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Theorem List for Intuitionistic Logic Explorer - 3201-3300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremin32 3201 A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( ( A  i^i  B )  i^i  C )  =  ( ( A  i^i  C )  i^i 
 B )
 
Theoremin13 3202 A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)
 |-  ( A  i^i  ( B  i^i  C ) )  =  ( C  i^i  ( B  i^i  A ) )
 
Theoremin31 3203 A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)
 |-  ( ( A  i^i  B )  i^i  C )  =  ( ( C  i^i  B )  i^i 
 A )
 
Theoreminrot 3204 Rotate the intersection of 3 classes. (Contributed by NM, 27-Aug-2012.)
 |-  ( ( A  i^i  B )  i^i  C )  =  ( ( C  i^i  A )  i^i 
 B )
 
Theoremin4 3205 Rearrangement of intersection of 4 classes. (Contributed by NM, 21-Apr-2001.)
 |-  ( ( A  i^i  B )  i^i  ( C  i^i  D ) )  =  ( ( A  i^i  C )  i^i  ( B  i^i  D ) )
 
Theoreminindi 3206 Intersection distributes over itself. (Contributed by NM, 6-May-1994.)
 |-  ( A  i^i  ( B  i^i  C ) )  =  ( ( A  i^i  B )  i^i  ( A  i^i  C ) )
 
Theoreminindir 3207 Intersection distributes over itself. (Contributed by NM, 17-Aug-2004.)
 |-  ( ( A  i^i  B )  i^i  C )  =  ( ( A  i^i  C )  i^i  ( B  i^i  C ) )
 
Theoremsseqin2 3208 A relationship between subclass and intersection. Similar to Exercise 9 of [TakeutiZaring] p. 18. (Contributed by NM, 17-May-1994.)
 |-  ( A  C_  B  <->  ( B  i^i  A )  =  A )
 
Theoreminss1 3209 The intersection of two classes is a subset of one of them. Part of Exercise 12 of [TakeutiZaring] p. 18. (Contributed by NM, 27-Apr-1994.)
 |-  ( A  i^i  B )  C_  A
 
Theoreminss2 3210 The intersection of two classes is a subset of one of them. Part of Exercise 12 of [TakeutiZaring] p. 18. (Contributed by NM, 27-Apr-1994.)
 |-  ( A  i^i  B )  C_  B
 
Theoremssin 3211 Subclass of intersection. Theorem 2.8(vii) of [Monk1] p. 26. (Contributed by NM, 15-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( ( A  C_  B  /\  A  C_  C ) 
 <->  A  C_  ( B  i^i  C ) )
 
Theoremssini 3212 An inference showing that a subclass of two classes is a subclass of their intersection. (Contributed by NM, 24-Nov-2003.)
 |-  A  C_  B   &    |-  A  C_  C   =>    |-  A  C_  ( B  i^i  C )
 
Theoremssind 3213 A deduction showing that a subclass of two classes is a subclass of their intersection. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  A 
 C_  C )   =>    |-  ( ph  ->  A 
 C_  ( B  i^i  C ) )
 
Theoremssrin 3214 Add right intersection to subclass relation. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( A  C_  B  ->  ( A  i^i  C )  C_  ( B  i^i  C ) )
 
Theoremsslin 3215 Add left intersection to subclass relation. (Contributed by NM, 19-Oct-1999.)
 |-  ( A  C_  B  ->  ( C  i^i  A )  C_  ( C  i^i  B ) )
 
Theoremss2in 3216 Intersection of subclasses. (Contributed by NM, 5-May-2000.)
 |-  ( ( A  C_  B  /\  C  C_  D )  ->  ( A  i^i  C )  C_  ( B  i^i  D ) )
 
Theoremssinss1 3217 Intersection preserves subclass relationship. (Contributed by NM, 14-Sep-1999.)
 |-  ( A  C_  C  ->  ( A  i^i  B )  C_  C )
 
Theoreminss 3218 Inclusion of an intersection of two classes. (Contributed by NM, 30-Oct-2014.)
 |-  ( ( A  C_  C  \/  B  C_  C )  ->  ( A  i^i  B )  C_  C )
 
2.1.13.4  Combinations of difference, union, and intersection of two classes
 
Theoremunabs 3219 Absorption law for union. (Contributed by NM, 16-Apr-2006.)
 |-  ( A  u.  ( A  i^i  B ) )  =  A
 
Theoreminabs 3220 Absorption law for intersection. (Contributed by NM, 16-Apr-2006.)
 |-  ( A  i^i  ( A  u.  B ) )  =  A
 
Theoremdfss4st 3221* Subclass defined in terms of class difference. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( A. xSTAB  x  e.  A  ->  ( A  C_  B  <->  ( B  \  ( B  \  A ) )  =  A ) )
 
Theoremssddif 3222 Double complement and subset. Similar to ddifss 3226 but inside a class  B instead of the universal class  _V. In classical logic the subset operation on the right hand side could be an equality (that is,  A  C_  B  <->  ( B  \  ( B 
\  A ) )  =  A). (Contributed by Jim Kingdon, 24-Jul-2018.)
 |-  ( A  C_  B  <->  A 
 C_  ( B  \  ( B  \  A ) ) )
 
Theoremunssdif 3223 Union of two classes and class difference. In classical logic this would be an equality. (Contributed by Jim Kingdon, 24-Jul-2018.)
 |-  ( A  u.  B )  C_  ( _V  \  (
 ( _V  \  A )  \  B ) )
 
Theoreminssdif 3224 Intersection of two classes and class difference. In classical logic this would be an equality. (Contributed by Jim Kingdon, 24-Jul-2018.)
 |-  ( A  i^i  B )  C_  ( A  \  ( _V  \  B ) )
 
Theoremdifin 3225 Difference with intersection. Theorem 33 of [Suppes] p. 29. (Contributed by NM, 31-Mar-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( A  \  ( A  i^i  B ) )  =  ( A  \  B )
 
Theoremddifss 3226 Double complement under universal class. In classical logic (or given an additional hypothesis, as in ddifnel 3120), this is equality rather than subset. (Contributed by Jim Kingdon, 24-Jul-2018.)
 |-  A  C_  ( _V  \  ( _V  \  A ) )
 
Theoremunssin 3227 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.)
 |-  ( A  u.  B )  C_  ( _V  \  (
 ( _V  \  A )  i^i  ( _V  \  B ) ) )
 
Theoreminssun 3228 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.)
 |-  ( A  i^i  B )  C_  ( _V  \  (
 ( _V  \  A )  u.  ( _V  \  B ) ) )
 
Theoreminssddif 3229 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.)
 |-  ( A  i^i  B )  C_  ( A  \  ( A  \  B ) )
 
Theoreminvdif 3230 Intersection with universal complement. Remark in [Stoll] p. 20. (Contributed by NM, 17-Aug-2004.)
 |-  ( A  i^i  ( _V  \  B ) )  =  ( A  \  B )
 
Theoremindif 3231 Intersection with class difference. Theorem 34 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
 |-  ( A  i^i  ( A  \  B ) )  =  ( A  \  B )
 
Theoremindif2 3232 Bring an intersection in and out of a class difference. (Contributed by Jeff Hankins, 15-Jul-2009.)
 |-  ( A  i^i  ( B  \  C ) )  =  ( ( A  i^i  B )  \  C )
 
Theoremindif1 3233 Bring an intersection in and out of a class difference. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ( A  \  C )  i^i  B )  =  ( ( A  i^i  B )  \  C )
 
Theoremindifcom 3234 Commutation law for intersection and difference. (Contributed by Scott Fenton, 18-Feb-2013.)
 |-  ( A  i^i  ( B  \  C ) )  =  ( B  i^i  ( A  \  C ) )
 
Theoremindi 3235 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.)
 |-  ( A  i^i  ( B  u.  C ) )  =  ( ( A  i^i  B )  u.  ( A  i^i  C ) )
 
Theoremundi 3236 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.)
 |-  ( A  u.  ( B  i^i  C ) )  =  ( ( A  u.  B )  i^i  ( A  u.  C ) )
 
Theoremindir 3237 Distributive law for intersection over union. Theorem 28 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)
 |-  ( ( A  u.  B )  i^i  C )  =  ( ( A  i^i  C )  u.  ( B  i^i  C ) )
 
Theoremundir 3238 Distributive law for union over intersection. Theorem 29 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)
 |-  ( ( A  i^i  B )  u.  C )  =  ( ( A  u.  C )  i^i  ( B  u.  C ) )
 
Theoremuneqin 3239 Equality of union and intersection implies equality of their arguments. (Contributed by NM, 16-Apr-2006.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( ( A  u.  B )  =  ( A  i^i  B )  <->  A  =  B )
 
Theoremdifundi 3240 Distributive law for class difference. Theorem 39 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
 |-  ( A  \  ( B  u.  C ) )  =  ( ( A 
 \  B )  i^i  ( A  \  C ) )
 
Theoremdifundir 3241 Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)
 |-  ( ( A  u.  B )  \  C )  =  ( ( A 
 \  C )  u.  ( B  \  C ) )
 
Theoremdifindiss 3242 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.)
 |-  ( ( A  \  B )  u.  ( A  \  C ) ) 
 C_  ( A  \  ( B  i^i  C ) )
 
Theoremdifindir 3243 Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)
 |-  ( ( A  i^i  B )  \  C )  =  ( ( A 
 \  C )  i^i  ( B  \  C ) )
 
Theoremindifdir 3244 Distribute intersection over difference. (Contributed by Scott Fenton, 14-Apr-2011.)
 |-  ( ( A  \  B )  i^i  C )  =  ( ( A  i^i  C )  \  ( B  i^i  C ) )
 
Theoremdifdif2ss 3245 Set difference with a set difference. In classical logic this would be equality rather than subset. (Contributed by Jim Kingdon, 27-Jul-2018.)
 |-  ( ( A  \  B )  u.  ( A  i^i  C ) ) 
 C_  ( A  \  ( B  \  C ) )
 
Theoremundm 3246 De Morgan's law for union. Theorem 5.2(13) of [Stoll] p. 19. (Contributed by NM, 18-Aug-2004.)
 |-  ( _V  \  ( A  u.  B ) )  =  ( ( _V  \  A )  i^i  ( _V  \  B ) )
 
Theoremindmss 3247 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.)
 |-  ( ( _V  \  A )  u.  ( _V  \  B ) )  C_  ( _V  \  ( A  i^i  B ) )
 
Theoremdifun1 3248 A relationship involving double difference and union. (Contributed by NM, 29-Aug-2004.)
 |-  ( A  \  ( B  u.  C ) )  =  ( ( A 
 \  B )  \  C )
 
Theoremundif3ss 3249 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.)
 |-  ( A  u.  ( B  \  C ) ) 
 C_  ( ( A  u.  B )  \  ( C  \  A ) )
 
Theoremdifin2 3250 Represent a set difference as an intersection with a larger difference. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( A  C_  C  ->  ( A  \  B )  =  ( ( C  \  B )  i^i 
 A ) )
 
Theoremdif32 3251 Swap second and third argument of double difference. (Contributed by NM, 18-Aug-2004.)
 |-  ( ( A  \  B )  \  C )  =  ( ( A 
 \  C )  \  B )
 
Theoremdifabs 3252 Absorption-like law for class difference: you can remove a class only once. (Contributed by FL, 2-Aug-2009.)
 |-  ( ( A  \  B )  \  B )  =  ( A  \  B )
 
Theoremsymdif1 3253 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.)
 |-  ( ( A  \  B )  u.  ( B  \  A ) )  =  ( ( A  u.  B )  \  ( A  i^i  B ) )
 
2.1.13.5  Class abstractions with difference, union, and intersection of two classes
 
Theoremsymdifxor 3254* Expressing symmetric difference with exclusive-or or two differences. (Contributed by Jim Kingdon, 28-Jul-2018.)
 |-  ( ( A  \  B )  u.  ( B  \  A ) )  =  { x  |  ( x  e.  A  \/_  x  e.  B ) }
 
Theoremunab 3255 Union of two class abstractions. (Contributed by NM, 29-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( { x  |  ph
 }  u.  { x  |  ps } )  =  { x  |  (
 ph  \/  ps ) }
 
Theoreminab 3256 Intersection of two class abstractions. (Contributed by NM, 29-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( { x  |  ph
 }  i^i  { x  |  ps } )  =  { x  |  (
 ph  /\  ps ) }
 
Theoremdifab 3257 Difference of two class abstractions. (Contributed by NM, 23-Oct-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( { x  |  ph
 }  \  { x  |  ps } )  =  { x  |  (
 ph  /\  -.  ps ) }
 
Theoremnotab 3258 A class builder defined by a negation. (Contributed by FL, 18-Sep-2010.)
 |- 
 { x  |  -.  ph
 }  =  ( _V  \  { x  |  ph } )
 
Theoremunrab 3259 Union of two restricted class abstractions. (Contributed by NM, 25-Mar-2004.)
 |-  ( { x  e.  A  |  ph }  u.  { x  e.  A  |  ps } )  =  { x  e.  A  |  ( ph  \/  ps ) }
 
Theoreminrab 3260 Intersection of two restricted class abstractions. (Contributed by NM, 1-Sep-2006.)
 |-  ( { x  e.  A  |  ph }  i^i  { x  e.  A  |  ps } )  =  { x  e.  A  |  ( ph  /\  ps ) }
 
Theoreminrab2 3261* Intersection with a restricted class abstraction. (Contributed by NM, 19-Nov-2007.)
 |-  ( { x  e.  A  |  ph }  i^i  B )  =  { x  e.  ( A  i^i  B )  |  ph }
 
Theoremdifrab 3262 Difference of two restricted class abstractions. (Contributed by NM, 23-Oct-2004.)
 |-  ( { x  e.  A  |  ph }  \  { x  e.  A  |  ps } )  =  { x  e.  A  |  ( ph  /\  -.  ps ) }
 
Theoremdfrab2 3263* Alternate definition of restricted class abstraction. (Contributed by NM, 20-Sep-2003.)
 |- 
 { x  e.  A  |  ph }  =  ( { x  |  ph }  i^i  A )
 
Theoremdfrab3 3264* Alternate definition of restricted class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)
 |- 
 { x  e.  A  |  ph }  =  ( A  i^i  { x  |  ph } )
 
Theoremnotrab 3265* Complementation of restricted class abstractions. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  ( A  \  { x  e.  A  |  ph
 } )  =  { x  e.  A  |  -.  ph }
 
Theoremdfrab3ss 3266* Restricted class abstraction with a common superset. (Contributed by Stefan O'Rear, 12-Sep-2015.) (Proof shortened by Mario Carneiro, 8-Nov-2015.)
 |-  ( A  C_  B  ->  { x  e.  A  |  ph }  =  ( A  i^i  { x  e.  B  |  ph } )
 )
 
Theoremrabun2 3267 Abstraction restricted to a union. (Contributed by Stefan O'Rear, 5-Feb-2015.)
 |- 
 { x  e.  ( A  u.  B )  | 
 ph }  =  ( { x  e.  A  |  ph }  u.  { x  e.  B  |  ph
 } )
 
2.1.13.6  Restricted uniqueness with difference, union, and intersection
 
Theoremreuss2 3268* Transfer uniqueness to a smaller subclass. (Contributed by NM, 20-Oct-2005.)
 |-  ( ( ( A 
 C_  B  /\  A. x  e.  A  ( ph  ->  ps ) )  /\  ( E. x  e.  A  ph 
 /\  E! x  e.  B  ps ) )  ->  E! x  e.  A  ph )
 
Theoremreuss 3269* Transfer uniqueness to a smaller subclass. (Contributed by NM, 21-Aug-1999.)
 |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\  E! x  e.  B  ph )  ->  E! x  e.  A  ph )
 
Theoremreuun1 3270* Transfer uniqueness to a smaller class. (Contributed by NM, 21-Oct-2005.)
 |-  ( ( E. x  e.  A  ph  /\  E! x  e.  ( A  u.  B ) ( ph  \/  ps ) )  ->  E! x  e.  A  ph )
 
Theoremreuun2 3271* Transfer uniqueness to a smaller or larger class. (Contributed by NM, 21-Oct-2005.)
 |-  ( -.  E. x  e.  B  ph  ->  ( E! x  e.  ( A  u.  B ) ph  <->  E! x  e.  A  ph )
 )
 
Theoremreupick 3272* Restricted uniqueness "picks" a member of a subclass. (Contributed by NM, 21-Aug-1999.)
 |-  ( ( ( A 
 C_  B  /\  ( E. x  e.  A  ph 
 /\  E! x  e.  B  ph ) )  /\  ph )  ->  ( x  e.  A  <->  x  e.  B ) )
 
Theoremreupick3 3273* Restricted uniqueness "picks" a member of a subclass. (Contributed by Mario Carneiro, 19-Nov-2016.)
 |-  ( ( E! x  e.  A  ph  /\  E. x  e.  A  ( ph  /\  ps )  /\  x  e.  A )  ->  ( ph  ->  ps ) )
 
Theoremreupick2 3274* Restricted uniqueness "picks" a member of a subclass. (Contributed by Mario Carneiro, 15-Dec-2013.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
 |-  ( ( ( A. x  e.  A  ( ps  ->  ph )  /\  E. x  e.  A  ps  /\ 
 E! x  e.  A  ph )  /\  x  e.  A )  ->  ( ph 
 <->  ps ) )
 
2.1.14  The empty set
 
Syntaxc0 3275 Extend class notation to include the empty set.
 class  (/)
 
Definitiondf-nul 3276 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 3277. (Contributed by NM, 5-Aug-1993.)
 |-  (/)  =  ( _V  \  _V )
 
Theoremdfnul2 3277 Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring] p. 20. (Contributed by NM, 26-Dec-1996.)
 |-  (/)  =  { x  |  -.  x  =  x }
 
Theoremdfnul3 3278 Alternate definition of the empty set. (Contributed by NM, 25-Mar-2004.)
 |-  (/)  =  { x  e.  A  |  -.  x  e.  A }
 
Theoremnoel 3279 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.)
 |- 
 -.  A  e.  (/)
 
Theoremn0i 3280 If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2629. (Contributed by NM, 31-Dec-1993.)
 |-  ( B  e.  A  ->  -.  A  =  (/) )
 
Theoremne0i 3281 If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2629. (Contributed by NM, 31-Dec-1993.)
 |-  ( B  e.  A  ->  A  =/=  (/) )
 
Theoremvn0 3282 The universal class is not equal to the empty set. (Contributed by NM, 11-Sep-2008.)
 |- 
 _V  =/=  (/)
 
Theoremvn0m 3283 The universal class is inhabited. (Contributed by Jim Kingdon, 17-Dec-2018.)
 |- 
 E. x  x  e. 
 _V
 
Theoremn0rf 3284 An inhabited class is nonempty. Following the Definition of [Bauer], p. 483, we call a class  A nonempty if  A  =/=  (/) 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 3285 requires only that  x not be free in, rather than not occur in,  A. (Contributed by Jim Kingdon, 31-Jul-2018.)
 |-  F/_ x A   =>    |-  ( E. x  x  e.  A  ->  A  =/= 
 (/) )
 
Theoremn0r 3285* An inhabited class is nonempty. See n0rf 3284 for more discussion. (Contributed by Jim Kingdon, 31-Jul-2018.)
 |-  ( E. x  x  e.  A  ->  A  =/= 
 (/) )
 
Theoremneq0r 3286* An inhabited class is nonempty. See n0rf 3284 for more discussion. (Contributed by Jim Kingdon, 31-Jul-2018.)
 |-  ( E. x  x  e.  A  ->  -.  A  =  (/) )
 
Theoremreximdva0m 3287* Restricted existence deduced from inhabited class. (Contributed by Jim Kingdon, 31-Jul-2018.)
 |-  ( ( ph  /\  x  e.  A )  ->  ps )   =>    |-  (
 ( ph  /\  E. x  x  e.  A )  ->  E. x  e.  A  ps )
 
Theoremn0mmoeu 3288* 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.)
 |-  ( E. x  x  e.  A  ->  ( E* x  x  e.  A 
 <->  E! x  x  e.  A ) )
 
Theoremrex0 3289 Vacuous existential quantification is false. (Contributed by NM, 15-Oct-2003.)
 |- 
 -.  E. x  e.  (/)  ph
 
Theoremeq0 3290* The empty set has no elements. Theorem 2 of [Suppes] p. 22. (Contributed by NM, 29-Aug-1993.)
 |-  ( A  =  (/)  <->  A. x  -.  x  e.  A )
 
Theoremeqv 3291* The universe contains every set. (Contributed by NM, 11-Sep-2006.)
 |-  ( A  =  _V  <->  A. x  x  e.  A )
 
Theoremnotm0 3292* A class is not inhabited if and only if it is empty. (Contributed by Jim Kingdon, 1-Jul-2022.)
 |-  ( -.  E. x  x  e.  A  <->  A  =  (/) )
 
Theoremnel0 3293* From the general negation of membership in  A, infer that  A is the empty set. (Contributed by BJ, 6-Oct-2018.)
 |- 
 -.  x  e.  A   =>    |-  A  =  (/)
 
Theorem0el 3294* Membership of the empty set in another class. (Contributed by NM, 29-Jun-2004.)
 |-  ( (/)  e.  A  <->  E. x  e.  A  A. y  -.  y  e.  x )
 
Theoremabvor0dc 3295* 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 
 ph  ->  ( { x  |  ph }  =  _V  \/  { x  |  ph }  =  (/) ) )
 
Theoremabn0r 3296 Nonempty class abstraction. (Contributed by Jim Kingdon, 1-Aug-2018.)
 |-  ( E. x ph  ->  { x  |  ph }  =/=  (/) )
 
Theoremabn0m 3297* Inhabited class abstraction. (Contributed by Jim Kingdon, 8-Jul-2022.)
 |-  ( E. y  y  e.  { x  |  ph
 } 
 <-> 
 E. x ph )
 
Theoremrabn0r 3298 Nonempty restricted class abstraction. (Contributed by Jim Kingdon, 1-Aug-2018.)
 |-  ( E. x  e.  A  ph  ->  { x  e.  A  |  ph }  =/=  (/) )
 
Theoremrabn0m 3299* Inhabited restricted class abstraction. (Contributed by Jim Kingdon, 18-Sep-2018.)
 |-  ( E. y  y  e.  { x  e.  A  |  ph }  <->  E. x  e.  A  ph )
 
Theoremrab0 3300 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.)
 |- 
 { x  e.  (/)  |  ph }  =  (/)
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