![]() |
Metamath
Proof Explorer Theorem List (p. 53 of 454) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | ![]() (1-28701) |
![]() (28702-30224) |
![]() (30225-45333) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | rabex2 5201* | 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 5202* | 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 5203* | Membership in a class abstraction involving a subset. Unlike elabg 3614, 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 5204 | 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 5205 | 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 5206 | The intersection of a nonempty class abstraction exists. (Contributed by NM, 21-Oct-2003.) |
⊢ (∃𝑥𝜑 ↔ ∩ {𝑥 ∣ 𝜑} ∈ V) | ||
Theorem | intexrab 5207 | The intersection of a nonempty restricted class abstraction exists. (Contributed by NM, 21-Oct-2003.) |
⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∩ {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V) | ||
Theorem | iinexg 5208* | 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 5209* | 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 5210* | 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 5211 | 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 5212 | 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 5213 | 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 5214 | Obsolete version of elpwi2 5213 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 | pwnss 5215 | 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 5216 | No set equals its power set. The sethood antecedent is necessary; compare pwv 4797. (Contributed by NM, 17-Nov-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.) |
⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ≠ 𝐴) | ||
Theorem | difelpw 5217 | 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 5218* | 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 5219* | 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 5220* | Equality theorem based on class2set 5219. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Raph Levien, 30-Jun-2006.) |
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} = 𝐴) | ||
Theorem | 0elpw 5221 | Every power class contains the empty set. (Contributed by NM, 25-Oct-2007.) |
⊢ ∅ ∈ 𝒫 𝐴 | ||
Theorem | pwne0 5222 | A power class is never empty. (Contributed by NM, 3-Sep-2018.) |
⊢ 𝒫 𝐴 ≠ ∅ | ||
Theorem | 0nep0 5223 | The empty set and its power set are not equal. (Contributed by NM, 23-Dec-1993.) |
⊢ ∅ ≠ {∅} | ||
Theorem | 0inp0 5224 | 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 5225 | The removal of the empty set from a class does not affect its union. (Contributed by NM, 22-Mar-2004.) |
⊢ ∪ (𝐴 ∖ {∅}) = ∪ 𝐴 | ||
Theorem | iin0 5226* | 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 5227* | In the Separation Scheme zfauscl 5169, 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 5228* | Obsolete proof of notzfaus 5227 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 5229 | The intersection of the universal class is empty. (Contributed by NM, 11-Sep-2008.) |
⊢ ∩ V = ∅ | ||
Theorem | axpweq 5230* | Two equivalent ways to express the Power Set Axiom. Note that ax-pow 5231 is not used by the proof. When ax-pow 5231 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 7467). (Contributed by NM, 22-Jun-2009.) |
⊢ (𝒫 𝐴 ∈ V ↔ ∃𝑥∀𝑦(∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴) → 𝑦 ∈ 𝑥)) | ||
Axiom | ax-pow 5231* | 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 5233 uses explicit subset notation. A version using class notation is pwex 5246. (Contributed by NM, 21-Jun-1993.) |
⊢ ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦) | ||
Theorem | zfpow 5232* | 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 5233* | A variant of the Axiom of Power Sets ax-pow 5231 using subset notation. Problem in [BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.) |
⊢ ∃𝑦∀𝑧(𝑧 ⊆ 𝑥 → 𝑧 ∈ 𝑦) | ||
Theorem | axpow3 5234* | A variant of the Axiom of Power Sets ax-pow 5231. 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 5235* | Every set is an element of some other set. See elALT 5300 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 5236* |
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 2035.
This theorem is proved directly from set theory axioms (no set theory definitions) and does not use ax-ext 2770 or ax-sep 5167. See dtruALT 5254 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 2379. (Revised by Gino Giotto, 5-Sep-2023.) |
⊢ ¬ ∀𝑥 𝑥 = 𝑦 | ||
Theorem | dtrucor 5237* | Corollary of dtru 5236. 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 5238. (Contributed by NM, 27-Jun-2002.) |
⊢ 𝑥 = 𝑦 ⇒ ⊢ 𝑥 ≠ 𝑦 | ||
Theorem | dtrucor2 5238 | The theorem form of the deduction dtrucor 5237 leads to a contradiction, as mentioned in the "Wrong!" example at mmdeduction.html#bad 5237. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by NM, 20-Oct-2007.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → 𝑥 ≠ 𝑦) ⇒ ⊢ (𝜑 ∧ ¬ 𝜑) | ||
Theorem | dvdemo1 5239* |
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 5240, 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 5240 for details on the "disjoint variable" mechanism. (The verb "bundle" to express this phenomenon was introduced by Raph Levien.) Note that dvdemo1 5239 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 2158 nor ax-13 2379. (Contributed by NM, 1-Dec-2006.) (Revised by BJ, 13-Jan-2024.) |
⊢ ∃𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝑥) | ||
Theorem | dvdemo2 5240* |
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 5239, 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 5239 for details on the "disjoint variable" mechanism. Note that dvdemo2 5240 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 2142, ax-11 2158, ax-12 2175, ax-13 2379. (Contributed by NM, 1-Dec-2006.) (Revised by BJ, 13-Jan-2024.) |
⊢ ∃𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝑥) | ||
Theorem | nfnid 5241 | 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 5236 in its proof. (Contributed by Mario Carneiro, 8-Oct-2016.) |
⊢ ¬ Ⅎ𝑥𝑥 | ||
Theorem | nfcvb 5242 | 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. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by Mario Carneiro, 8-Oct-2016.) (New usage is discouraged.) |
⊢ (Ⅎ𝑥𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦) | ||
Theorem | vpwex 5243 | 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 5244 from vpwex 5243. (Revised by BJ, 10-Aug-2022.) |
⊢ 𝒫 𝑥 ∈ V | ||
Theorem | pwexg 5244 | 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 5245 | Deduction version of the power set axiom. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝒫 𝐴 ∈ V) | ||
Theorem | pwex 5246 | Power set axiom expressed in class notation. (Contributed by NM, 21-Jun-1993.) |
⊢ 𝐴 ∈ V ⇒ ⊢ 𝒫 𝐴 ∈ V | ||
Theorem | pwel 5247 | Quantitative version of pwexg 5244: 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 5174 and ax-pr 5295 and shorten proof. (Revised by BJ, 13-Apr-2024.) |
⊢ (𝐴 ∈ 𝐵 → 𝒫 𝐴 ∈ 𝒫 𝒫 ∪ 𝐵) | ||
Theorem | abssexg 5248* | 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 5249 | Alternate proof of snex 5297 using Power Set (ax-pow 5231) instead of Pairing (ax-pr 5295). Unlike in the proof of zfpair 5287, Replacement (ax-rep 5154) 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 5250 | The power set of the empty set (the ordinal 1) is a set. See also p0exALT 5251. (Contributed by NM, 23-Dec-1993.) |
⊢ {∅} ∈ V | ||
Theorem | p0exALT 5251 | Alternate proof of p0ex 5250 which is quite different and longer if snexALT 5249 is expanded. (Contributed by NM, 23-Dec-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ {∅} ∈ V | ||
Theorem | pp0ex 5252 | 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 5253 | The ordinal number 3 is a set, proved without the Axiom of Union ax-un 7441. (Contributed by NM, 2-May-2009.) |
⊢ {∅, {∅}, {∅, {∅}}} ∈ V | ||
Theorem | dtruALT 5254* |
Alternate proof of dtru 5236 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 3555 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 5255* | This theorem shows that axiom ax-c16 36188 is redundant in the presence of theorem dtru 5236, which states simply that at least two things exist. This justifies the remark at mmzfcnd.html#twoness 5236 (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 5256 | 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 5257* | 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 5258* | 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 5259* | 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 5260* | 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 5261* | 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 5262* | 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 5263* | 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 5264* | Lemma for reusv2 5269. (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 5265* | Lemma for reusv2 5269. (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 5266* | Lemma for reusv2 5269. (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 5267* | Lemma for reusv2 5269. (Contributed by NM, 13-Dec-2012.) |
⊢ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝑥 = 𝐶) ↔ ∃!𝑥∀𝑦 ∈ 𝐵 ((𝐶 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝐶)) | ||
Theorem | reusv2lem5 5268* | Lemma for reusv2 5269. (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 5269* | 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 5270* | 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 5271* | Two ways to express single-valuedness of a class expression 𝐶(𝑦). See reusv1 5263 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 5272* | 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 5273* | 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 5274* | 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)) | ||
Theorem | rexxfrd 5275* | Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by FL, 10-Apr-2007.) (Revised by Mario Carneiro, 15-Aug-2014.) |
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 ↔ ∃𝑦 ∈ 𝐶 𝜒)) | ||
Theorem | ralxfr2d 5276* | Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by Mario Carneiro, 20-Aug-2014.) |
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐶 𝑥 = 𝐴)) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑦 ∈ 𝐶 𝜒)) | ||
Theorem | rexxfr2d 5277* | Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by Mario Carneiro, 20-Aug-2014.) (Proof shortened by Mario Carneiro, 19-Nov-2016.) |
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐶 𝑥 = 𝐴)) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 ↔ ∃𝑦 ∈ 𝐶 𝜒)) | ||
Theorem | ralxfrd2 5278* | Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Variant of ralxfrd 5274. (Contributed by Alexander van der Vekens, 25-Apr-2018.) |
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑦 ∈ 𝐶 𝜒)) | ||
Theorem | rexxfrd2 5279* | Transfer existence from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Variant of rexxfrd 5275. (Contributed by Alexander van der Vekens, 25-Apr-2018.) |
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 ↔ ∃𝑦 ∈ 𝐶 𝜒)) | ||
Theorem | ralxfr 5280* | Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.) |
⊢ (𝑦 ∈ 𝐶 → 𝐴 ∈ 𝐵) & ⊢ (𝑥 ∈ 𝐵 → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴) & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐶 𝜓) | ||
Theorem | ralxfrALT 5281* | Alternate proof of ralxfr 5280 which does not use ralxfrd 5274. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝑦 ∈ 𝐶 → 𝐴 ∈ 𝐵) & ⊢ (𝑥 ∈ 𝐵 → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴) & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐶 𝜓) | ||
Theorem | rexxfr 5282* | Transfer existence from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.) |
⊢ (𝑦 ∈ 𝐶 → 𝐴 ∈ 𝐵) & ⊢ (𝑥 ∈ 𝐵 → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴) & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐶 𝜓) | ||
Theorem | rabxfrd 5283* | Membership in a restricted class abstraction after substituting an expression 𝐴 (containing 𝑦) for 𝑥 in the formula defining the class abstraction. (Contributed by NM, 16-Jan-2012.) |
⊢ Ⅎ𝑦𝐵 & ⊢ Ⅎ𝑦𝐶 & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → 𝐴 ∈ 𝐷) & ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒)) & ⊢ (𝑦 = 𝐵 → 𝐴 = 𝐶) ⇒ ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐷) → (𝐶 ∈ {𝑥 ∈ 𝐷 ∣ 𝜓} ↔ 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝜒})) | ||
Theorem | rabxfr 5284* | Membership in a restricted class abstraction after substituting an expression 𝐴 (containing 𝑦) for 𝑥 in the the formula defining the class abstraction. (Contributed by NM, 10-Jun-2005.) |
⊢ Ⅎ𝑦𝐵 & ⊢ Ⅎ𝑦𝐶 & ⊢ (𝑦 ∈ 𝐷 → 𝐴 ∈ 𝐷) & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → 𝐴 = 𝐶) ⇒ ⊢ (𝐵 ∈ 𝐷 → (𝐶 ∈ {𝑥 ∈ 𝐷 ∣ 𝜑} ↔ 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝜓})) | ||
Theorem | reuhypd 5285* | A theorem useful for eliminating the restricted existential uniqueness hypotheses in riotaxfrd 7127. (Contributed by NM, 16-Jan-2012.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐵 ∈ 𝐶) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (𝑥 = 𝐴 ↔ 𝑦 = 𝐵)) ⇒ ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴) | ||
Theorem | reuhyp 5286* | A theorem useful for eliminating the restricted existential uniqueness hypotheses in reuxfr1 3691. (Contributed by NM, 15-Nov-2004.) |
⊢ (𝑥 ∈ 𝐶 → 𝐵 ∈ 𝐶) & ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (𝑥 = 𝐴 ↔ 𝑦 = 𝐵)) ⇒ ⊢ (𝑥 ∈ 𝐶 → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴) | ||
Theorem | zfpair 5287 |
The Axiom of Pairing of Zermelo-Fraenkel set theory. Axiom 2 of
[TakeutiZaring] p. 15. In some
textbooks this is stated as a separate
axiom; here we show it is redundant since it can be derived from the
other axioms.
This theorem should not be referenced by any proof other than axprALT 5288. Instead, use zfpair2 5296 below so that the uses of the Axiom of Pairing can be more easily identified. (Contributed by NM, 18-Oct-1995.) (New usage is discouraged.) |
⊢ {𝑥, 𝑦} ∈ V | ||
Theorem | axprALT 5288* | Alternate proof of axpr 5294. (Contributed by NM, 14-Nov-2006.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) | ||
Theorem | axprlem1 5289* | Lemma for axpr 5294. There exists a set to which all empty sets belong. (Contributed by Rohan Ridenour, 10-Aug-2023.) (Revised by BJ, 13-Aug-2023.) |
⊢ ∃𝑥∀𝑦(∀𝑧 ¬ 𝑧 ∈ 𝑦 → 𝑦 ∈ 𝑥) | ||
Theorem | axprlem2 5290* | Lemma for axpr 5294. There exists a set to which all sets whose only members are empty sets belong. (Contributed by Rohan Ridenour, 9-Aug-2023.) (Revised by BJ, 13-Aug-2023.) |
⊢ ∃𝑥∀𝑦(∀𝑧 ∈ 𝑦 ∀𝑤 ¬ 𝑤 ∈ 𝑧 → 𝑦 ∈ 𝑥) | ||
Theorem | axprlem3 5291* | Lemma for axpr 5294. Eliminate the antecedent of the relevant replacement instance. (Contributed by Rohan Ridenour, 10-Aug-2023.) |
⊢ ∃𝑧∀𝑤(𝑤 ∈ 𝑧 ↔ ∃𝑠(𝑠 ∈ 𝑝 ∧ if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦))) | ||
Theorem | axprlem4 5292* | Lemma for axpr 5294. The first element of the pair is included in any superset of the set whose existence is asserted by the axiom of replacement. (Contributed by Rohan Ridenour, 10-Aug-2023.) (Revised by BJ, 13-Aug-2023.) |
⊢ ((∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑥) → ∃𝑠(𝑠 ∈ 𝑝 ∧ if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦))) | ||
Theorem | axprlem5 5293* | Lemma for axpr 5294. The second element of the pair is included in any superset of the set whose existence is asserted by the axiom of replacement. (Contributed by Rohan Ridenour, 10-Aug-2023.) (Revised by BJ, 13-Aug-2023.) |
⊢ ((∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑦) → ∃𝑠(𝑠 ∈ 𝑝 ∧ if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦))) | ||
Theorem | axpr 5294* |
Unabbreviated version of the Axiom of Pairing of ZF set theory, derived
as a theorem from the other axioms.
This theorem should not be referenced by any proof. Instead, use ax-pr 5295 below so that the uses of the Axiom of Pairing can be more easily identified. For a shorter proof using ax-ext 2770, see axprALT 5288. (Contributed by NM, 14-Nov-2006.) Remove dependency on ax-ext 2770. (Revised by Rohan Ridenour, 10-Aug-2023.) (Proof shortened by BJ, 13-Aug-2023.) Use ax-pr 5295 instead. (New usage is discouraged.) |
⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) | ||
Axiom | ax-pr 5295* | The Axiom of Pairing of ZF set theory. It was derived as theorem axpr 5294 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, 14-Nov-2006.) |
⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) | ||
Theorem | zfpair2 5296 | Derive the abbreviated version of the Axiom of Pairing from ax-pr 5295. See zfpair 5287 for its derivation from the other axioms. (Contributed by NM, 14-Nov-2006.) |
⊢ {𝑥, 𝑦} ∈ V | ||
Theorem | snex 5297 | A singleton is a set. Theorem 7.12 of [Quine] p. 51, proved using Extensionality, Separation, Null Set, and Pairing. See also snexALT 5249. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 19-May-2013.) (Proof modification is discouraged.) |
⊢ {𝐴} ∈ V | ||
Theorem | prex 5298 | The Axiom of Pairing using class variables. Theorem 7.13 of [Quine] p. 51. By virtue of its definition, an unordered pair remains a set (even though no longer a pair) even when its components are proper classes (see prprc 4663), so we can dispense with hypotheses requiring them to be sets. (Contributed by NM, 15-Jul-1993.) |
⊢ {𝐴, 𝐵} ∈ V | ||
Theorem | sels 5299* | If a class is a set, then it is a member of a set. (Contributed by BJ, 3-Apr-2019.) |
⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝐴 ∈ 𝑥) | ||
Theorem | elALT 5300* | Alternate proof of el 5235, shorter but requiring more axioms. (Contributed by NM, 4-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ∃𝑦 𝑥 ∈ 𝑦 |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |