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Theorem List for Metamath Proof Explorer - 3301-3400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremssrin 3301 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 3302 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 3303 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 3304 Intersection preserves subclass relationship. (Contributed by NM, 14-Sep-1999.)
 |-  ( A  C_  C  ->  ( A  i^i  B )  C_  C )
 
Theoreminss 3305 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 )
 
Theoremunabs 3306 Absorption law for union. (Contributed by NM, 16-Apr-2006.)
 |-  ( A  u.  ( A  i^i  B ) )  =  A
 
Theoreminabs 3307 Absorption law for intersection. (Contributed by NM, 16-Apr-2006.)
 |-  ( A  i^i  ( A  u.  B ) )  =  A
 
Theoremnssinpss 3308 Negation of subclass expressed in terms of intersection and proper subclass. (Contributed by NM, 30-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( -.  A  C_  B 
 <->  ( A  i^i  B )  C.  A )
 
Theoremnsspssun 3309 Negation of subclass expressed in terms of proper subclass and union. (Contributed by NM, 15-Sep-2004.)
 |-  ( -.  A  C_  B 
 <->  B  C.  ( A  u.  B ) )
 
Theoremdfss4 3310 Subclass defined in terms of class difference. See comments under dfun2 3311. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( A  C_  B  <->  ( B  \  ( B 
 \  A ) )  =  A )
 
Theoremdfun2 3311 An alternate definition of the union of two classes in terms of class difference, requiring no dummy variables. Along with dfin2 3312 and dfss4 3310 it shows we can express union, intersection, and subset directly in terms of the single "primitive" operation  \ (class difference). (Contributed by NM, 10-Jun-2004.)
 |-  ( A  u.  B )  =  ( _V  \  ( ( _V  \  A )  \  B ) )
 
Theoremdfin2 3312 An alternate definition of the intersection of two classes in terms of class difference, requiring no dummy variables. See comments under dfun2 3311. Another version is given by dfin4 3316. (Contributed by NM, 10-Jun-2004.)
 |-  ( A  i^i  B )  =  ( A  \  ( _V  \  B ) )
 
Theoremdifin 3313 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 )
 
Theoremdfun3 3314 Union defined in terms of intersection (DeMorgan's law). Definition of union in [Mendelson] p. 231. (Contributed by NM, 8-Jan-2002.)
 |-  ( A  u.  B )  =  ( _V  \  ( ( _V  \  A )  i^i  ( _V  \  B ) ) )
 
Theoremdfin3 3315 Intersection defined in terms of union (DeMorgan's law. Similar to Exercise 4.10(n) of [Mendelson] p. 231. (Contributed by NM, 8-Jan-2002.)
 |-  ( A  i^i  B )  =  ( _V  \  ( ( _V  \  A )  u.  ( _V  \  B ) ) )
 
Theoremdfin4 3316 Alternate definition of the intersection of two classes. Exercise 4.10(q) of [Mendelson] p. 231. (Contributed by NM, 25-Nov-2003.)
 |-  ( A  i^i  B )  =  ( A  \  ( A  \  B ) )
 
Theoreminvdif 3317 Intersection with universal complement. Remark in [Stoll] p. 20. (Contributed by NM, 17-Aug-2004.)
 |-  ( A  i^i  ( _V  \  B ) )  =  ( A  \  B )
 
Theoremindif 3318 Intersection with class difference. Theorem 34 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
 |-  ( A  i^i  ( A  \  B ) )  =  ( A  \  B )
 
Theoremindif2 3319 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 3320 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 3321 Commutation law for intersection and difference. (Contributed by Scott Fenton, 18-Feb-2013.)
 |-  ( A  i^i  ( B  \  C ) )  =  ( B  i^i  ( A  \  C ) )
 
Theoremindi 3322 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 3323 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 3324 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 3325 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 ) )
 
Theoremunineq 3326 Infer equality from equalities of union and intersection. Exercise 20 of [Enderton] p. 32 and its converse. (Contributed by NM, 10-Aug-2004.)
 |-  ( ( ( A  u.  C )  =  ( B  u.  C )  /\  ( A  i^i  C )  =  ( B  i^i  C ) )  <->  A  =  B )
 
Theoremuneqin 3327 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 3328 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 3329 Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)
 |-  ( ( A  u.  B )  \  C )  =  ( ( A 
 \  C )  u.  ( B  \  C ) )
 
Theoremdifindi 3330 Distributive law for class difference. Theorem 40 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
 |-  ( A  \  ( B  i^i  C ) )  =  ( ( A 
 \  B )  u.  ( A  \  C ) )
 
Theoremdifindir 3331 Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)
 |-  ( ( A  i^i  B )  \  C )  =  ( ( A 
 \  C )  i^i  ( B  \  C ) )
 
Theoremindifdir 3332 Distribute intersection over difference. (Contributed by Scott Fenton, 14-Apr-2011.)
 |-  ( ( A  \  B )  i^i  C )  =  ( ( A  i^i  C )  \  ( B  i^i  C ) )
 
Theoremundm 3333 DeMorgan'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 ) )
 
Theoremindm 3334 DeMorgan's law for intersection. Theorem 5.2(13') of [Stoll] p. 19. (Contributed by NM, 18-Aug-2004.)
 |-  ( _V  \  ( A  i^i  B ) )  =  ( ( _V  \  A )  u.  ( _V  \  B ) )
 
Theoremdifun1 3335 A relationship involving double difference and union. (Contributed by NM, 29-Aug-2004.)
 |-  ( A  \  ( B  u.  C ) )  =  ( ( A 
 \  B )  \  C )
 
Theoremundif3 3336 An equality involving class union and class difference. The first equality of Exercise 13 of [TakeutiZaring] p. 22. (Contributed by Alan Sare, 17-Apr-2012.)
 |-  ( A  u.  ( B  \  C ) )  =  ( ( A  u.  B )  \  ( C  \  A ) )
 
Theoremdifin2 3337 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 3338 Swap second and third argument of double difference. (Contributed by NM, 18-Aug-2004.)
 |-  ( ( A  \  B )  \  C )  =  ( ( A 
 \  C )  \  B )
 
Theoremdifabs 3339 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 3340 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 ) )
 
Theoremsymdif2 3341* Two ways to express symmetric difference. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( ( A  \  B )  u.  ( B  \  A ) )  =  { x  |  -.  ( x  e.  A  <->  x  e.  B ) }
 
Theoremunab 3342 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 3343 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 3344 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 3345 A class builder defined by a negation. (Contributed by FL, 18-Sep-2010.)
 |- 
 { x  |  -.  ph
 }  =  ( _V  \  { x  |  ph } )
 
Theoremunrab 3346 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 3347 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 3348* 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 3349 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 3350* Alternate definition of restricted class abstraction. (Contributed by NM, 20-Sep-2003.)
 |- 
 { x  e.  A  |  ph }  =  ( { x  |  ph }  i^i  A )
 
Theoremdfrab3 3351* Alternate definition of restricted class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)
 |- 
 { x  e.  A  |  ph }  =  ( A  i^i  { x  |  ph } )
 
Theoremnotrab 3352* Complementation of a restricted class abstractions. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  ( A  \  { x  e.  A  |  ph
 } )  =  { x  e.  A  |  -.  ph }
 
Theoremdfrab3ss 3353* 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 3354 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
 } )
 
Theoremreuss2 3355* 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 3356* 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 3357* 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 3358* 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 3359* 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 3360* 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 3361* 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 3362 Extend class notation to include the empty set.
 class  (/)
 
Definitiondf-nul 3363 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 3364. (Contributed by NM, 5-Aug-1993.)
 |-  (/)  =  ( _V  \  _V )
 
Theoremdfnul2 3364 Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring] p. 20. (Contributed by NM, 26-Dec-1996.)
 |-  (/)  =  { x  |  -.  x  =  x }
 
Theoremdfnul3 3365 Alternate definition of the empty set.. (Contributed by NM, 25-Mar-2004.)
 |-  (/)  =  { x  e.  A  |  -.  x  e.  A }
 
Theoremnoel 3366 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 3367 If a set has elements, it is not empty. (Contributed by NM, 31-Dec-1993.)
 |-  ( B  e.  A  ->  -.  A  =  (/) )
 
Theoremne0i 3368 If a set has elements, it is not empty. (Contributed by NM, 31-Dec-1993.)
 |-  ( B  e.  A  ->  A  =/=  (/) )
 
Theoremvn0 3369 The universal class is not equal to the empty set. (Contributed by NM, 11-Sep-2008.)
 |- 
 _V  =/=  (/)
 
Theoremn0f 3370 A nonempty class has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. This version of n0 3371 requires only that  x not be free in, rather than not occur in,  A. (Contributed by NM, 17-Oct-2003.)
 |-  F/_ x A   =>    |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
 
Theoremn0 3371* A nonempty class has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. (Contributed by NM, 29-Sep-2006.)
 |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
 
Theoremneq0 3372* A nonempty class has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. (Contributed by NM, 5-Aug-1993.)
 |-  ( -.  A  =  (/)  <->  E. x  x  e.  A )
 
Theoremreximdva0 3373* Restricted existence deduced from non-empty class. (Contributed by NM, 1-Feb-2012.)
 |-  ( ( ph  /\  x  e.  A )  ->  ps )   =>    |-  (
 ( ph  /\  A  =/=  (/) )  ->  E. x  e.  A  ps )
 
Theoremn0moeu 3374* A case of equivalence of "at most one" and "only one". (Contributed by FL, 6-Dec-2010.)
 |-  ( A  =/=  (/)  ->  ( E* x  x  e.  A 
 <->  E! x  x  e.  A ) )
 
Theoremrex0 3375 Vacuous existential quantification is false. (Contributed by NM, 15-Oct-2003.)
 |- 
 -.  E. x  e.  (/)  ph
 
Theoremeq0 3376* 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 3377* The universe contains every set. (Contributed by NM, 11-Sep-2006.)
 |-  ( A  =  _V  <->  A. x  x  e.  A )
 
Theorem0el 3378* 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 )
 
Theoremabvor0 3379* The class builder of a wff not containing the abstraction variable is either the universal class or the empty set. (Contributed by Mario Carneiro, 29-Aug-2013.)
 |-  ( { x  |  ph
 }  =  _V  \/  { x  |  ph }  =  (/) )
 
Theoremabn0 3380 Nonempty class abstraction. (Contributed by NM, 26-Dec-1996.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)
 |-  ( { x  |  ph
 }  =/=  (/)  <->  E. x ph )
 
Theoremrabn0 3381 Non-empty restricted class abstraction. (Contributed by NM, 29-Aug-1999.)
 |-  ( { x  e.  A  |  ph }  =/=  (/)  <->  E. x  e.  A  ph )
 
Theoremrab0 3382 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 3383 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 )
 
Theoremrabxm 3384* Law of excluded middle, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.)
 |-  A  =  ( { x  e.  A  |  ph
 }  u.  { x  e.  A  |  -.  ph } )
 
Theoremrabnc 3385* 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 3386 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 3387 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 3388 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 3389 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 3390 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 3391 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 3392 Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23. (Contributed by NM, 13-Aug-1994.)
 |-  ( A  C_  (/)  ->  A  =  (/) )
 
Theoremsseq0 3393 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 3394 A class with a nonempty subclass is nonempty. (Contributed by NM, 17-Feb-2007.)
 |-  ( ( A  C_  B  /\  A  =/=  (/) )  ->  B  =/=  (/) )
 
Theoremabf 3395 A class builder with a false argument is empty. (Contributed by NM, 20-Jan-2012.)
 |- 
 -.  ph   =>    |- 
 { x  |  ph }  =  (/)
 
Theoremeq0rdv 3396* Deduction rule for equality to the empty set. (Contributed by NM, 11-Jul-2014.)
 |-  ( ph  ->  -.  x  e.  A )   =>    |-  ( ph  ->  A  =  (/) )
 
Theoremun00 3397 Two classes are empty iff their union is empty. (Contributed by NM, 11-Aug-2004.)
 |-  ( ( A  =  (/)  /\  B  =  (/) )  <->  ( A  u.  B )  =  (/) )
 
Theoremvss 3398 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 )
 
Theorem0pss 3399 The null set is a proper subset of any non-empty set. (Contributed by NM, 27-Feb-1996.)
 |-  ( (/)  C.  A  <->  A  =/=  (/) )
 
Theoremnpss0 3400 No set is a proper subset of the empty set. (Contributed by NM, 17-Jun-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |- 
 -.  A  C.  (/)
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