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Theorem List for Metamath Proof Explorer - 31001-31100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembrsset 31001 For sets, the SSet binary relationship is equivalent to the subset relationship. (Contributed by Scott Fenton, 31-Mar-2012.)
𝐵 ∈ V       (𝐴 SSet 𝐵𝐴𝐵)
 
Theoremidsset 31002 I is equal to SSet and its converse. (Contributed by Scott Fenton, 31-Mar-2012.)
I = ( SSet SSet )
 
Theoremeltrans 31003 Membership in the class of all transitive sets. (Contributed by Scott Fenton, 31-Mar-2012.)
𝐴 ∈ V       (𝐴 Trans ↔ Tr 𝐴)
 
Theoremdfon3 31004 A quantifier-free definition of On. (Contributed by Scott Fenton, 5-Apr-2012.)
On = (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))
 
Theoremdfon4 31005 Another quantifier-free definition of On. (Contributed by Scott Fenton, 4-May-2014.)
On = (V ∖ (( SSet ∖ ( I ∪ E )) “ Trans ))
 
Theorembrtxpsd 31006* Expansion of a common form used in quantifier-free definitions. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
𝐴 ∈ V    &   𝐵 ∈ V       𝐴ran ((V ⊗ E ) △ (𝑅 ⊗ V))𝐵 ↔ ∀𝑥(𝑥𝐵𝑥𝑅𝐴))
 
Theorembrtxpsd2 31007* Another common abbreviation for quantifier-free definitions. (Contributed by Scott Fenton, 21-Apr-2014.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝑅 = (𝐶 ∖ ran ((V ⊗ E ) △ (𝑆 ⊗ V)))    &   𝐴𝐶𝐵       (𝐴𝑅𝐵 ↔ ∀𝑥(𝑥𝐵𝑥𝑆𝐴))
 
Theorembrtxpsd3 31008* A third common abbreviation for quantifier-free definitions. (Contributed by Scott Fenton, 3-May-2014.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝑅 = (𝐶 ∖ ran ((V ⊗ E ) △ (𝑆 ⊗ V)))    &   𝐴𝐶𝐵    &   (𝑥𝑋𝑥𝑆𝐴)       (𝐴𝑅𝐵𝐵 = 𝑋)
 
Theoremrelbigcup 31009 The Bigcup relationship is a relationship. (Contributed by Scott Fenton, 11-Apr-2012.)
Rel Bigcup
 
Theorembrbigcup 31010 Binary relationship over Bigcup . (Contributed by Scott Fenton, 11-Apr-2012.)
𝐵 ∈ V       (𝐴 Bigcup 𝐵 𝐴 = 𝐵)
 
Theoremdfbigcup2 31011 Bigcup using maps-to notation. (Contributed by Scott Fenton, 16-Apr-2012.)
Bigcup = (𝑥 ∈ V ↦ 𝑥)
 
Theoremfobigcup 31012 Bigcup maps the universe onto itself. (Contributed by Scott Fenton, 16-Apr-2012.)
Bigcup :V–onto→V
 
Theoremfnbigcup 31013 Bigcup is a function over the universal class. (Contributed by Scott Fenton, 11-Apr-2012.)
Bigcup Fn V
 
Theoremfvbigcup 31014 For sets, Bigcup yields union. (Contributed by Scott Fenton, 11-Apr-2012.)
𝐴 ∈ V       ( Bigcup 𝐴) = 𝐴
 
Theoremelfix 31015 Membership in the fixpoints of a class. (Contributed by Scott Fenton, 11-Apr-2012.)
𝐴 ∈ V       (𝐴 Fix 𝑅𝐴𝑅𝐴)
 
Theoremelfix2 31016 Alternative membership in the fixpoint of a class. (Contributed by Scott Fenton, 11-Apr-2012.)
Rel 𝑅       (𝐴 Fix 𝑅𝐴𝑅𝐴)
 
Theoremdffix2 31017 The fixpoints of a class in terms of its range. (Contributed by Scott Fenton, 16-Apr-2012.)
Fix 𝐴 = ran (𝐴 ∩ I )
 
Theoremfixssdm 31018 The fixpoints of a class are a subset of its domain. (Contributed by Scott Fenton, 16-Apr-2012.)
Fix 𝐴 ⊆ dom 𝐴
 
Theoremfixssrn 31019 The fixpoints of a class are a subset of its range. (Contributed by Scott Fenton, 16-Apr-2012.)
Fix 𝐴 ⊆ ran 𝐴
 
Theoremfixcnv 31020 The fixpoints of a class are the same as those of its converse. (Contributed by Scott Fenton, 16-Apr-2012.)
Fix 𝐴 = Fix 𝐴
 
Theoremfixun 31021 The fixpoint operator distributes over union. (Contributed by Scott Fenton, 16-Apr-2012.)
Fix (𝐴𝐵) = ( Fix 𝐴 Fix 𝐵)
 
Theoremellimits 31022 Membership in the class of all limit ordinals. (Contributed by Scott Fenton, 11-Apr-2012.)
𝐴 ∈ V       (𝐴 Limits ↔ Lim 𝐴)
 
Theoremlimitssson 31023 The class of all limit ordinals is a subclass of the class of all ordinals. (Contributed by Scott Fenton, 11-Apr-2012.)
Limits ⊆ On
 
Theoremdfom5b 31024 A quantifier-free definition of ω that does not depend on ax-inf 8294. (Note: label was changed from dfom5 8306 to dfom5b 31024 to prevent naming conflict. NM, 12-Feb-2013.) (Contributed by Scott Fenton, 11-Apr-2012.)
ω = (On ∩ Limits )
 
Theoremsscoid 31025 A condition for subset and composition with identity. (Contributed by Scott Fenton, 13-Apr-2018.)
(𝐴 ⊆ ( I ∘ 𝐵) ↔ (Rel 𝐴𝐴𝐵))
 
Theoremdffun10 31026 Another potential definition of functionhood. Based on statements in http://people.math.gatech.edu/~belinfan/research/autoreas/otter/sum/fs/. (Contributed by Scott Fenton, 30-Aug-2017.)
(Fun 𝐹𝐹 ⊆ ( I ∘ (V ∖ ((V ∖ I ) ∘ 𝐹))))
 
Theoremelfuns 31027 Membership in the class of all functions. (Contributed by Scott Fenton, 18-Feb-2013.)
𝐹 ∈ V       (𝐹 Funs ↔ Fun 𝐹)
 
Theoremelfunsg 31028 Closed form of elfuns 31027. (Contributed by Scott Fenton, 2-May-2014.)
(𝐹𝑉 → (𝐹 Funs ↔ Fun 𝐹))
 
Theorembrsingle 31029 The binary relationship form of the singleton function. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴Singleton𝐵𝐵 = {𝐴})
 
Theoremelsingles 31030* Membership in the class of all singletons. (Contributed by Scott Fenton, 19-Feb-2013.)
(𝐴 Singletons ↔ ∃𝑥 𝐴 = {𝑥})
 
Theoremfnsingle 31031 The singleton relationship is a function over the universe. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Singleton Fn V
 
Theoremfvsingle 31032 The value of the singleton function. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Revised by Scott Fenton, 13-Apr-2018.)
(Singleton‘𝐴) = {𝐴}
 
Theoremdfsingles2 31033* Alternate definition of the class of all singletons. (Contributed by Scott Fenton, 20-Nov-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Singletons = {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}}
 
Theoremsnelsingles 31034 A singleton is a member of the class of all singletons. (Contributed by Scott Fenton, 19-Feb-2013.)
𝐴 ∈ V       {𝐴} ∈ Singletons
 
Theoremdfiota3 31035 A definiton of iota using minimal quantifiers. (Contributed by Scott Fenton, 19-Feb-2013.)
(℩𝑥𝜑) = ({{𝑥𝜑}} ∩ Singletons )
 
Theoremdffv5 31036 Another quantifier free definition of function value. (Contributed by Scott Fenton, 19-Feb-2013.)
(𝐹𝐴) = ({(𝐹 “ {𝐴})} ∩ Singletons )
 
Theoremunisnif 31037 Express union of singleton in terms of if. (Contributed by Scott Fenton, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
{𝐴} = if(𝐴 ∈ V, 𝐴, ∅)
 
Theorembrimage 31038 Binary relationship form of the Image functor. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴Image𝑅𝐵𝐵 = (𝑅𝐴))
 
Theorembrimageg 31039 Closed form of brimage 31038. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
((𝐴𝑉𝐵𝑊) → (𝐴Image𝑅𝐵𝐵 = (𝑅𝐴)))
 
Theoremfunimage 31040 Image𝐴 is a function. (Contributed by Scott Fenton, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Fun Image𝐴
 
Theoremfnimage 31041* Image𝑅 is a function over the set-like portion of 𝑅. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Image𝑅 Fn {𝑥 ∣ (𝑅𝑥) ∈ V}
 
Theoremimageval 31042* The image functor in maps-to notation. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Image𝑅 = (𝑥 ∈ V ↦ (𝑅𝑥))
 
Theoremfvimage 31043 The value of the image functor. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
((𝐴𝑉 ∧ (𝑅𝐴) ∈ 𝑊) → (Image𝑅𝐴) = (𝑅𝐴))
 
Theorembrcart 31044 Binary relationship form of the cartesian product operator. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V       (⟨𝐴, 𝐵⟩Cart𝐶𝐶 = (𝐴 × 𝐵))
 
Theorembrdomain 31045 The binary relationship form of the domain function. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴Domain𝐵𝐵 = dom 𝐴)
 
Theorembrrange 31046 The binary relationship form of the range function. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴Range𝐵𝐵 = ran 𝐴)
 
Theorembrdomaing 31047 Closed form of brdomain 31045. (Contributed by Scott Fenton, 2-May-2014.)
((𝐴𝑉𝐵𝑊) → (𝐴Domain𝐵𝐵 = dom 𝐴))
 
Theorembrrangeg 31048 Closed form of brrange 31046. (Contributed by Scott Fenton, 3-May-2014.)
((𝐴𝑉𝐵𝑊) → (𝐴Range𝐵𝐵 = ran 𝐴))
 
Theorembrimg 31049 The binary relationship form of the Img function. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V       (⟨𝐴, 𝐵⟩Img𝐶𝐶 = (𝐴𝐵))
 
Theorembrapply 31050 The binary relationship form of the Apply function. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V       (⟨𝐴, 𝐵⟩Apply𝐶𝐶 = (𝐴𝐵))
 
Theorembrcup 31051 Binary relationship form of the Cup function. (Contributed by Scott Fenton, 14-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V       (⟨𝐴, 𝐵⟩Cup𝐶𝐶 = (𝐴𝐵))
 
Theorembrcap 31052 Binary relationship form of the Cap function. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V       (⟨𝐴, 𝐵⟩Cap𝐶𝐶 = (𝐴𝐵))
 
Theorembrsuccf 31053 Binary relationship form of the Succ function. (Contributed by Scott Fenton, 14-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴Succ𝐵𝐵 = suc 𝐴)
 
Theoremfunpartlem 31054* Lemma for funpartfun 31055. Show membership in the restriction. (Contributed by Scott Fenton, 4-Dec-2017.)
(𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑥(𝐹 “ {𝐴}) = {𝑥})
 
Theoremfunpartfun 31055 The functional part of 𝐹 is a function. (Contributed by Scott Fenton, 16-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Fun Funpart𝐹
 
Theoremfunpartss 31056 The functional part of 𝐹 is a subset of 𝐹. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Funpart𝐹𝐹
 
Theoremfunpartfv 31057 The function value of the functional part is identical to the original functional value. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
(Funpart𝐹𝐴) = (𝐹𝐴)
 
Theoremfullfunfnv 31058 The full functional part of 𝐹 is a function over V. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
FullFun𝐹 Fn V
 
Theoremfullfunfv 31059 The function value of the full function of 𝐹 agrees with 𝐹. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
(FullFun𝐹𝐴) = (𝐹𝐴)
 
Theorembrfullfun 31060 A binary relationship form condition for the full function. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴FullFun𝐹𝐵𝐵 = (𝐹𝐴))
 
Theorembrrestrict 31061 The binary relationship form of the Restrict function. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V       (⟨𝐴, 𝐵⟩Restrict𝐶𝐶 = (𝐴𝐵))
 
Theoremdfrecs2 31062 A quantifier-free definition of recs. (Contributed by Scott Fenton, 17-Jul-2020.)
recs(𝐹) = (( Funs ∩ (Domain “ On)) ∖ dom (( E ∘ Domain) ∖ Fix (Apply ∘ (FullFun𝐹 ∘ Restrict))))
 
Theoremdfrdg4 31063 A quantifier-free definition of the recursive definition generator. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
rec(𝐹, 𝐴) = (( Funs ∩ (Domain “ On)) ∖ dom (( E ∘ Domain) ∖ Fix (Apply ∘ (((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))))
 
Theoremdfint3 31064 Quantifier-free definition of class intersection. (Contributed by Scott Fenton, 13-Apr-2018.)
𝐴 = (V ∖ ((V ∖ E ) “ 𝐴))
 
Theoremimagesset 31065 The Image functor applied to the converse of the subset relationship yields a subset of the subset relationship. (Contributed by Scott Fenton, 14-Apr-2018.)
Image SSet SSet
 
Theorembrub 31066* Binary relationship form of the upper bound functor. (Contributed by Scott Fenton, 3-May-2018.)
𝑆 ∈ V    &   𝐴 ∈ V       (𝑆UB𝑅𝐴 ↔ ∀𝑥𝑆 𝑥𝑅𝐴)
 
Theorembrlb 31067* Binary relationship form of the lower bound functor. (Contributed by Scott Fenton, 3-May-2018.)
𝑆 ∈ V    &   𝐴 ∈ V       (𝑆LB𝑅𝐴 ↔ ∀𝑥𝑆 𝐴𝑅𝑥)
 
20.8.28  Alternate ordered pairs
 
Syntaxcaltop 31068 Declare the syntax for an alternate ordered pair.
class 𝐴, 𝐵
 
Syntaxcaltxp 31069 Declare the syntax for an alternate Cartesian product.
class (𝐴 ×× 𝐵)
 
Definitiondf-altop 31070 An alternative definition of ordered pairs. This definition removes a hypothesis from its defining theorem (see altopth 31081), making it more convenient in some circumstances. (Contributed by Scott Fenton, 22-Mar-2012.)
𝐴, 𝐵⟫ = {{𝐴}, {𝐴, {𝐵}}}
 
Definitiondf-altxp 31071* Define Cartesian products of alternative ordered pairs. (Contributed by Scott Fenton, 23-Mar-2012.)
(𝐴 ×× 𝐵) = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = ⟪𝑥, 𝑦⟫}
 
Theoremaltopex 31072 Alternative ordered pairs always exist. (Contributed by Scott Fenton, 22-Mar-2012.)
𝐴, 𝐵⟫ ∈ V
 
Theoremaltopthsn 31073 Two alternate ordered pairs are equal iff the singletons of their respective elements are equal. Note that this holds regardless of sethood of any of the elements. (Contributed by Scott Fenton, 16-Apr-2012.)
(⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ ({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}))
 
Theoremaltopeq12 31074 Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.)
((𝐴 = 𝐵𝐶 = 𝐷) → ⟪𝐴, 𝐶⟫ = ⟪𝐵, 𝐷⟫)
 
Theoremaltopeq1 31075 Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.)
(𝐴 = 𝐵 → ⟪𝐴, 𝐶⟫ = ⟪𝐵, 𝐶⟫)
 
Theoremaltopeq2 31076 Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.)
(𝐴 = 𝐵 → ⟪𝐶, 𝐴⟫ = ⟪𝐶, 𝐵⟫)
 
Theoremaltopth1 31077 Equality of the first members of equal alternate ordered pairs, which holds regardless of the second members' sethood. (Contributed by Scott Fenton, 22-Mar-2012.)
(𝐴𝑉 → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ → 𝐴 = 𝐶))
 
Theoremaltopth2 31078 Equality of the second members of equal alternate ordered pairs, which holds regardless of the first members' sethood. (Contributed by Scott Fenton, 22-Mar-2012.)
(𝐵𝑉 → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ → 𝐵 = 𝐷))
 
Theoremaltopthg 31079 Alternate ordered pair theorem. (Contributed by Scott Fenton, 22-Mar-2012.)
((𝐴𝑉𝐵𝑊) → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
 
Theoremaltopthbg 31080 Alternate ordered pair theorem. (Contributed by Scott Fenton, 14-Apr-2012.)
((𝐴𝑉𝐷𝑊) → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
 
Theoremaltopth 31081 The alternate ordered pair theorem. If two alternate ordered pairs are equal, their first elements are equal and their second elements are equal. Note that 𝐶 and 𝐷 are not required to be a set due to a peculiarity of our specific ordered pair definition, as opposed to the regular ordered pairs used here, which (as in opth 4769), requires 𝐷 to be a set. (Contributed by Scott Fenton, 23-Mar-2012.)
𝐴 ∈ V    &   𝐵 ∈ V       (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶𝐵 = 𝐷))
 
Theoremaltopthb 31082 Alternate ordered pair theorem with different sethood requirements. See altopth 31081 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.)
𝐴 ∈ V    &   𝐷 ∈ V       (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶𝐵 = 𝐷))
 
Theoremaltopthc 31083 Alternate ordered pair theorem with different sethood requirements. See altopth 31081 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.)
𝐵 ∈ V    &   𝐶 ∈ V       (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶𝐵 = 𝐷))
 
Theoremaltopthd 31084 Alternate ordered pair theorem with different sethood requirements. See altopth 31081 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.)
𝐶 ∈ V    &   𝐷 ∈ V       (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶𝐵 = 𝐷))
 
Theoremaltxpeq1 31085 Equality for alternate Cartesian products. (Contributed by Scott Fenton, 24-Mar-2012.)
(𝐴 = 𝐵 → (𝐴 ×× 𝐶) = (𝐵 ×× 𝐶))
 
Theoremaltxpeq2 31086 Equality for alternate Cartesian products. (Contributed by Scott Fenton, 24-Mar-2012.)
(𝐴 = 𝐵 → (𝐶 ×× 𝐴) = (𝐶 ×× 𝐵))
 
Theoremelaltxp 31087* Membership in alternate Cartesian products. (Contributed by Scott Fenton, 23-Mar-2012.)
(𝑋 ∈ (𝐴 ×× 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝑋 = ⟪𝑥, 𝑦⟫)
 
Theoremaltopelaltxp 31088 Alternate ordered pair membership in a Cartesian product. Note that, unlike opelxp 4964, there is no sethood requirement here. (Contributed by Scott Fenton, 22-Mar-2012.)
(⟪𝑋, 𝑌⟫ ∈ (𝐴 ×× 𝐵) ↔ (𝑋𝐴𝑌𝐵))
 
Theoremaltxpsspw 31089 An inclusion rule for alternate Cartesian products. (Contributed by Scott Fenton, 24-Mar-2012.)
(𝐴 ×× 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵)
 
Theoremaltxpexg 31090 The alternate Cartesian product of two sets is a set. (Contributed by Scott Fenton, 24-Mar-2012.)
((𝐴𝑉𝐵𝑊) → (𝐴 ×× 𝐵) ∈ V)
 
Theoremrankaltopb 31091 Compute the rank of an alternate ordered pair. (Contributed by Scott Fenton, 18-Dec-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (rank‘⟪𝐴, 𝐵⟫) = suc suc ((rank‘𝐴) ∪ suc (rank‘𝐵)))
 
Theoremnfaltop 31092 Bound-variable hypothesis builder for alternate ordered pairs. (Contributed by Scott Fenton, 25-Sep-2015.)
𝑥𝐴    &   𝑥𝐵       𝑥𝐴, 𝐵
 
Theoremsbcaltop 31093* Distribution of class substitution over alternate ordered pairs. (Contributed by Scott Fenton, 25-Sep-2015.)
(𝐴 ∈ V → 𝐴 / 𝑥𝐶, 𝐷⟫ = ⟪𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷⟫)
 
20.8.29  Geometry in the Euclidean space
 
20.8.29.1  Congruence properties
 
Syntaxcofs 31094 Declare the syntax for the outer five segment configuration.
class OuterFiveSeg
 
Definitiondf-ofs 31095* The outer five segment configuration is an abbreviation for the conditions of the Five Segment Axiom (ax5seg 25508). See brofs 31117 and 5segofs 31118 for how it is used. Definition 2.10 of [Schwabhauser] p. 28. (Contributed by Scott Fenton, 21-Sep-2013.)
OuterFiveSeg = {⟨𝑝, 𝑞⟩ ∣ ∃𝑛 ∈ ℕ ∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)∃𝑥 ∈ (𝔼‘𝑛)∃𝑦 ∈ (𝔼‘𝑛)∃𝑧 ∈ (𝔼‘𝑛)∃𝑤 ∈ (𝔼‘𝑛)(𝑝 = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ 𝑞 = ⟨⟨𝑥, 𝑦⟩, ⟨𝑧, 𝑤⟩⟩ ∧ ((𝑏 Btwn ⟨𝑎, 𝑐⟩ ∧ 𝑦 Btwn ⟨𝑥, 𝑧⟩) ∧ (⟨𝑎, 𝑏⟩Cgr⟨𝑥, 𝑦⟩ ∧ ⟨𝑏, 𝑐⟩Cgr⟨𝑦, 𝑧⟩) ∧ (⟨𝑎, 𝑑⟩Cgr⟨𝑥, 𝑤⟩ ∧ ⟨𝑏, 𝑑⟩Cgr⟨𝑦, 𝑤⟩)))}
 
Theoremcgrrflx2d 31096 Deduction form of axcgrrflx 25484. (Contributed by Scott Fenton, 13-Oct-2013.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴 ∈ (𝔼‘𝑁))    &   (𝜑𝐵 ∈ (𝔼‘𝑁))       (𝜑 → ⟨𝐴, 𝐵⟩Cgr⟨𝐵, 𝐴⟩)
 
Theoremcgrtr4d 31097 Deduction form of axcgrtr 25485. (Contributed by Scott Fenton, 13-Oct-2013.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴 ∈ (𝔼‘𝑁))    &   (𝜑𝐵 ∈ (𝔼‘𝑁))    &   (𝜑𝐶 ∈ (𝔼‘𝑁))    &   (𝜑𝐷 ∈ (𝔼‘𝑁))    &   (𝜑𝐸 ∈ (𝔼‘𝑁))    &   (𝜑𝐹 ∈ (𝔼‘𝑁))    &   (𝜑 → ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝐷⟩)    &   (𝜑 → ⟨𝐴, 𝐵⟩Cgr⟨𝐸, 𝐹⟩)       (𝜑 → ⟨𝐶, 𝐷⟩Cgr⟨𝐸, 𝐹⟩)
 
Theoremcgrtr4and 31098 Deduction form of axcgrtr 25485. (Contributed by Scott Fenton, 13-Oct-2013.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴 ∈ (𝔼‘𝑁))    &   (𝜑𝐵 ∈ (𝔼‘𝑁))    &   (𝜑𝐶 ∈ (𝔼‘𝑁))    &   (𝜑𝐷 ∈ (𝔼‘𝑁))    &   (𝜑𝐸 ∈ (𝔼‘𝑁))    &   (𝜑𝐹 ∈ (𝔼‘𝑁))    &   ((𝜑𝜓) → ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝐷⟩)    &   ((𝜑𝜓) → ⟨𝐴, 𝐵⟩Cgr⟨𝐸, 𝐹⟩)       ((𝜑𝜓) → ⟨𝐶, 𝐷⟩Cgr⟨𝐸, 𝐹⟩)
 
Theoremcgrrflx 31099 Reflexivity law for congruence. Theorem 2.1 of [Schwabhauser] p. 27. (Contributed by Scott Fenton, 12-Jun-2013.)
((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → ⟨𝐴, 𝐵⟩Cgr⟨𝐴, 𝐵⟩)
 
Theoremcgrrflxd 31100 Deduction form of cgrrflx 31099. (Contributed by Scott Fenton, 13-Oct-2013.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴 ∈ (𝔼‘𝑁))    &   (𝜑𝐵 ∈ (𝔼‘𝑁))       (𝜑 → ⟨𝐴, 𝐵⟩Cgr⟨𝐴, 𝐵⟩)
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