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

Theoremdifabs 3201 Absorption-like law for class difference: you can remove a class only once. (Contributed by FL, 2-Aug-2009.)

Theoremsymdif1 3202 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 3203* Expressing symmetric difference with exclusive-or or two differences. (Contributed by Jim Kingdon, 28-Jul-2018.)

Theoremunab 3204 Union of two class abstractions. (Contributed by NM, 29-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoreminab 3205 Intersection of two class abstractions. (Contributed by NM, 29-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremdifab 3206 Difference of two class abstractions. (Contributed by NM, 23-Oct-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremnotab 3207 A class builder defined by a negation. (Contributed by FL, 18-Sep-2010.)

Theoremunrab 3208 Union of two restricted class abstractions. (Contributed by NM, 25-Mar-2004.)

Theoreminrab 3209 Intersection of two restricted class abstractions. (Contributed by NM, 1-Sep-2006.)

Theoreminrab2 3210* Intersection with a restricted class abstraction. (Contributed by NM, 19-Nov-2007.)

Theoremdifrab 3211 Difference of two restricted class abstractions. (Contributed by NM, 23-Oct-2004.)

Theoremdfrab2 3212* Alternate definition of restricted class abstraction. (Contributed by NM, 20-Sep-2003.)

Theoremdfrab3 3213* Alternate definition of restricted class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)

Theoremnotrab 3214* Complementation of restricted class abstractions. (Contributed by Mario Carneiro, 3-Sep-2015.)

Theoremdfrab3ss 3215* Restricted class abstraction with a common superset. (Contributed by Stefan O'Rear, 12-Sep-2015.) (Proof shortened by Mario Carneiro, 8-Nov-2015.)

Theoremrabun2 3216 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 3217* Transfer uniqueness to a smaller subclass. (Contributed by NM, 20-Oct-2005.)

Theoremreuss 3218* Transfer uniqueness to a smaller subclass. (Contributed by NM, 21-Aug-1999.)

Theoremreuun1 3219* Transfer uniqueness to a smaller class. (Contributed by NM, 21-Oct-2005.)

Theoremreuun2 3220* Transfer uniqueness to a smaller or larger class. (Contributed by NM, 21-Oct-2005.)

Theoremreupick 3221* Restricted uniqueness "picks" a member of a subclass. (Contributed by NM, 21-Aug-1999.)

Theoremreupick3 3222* Restricted uniqueness "picks" a member of a subclass. (Contributed by Mario Carneiro, 19-Nov-2016.)

Theoremreupick2 3223* 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 3224 Extend class notation to include the empty set.

Definitiondf-nul 3225 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 3226. (Contributed by NM, 5-Aug-1993.)

Theoremdfnul2 3226 Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring] p. 20. (Contributed by NM, 26-Dec-1996.)

Theoremdfnul3 3227 Alternate definition of the empty set. (Contributed by NM, 25-Mar-2004.)

Theoremnoel 3228 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 3229 If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2570. (Contributed by NM, 31-Dec-1993.)

Theoremne0i 3230 If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2570. (Contributed by NM, 31-Dec-1993.)

Theoremvn0 3231 The universal class is not equal to the empty set. (Contributed by NM, 11-Sep-2008.)

Theoremvn0m 3232 The universal class is inhabited. (Contributed by Jim Kingdon, 17-Dec-2018.)

Theoremn0rf 3233 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 3234 requires only that not be free in, rather than not occur in, . (Contributed by Jim Kingdon, 31-Jul-2018.)

Theoremn0r 3234* An inhabited class is nonempty. See n0rf 3233 for more discussion. (Contributed by Jim Kingdon, 31-Jul-2018.)

Theoremneq0r 3235* An inhabited class is nonempty. See n0rf 3233 for more discussion. (Contributed by Jim Kingdon, 31-Jul-2018.)

Theoremreximdva0m 3236* Restricted existence deduced from inhabited class. (Contributed by Jim Kingdon, 31-Jul-2018.)

Theoremn0mmoeu 3237* 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 3238 Vacuous existential quantification is false. (Contributed by NM, 15-Oct-2003.)

Theoremeq0 3239* The empty set has no elements. Theorem 2 of [Suppes] p. 22. (Contributed by NM, 29-Aug-1993.)

Theoremeqv 3240* The universe contains every set. (Contributed by NM, 11-Sep-2006.)

Theorem0el 3241* Membership of the empty set in another class. (Contributed by NM, 29-Jun-2004.)

Theoremabvor0dc 3242* 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 3243 Nonempty class abstraction. (Contributed by Jim Kingdon, 1-Aug-2018.)

Theoremrabn0r 3244 Non-empty restricted class abstraction. (Contributed by Jim Kingdon, 1-Aug-2018.)

Theoremrabn0m 3245* Inhabited restricted class abstraction. (Contributed by Jim Kingdon, 18-Sep-2018.)

Theoremrab0 3246 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 3247 Condition for a restricted class abstraction to be empty. (Contributed by Jeff Madsen, 7-Jun-2010.)

Theoremabeq0 3248 Condition for a class abstraction to be empty. (Contributed by Jim Kingdon, 12-Aug-2018.)

Theoremrabxmdc 3249* Law of excluded middle given decidability, in terms of restricted class abstractions. (Contributed by Jim Kingdon, 2-Aug-2018.)
DECID

Theoremrabnc 3250* Law of noncontradiction, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.)

Theoremun0 3251 The union of a class with the empty set is itself. Theorem 24 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)

Theoremin0 3252 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 3253 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 3254 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 3255 The null set is a subset of any class. Part of Exercise 1 of [TakeutiZaring] p. 22. (Contributed by NM, 5-Aug-1993.)

Theoremss0b 3256 Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23 and its converse. (Contributed by NM, 17-Sep-2003.)

Theoremss0 3257 Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23. (Contributed by NM, 13-Aug-1994.)

Theoremsseq0 3258 A subclass of an empty class is empty. (Contributed by NM, 7-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremssn0 3259 A class with a nonempty subclass is nonempty. (Contributed by NM, 17-Feb-2007.)

Theoremabf 3260 A class builder with a false argument is empty. (Contributed by NM, 20-Jan-2012.)

Theoremeq0rdv 3261* Deduction rule for equality to the empty set. (Contributed by NM, 11-Jul-2014.)

Theoremcsbprc 3262 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 3263 Two classes are empty iff their union is empty. (Contributed by NM, 11-Aug-2004.)

Theoremvss 3264 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.)

Theorem0pss 3265 The null set is a proper subset of any non-empty set. (Contributed by NM, 27-Feb-1996.)

Theoremnpss0 3266 No set is a proper subset of the empty set. (Contributed by NM, 17-Jun-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theorempssv 3267 Any non-universal class is a proper subclass of the universal class. (Contributed by NM, 17-May-1998.)

Theoremdisj 3268* Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 17-Feb-2004.)

Theoremdisjr 3269* Two ways of saying that two classes are disjoint. (Contributed by Jeff Madsen, 19-Jun-2011.)

Theoremdisj1 3270* Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 19-Aug-1993.)

Theoremreldisj 3271 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 3272 Two ways of saying that two classes are disjoint. (Contributed by NM, 19-May-1998.)

Theoremdisjne 3273 Members of disjoint sets are not equal. (Contributed by NM, 28-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremdisjel 3274 A set can't belong to both members of disjoint classes. (Contributed by NM, 28-Feb-2015.)

Theoremdisj2 3275 Two ways of saying that two classes are disjoint. (Contributed by NM, 17-May-1998.)

Theoremdisj4im 3276 A consequence of two classes being disjoint. In classical logic this would be a biconditional. (Contributed by Jim Kingdon, 2-Aug-2018.)

Theoremssdisj 3277 Intersection with a subclass of a disjoint class. (Contributed by FL, 24-Jan-2007.)

Theoremdisjpss 3278 A class is a proper subset of its union with a disjoint nonempty class. (Contributed by NM, 15-Sep-2004.)

Theoremundisj1 3279 The union of disjoint classes is disjoint. (Contributed by NM, 26-Sep-2004.)

Theoremundisj2 3280 The union of disjoint classes is disjoint. (Contributed by NM, 13-Sep-2004.)

Theoremssindif0im 3281 Subclass implies empty intersection with difference from the universal class. (Contributed by NM, 17-Sep-2003.)

Theoreminelcm 3282 The intersection of classes with a common member is nonempty. (Contributed by NM, 7-Apr-1994.)

Theoremminel 3283 A minimum element of a class has no elements in common with the class. (Contributed by NM, 22-Jun-1994.)

Theoremundif4 3284 Distribute union over difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremdisjssun 3285 Subset relation for disjoint classes. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremssdif0im 3286 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 3287 Universal class equality in terms of empty difference. (Contributed by Jim Kingdon, 3-Aug-2018.)

Theoremdifrab0eqim 3288* 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.)

Theoremssnelpss 3289 A subclass missing a member is a proper subclass. (Contributed by NM, 12-Jan-2002.)

Theoremssnelpssd 3290 Subclass inclusion with one element of the superclass missing is proper subclass inclusion. Deduction form of ssnelpss 3289. (Contributed by David Moews, 1-May-2017.)

Theoreminssdif0im 3291 Intersection, subclass, and difference relationship. In classical logic the converse would also hold. (Contributed by Jim Kingdon, 3-Aug-2018.)

Theoremdifid 3292 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 3293 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 3292. (Contributed by David Abernethy, 17-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremdif0 3294 The difference between a class and the empty set. Part of Exercise 4.4 of [Stoll] p. 16. (Contributed by NM, 17-Aug-2004.)

Theorem0dif 3295 The difference between the empty set and a class. Part of Exercise 4.4 of [Stoll] p. 16. (Contributed by NM, 17-Aug-2004.)

Theoremdisjdif 3296 A class and its relative complement are disjoint. Theorem 38 of [Suppes] p. 29. (Contributed by NM, 24-Mar-1998.)

Theoremdifin0 3297 The difference of a class from its intersection is empty. Theorem 37 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremundif1ss 3298 Absorption of difference by union. In classical logic, as Theorem 35 of [Suppes] p. 29, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.)

Theoremundif2ss 3299 Absorption of difference by union. In classical logic, as in Part of proof of Corollary 6K of [Enderton] p. 144, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.)

Theoremundifabs 3300 Absorption of difference by union. (Contributed by NM, 18-Aug-2013.)

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