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Theorem List for Metamath Proof Explorer - 4701-4800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremreusv3 4701* Two ways to express single-valuedness of a class expression 𝐶(𝑦). See reusv1 4691 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 4702* Two ways to express single-valuedness of a class expression 𝐵(𝑥). (Contributed by NM, 27-Oct-2010.)
𝐵 ∈ V       (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃!𝑥𝑦𝐴 𝑥 = 𝐵)
 
Theoremalxfr 4703* 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 4704* 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))
 
TheoremralxfrdOLD 4705* Obsolete proof of ralxfrd 4704 as of 7-Aug-2021. (Contributed by NM, 15-Aug-2014.) (Proof shortened by Mario Carneiro, 19-Nov-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝜑𝑦𝐶) → 𝐴𝐵)    &   ((𝜑𝑥𝐵) → ∃𝑦𝐶 𝑥 = 𝐴)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐵 𝜓 ↔ ∀𝑦𝐶 𝜒))
 
Theoremrexxfrd 4706* 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 4707* 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 4708* 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 4709* Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Variant of ralxfrd 4704. (Contributed by Alexander van der Vekens, 25-Apr-2018.)
((𝜑𝑦𝐶) → 𝐴𝐵)    &   ((𝜑𝑥𝐵) → ∃𝑦𝐶 𝑥 = 𝐴)    &   ((𝜑𝑦𝐶𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐵 𝜓 ↔ ∀𝑦𝐶 𝜒))
 
Theoremrexxfrd2 4710* Transfer existence from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Variant of rexxfrd 4706. (Contributed by Alexander van der Vekens, 25-Apr-2018.)
((𝜑𝑦𝐶) → 𝐴𝐵)    &   ((𝜑𝑥𝐵) → ∃𝑦𝐶 𝑥 = 𝐴)    &   ((𝜑𝑦𝐶𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐵 𝜓 ↔ ∃𝑦𝐶 𝜒))
 
Theoremralxfr 4711* 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 4712* Alternate proof of ralxfr 4711 which does not use ralxfrd 4704. (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 4713* 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 4714* Class builder membership after substituting an expression 𝐴 (containing 𝑦) for 𝑥 in the class expression 𝜒. (Contributed by NM, 16-Jan-2012.)
𝑦𝐵    &   𝑦𝐶    &   ((𝜑𝑦𝐷) → 𝐴𝐷)    &   (𝑥 = 𝐴 → (𝜓𝜒))    &   (𝑦 = 𝐵𝐴 = 𝐶)       ((𝜑𝐵𝐷) → (𝐶 ∈ {𝑥𝐷𝜓} ↔ 𝐵 ∈ {𝑦𝐷𝜒}))
 
Theoremrabxfr 4715* Class builder membership after substituting an expression 𝐴 (containing 𝑦) for 𝑥 in the class expression 𝜑. (Contributed by NM, 10-Jun-2005.)
𝑦𝐵    &   𝑦𝐶    &   (𝑦𝐷𝐴𝐷)    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵𝐴 = 𝐶)       (𝐵𝐷 → (𝐶 ∈ {𝑥𝐷𝜑} ↔ 𝐵 ∈ {𝑦𝐷𝜓}))
 
Theoremreuxfr2d 4716* Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 16-Jan-2012.) (Revised by NM, 16-Jun-2017.)
((𝜑𝑦𝐵) → 𝐴𝐵)    &   ((𝜑𝑥𝐵) → ∃*𝑦𝐵 𝑥 = 𝐴)       (𝜑 → (∃!𝑥𝐵𝑦𝐵 (𝑥 = 𝐴𝜓) ↔ ∃!𝑦𝐵 𝜓))
 
Theoremreuxfr2 4717* Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 14-Nov-2004.) (Revised by NM, 16-Jun-2017.)
(𝑦𝐵𝐴𝐵)    &   (𝑥𝐵 → ∃*𝑦𝐵 𝑥 = 𝐴)       (∃!𝑥𝐵𝑦𝐵 (𝑥 = 𝐴𝜑) ↔ ∃!𝑦𝐵 𝜑)
 
Theoremreuxfrd 4718* Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Use reuhypd 4720 to eliminate the second hypothesis. (Contributed by NM, 16-Jan-2012.)
((𝜑𝑦𝐵) → 𝐴𝐵)    &   ((𝜑𝑥𝐵) → ∃!𝑦𝐵 𝑥 = 𝐴)    &   (𝑥 = 𝐴 → (𝜓𝜒))       (𝜑 → (∃!𝑥𝐵 𝜓 ↔ ∃!𝑦𝐵 𝜒))
 
Theoremreuxfr 4719* Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Use reuhyp 4721 to eliminate the second hypothesis. (Contributed by NM, 14-Nov-2004.)
(𝑦𝐵𝐴𝐵)    &   (𝑥𝐵 → ∃!𝑦𝐵 𝑥 = 𝐴)    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∃!𝑥𝐵 𝜑 ↔ ∃!𝑦𝐵 𝜓)
 
Theoremreuhypd 4720* A theorem useful for eliminating the restricted existential uniqueness hypotheses in riotaxfrd 6418. (Contributed by NM, 16-Jan-2012.)
((𝜑𝑥𝐶) → 𝐵𝐶)    &   ((𝜑𝑥𝐶𝑦𝐶) → (𝑥 = 𝐴𝑦 = 𝐵))       ((𝜑𝑥𝐶) → ∃!𝑦𝐶 𝑥 = 𝐴)
 
Theoremreuhyp 4721* A theorem useful for eliminating the restricted existential uniqueness hypotheses in reuxfr 4719. (Contributed by NM, 15-Nov-2004.)
(𝑥𝐶𝐵𝐶)    &   ((𝑥𝐶𝑦𝐶) → (𝑥 = 𝐴𝑦 = 𝐵))       (𝑥𝐶 → ∃!𝑦𝐶 𝑥 = 𝐴)
 
Theoremnfnid 4722 A setvar variable is not free from itself. The proof relies on dtru 4682, that is, it is not true in a one-element domain. (Contributed by Mario Carneiro, 8-Oct-2016.)
¬ 𝑥𝑥
 
Theoremnfcvb 4723 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.)
(𝑥𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
 
Theorempwuni 4724 A class is a subclass of the power class of its union. Exercise 6(b) of [Enderton] p. 38. (Contributed by NM, 14-Oct-1996.)
𝐴 ⊆ 𝒫 𝐴
 
TheoremdtruALT 4725* Alternate proof of dtru 4682 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 3177 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
 
Theoremdtrucor 4726* Corollary of dtru 4682. 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 4727. (Contributed by NM, 27-Jun-2002.)
𝑥 = 𝑦       𝑥𝑦
 
Theoremdtrucor2 4727 The theorem form of the deduction dtrucor 4726 leads to a contradiction, as mentioned in the "Wrong!" example at mmdeduction.html#bad. (Contributed by NM, 20-Oct-2007.)
(𝑥 = 𝑦𝑥𝑦)       (𝜑 ∧ ¬ 𝜑)
 
Theoremdvdemo1 4728* Demonstration of a theorem (scheme) that requires (meta)variables 𝑥 and 𝑦 to be distinct, but no others. It bundles the theorem schemes 𝑥(𝑥 = 𝑦𝑥𝑥) and 𝑥(𝑥 = 𝑦𝑦𝑥). Compare dvdemo2 4729. ("Bundles" is a term introduced by Raph Levien.) (Contributed by NM, 1-Dec-2006.)
𝑥(𝑥 = 𝑦𝑧𝑥)
 
Theoremdvdemo2 4729* Demonstration of a theorem (scheme) that requires (meta)variables 𝑥 and 𝑧 to be distinct, but no others. It bundles the theorem schemes 𝑥(𝑥 = 𝑥𝑧𝑥) and 𝑥(𝑥 = 𝑦𝑦𝑥). Compare dvdemo1 4728. (Contributed by NM, 1-Dec-2006.)
𝑥(𝑥 = 𝑦𝑧𝑥)
 
2.3.2  Derive the Axiom of Pairing
 
Theoremzfpair 4730 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 axpr 4731. Instead, use zfpair2 4733 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
 
Theoremaxpr 4731* 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 4732 below so that the uses of the Axiom of Pairing can be more easily identified. (Contributed by NM, 14-Nov-2006.) (New usage is discouraged.)

𝑧𝑤((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧)
 
Axiomax-pr 4732* The Axiom of Pairing of ZF set theory. It was derived as theorem axpr 4731 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 4733 Derive the abbreviated version of the Axiom of Pairing from ax-pr 4732. See zfpair 4730 for its derivation from the other axioms. (Contributed by NM, 14-Nov-2006.)
{𝑥, 𝑦} ∈ V
 
Theoremsnex 4734 A singleton is a set. Theorem 7.12 of [Quine] p. 51, proved using Extensionality, Separation, Null Set, and Pairing. See also snexALT 4677. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 19-May-2013.) (Proof modification is discouraged.)
{𝐴} ∈ V
 
Theoremprex 4735 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 4148), so we can dispense with hypotheses requiring them to be sets. (Contributed by NM, 15-Jul-1993.)
{𝐴, 𝐵} ∈ V
 
TheoremelALT 4736* Alternate proof of el 4672, shorter but requiring more axioms. (Contributed by NM, 4-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑦 𝑥𝑦
 
TheoremdtruALT2 4737* Alternate proof of dtru 4682 using ax-pr 4732 instead of ax-pow 4668. (Contributed by Mario Carneiro, 31-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
¬ ∀𝑥 𝑥 = 𝑦
 
Theoremsnelpwi 4738 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 4739 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)
 
Theoremprelpwi 4740 A pair of two sets belongs to the power class of a class containing those two sets. (Contributed by Thierry Arnoux, 10-Mar-2017.)
((𝐴𝐶𝐵𝐶) → {𝐴, 𝐵} ∈ 𝒫 𝐶)
 
Theoremrext 4741* 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 4742 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 4743 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 4744 The union of the universe is the universe. Exercise 4.12(c) of [Mendelson] p. 235. (Contributed by NM, 14-Sep-2003.)
V = V
 
Theorempwel 4745 Membership of a power class. Exercise 10 of [Enderton] p. 26. (Contributed by NM, 13-Jan-2007.)
(𝐴𝐵 → 𝒫 𝐴 ∈ 𝒫 𝒫 𝐵)
 
Theorempwtr 4746 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 4747* An extensionality-like principle defining subclass in terms of subsets. (Contributed by NM, 30-Jun-2004.)
(𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
 
Theoremssext 4748* 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 4749* Negation of subclass relationship. Compare nss 3530. (Contributed by NM, 30-Jun-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
𝐴𝐵 ↔ ∃𝑥(𝑥𝐴 ∧ ¬ 𝑥𝐵))
 
Theorempweqb 4750 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 4751* 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 4752 "At most one" existence implies a class abstraction exists. (Contributed by NM, 30-Dec-1996.)
(∃*𝑥𝜑 → {𝑥𝜑} ∈ V)
 
Theoremrmorabex 4753 Restricted "at most one" existence implies a restricted class abstraction exists. (Contributed by NM, 17-Jun-2017.)
(∃*𝑥𝐴 𝜑 → {𝑥𝐴𝜑} ∈ V)
 
Theoremeuabex 4754 The abstraction of a wff with existential uniqueness exists. (Contributed by NM, 25-Nov-1994.)
(∃!𝑥𝜑 → {𝑥𝜑} ∈ V)
 
Theoremnnullss 4755* A nonempty class (even if proper) has a nonempty subset. (Contributed by NM, 23-Aug-2003.)
(𝐴 ≠ ∅ → ∃𝑥(𝑥𝐴𝑥 ≠ ∅))
 
Theoremexss 4756* 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 4757 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 4758 An ordered triple of classes is a set. (Contributed by NM, 3-Apr-2015.)
𝐴, 𝐵, 𝐶⟩ ∈ V
 
Theoremelopg 4759 Characterization of the elements of an ordered pair. Closed form of elop 4760. (Contributed by BJ, 22-Jun-2019.) (Avoid depending on this detail.)
((𝐴𝑉𝐵𝑊) → (𝐶 ∈ ⟨𝐴, 𝐵⟩ ↔ (𝐶 = {𝐴} ∨ 𝐶 = {𝐴, 𝐵})))
 
Theoremelop 4760 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}))
 
TheoremelopOLD 4761 Obsolete version of elop 4760, with one extraneous hypothesis. Obsolete as of 25-Dec-2020 . (Contributed by NM, 15-Jul-1993.) (Revised by Mario Carneiro, 26-Apr-2015.) (Avoid depending on this detail.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐵 ∈ V    &   𝐶 ∈ V    &   𝐴 ∈ V       (𝐴 ∈ ⟨𝐵, 𝐶⟩ ↔ (𝐴 = {𝐵} ∨ 𝐴 = {𝐵, 𝐶}))
 
Theoremopi1 4762 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 4763 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 4764 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 4765 Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (See op2ndb 5427 to extract the second member, op1sta 5425 for an alternate version, and op1st 6942 for the preferred version.) (Contributed by NM, 25-Nov-2003.)
𝐴 ∈ V    &   𝐵 ∈ V        𝐴, 𝐵⟩ = 𝐴
 
2.3.3  Ordered pair theorem
 
Theoremopnz 4766 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 4767 An ordered pair is nonempty if the arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.)
𝐴 ∈ V    &   𝐵 ∈ V       𝐴, 𝐵⟩ ≠ ∅
 
Theoremopth1 4768 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 4769 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 4770 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 4771 Equality of the first members of equal ordered pairs. Closed form of opth1 4768. (Contributed by AV, 14-Oct-2018.)
((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐴 = 𝐶))
 
Theoremopthg2 4772 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 4773 Ordered pair theorem. (Contributed by NM, 21-Sep-2014.)
𝐶 ∈ V    &   𝐷 ∈ V       (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷))
 
Theoremopthneg 4774 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)))
 
Theoremopthne 4775 Two ordered pairs are not equal iff their first components or their second components are not equal. (Contributed by AV, 13-Dec-2018.)
𝐴 ∈ V    &   𝐵 ∈ V       (⟨𝐴, 𝐵⟩ ≠ ⟨𝐶, 𝐷⟩ ↔ (𝐴𝐶𝐵𝐷))
 
Theoremotth2 4776 Ordered triple theorem, with triple expressed with ordered pairs. (Contributed by NM, 1-May-1995.) (Revised by Mario Carneiro, 26-Apr-2015.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝑅 ∈ V       (⟨⟨𝐴, 𝐵⟩, 𝑅⟩ = ⟨⟨𝐶, 𝐷⟩, 𝑆⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷𝑅 = 𝑆))
 
Theoremotth 4777 Ordered triple theorem. (Contributed by NM, 25-Sep-2014.) (Revised by Mario Carneiro, 26-Apr-2015.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝑅 ∈ V       (⟨𝐴, 𝐵, 𝑅⟩ = ⟨𝐶, 𝐷, 𝑆⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷𝑅 = 𝑆))
 
Theoremotthg 4778 Ordered triple theorem, closed form. (Contributed by Alexander van der Vekens, 10-Mar-2018.)
((𝐴𝑈𝐵𝑉𝐶𝑊) → (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝐷, 𝐸, 𝐹⟩ ↔ (𝐴 = 𝐷𝐵 = 𝐸𝐶 = 𝐹)))
 
Theoremeqvinop 4779* A variable introduction law for ordered pairs. Analogue of Lemma 15 of [Monk2] p. 109. (Contributed by NM, 28-May-1995.)
𝐵 ∈ V    &   𝐶 ∈ V       (𝐴 = ⟨𝐵, 𝐶⟩ ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩))
 
Theoremcopsexg 4780* Substitution of class 𝐴 for ordered pair 𝑥, 𝑦. (Contributed by NM, 27-Dec-1996.) (Revised by Andrew Salmon, 11-Jul-2011.) (Proof shortened by Wolf Lammen, 25-Aug-2019.)
(𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
 
Theoremcopsex2t 4781* Closed theorem form of copsex2g 4782. (Contributed by NM, 17-Feb-2013.)
((∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓)) ∧ (𝐴𝑉𝐵𝑊)) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓))
 
Theoremcopsex2g 4782* Implicit substitution inference for ordered pairs. (Contributed by NM, 28-May-1995.)
((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))       ((𝐴𝑉𝐵𝑊) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓))
 
Theoremcopsex4g 4783* An implicit substitution inference for 2 ordered pairs. (Contributed by NM, 5-Aug-1995.)
(((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) → (𝜑𝜓))       (((𝐴𝑅𝐵𝑆) ∧ (𝐶𝑅𝐷𝑆)) → (∃𝑥𝑦𝑧𝑤((⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑧, 𝑤⟩) ∧ 𝜑) ↔ 𝜓))
 
Theorem0nelop 4784 A property of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.)
¬ ∅ ∈ ⟨𝐴, 𝐵
 
Theoremopeqex 4785 Equivalence of existence implied by equality of ordered pairs. (Contributed by NM, 28-May-2008.)
(⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐶 ∈ V ∧ 𝐷 ∈ V)))
 
Theoremoteqex2 4786 Equivalence of existence implied by equality of ordered triples. (Contributed by NM, 26-Apr-2015.)
(⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝑅, 𝑆⟩, 𝑇⟩ → (𝐶 ∈ V ↔ 𝑇 ∈ V))
 
Theoremoteqex 4787 Equivalence of existence implied by equality of ordered triples. (Contributed by NM, 28-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
(⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝑅, 𝑆⟩, 𝑇⟩ → ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) ↔ (𝑅 ∈ V ∧ 𝑆 ∈ V ∧ 𝑇 ∈ V)))
 
Theoremopcom 4788 An ordered pair commutes iff its members are equal. (Contributed by NM, 28-May-2009.)
𝐴 ∈ V    &   𝐵 ∈ V       (⟨𝐴, 𝐵⟩ = ⟨𝐵, 𝐴⟩ ↔ 𝐴 = 𝐵)
 
Theoremmoop2 4789* "At most one" property of an ordered pair. (Contributed by NM, 11-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
𝐵 ∈ V       ∃*𝑥 𝐴 = ⟨𝐵, 𝑥
 
Theoremopeqsn 4790 Equivalence for an ordered pair equal to a singleton. (Contributed by NM, 3-Jun-2008.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V       (⟨𝐴, 𝐵⟩ = {𝐶} ↔ (𝐴 = 𝐵𝐶 = {𝐴}))
 
Theoremopeqpr 4791 Equivalence for an ordered pair equal to an unordered pair. (Contributed by NM, 3-Jun-2008.) (Avoid depending on this detail.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   𝐷 ∈ V       (⟨𝐴, 𝐵⟩ = {𝐶, 𝐷} ↔ ((𝐶 = {𝐴} ∧ 𝐷 = {𝐴, 𝐵}) ∨ (𝐶 = {𝐴, 𝐵} ∧ 𝐷 = {𝐴})))
 
Theoremmosubopt 4792* "At most one" remains true inside ordered pair quantification. (Contributed by NM, 28-Aug-2007.)
(∀𝑦𝑧∃*𝑥𝜑 → ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑))
 
Theoremmosubop 4793* "At most one" remains true inside ordered pair quantification. (Contributed by NM, 28-May-1995.)
∃*𝑥𝜑       ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)
 
Theoremeuop2 4794* Transfer existential uniqueness to second member of an ordered pair. (Contributed by NM, 10-Apr-2004.)
𝐴 ∈ V       (∃!𝑥𝑦(𝑥 = ⟨𝐴, 𝑦⟩ ∧ 𝜑) ↔ ∃!𝑦𝜑)
 
Theoremeuotd 4795* Prove existential uniqueness for an ordered triple. (Contributed by Mario Carneiro, 20-May-2015.)
(𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐶 ∈ V)    &   (𝜑 → (𝜓 ↔ (𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶)))       (𝜑 → ∃!𝑥𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))
 
Theoremopthwiener 4796 Justification theorem for the ordered pair definition in Norbert Wiener, "A simplification of the logic of relations," Proc. of the Cambridge Philos. Soc., 1914, vol. 17, pp.387-390. It is also shown as a definition in [Enderton] p. 36 and as Exercise 4.8(b) of [Mendelson] p. 230. It is meaningful only for classes that exist as sets (i.e. are not proper classes). See df-op 4035 for other ordered pair definitions. (Contributed by NM, 28-Sep-2003.)
𝐴 ∈ V    &   𝐵 ∈ V       ({{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}} ↔ (𝐴 = 𝐶𝐵 = 𝐷))
 
Theoremuniop 4797 The union of an ordered pair. Theorem 65 of [Suppes] p. 39. (Contributed by NM, 17-Aug-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
𝐴 ∈ V    &   𝐵 ∈ V       𝐴, 𝐵⟩ = {𝐴, 𝐵}
 
Theoremuniopel 4798 Ordered pair membership is inherited by class union. (Contributed by NM, 13-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
𝐴 ∈ V    &   𝐵 ∈ V       (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴, 𝐵⟩ ∈ 𝐶)
 
Theoremotsndisj 4799* The singletons consisting of ordered triples which have distinct third components are disjoint. (Contributed by Alexander van der Vekens, 10-Mar-2018.)
((𝐴𝑋𝐵𝑌) → Disj 𝑐𝑉 {⟨𝐴, 𝐵, 𝑐⟩})
 
Theoremotiunsndisj 4800* The union of singletons consisting of ordered triples which have distinct first and third components are disjoint. (Contributed by Alexander van der Vekens, 10-Mar-2018.)
(𝐵𝑋Disj 𝑎𝑉 𝑐 ∈ (𝑊 ∖ {𝑎}){⟨𝑎, 𝐵, 𝑐⟩})
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