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Theorem List for Metamath Proof Explorer - 5301-5400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdvdemo2 5301* 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 5300, 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 5300 for details on the "disjoint variable" mechanism.

Note that dvdemo2 5301 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 2158, ax-12 2175, ax-13 2374. (Contributed by NM, 1-Dec-2006.) (Revised by BJ, 13-Jan-2024.)

𝑥(𝑥 = 𝑦𝑧𝑥)
 
Theoremnfnid 5302 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 5297 in its proof. (Contributed by Mario Carneiro, 8-Oct-2016.)
¬ 𝑥𝑥
 
Theoremnfcvb 5303 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. (Contributed by Mario Carneiro, 8-Oct-2016.) Usage of this theorem is discouraged because it depends on ax-13 2374. (New usage is discouraged.)
(𝑥𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
 
Theoremvpwex 5304 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 5305 from vpwex 5304. (Revised by BJ, 10-Aug-2022.)
𝒫 𝑥 ∈ V
 
Theorempwexg 5305 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 5306 Deduction version of the power set axiom. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴𝑉)       (𝜑 → 𝒫 𝐴 ∈ V)
 
Theorempwex 5307 Power set axiom expressed in class notation. (Contributed by NM, 21-Jun-1993.)
𝐴 ∈ V       𝒫 𝐴 ∈ V
 
Theorempwel 5308 Quantitative version of pwexg 5305: 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 5234 and ax-pr 5356 and shorten proof. (Revised by BJ, 13-Apr-2024.)
(𝐴𝐵 → 𝒫 𝐴 ∈ 𝒫 𝒫 𝐵)
 
Theoremabssexg 5309* 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 5310 Alternate proof of snex 5358 using Power Set (ax-pow 5292) instead of Pairing (ax-pr 5356). Unlike in the proof of zfpair 5348, Replacement (ax-rep 5214) 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 5311 The power set of the empty set (the ordinal 1) is a set. See also p0exALT 5312. (Contributed by NM, 23-Dec-1993.)
{∅} ∈ V
 
Theoremp0exALT 5312 Alternate proof of p0ex 5311 which is quite different and longer if snexALT 5310 is expanded. (Contributed by NM, 23-Dec-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
{∅} ∈ V
 
Theorempp0ex 5313 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 5314 The ordinal number 3 is a set, proved without the Axiom of Union ax-un 7583. (Contributed by NM, 2-May-2009.)
{∅, {∅}, {∅, {∅}}} ∈ V
 
TheoremdtruALT 5315* Alternate proof of dtru 5363 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 3544 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 5316* This theorem shows that Axiom ax-c16 36915 is redundant in the presence of Theorem dtruALT2 5297, which states simply that at least two things exist. This justifies the remark at mmzfcnd.html#twoness 5297 (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 5317 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 5318* 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 5319* 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 5320* 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 5321* 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 5322* 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 5323* 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 5324* 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 5325* Lemma for reusv2 5330. (Contributed by NM, 22-Oct-2010.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
(𝐴 ≠ ∅ → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃𝑥𝑦𝐴 𝑥 = 𝐵))
 
Theoremreusv2lem2 5326* Lemma for reusv2 5330. (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 5327* Lemma for reusv2 5330. (Contributed by NM, 14-Dec-2012.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
(∀𝑦𝐴 𝐵 ∈ V → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃!𝑥𝑦𝐴 𝑥 = 𝐵))
 
Theoremreusv2lem4 5328* Lemma for reusv2 5330. (Contributed by NM, 13-Dec-2012.)
(∃!𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) ↔ ∃!𝑥𝑦𝐵 ((𝐶𝐴𝜑) → 𝑥 = 𝐶))
 
Theoremreusv2lem5 5329* Lemma for reusv2 5330. (Contributed by NM, 4-Jan-2013.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
((∀𝑦𝐵 𝐶𝐴𝐵 ≠ ∅) → (∃!𝑥𝐴𝑦𝐵 𝑥 = 𝐶 ↔ ∃!𝑥𝐴𝑦𝐵 𝑥 = 𝐶))
 
Theoremreusv2 5330* 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 5331* 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 5332* Two ways to express single-valuedness of a class expression 𝐶(𝑦). See reusv1 5324 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 5333* 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 5334* 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 5335* 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 5336* 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 5337* 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 5338* 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 5339* Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Variant of ralxfrd 5335. (Contributed by Alexander van der Vekens, 25-Apr-2018.)
((𝜑𝑦𝐶) → 𝐴𝐵)    &   ((𝜑𝑥𝐵) → ∃𝑦𝐶 𝑥 = 𝐴)    &   ((𝜑𝑦𝐶𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐵 𝜓 ↔ ∀𝑦𝐶 𝜒))
 
Theoremrexxfrd2 5340* Transfer existence from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Variant of rexxfrd 5336. (Contributed by Alexander van der Vekens, 25-Apr-2018.)
((𝜑𝑦𝐶) → 𝐴𝐵)    &   ((𝜑𝑥𝐵) → ∃𝑦𝐶 𝑥 = 𝐴)    &   ((𝜑𝑦𝐶𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐵 𝜓 ↔ ∃𝑦𝐶 𝜒))
 
Theoremralxfr 5341* 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 5342* Alternate proof of ralxfr 5341 which does not use ralxfrd 5335. (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 5343* 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 5344* 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 5345* 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 5346* A theorem useful for eliminating the restricted existential uniqueness hypotheses in riotaxfrd 7264. (Contributed by NM, 16-Jan-2012.)
((𝜑𝑥𝐶) → 𝐵𝐶)    &   ((𝜑𝑥𝐶𝑦𝐶) → (𝑥 = 𝐴𝑦 = 𝐵))       ((𝜑𝑥𝐶) → ∃!𝑦𝐶 𝑥 = 𝐴)
 
Theoremreuhyp 5347* A theorem useful for eliminating the restricted existential uniqueness hypotheses in reuxfr1 3691. (Contributed by NM, 15-Nov-2004.)
(𝑥𝐶𝐵𝐶)    &   ((𝑥𝐶𝑦𝐶) → (𝑥 = 𝐴𝑦 = 𝐵))       (𝑥𝐶 → ∃!𝑦𝐶 𝑥 = 𝐴)
 
Theoremzfpair 5348 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 5349. Instead, use zfpair2 5357 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 5349* Alternate proof of axpr 5355. (Contributed by NM, 14-Nov-2006.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑧𝑤((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧)
 
2.3.2  Derive the Axiom of Pairing
 
Theoremaxprlem1 5350* Lemma for axpr 5355. There exists a set to which all empty sets belong. (Contributed by Rohan Ridenour, 10-Aug-2023.) (Revised by BJ, 13-Aug-2023.)
𝑥𝑦(∀𝑧 ¬ 𝑧𝑦𝑦𝑥)
 
Theoremaxprlem2 5351* Lemma for axpr 5355. 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.)
𝑥𝑦(∀𝑧𝑦𝑤 ¬ 𝑤𝑧𝑦𝑥)
 
Theoremaxprlem3 5352* Lemma for axpr 5355. Eliminate the antecedent of the relevant replacement instance. (Contributed by Rohan Ridenour, 10-Aug-2023.)
𝑧𝑤(𝑤𝑧 ↔ ∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)))
 
Theoremaxprlem4 5353* Lemma for axpr 5355. 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(∀𝑛𝑠𝑡 ¬ 𝑡𝑛𝑠𝑝) ∧ 𝑤 = 𝑥) → ∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)))
 
Theoremaxprlem5 5354* Lemma for axpr 5355. 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(∀𝑛𝑠𝑡 ¬ 𝑡𝑛𝑠𝑝) ∧ 𝑤 = 𝑦) → ∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)))
 
Theoremaxpr 5355* 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 5356 below so that the uses of the Axiom of Pairing can be more easily identified.

For a shorter proof using ax-ext 2711, see axprALT 5349. (Contributed by NM, 14-Nov-2006.) Remove dependency on ax-ext 2711. (Revised by Rohan Ridenour, 10-Aug-2023.) (Proof shortened by BJ, 13-Aug-2023.) Use ax-pr 5356 instead. (New usage is discouraged.)

𝑧𝑤((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧)
 
Axiomax-pr 5356* The Axiom of Pairing of ZF set theory. It was derived as Theorem axpr 5355 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.)
𝑧𝑤((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧)
 
Theoremzfpair2 5357 Derive the abbreviated version of the Axiom of Pairing from ax-pr 5356. See zfpair 5348 for its derivation from the other axioms. (Contributed by NM, 14-Nov-2006.)
{𝑥, 𝑦} ∈ V
 
Theoremsnex 5358 A singleton is a set. Theorem 7.12 of [Quine] p. 51, proved using Extensionality, Separation, Null Set, and Pairing. See also snexALT 5310. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 19-May-2013.) (Proof modification is discouraged.)
{𝐴} ∈ V
 
Theoremprex 5359 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 4709), so we can dispense with hypotheses requiring them to be sets. (Contributed by NM, 15-Jul-1993.)
{𝐴, 𝐵} ∈ V
 
Theoremsels 5360* If a class is a set, then it is a member of a set. (Contributed by BJ, 3-Apr-2019.)
(𝐴𝑉 → ∃𝑥 𝐴𝑥)
 
Theoremel 5361* Every set is an element of some other set. See elALT 5362 for a shorter proof using more axioms, and see elALT2 5296 for a proof that uses ax-9 2120 and ax-pow 5292 instead of ax-pr 5356. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) Avoid ax-9 2120, ax-pow 5292. (Revised by BTernaryTau, 2-Dec-2024.)
𝑦 𝑥𝑦
 
TheoremelALT 5362* Alternate proof of el 5361, shorter but requiring more axioms. (Contributed by NM, 4-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑦 𝑥𝑦
 
Theoremdtru 5363* 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 2711, ax-sep 5227, or ax-pow 5292. See dtruALT 5315 for a shorter proof using these axioms, and see dtruALT2 5297 for a proof that uses ax-pow 5292 instead of ax-pr 5356.

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 2374. (Revised by Gino Giotto, 5-Sep-2023.) Avoid ax-12 2175. (Revised by Rohan Ridenour, 9-Oct-2024.) Use ax-pr 5356 instead of ax-pow 5292. (Revised by BTernaryTau, 3-Dec-2024.)

¬ ∀𝑥 𝑥 = 𝑦
 
Theoremsnelpwi 5364 A singleton of a set belongs to the power class of a class containing the set. (Contributed by Alan Sare, 25-Aug-2011.)
(𝐴𝐵 → {𝐴} ∈ 𝒫 𝐵)
 
Theoremsnelpw 5365 A singleton of a set belongs to the power class of a class containing the set. (Contributed by NM, 1-Apr-1998.)
𝐴 ∈ V       (𝐴𝐵 ↔ {𝐴} ∈ 𝒫 𝐵)
 
Theoremprelpw 5366 A pair of two sets belongs to the power class of a class containing those two sets and vice versa. (Contributed by AV, 8-Jan-2020.)
((𝐴𝑉𝐵𝑊) → ((𝐴𝐶𝐵𝐶) ↔ {𝐴, 𝐵} ∈ 𝒫 𝐶))
 
Theoremprelpwi 5367 A pair of two sets belongs to the power class of a class containing those two sets. (Contributed by Thierry Arnoux, 10-Mar-2017.) (Proof shortened by AV, 23-Oct-2021.)
((𝐴𝐶𝐵𝐶) → {𝐴, 𝐵} ∈ 𝒫 𝐶)
 
Theoremrext 5368* A theorem similar to extensionality, requiring the existence of a singleton. Exercise 8 of [TakeutiZaring] p. 16. (Contributed by NM, 10-Aug-1993.)
(∀𝑧(𝑥𝑧𝑦𝑧) → 𝑥 = 𝑦)
 
Theoremsspwb 5369 The powerclass construction preserves and reflects inclusion. Classes are subclasses if and only if their power classes are subclasses. Exercise 18 of [TakeutiZaring] p. 18. (Contributed by NM, 13-Oct-1996.)
(𝐴𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵)
 
Theoremunipw 5370 A class equals the union of its power class. Exercise 6(a) of [Enderton] p. 38. (Contributed by NM, 14-Oct-1996.) (Proof shortened by Alan Sare, 28-Dec-2008.)
𝒫 𝐴 = 𝐴
 
Theoremuniv 5371 The union of the universe is the universe. Exercise 4.12(c) of [Mendelson] p. 235. (Contributed by NM, 14-Sep-2003.)
V = V
 
Theorempwtr 5372 A class is transitive iff its power class is transitive. (Contributed by Alan Sare, 25-Aug-2011.) (Revised by Mario Carneiro, 15-Jun-2014.)
(Tr 𝐴 ↔ Tr 𝒫 𝐴)
 
Theoremssextss 5373* An extensionality-like principle defining subclass in terms of subsets. (Contributed by NM, 30-Jun-2004.)
(𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
 
Theoremssext 5374* An extensionality-like principle that uses the subset instead of the membership relation: two classes are equal iff they have the same subsets. (Contributed by NM, 30-Jun-2004.)
(𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
 
Theoremnssss 5375* Negation of subclass relationship. Compare nss 3988. (Contributed by NM, 30-Jun-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
𝐴𝐵 ↔ ∃𝑥(𝑥𝐴 ∧ ¬ 𝑥𝐵))
 
Theorempweqb 5376 Classes are equal if and only if their power classes are equal. Exercise 19 of [TakeutiZaring] p. 18. (Contributed by NM, 13-Oct-1996.)
(𝐴 = 𝐵 ↔ 𝒫 𝐴 = 𝒫 𝐵)
 
Theoremintid 5377* The intersection of all sets to which a set belongs is the singleton of that set. (Contributed by NM, 5-Jun-2009.)
𝐴 ∈ V        {𝑥𝐴𝑥} = {𝐴}
 
Theoremmoabex 5378 "At most one" existence implies a class abstraction exists. (Contributed by NM, 30-Dec-1996.)
(∃*𝑥𝜑 → {𝑥𝜑} ∈ V)
 
Theoremrmorabex 5379 Restricted "at most one" existence implies a restricted class abstraction exists. (Contributed by NM, 17-Jun-2017.)
(∃*𝑥𝐴 𝜑 → {𝑥𝐴𝜑} ∈ V)
 
Theoremeuabex 5380 The abstraction of a wff with existential uniqueness exists. (Contributed by NM, 25-Nov-1994.)
(∃!𝑥𝜑 → {𝑥𝜑} ∈ V)
 
Theoremnnullss 5381* A nonempty class (even if proper) has a nonempty subset. (Contributed by NM, 23-Aug-2003.)
(𝐴 ≠ ∅ → ∃𝑥(𝑥𝐴𝑥 ≠ ∅))
 
Theoremexss 5382* Restricted existence in a class (even if proper) implies restricted existence in a subset. (Contributed by NM, 23-Aug-2003.)
(∃𝑥𝐴 𝜑 → ∃𝑦(𝑦𝐴 ∧ ∃𝑥𝑦 𝜑))
 
Theoremopex 5383 An ordered pair of classes is a set. Exercise 7 of [TakeutiZaring] p. 16. (Contributed by NM, 18-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.)
𝐴, 𝐵⟩ ∈ V
 
Theoremotex 5384 An ordered triple of classes is a set. (Contributed by NM, 3-Apr-2015.)
𝐴, 𝐵, 𝐶⟩ ∈ V
 
Theoremelopg 5385 Characterization of the elements of an ordered pair. Closed form of elop 5386. (Contributed by BJ, 22-Jun-2019.) (Avoid depending on this detail.)
((𝐴𝑉𝐵𝑊) → (𝐶 ∈ ⟨𝐴, 𝐵⟩ ↔ (𝐶 = {𝐴} ∨ 𝐶 = {𝐴, 𝐵})))
 
Theoremelop 5386 Characterization of the elements of an ordered pair. Exercise 3 of [TakeutiZaring] p. 15. (Contributed by NM, 15-Jul-1993.) (Revised by Mario Carneiro, 26-Apr-2015.) Remove an extraneous hypothesis. (Revised by BJ, 25-Dec-2020.) (Avoid depending on this detail.)
𝐵 ∈ V    &   𝐶 ∈ V       (𝐴 ∈ ⟨𝐵, 𝐶⟩ ↔ (𝐴 = {𝐵} ∨ 𝐴 = {𝐵, 𝐶}))
 
Theoremopi1 5387 One of the two elements in an ordered pair. (Contributed by NM, 15-Jul-1993.) (Revised by Mario Carneiro, 26-Apr-2015.) (Avoid depending on this detail.)
𝐴 ∈ V    &   𝐵 ∈ V       {𝐴} ∈ ⟨𝐴, 𝐵
 
Theoremopi2 5388 One of the two elements of an ordered pair. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.) (Avoid depending on this detail.)
𝐴 ∈ V    &   𝐵 ∈ V       {𝐴, 𝐵} ∈ ⟨𝐴, 𝐵
 
Theoremopeluu 5389 Each member of an ordered pair belongs to the union of the union of a class to which the ordered pair belongs. Lemma 3D of [Enderton] p. 41. (Contributed by NM, 31-Mar-1995.) (Revised by Mario Carneiro, 27-Feb-2016.)
𝐴 ∈ V    &   𝐵 ∈ V       (⟨𝐴, 𝐵⟩ ∈ 𝐶 → (𝐴 𝐶𝐵 𝐶))
 
Theoremop1stb 5390 Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (See op2ndb 6129 to extract the second member, op1sta 6127 for an alternate version, and op1st 7833 for the preferred version.) (Contributed by NM, 25-Nov-2003.)
𝐴 ∈ V    &   𝐵 ∈ V        𝐴, 𝐵⟩ = 𝐴
 
Theorembrv 5391 Two classes are always in relation by V. This is simply equivalent to 𝐴, 𝐵⟩ ∈ V, and does not imply that V is a relation: see nrelv 5709. (Contributed by Scott Fenton, 11-Apr-2012.)
𝐴V𝐵
 
2.3.3  Ordered pair theorem
 
Theoremopnz 5392 An ordered pair is nonempty iff the arguments are sets. (Contributed by NM, 24-Jan-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
(⟨𝐴, 𝐵⟩ ≠ ∅ ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
 
Theoremopnzi 5393 An ordered pair is nonempty if the arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.)
𝐴 ∈ V    &   𝐵 ∈ V       𝐴, 𝐵⟩ ≠ ∅
 
Theoremopth1 5394 Equality of the first members of equal ordered pairs. (Contributed by NM, 28-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
𝐴 ∈ V    &   𝐵 ∈ V       (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐴 = 𝐶)
 
Theoremopth 5395 The ordered pair theorem. If two ordered pairs are equal, their first elements are equal and their second elements are equal. Exercise 6 of [TakeutiZaring] p. 16. Note that 𝐶 and 𝐷 are not required to be sets due our specific ordered pair definition. (Contributed by NM, 28-May-1995.)
𝐴 ∈ V    &   𝐵 ∈ V       (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷))
 
Theoremopthg 5396 Ordered pair theorem. 𝐶 and 𝐷 are not required to be sets under our specific ordered pair definition. (Contributed by NM, 14-Oct-2005.) (Revised by Mario Carneiro, 26-Apr-2015.)
((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
 
Theoremopth1g 5397 Equality of the first members of equal ordered pairs. Closed form of opth1 5394. (Contributed by AV, 14-Oct-2018.)
((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐴 = 𝐶))
 
Theoremopthg2 5398 Ordered pair theorem. (Contributed by NM, 14-Oct-2005.) (Revised by Mario Carneiro, 26-Apr-2015.)
((𝐶𝑉𝐷𝑊) → (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
 
Theoremopth2 5399 Ordered pair theorem. (Contributed by NM, 21-Sep-2014.)
𝐶 ∈ V    &   𝐷 ∈ V       (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷))
 
Theoremopthneg 5400 Two ordered pairs are not equal iff their first components or their second components are not equal. (Contributed by AV, 13-Dec-2018.)
((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ ≠ ⟨𝐶, 𝐷⟩ ↔ (𝐴𝐶𝐵𝐷)))
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330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500
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