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Theorem List for Metamath Proof Explorer - 4801-4900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremreusv3i 4801* 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 4802* Two ways to express single-valuedness of a class expression 𝐶(𝑦). See reusv1 4792 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 4803* 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 4804* 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 4805* 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 4806* Obsolete proof of ralxfrd 4805 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 4807* 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 4808* 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 4809* 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 4810* Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Variant of ralxfrd 4805. (Contributed by Alexander van der Vekens, 25-Apr-2018.)
((𝜑𝑦𝐶) → 𝐴𝐵)    &   ((𝜑𝑥𝐵) → ∃𝑦𝐶 𝑥 = 𝐴)    &   ((𝜑𝑦𝐶𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐵 𝜓 ↔ ∀𝑦𝐶 𝜒))

Theoremrexxfrd2 4811* Transfer existence from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Variant of rexxfrd 4807. (Contributed by Alexander van der Vekens, 25-Apr-2018.)
((𝜑𝑦𝐶) → 𝐴𝐵)    &   ((𝜑𝑥𝐵) → ∃𝑦𝐶 𝑥 = 𝐴)    &   ((𝜑𝑦𝐶𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐵 𝜓 ↔ ∃𝑦𝐶 𝜒))

Theoremralxfr 4812* 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 4813* Alternate proof of ralxfr 4812 which does not use ralxfrd 4805. (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 4814* 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 4815* 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 4816* 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 4817* 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 4818* 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 4819* Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Use reuhypd 4821 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 4820* Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Use reuhyp 4822 to eliminate the second hypothesis. (Contributed by NM, 14-Nov-2004.)
(𝑦𝐵𝐴𝐵)    &   (𝑥𝐵 → ∃!𝑦𝐵 𝑥 = 𝐴)    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∃!𝑥𝐵 𝜑 ↔ ∃!𝑦𝐵 𝜓)

Theoremreuhypd 4821* A theorem useful for eliminating the restricted existential uniqueness hypotheses in riotaxfrd 6541. (Contributed by NM, 16-Jan-2012.)
((𝜑𝑥𝐶) → 𝐵𝐶)    &   ((𝜑𝑥𝐶𝑦𝐶) → (𝑥 = 𝐴𝑦 = 𝐵))       ((𝜑𝑥𝐶) → ∃!𝑦𝐶 𝑥 = 𝐴)

Theoremreuhyp 4822* A theorem useful for eliminating the restricted existential uniqueness hypotheses in reuxfr 4820. (Contributed by NM, 15-Nov-2004.)
(𝑥𝐶𝐵𝐶)    &   ((𝑥𝐶𝑦𝐶) → (𝑥 = 𝐴𝑦 = 𝐵))       (𝑥𝐶 → ∃!𝑦𝐶 𝑥 = 𝐴)

Theoremnfnid 4823 A setvar variable is not free from itself. The proof relies on dtru 4783, that is, it is not true in a one-element domain. (Contributed by Mario Carneiro, 8-Oct-2016.)
¬ 𝑥𝑥

Theoremnfcvb 4824 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 4825 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 4826* Alternate proof of dtru 4783 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 3273 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 4827* Corollary of dtru 4783. 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 4828. (Contributed by NM, 27-Jun-2002.)
𝑥 = 𝑦       𝑥𝑦

Theoremdtrucor2 4828 The theorem form of the deduction dtrucor 4827 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 4829* 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 4830. ("Bundles" is a term introduced by Raph Levien.) (Contributed by NM, 1-Dec-2006.)
𝑥(𝑥 = 𝑦𝑧𝑥)

Theoremdvdemo2 4830* 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 4829. (Contributed by NM, 1-Dec-2006.)
𝑥(𝑥 = 𝑦𝑧𝑥)

2.3.2  Derive the Axiom of Pairing

Theoremzfpair 4831 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 4832. Instead, use zfpair2 4834 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 4832* 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 4833 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 4833* The Axiom of Pairing of ZF set theory. It was derived as theorem axpr 4832 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 4834 Derive the abbreviated version of the Axiom of Pairing from ax-pr 4833. See zfpair 4831 for its derivation from the other axioms. (Contributed by NM, 14-Nov-2006.)
{𝑥, 𝑦} ∈ V

Theoremsnex 4835 A singleton is a set. Theorem 7.12 of [Quine] p. 51, proved using Extensionality, Separation, Null Set, and Pairing. See also snexALT 4778. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 19-May-2013.) (Proof modification is discouraged.)
{𝐴} ∈ V

Theoremprex 4836 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 4245), so we can dispense with hypotheses requiring them to be sets. (Contributed by NM, 15-Jul-1993.)
{𝐴, 𝐵} ∈ V

TheoremelALT 4837* Alternate proof of el 4773, shorter but requiring more axioms. (Contributed by NM, 4-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑦 𝑥𝑦

TheoremdtruALT2 4838* Alternate proof of dtru 4783 using ax-pr 4833 instead of ax-pow 4769. (Contributed by Mario Carneiro, 31-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
¬ ∀𝑥 𝑥 = 𝑦

Theoremsnelpwi 4839 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 4840 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 4841 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 4842 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 4843* 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 4844 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 4845 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 4846 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 4847 Membership of a power class. Exercise 10 of [Enderton] p. 26. (Contributed by NM, 13-Jan-2007.)
(𝐴𝐵 → 𝒫 𝐴 ∈ 𝒫 𝒫 𝐵)

Theorempwtr 4848 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 4849* An extensionality-like principle defining subclass in terms of subsets. (Contributed by NM, 30-Jun-2004.)
(𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))

Theoremssext 4850* 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 4851* Negation of subclass relationship. Compare nss 3626. (Contributed by NM, 30-Jun-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
𝐴𝐵 ↔ ∃𝑥(𝑥𝐴 ∧ ¬ 𝑥𝐵))

Theorempweqb 4852 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 4853* 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 4854 "At most one" existence implies a class abstraction exists. (Contributed by NM, 30-Dec-1996.)
(∃*𝑥𝜑 → {𝑥𝜑} ∈ V)

Theoremrmorabex 4855 Restricted "at most one" existence implies a restricted class abstraction exists. (Contributed by NM, 17-Jun-2017.)
(∃*𝑥𝐴 𝜑 → {𝑥𝐴𝜑} ∈ V)

Theoremeuabex 4856 The abstraction of a wff with existential uniqueness exists. (Contributed by NM, 25-Nov-1994.)
(∃!𝑥𝜑 → {𝑥𝜑} ∈ V)

Theoremnnullss 4857* A nonempty class (even if proper) has a nonempty subset. (Contributed by NM, 23-Aug-2003.)
(𝐴 ≠ ∅ → ∃𝑥(𝑥𝐴𝑥 ≠ ∅))

Theoremexss 4858* 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 4859 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 4860 An ordered triple of classes is a set. (Contributed by NM, 3-Apr-2015.)
𝐴, 𝐵, 𝐶⟩ ∈ V

Theoremelopg 4861 Characterization of the elements of an ordered pair. Closed form of elop 4862. (Contributed by BJ, 22-Jun-2019.) (Avoid depending on this detail.)
((𝐴𝑉𝐵𝑊) → (𝐶 ∈ ⟨𝐴, 𝐵⟩ ↔ (𝐶 = {𝐴} ∨ 𝐶 = {𝐴, 𝐵})))

Theoremelop 4862 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 4863 Obsolete version of elop 4862, 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 4864 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 4865 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 4866 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 4867 Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (See op2ndb 5537 to extract the second member, op1sta 5535 for an alternate version, and op1st 7067 for the preferred version.) (Contributed by NM, 25-Nov-2003.)
𝐴 ∈ V    &   𝐵 ∈ V        𝐴, 𝐵⟩ = 𝐴

2.3.3  Ordered pair theorem

Theoremopnz 4868 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 4869 An ordered pair is nonempty if the arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.)
𝐴 ∈ V    &   𝐵 ∈ V       𝐴, 𝐵⟩ ≠ ∅

Theoremopth1 4870 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 4871 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 4872 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 4873 Equality of the first members of equal ordered pairs. Closed form of opth1 4870. (Contributed by AV, 14-Oct-2018.)
((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐴 = 𝐶))

Theoremopthg2 4874 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 4875 Ordered pair theorem. (Contributed by NM, 21-Sep-2014.)
𝐶 ∈ V    &   𝐷 ∈ V       (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷))

Theoremopthneg 4876 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 4877 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 4878 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 4879 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 4880 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 4881* 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 4882* 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 4883* Closed theorem form of copsex2g 4884. (Contributed by NM, 17-Feb-2013.)
((∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓)) ∧ (𝐴𝑉𝐵𝑊)) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓))

Theoremcopsex2g 4884* 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 4885* 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 4886 A property of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.)
¬ ∅ ∈ ⟨𝐴, 𝐵

Theoremopeqex 4887 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 4888 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 4889 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 4890 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 4891* "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 4892 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 4893 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})))

Theoremsnopeqop 4894 Equivalence for an ordered pair equal to a singleton of an ordered pair. (Contributed by AV, 18-Sep-2020.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   𝐷 ∈ V       ({⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐵𝐶 = 𝐷𝐶 = {𝐴}))

Theorempropeqop 4895 Equivalence for an ordered pair equal to a pair of ordered pairs. (Contributed by AV, 18-Sep-2020.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   𝐷 ∈ V    &   𝐸 ∈ V    &   𝐹 ∈ V       ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ⟨𝐸, 𝐹⟩ ↔ ((𝐴 = 𝐶𝐸 = {𝐴}) ∧ ((𝐴 = 𝐵𝐹 = {𝐴, 𝐷}) ∨ (𝐴 = 𝐷𝐹 = {𝐴, 𝐵}))))

Theorempropssopi 4896 If a pair of ordered pairs is a subset of an ordered pair, their first components are equal. (Contributed by AV, 20-Sep-2020.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   𝐷 ∈ V    &   𝐸 ∈ V    &   𝐹 ∈ V       ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ⊆ ⟨𝐸, 𝐹⟩ → 𝐴 = 𝐶)

Theoremmosubopt 4897* "At most one" remains true inside ordered pair quantification. (Contributed by NM, 28-Aug-2007.)
(∀𝑦𝑧∃*𝑥𝜑 → ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑))

Theoremmosubop 4898* "At most one" remains true inside ordered pair quantification. (Contributed by NM, 28-May-1995.)
∃*𝑥𝜑       ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)

Theoremeuop2 4899* 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 4900* 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))

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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 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 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