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Theorem List for Metamath Proof Explorer - 5201-5300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremssex 5201 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 5179 (a.k.a. Subset Axiom). (Contributed by NM, 27-Apr-1994.)
𝐵 ∈ V       (𝐴𝐵𝐴 ∈ V)

Theoremssexi 5202 The subset of a set is also a set. (Contributed by NM, 9-Sep-1993.)
𝐵 ∈ V    &   𝐴𝐵       𝐴 ∈ V

Theoremssexg 5203 The subset of a set is also a set. Exercise 3 of [TakeutiZaring] p. 22 (generalized). (Contributed by NM, 14-Aug-1994.)
((𝐴𝐵𝐵𝐶) → 𝐴 ∈ V)

Theoremssexd 5204 A subclass of a set is a set. Deduction form of ssexg 5203. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐵𝐶)    &   (𝜑𝐴𝐵)       (𝜑𝐴 ∈ V)

Theoremprcssprc 5205 The superclass of a proper class is a proper class. (Contributed by AV, 27-Dec-2020.)
((𝐴𝐵𝐴 ∉ V) → 𝐵 ∉ V)

Theoremsselpwd 5206 Elementhood to a power set. (Contributed by Thierry Arnoux, 18-May-2020.)
(𝜑𝐵𝑉)    &   (𝜑𝐴𝐵)       (𝜑𝐴 ∈ 𝒫 𝐵)

Theoremdifexg 5207 Existence of a difference. (Contributed by NM, 26-May-1998.)
(𝐴𝑉 → (𝐴𝐵) ∈ V)

Theoremdifexi 5208 Existence of a difference, inference version of difexg 5207. (Contributed by Glauco Siliprandi, 3-Mar-2021.) (Revised by AV, 26-Mar-2021.)
𝐴 ∈ V       (𝐴𝐵) ∈ V

Theoremzfausab 5209* Separation Scheme (Aussonderung) in terms of a class abstraction. (Contributed by NM, 8-Jun-1994.)
𝐴 ∈ V       {𝑥 ∣ (𝑥𝐴𝜑)} ∈ V

Theoremrabexg 5210* Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 23-Oct-1999.)
(𝐴𝑉 → {𝑥𝐴𝜑} ∈ V)

Theoremrabex 5211* Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 19-Jul-1996.)
𝐴 ∈ V       {𝑥𝐴𝜑} ∈ V

Theoremrabexd 5212* Separation Scheme in terms of a restricted class abstraction, deduction form of rabex2 5213. (Contributed by AV, 16-Jul-2019.)
𝐵 = {𝑥𝐴𝜓}    &   (𝜑𝐴𝑉)       (𝜑𝐵 ∈ V)

Theoremrabex2 5213* Separation Scheme in terms of a restricted class abstraction. (Contributed by AV, 16-Jul-2019.) (Revised by AV, 26-Mar-2021.)
𝐵 = {𝑥𝐴𝜓}    &   𝐴 ∈ V       𝐵 ∈ V

Theoremrab2ex 5214* 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 = {𝑦𝐴𝜓}    &   𝐴 ∈ V       {𝑥𝐵𝜑} ∈ V

Theoremelssabg 5215* Membership in a class abstraction involving a subset. Unlike elabg 3641, 𝐴 does not have to be a set. (Contributed by NM, 29-Aug-2006.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐵𝑉 → (𝐴 ∈ {𝑥 ∣ (𝑥𝐵𝜑)} ↔ (𝐴𝐵𝜓)))

Theoremintex 5216 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)

Theoremintnex 5217 If a class intersection is not a set, it must be the universe. (Contributed by NM, 3-Jul-2005.)
𝐴 ∈ V ↔ 𝐴 = V)

Theoremintexab 5218 The intersection of a nonempty class abstraction exists. (Contributed by NM, 21-Oct-2003.)
(∃𝑥𝜑 {𝑥𝜑} ∈ V)

Theoremintexrab 5219 The intersection of a nonempty restricted class abstraction exists. (Contributed by NM, 21-Oct-2003.)
(∃𝑥𝐴 𝜑 {𝑥𝐴𝜑} ∈ V)

Theoremiinexg 5220* 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 ≠ ∅ ∧ ∀𝑥𝐴 𝐵𝐶) → 𝑥𝐴 𝐵 ∈ V)

Theoremintabs 5221* Absorption of a redundant conjunct in the intersection of a class abstraction. (Contributed by NM, 3-Jul-2005.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   (𝑥 = {𝑦𝜓} → (𝜑𝜒))    &   ( {𝑦𝜓} ⊆ 𝐴𝜒)        {𝑥 ∣ (𝑥𝐴𝜑)} = {𝑥𝜑}

Theoreminuni 5222* 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.)
( 𝐴𝐵) = {𝑥 ∣ ∃𝑦𝐴 𝑥 = (𝑦𝐵)}

Theoremelpw2g 5223 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 7-Aug-2000.)
(𝐵𝑉 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))

Theoremelpw2 5224 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 11-Oct-2007.)
𝐵 ∈ V       (𝐴 ∈ 𝒫 𝐵𝐴𝐵)

Theoremelpwi2 5225 Membership in a power class. (Contributed by Glauco Siliprandi, 3-Mar-2021.) (Proof shortened by Wolf Lammen, 26-May-2024.)
𝐵𝑉    &   𝐴𝐵       𝐴 ∈ 𝒫 𝐵

Theoremelpwi2OLD 5226 Obsolete version of elpwi2 5225 as of 26-May-2024. (Contributed by Glauco Siliprandi, 3-Mar-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐵𝑉    &   𝐴𝐵       𝐴 ∈ 𝒫 𝐵

Theorempwnss 5227 The power set of a set is never a subset. (Contributed by Stefan O'Rear, 22-Feb-2015.)
(𝐴𝑉 → ¬ 𝒫 𝐴𝐴)

Theorempwne 5228 No set equals its power set. The sethood antecedent is necessary; compare pwv 4810. (Contributed by NM, 17-Nov-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
(𝐴𝑉 → 𝒫 𝐴𝐴)

Theoremdifelpw 5229 A difference is an element of the power set of its minuend. (Contributed by AV, 9-Oct-2023.)
(𝐴𝑉 → (𝐴𝐵) ∈ 𝒫 𝐴)

Theoremrabelpw 5230* A restricted class abstraction is an element of the power set of its restricting set. (Contributed by AV, 9-Oct-2023.)
(𝐴𝑉 → {𝑥𝐴𝜑} ∈ 𝒫 𝐴)

2.2.5  Theorems requiring empty set existence

Theoremclass2set 5231* 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.)
{𝑥𝐴𝐴 ∈ V} ∈ V

Theoremclass2seteq 5232* Equality theorem based on class2set 5231. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Raph Levien, 30-Jun-2006.)
(𝐴𝑉 → {𝑥𝐴𝐴 ∈ V} = 𝐴)

Theorem0elpw 5233 Every power class contains the empty set. (Contributed by NM, 25-Oct-2007.)
∅ ∈ 𝒫 𝐴

Theorempwne0 5234 A power class is never empty. (Contributed by NM, 3-Sep-2018.)
𝒫 𝐴 ≠ ∅

Theorem0nep0 5235 The empty set and its power set are not equal. (Contributed by NM, 23-Dec-1993.)
∅ ≠ {∅}

Theorem0inp0 5236 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 = {∅})

Theoremunidif0 5237 The removal of the empty set from a class does not affect its union. (Contributed by NM, 22-Mar-2004.)
(𝐴 ∖ {∅}) = 𝐴

Theoremiin0 5238* An indexed intersection of the empty set, with a nonempty index set, is empty. (Contributed by NM, 20-Oct-2005.)
(𝐴 ≠ ∅ ↔ 𝑥𝐴 ∅ = ∅)

Theoremnotzfaus 5239* In the Separation Scheme zfauscl 5181, 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 ↔ ¬ 𝑥𝑦)        ¬ ∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))

TheoremnotzfausOLD 5240* Obsolete proof of notzfaus 5239 as of 18-Nov-2023. (Contributed by NM, 8-Feb-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐴 = {∅}    &   (𝜑 ↔ ¬ 𝑥𝑦)        ¬ ∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))

Theoremintv 5241 The intersection of the universal class is empty. (Contributed by NM, 11-Sep-2008.)
V = ∅

Theoremaxpweq 5242* Two equivalent ways to express the Power Set Axiom. Note that ax-pow 5243 is not used by the proof. When ax-pow 5243 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 7472). (Contributed by NM, 22-Jun-2009.)
(𝒫 𝐴 ∈ V ↔ ∃𝑥𝑦(∀𝑧(𝑧𝑦𝑧𝐴) → 𝑦𝑥))

2.3  ZF Set Theory - add the Axiom of Power Sets

2.3.1  Introduce the Axiom of Power Sets

Axiomax-pow 5243* 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 5245 uses explicit subset notation. A version using class notation is pwex 5258. (Contributed by NM, 21-Jun-1993.)
𝑦𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦)

Theoremzfpow 5244* Axiom of Power Sets expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.)
𝑥𝑦(∀𝑥(𝑥𝑦𝑥𝑧) → 𝑦𝑥)

Theoremaxpow2 5245* A variant of the Axiom of Power Sets ax-pow 5243 using subset notation. Problem in [BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)
𝑦𝑧(𝑧𝑥𝑧𝑦)

Theoremaxpow3 5246* A variant of the Axiom of Power Sets ax-pow 5243. 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.)
𝑦𝑧(𝑧𝑥𝑧𝑦)

Theoremel 5247* Every set is an element of some other set. See elALT 5312 for a shorter proof using more axioms. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
𝑦 𝑥𝑦

Theoremdtru 5248* 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 2794 or ax-sep 5179. See dtruALT 5266 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 2391. (Revised by Gino Giotto, 5-Sep-2023.)

¬ ∀𝑥 𝑥 = 𝑦

Theoremdtrucor 5249* Corollary of dtru 5248. 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 5250. (Contributed by NM, 27-Jun-2002.)
𝑥 = 𝑦       𝑥𝑦

Theoremdtrucor2 5250 The theorem form of the deduction dtrucor 5249 leads to a contradiction, as mentioned in the "Wrong!" example at mmdeduction.html#bad 5249. Usage of this theorem is discouraged because it depends on ax-13 2391. (Contributed by NM, 20-Oct-2007.) (New usage is discouraged.)
(𝑥 = 𝑦𝑥𝑦)       (𝜑 ∧ ¬ 𝜑)

Theoremdvdemo1 5251* 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 = 𝑦𝑧𝑥) with 𝑥, 𝑦, 𝑧 disjoint), often called its "principal instance", and the two "degenerate instances" (𝑥(𝑥 = 𝑦𝑥𝑥) with 𝑥, 𝑦 disjoint) and (𝑥(𝑥 = 𝑦𝑦𝑥) with 𝑥, 𝑦 disjoint).

Compare with dvdemo2 5252, 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 5252 for details on the "disjoint variable" mechanism. (The verb "bundle" to express this phenomenon was introduced by Raph Levien.)

Note that dvdemo1 5251 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 2161 nor ax-13 2391. (Contributed by NM, 1-Dec-2006.) (Revised by BJ, 13-Jan-2024.)

𝑥(𝑥 = 𝑦𝑧𝑥)

Theoremdvdemo2 5252* 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 = 𝑦𝑧𝑥) with 𝑥, 𝑦, 𝑧 disjoint), often called its "principal instance", and the two "degenerate instances" (𝑥(𝑥 = 𝑥𝑧𝑥) with 𝑥, 𝑧 disjoint) and (𝑥(𝑥 = 𝑧𝑧𝑥) with 𝑥, 𝑧 disjoint).

Compare with dvdemo1 5251, 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 5251 for details on the "disjoint variable" mechanism.

Note that dvdemo2 5252 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 2145, ax-11 2161, ax-12 2178, ax-13 2391. (Contributed by NM, 1-Dec-2006.) (Revised by BJ, 13-Jan-2024.)

𝑥(𝑥 = 𝑦𝑧𝑥)

Theoremnfnid 5253 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 5248 in its proof. (Contributed by Mario Carneiro, 8-Oct-2016.)
¬ 𝑥𝑥

Theoremnfcvb 5254 The "distinctor" expression ¬ ∀𝑥𝑥 = 𝑦, stating that 𝑥 and 𝑦 are not the same variable, can be written in terms of in the obvious way. This theorem is not true in a one-element domain, because then 𝑥𝑦 and 𝑥𝑥 = 𝑦 will both be true. Usage of this theorem is discouraged because it depends on ax-13 2391. (Contributed by Mario Carneiro, 8-Oct-2016.) (New usage is discouraged.)
(𝑥𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)

Theoremvpwex 5255 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 5256 from vpwex 5255. (Revised by BJ, 10-Aug-2022.)
𝒫 𝑥 ∈ V

Theorempwexg 5256 Power set axiom expressed in class notation, with the sethood requirement as an antecedent. (Contributed by NM, 30-Oct-2003.)
(𝐴𝑉 → 𝒫 𝐴 ∈ V)

Theorempwexd 5257 Deduction version of the power set axiom. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴𝑉)       (𝜑 → 𝒫 𝐴 ∈ V)

Theorempwex 5258 Power set axiom expressed in class notation. (Contributed by NM, 21-Jun-1993.)
𝐴 ∈ V       𝒫 𝐴 ∈ V

Theorempwel 5259 Quantitative version of pwexg 5256: 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 5186 and ax-pr 5307 and shorten proof. (Revised by BJ, 13-Apr-2024.)
(𝐴𝐵 → 𝒫 𝐴 ∈ 𝒫 𝒫 𝐵)

Theoremabssexg 5260* Existence of a class of subsets. (Contributed by NM, 15-Jul-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(𝐴𝑉 → {𝑥 ∣ (𝑥𝐴𝜑)} ∈ V)

TheoremsnexALT 5261 Alternate proof of snex 5309 using Power Set (ax-pow 5243) instead of Pairing (ax-pr 5307). Unlike in the proof of zfpair 5299, Replacement (ax-rep 5166) 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

Theoremp0ex 5262 The power set of the empty set (the ordinal 1) is a set. See also p0exALT 5263. (Contributed by NM, 23-Dec-1993.)
{∅} ∈ V

Theoremp0exALT 5263 Alternate proof of p0ex 5262 which is quite different and longer if snexALT 5261 is expanded. (Contributed by NM, 23-Dec-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
{∅} ∈ V

Theorempp0ex 5264 The power set of the power set of the empty set (the ordinal 2) is a set. (Contributed by NM, 24-Jun-1993.)
{∅, {∅}} ∈ V

Theoremord3ex 5265 The ordinal number 3 is a set, proved without the Axiom of Union ax-un 7446. (Contributed by NM, 2-May-2009.)
{∅, {∅}, {∅, {∅}}} ∈ V

TheoremdtruALT 5266* Alternate proof of dtru 5248 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 3582 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

Theoremaxc16b 5267* This theorem shows that axiom ax-c16 36147 is redundant in the presence of theorem dtru 5248, which states simply that at least two things exist. This justifies the remark at mmzfcnd.html#twoness 5248 (which links to this theorem). (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by NM, 7-Nov-2006.)
(∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))

Theoremeunex 5268 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)

Theoremeusv1 5269* Two ways to express single-valuedness of a class expression 𝐴(𝑥). (Contributed by NM, 14-Oct-2010.)
(∃!𝑦𝑥 𝑦 = 𝐴 ↔ ∃𝑦𝑥 𝑦 = 𝐴)

Theoremeusvnf 5270* 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.)
(∃!𝑦𝑥 𝑦 = 𝐴𝑥𝐴)

Theoremeusvnfb 5271* Two ways to say that 𝐴(𝑥) is a set expression that does not depend on 𝑥. (Contributed by Mario Carneiro, 18-Nov-2016.)
(∃!𝑦𝑥 𝑦 = 𝐴 ↔ (𝑥𝐴𝐴 ∈ V))

Theoremeusv2i 5272* 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.)
(∃!𝑦𝑥 𝑦 = 𝐴 → ∃!𝑦𝑥 𝑦 = 𝐴)

Theoremeusv2nf 5273* Two ways to express single-valuedness of a class expression 𝐴(𝑥). (Contributed by Mario Carneiro, 18-Nov-2016.)
𝐴 ∈ V       (∃!𝑦𝑥 𝑦 = 𝐴𝑥𝐴)

Theoremeusv2 5274* 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       (∃!𝑦𝑥 𝑦 = 𝐴 ↔ ∃!𝑦𝑥 𝑦 = 𝐴)

Theoremreusv1 5275* 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.)
(∃𝑦𝐵 𝜑 → (∃!𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) ↔ ∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)))

Theoremreusv2lem1 5276* Lemma for reusv2 5281. (Contributed by NM, 22-Oct-2010.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
(𝐴 ≠ ∅ → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃𝑥𝑦𝐴 𝑥 = 𝐵))

Theoremreusv2lem2 5277* Lemma for reusv2 5281. (Contributed by NM, 27-Oct-2010.) (Proof shortened by Mario Carneiro, 19-Nov-2016.) (Proof shortened by JJ, 7-Aug-2021.)
(∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃!𝑥𝑦𝐴 𝑥 = 𝐵)

Theoremreusv2lem3 5278* Lemma for reusv2 5281. (Contributed by NM, 14-Dec-2012.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
(∀𝑦𝐴 𝐵 ∈ V → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃!𝑥𝑦𝐴 𝑥 = 𝐵))

Theoremreusv2lem4 5279* Lemma for reusv2 5281. (Contributed by NM, 13-Dec-2012.)
(∃!𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) ↔ ∃!𝑥𝑦𝐵 ((𝐶𝐴𝜑) → 𝑥 = 𝐶))

Theoremreusv2lem5 5280* Lemma for reusv2 5281. (Contributed by NM, 4-Jan-2013.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
((∀𝑦𝐵 𝐶𝐴𝐵 ≠ ∅) → (∃!𝑥𝐴𝑦𝐵 𝑥 = 𝐶 ↔ ∃!𝑥𝐴𝑦𝐵 𝑥 = 𝐶))

Theoremreusv2 5281* Two ways to express single-valuedness of a class expression 𝐶(𝑦) that is constant for those 𝑦𝐵 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.)
((∀𝑦𝐵 (𝜑𝐶𝐴) ∧ ∃𝑦𝐵 𝜑) → (∃!𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) ↔ ∃!𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)))

Theoremreusv3i 5282* Two ways of expressing existential uniqueness via an indirect equality. (Contributed by NM, 23-Dec-2012.)
(𝑦 = 𝑧 → (𝜑𝜓))    &   (𝑦 = 𝑧𝐶 = 𝐷)       (∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) → ∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷))

Theoremreusv3 5283* Two ways to express single-valuedness of a class expression 𝐶(𝑦). See reusv1 5275 for the connection to uniqueness. (Contributed by NM, 27-Dec-2012.)
(𝑦 = 𝑧 → (𝜑𝜓))    &   (𝑦 = 𝑧𝐶 = 𝐷)       (∃𝑦𝐵 (𝜑𝐶𝐴) → (∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷) ↔ ∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)))

Theoremeusv4 5284* Two ways to express single-valuedness of a class expression 𝐵(𝑦). (Contributed by NM, 27-Oct-2010.)
𝐵 ∈ V       (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃!𝑥𝑦𝐴 𝑥 = 𝐵)

Theoremalxfr 5285* Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 18-Feb-2007.)
(𝑥 = 𝐴 → (𝜑𝜓))       ((∀𝑦 𝐴𝐵 ∧ ∀𝑥𝑦 𝑥 = 𝐴) → (∀𝑥𝜑 ↔ ∀𝑦𝜓))

Theoremralxfrd 5286* 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.)
((𝜑𝑦𝐶) → 𝐴𝐵)    &   ((𝜑𝑥𝐵) → ∃𝑦𝐶 𝑥 = 𝐴)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐵 𝜓 ↔ ∀𝑦𝐶 𝜒))

Theoremrexxfrd 5287* 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.)
((𝜑𝑦𝐶) → 𝐴𝐵)    &   ((𝜑𝑥𝐵) → ∃𝑦𝐶 𝑥 = 𝐴)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐵 𝜓 ↔ ∃𝑦𝐶 𝜒))

Theoremralxfr2d 5288* Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by Mario Carneiro, 20-Aug-2014.)
((𝜑𝑦𝐶) → 𝐴𝑉)    &   (𝜑 → (𝑥𝐵 ↔ ∃𝑦𝐶 𝑥 = 𝐴))    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐵 𝜓 ↔ ∀𝑦𝐶 𝜒))

Theoremrexxfr2d 5289* 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.)
((𝜑𝑦𝐶) → 𝐴𝑉)    &   (𝜑 → (𝑥𝐵 ↔ ∃𝑦𝐶 𝑥 = 𝐴))    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐵 𝜓 ↔ ∃𝑦𝐶 𝜒))

Theoremralxfrd2 5290* Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Variant of ralxfrd 5286. (Contributed by Alexander van der Vekens, 25-Apr-2018.)
((𝜑𝑦𝐶) → 𝐴𝐵)    &   ((𝜑𝑥𝐵) → ∃𝑦𝐶 𝑥 = 𝐴)    &   ((𝜑𝑦𝐶𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐵 𝜓 ↔ ∀𝑦𝐶 𝜒))

Theoremrexxfrd2 5291* Transfer existence from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Variant of rexxfrd 5287. (Contributed by Alexander van der Vekens, 25-Apr-2018.)
((𝜑𝑦𝐶) → 𝐴𝐵)    &   ((𝜑𝑥𝐵) → ∃𝑦𝐶 𝑥 = 𝐴)    &   ((𝜑𝑦𝐶𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐵 𝜓 ↔ ∃𝑦𝐶 𝜒))

Theoremralxfr 5292* 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.)
(𝑦𝐶𝐴𝐵)    &   (𝑥𝐵 → ∃𝑦𝐶 𝑥 = 𝐴)    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∀𝑥𝐵 𝜑 ↔ ∀𝑦𝐶 𝜓)

TheoremralxfrALT 5293* Alternate proof of ralxfr 5292 which does not use ralxfrd 5286. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑦𝐶𝐴𝐵)    &   (𝑥𝐵 → ∃𝑦𝐶 𝑥 = 𝐴)    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∀𝑥𝐵 𝜑 ↔ ∀𝑦𝐶 𝜓)

Theoremrexxfr 5294* 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.)
(𝑦𝐶𝐴𝐵)    &   (𝑥𝐵 → ∃𝑦𝐶 𝑥 = 𝐴)    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∃𝑥𝐵 𝜑 ↔ ∃𝑦𝐶 𝜓)

Theoremrabxfrd 5295* 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.)
𝑦𝐵    &   𝑦𝐶    &   ((𝜑𝑦𝐷) → 𝐴𝐷)    &   (𝑥 = 𝐴 → (𝜓𝜒))    &   (𝑦 = 𝐵𝐴 = 𝐶)       ((𝜑𝐵𝐷) → (𝐶 ∈ {𝑥𝐷𝜓} ↔ 𝐵 ∈ {𝑦𝐷𝜒}))

Theoremrabxfr 5296* 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.)
𝑦𝐵    &   𝑦𝐶    &   (𝑦𝐷𝐴𝐷)    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵𝐴 = 𝐶)       (𝐵𝐷 → (𝐶 ∈ {𝑥𝐷𝜑} ↔ 𝐵 ∈ {𝑦𝐷𝜓}))

Theoremreuhypd 5297* A theorem useful for eliminating the restricted existential uniqueness hypotheses in riotaxfrd 7132. (Contributed by NM, 16-Jan-2012.)
((𝜑𝑥𝐶) → 𝐵𝐶)    &   ((𝜑𝑥𝐶𝑦𝐶) → (𝑥 = 𝐴𝑦 = 𝐵))       ((𝜑𝑥𝐶) → ∃!𝑦𝐶 𝑥 = 𝐴)

Theoremreuhyp 5298* A theorem useful for eliminating the restricted existential uniqueness hypotheses in reuxfr1 3718. (Contributed by NM, 15-Nov-2004.)
(𝑥𝐶𝐵𝐶)    &   ((𝑥𝐶𝑦𝐶) → (𝑥 = 𝐴𝑦 = 𝐵))       (𝑥𝐶 → ∃!𝑦𝐶 𝑥 = 𝐴)

Theoremzfpair 5299 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 5300. Instead, use zfpair2 5308 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

TheoremaxprALT 5300* Alternate proof of axpr 5306. (Contributed by NM, 14-Nov-2006.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑧𝑤((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧)

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