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Theorem List for Intuitionistic Logic Explorer - 3201-3300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremssinss1 3201 Intersection preserves subclass relationship. (Contributed by NM, 14-Sep-1999.)

Theoreminss 3202 Inclusion of an intersection of two classes. (Contributed by NM, 30-Oct-2014.)

2.1.13.4  Combinations of difference, union, and intersection of two classes

Theoremunabs 3203 Absorption law for union. (Contributed by NM, 16-Apr-2006.)

Theoreminabs 3204 Absorption law for intersection. (Contributed by NM, 16-Apr-2006.)

Theoremssddif 3205 Double complement and subset. Similar to ddifss 3209 but inside a class instead of the universal class . In classical logic the subset operation on the right hand side could be an equality (that is, ). (Contributed by Jim Kingdon, 24-Jul-2018.)

Theoremunssdif 3206 Union of two classes and class difference. In classical logic this would be an equality. (Contributed by Jim Kingdon, 24-Jul-2018.)

Theoreminssdif 3207 Intersection of two classes and class difference. In classical logic this would be an equality. (Contributed by Jim Kingdon, 24-Jul-2018.)

Theoremdifin 3208 Difference with intersection. Theorem 33 of [Suppes] p. 29. (Contributed by NM, 31-Mar-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremddifss 3209 Double complement under universal class. In classical logic (or given an additional hypothesis, as in ddifnel 3104), this is equality rather than subset. (Contributed by Jim Kingdon, 24-Jul-2018.)

Theoremunssin 3210 Union as a subset of class complement and intersection (De Morgan's law). One direction of the definition of union in [Mendelson] p. 231. This would be an equality, rather than subset, in classical logic. (Contributed by Jim Kingdon, 25-Jul-2018.)

Theoreminssun 3211 Intersection in terms of class difference and union (De Morgan's law). Similar to Exercise 4.10(n) of [Mendelson] p. 231. This would be an equality, rather than subset, in classical logic. (Contributed by Jim Kingdon, 25-Jul-2018.)

Theoreminssddif 3212 Intersection of two classes and class difference. In classical logic, such as Exercise 4.10(q) of [Mendelson] p. 231, this is an equality rather than subset. (Contributed by Jim Kingdon, 26-Jul-2018.)

Theoreminvdif 3213 Intersection with universal complement. Remark in [Stoll] p. 20. (Contributed by NM, 17-Aug-2004.)

Theoremindif 3214 Intersection with class difference. Theorem 34 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)

Theoremindif2 3215 Bring an intersection in and out of a class difference. (Contributed by Jeff Hankins, 15-Jul-2009.)

Theoremindif1 3216 Bring an intersection in and out of a class difference. (Contributed by Mario Carneiro, 15-May-2015.)

Theoremindifcom 3217 Commutation law for intersection and difference. (Contributed by Scott Fenton, 18-Feb-2013.)

Theoremindi 3218 Distributive law for intersection over union. Exercise 10 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremundi 3219 Distributive law for union over intersection. Exercise 11 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremindir 3220 Distributive law for intersection over union. Theorem 28 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)

Theoremundir 3221 Distributive law for union over intersection. Theorem 29 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)

Theoremuneqin 3222 Equality of union and intersection implies equality of their arguments. (Contributed by NM, 16-Apr-2006.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremdifundi 3223 Distributive law for class difference. Theorem 39 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)

Theoremdifundir 3224 Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)

Theoremdifindiss 3225 Distributive law for class difference. In classical logic, for example, theorem 40 of [Suppes] p. 29, this is an equality instead of subset. (Contributed by Jim Kingdon, 26-Jul-2018.)

Theoremdifindir 3226 Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)

Theoremindifdir 3227 Distribute intersection over difference. (Contributed by Scott Fenton, 14-Apr-2011.)

Theoremdifdif2ss 3228 Set difference with a set difference. In classical logic this would be equality rather than subset. (Contributed by Jim Kingdon, 27-Jul-2018.)

Theoremundm 3229 De Morgan's law for union. Theorem 5.2(13) of [Stoll] p. 19. (Contributed by NM, 18-Aug-2004.)

Theoremindmss 3230 De Morgan's law for intersection. In classical logic, this would be equality rather than subset, as in Theorem 5.2(13') of [Stoll] p. 19. (Contributed by Jim Kingdon, 27-Jul-2018.)

Theoremdifun1 3231 A relationship involving double difference and union. (Contributed by NM, 29-Aug-2004.)

Theoremundif3ss 3232 A subset relationship involving class union and class difference. In classical logic, this would be equality rather than subset, as in the first equality of Exercise 13 of [TakeutiZaring] p. 22. (Contributed by Jim Kingdon, 28-Jul-2018.)

Theoremdifin2 3233 Represent a set difference as an intersection with a larger difference. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremdif32 3234 Swap second and third argument of double difference. (Contributed by NM, 18-Aug-2004.)

Theoremdifabs 3235 Absorption-like law for class difference: you can remove a class only once. (Contributed by FL, 2-Aug-2009.)

Theoremsymdif1 3236 Two ways to express symmetric difference. This theorem shows the equivalence of the definition of symmetric difference in [Stoll] p. 13 and the restated definition in Example 4.1 of [Stoll] p. 262. (Contributed by NM, 17-Aug-2004.)

2.1.13.5  Class abstractions with difference, union, and intersection of two classes

Theoremsymdifxor 3237* Expressing symmetric difference with exclusive-or or two differences. (Contributed by Jim Kingdon, 28-Jul-2018.)

Theoremunab 3238 Union of two class abstractions. (Contributed by NM, 29-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoreminab 3239 Intersection of two class abstractions. (Contributed by NM, 29-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremdifab 3240 Difference of two class abstractions. (Contributed by NM, 23-Oct-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremnotab 3241 A class builder defined by a negation. (Contributed by FL, 18-Sep-2010.)

Theoremunrab 3242 Union of two restricted class abstractions. (Contributed by NM, 25-Mar-2004.)

Theoreminrab 3243 Intersection of two restricted class abstractions. (Contributed by NM, 1-Sep-2006.)

Theoreminrab2 3244* Intersection with a restricted class abstraction. (Contributed by NM, 19-Nov-2007.)

Theoremdifrab 3245 Difference of two restricted class abstractions. (Contributed by NM, 23-Oct-2004.)

Theoremdfrab2 3246* Alternate definition of restricted class abstraction. (Contributed by NM, 20-Sep-2003.)

Theoremdfrab3 3247* Alternate definition of restricted class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)

Theoremnotrab 3248* Complementation of restricted class abstractions. (Contributed by Mario Carneiro, 3-Sep-2015.)

Theoremdfrab3ss 3249* Restricted class abstraction with a common superset. (Contributed by Stefan O'Rear, 12-Sep-2015.) (Proof shortened by Mario Carneiro, 8-Nov-2015.)

Theoremrabun2 3250 Abstraction restricted to a union. (Contributed by Stefan O'Rear, 5-Feb-2015.)

2.1.13.6  Restricted uniqueness with difference, union, and intersection

Theoremreuss2 3251* Transfer uniqueness to a smaller subclass. (Contributed by NM, 20-Oct-2005.)

Theoremreuss 3252* Transfer uniqueness to a smaller subclass. (Contributed by NM, 21-Aug-1999.)

Theoremreuun1 3253* Transfer uniqueness to a smaller class. (Contributed by NM, 21-Oct-2005.)

Theoremreuun2 3254* Transfer uniqueness to a smaller or larger class. (Contributed by NM, 21-Oct-2005.)

Theoremreupick 3255* Restricted uniqueness "picks" a member of a subclass. (Contributed by NM, 21-Aug-1999.)

Theoremreupick3 3256* Restricted uniqueness "picks" a member of a subclass. (Contributed by Mario Carneiro, 19-Nov-2016.)

Theoremreupick2 3257* Restricted uniqueness "picks" a member of a subclass. (Contributed by Mario Carneiro, 15-Dec-2013.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)

2.1.14  The empty set

Syntaxc0 3258 Extend class notation to include the empty set.

Definitiondf-nul 3259 Define the empty set. Special case of Exercise 4.10(o) of [Mendelson] p. 231. For a more traditional definition, but requiring a dummy variable, see dfnul2 3260. (Contributed by NM, 5-Aug-1993.)

Theoremdfnul2 3260 Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring] p. 20. (Contributed by NM, 26-Dec-1996.)

Theoremdfnul3 3261 Alternate definition of the empty set. (Contributed by NM, 25-Mar-2004.)

Theoremnoel 3262 The empty set has no elements. Theorem 6.14 of [Quine] p. 44. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 1-Sep-2015.)

Theoremn0i 3263 If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2616. (Contributed by NM, 31-Dec-1993.)

Theoremne0i 3264 If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2616. (Contributed by NM, 31-Dec-1993.)

Theoremvn0 3265 The universal class is not equal to the empty set. (Contributed by NM, 11-Sep-2008.)

Theoremvn0m 3266 The universal class is inhabited. (Contributed by Jim Kingdon, 17-Dec-2018.)

Theoremn0rf 3267 An inhabited class is nonempty. Following the Definition of [Bauer], p. 483, we call a class nonempty if and inhabited if it has at least one element. In classical logic these two concepts are equivalent, for example see Proposition 5.17(1) of [TakeutiZaring] p. 20. This version of n0r 3268 requires only that not be free in, rather than not occur in, . (Contributed by Jim Kingdon, 31-Jul-2018.)

Theoremn0r 3268* An inhabited class is nonempty. See n0rf 3267 for more discussion. (Contributed by Jim Kingdon, 31-Jul-2018.)

Theoremneq0r 3269* An inhabited class is nonempty. See n0rf 3267 for more discussion. (Contributed by Jim Kingdon, 31-Jul-2018.)

Theoremreximdva0m 3270* Restricted existence deduced from inhabited class. (Contributed by Jim Kingdon, 31-Jul-2018.)

Theoremn0mmoeu 3271* A case of equivalence of "at most one" and "only one". If a class is inhabited, that class having at most one element is equivalent to it having only one element. (Contributed by Jim Kingdon, 31-Jul-2018.)

Theoremrex0 3272 Vacuous existential quantification is false. (Contributed by NM, 15-Oct-2003.)

Theoremeq0 3273* The empty set has no elements. Theorem 2 of [Suppes] p. 22. (Contributed by NM, 29-Aug-1993.)

Theoremeqv 3274* The universe contains every set. (Contributed by NM, 11-Sep-2006.)

Theorem0el 3275* Membership of the empty set in another class. (Contributed by NM, 29-Jun-2004.)

Theoremabvor0dc 3276* The class builder of a decidable proposition not containing the abstraction variable is either the universal class or the empty set. (Contributed by Jim Kingdon, 1-Aug-2018.)
DECID

Theoremabn0r 3277 Nonempty class abstraction. (Contributed by Jim Kingdon, 1-Aug-2018.)

Theoremrabn0r 3278 Non-empty restricted class abstraction. (Contributed by Jim Kingdon, 1-Aug-2018.)

Theoremrabn0m 3279* Inhabited restricted class abstraction. (Contributed by Jim Kingdon, 18-Sep-2018.)

Theoremrab0 3280 Any restricted class abstraction restricted to the empty set is empty. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremrabeq0 3281 Condition for a restricted class abstraction to be empty. (Contributed by Jeff Madsen, 7-Jun-2010.)

Theoremabeq0 3282 Condition for a class abstraction to be empty. (Contributed by Jim Kingdon, 12-Aug-2018.)

Theoremrabxmdc 3283* Law of excluded middle given decidability, in terms of restricted class abstractions. (Contributed by Jim Kingdon, 2-Aug-2018.)
DECID

Theoremrabnc 3284* Law of noncontradiction, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.)

Theoremun0 3285 The union of a class with the empty set is itself. Theorem 24 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)

Theoremin0 3286 The intersection of a class with the empty set is the empty set. Theorem 16 of [Suppes] p. 26. (Contributed by NM, 5-Aug-1993.)

Theoreminv1 3287 The intersection of a class with the universal class is itself. Exercise 4.10(k) of [Mendelson] p. 231. (Contributed by NM, 17-May-1998.)

Theoremunv 3288 The union of a class with the universal class is the universal class. Exercise 4.10(l) of [Mendelson] p. 231. (Contributed by NM, 17-May-1998.)

Theorem0ss 3289 The null set is a subset of any class. Part of Exercise 1 of [TakeutiZaring] p. 22. (Contributed by NM, 5-Aug-1993.)

Theoremss0b 3290 Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23 and its converse. (Contributed by NM, 17-Sep-2003.)

Theoremss0 3291 Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23. (Contributed by NM, 13-Aug-1994.)

Theoremsseq0 3292 A subclass of an empty class is empty. (Contributed by NM, 7-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremssn0 3293 A class with a nonempty subclass is nonempty. (Contributed by NM, 17-Feb-2007.)

Theoremabf 3294 A class builder with a false argument is empty. (Contributed by NM, 20-Jan-2012.)

Theoremeq0rdv 3295* Deduction rule for equality to the empty set. (Contributed by NM, 11-Jul-2014.)

Theoremcsbprc 3296 The proper substitution of a proper class for a set into a class results in the empty set. (Contributed by NM, 17-Aug-2018.)

Theoremun00 3297 Two classes are empty iff their union is empty. (Contributed by NM, 11-Aug-2004.)

Theoremvss 3298 Only the universal class has the universal class as a subclass. (Contributed by NM, 17-Sep-2003.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremdisj 3299* Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 17-Feb-2004.)

Theoremdisjr 3300* Two ways of saying that two classes are disjoint. (Contributed by Jeff Madsen, 19-Jun-2011.)

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