HomeHome Metamath Proof Explorer
Theorem List (p. 51 of 449)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-28623)
  Hilbert Space Explorer  Hilbert Space Explorer
(28624-30146)
  Users' Mathboxes  Users' Mathboxes
(30147-44804)
 

Theorem List for Metamath Proof Explorer - 5001-5100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsymdifid 5001 The symmetric difference of a class with itself is the empty class. (Contributed by Scott Fenton, 25-Apr-2012.)
(𝐴𝐴) = ∅
 
Theoremiinxsng 5002* A singleton index picks out an instance of an indexed intersection's argument. (Contributed by NM, 15-Jan-2012.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
(𝑥 = 𝐴𝐵 = 𝐶)       (𝐴𝑉 𝑥 ∈ {𝐴}𝐵 = 𝐶)
 
Theoremiinxprg 5003* Indexed intersection with an unordered pair index. (Contributed by NM, 25-Jan-2012.)
(𝑥 = 𝐴𝐶 = 𝐷)    &   (𝑥 = 𝐵𝐶 = 𝐸)       ((𝐴𝑉𝐵𝑊) → 𝑥 ∈ {𝐴, 𝐵}𝐶 = (𝐷𝐸))
 
Theoremiunxsng 5004* A singleton index picks out an instance of an indexed union's argument. (Contributed by Mario Carneiro, 25-Jun-2016.)
(𝑥 = 𝐴𝐵 = 𝐶)       (𝐴𝑉 𝑥 ∈ {𝐴}𝐵 = 𝐶)
 
Theoremiunxsn 5005* A singleton index picks out an instance of an indexed union's argument. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Mario Carneiro, 25-Jun-2016.)
𝐴 ∈ V    &   (𝑥 = 𝐴𝐵 = 𝐶)        𝑥 ∈ {𝐴}𝐵 = 𝐶
 
Theoremiunxsngf 5006* A singleton index picks out an instance of an indexed union's argument. (Contributed by Mario Carneiro, 25-Jun-2016.) (Revised by Thierry Arnoux, 2-May-2020.) Avoid ax-13 2383. (Revised by Gino Giotto, 19-May-2023.)
𝑥𝐶    &   (𝑥 = 𝐴𝐵 = 𝐶)       (𝐴𝑉 𝑥 ∈ {𝐴}𝐵 = 𝐶)
 
Theoremiunun 5007 Separate a union in an indexed union. (Contributed by NM, 27-Dec-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
𝑥𝐴 (𝐵𝐶) = ( 𝑥𝐴 𝐵 𝑥𝐴 𝐶)
 
Theoremiunxun 5008 Separate a union in the index of an indexed union. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
𝑥 ∈ (𝐴𝐵)𝐶 = ( 𝑥𝐴 𝐶 𝑥𝐵 𝐶)
 
Theoremiunxdif3 5009* An indexed union where some terms are the empty set. See iunxdif2 4969. (Contributed by Thierry Arnoux, 4-May-2020.)
𝑥𝐸       (∀𝑥𝐸 𝐵 = ∅ → 𝑥 ∈ (𝐴𝐸)𝐵 = 𝑥𝐴 𝐵)
 
Theoremiunxprg 5010* A pair index picks out two instances of an indexed union's argument. (Contributed by Alexander van der Vekens, 2-Feb-2018.)
(𝑥 = 𝐴𝐶 = 𝐷)    &   (𝑥 = 𝐵𝐶 = 𝐸)       ((𝐴𝑉𝐵𝑊) → 𝑥 ∈ {𝐴, 𝐵}𝐶 = (𝐷𝐸))
 
Theoremiunxiun 5011* Separate an indexed union in the index of an indexed union. (Contributed by Mario Carneiro, 5-Dec-2016.)
𝑥 𝑦𝐴 𝐵𝐶 = 𝑦𝐴 𝑥𝐵 𝐶
 
Theoremiinuni 5012* A relationship involving union and indexed intersection. Exercise 23 of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
(𝐴 𝐵) = 𝑥𝐵 (𝐴𝑥)
 
Theoremiununi 5013* A relationship involving union and indexed union. Exercise 25 of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
((𝐵 = ∅ → 𝐴 = ∅) ↔ (𝐴 𝐵) = 𝑥𝐵 (𝐴𝑥))
 
Theoremsspwuni 5014 Subclass relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
(𝐴 ⊆ 𝒫 𝐵 𝐴𝐵)
 
Theorempwssb 5015* Two ways to express a collection of subclasses. (Contributed by NM, 19-Jul-2006.)
(𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
 
Theoremelpwpw 5016 Characterization of the elements of a double power class: they are exactly the sets whose union is included in that class. (Contributed by BJ, 29-Apr-2021.)
(𝐴 ∈ 𝒫 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴𝐵))
 
Theorempwpwab 5017* The double power class written as a class abstraction: the class of sets whose union is included in the given class. (Contributed by BJ, 29-Apr-2021.)
𝒫 𝒫 𝐴 = {𝑥 𝑥𝐴}
 
Theorempwpwssunieq 5018* The class of sets whose union is equal to a given class is included in the double power class of that class. (Contributed by BJ, 29-Apr-2021.)
{𝑥 𝑥 = 𝐴} ⊆ 𝒫 𝒫 𝐴
 
Theoremelpwuni 5019 Relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
(𝐵𝐴 → (𝐴 ⊆ 𝒫 𝐵 𝐴 = 𝐵))
 
Theoremiinpw 5020* The power class of an intersection in terms of indexed intersection. Exercise 24(a) of [Enderton] p. 33. (Contributed by NM, 29-Nov-2003.)
𝒫 𝐴 = 𝑥𝐴 𝒫 𝑥
 
Theoremiunpwss 5021* Inclusion of an indexed union of a power class in the power class of the union of its index. Part of Exercise 24(b) of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.)
𝑥𝐴 𝒫 𝑥 ⊆ 𝒫 𝐴
 
Theoremrintn0 5022 Relative intersection of a nonempty set. (Contributed by Stefan O'Rear, 3-Apr-2015.) (Revised by Mario Carneiro, 5-Jun-2015.)
((𝑋 ⊆ 𝒫 𝐴𝑋 ≠ ∅) → (𝐴 𝑋) = 𝑋)
 
2.1.22  Disjointness
 
Syntaxwdisj 5023 Extend wff notation to include the statement that a family of classes 𝐵(𝑥), for 𝑥𝐴, is a disjoint family.
wff Disj 𝑥𝐴 𝐵
 
Definitiondf-disj 5024* A collection of classes 𝐵(𝑥) is disjoint when for each element 𝑦, it is in 𝐵(𝑥) for at most one 𝑥. (Contributed by Mario Carneiro, 14-Nov-2016.) (Revised by NM, 16-Jun-2017.)
(Disj 𝑥𝐴 𝐵 ↔ ∀𝑦∃*𝑥𝐴 𝑦𝐵)
 
Theoremdfdisj2 5025* Alternate definition for disjoint classes. (Contributed by NM, 17-Jun-2017.)
(Disj 𝑥𝐴 𝐵 ↔ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝐵))
 
Theoremdisjss2 5026 If each element of a collection is contained in a disjoint collection, the original collection is also disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
(∀𝑥𝐴 𝐵𝐶 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐴 𝐵))
 
Theoremdisjeq2 5027 Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
(∀𝑥𝐴 𝐵 = 𝐶 → (Disj 𝑥𝐴 𝐵Disj 𝑥𝐴 𝐶))
 
Theoremdisjeq2dv 5028* Equality deduction for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
((𝜑𝑥𝐴) → 𝐵 = 𝐶)       (𝜑 → (Disj 𝑥𝐴 𝐵Disj 𝑥𝐴 𝐶))
 
Theoremdisjss1 5029* A subset of a disjoint collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
(𝐴𝐵 → (Disj 𝑥𝐵 𝐶Disj 𝑥𝐴 𝐶))
 
Theoremdisjeq1 5030* Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
(𝐴 = 𝐵 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
 
Theoremdisjeq1d 5031* Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
(𝜑𝐴 = 𝐵)       (𝜑 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
 
Theoremdisjeq12d 5032* Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐷))
 
Theoremcbvdisj 5033* Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
𝑦𝐵    &   𝑥𝐶    &   (𝑥 = 𝑦𝐵 = 𝐶)       (Disj 𝑥𝐴 𝐵Disj 𝑦𝐴 𝐶)
 
Theoremcbvdisjv 5034* Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 11-Dec-2016.)
(𝑥 = 𝑦𝐵 = 𝐶)       (Disj 𝑥𝐴 𝐵Disj 𝑦𝐴 𝐶)
 
Theoremnfdisjw 5035* Version of nfdisj 5036 with a disjoint variable condition, which does not require ax-13 2383. (Contributed by Gino Giotto, 26-Jan-2024.)
𝑦𝐴    &   𝑦𝐵       𝑦Disj 𝑥𝐴 𝐵
 
Theoremnfdisj 5036 Bound-variable hypothesis builder for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
𝑦𝐴    &   𝑦𝐵       𝑦Disj 𝑥𝐴 𝐵
 
Theoremnfdisj1 5037 Bound-variable hypothesis builder for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
𝑥Disj 𝑥𝐴 𝐵
 
Theoremdisjor 5038* Two ways to say that a collection 𝐵(𝑖) for 𝑖𝐴 is disjoint. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by Mario Carneiro, 14-Nov-2016.)
(𝑖 = 𝑗𝐵 = 𝐶)       (Disj 𝑖𝐴 𝐵 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ (𝐵𝐶) = ∅))
 
Theoremdisjors 5039* Two ways to say that a collection 𝐵(𝑖) for 𝑖𝐴 is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
(Disj 𝑥𝐴 𝐵 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅))
 
Theoremdisji2 5040* Property of a disjoint collection: if 𝐵(𝑋) = 𝐶 and 𝐵(𝑌) = 𝐷, and 𝑋𝑌, then 𝐶 and 𝐷 are disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
(𝑥 = 𝑋𝐵 = 𝐶)    &   (𝑥 = 𝑌𝐵 = 𝐷)       ((Disj 𝑥𝐴 𝐵 ∧ (𝑋𝐴𝑌𝐴) ∧ 𝑋𝑌) → (𝐶𝐷) = ∅)
 
Theoremdisji 5041* Property of a disjoint collection: if 𝐵(𝑋) = 𝐶 and 𝐵(𝑌) = 𝐷 have a common element 𝑍, then 𝑋 = 𝑌. (Contributed by Mario Carneiro, 14-Nov-2016.)
(𝑥 = 𝑋𝐵 = 𝐶)    &   (𝑥 = 𝑌𝐵 = 𝐷)       ((Disj 𝑥𝐴 𝐵 ∧ (𝑋𝐴𝑌𝐴) ∧ (𝑍𝐶𝑍𝐷)) → 𝑋 = 𝑌)
 
Theoreminvdisj 5042* If there is a function 𝐶(𝑦) such that 𝐶(𝑦) = 𝑥 for all 𝑦𝐵(𝑥), then the sets 𝐵(𝑥) for distinct 𝑥𝐴 are disjoint. (Contributed by Mario Carneiro, 10-Dec-2016.)
(∀𝑥𝐴𝑦𝐵 𝐶 = 𝑥Disj 𝑥𝐴 𝐵)
 
Theoreminvdisjrabw 5043* Version of invdisjrab 5044 with a disjoint variable condition, which does not require ax-13 2383. (Contributed by Gino Giotto, 26-Jan-2024.)
Disj 𝑦𝐴 {𝑥𝐵𝐶 = 𝑦}
 
Theoreminvdisjrab 5044* The restricted class abstractions {𝑥𝐵𝐶 = 𝑦} for distinct 𝑦𝐴 are disjoint. (Contributed by AV, 6-May-2020.)
Disj 𝑦𝐴 {𝑥𝐵𝐶 = 𝑦}
 
Theoremdisjiun 5045* A disjoint collection yields disjoint indexed unions for disjoint index sets. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by Mario Carneiro, 14-Nov-2016.)
((Disj 𝑥𝐴 𝐵 ∧ (𝐶𝐴𝐷𝐴 ∧ (𝐶𝐷) = ∅)) → ( 𝑥𝐶 𝐵 𝑥𝐷 𝐵) = ∅)
 
Theoremdisjord 5046* Conditions for a collection of sets 𝐴(𝑎) for 𝑎𝑉 to be disjoint. (Contributed by AV, 9-Jan-2022.)
(𝑎 = 𝑏𝐴 = 𝐵)    &   ((𝜑𝑥𝐴𝑥𝐵) → 𝑎 = 𝑏)       (𝜑Disj 𝑎𝑉 𝐴)
 
Theoremdisjiunb 5047* Two ways to say that a collection of index unions 𝐶(𝑖, 𝑥) for 𝑖𝐴 and 𝑥𝐵 is disjoint. (Contributed by AV, 9-Jan-2022.)
(𝑖 = 𝑗𝐵 = 𝐷)    &   (𝑖 = 𝑗𝐶 = 𝐸)       (Disj 𝑖𝐴 𝑥𝐵 𝐶 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ ( 𝑥𝐵 𝐶 𝑥𝐷 𝐸) = ∅))
 
Theoremdisjiund 5048* Conditions for a collection of index unions of sets 𝐴(𝑎, 𝑏) for 𝑎𝑉 and 𝑏𝑊 to be disjoint. (Contributed by AV, 9-Jan-2022.)
(𝑎 = 𝑐𝐴 = 𝐶)    &   (𝑏 = 𝑑𝐶 = 𝐷)    &   (𝑎 = 𝑐𝑊 = 𝑋)    &   ((𝜑𝑥𝐴𝑥𝐷) → 𝑎 = 𝑐)       (𝜑Disj 𝑎𝑉 𝑏𝑊 𝐴)
 
Theoremsndisj 5049 Any collection of singletons is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj 𝑥𝐴 {𝑥}
 
Theorem0disj 5050 Any collection of empty sets is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj 𝑥𝐴
 
Theoremdisjxsn 5051* A singleton collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj 𝑥 ∈ {𝐴}𝐵
 
Theoremdisjx0 5052 An empty collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj 𝑥 ∈ ∅ 𝐵
 
Theoremdisjprgw 5053* Version of disjprg 5054 with a disjoint variable condition, which does not require ax-13 2383. (Contributed by Gino Giotto, 26-Jan-2024.)
(𝑥 = 𝐴𝐶 = 𝐷)    &   (𝑥 = 𝐵𝐶 = 𝐸)       ((𝐴𝑉𝐵𝑉𝐴𝐵) → (Disj 𝑥 ∈ {𝐴, 𝐵}𝐶 ↔ (𝐷𝐸) = ∅))
 
Theoremdisjprg 5054* A pair collection is disjoint iff the two sets in the family have empty intersection. (Contributed by Mario Carneiro, 14-Nov-2016.)
(𝑥 = 𝐴𝐶 = 𝐷)    &   (𝑥 = 𝐵𝐶 = 𝐸)       ((𝐴𝑉𝐵𝑉𝐴𝐵) → (Disj 𝑥 ∈ {𝐴, 𝐵}𝐶 ↔ (𝐷𝐸) = ∅))
 
Theoremdisjxiun 5055* An indexed union of a disjoint collection of disjoint collections is disjoint if each component is disjoint, and the disjoint unions in the collection are also disjoint. Note that 𝐵(𝑦) and 𝐶(𝑥) may have the displayed free variables. (Contributed by Mario Carneiro, 14-Nov-2016.) (Proof shortened by JJ, 27-May-2021.)
(Disj 𝑦𝐴 𝐵 → (Disj 𝑥 𝑦𝐴 𝐵𝐶 ↔ (∀𝑦𝐴 Disj 𝑥𝐵 𝐶Disj 𝑦𝐴 𝑥𝐵 𝐶)))
 
Theoremdisjxun 5056* The union of two disjoint collections. (Contributed by Mario Carneiro, 14-Nov-2016.)
(𝑥 = 𝑦𝐶 = 𝐷)       ((𝐴𝐵) = ∅ → (Disj 𝑥 ∈ (𝐴𝐵)𝐶 ↔ (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶 ∧ ∀𝑥𝐴𝑦𝐵 (𝐶𝐷) = ∅)))
 
Theoremdisjss3 5057* Expand a disjoint collection with any number of empty sets. (Contributed by Mario Carneiro, 15-Nov-2016.)
((𝐴𝐵 ∧ ∀𝑥 ∈ (𝐵𝐴)𝐶 = ∅) → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
 
2.1.23  Binary relations
 
Syntaxwbr 5058 Extend wff notation to include the general binary relation predicate. Note that the syntax is simply three class symbols in a row. Since binary relations are the only possible wff expressions consisting of three class expressions in a row, the syntax is unambiguous. (For an example of how syntax could become ambiguous if we are not careful, see the comment in cneg 10860.)
wff 𝐴𝑅𝐵
 
Definitiondf-br 5059 Define a general binary relation. Note that the syntax is simply three class symbols in a row. Definition 6.18 of [TakeutiZaring] p. 29 generalized to arbitrary classes. Class 𝑅 often denotes a relation such as "< " that compares two classes 𝐴 and 𝐵, which might be numbers such as 1 and 2 (see df-ltxr 10669 for the specific definition of <). As a wff, relations are true or false. For example, (𝑅 = {⟨2, 6⟩, ⟨3, 9⟩} → 3𝑅9) (ex-br 28138). Often class 𝑅 meets the Rel criteria to be defined in df-rel 5556, and in particular 𝑅 may be a function (see df-fun 6351). This definition of relations is well-defined, although not very meaningful, when classes 𝐴 and/or 𝐵 are proper classes (i.e. are not sets). On the other hand, we often find uses for this definition when 𝑅 is a proper class (see for example iprc 7606). (Contributed by NM, 31-Dec-1993.)
(𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
 
Theorembreq 5060 Equality theorem for binary relations. (Contributed by NM, 4-Jun-1995.)
(𝑅 = 𝑆 → (𝐴𝑅𝐵𝐴𝑆𝐵))
 
Theorembreq1 5061 Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)
(𝐴 = 𝐵 → (𝐴𝑅𝐶𝐵𝑅𝐶))
 
Theorembreq2 5062 Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)
(𝐴 = 𝐵 → (𝐶𝑅𝐴𝐶𝑅𝐵))
 
Theorembreq12 5063 Equality theorem for a binary relation. (Contributed by NM, 8-Feb-1996.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝑅𝐶𝐵𝑅𝐷))
 
Theorembreqi 5064 Equality inference for binary relations. (Contributed by NM, 19-Feb-2005.)
𝑅 = 𝑆       (𝐴𝑅𝐵𝐴𝑆𝐵)
 
Theorembreq1i 5065 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)
𝐴 = 𝐵       (𝐴𝑅𝐶𝐵𝑅𝐶)
 
Theorembreq2i 5066 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)
𝐴 = 𝐵       (𝐶𝑅𝐴𝐶𝑅𝐵)
 
Theorembreq12i 5067 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)
𝐴 = 𝐵    &   𝐶 = 𝐷       (𝐴𝑅𝐶𝐵𝑅𝐷)
 
Theorembreq1d 5068 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝑅𝐶𝐵𝑅𝐶))
 
Theorembreqd 5069 Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴𝐷𝐶𝐵𝐷))
 
Theorembreq2d 5070 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝑅𝐴𝐶𝑅𝐵))
 
Theorembreq12d 5071 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝑅𝐶𝐵𝑅𝐷))
 
Theorembreq123d 5072 Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝑅 = 𝑆)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝑅𝐶𝐵𝑆𝐷))
 
Theorembreqdi 5073 Equality deduction for a binary relation. (Contributed by Thierry Arnoux, 5-Oct-2020.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶𝐴𝐷)       (𝜑𝐶𝐵𝐷)
 
Theorembreqan12d 5074 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
(𝜑𝐴 = 𝐵)    &   (𝜓𝐶 = 𝐷)       ((𝜑𝜓) → (𝐴𝑅𝐶𝐵𝑅𝐷))
 
Theorembreqan12rd 5075 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
(𝜑𝐴 = 𝐵)    &   (𝜓𝐶 = 𝐷)       ((𝜓𝜑) → (𝐴𝑅𝐶𝐵𝑅𝐷))
 
Theoremeqnbrtrd 5076 Substitution of equal classes into the negation of a binary relation. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝜑𝐴 = 𝐵)    &   (𝜑 → ¬ 𝐵𝑅𝐶)       (𝜑 → ¬ 𝐴𝑅𝐶)
 
Theoremnbrne1 5077 Two classes are different if they don't have the same relationship to a third class. (Contributed by NM, 3-Jun-2012.)
((𝐴𝑅𝐵 ∧ ¬ 𝐴𝑅𝐶) → 𝐵𝐶)
 
Theoremnbrne2 5078 Two classes are different if they don't have the same relationship to a third class. (Contributed by NM, 3-Jun-2012.)
((𝐴𝑅𝐶 ∧ ¬ 𝐵𝑅𝐶) → 𝐴𝐵)
 
Theoremeqbrtri 5079 Substitution of equal classes into a binary relation. (Contributed by NM, 1-Aug-1999.)
𝐴 = 𝐵    &   𝐵𝑅𝐶       𝐴𝑅𝐶
 
Theoremeqbrtrd 5080 Substitution of equal classes into a binary relation. (Contributed by NM, 8-Oct-1999.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐵𝑅𝐶)       (𝜑𝐴𝑅𝐶)
 
Theoremeqbrtrri 5081 Substitution of equal classes into a binary relation. (Contributed by NM, 1-Aug-1999.)
𝐴 = 𝐵    &   𝐴𝑅𝐶       𝐵𝑅𝐶
 
Theoremeqbrtrrd 5082 Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐴𝑅𝐶)       (𝜑𝐵𝑅𝐶)
 
Theorembreqtri 5083 Substitution of equal classes into a binary relation. (Contributed by NM, 1-Aug-1999.)
𝐴𝑅𝐵    &   𝐵 = 𝐶       𝐴𝑅𝐶
 
Theorembreqtrd 5084 Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
(𝜑𝐴𝑅𝐵)    &   (𝜑𝐵 = 𝐶)       (𝜑𝐴𝑅𝐶)
 
Theorembreqtrri 5085 Substitution of equal classes into a binary relation. (Contributed by NM, 1-Aug-1999.)
𝐴𝑅𝐵    &   𝐶 = 𝐵       𝐴𝑅𝐶
 
Theorembreqtrrd 5086 Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
(𝜑𝐴𝑅𝐵)    &   (𝜑𝐶 = 𝐵)       (𝜑𝐴𝑅𝐶)
 
Theorem3brtr3i 5087 Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)
𝐴𝑅𝐵    &   𝐴 = 𝐶    &   𝐵 = 𝐷       𝐶𝑅𝐷
 
Theorem3brtr4i 5088 Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)
𝐴𝑅𝐵    &   𝐶 = 𝐴    &   𝐷 = 𝐵       𝐶𝑅𝐷
 
Theorem3brtr3d 5089 Substitution of equality into both sides of a binary relation. (Contributed by NM, 18-Oct-1999.)
(𝜑𝐴𝑅𝐵)    &   (𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑𝐶𝑅𝐷)
 
Theorem3brtr4d 5090 Substitution of equality into both sides of a binary relation. (Contributed by NM, 21-Feb-2005.)
(𝜑𝐴𝑅𝐵)    &   (𝜑𝐶 = 𝐴)    &   (𝜑𝐷 = 𝐵)       (𝜑𝐶𝑅𝐷)
 
Theorem3brtr3g 5091 Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)
(𝜑𝐴𝑅𝐵)    &   𝐴 = 𝐶    &   𝐵 = 𝐷       (𝜑𝐶𝑅𝐷)
 
Theorem3brtr4g 5092 Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)
(𝜑𝐴𝑅𝐵)    &   𝐶 = 𝐴    &   𝐷 = 𝐵       (𝜑𝐶𝑅𝐷)
 
Theoremeqbrtrid 5093 A chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
𝐴 = 𝐵    &   (𝜑𝐵𝑅𝐶)       (𝜑𝐴𝑅𝐶)
 
Theoremeqbrtrrid 5094 A chained equality inference for a binary relation. (Contributed by NM, 17-Sep-2004.)
𝐵 = 𝐴    &   (𝜑𝐵𝑅𝐶)       (𝜑𝐴𝑅𝐶)
 
Theorembreqtrid 5095 A chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
𝐴𝑅𝐵    &   (𝜑𝐵 = 𝐶)       (𝜑𝐴𝑅𝐶)
 
Theorembreqtrrid 5096 A chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)
𝐴𝑅𝐵    &   (𝜑𝐶 = 𝐵)       (𝜑𝐴𝑅𝐶)
 
Theoremeqbrtrdi 5097 A chained equality inference for a binary relation. (Contributed by NM, 12-Oct-1999.)
(𝜑𝐴 = 𝐵)    &   𝐵𝑅𝐶       (𝜑𝐴𝑅𝐶)
 
Theoremeqbrtrrdi 5098 A chained equality inference for a binary relation. (Contributed by NM, 4-Jan-2006.)
(𝜑𝐵 = 𝐴)    &   𝐵𝑅𝐶       (𝜑𝐴𝑅𝐶)
 
Theorembreqtrdi 5099 A chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
(𝜑𝐴𝑅𝐵)    &   𝐵 = 𝐶       (𝜑𝐴𝑅𝐶)
 
Theorembreqtrrdi 5100 A chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)
(𝜑𝐴𝑅𝐵)    &   𝐶 = 𝐵       (𝜑𝐴𝑅𝐶)
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 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 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 44801-44804
  Copyright terms: Public domain < Previous  Next >