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Theorem List for Metamath Proof Explorer - 32101-32200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-cup 32101 Define the little cup function. See brcup 32171 for its value. (Contributed by Scott Fenton, 14-Apr-2014.)
Cup = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (((1st ∘ E ) ∪ (2nd ∘ E )) ⊗ V)))

Definitiondf-cap 32102 Define the little cap function. See brcap 32172 for its value. (Contributed by Scott Fenton, 17-Apr-2014.)
Cap = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (((1st ∘ E ) ∩ (2nd ∘ E )) ⊗ V)))

Definitiondf-restrict 32103 Define the restriction function. See brrestrict 32181 for its value. (Contributed by Scott Fenton, 17-Apr-2014.)
Restrict = (Cap ∘ (1st ⊗ (Cart ∘ (2nd ⊗ (Range ∘ 1st )))))

Definitiondf-succf 32104 Define the successor function. See brsuccf 32173 for its value. (Contributed by Scott Fenton, 14-Apr-2014.)
Succ = (Cup ∘ ( I ⊗ Singleton))

Definitiondf-apply 32105 Define the application function. See brapply 32170 for its value. (Contributed by Scott Fenton, 12-Apr-2014.)
Apply = (( Bigcup Bigcup ) ∘ (((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V))) ∘ ((Singleton ∘ Img) ∘ pprod( I , Singleton))))

Definitiondf-funpart 32106 Define the functional part of a class 𝐹. This is the maximal part of 𝐹 that is a function. See funpartfun 32175 and funpartfv 32177 for the meaning of this statement. (Contributed by Scott Fenton, 16-Apr-2014.)
Funpart𝐹 = (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))

Definitiondf-fullfun 32107 Define the full function over 𝐹. This is a function with domain V that always agrees with 𝐹 for its value. (Contributed by Scott Fenton, 17-Apr-2014.)
FullFun𝐹 = (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))

Definitiondf-ub 32108 Define the upper bound relationship functor. See brub 32186 for value. (Contributed by Scott Fenton, 3-May-2018.)
UB𝑅 = ((V × V) ∖ ((V ∖ 𝑅) ∘ E ))

Definitiondf-lb 32109 Define the lower bound relationship functor. See brlb 32187 for value. (Contributed by Scott Fenton, 3-May-2018.)
LB𝑅 = UB𝑅

Theoremtxpss3v 32110 A tail Cartesian product is a subset of the class of ordered triples. (Contributed by Scott Fenton, 31-Mar-2012.)
(𝐴𝐵) ⊆ (V × (V × V))

Theoremtxprel 32111 A tail Cartesian product is a relationship. (Contributed by Scott Fenton, 31-Mar-2012.)
Rel (𝐴𝐵)

Theorembrtxp 32112 Characterize a ternary relation over a tail Cartesian product. Together with txpss3v 32110, this completely defines membership in a tail cross. (Contributed by Scott Fenton, 31-Mar-2012.)
𝑋 ∈ V    &   𝑌 ∈ V    &   𝑍 ∈ V       (𝑋(𝐴𝐵)⟨𝑌, 𝑍⟩ ↔ (𝑋𝐴𝑌𝑋𝐵𝑍))

Theorembrtxp2 32113* The binary relation over a tail cross when the second argument is not an ordered pair. (Contributed by Scott Fenton, 14-Apr-2014.) (Revised by Mario Carneiro, 3-May-2015.)
𝐴 ∈ V       (𝐴(𝑅𝑆)𝐵 ↔ ∃𝑥𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥𝐴𝑆𝑦))

Theoremdfpprod2 32114 Expanded definition of parallel product. (Contributed by Scott Fenton, 3-May-2014.)
pprod(𝐴, 𝐵) = (((1st ↾ (V × V)) ∘ (𝐴 ∘ (1st ↾ (V × V)))) ∩ ((2nd ↾ (V × V)) ∘ (𝐵 ∘ (2nd ↾ (V × V)))))

Theorempprodcnveq 32115 A converse law for parallel product. (Contributed by Scott Fenton, 3-May-2014.)
pprod(𝑅, 𝑆) = pprod(𝑅, 𝑆)

Theorempprodss4v 32116 The parallel product is a subclass of ((V × V) × (V × V)). (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
pprod(𝐴, 𝐵) ⊆ ((V × V) × (V × V))

Theorembrpprod 32117 Characterize a quaternary relation over a tail Cartesian product. Together with pprodss4v 32116, this completely defines membership in a parallel product. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
𝑋 ∈ V    &   𝑌 ∈ V    &   𝑍 ∈ V    &   𝑊 ∈ V       (⟨𝑋, 𝑌⟩pprod(𝐴, 𝐵)⟨𝑍, 𝑊⟩ ↔ (𝑋𝐴𝑍𝑌𝐵𝑊))

Theorembrpprod3a 32118* Condition for parallel product when the last argument is not an ordered pair. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
𝑋 ∈ V    &   𝑌 ∈ V    &   𝑍 ∈ V       (⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍 ↔ ∃𝑧𝑤(𝑍 = ⟨𝑧, 𝑤⟩ ∧ 𝑋𝑅𝑧𝑌𝑆𝑤))

Theorembrpprod3b 32119* Condition for parallel product when the first argument is not an ordered pair. (Contributed by Scott Fenton, 3-May-2014.)
𝑋 ∈ V    &   𝑌 ∈ V    &   𝑍 ∈ V       (𝑋pprod(𝑅, 𝑆)⟨𝑌, 𝑍⟩ ↔ ∃𝑧𝑤(𝑋 = ⟨𝑧, 𝑤⟩ ∧ 𝑧𝑅𝑌𝑤𝑆𝑍))

Theoremrelsset 32120 The subset class is a binary relation. (Contributed by Scott Fenton, 31-Mar-2012.)
Rel SSet

Theorembrsset 32121 For sets, the SSet binary relation is equivalent to the subset relationship. (Contributed by Scott Fenton, 31-Mar-2012.)
𝐵 ∈ V       (𝐴 SSet 𝐵𝐴𝐵)

Theoremidsset 32122 I is equal to the intersection of SSet and its converse. (Contributed by Scott Fenton, 31-Mar-2012.)
I = ( SSet SSet )

Theoremeltrans 32123 Membership in the class of all transitive sets. (Contributed by Scott Fenton, 31-Mar-2012.)
𝐴 ∈ V       (𝐴 Trans ↔ Tr 𝐴)

Theoremdfon3 32124 A quantifier-free definition of On. (Contributed by Scott Fenton, 5-Apr-2012.)
On = (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))

Theoremdfon4 32125 Another quantifier-free definition of On. (Contributed by Scott Fenton, 4-May-2014.)
On = (V ∖ (( SSet ∖ ( I ∪ E )) “ Trans ))

Theorembrtxpsd 32126* 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 32127* 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 32128* 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 32129 The Bigcup relationship is a relationship. (Contributed by Scott Fenton, 11-Apr-2012.)
Rel Bigcup

Theorembrbigcup 32130 Binary relation over Bigcup . (Contributed by Scott Fenton, 11-Apr-2012.)
𝐵 ∈ V       (𝐴 Bigcup 𝐵 𝐴 = 𝐵)

Theoremdfbigcup2 32131 Bigcup using maps-to notation. (Contributed by Scott Fenton, 16-Apr-2012.)
Bigcup = (𝑥 ∈ V ↦ 𝑥)

Theoremfobigcup 32132 Bigcup maps the universe onto itself. (Contributed by Scott Fenton, 16-Apr-2012.)
Bigcup :V–onto→V

Theoremfnbigcup 32133 Bigcup is a function over the universal class. (Contributed by Scott Fenton, 11-Apr-2012.)
Bigcup Fn V

Theoremfvbigcup 32134 For sets, Bigcup yields union. (Contributed by Scott Fenton, 11-Apr-2012.)
𝐴 ∈ V       ( Bigcup 𝐴) = 𝐴

Theoremelfix 32135 Membership in the fixpoints of a class. (Contributed by Scott Fenton, 11-Apr-2012.)
𝐴 ∈ V       (𝐴 Fix 𝑅𝐴𝑅𝐴)

Theoremelfix2 32136 Alternative membership in the fixpoint of a class. (Contributed by Scott Fenton, 11-Apr-2012.)
Rel 𝑅       (𝐴 Fix 𝑅𝐴𝑅𝐴)

Theoremdffix2 32137 The fixpoints of a class in terms of its range. (Contributed by Scott Fenton, 16-Apr-2012.)
Fix 𝐴 = ran (𝐴 ∩ I )

Theoremfixssdm 32138 The fixpoints of a class are a subset of its domain. (Contributed by Scott Fenton, 16-Apr-2012.)
Fix 𝐴 ⊆ dom 𝐴

Theoremfixssrn 32139 The fixpoints of a class are a subset of its range. (Contributed by Scott Fenton, 16-Apr-2012.)
Fix 𝐴 ⊆ ran 𝐴

Theoremfixcnv 32140 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 32141 The fixpoint operator distributes over union. (Contributed by Scott Fenton, 16-Apr-2012.)
Fix (𝐴𝐵) = ( Fix 𝐴 Fix 𝐵)

Theoremellimits 32142 Membership in the class of all limit ordinals. (Contributed by Scott Fenton, 11-Apr-2012.)
𝐴 ∈ V       (𝐴 Limits ↔ Lim 𝐴)

Theoremlimitssson 32143 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 32144 A quantifier-free definition of ω that does not depend on ax-inf 8573. (Note: label was changed from dfom5 8585 to dfom5b 32144 to prevent naming conflict. NM, 12-Feb-2013.) (Contributed by Scott Fenton, 11-Apr-2012.)
ω = (On ∩ Limits )

Theoremsscoid 32145 A condition for subset and composition with identity. (Contributed by Scott Fenton, 13-Apr-2018.)
(𝐴 ⊆ ( I ∘ 𝐵) ↔ (Rel 𝐴𝐴𝐵))

Theoremdffun10 32146 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 32147 Membership in the class of all functions. (Contributed by Scott Fenton, 18-Feb-2013.)
𝐹 ∈ V       (𝐹 Funs ↔ Fun 𝐹)

Theoremelfunsg 32148 Closed form of elfuns 32147. (Contributed by Scott Fenton, 2-May-2014.)
(𝐹𝑉 → (𝐹 Funs ↔ Fun 𝐹))

Theorembrsingle 32149 The binary relation 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 32150* Membership in the class of all singletons. (Contributed by Scott Fenton, 19-Feb-2013.)
(𝐴 Singletons ↔ ∃𝑥 𝐴 = {𝑥})

Theoremfnsingle 32151 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 32152 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 32153* 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 32154 A singleton is a member of the class of all singletons. (Contributed by Scott Fenton, 19-Feb-2013.)
𝐴 ∈ V       {𝐴} ∈ Singletons

Theoremdfiota3 32155 A definiton of iota using minimal quantifiers. (Contributed by Scott Fenton, 19-Feb-2013.)
(℩𝑥𝜑) = ({{𝑥𝜑}} ∩ Singletons )

Theoremdffv5 32156 Another quantifier free definition of function value. (Contributed by Scott Fenton, 19-Feb-2013.)
(𝐹𝐴) = ({(𝐹 “ {𝐴})} ∩ Singletons )

Theoremunisnif 32157 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 32158 Binary relation 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 32159 Closed form of brimage 32158. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
((𝐴𝑉𝐵𝑊) → (𝐴Image𝑅𝐵𝐵 = (𝑅𝐴)))

Theoremfunimage 32160 Image𝐴 is a function. (Contributed by Scott Fenton, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Fun Image𝐴

Theoremfnimage 32161* 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 32162* 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 32163 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 32164 Binary relation 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 32165 Binary relation 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 32166 Binary relation 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 32167 Closed form of brdomain 32165. (Contributed by Scott Fenton, 2-May-2014.)
((𝐴𝑉𝐵𝑊) → (𝐴Domain𝐵𝐵 = dom 𝐴))

Theorembrrangeg 32168 Closed form of brrange 32166. (Contributed by Scott Fenton, 3-May-2014.)
((𝐴𝑉𝐵𝑊) → (𝐴Range𝐵𝐵 = ran 𝐴))

Theorembrimg 32169 Binary relation 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 32170 Binary relation 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 32171 Binary relation 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 32172 Binary relation 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 32173 Binary relation 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 32174* Lemma for funpartfun 32175. Show membership in the restriction. (Contributed by Scott Fenton, 4-Dec-2017.)
(𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑥(𝐹 “ {𝐴}) = {𝑥})

Theoremfunpartfun 32175 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 32176 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 32177 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 32178 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 32179 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 32180 A binary relation 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 32181 Binary relation 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 32182 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 32183 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 32184 Quantifier-free definition of class intersection. (Contributed by Scott Fenton, 13-Apr-2018.)
𝐴 = (V ∖ ((V ∖ E ) “ 𝐴))

Theoremimagesset 32185 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 32186* Binary relation form of the upper bound functor. (Contributed by Scott Fenton, 3-May-2018.)
𝑆 ∈ V    &   𝐴 ∈ V       (𝑆UB𝑅𝐴 ↔ ∀𝑥𝑆 𝑥𝑅𝐴)

Theorembrlb 32187* Binary relation form of the lower bound functor. (Contributed by Scott Fenton, 3-May-2018.)
𝑆 ∈ V    &   𝐴 ∈ V       (𝑆LB𝑅𝐴 ↔ ∀𝑥𝑆 𝐴𝑅𝑥)

20.8.31  Alternate ordered pairs

Syntaxcaltop 32188 Declare the syntax for an alternate ordered pair.
class 𝐴, 𝐵

Syntaxcaltxp 32189 Declare the syntax for an alternate Cartesian product.
class (𝐴 ×× 𝐵)

Definitiondf-altop 32190 An alternative definition of ordered pairs. This definition removes a hypothesis from its defining theorem (see altopth 32201), making it more convenient in some circumstances. (Contributed by Scott Fenton, 22-Mar-2012.)
𝐴, 𝐵⟫ = {{𝐴}, {𝐴, {𝐵}}}

Definitiondf-altxp 32191* Define Cartesian products of alternative ordered pairs. (Contributed by Scott Fenton, 23-Mar-2012.)
(𝐴 ×× 𝐵) = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = ⟪𝑥, 𝑦⟫}

Theoremaltopex 32192 Alternative ordered pairs always exist. (Contributed by Scott Fenton, 22-Mar-2012.)
𝐴, 𝐵⟫ ∈ V

Theoremaltopthsn 32193 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 32194 Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.)
((𝐴 = 𝐵𝐶 = 𝐷) → ⟪𝐴, 𝐶⟫ = ⟪𝐵, 𝐷⟫)

Theoremaltopeq1 32195 Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.)
(𝐴 = 𝐵 → ⟪𝐴, 𝐶⟫ = ⟪𝐵, 𝐶⟫)

Theoremaltopeq2 32196 Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.)
(𝐴 = 𝐵 → ⟪𝐶, 𝐴⟫ = ⟪𝐶, 𝐵⟫)

Theoremaltopth1 32197 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 32198 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 32199 Alternate ordered pair theorem. (Contributed by Scott Fenton, 22-Mar-2012.)
((𝐴𝑉𝐵𝑊) → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶𝐵 = 𝐷)))

Theoremaltopthbg 32200 Alternate ordered pair theorem. (Contributed by Scott Fenton, 14-Apr-2012.)
((𝐴𝑉𝐷𝑊) → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶𝐵 = 𝐷)))

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