HomeHome Intuitionistic Logic Explorer
Theorem List (p. 33 of 110)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 3201-3300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremssinss1 3201 Intersection preserves subclass relationship. (Contributed by NM, 14-Sep-1999.)
 |-  ( A  C_  C  ->  ( A  i^i  B )  C_  C )
 
Theoreminss 3202 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 3203 Absorption law for union. (Contributed by NM, 16-Apr-2006.)
 |-  ( A  u.  ( A  i^i  B ) )  =  A
 
Theoreminabs 3204 Absorption law for intersection. (Contributed by NM, 16-Apr-2006.)
 |-  ( A  i^i  ( A  u.  B ) )  =  A
 
Theoremssddif 3205 Double complement and subset. Similar to ddifss 3209 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 3206 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 3207 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 3208 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 3209 Double complement under universal class. In classical logic (or given an additional hypothesis, as in ddifnel 3104), this is equality rather than subset. (Contributed by Jim Kingdon, 24-Jul-2018.)
 |-  A  C_  ( _V  \  ( _V  \  A ) )
 
Theoremunssin 3210 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 3211 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 3212 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 3213 Intersection with universal complement. Remark in [Stoll] p. 20. (Contributed by NM, 17-Aug-2004.)
 |-  ( A  i^i  ( _V  \  B ) )  =  ( A  \  B )
 
Theoremindif 3214 Intersection with class difference. Theorem 34 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
 |-  ( A  i^i  ( A  \  B ) )  =  ( A  \  B )
 
Theoremindif2 3215 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 3216 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 3217 Commutation law for intersection and difference. (Contributed by Scott Fenton, 18-Feb-2013.)
 |-  ( A  i^i  ( B  \  C ) )  =  ( B  i^i  ( A  \  C ) )
 
Theoremindi 3218 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 3219 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 3220 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 3221 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 3222 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 3223 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 3224 Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)
 |-  ( ( A  u.  B )  \  C )  =  ( ( A 
 \  C )  u.  ( B  \  C ) )
 
Theoremdifindiss 3225 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 3226 Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)
 |-  ( ( A  i^i  B )  \  C )  =  ( ( A 
 \  C )  i^i  ( B  \  C ) )
 
Theoremindifdir 3227 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 3228 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 3229 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 3230 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 3231 A relationship involving double difference and union. (Contributed by NM, 29-Aug-2004.)
 |-  ( A  \  ( B  u.  C ) )  =  ( ( A 
 \  B )  \  C )
 
Theoremundif3ss 3232 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 3233 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 3234 Swap second and third argument of double difference. (Contributed by NM, 18-Aug-2004.)
 |-  ( ( A  \  B )  \  C )  =  ( ( A 
 \  C )  \  B )
 
Theoremdifabs 3235 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 3236 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 3237* 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 3238 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 3239 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 3240 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 3241 A class builder defined by a negation. (Contributed by FL, 18-Sep-2010.)
 |- 
 { x  |  -.  ph
 }  =  ( _V  \  { x  |  ph } )
 
Theoremunrab 3242 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 3243 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 3244* 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 3245 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 3246* Alternate definition of restricted class abstraction. (Contributed by NM, 20-Sep-2003.)
 |- 
 { x  e.  A  |  ph }  =  ( { x  |  ph }  i^i  A )
 
Theoremdfrab3 3247* Alternate definition of restricted class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)
 |- 
 { x  e.  A  |  ph }  =  ( A  i^i  { x  |  ph } )
 
Theoremnotrab 3248* Complementation of restricted class abstractions. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  ( A  \  { x  e.  A  |  ph
 } )  =  { x  e.  A  |  -.  ph }
 
Theoremdfrab3ss 3249* 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 3250 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 3251* 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 3252* 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 3253* 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 3254* 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 3255* 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 3256* 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 3257* 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 3258 Extend class notation to include the empty set.
 class  (/)
 
Definitiondf-nul 3259 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 3260. (Contributed by NM, 5-Aug-1993.)
 |-  (/)  =  ( _V  \  _V )
 
Theoremdfnul2 3260 Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring] p. 20. (Contributed by NM, 26-Dec-1996.)
 |-  (/)  =  { x  |  -.  x  =  x }
 
Theoremdfnul3 3261 Alternate definition of the empty set. (Contributed by NM, 25-Mar-2004.)
 |-  (/)  =  { x  e.  A  |  -.  x  e.  A }
 
Theoremnoel 3262 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 3263 If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2616. (Contributed by NM, 31-Dec-1993.)
 |-  ( B  e.  A  ->  -.  A  =  (/) )
 
Theoremne0i 3264 If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2616. (Contributed by NM, 31-Dec-1993.)
 |-  ( B  e.  A  ->  A  =/=  (/) )
 
Theoremvn0 3265 The universal class is not equal to the empty set. (Contributed by NM, 11-Sep-2008.)
 |- 
 _V  =/=  (/)
 
Theoremvn0m 3266 The universal class is inhabited. (Contributed by Jim Kingdon, 17-Dec-2018.)
 |- 
 E. x  x  e. 
 _V
 
Theoremn0rf 3267 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 3268 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 3268* An inhabited class is nonempty. See n0rf 3267 for more discussion. (Contributed by Jim Kingdon, 31-Jul-2018.)
 |-  ( E. x  x  e.  A  ->  A  =/= 
 (/) )
 
Theoremneq0r 3269* An inhabited class is nonempty. See n0rf 3267 for more discussion. (Contributed by Jim Kingdon, 31-Jul-2018.)
 |-  ( E. x  x  e.  A  ->  -.  A  =  (/) )
 
Theoremreximdva0m 3270* 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 3271* 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 3272 Vacuous existential quantification is false. (Contributed by NM, 15-Oct-2003.)
 |- 
 -.  E. x  e.  (/)  ph
 
Theoremeq0 3273* 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 3274* The universe contains every set. (Contributed by NM, 11-Sep-2006.)
 |-  ( A  =  _V  <->  A. x  x  e.  A )
 
Theorem0el 3275* 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 3276* 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 3277 Nonempty class abstraction. (Contributed by Jim Kingdon, 1-Aug-2018.)
 |-  ( E. x ph  ->  { x  |  ph }  =/=  (/) )
 
Theoremrabn0r 3278 Non-empty restricted class abstraction. (Contributed by Jim Kingdon, 1-Aug-2018.)
 |-  ( E. x  e.  A  ph  ->  { x  e.  A  |  ph }  =/=  (/) )
 
Theoremrabn0m 3279* 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 3280 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 }  =  (/)
 
Theoremrabeq0 3281 Condition for a restricted class abstraction to be empty. (Contributed by Jeff Madsen, 7-Jun-2010.)
 |-  ( { x  e.  A  |  ph }  =  (/)  <->  A. x  e.  A  -.  ph )
 
Theoremabeq0 3282 Condition for a class abstraction to be empty. (Contributed by Jim Kingdon, 12-Aug-2018.)
 |-  ( { x  |  ph
 }  =  (/)  <->  A. x  -.  ph )
 
Theoremrabxmdc 3283* Law of excluded middle given decidability, in terms of restricted class abstractions. (Contributed by Jim Kingdon, 2-Aug-2018.)
 |-  ( A. xDECID  ph  ->  A  =  ( { x  e.  A  |  ph }  u.  { x  e.  A  |  -.  ph } ) )
 
Theoremrabnc 3284* Law of noncontradiction, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.)
 |-  ( { x  e.  A  |  ph }  i^i  { x  e.  A  |  -.  ph } )  =  (/)
 
Theoremun0 3285 The union of a class with the empty set is itself. Theorem 24 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  u.  (/) )  =  A
 
Theoremin0 3286 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.)
 |-  ( A  i^i  (/) )  =  (/)
 
Theoreminv1 3287 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.)
 |-  ( A  i^i  _V )  =  A
 
Theoremunv 3288 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.)
 |-  ( A  u.  _V )  =  _V
 
Theorem0ss 3289 The null set is a subset of any class. Part of Exercise 1 of [TakeutiZaring] p. 22. (Contributed by NM, 5-Aug-1993.)
 |-  (/)  C_  A
 
Theoremss0b 3290 Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23 and its converse. (Contributed by NM, 17-Sep-2003.)
 |-  ( A  C_  (/)  <->  A  =  (/) )
 
Theoremss0 3291 Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23. (Contributed by NM, 13-Aug-1994.)
 |-  ( A  C_  (/)  ->  A  =  (/) )
 
Theoremsseq0 3292 A subclass of an empty class is empty. (Contributed by NM, 7-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( ( A  C_  B  /\  B  =  (/) )  ->  A  =  (/) )
 
Theoremssn0 3293 A class with a nonempty subclass is nonempty. (Contributed by NM, 17-Feb-2007.)
 |-  ( ( A  C_  B  /\  A  =/=  (/) )  ->  B  =/=  (/) )
 
Theoremabf 3294 A class builder with a false argument is empty. (Contributed by NM, 20-Jan-2012.)
 |- 
 -.  ph   =>    |- 
 { x  |  ph }  =  (/)
 
Theoremeq0rdv 3295* Deduction rule for equality to the empty set. (Contributed by NM, 11-Jul-2014.)
 |-  ( ph  ->  -.  x  e.  A )   =>    |-  ( ph  ->  A  =  (/) )
 
Theoremcsbprc 3296 The proper substitution of a proper class for a set into a class results in the empty set. (Contributed by NM, 17-Aug-2018.)
 |-  ( -.  A  e.  _V 
 ->  [_ A  /  x ]_ B  =  (/) )
 
Theoremun00 3297 Two classes are empty iff their union is empty. (Contributed by NM, 11-Aug-2004.)
 |-  ( ( A  =  (/)  /\  B  =  (/) )  <->  ( A  u.  B )  =  (/) )
 
Theoremvss 3298 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_  A  <->  A  =  _V )
 
Theoremdisj 3299* Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 17-Feb-2004.)
 |-  ( ( A  i^i  B )  =  (/)  <->  A. x  e.  A  -.  x  e.  B )
 
Theoremdisjr 3300* Two ways of saying that two classes are disjoint. (Contributed by Jeff Madsen, 19-Jun-2011.)
 |-  ( ( A  i^i  B )  =  (/)  <->  A. x  e.  B  -.  x  e.  A )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-10980
  Copyright terms: Public domain < Previous  Next >