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Theorem List for Metamath Proof Explorer - 30901-31000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremslttri 30901 Surreal less than obeys trichotomy. (Contributed by Scott Fenton, 16-Jun-2011.)
((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵𝐴 = 𝐵𝐵 <s 𝐴))
 
Theoremslttrieq2 30902 Trichotomy law for surreal less than. (Contributed by Scott Fenton, 22-Apr-2012.)
((𝐴 No 𝐵 No ) → (𝐴 = 𝐵 ↔ (¬ 𝐴 <s 𝐵 ∧ ¬ 𝐵 <s 𝐴)))
 
20.8.23  Surreal Numbers: Birthday Function
 
Theorembdayfo 30903 The birthday function maps the surreals onto the ordinals. Alling's axiom (B). (Shortened proof on 2012-Apr-14, SF). (Contributed by Scott Fenton, 11-Jun-2011.)
bday : No onto→On
 
Theorembdayfun 30904 The birthday function is a function. (Contributed by Scott Fenton, 14-Jun-2011.)
Fun bday
 
Theorembdayrn 30905 The birthday function's range is On. (Contributed by Scott Fenton, 14-Jun-2011.)
ran bday = On
 
Theorembdaydm 30906 The birthday function's domain is No . (Contributed by Scott Fenton, 14-Jun-2011.)
dom bday = No
 
Theorembdayfn 30907 The birthday function is a function over No . (Contributed by Scott Fenton, 30-Jun-2011.)
bday Fn No
 
Theorembdayelon 30908 The value of the birthday function is always an ordinal. (Contributed by Scott Fenton, 14-Jun-2011.)
( bday 𝐴) ∈ On
 
Theoremnoprc 30909 The surreal numbers are a proper class. (Contributed by Scott Fenton, 16-Jun-2011.)
¬ No ∈ V
 
20.8.24  Surreal Numbers: Density
 
Theoremfvnobday 30910 The value of a surreal at its birthday is . (Contributed by Scott Fenton, 14-Jun-2011.) (Proof shortened by SF, 14-Apr-2012.)
(𝐴 No → (𝐴‘( bday 𝐴)) = ∅)
 
Theoremnodenselem3 30911* Lemma for nodense 30917. If one surreal is older than another, then there is an ordinal at which they are not equal. (Contributed by Scott Fenton, 16-Jun-2011.)
((𝐴 No 𝐵 No ) → (( bday 𝐴) ∈ ( bday 𝐵) → ∃𝑎 ∈ On (𝐴𝑎) ≠ (𝐵𝑎)))
 
Theoremnodenselem4 30912* Lemma for nodense 30917. Show that a particular abstraction is an ordinal. (Contributed by Scott Fenton, 16-Jun-2011.)
(((𝐴 No 𝐵 No ) ∧ 𝐴 <s 𝐵) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On)
 
Theoremnodenselem5 30913* Lemma for nodense 30917. If the birthdays of two distinct surreals are equal, then the ordinal from nodenselem4 30912 is an element of that birthday. (Contributed by Scott Fenton, 16-Jun-2011.)
(((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐴))
 
Theoremnodenselem6 30914* The restriction of a surreal to the abstraction from nodenselem4 30912 is still a surreal. (Contributed by Scott Fenton, 16-Jun-2011.)
(((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) ∈ No )
 
Theoremnodenselem7 30915* Lemma for nodense 30917. 𝐴 and 𝐵 are equal at all elements of the abstraction. (Contributed by Scott Fenton, 17-Jun-2011.)
(((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (𝐶 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → (𝐴𝐶) = (𝐵𝐶)))
 
Theoremnodenselem8 30916* Lemma for nodense 30917. Give a condition for surreal less than when two surreals have the same birthday. (Contributed by Scott Fenton, 19-Jun-2011.)
((𝐴 No 𝐵 No ∧ ( bday 𝐴) = ( bday 𝐵)) → (𝐴 <s 𝐵 ↔ ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1𝑜 ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2𝑜)))
 
Theoremnodense 30917* Given two distinct surreals with the same birthday, there is an older surreal lying between the two of them. Alling's axiom (SD). (Contributed by Scott Fenton, 16-Jun-2011.)
(((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → ∃𝑥 No (( bday 𝑥) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑥𝑥 <s 𝐵))
 
Theoremnocvxminlem 30918* Lemma for nocvxmin 30919. Given two birthday-minimal elements of a convex class of surreals, they are not comparable. (Contributed by Scott Fenton, 30-Jun-2011.)
((𝐴 No ∧ ∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)) → (((𝑋𝐴𝑌𝐴) ∧ (( bday 𝑋) = ( bday 𝐴) ∧ ( bday 𝑌) = ( bday 𝐴))) → ¬ 𝑋 <s 𝑌))
 
Theoremnocvxmin 30919* Given a nonempty convex class of surreals, there is a unique birthday-minimal element of that class. (Contributed by Scott Fenton, 30-Jun-2011.)
((𝐴 ≠ ∅ ∧ 𝐴 No ∧ ∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)) → ∃!𝑤𝐴 ( bday 𝑤) = ( bday 𝐴))
 
20.8.25  Surreal Numbers: Upper and Lower Bounds
 
Theoremnobndlem1 30920 Lemma for nobndup 30928 and nobnddown 30929. The successor of the union of the image of the birthday function under a set is an ordinal. (Contributed by Scott Fenton, 20-Aug-2011.)
(𝐴𝑉 → suc ( bday 𝐴) ∈ On)
 
Theoremnobndlem2 30921* Lemma for nobndup 30928 and nobnddown 30929. Show a particular abstraction is an ordinal. (Contributed by Scott Fenton, 17-Aug-2011.)
𝑋 ∈ {1𝑜, 2𝑜}    &   𝐶 = {𝑎 ∈ On ∣ ∀𝑛𝐹𝑏𝑎 (𝑛𝑏) ≠ 𝑋}       ((𝐹 No 𝐹𝐴) → 𝐶 ∈ On)
 
Theoremnobndlem3 30922* Lemma for nobndup 30928 and nobnddown 30929. Calculate the birthday of (𝐶 × {𝑋}). (Contributed by Scott Fenton, 17-Aug-2011.)
𝑋 ∈ {1𝑜, 2𝑜}    &   𝐶 = {𝑎 ∈ On ∣ ∀𝑛𝐹𝑏𝑎 (𝑛𝑏) ≠ 𝑋}       ((𝐹 No 𝐹𝐴) → ( bday ‘(𝐶 × {𝑋})) = 𝐶)
 
Theoremnobndlem4 30923* Lemma for nobndup 30928 and nobnddown 30929. The infimum of the class of all ordinals such that 𝐴 is not 𝑋 is an ordinal. (Contributed by Scott Fenton, 17-Aug-2011.)
𝑋 ∈ {1𝑜, 2𝑜}       (𝐴 No {𝑥 ∈ On ∣ (𝐴𝑥) ≠ 𝑋} ∈ On)
 
Theoremnobndlem5 30924* Lemma for nobndup 30928 and nobnddown 30929. There is always a minimal value of a surreal that is not a given sign. (Contributed by Scott Fenton, 3-Aug-2011.)
𝑋 ∈ {1𝑜, 2𝑜}       (𝐴 No → (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ 𝑋}) ≠ 𝑋)
 
Theoremnobndlem6 30925* Lemma for nobndup 30928 and nobnddown 30929. Given an element 𝐴 of 𝐹, then the first position where it differs from 𝑋 is strictly less than 𝐶. (Contributed by Scott Fenton, 3-Aug-2011.)
𝑋 ∈ {1𝑜, 2𝑜}    &   𝐶 = {𝑎 ∈ On ∣ ∀𝑛𝐹𝑏𝑎 (𝑛𝑏) ≠ 𝑋}       ((𝐹 No 𝐴𝐹) → {𝑥 ∈ On ∣ (𝐴𝑥) ≠ 𝑋} ∈ 𝐶)
 
Theoremnobndlem7 30926* Lemma for nobndup 30928 and nobnddown 30929. Calculate the value of (𝐶 × {𝑋}) at the minimal ordinal whose value is different from 𝑋. (Contributed by Scott Fenton, 3-Aug-2011.)
𝑋 ∈ {1𝑜, 2𝑜}    &   𝐶 = {𝑎 ∈ On ∣ ∀𝑛𝐹𝑏𝑎 (𝑛𝑏) ≠ 𝑋}       ((𝐹 No 𝐴𝐹) → ((𝐶 × {𝑋})‘ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ 𝑋}) = 𝑋)
 
Theoremnobndlem8 30927* Lemma for nobndup 30928 and nobnddown 30929. Bound the birthday of (𝐶 × {𝑆}) above. (Contributed by Scott Fenton, 10-Apr-2017.)
𝑆 ∈ {1𝑜, 2𝑜}    &   𝐶 = {𝑎 ∈ On ∣ ∀𝑛𝐹𝑏𝑎 (𝑛𝑏) ≠ 𝑆}       ((𝐹 No 𝐹𝐴) → ( bday ‘(𝐶 × {𝑆})) ⊆ suc ( bday 𝐹))
 
Theoremnobndup 30928* Any set of surreals is bounded above by a surreal with a birthday no greater than the successor of their maximum birthday. (Contributed by Scott Fenton, 10-Apr-2017.)
((𝐴 No 𝐴𝑉) → ∃𝑥 No (∀𝑦𝐴 𝑦 <s 𝑥 ∧ ( bday 𝑥) ⊆ suc ( bday 𝐴)))
 
Theoremnobnddown 30929* Any set of surreals is bounded below by a surreal with a birthday no greater than the successor of their maximum birthday. (Contributed by Scott Fenton, 10-Apr-2017.)
((𝐴 No 𝐴𝑉) → ∃𝑥 No (∀𝑦𝐴 𝑥 <s 𝑦 ∧ ( bday 𝑥) ⊆ suc ( bday 𝐴)))
 
20.8.26  Surreal Numbers: Full-Eta Property
 
Theoremnofulllem1 30930* Lemma for nofull (future) . The full statement of the axiom when 𝑅 is empty. (Contributed by Scott Fenton, 3-Aug-2011.)
(𝑅 = ∅ → (((𝐿 No 𝐿𝑉) ∧ (𝑅 No 𝑅𝑊) ∧ ∀𝑥𝐿𝑦𝑅 𝑥 <s 𝑦) → ∃𝑧 No (∀𝑥𝐿 𝑥 <s 𝑧 ∧ ∀𝑦𝑅 𝑧 <s 𝑦 ∧ ( bday 𝑧) ⊆ suc ( bday “ (𝐿𝑅)))))
 
Theoremnofulllem2 30931* Lemma for nofull (future) . The full statement of the axiom when 𝐿 is empty. (Contributed by Scott Fenton, 3-Aug-2011.)
(𝐿 = ∅ → (((𝐿 No 𝐿𝑉) ∧ (𝑅 No 𝑅𝑊) ∧ ∀𝑥𝐿𝑦𝑅 𝑥 <s 𝑦) → ∃𝑧 No (∀𝑥𝐿 𝑥 <s 𝑧 ∧ ∀𝑦𝑅 𝑧 <s 𝑦 ∧ ( bday 𝑧) ⊆ suc ( bday “ (𝐿𝑅)))))
 
Theoremnofulllem3 30932 Lemma for nofull (future) . Restriction of surreal number to a superset of its birthday does not change anything. (Contributed by Scott Fenton, 25-Apr-2017.)
((𝐴 No 𝑋𝐴𝐴𝑆) → (𝑋 ( bday 𝑆)) = 𝑋)
 
Theoremnofulllem4 30933* Lemma for nofull (future) . Show a particular abstraction is an ordinal. (Contributed by Scott Fenton, 25-Apr-2017.)
𝑀 = {𝑎 ∈ On ∣ ∀𝑥𝐿𝑦𝑅 (𝑥𝑎) ≠ (𝑦𝑎)}       (((𝐿 No 𝐿𝑉) ∧ (𝑅 No 𝑅𝑊) ∧ ∀𝑥𝐿𝑦𝑅 𝑥 <s 𝑦) → 𝑀 ∈ On)
 
Theoremnofulllem5 30934* Lemma for nofull (future) . Here, we introduce a new surreal number 𝑋. Eventually, we will show that either 𝑋 or a related surreal number has the required properties for the final theorem. We begin by calculating the domain of 𝑋. (Contributed by Scott Fenton, 1-May-2017.)
𝑀 = {𝑎 ∈ On ∣ ∀𝑥𝐿𝑦𝑅 (𝑥𝑎) ≠ (𝑦𝑎)}    &   𝑆 = {𝑓 ∣ ∃𝑔𝐿𝑅𝑎𝑀 ((𝑔𝑎) = 𝑓 ∧ (𝑎) = 𝑓)}    &   𝑋 = 𝑆       (((𝐿 No 𝐿𝑉) ∧ (𝑅 No 𝑅𝑊) ∧ ∀𝑥𝐿𝑦𝑅 𝑥 <s 𝑦) → dom 𝑋 = 𝑀)
 
20.8.27  Quantifier-free definitions
 
Syntaxctxp 30935 Declare the syntax for tail Cartesian product.
class (𝐴𝐵)
 
Syntaxcpprod 30936 Declare the syntax for the parallel product.
class pprod(𝑅, 𝑆)
 
Syntaxcsset 30937 Declare the subset relationship class.
class SSet
 
Syntaxctrans 30938 Declare the transitive set class.
class Trans
 
Syntaxcbigcup 30939 Declare the set union relationship.
class Bigcup
 
Syntaxcfix 30940 Declare the syntax for the fixpoints of a class.
class Fix 𝐴
 
Syntaxclimits 30941 Declare the class of limit ordinals.
class Limits
 
Syntaxcfuns 30942 Declare the syntax for the class of all function.
class Funs
 
Syntaxcsingle 30943 Declare the syntax for the singleton function.
class Singleton
 
Syntaxcsingles 30944 Declare the syntax for the class of all singletons.
class Singletons
 
Syntaxcimage 30945 Declare the syntax for the image functor.
class Image𝐴
 
Syntaxccart 30946 Declare the syntax for the cartesian function.
class Cart
 
Syntaxcimg 30947 Declare the syntax for the image function.
class Img
 
Syntaxcdomain 30948 Declare the syntax for the domain function.
class Domain
 
Syntaxcrange 30949 Declare the syntax for the range function.
class Range
 
Syntaxcapply 30950 Declare the syntax for the application function.
class Apply
 
Syntaxccup 30951 Declare the syntax for the cup function.
class Cup
 
Syntaxccap 30952 Declare the syntax for the cap function.
class Cap
 
Syntaxcsuccf 30953 Declare the syntax for the successor function.
class Succ
 
Syntaxcfunpart 30954 Declare the syntax for the functional part functor.
class Funpart𝐹
 
Syntaxcfullfn 30955 Declare the syntax for the full function functor.
class FullFun𝐹
 
Syntaxcrestrict 30956 Declare the syntax for the restriction function.
class Restrict
 
Syntaxcub 30957 Declare the syntax for the upper bound relationship functor.
class UB𝑅
 
Syntaxclb 30958 Declare the syntax for the lower bound relationship functor.
class LB𝑅
 
Definitiondf-txp 30959 Define the tail cross of two classes. Membership in this class is defined by txpss3v 30984 and brtxp 30986. (Contributed by Scott Fenton, 31-Mar-2012.)
(𝐴𝐵) = (((1st ↾ (V × V)) ∘ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝐵))
 
Definitiondf-pprod 30960 Define the parallel product of two classes. Membership in this class is defined by pprodss4v 30990 and brpprod 30991. (Contributed by Scott Fenton, 11-Apr-2014.)
pprod(𝐴, 𝐵) = ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))
 
Definitiondf-sset 30961 Define the subset class. For the value, see brsset 30995. (Contributed by Scott Fenton, 31-Mar-2012.)
SSet = ((V × V) ∖ ran ( E ⊗ (V ∖ E )))
 
Definitiondf-trans 30962 Define the class of all transitive sets. (Contributed by Scott Fenton, 31-Mar-2012.)
Trans = (V ∖ ran (( E ∘ E ) ∖ E ))
 
Definitiondf-bigcup 30963 Define the Bigcup function, which, per fvbigcup 31008, carries a set to its union. (Contributed by Scott Fenton, 11-Apr-2012.)
Bigcup = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V)))
 
Definitiondf-fix 30964 Define the class of all fixpoints of a relationship. (Contributed by Scott Fenton, 11-Apr-2012.)
Fix 𝐴 = dom (𝐴 ∩ I )
 
Definitiondf-limits 30965 Define the class of all limit ordinals. (Contributed by Scott Fenton, 11-Apr-2012.)
Limits = ((On ∩ Fix Bigcup ) ∖ {∅})
 
Definitiondf-funs 30966 Define the class of all functions. See elfuns 31021 for membership. (Contributed by Scott Fenton, 18-Feb-2013.)
Funs = (𝒫 (V × V) ∖ Fix ( E ∘ ((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ E )))
 
Definitiondf-singleton 30967 Define the singleton function. See brsingle 31023 for its value. (Contributed by Scott Fenton, 4-Apr-2014.)
Singleton = ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V)))
 
Definitiondf-singles 30968 Define the class of all singletons. See elsingles 31024 for membership. (Contributed by Scott Fenton, 19-Feb-2013.)
Singletons = ran Singleton
 
Definitiondf-image 30969 Define the image functor. This function takes a set 𝐴 to a function 𝑥 ↦ (𝐴𝑥), providing that the latter exists. See imageval 31036 for the derivation. (Contributed by Scott Fenton, 27-Mar-2014.)
Image𝐴 = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ 𝐴) ⊗ V)))
 
Definitiondf-cart 30970 Define the cartesian product function. See brcart 31038 for its value. (Contributed by Scott Fenton, 11-Apr-2014.)
Cart = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (pprod( E , E ) ⊗ V)))
 
Definitiondf-img 30971 Define the image function. See brimg 31043 for its value. (Contributed by Scott Fenton, 12-Apr-2014.)
Img = (Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V))) ∘ Cart)
 
Definitiondf-domain 30972 Define the domain function. See brdomain 31039 for its value. (Contributed by Scott Fenton, 11-Apr-2014.)
Domain = Image(1st ↾ (V × V))
 
Definitiondf-range 30973 Define the range function. See brrange 31040 for its value. (Contributed by Scott Fenton, 11-Apr-2014.)
Range = Image(2nd ↾ (V × V))
 
Definitiondf-cup 30974 Define the little cup function. See brcup 31045 for its value. (Contributed by Scott Fenton, 14-Apr-2014.)
Cup = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (((1st ∘ E ) ∪ (2nd ∘ E )) ⊗ V)))
 
Definitiondf-cap 30975 Define the little cap function. See brcap 31046 for its value. (Contributed by Scott Fenton, 17-Apr-2014.)
Cap = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (((1st ∘ E ) ∩ (2nd ∘ E )) ⊗ V)))
 
Definitiondf-restrict 30976 Define the restriction function. See brrestrict 31055 for its value. (Contributed by Scott Fenton, 17-Apr-2014.)
Restrict = (Cap ∘ (1st ⊗ (Cart ∘ (2nd ⊗ (Range ∘ 1st )))))
 
Definitiondf-succf 30977 Define the successor function. See brsuccf 31047 for its value. (Contributed by Scott Fenton, 14-Apr-2014.)
Succ = (Cup ∘ ( I ⊗ Singleton))
 
Definitiondf-apply 30978 Define the application function. See brapply 31044 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 30979 Define the functional part of a class 𝐹. This is the maximal part of 𝐹 that is a function. See funpartfun 31049 and funpartfv 31051 for the meaning of this statement. (Contributed by Scott Fenton, 16-Apr-2014.)
Funpart𝐹 = (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))
 
Definitiondf-fullfun 30980 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 30981 Define the upper bound relationship functor. See brub 31060 for value. (Contributed by Scott Fenton, 3-May-2018.)
UB𝑅 = ((V × V) ∖ ((V ∖ 𝑅) ∘ E ))
 
Definitiondf-lb 30982 Define the lower bound relationship functor. See brlb 31061 for value. (Contributed by Scott Fenton, 3-May-2018.)
LB𝑅 = UB𝑅
 
Theorembrv 30983 The binary relationship over V always holds. (Contributed by Scott Fenton, 11-Apr-2012.)
𝐴V𝐵
 
Theoremtxpss3v 30984 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 30985 A tail Cartesian product is a relationship. (Contributed by Scott Fenton, 31-Mar-2012.)
Rel (𝐴𝐵)
 
Theorembrtxp 30986 Characterize a trinary relationship over a tail Cartesian product. Together with txpss3v 30984, 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 30987* The binary relationship 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 30988 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 30989 A converse law for parallel product. (Contributed by Scott Fenton, 3-May-2014.)
pprod(𝑅, 𝑆) = pprod(𝑅, 𝑆)
 
Theorempprodss4v 30990 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 30991 Characterize a quatary relationship over a tail Cartesian product. Together with pprodss4v 30990, 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 30992* 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 30993* 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 30994 The subset class is a relationship. (Contributed by Scott Fenton, 31-Mar-2012.)
Rel SSet
 
Theorembrsset 30995 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 30996 I is equal to SSet and its converse. (Contributed by Scott Fenton, 31-Mar-2012.)
I = ( SSet SSet )
 
Theoremeltrans 30997 Membership in the class of all transitive sets. (Contributed by Scott Fenton, 31-Mar-2012.)
𝐴 ∈ V       (𝐴 Trans ↔ Tr 𝐴)
 
Theoremdfon3 30998 A quantifier-free definition of On. (Contributed by Scott Fenton, 5-Apr-2012.)
On = (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))
 
Theoremdfon4 30999 Another quantifier-free definition of On. (Contributed by Scott Fenton, 4-May-2014.)
On = (V ∖ (( SSet ∖ ( I ∪ E )) “ Trans ))
 
Theorembrtxpsd 31000* 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))
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