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Theorem List for Metamath Proof Explorer - 43401-43500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnleltd 43401 'Not less than or equal to' implies 'grater than'. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ Β¬ 𝐡 ≀ 𝐴)    β‡’   (πœ‘ β†’ 𝐴 < 𝐡)
 
Theoremzxrd 43402 An integer is an extended real number. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(πœ‘ β†’ 𝐴 ∈ β„€)    β‡’   (πœ‘ β†’ 𝐴 ∈ ℝ*)
 
Theoreminfxrgelbrnmpt 43403* The infimum of an indexed set of extended reals is greater than or equal to a lower bound. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
β„²π‘₯πœ‘    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 ∈ ℝ*)    β‡’   (πœ‘ β†’ (𝐢 ≀ inf(ran (π‘₯ ∈ 𝐴 ↦ 𝐡), ℝ*, < ) ↔ βˆ€π‘₯ ∈ 𝐴 𝐢 ≀ 𝐡))
 
Theoremrphalfltd 43404 Half of a positive real is less than the original number. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(πœ‘ β†’ 𝐴 ∈ ℝ+)    β‡’   (πœ‘ β†’ (𝐴 / 2) < 𝐴)
 
Theoremuzssz2 43405 An upper set of integers is a subset of all integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑍 = (β„€β‰₯β€˜π‘€)    β‡’   π‘ βŠ† β„€
 
Theoremleneg3d 43406 Negative of one side of 'less than or equal to'. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    β‡’   (πœ‘ β†’ (-𝐴 ≀ 𝐡 ↔ -𝐡 ≀ 𝐴))
 
Theoremmax2d 43407 A number is less than or equal to the maximum of it and another. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    β‡’   (πœ‘ β†’ 𝐡 ≀ if(𝐴 ≀ 𝐡, 𝐡, 𝐴))
 
Theoremuzn0bi 43408 The upper integers function needs to be applied to an integer, in order to return a nonempty set. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
((β„€β‰₯β€˜π‘€) β‰  βˆ… ↔ 𝑀 ∈ β„€)
 
Theoremxnegrecl2 43409 If the extended real negative is real, then the number itself is real. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
((𝐴 ∈ ℝ* ∧ -𝑒𝐴 ∈ ℝ) β†’ 𝐴 ∈ ℝ)
 
Theoremnfxneg 43410 Bound-variable hypothesis builder for the negative of an extended real number. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
β„²π‘₯𝐴    β‡’   β„²π‘₯-𝑒𝐴
 
Theoremuzxrd 43411 An upper integer is an extended real. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑍 = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝐴 ∈ 𝑍)    β‡’   (πœ‘ β†’ 𝐴 ∈ ℝ*)
 
Theoreminfxrpnf2 43412 Removing plus infinity from a set does not affect its infimum. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝐴 βŠ† ℝ* β†’ inf((𝐴 βˆ– {+∞}), ℝ*, < ) = inf(𝐴, ℝ*, < ))
 
Theoremsupminfxr 43413* The extended real suprema of a set of reals is the extended real negative of the extended real infima of that set's image under negation. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(πœ‘ β†’ 𝐴 βŠ† ℝ)    β‡’   (πœ‘ β†’ sup(𝐴, ℝ*, < ) = -𝑒inf({π‘₯ ∈ ℝ ∣ -π‘₯ ∈ 𝐴}, ℝ*, < ))
 
Theoreminfrpgernmpt 43414* The infimum of a nonempty, bounded below, indexed subset of extended reals can be approximated from above by an element of the set. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
β„²π‘₯πœ‘    &   (πœ‘ β†’ 𝐴 β‰  βˆ…)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ βˆƒπ‘¦ ∈ ℝ βˆ€π‘₯ ∈ 𝐴 𝑦 ≀ 𝐡)    &   (πœ‘ β†’ 𝐢 ∈ ℝ+)    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ 𝐴 𝐡 ≀ (inf(ran (π‘₯ ∈ 𝐴 ↦ 𝐡), ℝ*, < ) +𝑒 𝐢))
 
Theoremxnegre 43415 An extended real is real if and only if its extended negative is real. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝐴 ∈ ℝ* β†’ (𝐴 ∈ ℝ ↔ -𝑒𝐴 ∈ ℝ))
 
Theoremxnegrecl2d 43416 If the extended real negative is real, then the number itself is real. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ -𝑒𝐴 ∈ ℝ)    β‡’   (πœ‘ β†’ 𝐴 ∈ ℝ)
 
Theoremuzxr 43417 An upper integer is an extended real. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝐴 ∈ (β„€β‰₯β€˜π‘€) β†’ 𝐴 ∈ ℝ*)
 
Theoremsupminfxr2 43418* The extended real suprema of a set of extended reals is the extended real negative of the extended real infima of that set's image under extended real negation. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(πœ‘ β†’ 𝐴 βŠ† ℝ*)    β‡’   (πœ‘ β†’ sup(𝐴, ℝ*, < ) = -𝑒inf({π‘₯ ∈ ℝ* ∣ -𝑒π‘₯ ∈ 𝐴}, ℝ*, < ))
 
Theoremxnegred 43419 An extended real is real if and only if its extended negative is real. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    β‡’   (πœ‘ β†’ (𝐴 ∈ ℝ ↔ -𝑒𝐴 ∈ ℝ))
 
Theoremsupminfxrrnmpt 43420* The indexed supremum of a set of reals is the negation of the indexed infimum of that set's image under negation. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
β„²π‘₯πœ‘    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ ℝ*)    β‡’   (πœ‘ β†’ sup(ran (π‘₯ ∈ 𝐴 ↦ 𝐡), ℝ*, < ) = -𝑒inf(ran (π‘₯ ∈ 𝐴 ↦ -𝑒𝐡), ℝ*, < ))
 
Theoremmin1d 43421 The minimum of two numbers is less than or equal to the first. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    β‡’   (πœ‘ β†’ if(𝐴 ≀ 𝐡, 𝐴, 𝐡) ≀ 𝐴)
 
Theoremmin2d 43422 The minimum of two numbers is less than or equal to the second. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    β‡’   (πœ‘ β†’ if(𝐴 ≀ 𝐡, 𝐴, 𝐡) ≀ 𝐡)
 
Theorempnfged 43423 Plus infinity is an upper bound for extended reals. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    β‡’   (πœ‘ β†’ 𝐴 ≀ +∞)
 
Theoremxrnpnfmnf 43424 An extended real that is neither real nor plus infinity, is minus infinity. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ Β¬ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 β‰  +∞)    β‡’   (πœ‘ β†’ 𝐴 = -∞)
 
Theoremuzsscn 43425 An upper set of integers is a subset of the complex numbers. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(β„€β‰₯β€˜π‘€) βŠ† β„‚
 
Theoremabsimnre 43426 The absolute value of the imaginary part of a non-real, complex number, is strictly positive. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ Β¬ 𝐴 ∈ ℝ)    β‡’   (πœ‘ β†’ (absβ€˜(β„‘β€˜π΄)) ∈ ℝ+)
 
Theoremuzsscn2 43427 An upper set of integers is a subset of the complex numbers. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
𝑍 = (β„€β‰₯β€˜π‘€)    β‡’   π‘ βŠ† β„‚
 
Theoremxrtgcntopre 43428 The standard topologies on the extended reals and on the complex numbers, coincide when restricted to the reals. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
((ordTopβ€˜ ≀ ) β†Ύt ℝ) = ((TopOpenβ€˜β„‚fld) β†Ύt ℝ)
 
Theoremabsimlere 43429 The absolute value of the imaginary part of a complex number is a lower bound of the distance to any real number. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    β‡’   (πœ‘ β†’ (absβ€˜(β„‘β€˜π΄)) ≀ (absβ€˜(𝐡 βˆ’ 𝐴)))
 
Theoremrpssxr 43430 The positive reals are a subset of the extended reals. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
ℝ+ βŠ† ℝ*
 
Theoremmonoordxrv 43431* Ordering relation for a monotonic sequence, increasing case. (Contributed by Glauco Siliprandi, 13-Feb-2022.)
(πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    &   ((πœ‘ ∧ π‘˜ ∈ (𝑀...𝑁)) β†’ (πΉβ€˜π‘˜) ∈ ℝ*)    &   ((πœ‘ ∧ π‘˜ ∈ (𝑀...(𝑁 βˆ’ 1))) β†’ (πΉβ€˜π‘˜) ≀ (πΉβ€˜(π‘˜ + 1)))    β‡’   (πœ‘ β†’ (πΉβ€˜π‘€) ≀ (πΉβ€˜π‘))
 
Theoremmonoordxr 43432* Ordering relation for a monotonic sequence, increasing case. (Contributed by Glauco Siliprandi, 13-Feb-2022.)
β„²π‘˜πœ‘    &   β„²π‘˜πΉ    &   (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    &   ((πœ‘ ∧ π‘˜ ∈ (𝑀...𝑁)) β†’ (πΉβ€˜π‘˜) ∈ ℝ*)    &   ((πœ‘ ∧ π‘˜ ∈ (𝑀...(𝑁 βˆ’ 1))) β†’ (πΉβ€˜π‘˜) ≀ (πΉβ€˜(π‘˜ + 1)))    β‡’   (πœ‘ β†’ (πΉβ€˜π‘€) ≀ (πΉβ€˜π‘))
 
Theoremmonoord2xrv 43433* Ordering relation for a monotonic sequence, decreasing case. (Contributed by Glauco Siliprandi, 13-Feb-2022.)
(πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    &   ((πœ‘ ∧ π‘˜ ∈ (𝑀...𝑁)) β†’ (πΉβ€˜π‘˜) ∈ ℝ*)    &   ((πœ‘ ∧ π‘˜ ∈ (𝑀...(𝑁 βˆ’ 1))) β†’ (πΉβ€˜(π‘˜ + 1)) ≀ (πΉβ€˜π‘˜))    β‡’   (πœ‘ β†’ (πΉβ€˜π‘) ≀ (πΉβ€˜π‘€))
 
Theoremmonoord2xr 43434* Ordering relation for a monotonic sequence, decreasing case. (Contributed by Glauco Siliprandi, 13-Feb-2022.)
β„²π‘˜πœ‘    &   β„²π‘˜πΉ    &   (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    &   ((πœ‘ ∧ π‘˜ ∈ (𝑀...𝑁)) β†’ (πΉβ€˜π‘˜) ∈ ℝ*)    &   ((πœ‘ ∧ π‘˜ ∈ (𝑀...(𝑁 βˆ’ 1))) β†’ (πΉβ€˜(π‘˜ + 1)) ≀ (πΉβ€˜π‘˜))    β‡’   (πœ‘ β†’ (πΉβ€˜π‘) ≀ (πΉβ€˜π‘€))
 
Theoremxrpnf 43435* An extended real is plus infinity iff it's larger than all real numbers. (Contributed by Glauco Siliprandi, 13-Feb-2022.)
(𝐴 ∈ ℝ* β†’ (𝐴 = +∞ ↔ βˆ€π‘₯ ∈ ℝ π‘₯ ≀ 𝐴))
 
Theoremxlenegcon1 43436 Extended real version of lenegcon1 11593. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ (-𝑒𝐴 ≀ 𝐡 ↔ -𝑒𝐡 ≀ 𝐴))
 
Theoremxlenegcon2 43437 Extended real version of lenegcon2 11594. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ (𝐴 ≀ -𝑒𝐡 ↔ 𝐡 ≀ -𝑒𝐴))
 
Theorempimxrneun 43438 The preimage of a set of extended reals that does not contain a value 𝐢 is the union of the preimage of the elements smaller than 𝐢 and the preimage of the subset of elements larger than 𝐢. (Contributed by Glauco Siliprandi, 21-Dec-2024.)
β„²π‘₯πœ‘    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ ℝ*)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐢 ∈ ℝ*)    β‡’   (πœ‘ β†’ {π‘₯ ∈ 𝐴 ∣ 𝐡 β‰  𝐢} = ({π‘₯ ∈ 𝐴 ∣ 𝐡 < 𝐢} βˆͺ {π‘₯ ∈ 𝐴 ∣ 𝐢 < 𝐡}))
 
21.38.4  Real intervals
 
Theoremgtnelioc 43439 A real number larger than the upper bound of a left-open right-closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐢 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 < 𝐢)    β‡’   (πœ‘ β†’ Β¬ 𝐢 ∈ (𝐴(,]𝐡))
 
Theoremioossioc 43440 An open interval is a subset of its right closure. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴(,)𝐡) βŠ† (𝐴(,]𝐡)
 
Theoremioondisj2 43441 A condition for two open intervals not to be disjoint. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐴 < 𝐡) ∧ (𝐢 ∈ ℝ* ∧ 𝐷 ∈ ℝ* ∧ 𝐢 < 𝐷)) ∧ (𝐴 < 𝐷 ∧ 𝐷 ≀ 𝐡)) β†’ ((𝐴(,)𝐡) ∩ (𝐢(,)𝐷)) β‰  βˆ…)
 
Theoremioondisj1 43442 A condition for two open intervals not to be disjoint. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐴 < 𝐡) ∧ (𝐢 ∈ ℝ* ∧ 𝐷 ∈ ℝ* ∧ 𝐢 < 𝐷)) ∧ (𝐴 ≀ 𝐢 ∧ 𝐢 < 𝐡)) β†’ ((𝐴(,)𝐡) ∩ (𝐢(,)𝐷)) β‰  βˆ…)
 
Theoremioogtlb 43443 An element of a closed interval is greater than its lower bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐢 ∈ (𝐴(,)𝐡)) β†’ 𝐴 < 𝐢)
 
Theoremevthiccabs 43444* Extreme Value Theorem on y closed interval, for the absolute value of y continuous function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 ≀ 𝐡)    &   (πœ‘ β†’ 𝐹 ∈ ((𝐴[,]𝐡)–cn→ℝ))    β‡’   (πœ‘ β†’ (βˆƒπ‘₯ ∈ (𝐴[,]𝐡)βˆ€π‘¦ ∈ (𝐴[,]𝐡)(absβ€˜(πΉβ€˜π‘¦)) ≀ (absβ€˜(πΉβ€˜π‘₯)) ∧ βˆƒπ‘§ ∈ (𝐴[,]𝐡)βˆ€π‘€ ∈ (𝐴[,]𝐡)(absβ€˜(πΉβ€˜π‘§)) ≀ (absβ€˜(πΉβ€˜π‘€))))
 
Theoremltnelicc 43445 A real number smaller than the lower bound of a closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 < 𝐴)    β‡’   (πœ‘ β†’ Β¬ 𝐢 ∈ (𝐴[,]𝐡))
 
Theoremeliood 43446 Membership in an open real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 < 𝐢)    &   (πœ‘ β†’ 𝐢 < 𝐡)    β‡’   (πœ‘ β†’ 𝐢 ∈ (𝐴(,)𝐡))
 
Theoremiooabslt 43447 An upper bound for the distance from the center of an open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐢 ∈ ((𝐴 βˆ’ 𝐡)(,)(𝐴 + 𝐡)))    β‡’   (πœ‘ β†’ (absβ€˜(𝐴 βˆ’ 𝐢)) < 𝐡)
 
Theoremgtnelicc 43448 A real number greater than the upper bound of a closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐢 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 < 𝐢)    β‡’   (πœ‘ β†’ Β¬ 𝐢 ∈ (𝐴[,]𝐡))
 
Theoremiooinlbub 43449 An open interval has empty intersection with its bounds. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴(,)𝐡) ∩ {𝐴, 𝐡}) = βˆ…
 
Theoremiocgtlb 43450 An element of a left-open right-closed interval is larger than its lower bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐢 ∈ (𝐴(,]𝐡)) β†’ 𝐴 < 𝐢)
 
Theoremiocleub 43451 An element of a left-open right-closed interval is smaller than or equal to its upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐢 ∈ (𝐴(,]𝐡)) β†’ 𝐢 ≀ 𝐡)
 
Theoremeliccd 43452 Membership in a closed real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐢 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 ≀ 𝐢)    &   (πœ‘ β†’ 𝐢 ≀ 𝐡)    β‡’   (πœ‘ β†’ 𝐢 ∈ (𝐴[,]𝐡))
 
Theoremeliccre 43453 A member of a closed interval of reals is real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ ∧ 𝐢 ∈ (𝐴[,]𝐡)) β†’ 𝐢 ∈ ℝ)
 
Theoremeliooshift 43454 Element of an open interval shifted by a displacement. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐢 ∈ ℝ)    &   (πœ‘ β†’ 𝐷 ∈ ℝ)    β‡’   (πœ‘ β†’ (𝐴 ∈ (𝐡(,)𝐢) ↔ (𝐴 + 𝐷) ∈ ((𝐡 + 𝐷)(,)(𝐢 + 𝐷))))
 
Theoremeliocd 43455 Membership in a left-open right-closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 ∈ ℝ*)    &   (πœ‘ β†’ 𝐴 < 𝐢)    &   (πœ‘ β†’ 𝐢 ≀ 𝐡)    β‡’   (πœ‘ β†’ 𝐢 ∈ (𝐴(,]𝐡))
 
Theoremicoltub 43456 An element of a left-closed right-open interval is less than its upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐢 ∈ (𝐴[,)𝐡)) β†’ 𝐢 < 𝐡)
 
Theoremeliocre 43457 A member of a left-open right-closed interval of reals is real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐡 ∈ ℝ ∧ 𝐢 ∈ (𝐴(,]𝐡)) β†’ 𝐢 ∈ ℝ)
 
Theoremiooltub 43458 An element of an open interval is less than its upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐢 ∈ (𝐴(,)𝐡)) β†’ 𝐢 < 𝐡)
 
Theoremioontr 43459 The interior of an interval in the standard topology on ℝ is the open interval itself. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((intβ€˜(topGenβ€˜ran (,)))β€˜(𝐴(,)𝐡)) = (𝐴(,)𝐡)
 
Theoremsnunioo1 43460 The closure of one end of an open real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐴 < 𝐡) β†’ ((𝐴(,)𝐡) βˆͺ {𝐴}) = (𝐴[,)𝐡))
 
Theoremlbioc 43461 A left-open right-closed interval does not contain its left endpoint. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
¬ 𝐴 ∈ (𝐴(,]𝐡)
 
Theoremioomidp 43462 The midpoint is an element of the open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ ∧ 𝐴 < 𝐡) β†’ ((𝐴 + 𝐡) / 2) ∈ (𝐴(,)𝐡))
 
Theoremiccdifioo 43463 If the open inverval is removed from the closed interval, only the bounds are left. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐴 ≀ 𝐡) β†’ ((𝐴[,]𝐡) βˆ– (𝐴(,)𝐡)) = {𝐴, 𝐡})
 
Theoremiccdifprioo 43464 An open interval is the closed interval without the bounds. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ ((𝐴[,]𝐡) βˆ– {𝐴, 𝐡}) = (𝐴(,)𝐡))
 
Theoremioossioobi 43465 Biconditional form of ioossioo 13287. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 ∈ ℝ*)    &   (πœ‘ β†’ 𝐷 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 < 𝐷)    β‡’   (πœ‘ β†’ ((𝐢(,)𝐷) βŠ† (𝐴(,)𝐡) ↔ (𝐴 ≀ 𝐢 ∧ 𝐷 ≀ 𝐡)))
 
Theoremiccshift 43466* A closed interval shifted by a real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝑇 ∈ ℝ)    β‡’   (πœ‘ β†’ ((𝐴 + 𝑇)[,](𝐡 + 𝑇)) = {𝑀 ∈ β„‚ ∣ βˆƒπ‘§ ∈ (𝐴[,]𝐡)𝑀 = (𝑧 + 𝑇)})
 
Theoremiccsuble 43467 An upper bound to the distance of two elements in a closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐢 ∈ (𝐴[,]𝐡))    &   (πœ‘ β†’ 𝐷 ∈ (𝐴[,]𝐡))    β‡’   (πœ‘ β†’ (𝐢 βˆ’ 𝐷) ≀ (𝐡 βˆ’ 𝐴))
 
Theoremiocopn 43468 A left-open right-closed interval is an open set of the standard topology restricted to an interval that contains the original interval and has the same upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   πΎ = (topGenβ€˜ran (,))    &   π½ = (𝐾 β†Ύt (𝐴(,]𝐡))    &   (πœ‘ β†’ 𝐴 ≀ 𝐢)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    β‡’   (πœ‘ β†’ (𝐢(,]𝐡) ∈ 𝐽)
 
Theoremeliccelioc 43469 Membership in a closed interval and in a left-open right-closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐢 ∈ ℝ*)    β‡’   (πœ‘ β†’ (𝐢 ∈ (𝐴(,]𝐡) ↔ (𝐢 ∈ (𝐴[,]𝐡) ∧ 𝐢 β‰  𝐴)))
 
Theoremiooshift 43470* An open interval shifted by a real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝑇 ∈ ℝ)    β‡’   (πœ‘ β†’ ((𝐴 + 𝑇)(,)(𝐡 + 𝑇)) = {𝑀 ∈ β„‚ ∣ βˆƒπ‘§ ∈ (𝐴(,)𝐡)𝑀 = (𝑧 + 𝑇)})
 
Theoremiccintsng 43471 Intersection of two adiacent closed intervals is a singleton. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐢 ∈ ℝ*) ∧ (𝐴 ≀ 𝐡 ∧ 𝐡 ≀ 𝐢)) β†’ ((𝐴[,]𝐡) ∩ (𝐡[,]𝐢)) = {𝐡})
 
Theoremicoiccdif 43472 Left-closed right-open interval gotten by a closed iterval taking away the upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ (𝐴[,)𝐡) = ((𝐴[,]𝐡) βˆ– {𝐡}))
 
Theoremicoopn 43473 A left-closed right-open interval is an open set of the standard topology restricted to an interval that contains the original interval and has the same lower bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐢 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   πΎ = (topGenβ€˜ran (,))    &   π½ = (𝐾 β†Ύt (𝐴[,)𝐡))    &   (πœ‘ β†’ 𝐢 ≀ 𝐡)    β‡’   (πœ‘ β†’ (𝐴[,)𝐢) ∈ 𝐽)
 
Theoremicoub 43474 A left-closed, right-open interval does not contain its upper bound. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝐴 ∈ ℝ* β†’ Β¬ 𝐡 ∈ (𝐴[,)𝐡))
 
Theoremeliccxrd 43475 Membership in a closed real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 ∈ ℝ*)    &   (πœ‘ β†’ 𝐴 ≀ 𝐢)    &   (πœ‘ β†’ 𝐢 ≀ 𝐡)    β‡’   (πœ‘ β†’ 𝐢 ∈ (𝐴[,]𝐡))
 
Theorempnfel0pnf 43476 +∞ is a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
+∞ ∈ (0[,]+∞)
 
Theoremeliccnelico 43477 An element of a closed interval that is not a member of the left-closed right-open interval, must be the upper bound. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 ∈ (𝐴[,]𝐡))    &   (πœ‘ β†’ Β¬ 𝐢 ∈ (𝐴[,)𝐡))    β‡’   (πœ‘ β†’ 𝐢 = 𝐡)
 
Theoremeliccelicod 43478 A member of a closed interval that is not the upper bound, is a member of the left-closed, right-open interval. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 ∈ (𝐴[,]𝐡))    &   (πœ‘ β†’ 𝐢 β‰  𝐡)    β‡’   (πœ‘ β†’ 𝐢 ∈ (𝐴[,)𝐡))
 
Theoremge0xrre 43479 A nonnegative extended real that is not +∞ is a real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
((𝐴 ∈ (0[,]+∞) ∧ 𝐴 β‰  +∞) β†’ 𝐴 ∈ ℝ)
 
Theoremge0lere 43480 A nonnegative extended Real number smaller than or equal to a Real number, is a Real number. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ (0[,]+∞))    &   (πœ‘ β†’ 𝐡 ≀ 𝐴)    β‡’   (πœ‘ β†’ 𝐡 ∈ ℝ)
 
Theoremelicores 43481* Membership in a left-closed, right-open interval with real bounds. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝐴 ∈ ran ([,) β†Ύ (ℝ Γ— ℝ)) ↔ βˆƒπ‘₯ ∈ ℝ βˆƒπ‘¦ ∈ ℝ 𝐴 = (π‘₯[,)𝑦))
 
Theoreminficc 43482 The infimum of a nonempty set, included in a closed interval, is a member of the interval. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝑆 βŠ† (𝐴[,]𝐡))    &   (πœ‘ β†’ 𝑆 β‰  βˆ…)    β‡’   (πœ‘ β†’ inf(𝑆, ℝ*, < ) ∈ (𝐴[,]𝐡))
 
Theoremqinioo 43483 The rational numbers are dense in ℝ. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    β‡’   (πœ‘ β†’ ((β„š ∩ (𝐴(,)𝐡)) = βˆ… ↔ 𝐡 ≀ 𝐴))
 
Theoremlenelioc 43484 A real number smaller than or equal to the lower bound of a left-open right-closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 ≀ 𝐴)    β‡’   (πœ‘ β†’ Β¬ 𝐢 ∈ (𝐴(,]𝐡))
 
Theoremioonct 43485 A nonempty open interval is uncountable. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐴 < 𝐡)    &   πΆ = (𝐴(,)𝐡)    β‡’   (πœ‘ β†’ Β¬ 𝐢 β‰Ό Ο‰)
 
Theoremxrgtnelicc 43486 A real number greater than the upper bound of a closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 < 𝐢)    β‡’   (πœ‘ β†’ Β¬ 𝐢 ∈ (𝐴[,]𝐡))
 
Theoremiccdificc 43487 The difference of two closed intervals with the same lower bound. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 ∈ ℝ*)    &   (πœ‘ β†’ 𝐴 ≀ 𝐡)    β‡’   (πœ‘ β†’ ((𝐴[,]𝐢) βˆ– (𝐴[,]𝐡)) = (𝐡(,]𝐢))
 
Theoremiocnct 43488 A nonempty left-open, right-closed interval is uncountable. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐴 < 𝐡)    &   πΆ = (𝐴(,]𝐡)    β‡’   (πœ‘ β†’ Β¬ 𝐢 β‰Ό Ο‰)
 
Theoremiccnct 43489 A closed interval, with more than one element is uncountable. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐴 < 𝐡)    &   πΆ = (𝐴[,]𝐡)    β‡’   (πœ‘ β†’ Β¬ 𝐢 β‰Ό Ο‰)
 
Theoremiooiinicc 43490* A closed interval expressed as the indexed intersection of open intervals. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    β‡’   (πœ‘ β†’ ∩ 𝑛 ∈ β„• ((𝐴 βˆ’ (1 / 𝑛))(,)(𝐡 + (1 / 𝑛))) = (𝐴[,]𝐡))
 
Theoremiccgelbd 43491 An element of a closed interval is more than or equal to its lower bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 ∈ (𝐴[,]𝐡))    β‡’   (πœ‘ β†’ 𝐴 ≀ 𝐢)
 
Theoremiooltubd 43492 An element of an open interval is less than its upper bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 ∈ (𝐴(,)𝐡))    β‡’   (πœ‘ β†’ 𝐢 < 𝐡)
 
Theoremicoltubd 43493 An element of a left-closed right-open interval is less than its upper bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 ∈ (𝐴[,)𝐡))    β‡’   (πœ‘ β†’ 𝐢 < 𝐡)
 
Theoremqelioo 43494* The rational numbers are dense in ℝ*: any two extended real numbers have a rational between them. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐴 < 𝐡)    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ β„š π‘₯ ∈ (𝐴(,)𝐡))
 
Theoremtgqioo2 43495* Every open set of reals is the (countable) union of open interval with rational bounds. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝐽 = (topGenβ€˜ran (,))    &   (πœ‘ β†’ 𝐴 ∈ 𝐽)    β‡’   (πœ‘ β†’ βˆƒπ‘ž(π‘ž βŠ† ((,) β€œ (β„š Γ— β„š)) ∧ 𝐴 = βˆͺ π‘ž))
 
Theoremiccleubd 43496 An element of a closed interval is less than or equal to its upper bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 ∈ (𝐴[,]𝐡))    β‡’   (πœ‘ β†’ 𝐢 ≀ 𝐡)
 
Theoremelioored 43497 A member of an open interval of reals is a real. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(πœ‘ β†’ 𝐴 ∈ (𝐡(,)𝐢))    β‡’   (πœ‘ β†’ 𝐴 ∈ ℝ)
 
Theoremioogtlbd 43498 An element of a closed interval is greater than its lower bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 ∈ (𝐴(,)𝐡))    β‡’   (πœ‘ β†’ 𝐴 < 𝐢)
 
Theoremioofun 43499 (,) is a function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Fun (,)
 
Theoremicomnfinre 43500 A left-closed, right-open, interval of extended reals, intersected with the Reals. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    β‡’   (πœ‘ β†’ ((-∞[,)𝐴) ∩ ℝ) = (-∞(,)𝐴))
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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 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454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-46948
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