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Theorem List for Intuitionistic Logic Explorer - 3201-3300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnotab 3201 A class builder defined by a negation. (Contributed by FL, 18-Sep-2010.)

 {  | 
 }  _V  \  {  |  }
 
Theoremunrab 3202 Union of two restricted class abstractions. (Contributed by NM, 25-Mar-2004.)
 {  |  }  u.  {  |  }  {  |  }
 
Theoreminrab 3203 Intersection of two restricted class abstractions. (Contributed by NM, 1-Sep-2006.)
 {  |  }  i^i  {  |  }  {  |  }
 
Theoreminrab2 3204* Intersection with a restricted class abstraction. (Contributed by NM, 19-Nov-2007.)
 {  |  }  i^i  {  i^i  |  }
 
Theoremdifrab 3205 Difference of two restricted class abstractions. (Contributed by NM, 23-Oct-2004.)
 {  |  }  \  {  |  } 
 {  |  }
 
Theoremdfrab2 3206* Alternate definition of restricted class abstraction. (Contributed by NM, 20-Sep-2003.)

 {  |  }  {  |  }  i^i
 
Theoremdfrab3 3207* Alternate definition of restricted class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)

 {  |  }  i^i  {  |  }
 
Theoremnotrab 3208* Complementation of restricted class abstractions. (Contributed by Mario Carneiro, 3-Sep-2015.)
 \  {  |  }  {  |  }
 
Theoremdfrab3ss 3209* Restricted class abstraction with a common superset. (Contributed by Stefan O'Rear, 12-Sep-2015.) (Proof shortened by Mario Carneiro, 8-Nov-2015.)
 C_  {  |  }  i^i  {  |  }
 
Theoremrabun2 3210 Abstraction restricted to a union. (Contributed by Stefan O'Rear, 5-Feb-2015.)

 {  u.  | 
 }  {  |  }  u.  {  |  }
 
2.1.13.6  Restricted uniqueness with difference, union, and intersection
 
Theoremreuss2 3211* Transfer uniqueness to a smaller subclass. (Contributed by NM, 20-Oct-2005.)
 C_
 
Theoremreuss 3212* Transfer uniqueness to a smaller subclass. (Contributed by NM, 21-Aug-1999.)
 C_
 
Theoremreuun1 3213* Transfer uniqueness to a smaller class. (Contributed by NM, 21-Oct-2005.)
 u.
 
Theoremreuun2 3214* Transfer uniqueness to a smaller or larger class. (Contributed by NM, 21-Oct-2005.)
 u.
 
Theoremreupick 3215* Restricted uniqueness "picks" a member of a subclass. (Contributed by NM, 21-Aug-1999.)
 C_
 
Theoremreupick3 3216* Restricted uniqueness "picks" a member of a subclass. (Contributed by Mario Carneiro, 19-Nov-2016.)
 
Theoremreupick2 3217* 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 3218 Extend class notation to include the empty set.
 (/)
 
Definitiondf-nul 3219 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 3220. (Contributed by NM, 5-Aug-1993.)
 (/)  _V  \  _V
 
Theoremdfnul2 3220 Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring] p. 20. (Contributed by NM, 26-Dec-1996.)
 (/)  {  |  }
 
Theoremdfnul3 3221 Alternate definition of the empty set. (Contributed by NM, 25-Mar-2004.)
 (/)  {  |  }
 
Theoremnoel 3222 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 3223 If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2564. (Contributed by NM, 31-Dec-1993.)
 (/)
 
Theoremne0i 3224 If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2564. (Contributed by NM, 31-Dec-1993.)
 =/=  (/)
 
Theoremvn0 3225 The universal class is not equal to the empty set. (Contributed by NM, 11-Sep-2008.)

 _V  =/=  (/)
 
Theoremvn0m 3226 The universal class is inhabited. (Contributed by Jim Kingdon, 17-Dec-2018.)
 _V
 
Theoremn0rf 3227 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 3228 requires only that not be free in, rather than not occur in, . (Contributed by Jim Kingdon, 31-Jul-2018.)
 F/_   =>     =/=  (/)
 
Theoremn0r 3228* An inhabited class is nonempty. See n0rf 3227 for more discussion. (Contributed by Jim Kingdon, 31-Jul-2018.)
 =/=  (/)
 
Theoremneq0r 3229* An inhabited class is nonempty. See n0rf 3227 for more discussion. (Contributed by Jim Kingdon, 31-Jul-2018.)
 (/)
 
Theoremreximdva0m 3230* Restricted existence deduced from inhabited class. (Contributed by Jim Kingdon, 31-Jul-2018.)
   =>   
 
Theoremn0mmoeu 3231* 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 3232 Vacuous existential quantification is false. (Contributed by NM, 15-Oct-2003.)
 (/)
 
Theoremeq0 3233* The empty set has no elements. Theorem 2 of [Suppes] p. 22. (Contributed by NM, 29-Aug-1993.)
 (/)
 
Theoremeqv 3234* The universe contains every set. (Contributed by NM, 11-Sep-2006.)
 _V
 
Theorem0el 3235* Membership of the empty set in another class. (Contributed by NM, 29-Jun-2004.)
 (/)
 
Theoremabvor0dc 3236* 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  {  |  }  _V  {  |  }  (/)
 
Theoremabn0r 3237 Nonempty class abstraction. (Contributed by Jim Kingdon, 1-Aug-2018.)
 {  |  }  =/=  (/)
 
Theoremrabn0r 3238 Non-empty restricted class abstraction. (Contributed by Jim Kingdon, 1-Aug-2018.)
 {  |  }  =/=  (/)
 
Theoremrabn0m 3239* Inhabited restricted class abstraction. (Contributed by Jim Kingdon, 18-Sep-2018.)
 {  |  }
 
Theoremrab0 3240 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 3241 Condition for a restricted class abstraction to be empty. (Contributed by Jeff Madsen, 7-Jun-2010.)
 {  |  }  (/)
 
Theoremabeq0 3242 Condition for a class abstraction to be empty. (Contributed by Jim Kingdon, 12-Aug-2018.)
 {  | 
 }  (/)
 
Theoremrabxmdc 3243* Law of excluded middle given decidability, in terms of restricted class abstractions. (Contributed by Jim Kingdon, 2-Aug-2018.)
DECID  {  |  }  u.  {  |  }
 
Theoremrabnc 3244* Law of noncontradiction, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.)
 {  |  }  i^i  {  |  }  (/)
 
Theoremun0 3245 The union of a class with the empty set is itself. Theorem 24 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)
 u.  (/)
 
Theoremin0 3246 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.)
 i^i  (/)  (/)
 
Theoreminv1 3247 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.)
 i^i  _V
 
Theoremunv 3248 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.)
 u.  _V  _V
 
Theorem0ss 3249 The null set is a subset of any class. Part of Exercise 1 of [TakeutiZaring] p. 22. (Contributed by NM, 5-Aug-1993.)
 (/)  C_
 
Theoremss0b 3250 Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23 and its converse. (Contributed by NM, 17-Sep-2003.)
 C_  (/)  (/)
 
Theoremss0 3251 Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23. (Contributed by NM, 13-Aug-1994.)
 C_  (/)  (/)
 
Theoremsseq0 3252 A subclass of an empty class is empty. (Contributed by NM, 7-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 C_  (/)  (/)
 
Theoremssn0 3253 A class with a nonempty subclass is nonempty. (Contributed by NM, 17-Feb-2007.)
 C_  =/=  (/)  =/=  (/)
 
Theoremabf 3254 A class builder with a false argument is empty. (Contributed by NM, 20-Jan-2012.)
   =>    
 {  |  }  (/)
 
Theoremeq0rdv 3255* Deduction rule for equality to the empty set. (Contributed by NM, 11-Jul-2014.)
   =>     (/)
 
Theoremcsbprc 3256 The proper substitution of a proper class for a set into a class results in the empty set. (Contributed by NM, 17-Aug-2018.)
 _V  [_  ]_  (/)
 
Theoremun00 3257 Two classes are empty iff their union is empty. (Contributed by NM, 11-Aug-2004.)
 (/)  (/)  u.  (/)
 
Theoremvss 3258 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.)
 _V  C_  _V
 
Theorem0pss 3259 The null set is a proper subset of any non-empty set. (Contributed by NM, 27-Feb-1996.)
 (/)  C.  =/=  (/)
 
Theoremnpss0 3260 No set is a proper subset of the empty set. (Contributed by NM, 17-Jun-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 C.  (/)
 
Theorempssv 3261 Any non-universal class is a proper subclass of the universal class. (Contributed by NM, 17-May-1998.)
 C.  _V  _V
 
Theoremdisj 3262* Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 17-Feb-2004.)
 i^i  (/)
 
Theoremdisjr 3263* Two ways of saying that two classes are disjoint. (Contributed by Jeff Madsen, 19-Jun-2011.)
 i^i  (/)
 
Theoremdisj1 3264* Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 19-Aug-1993.)
 i^i  (/)
 
Theoremreldisj 3265 Two ways of saying that two classes are disjoint, using the complement of relative to a universe  C. (Contributed by NM, 15-Feb-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 C_  C  i^i  (/)  C_  C  \
 
Theoremdisj3 3266 Two ways of saying that two classes are disjoint. (Contributed by NM, 19-May-1998.)
 i^i  (/)  \
 
Theoremdisjne 3267 Members of disjoint sets are not equal. (Contributed by NM, 28-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

 i^i  (/)  C  D  C  =/=  D
 
Theoremdisjel 3268 A set can't belong to both members of disjoint classes. (Contributed by NM, 28-Feb-2015.)

 i^i  (/)  C  C
 
Theoremdisj2 3269 Two ways of saying that two classes are disjoint. (Contributed by NM, 17-May-1998.)
 i^i  (/)  C_  _V  \
 
Theoremdisj4im 3270 A consequence of two classes being disjoint. In classical logic this would be a biconditional. (Contributed by Jim Kingdon, 2-Aug-2018.)
 i^i  (/)  \  C.
 
Theoremssdisj 3271 Intersection with a subclass of a disjoint class. (Contributed by FL, 24-Jan-2007.)
 C_  i^i  C  (/)  i^i  C  (/)
 
Theoremdisjpss 3272 A class is a proper subset of its union with a disjoint nonempty class. (Contributed by NM, 15-Sep-2004.)

 i^i  (/)  =/=  (/)  C.  u.
 
Theoremundisj1 3273 The union of disjoint classes is disjoint. (Contributed by NM, 26-Sep-2004.)

 i^i  C  (/)  i^i  C  (/)  u. 
 i^i  C  (/)
 
Theoremundisj2 3274 The union of disjoint classes is disjoint. (Contributed by NM, 13-Sep-2004.)

 i^i  (/)  i^i  C  (/) 
 i^i  u.  C  (/)
 
Theoremssindif0im 3275 Subclass implies empty intersection with difference from the universal class. (Contributed by NM, 17-Sep-2003.)
 C_  i^i  _V  \  (/)
 
Theoreminelcm 3276 The intersection of classes with a common member is nonempty. (Contributed by NM, 7-Apr-1994.)
 C  i^i  C  =/=  (/)
 
Theoremminel 3277 A minimum element of a class has no elements in common with the class. (Contributed by NM, 22-Jun-1994.)
 C  i^i  (/)  C
 
Theoremundif4 3278 Distribute union over difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 i^i  C  (/)  u.  \  C  u.  \  C
 
Theoremdisjssun 3279 Subset relation for disjoint classes. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 i^i  (/)  C_  u.  C  C_  C
 
Theoremssdif0im 3280 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.)
 C_  \  (/)
 
Theoremvdif0im 3281 Universal class equality in terms of empty difference. (Contributed by Jim Kingdon, 3-Aug-2018.)
 _V  _V  \  (/)
 
Theoremdifrab0eqim 3282* 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.)
 V  {  V  |  }  V  \  {  V  |  }  (/)
 
Theoremssnelpss 3283 A subclass missing a member is a proper subclass. (Contributed by NM, 12-Jan-2002.)
 C_  C  C  C.
 
Theoremssnelpssd 3284 Subclass inclusion with one element of the superclass missing is proper subclass inclusion. Deduction form of ssnelpss 3283. (Contributed by David Moews, 1-May-2017.)
 C_    &     C    &     C    =>     C.
 
Theoreminssdif0im 3285 Intersection, subclass, and difference relationship. In classical logic the converse would also hold. (Contributed by Jim Kingdon, 3-Aug-2018.)
 i^i  C_  C  i^i 
 \  C  (/)
 
Theoremdifid 3286 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 3287 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 3286. (Contributed by David Abernethy, 17-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 \  (/)
 
Theoremdif0 3288 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 3289 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 3290 A class and its relative complement are disjoint. Theorem 38 of [Suppes] p. 29. (Contributed by NM, 24-Mar-1998.)
 i^i  \  (/)
 
Theoremdifin0 3291 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.)
 i^i  \  (/)
 
Theoremundif1ss 3292 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.)
 \  u.  C_  u.
 
Theoremundif2ss 3293 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.)
 u.  \  C_  u.
 
Theoremundifabs 3294 Absorption of difference by union. (Contributed by NM, 18-Aug-2013.)
 u.  \
 
Theoreminundifss 3295 The intersection and class difference of a class with another class are contained in the original class. In classical logic we'd be able to make a stronger statement: that everything in the original class is in the intersection or the difference (that is, this theorem would be equality rather than subset). (Contributed by Jim Kingdon, 4-Aug-2018.)
 i^i  u. 
 \  C_
 
Theoremdifun2 3296 Absorption of union by difference. Theorem 36 of [Suppes] p. 29. (Contributed by NM, 19-May-1998.)
 u.  \  \
 
Theoremundifss 3297 Union of complementary parts into whole. (Contributed by Jim Kingdon, 4-Aug-2018.)
 C_  u.  \  C_
 
Theoremssdifin0 3298 A subset of a difference does not intersect the subtrahend. (Contributed by Jeff Hankins, 1-Sep-2013.) (Proof shortened by Mario Carneiro, 24-Aug-2015.)
 C_  \  C  i^i  C  (/)
 
Theoremssdifeq0 3299 A class is a subclass of itself subtracted from another iff it is the empty set. (Contributed by Steve Rodriguez, 20-Nov-2015.)
 C_  \  (/)
 
Theoremssundifim 3300 A consequence of inclusion in the union of two classes. In classical logic this would be a biconditional. (Contributed by Jim Kingdon, 4-Aug-2018.)
 C_  u.  C  \  C_  C
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