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Theorem List for Intuitionistic Logic Explorer - 3301-3400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdifdif2ss 3301 Set difference with a set difference. In classical logic this would be equality rather than subset. (Contributed by Jim Kingdon, 27-Jul-2018.)

Theoremundm 3302 De Morgan's law for union. Theorem 5.2(13) of [Stoll] p. 19. (Contributed by NM, 18-Aug-2004.)

Theoremindmss 3303 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 3304 A relationship involving double difference and union. (Contributed by NM, 29-Aug-2004.)

Theoremundif3ss 3305 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 3306 Represent a set difference as an intersection with a larger difference. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremdif32 3307 Swap second and third argument of double difference. (Contributed by NM, 18-Aug-2004.)

Theoremdifabs 3308 Absorption-like law for class difference: you can remove a class only once. (Contributed by FL, 2-Aug-2009.)

Theoremsymdif1 3309 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 3310* Expressing symmetric difference with exclusive-or or two differences. (Contributed by Jim Kingdon, 28-Jul-2018.)

Theoremunab 3311 Union of two class abstractions. (Contributed by NM, 29-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoreminab 3312 Intersection of two class abstractions. (Contributed by NM, 29-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremdifab 3313 Difference of two class abstractions. (Contributed by NM, 23-Oct-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremnotab 3314 A class builder defined by a negation. (Contributed by FL, 18-Sep-2010.)

Theoremunrab 3315 Union of two restricted class abstractions. (Contributed by NM, 25-Mar-2004.)

Theoreminrab 3316 Intersection of two restricted class abstractions. (Contributed by NM, 1-Sep-2006.)

Theoreminrab2 3317* Intersection with a restricted class abstraction. (Contributed by NM, 19-Nov-2007.)

Theoremdifrab 3318 Difference of two restricted class abstractions. (Contributed by NM, 23-Oct-2004.)

Theoremdfrab2 3319* Alternate definition of restricted class abstraction. (Contributed by NM, 20-Sep-2003.)

Theoremdfrab3 3320* Alternate definition of restricted class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)

Theoremnotrab 3321* Complementation of restricted class abstractions. (Contributed by Mario Carneiro, 3-Sep-2015.)

Theoremdfrab3ss 3322* Restricted class abstraction with a common superset. (Contributed by Stefan O'Rear, 12-Sep-2015.) (Proof shortened by Mario Carneiro, 8-Nov-2015.)

Theoremrabun2 3323 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 3324* Transfer uniqueness to a smaller subclass. (Contributed by NM, 20-Oct-2005.)

Theoremreuss 3325* Transfer uniqueness to a smaller subclass. (Contributed by NM, 21-Aug-1999.)

Theoremreuun1 3326* Transfer uniqueness to a smaller class. (Contributed by NM, 21-Oct-2005.)

Theoremreuun2 3327* Transfer uniqueness to a smaller or larger class. (Contributed by NM, 21-Oct-2005.)

Theoremreupick 3328* Restricted uniqueness "picks" a member of a subclass. (Contributed by NM, 21-Aug-1999.)

Theoremreupick3 3329* Restricted uniqueness "picks" a member of a subclass. (Contributed by Mario Carneiro, 19-Nov-2016.)

Theoremreupick2 3330* 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 3331 Extend class notation to include the empty set.

Definitiondf-nul 3332 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 3333. (Contributed by NM, 5-Aug-1993.)

Theoremdfnul2 3333 Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring] p. 20. (Contributed by NM, 26-Dec-1996.)

Theoremdfnul3 3334 Alternate definition of the empty set. (Contributed by NM, 25-Mar-2004.)

Theoremnoel 3335 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 3336 If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2674. (Contributed by NM, 31-Dec-1993.)

Theoremne0i 3337 If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2674. (Contributed by NM, 31-Dec-1993.)

Theoremne0d 3338 Deduction form of ne0i 3337. If a class has elements, then it is nonempty. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Theoremn0ii 3339 If a class has elements, then it is not empty. Inference associated with n0i 3336. (Contributed by BJ, 15-Jul-2021.)

Theoremne0ii 3340 If a class has elements, then it is nonempty. Inference associated with ne0i 3337. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremvn0 3341 The universal class is not equal to the empty set. (Contributed by NM, 11-Sep-2008.)

Theoremvn0m 3342 The universal class is inhabited. (Contributed by Jim Kingdon, 17-Dec-2018.)

Theoremn0rf 3343 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 3344 requires only that not be free in, rather than not occur in, . (Contributed by Jim Kingdon, 31-Jul-2018.)

Theoremn0r 3344* An inhabited class is nonempty. See n0rf 3343 for more discussion. (Contributed by Jim Kingdon, 31-Jul-2018.)

Theoremneq0r 3345* An inhabited class is nonempty. See n0rf 3343 for more discussion. (Contributed by Jim Kingdon, 31-Jul-2018.)

Theoremreximdva0m 3346* Restricted existence deduced from inhabited class. (Contributed by Jim Kingdon, 31-Jul-2018.)

Theoremn0mmoeu 3347* 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 3348 Vacuous existential quantification is false. (Contributed by NM, 15-Oct-2003.)

Theoremeq0 3349* The empty set has no elements. Theorem 2 of [Suppes] p. 22. (Contributed by NM, 29-Aug-1993.)

Theoremeqv 3350* The universe contains every set. (Contributed by NM, 11-Sep-2006.)

Theoremnotm0 3351* A class is not inhabited if and only if it is empty. (Contributed by Jim Kingdon, 1-Jul-2022.)

Theoremnel0 3352* From the general negation of membership in , infer that is the empty set. (Contributed by BJ, 6-Oct-2018.)

Theorem0el 3353* Membership of the empty set in another class. (Contributed by NM, 29-Jun-2004.)

Theoremabvor0dc 3354* 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 3355 Nonempty class abstraction. (Contributed by Jim Kingdon, 1-Aug-2018.)

Theoremabn0m 3356* Inhabited class abstraction. (Contributed by Jim Kingdon, 8-Jul-2022.)

Theoremrabn0r 3357 Nonempty restricted class abstraction. (Contributed by Jim Kingdon, 1-Aug-2018.)

Theoremrabn0m 3358* Inhabited restricted class abstraction. (Contributed by Jim Kingdon, 18-Sep-2018.)

Theoremrab0 3359 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 3360 Condition for a restricted class abstraction to be empty. (Contributed by Jeff Madsen, 7-Jun-2010.)

Theoremabeq0 3361 Condition for a class abstraction to be empty. (Contributed by Jim Kingdon, 12-Aug-2018.)

Theoremrabxmdc 3362* Law of excluded middle given decidability, in terms of restricted class abstractions. (Contributed by Jim Kingdon, 2-Aug-2018.)
DECID

Theoremrabnc 3363* Law of noncontradiction, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.)

Theoremun0 3364 The union of a class with the empty set is itself. Theorem 24 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)

Theoremin0 3365 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.)

Theorem0in 3366 The intersection of the empty set with a class is the empty set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Theoreminv1 3367 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 3368 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 3369 The null set is a subset of any class. Part of Exercise 1 of [TakeutiZaring] p. 22. (Contributed by NM, 5-Aug-1993.)

Theoremss0b 3370 Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23 and its converse. (Contributed by NM, 17-Sep-2003.)

Theoremss0 3371 Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23. (Contributed by NM, 13-Aug-1994.)

Theoremsseq0 3372 A subclass of an empty class is empty. (Contributed by NM, 7-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremssn0 3373 A class with a nonempty subclass is nonempty. (Contributed by NM, 17-Feb-2007.)

Theoremabf 3374 A class builder with a false argument is empty. (Contributed by NM, 20-Jan-2012.)

Theoremeq0rdv 3375* Deduction for equality to the empty set. (Contributed by NM, 11-Jul-2014.)

Theoremcsbprc 3376 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 3377 Two classes are empty iff their union is empty. (Contributed by NM, 11-Aug-2004.)

Theoremvss 3378 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 3379* Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 17-Feb-2004.)

Theoremdisjr 3380* Two ways of saying that two classes are disjoint. (Contributed by Jeff Madsen, 19-Jun-2011.)

Theoremdisj1 3381* Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 19-Aug-1993.)

Theoremreldisj 3382 Two ways of saying that two classes are disjoint, using the complement of relative to a universe . (Contributed by NM, 15-Feb-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremdisj3 3383 Two ways of saying that two classes are disjoint. (Contributed by NM, 19-May-1998.)

Theoremdisjne 3384 Members of disjoint sets are not equal. (Contributed by NM, 28-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremdisjel 3385 A set can't belong to both members of disjoint classes. (Contributed by NM, 28-Feb-2015.)

Theoremdisj2 3386 Two ways of saying that two classes are disjoint. (Contributed by NM, 17-May-1998.)

Theoremssdisj 3387 Intersection with a subclass of a disjoint class. (Contributed by FL, 24-Jan-2007.)

Theoremundisj1 3388 The union of disjoint classes is disjoint. (Contributed by NM, 26-Sep-2004.)

Theoremundisj2 3389 The union of disjoint classes is disjoint. (Contributed by NM, 13-Sep-2004.)

Theoremssindif0im 3390 Subclass implies empty intersection with difference from the universal class. (Contributed by NM, 17-Sep-2003.)

Theoreminelcm 3391 The intersection of classes with a common member is nonempty. (Contributed by NM, 7-Apr-1994.)

Theoremminel 3392 A minimum element of a class has no elements in common with the class. (Contributed by NM, 22-Jun-1994.)

Theoremundif4 3393 Distribute union over difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremdisjssun 3394 Subset relation for disjoint classes. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremssdif0im 3395 Subclass implies empty difference. One direction of Exercise 7 of [TakeutiZaring] p. 22. In classical logic this would be an equivalence. (Contributed by Jim Kingdon, 2-Aug-2018.)

Theoremvdif0im 3396 Universal class equality in terms of empty difference. (Contributed by Jim Kingdon, 3-Aug-2018.)

Theoremdifrab0eqim 3397* If the difference between the restricting class of a restricted class abstraction and the restricted class abstraction is empty, the restricting class is equal to this restricted class abstraction. (Contributed by Jim Kingdon, 3-Aug-2018.)

Theoreminssdif0im 3398 Intersection, subclass, and difference relationship. In classical logic the converse would also hold. (Contributed by Jim Kingdon, 3-Aug-2018.)

Theoremdifid 3399 The difference between a class and itself is the empty set. Proposition 5.15 of [TakeutiZaring] p. 20. Also Theorem 32 of [Suppes] p. 28. (Contributed by NM, 22-Apr-2004.)

TheoremdifidALT 3400 The difference between a class and itself is the empty set. Proposition 5.15 of [TakeutiZaring] p. 20. Also Theorem 32 of [Suppes] p. 28. Alternate proof of difid 3399. (Contributed by David Abernethy, 17-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

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