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Theorem List for Metamath Proof Explorer - 3401-3500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremundm 3401 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 3402 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 3403 A relationship involving double difference and union. (Contributed by NM, 29-Aug-2004.)
 |-  ( A  \  ( B  u.  C ) )  =  ( ( A 
 \  B )  \  C )
 
Theoremundif3 3404 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 3405 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 3406 Swap second and third argument of double difference. (Contributed by NM, 18-Aug-2004.)
 |-  ( ( A  \  B )  \  C )  =  ( ( A 
 \  C )  \  B )
 
Theoremdifabs 3407 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 3408 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 3409* 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 3410 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 3411 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 3412 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 3413 A class builder defined by a negation. (Contributed by FL, 18-Sep-2010.)
 |- 
 { x  |  -.  ph
 }  =  ( _V  \  { x  |  ph } )
 
Theoremunrab 3414 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 3415 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 3416* 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 3417 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 3418* Alternate definition of restricted class abstraction. (Contributed by NM, 20-Sep-2003.)
 |- 
 { x  e.  A  |  ph }  =  ( { x  |  ph }  i^i  A )
 
Theoremdfrab3 3419* Alternate definition of restricted class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)
 |- 
 { x  e.  A  |  ph }  =  ( A  i^i  { x  |  ph } )
 
Theoremnotrab 3420* Complementation of a restricted class abstractions. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  ( A  \  { x  e.  A  |  ph
 } )  =  { x  e.  A  |  -.  ph }
 
Theoremdfrab3ss 3421* 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 3422 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 3423* 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 3424* 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 3425* 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 3426* 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 3427* 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 3428* 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 3429* 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 3430 Extend class notation to include the empty set.
 class  (/)
 
Definitiondf-nul 3431 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 3432. (Contributed by NM, 5-Aug-1993.)
 |-  (/)  =  ( _V  \  _V )
 
Theoremdfnul2 3432 Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring] p. 20. (Contributed by NM, 26-Dec-1996.)
 |-  (/)  =  { x  |  -.  x  =  x }
 
Theoremdfnul3 3433 Alternate definition of the empty set.. (Contributed by NM, 25-Mar-2004.)
 |-  (/)  =  { x  e.  A  |  -.  x  e.  A }
 
Theoremnoel 3434 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 3435 If a set has elements, it is not empty. (Contributed by NM, 31-Dec-1993.)
 |-  ( B  e.  A  ->  -.  A  =  (/) )
 
Theoremne0i 3436 If a set has elements, it is not empty. (Contributed by NM, 31-Dec-1993.)
 |-  ( B  e.  A  ->  A  =/=  (/) )
 
Theoremvn0 3437 The universal class is not equal to the empty set. (Contributed by NM, 11-Sep-2008.)
 |- 
 _V  =/=  (/)
 
Theoremn0f 3438 A nonempty class has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. This version of n0 3439 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 3439* 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 3440* 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 3441* 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 3442* 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 3443 Vacuous existential quantification is false. (Contributed by NM, 15-Oct-2003.)
 |- 
 -.  E. x  e.  (/)  ph
 
Theoremeq0 3444* 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 3445* The universe contains every set. (Contributed by NM, 11-Sep-2006.)
 |-  ( A  =  _V  <->  A. x  x  e.  A )
 
Theorem0el 3446* 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 3447* 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 3448 Nonempty class abstraction. (Contributed by NM, 26-Dec-1996.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)
 |-  ( { x  |  ph
 }  =/=  (/)  <->  E. x ph )
 
Theoremrabn0 3449 Non-empty restricted class abstraction. (Contributed by NM, 29-Aug-1999.)
 |-  ( { x  e.  A  |  ph }  =/=  (/)  <->  E. x  e.  A  ph )
 
Theoremrab0 3450 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 3451 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 3452* 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 3453* 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 3454 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 3455 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 3456 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 3457 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 3458 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 3459 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 3460 Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23. (Contributed by NM, 13-Aug-1994.)
 |-  ( A  C_  (/)  ->  A  =  (/) )
 
Theoremsseq0 3461 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 3462 A class with a nonempty subclass is nonempty. (Contributed by NM, 17-Feb-2007.)
 |-  ( ( A  C_  B  /\  A  =/=  (/) )  ->  B  =/=  (/) )
 
Theoremabf 3463 A class builder with a false argument is empty. (Contributed by NM, 20-Jan-2012.)
 |- 
 -.  ph   =>    |- 
 { x  |  ph }  =  (/)
 
Theoremeq0rdv 3464* Deduction rule for equality to the empty set. (Contributed by NM, 11-Jul-2014.)
 |-  ( ph  ->  -.  x  e.  A )   =>    |-  ( ph  ->  A  =  (/) )
 
Theoremun00 3465 Two classes are empty iff their union is empty. (Contributed by NM, 11-Aug-2004.)
 |-  ( ( A  =  (/)  /\  B  =  (/) )  <->  ( A  u.  B )  =  (/) )
 
Theoremvss 3466 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 3467 The null set is a proper subset of any non-empty set. (Contributed by NM, 27-Feb-1996.)
 |-  ( (/)  C.  A  <->  A  =/=  (/) )
 
Theoremnpss0 3468 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.  (/)
 
Theorempssv 3469 Any non-universal class is a proper subclass of the universal class. (Contributed by NM, 17-May-1998.)
 |-  ( A  C.  _V  <->  -.  A  =  _V )
 
Theoremdisj 3470* 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 3471* 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 3472* 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 3473 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 3474 Two ways of saying that two classes are disjoint. (Contributed by NM, 19-May-1998.)
 |-  ( ( A  i^i  B )  =  (/)  <->  A  =  ( A  \  B ) )
 
Theoremdisjne 3475 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 3476 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 3477 Two ways of saying that two classes are disjoint. (Contributed by NM, 17-May-1998.)
 |-  ( ( A  i^i  B )  =  (/)  <->  A  C_  ( _V  \  B ) )
 
Theoremdisj4 3478 Two ways of saying that two classes are disjoint. (Contributed by NM, 21-Mar-2004.)
 |-  ( ( A  i^i  B )  =  (/)  <->  -.  ( A  \  B )  C.  A )
 
Theoremssdisj 3479 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 )  =  (/) )
 
Theoremdisjpss 3480 A class is a proper subset of its union with a disjoint nonempty class. (Contributed by NM, 15-Sep-2004.)
 |-  ( ( ( A  i^i  B )  =  (/)  /\  B  =/=  (/) )  ->  A  C.  ( A  u.  B ) )
 
Theoremundisj1 3481 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 3482 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 ) )  =  (/) )
 
Theoremssindif0 3483 Subclass expressed in terms of intersection with difference from the universal class. (Contributed by NM, 17-Sep-2003.)
 |-  ( A  C_  B  <->  ( A  i^i  ( _V  \  B ) )  =  (/) )
 
Theoreminelcm 3484 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 3485 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 3486 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 3487 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 ) )
 
Theoremssdif0 3488 Subclass expressed in terms of difference. Exercise 7 of [TakeutiZaring] p. 22. (Contributed by NM, 29-Apr-1994.)
 |-  ( A  C_  B  <->  ( A  \  B )  =  (/) )
 
Theoremvdif0 3489 Universal class equality in terms of empty difference. (Contributed by NM, 17-Sep-2003.)
 |-  ( A  =  _V  <->  ( _V  \  A )  =  (/) )
 
Theorempssdifn0 3490 A proper subclass has a nonempty difference. (Contributed by NM, 3-May-1994.)
 |-  ( ( A  C_  B  /\  A  =/=  B )  ->  ( B  \  A )  =/=  (/) )
 
Theorempssdif 3491 A proper subclass has a nonempty difference. (Contributed by Mario Carneiro, 27-Apr-2016.)
 |-  ( A  C.  B  ->  ( B  \  A )  =/=  (/) )
 
Theoremssnelpss 3492 A subclass missing a member is a proper subclass. (Contributed by NM, 12-Jan-2002.)
 |-  ( A  C_  B  ->  ( ( C  e.  B  /\  -.  C  e.  A )  ->  A  C.  B ) )
 
Theoremssnelpssd 3493 Subclass inclusion with one element of the superclass missing is proper subclass inclusion. Deduction form of ssnelpss 3492. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  C  e.  B )   &    |-  ( ph  ->  -.  C  e.  A )   =>    |-  ( ph  ->  A  C.  B )
 
Theorempssnel 3494* A proper subclass has a member in one argument that's not in both. (Contributed by NM, 29-Feb-1996.)
 |-  ( A  C.  B  ->  E. x ( x  e.  B  /\  -.  x  e.  A )
 )
 
Theoremdifin0ss 3495 Difference, intersection, and subclass relationship. (Contributed by NM, 30-Apr-1994.) (Proof shortened by Wolf Lammen, 30-Sep-2014.)
 |-  ( ( ( A 
 \  B )  i^i 
 C )  =  (/)  ->  ( C  C_  A  ->  C 
 C_  B ) )
 
Theoreminssdif0 3496 Intersection, subclass, and difference relationship. (Contributed by NM, 27-Oct-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (Proof shortened by Wolf Lammen, 30-Sep-2014.)
 |-  ( ( A  i^i  B )  C_  C  <->  ( A  i^i  ( B  \  C ) )  =  (/) )
 
Theoremdifid 3497 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 3498 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 3497 suggested by David Abernethy, 17-Jun-2012.) (Contributed by NM, 17-Jun-2012.) (Proof modification is discouraged.)
 |-  ( A  \  A )  =  (/)
 
Theoremdif0 3499 The difference between a class and the empty set. Part of Exercise 4.4 of [Stoll] p. 16. (Contributed by NM, 17-Aug-2004.)
 |-  ( A  \  (/) )  =  A
 
Theorem0dif 3500 The difference between the empty set and a class. Part of Exercise 4.4 of [Stoll] p. 16. (Contributed by NM, 17-Aug-2004.)
 |-  ( (/)  \  A )  =  (/)
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