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Theorem List for Intuitionistic Logic Explorer - 3201-3300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnoel 3201 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 3202 If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2543. (Contributed by NM, 31-Dec-1993.)
 (/)
 
Theoremne0i 3203 If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2543. (Contributed by NM, 31-Dec-1993.)
 =/=  (/)
 
Theoremvn0 3204 The universal class is not equal to the empty set. (Contributed by NM, 11-Sep-2008.)

 _V  =/=  (/)
 
Theoremvn0m 3205 The universal class is inhabited. (Contributed by Jim Kingdon, 17-Dec-2018.)
 _V
 
Theoremn0rf 3206 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 3207 requires only that not be free in, rather than not occur in, . (Contributed by Jim Kingdon, 31-Jul-2018.)
 F/_   =>     =/=  (/)
 
Theoremn0r 3207* An inhabited class is nonempty. See n0rf 3206 for more discussion. (Contributed by Jim Kingdon, 31-Jul-2018.)
 =/=  (/)
 
Theoremneq0r 3208* An inhabited class is nonempty. See n0rf 3206 for more discussion. (Contributed by Jim Kingdon, 31-Jul-2018.)
 (/)
 
Theoremreximdva0m 3209* Restricted existence deduced from inhabited class. (Contributed by Jim Kingdon, 31-Jul-2018.)
   =>   
 
Theoremn0mmoeu 3210* 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 3211 Vacuous existential quantification is false. (Contributed by NM, 15-Oct-2003.)
 (/)
 
Theoremeq0 3212* The empty set has no elements. Theorem 2 of [Suppes] p. 22. (Contributed by NM, 29-Aug-1993.)
 (/)
 
Theoremeqv 3213* The universe contains every set. (Contributed by NM, 11-Sep-2006.)
 _V
 
Theorem0el 3214* Membership of the empty set in another class. (Contributed by NM, 29-Jun-2004.)
 (/)
 
Theoremabvor0dc 3215* 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 3216 Nonempty class abstraction. (Contributed by Jim Kingdon, 1-Aug-2018.)
 {  |  }  =/=  (/)
 
Theoremrabn0r 3217 Non-empty restricted class abstraction. (Contributed by Jim Kingdon, 1-Aug-2018.)
 {  |  }  =/=  (/)
 
Theoremrabn0m 3218* Inhabited restricted class abstraction. (Contributed by Jim Kingdon, 18-Sep-2018.)
 {  |  }
 
Theoremrab0 3219 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 3220 Condition for a restricted class abstraction to be empty. (Contributed by Jeff Madsen, 7-Jun-2010.)
 {  |  }  (/)
 
Theoremabeq0 3221 Condition for a class abstraction to be empty. (Contributed by Jim Kingdon, 12-Aug-2018.)
 {  | 
 }  (/)
 
Theoremrabxmdc 3222* Law of excluded middle given decidability, in terms of restricted class abstractions. (Contributed by Jim Kingdon, 2-Aug-2018.)
DECID  {  |  }  u.  {  |  }
 
Theoremrabnc 3223* Law of noncontradiction, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.)
 {  |  }  i^i  {  |  }  (/)
 
Theoremun0 3224 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 3225 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 3226 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 3227 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 3228 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 3229 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 3230 Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23. (Contributed by NM, 13-Aug-1994.)
 C_  (/)  (/)
 
Theoremsseq0 3231 A subclass of an empty class is empty. (Contributed by NM, 7-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 C_  (/)  (/)
 
Theoremssn0 3232 A class with a nonempty subclass is nonempty. (Contributed by NM, 17-Feb-2007.)
 C_  =/=  (/)  =/=  (/)
 
Theoremabf 3233 A class builder with a false argument is empty. (Contributed by NM, 20-Jan-2012.)
   =>    
 {  |  }  (/)
 
Theoremeq0rdv 3234* Deduction rule for equality to the empty set. (Contributed by NM, 11-Jul-2014.)
   =>     (/)
 
Theoremcsbprc 3235 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 3236 Two classes are empty iff their union is empty. (Contributed by NM, 11-Aug-2004.)
 (/)  (/)  u.  (/)
 
Theoremvss 3237 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 3238 The null set is a proper subset of any non-empty set. (Contributed by NM, 27-Feb-1996.)
 (/)  C.  =/=  (/)
 
Theoremnpss0 3239 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 3240 Any non-universal class is a proper subclass of the universal class. (Contributed by NM, 17-May-1998.)
 C.  _V  _V
 
Theoremdisj 3241* Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 17-Feb-2004.)
 i^i  (/)
 
Theoremdisjr 3242* Two ways of saying that two classes are disjoint. (Contributed by Jeff Madsen, 19-Jun-2011.)
 i^i  (/)
 
Theoremdisj1 3243* Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 19-Aug-1993.)
 i^i  (/)
 
Theoremreldisj 3244 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 3245 Two ways of saying that two classes are disjoint. (Contributed by NM, 19-May-1998.)
 i^i  (/)  \
 
Theoremdisjne 3246 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 3247 A set can't belong to both members of disjoint classes. (Contributed by NM, 28-Feb-2015.)

 i^i  (/)  C  C
 
Theoremdisj2 3248 Two ways of saying that two classes are disjoint. (Contributed by NM, 17-May-1998.)
 i^i  (/)  C_  _V  \
 
Theoremdisj4im 3249 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 3250 Intersection with a subclass of a disjoint class. (Contributed by FL, 24-Jan-2007.)
 C_  i^i  C  (/)  i^i  C  (/)
 
Theoremdisjpss 3251 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 3252 The union of disjoint classes is disjoint. (Contributed by NM, 26-Sep-2004.)

 i^i  C  (/)  i^i  C  (/)  u. 
 i^i  C  (/)
 
Theoremundisj2 3253 The union of disjoint classes is disjoint. (Contributed by NM, 13-Sep-2004.)

 i^i  (/)  i^i  C  (/) 
 i^i  u.  C  (/)
 
Theoremssindif0im 3254 Subclass implies empty intersection with difference from the universal class. (Contributed by NM, 17-Sep-2003.)
 C_  i^i  _V  \  (/)
 
Theoreminelcm 3255 The intersection of classes with a common member is nonempty. (Contributed by NM, 7-Apr-1994.)
 C  i^i  C  =/=  (/)
 
Theoremminel 3256 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 3257 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 3258 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 3259 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 3260 Universal class equality in terms of empty difference. (Contributed by Jim Kingdon, 3-Aug-2018.)
 _V  _V  \  (/)
 
Theoremdifrab0eqim 3261* 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 3262 A subclass missing a member is a proper subclass. (Contributed by NM, 12-Jan-2002.)
 C_  C  C  C.
 
Theoremssnelpssd 3263 Subclass inclusion with one element of the superclass missing is proper subclass inclusion. Deduction form of ssnelpss 3262. (Contributed by David Moews, 1-May-2017.)
 C_    &     C    &     C    =>     C.
 
Theoreminssdif0im 3264 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 3265 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 3266 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 3265. (Contributed by David Abernethy, 17-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 \  (/)
 
Theoremdif0 3267 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 3268 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 3269 A class and its relative complement are disjoint. Theorem 38 of [Suppes] p. 29. (Contributed by NM, 24-Mar-1998.)
 i^i  \  (/)
 
Theoremdifin0 3270 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 3271 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 3272 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 3273 Absorption of difference by union. (Contributed by NM, 18-Aug-2013.)
 u.  \
 
Theoreminundifss 3274 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 3275 Absorption of union by difference. Theorem 36 of [Suppes] p. 29. (Contributed by NM, 19-May-1998.)
 u.  \  \
 
Theoremundifss 3276 Union of complementary parts into whole. (Contributed by Jim Kingdon, 4-Aug-2018.)
 C_  u.  \  C_
 
Theoremssdifin0 3277 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 3278 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 3279 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
 
Theoremdifdifdirss 3280 Distributive law for class difference. In classical logic, as in Exercise 4.8 of [Stoll] p. 16, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.)
 \  \  C  C_  \  C  \ 
 \  C
 
Theoremuneqdifeqim 3281 Two ways that and can "partition"  C (when and don't overlap and is a part of  C). In classical logic, the second implication would be a biconditional. (Contributed by Jim Kingdon, 4-Aug-2018.)
 C_  C  i^i  (/)  u.  C  C  \
 
Theoremr19.2m 3282* Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1507). The restricted version is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.)
 
Theoremr19.3rmOLD 3283* Restricted quantification of wff not containing quantified variable. This is a special case of r19.3rm 3285 where the two setvar variables are both , so new proofs should just use r19.3rm 3285 instead. (Contributed by Jim Kingdon, 5-Aug-2018.) (New usage is discouraged.)

 F/   =>   
 
Theoremr19.28m 3284* Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.)

 F/   =>   
 
Theoremr19.3rm 3285* Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 19-Dec-2018.)

 F/   =>   
 
Theoremr19.3rmv 3286* Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 6-Aug-2018.)
 
Theoremr19.9rmv 3287* Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 5-Aug-2018.)
 
Theoremr19.9rmvOLD 3288* Restricted quantification of wff not containing quantified variable. This is a special case of r19.9rmv 3287 where and are the same variable, but new proofs should use r19.9rmv 3287 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 5-Aug-2018.)
 
Theoremr19.28mv 3289* Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 6-Aug-2018.)
 
Theoremr19.45mv 3290* Restricted version of Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)
 
Theoremr19.27m 3291* Restricted quantifier version of Theorem 19.27 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.)

 F/   =>   
 
Theoremr19.27mv 3292* Restricted quantifier version of Theorem 19.27 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.)
 
Theoremrzal 3293* Vacuous quantification is always true. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 (/)
 
Theoremrexn0 3294* Restricted existential quantification implies its restriction is nonempty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)
 =/=  (/)
 
Theoremrexm 3295* Restricted existential quantification implies its restriction is inhabited. (Contributed by Jim Kingdon, 16-Oct-2018.)
 
Theoremralidm 3296* Idempotent law for restricted quantifier. (Contributed by NM, 28-Mar-1997.)
 
Theoremral0 3297 Vacuous universal quantification is always true. (Contributed by NM, 20-Oct-2005.)
 (/)
 
Theoremrgenm 3298* Generalization rule that eliminates an inhabited class requirement. (Contributed by Jim Kingdon, 5-Aug-2018.)
   =>   
 
Theoremralf0 3299* The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.)
   =>     (/)
 
Theoremralm 3300 Inhabited classes and restricted quantification. (Contributed by Jim Kingdon, 6-Aug-2018.)
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