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Theorem List for Intuitionistic Logic Explorer - 3301-3400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdifdif2ss 3301 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 3302 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 3303 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 3304 A relationship involving double difference and union. (Contributed by NM, 29-Aug-2004.)
 |-  ( A  \  ( B  u.  C ) )  =  ( ( A 
 \  B )  \  C )
 
Theoremundif3ss 3305 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 3306 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 3307 Swap second and third argument of double difference. (Contributed by NM, 18-Aug-2004.)
 |-  ( ( A  \  B )  \  C )  =  ( ( A 
 \  C )  \  B )
 
Theoremdifabs 3308 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 3309 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 3310* 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 3311 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 3312 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 3313 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 3314 A class builder defined by a negation. (Contributed by FL, 18-Sep-2010.)
 |- 
 { x  |  -.  ph
 }  =  ( _V  \  { x  |  ph } )
 
Theoremunrab 3315 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 3316 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 3317* 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 3318 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 3319* Alternate definition of restricted class abstraction. (Contributed by NM, 20-Sep-2003.)
 |- 
 { x  e.  A  |  ph }  =  ( { x  |  ph }  i^i  A )
 
Theoremdfrab3 3320* Alternate definition of restricted class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)
 |- 
 { x  e.  A  |  ph }  =  ( A  i^i  { x  |  ph } )
 
Theoremnotrab 3321* Complementation of restricted class abstractions. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  ( A  \  { x  e.  A  |  ph
 } )  =  { x  e.  A  |  -.  ph }
 
Theoremdfrab3ss 3322* 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 3323 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 3324* 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 3325* 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 3326* 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 3327* 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 3328* 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 3329* 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 3330* 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 3331 Extend class notation to include the empty set.
 class  (/)
 
Definitiondf-nul 3332 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 3333. (Contributed by NM, 5-Aug-1993.)
 |-  (/)  =  ( _V  \  _V )
 
Theoremdfnul2 3333 Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring] p. 20. (Contributed by NM, 26-Dec-1996.)
 |-  (/)  =  { x  |  -.  x  =  x }
 
Theoremdfnul3 3334 Alternate definition of the empty set. (Contributed by NM, 25-Mar-2004.)
 |-  (/)  =  { x  e.  A  |  -.  x  e.  A }
 
Theoremnoel 3335 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 3336 If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2674. (Contributed by NM, 31-Dec-1993.)
 |-  ( B  e.  A  ->  -.  A  =  (/) )
 
Theoremne0i 3337 If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2674. (Contributed by NM, 31-Dec-1993.)
 |-  ( B  e.  A  ->  A  =/=  (/) )
 
Theoremne0d 3338 Deduction form of ne0i 3337. If a class has elements, then it is nonempty. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
 |-  ( ph  ->  B  e.  A )   =>    |-  ( ph  ->  A  =/= 
 (/) )
 
Theoremn0ii 3339 If a class has elements, then it is not empty. Inference associated with n0i 3336. (Contributed by BJ, 15-Jul-2021.)
 |-  A  e.  B   =>    |-  -.  B  =  (/)
 
Theoremne0ii 3340 If a class has elements, then it is nonempty. Inference associated with ne0i 3337. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  A  e.  B   =>    |-  B  =/=  (/)
 
Theoremvn0 3341 The universal class is not equal to the empty set. (Contributed by NM, 11-Sep-2008.)
 |- 
 _V  =/=  (/)
 
Theoremvn0m 3342 The universal class is inhabited. (Contributed by Jim Kingdon, 17-Dec-2018.)
 |- 
 E. x  x  e. 
 _V
 
Theoremn0rf 3343 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 3344 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 3344* An inhabited class is nonempty. See n0rf 3343 for more discussion. (Contributed by Jim Kingdon, 31-Jul-2018.)
 |-  ( E. x  x  e.  A  ->  A  =/= 
 (/) )
 
Theoremneq0r 3345* An inhabited class is nonempty. See n0rf 3343 for more discussion. (Contributed by Jim Kingdon, 31-Jul-2018.)
 |-  ( E. x  x  e.  A  ->  -.  A  =  (/) )
 
Theoremreximdva0m 3346* 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 3347* 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 3348 Vacuous existential quantification is false. (Contributed by NM, 15-Oct-2003.)
 |- 
 -.  E. x  e.  (/)  ph
 
Theoremeq0 3349* 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 3350* The universe contains every set. (Contributed by NM, 11-Sep-2006.)
 |-  ( A  =  _V  <->  A. x  x  e.  A )
 
Theoremnotm0 3351* 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 3352* 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 3353* 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 3354* 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 3355 Nonempty class abstraction. (Contributed by Jim Kingdon, 1-Aug-2018.)
 |-  ( E. x ph  ->  { x  |  ph }  =/=  (/) )
 
Theoremabn0m 3356* Inhabited class abstraction. (Contributed by Jim Kingdon, 8-Jul-2022.)
 |-  ( E. y  y  e.  { x  |  ph
 } 
 <-> 
 E. x ph )
 
Theoremrabn0r 3357 Nonempty restricted class abstraction. (Contributed by Jim Kingdon, 1-Aug-2018.)
 |-  ( E. x  e.  A  ph  ->  { x  e.  A  |  ph }  =/=  (/) )
 
Theoremrabn0m 3358* 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 3359 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 3360 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 3361 Condition for a class abstraction to be empty. (Contributed by Jim Kingdon, 12-Aug-2018.)
 |-  ( { x  |  ph
 }  =  (/)  <->  A. x  -.  ph )
 
Theoremrabxmdc 3362* 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 3363* 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 3364 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 3365 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  (/) )  =  (/)
 
Theorem0in 3366 The intersection of the empty set with a class is the empty set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( (/)  i^i  A )  =  (/)
 
Theoreminv1 3367 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 3368 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 3369 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 3370 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 3371 Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23. (Contributed by NM, 13-Aug-1994.)
 |-  ( A  C_  (/)  ->  A  =  (/) )
 
Theoremsseq0 3372 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 3373 A class with a nonempty subclass is nonempty. (Contributed by NM, 17-Feb-2007.)
 |-  ( ( A  C_  B  /\  A  =/=  (/) )  ->  B  =/=  (/) )
 
Theoremabf 3374 A class builder with a false argument is empty. (Contributed by NM, 20-Jan-2012.)
 |- 
 -.  ph   =>    |- 
 { x  |  ph }  =  (/)
 
Theoremeq0rdv 3375* Deduction for equality to the empty set. (Contributed by NM, 11-Jul-2014.)
 |-  ( ph  ->  -.  x  e.  A )   =>    |-  ( ph  ->  A  =  (/) )
 
Theoremcsbprc 3376 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 3377 Two classes are empty iff their union is empty. (Contributed by NM, 11-Aug-2004.)
 |-  ( ( A  =  (/)  /\  B  =  (/) )  <->  ( A  u.  B )  =  (/) )
 
Theoremvss 3378 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 3379* 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 3380* 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 )
 
Theoremdisj1 3381* Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 19-Aug-1993.)
 |-  ( ( A  i^i  B )  =  (/)  <->  A. x ( x  e.  A  ->  -.  x  e.  B ) )
 
Theoremreldisj 3382 Two ways of saying that two classes are disjoint, using the complement of  B relative to a universe  C. (Contributed by NM, 15-Feb-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( A  C_  C  ->  ( ( A  i^i  B )  =  (/)  <->  A  C_  ( C 
 \  B ) ) )
 
Theoremdisj3 3383 Two ways of saying that two classes are disjoint. (Contributed by NM, 19-May-1998.)
 |-  ( ( A  i^i  B )  =  (/)  <->  A  =  ( A  \  B ) )
 
Theoremdisjne 3384 Members of disjoint sets are not equal. (Contributed by NM, 28-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( ( ( A  i^i  B )  =  (/)  /\  C  e.  A  /\  D  e.  B ) 
 ->  C  =/=  D )
 
Theoremdisjel 3385 A set can't belong to both members of disjoint classes. (Contributed by NM, 28-Feb-2015.)
 |-  ( ( ( A  i^i  B )  =  (/)  /\  C  e.  A )  ->  -.  C  e.  B )
 
Theoremdisj2 3386 Two ways of saying that two classes are disjoint. (Contributed by NM, 17-May-1998.)
 |-  ( ( A  i^i  B )  =  (/)  <->  A  C_  ( _V  \  B ) )
 
Theoremssdisj 3387 Intersection with a subclass of a disjoint class. (Contributed by FL, 24-Jan-2007.)
 |-  ( ( A  C_  B  /\  ( B  i^i  C )  =  (/) )  ->  ( A  i^i  C )  =  (/) )
 
Theoremundisj1 3388 The union of disjoint classes is disjoint. (Contributed by NM, 26-Sep-2004.)
 |-  ( ( ( A  i^i  C )  =  (/)  /\  ( B  i^i  C )  =  (/) )  <->  ( ( A  u.  B )  i^i 
 C )  =  (/) )
 
Theoremundisj2 3389 The union of disjoint classes is disjoint. (Contributed by NM, 13-Sep-2004.)
 |-  ( ( ( A  i^i  B )  =  (/)  /\  ( A  i^i  C )  =  (/) )  <->  ( A  i^i  ( B  u.  C ) )  =  (/) )
 
Theoremssindif0im 3390 Subclass implies empty intersection with difference from the universal class. (Contributed by NM, 17-Sep-2003.)
 |-  ( A  C_  B  ->  ( A  i^i  ( _V  \  B ) )  =  (/) )
 
Theoreminelcm 3391 The intersection of classes with a common member is nonempty. (Contributed by NM, 7-Apr-1994.)
 |-  ( ( A  e.  B  /\  A  e.  C )  ->  ( B  i^i  C )  =/=  (/) )
 
Theoremminel 3392 A minimum element of a class has no elements in common with the class. (Contributed by NM, 22-Jun-1994.)
 |-  ( ( A  e.  B  /\  ( C  i^i  B )  =  (/) )  ->  -.  A  e.  C )
 
Theoremundif4 3393 Distribute union over difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( ( A  i^i  C )  =  (/)  ->  ( A  u.  ( B  \  C ) )  =  ( ( A  u.  B )  \  C ) )
 
Theoremdisjssun 3394 Subset relation for disjoint classes. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( ( A  i^i  B )  =  (/)  ->  ( A  C_  ( B  u.  C )  <->  A  C_  C ) )
 
Theoremssdif0im 3395 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.)
 |-  ( A  C_  B  ->  ( A  \  B )  =  (/) )
 
Theoremvdif0im 3396 Universal class equality in terms of empty difference. (Contributed by Jim Kingdon, 3-Aug-2018.)
 |-  ( A  =  _V  ->  ( _V  \  A )  =  (/) )
 
Theoremdifrab0eqim 3397* 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  =  { x  e.  V  |  ph
 }  ->  ( V  \  { x  e.  V  |  ph } )  =  (/) )
 
Theoreminssdif0im 3398 Intersection, subclass, and difference relationship. In classical logic the converse would also hold. (Contributed by Jim Kingdon, 3-Aug-2018.)
 |-  ( ( A  i^i  B )  C_  C  ->  ( A  i^i  ( B 
 \  C ) )  =  (/) )
 
Theoremdifid 3399 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.)
 |-  ( A  \  A )  =  (/)
 
TheoremdifidALT 3400 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 3399. (Contributed by David Abernethy, 17-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  \  A )  =  (/)
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