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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | rzalf 43701 | A version of rzal 4509 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
β’ β²π₯ π΄ = β β β’ (π΄ = β β βπ₯ β π΄ π) | ||
Theorem | fvelrnbf 43702 | A version of fvelrnb 6953 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
β’ β²π₯π΄ & β’ β²π₯π΅ & β’ β²π₯πΉ β β’ (πΉ Fn π΄ β (π΅ β ran πΉ β βπ₯ β π΄ (πΉβπ₯) = π΅)) | ||
Theorem | rfcnpre1 43703 | If F is a continuous function with respect to the standard topology, then the preimage A of the values greater than a given extended real B is an open set. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
β’ β²π₯π΅ & β’ β²π₯πΉ & β’ β²π₯π & β’ πΎ = (topGenβran (,)) & β’ π = βͺ π½ & β’ π΄ = {π₯ β π β£ π΅ < (πΉβπ₯)} & β’ (π β π΅ β β*) & β’ (π β πΉ β (π½ Cn πΎ)) β β’ (π β π΄ β π½) | ||
Theorem | ubelsupr 43704* | If U belongs to A and U is an upper bound, then U is the sup of A. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
β’ ((π΄ β β β§ π β π΄ β§ βπ₯ β π΄ π₯ β€ π) β π = sup(π΄, β, < )) | ||
Theorem | fsumcnf 43705* | A finite sum of functions to complex numbers from a common topological space is continuous, without disjoint var constraint x ph. The class expression for B normally contains free variables k and x to index it. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
β’ πΎ = (TopOpenββfld) & β’ (π β π½ β (TopOnβπ)) & β’ (π β π΄ β Fin) & β’ ((π β§ π β π΄) β (π₯ β π β¦ π΅) β (π½ Cn πΎ)) β β’ (π β (π₯ β π β¦ Ξ£π β π΄ π΅) β (π½ Cn πΎ)) | ||
Theorem | mulltgt0 43706 | The product of a negative and a positive number is negative. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
β’ (((π΄ β β β§ π΄ < 0) β§ (π΅ β β β§ 0 < π΅)) β (π΄ Β· π΅) < 0) | ||
Theorem | rspcegf 43707 | A version of rspcev 3613 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
β’ β²π₯π & β’ β²π₯π΄ & β’ β²π₯π΅ & β’ (π₯ = π΄ β (π β π)) β β’ ((π΄ β π΅ β§ π) β βπ₯ β π΅ π) | ||
Theorem | rabexgf 43708 | A version of rabexg 5332 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
β’ β²π₯π΄ β β’ (π΄ β π β {π₯ β π΄ β£ π} β V) | ||
Theorem | fcnre 43709 | A function continuous with respect to the standard topology, is a real mapping. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
β’ πΎ = (topGenβran (,)) & β’ π = βͺ π½ & β’ πΆ = (π½ Cn πΎ) & β’ (π β πΉ β πΆ) β β’ (π β πΉ:πβΆβ) | ||
Theorem | sumsnd 43710* | A sum of a singleton is the term. The deduction version of sumsn 15692. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
β’ (π β β²ππ΅) & β’ β²ππ & β’ ((π β§ π = π) β π΄ = π΅) & β’ (π β π β π) & β’ (π β π΅ β β) β β’ (π β Ξ£π β {π}π΄ = π΅) | ||
Theorem | evthf 43711* | A version of evth 24475 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
β’ β²π₯πΉ & β’ β²π¦πΉ & β’ β²π₯π & β’ β²π¦π & β’ β²π₯π & β’ β²π¦π & β’ π = βͺ π½ & β’ πΎ = (topGenβran (,)) & β’ (π β π½ β Comp) & β’ (π β πΉ β (π½ Cn πΎ)) & β’ (π β π β β ) β β’ (π β βπ₯ β π βπ¦ β π (πΉβπ¦) β€ (πΉβπ₯)) | ||
Theorem | cnfex 43712 | The class of continuous functions between two topologies is a set. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
β’ ((π½ β Top β§ πΎ β Top) β (π½ Cn πΎ) β V) | ||
Theorem | fnchoice 43713* | For a finite set, a choice function exists, without using the axiom of choice. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
β’ (π΄ β Fin β βπ(π Fn π΄ β§ βπ₯ β π΄ (π₯ β β β (πβπ₯) β π₯))) | ||
Theorem | refsumcn 43714* | A finite sum of continuous real functions, from a common topological space, is continuous. The class expression for B normally contains free variables k and x to index it. See fsumcn 24386 for the analogous theorem on continuous complex functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
β’ β²π₯π & β’ πΎ = (topGenβran (,)) & β’ (π β π½ β (TopOnβπ)) & β’ (π β π΄ β Fin) & β’ ((π β§ π β π΄) β (π₯ β π β¦ π΅) β (π½ Cn πΎ)) β β’ (π β (π₯ β π β¦ Ξ£π β π΄ π΅) β (π½ Cn πΎ)) | ||
Theorem | rfcnpre2 43715 | If πΉ is a continuous function with respect to the standard topology, then the preimage A of the values smaller than a given extended real π΅, is an open set. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
β’ β²π₯π΅ & β’ β²π₯πΉ & β’ β²π₯π & β’ πΎ = (topGenβran (,)) & β’ π = βͺ π½ & β’ π΄ = {π₯ β π β£ (πΉβπ₯) < π΅} & β’ (π β π΅ β β*) & β’ (π β πΉ β (π½ Cn πΎ)) β β’ (π β π΄ β π½) | ||
Theorem | cncmpmax 43716* | When the hypothesis for the extreme value theorem hold, then the sup of the range of the function belongs to the range, it is real and it an upper bound of the range. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
β’ π = βͺ π½ & β’ πΎ = (topGenβran (,)) & β’ (π β π½ β Comp) & β’ (π β πΉ β (π½ Cn πΎ)) & β’ (π β π β β ) β β’ (π β (sup(ran πΉ, β, < ) β ran πΉ β§ sup(ran πΉ, β, < ) β β β§ βπ‘ β π (πΉβπ‘) β€ sup(ran πΉ, β, < ))) | ||
Theorem | rfcnpre3 43717* | If F is a continuous function with respect to the standard topology, then the preimage A of the values greater than or equal to a given real B is a closed set. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
β’ β²π‘πΉ & β’ πΎ = (topGenβran (,)) & β’ π = βͺ π½ & β’ π΄ = {π‘ β π β£ π΅ β€ (πΉβπ‘)} & β’ (π β π΅ β β) & β’ (π β πΉ β (π½ Cn πΎ)) β β’ (π β π΄ β (Clsdβπ½)) | ||
Theorem | rfcnpre4 43718* | If F is a continuous function with respect to the standard topology, then the preimage A of the values less than or equal to a given real B is a closed set. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
β’ β²π‘πΉ & β’ πΎ = (topGenβran (,)) & β’ π = βͺ π½ & β’ π΄ = {π‘ β π β£ (πΉβπ‘) β€ π΅} & β’ (π β π΅ β β) & β’ (π β πΉ β (π½ Cn πΎ)) β β’ (π β π΄ β (Clsdβπ½)) | ||
Theorem | sumpair 43719* | Sum of two distinct complex values. The class expression for π΄ and π΅ normally contain free variable π to index it. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
β’ (π β β²ππ·) & β’ (π β β²ππΈ) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β π· β β) & β’ (π β πΈ β β) & β’ (π β π΄ β π΅) & β’ ((π β§ π = π΄) β πΆ = π·) & β’ ((π β§ π = π΅) β πΆ = πΈ) β β’ (π β Ξ£π β {π΄, π΅}πΆ = (π· + πΈ)) | ||
Theorem | rfcnnnub 43720* | Given a real continuous function πΉ defined on a compact topological space, there is always a positive integer that is a strict upper bound of its range. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
β’ β²π‘πΉ & β’ β²π‘π & β’ πΎ = (topGenβran (,)) & β’ (π β π½ β Comp) & β’ π = βͺ π½ & β’ (π β π β β ) & β’ πΆ = (π½ Cn πΎ) & β’ (π β πΉ β πΆ) β β’ (π β βπ β β βπ‘ β π (πΉβπ‘) < π) | ||
Theorem | refsum2cnlem1 43721* | This is the core Lemma for refsum2cn 43722: the sum of two continuous real functions (from a common topological space) is continuous. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
β’ β²π₯π΄ & β’ β²π₯πΉ & β’ β²π₯πΊ & β’ β²π₯π & β’ π΄ = (π β {1, 2} β¦ if(π = 1, πΉ, πΊ)) & β’ πΎ = (topGenβran (,)) & β’ (π β π½ β (TopOnβπ)) & β’ (π β πΉ β (π½ Cn πΎ)) & β’ (π β πΊ β (π½ Cn πΎ)) β β’ (π β (π₯ β π β¦ ((πΉβπ₯) + (πΊβπ₯))) β (π½ Cn πΎ)) | ||
Theorem | refsum2cn 43722* | The sum of two continuus real functions (from a common topological space) is continuous. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
β’ β²π₯πΉ & β’ β²π₯πΊ & β’ β²π₯π & β’ πΎ = (topGenβran (,)) & β’ (π β π½ β (TopOnβπ)) & β’ (π β πΉ β (π½ Cn πΎ)) & β’ (π β πΊ β (π½ Cn πΎ)) β β’ (π β (π₯ β π β¦ ((πΉβπ₯) + (πΊβπ₯))) β (π½ Cn πΎ)) | ||
Theorem | adantlllr 43723 | Deduction adding a conjunct to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ ((((π β§ π) β§ π) β§ π) β π) β β’ (((((π β§ π) β§ π) β§ π) β§ π) β π) | ||
Theorem | 3adantlr3 43724 | Deduction adding a conjunct to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ (((π β§ (π β§ π)) β§ π) β π) β β’ (((π β§ (π β§ π β§ π)) β§ π) β π) | ||
Theorem | 3adantll2 43725 | Deduction adding a conjunct to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ ((((π β§ π) β§ π) β§ π) β π) β β’ ((((π β§ π β§ π) β§ π) β§ π) β π) | ||
Theorem | 3adantll3 43726 | Deduction adding a conjunct to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ ((((π β§ π) β§ π) β§ π) β π) β β’ ((((π β§ π β§ π) β§ π) β§ π) β π) | ||
Theorem | ssnel 43727 | If not element of a set, then not element of a subset. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ ((π΄ β π΅ β§ Β¬ πΆ β π΅) β Β¬ πΆ β π΄) | ||
Theorem | elabrexg 43728* | Elementhood in an image set. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ ((π₯ β π΄ β§ π΅ β π) β π΅ β {π¦ β£ βπ₯ β π΄ π¦ = π΅}) | ||
Theorem | sncldre 43729 | A singleton is closed w.r.t. the standard topology on the reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ (π΄ β β β {π΄} β (Clsdβ(topGenβran (,)))) | ||
Theorem | n0p 43730 | A polynomial with a nonzero coefficient is not the zero polynomial. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
β’ ((π β (Polyββ€) β§ π β β0 β§ ((coeffβπ)βπ) β 0) β π β 0π) | ||
Theorem | pm2.65ni 43731 | Inference rule for proof by contradiction. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
β’ (Β¬ π β π) & β’ (Β¬ π β Β¬ π) β β’ π | ||
Theorem | pwssfi 43732 | Every element of the power set of π΄ is finite if and only if π΄ is finite. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
β’ (π΄ β π β (π΄ β Fin β π« π΄ β Fin)) | ||
Theorem | iuneq2df 43733 | Equality deduction for indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
β’ β²π₯π & β’ ((π β§ π₯ β π΄) β π΅ = πΆ) β β’ (π β βͺ π₯ β π΄ π΅ = βͺ π₯ β π΄ πΆ) | ||
Theorem | nnfoctb 43734* | There exists a mapping from β onto any (nonempty) countable set. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
β’ ((π΄ βΌ Ο β§ π΄ β β ) β βπ π:ββontoβπ΄) | ||
Theorem | ssinss1d 43735 | Intersection preserves subclass relationship. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
β’ (π β π΄ β πΆ) β β’ (π β (π΄ β© π΅) β πΆ) | ||
Theorem | elpwinss 43736 | An element of the powerset of π΅ intersected with anything, is a subset of π΅. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
β’ (π΄ β (π« π΅ β© πΆ) β π΄ β π΅) | ||
Theorem | unidmex 43737 | If πΉ is a set, then βͺ dom πΉ is a set. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
β’ (π β πΉ β π) & β’ π = βͺ dom πΉ β β’ (π β π β V) | ||
Theorem | ndisj2 43738* | A non-disjointness condition. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
β’ (π₯ = π¦ β π΅ = πΆ) β β’ (Β¬ Disj π₯ β π΄ π΅ β βπ₯ β π΄ βπ¦ β π΄ (π₯ β π¦ β§ (π΅ β© πΆ) β β )) | ||
Theorem | zenom 43739 | The set of integer numbers is equinumerous to omega (the set of finite ordinal numbers). (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
β’ β€ β Ο | ||
Theorem | uzwo4 43740* | Well-ordering principle: any nonempty subset of an upper set of integers has the least element. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
β’ β²ππ & β’ (π = π β (π β π)) β β’ ((π β (β€β₯βπ) β§ βπ β π π) β βπ β π (π β§ βπ β π (π < π β Β¬ π))) | ||
Theorem | unisn0 43741 | The union of the singleton of the empty set is the empty set. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
β’ βͺ {β } = β | ||
Theorem | ssin0 43742 | If two classes are disjoint, two respective subclasses are disjoint. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
β’ (((π΄ β© π΅) = β β§ πΆ β π΄ β§ π· β π΅) β (πΆ β© π·) = β ) | ||
Theorem | inabs3 43743 | Absorption law for intersection. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
β’ (πΆ β π΅ β ((π΄ β© π΅) β© πΆ) = (π΄ β© πΆ)) | ||
Theorem | pwpwuni 43744 | Relationship between power class and union. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
β’ (π΄ β π β (π΄ β π« π« π΅ β βͺ π΄ β π« π΅)) | ||
Theorem | disjiun2 43745* | In a disjoint collection, an indexed union is disjoint from an additional term. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
β’ (π β Disj π₯ β π΄ π΅) & β’ (π β πΆ β π΄) & β’ (π β π· β (π΄ β πΆ)) & β’ (π₯ = π· β π΅ = πΈ) β β’ (π β (βͺ π₯ β πΆ π΅ β© πΈ) = β ) | ||
Theorem | 0pwfi 43746 | The empty set is in any power set, and it's finite. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
β’ β β (π« π΄ β© Fin) | ||
Theorem | ssinss2d 43747 | Intersection preserves subclass relationship. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
β’ (π β π΅ β πΆ) β β’ (π β (π΄ β© π΅) β πΆ) | ||
Theorem | zct 43748 | The set of integer numbers is countable. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
β’ β€ βΌ Ο | ||
Theorem | pwfin0 43749 | A finite set always belongs to a power class. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
β’ (π« π΄ β© Fin) β β | ||
Theorem | uzct 43750 | An upper integer set is countable. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
β’ π = (β€β₯βπ) β β’ π βΌ Ο | ||
Theorem | iunxsnf 43751* | A singleton index picks out an instance of an indexed union's argument. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
β’ β²π₯πΆ & β’ π΄ β V & β’ (π₯ = π΄ β π΅ = πΆ) β β’ βͺ π₯ β {π΄}π΅ = πΆ | ||
Theorem | fiiuncl 43752* | 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.) |
β’ β²π₯π & β’ ((π β§ π₯ β π΄) β π΅ β π·) & β’ ((π β§ π¦ β π· β§ π§ β π·) β (π¦ βͺ π§) β π·) & β’ (π β π΄ β Fin) & β’ (π β π΄ β β ) β β’ (π β βͺ π₯ β π΄ π΅ β π·) | ||
Theorem | iunp1 43753* | The addition of the next set to a union indexed by a finite set of sequential integers. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
β’ β²ππ΅ & β’ (π β π β (β€β₯βπ)) & β’ (π = (π + 1) β π΄ = π΅) β β’ (π β βͺ π β (π...(π + 1))π΄ = (βͺ π β (π...π)π΄ βͺ π΅)) | ||
Theorem | fiunicl 43754* | 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.) |
β’ ((π β§ π₯ β π΄ β§ π¦ β π΄) β (π₯ βͺ π¦) β π΄) & β’ (π β π΄ β Fin) & β’ (π β π΄ β β ) β β’ (π β βͺ π΄ β π΄) | ||
Theorem | ixpeq2d 43755 | Equality theorem for infinite Cartesian product. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
β’ β²π₯π & β’ ((π β§ π₯ β π΄) β π΅ = πΆ) β β’ (π β Xπ₯ β π΄ π΅ = Xπ₯ β π΄ πΆ) | ||
Theorem | disjxp1 43756* | The sets of a cartesian product are disjoint if the sets in the first argument are disjoint. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
β’ (π β Disj π₯ β π΄ π΅) β β’ (π β Disj π₯ β π΄ (π΅ Γ πΆ)) | ||
Theorem | disjsnxp 43757* | The sets in the cartesian product of singletons with other sets, are disjoint. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
β’ Disj π β π΄ ({π} Γ π΅) | ||
Theorem | eliind 43758* | Membership in indexed intersection. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
β’ (π β π΄ β β© π₯ β π΅ πΆ) & β’ (π β πΎ β π΅) & β’ (π₯ = πΎ β (π΄ β πΆ β π΄ β π·)) β β’ (π β π΄ β π·) | ||
Theorem | rspcef 43759 | Restricted existential specialization, using implicit substitution. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
β’ β²π₯π & β’ β²π₯π΄ & β’ β²π₯π΅ & β’ (π₯ = π΄ β (π β π)) β β’ ((π΄ β π΅ β§ π) β βπ₯ β π΅ π) | ||
Theorem | inn0f 43760 | A nonempty intersection. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
β’ β²π₯π΄ & β’ β²π₯π΅ β β’ ((π΄ β© π΅) β β β βπ₯ β π΄ π₯ β π΅) | ||
Theorem | ixpssmapc 43761* | An infinite Cartesian product is a subset of set exponentiation. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
β’ β²π₯π & β’ (π β πΆ β π) & β’ ((π β§ π₯ β π΄) β π΅ β πΆ) β β’ (π β Xπ₯ β π΄ π΅ β (πΆ βm π΄)) | ||
Theorem | inn0 43762* | A nonempty intersection. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
β’ ((π΄ β© π΅) β β β βπ₯ β π΄ π₯ β π΅) | ||
Theorem | elintd 43763* | Membership in class intersection. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
β’ β²π₯π & β’ (π β π΄ β π) & β’ ((π β§ π₯ β π΅) β π΄ β π₯) β β’ (π β π΄ β β© π΅) | ||
Theorem | ssdf 43764* | A sufficient condition for a subclass relationship. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
β’ β²π₯π & β’ ((π β§ π₯ β π΄) β π₯ β π΅) β β’ (π β π΄ β π΅) | ||
Theorem | brneqtrd 43765 | Substitution of equal classes into the negation of a binary relation. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
β’ (π β Β¬ π΄π π΅) & β’ (π β π΅ = πΆ) β β’ (π β Β¬ π΄π πΆ) | ||
Theorem | ssnct 43766 | A set containing an uncountable set is itself uncountable. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
β’ (π β Β¬ π΄ βΌ Ο) & β’ (π β π΄ β π΅) β β’ (π β Β¬ π΅ βΌ Ο) | ||
Theorem | ssuniint 43767* | Sufficient condition for being a subclass of the union of an intersection. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
β’ β²π₯π & β’ (π β π΄ β π) & β’ ((π β§ π₯ β π΅) β π΄ β π₯) β β’ (π β π΄ β βͺ β© π΅) | ||
Theorem | elintdv 43768* | Membership in class intersection. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
β’ (π β π΄ β π) & β’ ((π β§ π₯ β π΅) β π΄ β π₯) β β’ (π β π΄ β β© π΅) | ||
Theorem | ssd 43769* | A sufficient condition for a subclass relationship. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
β’ ((π β§ π₯ β π΄) β π₯ β π΅) β β’ (π β π΄ β π΅) | ||
Theorem | ralimralim 43770 | Introducing any antecedent in a restricted universal quantification. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
β’ (βπ₯ β π΄ π β βπ₯ β π΄ (π β π)) | ||
Theorem | snelmap 43771 | Membership of the element in the range of a constant map. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β π΄ β β ) & β’ (π β (π΄ Γ {π₯}) β (π΅ βm π΄)) β β’ (π β π₯ β π΅) | ||
Theorem | xrnmnfpnf 43772 | An extended real that is neither real nor minus infinity, is plus infinity. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
β’ (π β π΄ β β*) & β’ (π β Β¬ π΄ β β) & β’ (π β π΄ β -β) β β’ (π β π΄ = +β) | ||
Theorem | nelrnmpt 43773* | Non-membership in the range of a function in maps-to notaion. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
β’ β²π₯π & β’ πΉ = (π₯ β π΄ β¦ π΅) & β’ (π β πΆ β π) & β’ ((π β§ π₯ β π΄) β πΆ β π΅) β β’ (π β Β¬ πΆ β ran πΉ) | ||
Theorem | iuneq1i 43774* | Equality theorem for indexed union. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
β’ π΄ = π΅ β β’ βͺ π₯ β π΄ πΆ = βͺ π₯ β π΅ πΆ | ||
Theorem | nssrex 43775* | Negation of subclass relationship. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
β’ (Β¬ π΄ β π΅ β βπ₯ β π΄ Β¬ π₯ β π΅) | ||
Theorem | ssinc 43776* | Inclusion relation for a monotonic sequence of sets. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
β’ (π β π β (β€β₯βπ)) & β’ ((π β§ π β (π..^π)) β (πΉβπ) β (πΉβ(π + 1))) β β’ (π β (πΉβπ) β (πΉβπ)) | ||
Theorem | ssdec 43777* | Inclusion relation for a monotonic sequence of sets. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
β’ (π β π β (β€β₯βπ)) & β’ ((π β§ π β (π..^π)) β (πΉβ(π + 1)) β (πΉβπ)) β β’ (π β (πΉβπ) β (πΉβπ)) | ||
Theorem | elixpconstg 43778* | Membership in an infinite Cartesian product of a constant π΅. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
β’ (πΉ β π β (πΉ β Xπ₯ β π΄ π΅ β πΉ:π΄βΆπ΅)) | ||
Theorem | iineq1d 43779* | Equality theorem for indexed intersection. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
β’ (π β π΄ = π΅) β β’ (π β β© π₯ β π΄ πΆ = β© π₯ β π΅ πΆ) | ||
Theorem | metpsmet 43780 | A metric is a pseudometric. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
β’ (π· β (Metβπ) β π· β (PsMetβπ)) | ||
Theorem | ixpssixp 43781 | Subclass theorem for infinite Cartesian product. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
β’ β²π₯π & β’ ((π β§ π₯ β π΄) β π΅ β πΆ) β β’ (π β Xπ₯ β π΄ π΅ β Xπ₯ β π΄ πΆ) | ||
Theorem | ballss3 43782* | A sufficient condition for a ball being a subset. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
β’ β²π₯π & β’ (π β π· β (PsMetβπ)) & β’ (π β π β π) & β’ (π β π β β*) & β’ ((π β§ π₯ β π β§ (ππ·π₯) < π ) β π₯ β π΄) β β’ (π β (π(ballβπ·)π ) β π΄) | ||
Theorem | iunincfi 43783* | Given a sequence of increasing sets, the union of a finite subsequence, is its last element. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
β’ (π β π β (β€β₯βπ)) & β’ ((π β§ π β (π..^π)) β (πΉβπ) β (πΉβ(π + 1))) β β’ (π β βͺ π β (π...π)(πΉβπ) = (πΉβπ)) | ||
Theorem | nsstr 43784 | If it's not a subclass, it's not a subclass of a smaller one. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
β’ ((Β¬ π΄ β π΅ β§ πΆ β π΅) β Β¬ π΄ β πΆ) | ||
Theorem | rexanuz3 43785* | Combine two different upper integer properties into one, for a single integer. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
β’ β²ππ & β’ π = (β€β₯βπ) & β’ (π β βπ β π βπ β (β€β₯βπ)π) & β’ (π β βπ β π βπ β (β€β₯βπ)π) & β’ (π = π β (π β π)) & β’ (π = π β (π β π)) β β’ (π β βπ β π (π β§ π)) | ||
Theorem | cbvmpo2 43786* | Rule to change the second bound variable in a maps-to function, using implicit substitution. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
β’ β²π¦π΄ & β’ β²π€π΄ & β’ β²π€πΆ & β’ β²π¦πΈ & β’ (π¦ = π€ β πΆ = πΈ) β β’ (π₯ β π΄, π¦ β π΅ β¦ πΆ) = (π₯ β π΄, π€ β π΅ β¦ πΈ) | ||
Theorem | cbvmpo1 43787* | Rule to change the first bound variable in a maps-to function, using implicit substitution. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
β’ β²π₯π΅ & β’ β²π§π΅ & β’ β²π§πΆ & β’ β²π₯πΈ & β’ (π₯ = π§ β πΆ = πΈ) β β’ (π₯ β π΄, π¦ β π΅ β¦ πΆ) = (π§ β π΄, π¦ β π΅ β¦ πΈ) | ||
Theorem | eliuniin 43788* | Indexed union of indexed intersections. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
β’ π΄ = βͺ π₯ β π΅ β© π¦ β πΆ π· β β’ (π β π β (π β π΄ β βπ₯ β π΅ βπ¦ β πΆ π β π·)) | ||
Theorem | ssabf 43789 | Subclass of a class abstraction. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
β’ β²π₯π΄ β β’ (π΄ β {π₯ β£ π} β βπ₯(π₯ β π΄ β π)) | ||
Theorem | pssnssi 43790 | A proper subclass does not include the other class. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
β’ π΄ β π΅ β β’ Β¬ π΅ β π΄ | ||
Theorem | rabidim2 43791 | Membership in a restricted abstraction, implication. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
β’ (π₯ β {π₯ β π΄ β£ π} β π) | ||
Theorem | eluni2f 43792* | Membership in class union. Restricted quantifier version. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
β’ β²π₯π΄ & β’ β²π₯π΅ β β’ (π΄ β βͺ π΅ β βπ₯ β π΅ π΄ β π₯) | ||
Theorem | eliin2f 43793* | Membership in indexed intersection. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
β’ β²π₯π΅ β β’ (π΅ β β β (π΄ β β© π₯ β π΅ πΆ β βπ₯ β π΅ π΄ β πΆ)) | ||
Theorem | nssd 43794 | Negation of subclass relationship. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
β’ (π β π β π΄) & β’ (π β Β¬ π β π΅) β β’ (π β Β¬ π΄ β π΅) | ||
Theorem | iineq12dv 43795* | Equality deduction for indexed intersection. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
β’ (π β π΄ = π΅) & β’ ((π β§ π₯ β π΅) β πΆ = π·) β β’ (π β β© π₯ β π΄ πΆ = β© π₯ β π΅ π·) | ||
Theorem | supxrcld 43796 | The supremum of an arbitrary set of extended reals is an extended real. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
β’ (π β π΄ β β*) β β’ (π β sup(π΄, β*, < ) β β*) | ||
Theorem | elrestd 43797 | A sufficient condition for being an open set of a subspace topology. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
β’ (π β π½ β π) & β’ (π β π΅ β π) & β’ (π β π β π½) & β’ π΄ = (π β© π΅) β β’ (π β π΄ β (π½ βΎt π΅)) | ||
Theorem | eliuniincex 43798* | Counterexample to show that the additional conditions in eliuniin 43788 and eliuniin2 43809 are actually needed. Notice that the definition of π΄ is not even needed (it can be any class). (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
β’ π΅ = {β } & β’ πΆ = β & β’ π· = β & β’ π = V β β’ Β¬ (π β π΄ β βπ₯ β π΅ βπ¦ β πΆ π β π·) | ||
Theorem | eliincex 43799* | Counterexample to show that the additional conditions in eliin 5003 and eliin2 43805 are actually needed. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
β’ π΄ = V & β’ π΅ = β β β’ Β¬ (π΄ β β© π₯ β π΅ πΆ β βπ₯ β π΅ π΄ β πΆ) | ||
Theorem | eliinid 43800* | Membership in an indexed intersection implies membership in any intersected set. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
β’ ((π΄ β β© π₯ β π΅ πΆ β§ π₯ β π΅) β π΄ β πΆ) |
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