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Type | Label | Description |
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Statement | ||
Theorem | elunnel2 37401 | A member of a union that is not a member of the second class, is a member of the first class. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | adantlllr 37402 | Deduction adding a conjunct to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | 3adantlr3 37403 | Deduction adding a conjunct to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | nnxrd 37404 | A natural number is an extended real. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | 3adantll2 37405 | Deduction adding a conjunct to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | 3adantll3 37406 | Deduction adding a conjunct to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | ssnel 37407 | If not element of a set, then not element of a subset. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | adant423 37408 | Deduction adding conjuncts to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | jcn 37409 | Inference joining the consequents of two premises. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | elabrexg 37410* | Elementhood in an image set. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | unicntop 37411 | The underlying set of the standard topology on the complex numers is the set of complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | ifeq123d 37412 | Equality deduction for conditional operator. (Contributed by Glauco Siliprandi, 11-Dec-2019.) AV: This theorem already exists as ifbieq12d 3920. TODO (NM): Please replace the usage of this theorem by ifbieq12d 3920 then delete this theorem. (New usage is discouraged.) |
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Theorem | sncldre 37413 | A singleton is closed w.r.t. the standard topology on the reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | cnopn 37414 | The set of complex numbers is open with respect to the standard topology on complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | n0p 37415 | A polynomial with a nonzero coefficient is not the zero polynomial. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
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Theorem | rabeqd 37416* | Equality theorem for restricted class abstractions. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
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Theorem | pm2.65ni 37417 | Inference rule for proof by contradiction. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
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Theorem | raleqdOLD 37418* | Equality theorem for restricted universal quantifier. (Contributed by Glauco Siliprandi, 5-Apr-2020.) Obsolete as of 18-Jul-2020. Use raleqdv 3005 instead, which is the same. (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | elini 37419 | Membership in an intersection of two classes. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | pwssfi 37420 |
Every element of the power set of ![]() ![]() |
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Theorem | iuneq2df 37421 | Equality deduction for indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | nnfoctb 37422* |
There exists a mapping from ![]() |
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Theorem | prssd 37423 | A pair of elements of a class is a subset of the class. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | ssinss1d 37424 | Intersection preserves subclass relationship. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | 0un 37425 | The union of the empty set with a class is itself. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | elpwinss 37426 |
An element of the powerset of ![]() ![]() |
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Theorem | unidmex 37427 |
If ![]() ![]() ![]() ![]() |
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Theorem | ndisj2 37428* | A non disjointness condition. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | zenom 37429 | The set of integer numbers is equinumerous to omega (the set of finite ordinal numbers). (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | rexsngf 37430* | Restricted existential quantification over a singleton. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | uzwo4 37431* | Well-ordering principle: any nonempty subset of an upper set of integers has the least element. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | 0in 37432 | The intersection of the empty set with a class is the empty set. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | unisn0 37433 | The union of the singleton of the empty set is the empty set. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | ssin0 37434 | If two classes are disjoint, two respective subclasses are disjoint. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | inabs3 37435 | Absorption law for intersection. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | pwpwuni 37436 | Relationship between power class and union. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | disjiun2 37437* | In a disjoint collection, an indexed union is disjoint from an additional term. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | 0pwfi 37438 | The empty set is in any power set, and it's finite. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | ssinss2d 37439 | Intersection preserves subclass relationship. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | zct 37440 | The set of integer numbers is countable. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | iunxsngf2 37441* | A singleton index picks out an instance of an indexed union's argument. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | pwfin0 37442 | A finite set always belongs to a power class. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | uzct 37443 | An upper integer set is countable. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | iunxsnf 37444* | A singleton index picks out an instance of an indexed union's argument. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | fiiuncl 37445* | If a set is closed under the union of two sets, then it is closed under finite indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | iunp1 37446* | The addition of the next set to a union indexed by a finite set of sequential integers. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | fiunicl 37447* | If a set is closed under the union of two sets, then it is closed under finite union. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | nnnfi 37448 | The set of positive integers is infinite. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
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Theorem | ixpeq2d 37449 | Equality theorem for infinite Cartesian product. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
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Theorem | disjxp1 37450* | The sets of a cartesian product are disjoint if the sets in the first argument are disjoint. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
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Theorem | elpwd 37451 | Membership in a power class. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
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Theorem | disjsnxp 37452* | The sets in the cartesian product of singletons with other sets, are disjoint. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
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Theorem | ssfid 37453 | A subset of a finite set is finite. (Contributed by Glauco Siliprandi, 21-Nov-2020.) |
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Theorem | eliind 37454* | Membership in indexed intersection. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
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Theorem | rspcef 37455 | Restricted existential specialization, using implicit substitution. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
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Theorem | inn0f 37456 | A non-empty intersection. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
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Theorem | ixpssmapc 37457* | An infinite Cartesian product is a subset of set exponentiation. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
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Theorem | inn0 37458* | A non-empty intersection. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
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Theorem | elintd 37459* | Membership in class intersection. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
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Theorem | eqneltri 37460 | If a class is not an element of another class, an equal class is also not an element. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
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Theorem | eqnbrtrd 37461 | Substitution of equal classes into the negation of a binary relation. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
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Theorem | ssdf 37462* | A sufficient condition for a subclass relationship. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
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Theorem | brneqtrd 37463 | Substitution of equal classes into the negation of a binary relation. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
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Theorem | ssnct 37464 | A set containing an uncountable set is itself uncountable. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
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Theorem | ssuniint 37465* | Sufficient condition for being a subclass of the union of an intersection. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
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Theorem | elintdv 37466* | Membership in class intersection. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
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Theorem | ssd 37467* | A sufficient condition for a subclass relationship. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
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Theorem | ralimralim 37468 | Introducing any antecedent in a restricted universal quantification. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
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Theorem | snelmap 37469 | Membership of the element in the range of a constant map. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
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Theorem | difex 37470 | Existence of a difference. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
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Theorem | dfcleqf 37471 | Equality connective between classes. Same as dfcleq 2456, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
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Theorem | xrnmnfpnf 37472 | An extended real that is neither real nor minus infinity, is plus infinity. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
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Theorem | nelrnmpt 37473* | Non-membership in the range of a function in maps-to notaion. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
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Theorem | snn0d 37474 | The singleton of a set is not empty. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
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Theorem | rabid3 37475 | Membership in a restricted abstraction (special case). (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
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Theorem | opelxpd 37476 | Ordered pair membership in a Cartesian product (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
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Theorem | iuneq1i 37477* | Equality theorem for indexed union. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
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Theorem | nssrex 37478* | Negation of subclass relationship. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
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Theorem | nelpr2 37479 | If a class is not an element of an unordered pair, it is not the second listed element. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
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Theorem | nelpr1 37480 | If a class is not an element of an unordered pair, it is not the first listed element. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
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Theorem | iunssf 37481 | Subset theorem for an indexed union. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
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Theorem | elpwi2 37482 | Membership in a power class. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
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Theorem | unima 37483 | Image of a union. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | feq1dd 37484 | Equality deduction for functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | fnresdmss 37485 | A function does not change when restricted to a set that contains its domain. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | fmptsnxp 37486* | Maps-to notation and cross product for a singleton function. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | mptex2 37487* | If a class given as a map-to notation is a set, it's image values are set. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | fvmpt2bd 37488* | Value of a function given by the "maps to" notation. Deduction version. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | rnmptfi 37489* | The range of a function with finite domain is finite. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | fresin2 37490 | Restriction of a function with respect to the intersection with its domain. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | rnmptc 37491* | Range of a constant function in map to notation. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | ffi 37492 | A function with finite domain is finite. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | suprnmpt 37493* | An explicit bound for the range of a bounded function. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | rnffi 37494 | The range of a function with finite domain is finite. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | mptelpm 37495* | A function in maps-to notation is a partial map . (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
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Theorem | f1oeq3d 37496 | Equality deduction for one-to-one onto functions. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | fimass 37497 | The image of a class is a subset of its codomain. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | rnmptpr 37498* | Range of a function defined on an unordered pair. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | resmpti 37499* | Restriction of the mapping operation. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | founiiun 37500* | Union expressed as an indexed union, when a map onto is given. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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