | Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 19-Jul-1996.) |
⊢ 𝐴 ∈ V ⇒ ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V | ||
Theorem | rabexd 5333* | Separation Scheme in terms of a restricted class abstraction, deduction form of rabex2 5334. (Contributed by AV, 16-Jul-2019.) |
⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜓} & ⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝐵 ∈ V) | ||
Theorem | rabex2 5334* | 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 5335* | 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 5336* | Membership in a class abstraction involving a subset. Unlike elabg 3666, 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 5337 | 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 5338 | 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 5339 | The intersection of a nonempty class abstraction exists. (Contributed by NM, 21-Oct-2003.) |
⊢ (∃𝑥𝜑 ↔ ∩ {𝑥 ∣ 𝜑} ∈ V) | ||
Theorem | intexrab 5340 | The intersection of a nonempty restricted class abstraction exists. (Contributed by NM, 21-Oct-2003.) |
⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∩ {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V) | ||
Theorem | iinexg 5341* | 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 5342* | 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 5343* | 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 5344 | 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 5345 | 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 5346 | 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 | elpwi2OLD 5347 | Obsolete version of elpwi2 5346 as of 26-May-2024. (Contributed by Glauco Siliprandi, 3-Mar-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝐵 ∈ 𝑉 & ⊢ 𝐴 ⊆ 𝐵 ⇒ ⊢ 𝐴 ∈ 𝒫 𝐵 | ||
Theorem | axpweq 5348* | Two equivalent ways to express the Power Set Axiom. Note that ax-pow 5363 is not used by the proof. When ax-pow 5363 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 7756). (Contributed by NM, 22-Jun-2009.) |
⊢ (𝒫 𝐴 ∈ V ↔ ∃𝑥∀𝑦(∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴) → 𝑦 ∈ 𝑥)) | ||
Theorem | pwnss 5349 | 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 5350 | No set equals its power set. The sethood antecedent is necessary; compare pwv 4905. (Contributed by NM, 17-Nov-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.) |
⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ≠ 𝐴) | ||
Theorem | difelpw 5351 | 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 5352* | 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 5353* | 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 3700) and to the empty set when 𝐴 is a proper class. (Contributed by NM, 16-Oct-2003.) |
⊢ {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} ∈ V | ||
Theorem | 0elpw 5354 | Every power class contains the empty set. (Contributed by NM, 25-Oct-2007.) |
⊢ ∅ ∈ 𝒫 𝐴 | ||
Theorem | pwne0 5355 | A power class is never empty. (Contributed by NM, 3-Sep-2018.) |
⊢ 𝒫 𝐴 ≠ ∅ | ||
Theorem | 0nep0 5356 | The empty set and its power set are not equal. (Contributed by NM, 23-Dec-1993.) |
⊢ ∅ ≠ {∅} | ||
Theorem | 0inp0 5357 | 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 5358 | The removal of the empty set from a class does not affect its union. (Contributed by NM, 22-Mar-2004.) |
⊢ ∪ (𝐴 ∖ {∅}) = ∪ 𝐴 | ||
Theorem | eqsnuniex 5359 | 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 5360* | 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 5361* | In the Separation Scheme zfauscl 5301, 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 5362 | The intersection of the universal class is empty. (Contributed by NM, 11-Sep-2008.) |
⊢ ∩ V = ∅ | ||
Axiom | ax-pow 5363* | 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 5365 uses explicit subset notation. A version using class notation is pwex 5378. (Contributed by NM, 21-Jun-1993.) |
⊢ ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦) | ||
Theorem | zfpow 5364* | 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 5365* | A variant of the Axiom of Power Sets ax-pow 5363 using subset notation. Problem in [BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.) |
⊢ ∃𝑦∀𝑧(𝑧 ⊆ 𝑥 → 𝑧 ∈ 𝑦) | ||
Theorem | axpow3 5366* | A variant of the Axiom of Power Sets ax-pow 5363. 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 5367* | Alternate proof of el 5437 using ax-9 2115 and ax-pow 5363 instead of ax-pr 5427. (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 5368* | Alternate proof of dtru 5436 using ax-pow 5363 instead of ax-pr 5427. (Contributed by NM, 7-Nov-2006.) Avoid ax-13 2370. (Revised by BJ, 31-May-2019.) Avoid ax-12 2170. (Revised by Rohan Ridenour, 9-Oct-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ¬ ∀𝑥 𝑥 = 𝑦 | ||
Theorem | dtrucor 5369* | Corollary of dtru 5436. 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 5370. (Contributed by NM, 27-Jun-2002.) |
⊢ 𝑥 = 𝑦 ⇒ ⊢ 𝑥 ≠ 𝑦 | ||
Theorem | dtrucor2 5370 | The theorem form of the deduction dtrucor 5369 leads to a contradiction, as mentioned in the "Wrong!" example at mmdeduction.html#bad 5369. Usage of this theorem is discouraged because it depends on ax-13 2370. (Contributed by NM, 20-Oct-2007.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → 𝑥 ≠ 𝑦) ⇒ ⊢ (𝜑 ∧ ¬ 𝜑) | ||
Theorem | dvdemo1 5371* |
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 5372, 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 5372 for details on the "disjoint variable" mechanism. (The verb "bundle" to express this phenomenon was introduced by Raph Levien.) Note that dvdemo1 5371 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 2153 nor ax-13 2370. (Contributed by NM, 1-Dec-2006.) (Revised by BJ, 13-Jan-2024.) |
⊢ ∃𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝑥) | ||
Theorem | dvdemo2 5372* |
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 5371, 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 5371 for details on the "disjoint variable" mechanism. Note that dvdemo2 5372 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 2153, ax-12 2170, ax-13 2370. (Contributed by NM, 1-Dec-2006.) Avoid ax-13 2370. (Revised by BJ, 13-Jan-2024.) |
⊢ ∃𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝑥) | ||
Theorem | nfnid 5373 | 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 5368 in its proof. (Contributed by Mario Carneiro, 8-Oct-2016.) |
⊢ ¬ Ⅎ𝑥𝑥 | ||
Theorem | nfcvb 5374 | 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 2370. (New usage is discouraged.) |
⊢ (Ⅎ𝑥𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦) | ||
Theorem | vpwex 5375 | 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 5376 from vpwex 5375. (Revised by BJ, 10-Aug-2022.) |
⊢ 𝒫 𝑥 ∈ V | ||
Theorem | pwexg 5376 | 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 5377 | Deduction version of the power set axiom. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝒫 𝐴 ∈ V) | ||
Theorem | pwex 5378 | Power set axiom expressed in class notation. (Contributed by NM, 21-Jun-1993.) |
⊢ 𝐴 ∈ V ⇒ ⊢ 𝒫 𝐴 ∈ V | ||
Theorem | pwel 5379 | Quantitative version of pwexg 5376: 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 5427 and shorten proof. (Revised by BJ, 13-Apr-2024.) |
⊢ (𝐴 ∈ 𝐵 → 𝒫 𝐴 ∈ 𝒫 𝒫 ∪ 𝐵) | ||
Theorem | abssexg 5380* | 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 5381 | Alternate proof of snex 5431 using Power Set (ax-pow 5363) instead of Pairing (ax-pr 5427). Unlike in the proof of zfpair 5419, Replacement (ax-rep 5285) 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 5382 | The power set of the empty set (the ordinal 1) is a set. See also p0exALT 5383. (Contributed by NM, 23-Dec-1993.) |
⊢ {∅} ∈ V | ||
Theorem | p0exALT 5383 | Alternate proof of p0ex 5382 which is quite different and longer if snexALT 5381 is expanded. (Contributed by NM, 23-Dec-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ {∅} ∈ V | ||
Theorem | pp0ex 5384 | 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 5385 | The ordinal number 3 is a set, proved without the Axiom of Union ax-un 7729. (Contributed by NM, 2-May-2009.) |
⊢ {∅, {∅}, {∅, {∅}}} ∈ V | ||
Theorem | dtruALT 5386* |
Alternate proof of dtru 5436 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 3596 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 5387* | This theorem shows that Axiom ax-c16 38066 is redundant in the presence of Theorem dtruALT2 5368, which states simply that at least two things exist. This justifies the remark at mmzfcnd.html#twoness 5368 (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 5388 | 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 5389* | 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 5390* | 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 5391* | 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 5392* | 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 5393* | 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 5394* | 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 5395* | 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 5396* | Lemma for reusv2 5401. (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 5397* | Lemma for reusv2 5401. (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 5398* | Lemma for reusv2 5401. (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 5399* | Lemma for reusv2 5401. (Contributed by NM, 13-Dec-2012.) |
⊢ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝑥 = 𝐶) ↔ ∃!𝑥∀𝑦 ∈ 𝐵 ((𝐶 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝐶)) | ||
Theorem | reusv2lem5 5400* | Lemma for reusv2 5401. (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)) |
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