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Theorem List for Intuitionistic Logic Explorer - 3301-3400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremunss 3301 The union of two subclasses is a subclass. Theorem 27 of [Suppes] p. 27 and its converse. (Contributed by NM, 11-Jun-2004.)
 |-  ( ( A  C_  C  /\  B  C_  C ) 
 <->  ( A  u.  B )  C_  C )
 
Theoremunssi 3302 An inference showing the union of two subclasses is a subclass. (Contributed by Raph Levien, 10-Dec-2002.)
 |-  A  C_  C   &    |-  B  C_  C   =>    |-  ( A  u.  B )  C_  C
 
Theoremunssd 3303 A deduction showing the union of two subclasses is a subclass. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  ( ph  ->  A  C_  C )   &    |-  ( ph  ->  B 
 C_  C )   =>    |-  ( ph  ->  ( A  u.  B ) 
 C_  C )
 
Theoremunssad 3304 If  ( A  u.  B ) is contained in  C, so is  A. One-way deduction form of unss 3301. Partial converse of unssd 3303. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  ( A  u.  B )  C_  C )   =>    |-  ( ph  ->  A  C_  C )
 
Theoremunssbd 3305 If  ( A  u.  B ) is contained in  C, so is  B. One-way deduction form of unss 3301. Partial converse of unssd 3303. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  ( A  u.  B )  C_  C )   =>    |-  ( ph  ->  B  C_  C )
 
Theoremssun 3306 A condition that implies inclusion in the union of two classes. (Contributed by NM, 23-Nov-2003.)
 |-  ( ( A  C_  B  \/  A  C_  C )  ->  A  C_  ( B  u.  C ) )
 
Theoremrexun 3307 Restricted existential quantification over union. (Contributed by Jeff Madsen, 5-Jan-2011.)
 |-  ( E. x  e.  ( A  u.  B ) ph  <->  ( E. x  e.  A  ph  \/  E. x  e.  B  ph ) )
 
Theoremralunb 3308 Restricted quantification over a union. (Contributed by Scott Fenton, 12-Apr-2011.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  ( A. x  e.  ( A  u.  B ) ph  <->  ( A. x  e.  A  ph  /\  A. x  e.  B  ph ) )
 
Theoremralun 3309 Restricted quantification over union. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ( A. x  e.  A  ph  /\  A. x  e.  B  ph )  ->  A. x  e.  ( A  u.  B ) ph )
 
2.1.13.3  The intersection of two classes
 
Theoremelin 3310 Expansion of membership in an intersection of two classes. Theorem 12 of [Suppes] p. 25. (Contributed by NM, 29-Apr-1994.)
 |-  ( A  e.  ( B  i^i  C )  <->  ( A  e.  B  /\  A  e.  C ) )
 
Theoremelini 3311 Membership in an intersection of two classes. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  A  e.  B   &    |-  A  e.  C   =>    |-  A  e.  ( B  i^i  C )
 
Theoremelind 3312 Deduce membership in an intersection of two classes. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  ( ph  ->  X  e.  A )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  X  e.  ( A  i^i  B ) )
 
Theoremelinel1 3313 Membership in an intersection implies membership in the first set. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( A  e.  ( B  i^i  C )  ->  A  e.  B )
 
Theoremelinel2 3314 Membership in an intersection implies membership in the second set. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( A  e.  ( B  i^i  C )  ->  A  e.  C )
 
Theoremelin2 3315 Membership in a class defined as an intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  X  =  ( B  i^i  C )   =>    |-  ( A  e.  X 
 <->  ( A  e.  B  /\  A  e.  C ) )
 
Theoremelin1d 3316 Elementhood in the first set of an intersection - deduction version. (Contributed by Thierry Arnoux, 3-May-2020.)
 |-  ( ph  ->  X  e.  ( A  i^i  B ) )   =>    |-  ( ph  ->  X  e.  A )
 
Theoremelin2d 3317 Elementhood in the first set of an intersection - deduction version. (Contributed by Thierry Arnoux, 3-May-2020.)
 |-  ( ph  ->  X  e.  ( A  i^i  B ) )   =>    |-  ( ph  ->  X  e.  B )
 
Theoremelin3 3318 Membership in a class defined as a ternary intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  X  =  ( ( B  i^i  C )  i^i  D )   =>    |-  ( A  e.  X 
 <->  ( A  e.  B  /\  A  e.  C  /\  A  e.  D )
 )
 
Theoremincom 3319 Commutative law for intersection of classes. Exercise 7 of [TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  i^i  B )  =  ( B  i^i  A )
 
Theoremineqri 3320* Inference from membership to intersection. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( x  e.  A  /\  x  e.  B )  <->  x  e.  C )   =>    |-  ( A  i^i  B )  =  C
 
Theoremineq1 3321 Equality theorem for intersection of two classes. (Contributed by NM, 14-Dec-1993.)
 |-  ( A  =  B  ->  ( A  i^i  C )  =  ( B  i^i  C ) )
 
Theoremineq2 3322 Equality theorem for intersection of two classes. (Contributed by NM, 26-Dec-1993.)
 |-  ( A  =  B  ->  ( C  i^i  A )  =  ( C  i^i  B ) )
 
Theoremineq12 3323 Equality theorem for intersection of two classes. (Contributed by NM, 8-May-1994.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  i^i  C )  =  ( B  i^i  D ) )
 
Theoremineq1i 3324 Equality inference for intersection of two classes. (Contributed by NM, 26-Dec-1993.)
 |-  A  =  B   =>    |-  ( A  i^i  C )  =  ( B  i^i  C )
 
Theoremineq2i 3325 Equality inference for intersection of two classes. (Contributed by NM, 26-Dec-1993.)
 |-  A  =  B   =>    |-  ( C  i^i  A )  =  ( C  i^i  B )
 
Theoremineq12i 3326 Equality inference for intersection of two classes. (Contributed by NM, 24-Jun-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
 |-  A  =  B   &    |-  C  =  D   =>    |-  ( A  i^i  C )  =  ( B  i^i  D )
 
Theoremineq1d 3327 Equality deduction for intersection of two classes. (Contributed by NM, 10-Apr-1994.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  i^i  C )  =  ( B  i^i  C ) )
 
Theoremineq2d 3328 Equality deduction for intersection of two classes. (Contributed by NM, 10-Apr-1994.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C  i^i  A )  =  ( C  i^i  B ) )
 
Theoremineq12d 3329 Equality deduction for intersection of two classes. (Contributed by NM, 24-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A  i^i  C )  =  ( B  i^i  D ) )
 
Theoremineqan12d 3330 Equality deduction for intersection of two classes. (Contributed by NM, 7-Feb-2007.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ps  ->  C  =  D )   =>    |-  ( ( ph  /\ 
 ps )  ->  ( A  i^i  C )  =  ( B  i^i  D ) )
 
Theoremdfss1 3331 A frequently-used variant of subclass definition df-ss 3134. (Contributed by NM, 10-Jan-2015.)
 |-  ( A  C_  B  <->  ( B  i^i  A )  =  A )
 
Theoremdfss5 3332 Another definition of subclasshood. Similar to df-ss 3134, dfss 3135, and dfss1 3331. (Contributed by David Moews, 1-May-2017.)
 |-  ( A  C_  B  <->  A  =  ( B  i^i  A ) )
 
Theoremnfin 3333 Bound-variable hypothesis builder for the intersection of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x ( A  i^i  B )
 
Theoremcsbing 3334 Distribute proper substitution through an intersection relation. (Contributed by Alan Sare, 22-Jul-2012.)
 |-  ( A  e.  B  -> 
 [_ A  /  x ]_ ( C  i^i  D )  =  ( [_ A  /  x ]_ C  i^i  [_ A  /  x ]_ D ) )
 
Theoremrabbi2dva 3335* Deduction from a wff to a restricted class abstraction. (Contributed by NM, 14-Jan-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  ( x  e.  B  <->  ps ) )   =>    |-  ( ph  ->  ( A  i^i  B )  =  { x  e.  A  |  ps }
 )
 
Theoreminidm 3336 Idempotent law for intersection of classes. Theorem 15 of [Suppes] p. 26. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  i^i  A )  =  A
 
Theoreminass 3337 Associative law for intersection of classes. Exercise 9 of [TakeutiZaring] p. 17. (Contributed by NM, 3-May-1994.)
 |-  ( ( A  i^i  B )  i^i  C )  =  ( A  i^i  ( B  i^i  C ) )
 
Theoremin12 3338 A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.)
 |-  ( A  i^i  ( B  i^i  C ) )  =  ( B  i^i  ( A  i^i  C ) )
 
Theoremin32 3339 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 3340 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 3341 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 3342 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 3343 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 3344 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 3345 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 3346 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 3347 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 3348 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 3349 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 3350 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 3351 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 3352 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 3353 Add left intersection to subclass relation. (Contributed by NM, 19-Oct-1999.)
 |-  ( A  C_  B  ->  ( C  i^i  A )  C_  ( C  i^i  B ) )
 
Theoremssrind 3354 Add right intersection to subclass relation. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
 |-  ( ph  ->  A  C_  B )   =>    |-  ( ph  ->  ( A  i^i  C )  C_  ( B  i^i  C ) )
 
Theoremss2in 3355 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 3356 Intersection preserves subclass relationship. (Contributed by NM, 14-Sep-1999.)
 |-  ( A  C_  C  ->  ( A  i^i  B )  C_  C )
 
Theoreminss 3357 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 3358 Absorption law for union. (Contributed by NM, 16-Apr-2006.)
 |-  ( A  u.  ( A  i^i  B ) )  =  A
 
Theoreminabs 3359 Absorption law for intersection. (Contributed by NM, 16-Apr-2006.)
 |-  ( A  i^i  ( A  u.  B ) )  =  A
 
Theoremdfss4st 3360* 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 3361 Double complement and subset. Similar to ddifss 3365 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 3362 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 3363 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 3364 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 3365 Double complement under universal class. In classical logic (or given an additional hypothesis, as in ddifnel 3258), this is equality rather than subset. (Contributed by Jim Kingdon, 24-Jul-2018.)
 |-  A  C_  ( _V  \  ( _V  \  A ) )
 
Theoremunssin 3366 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 3367 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 3368 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 3369 Intersection with universal complement. Remark in [Stoll] p. 20. (Contributed by NM, 17-Aug-2004.)
 |-  ( A  i^i  ( _V  \  B ) )  =  ( A  \  B )
 
Theoremindif 3370 Intersection with class difference. Theorem 34 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
 |-  ( A  i^i  ( A  \  B ) )  =  ( A  \  B )
 
Theoremindif2 3371 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 3372 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 3373 Commutation law for intersection and difference. (Contributed by Scott Fenton, 18-Feb-2013.)
 |-  ( A  i^i  ( B  \  C ) )  =  ( B  i^i  ( A  \  C ) )
 
Theoremindi 3374 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 3375 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 3376 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 3377 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 3378 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 3379 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 3380 Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)
 |-  ( ( A  u.  B )  \  C )  =  ( ( A 
 \  C )  u.  ( B  \  C ) )
 
Theoremdifindiss 3381 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 3382 Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)
 |-  ( ( A  i^i  B )  \  C )  =  ( ( A 
 \  C )  i^i  ( B  \  C ) )
 
Theoremindifdir 3383 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 3384 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 3385 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 3386 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 3387 A relationship involving double difference and union. (Contributed by NM, 29-Aug-2004.)
 |-  ( A  \  ( B  u.  C ) )  =  ( ( A 
 \  B )  \  C )
 
Theoremundif3ss 3388 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 3389 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 3390 Swap second and third argument of double difference. (Contributed by NM, 18-Aug-2004.)
 |-  ( ( A  \  B )  \  C )  =  ( ( A 
 \  C )  \  B )
 
Theoremdifabs 3391 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 3392 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 3393* 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 3394 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 3395 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 3396 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 3397 A class builder defined by a negation. (Contributed by FL, 18-Sep-2010.)
 |- 
 { x  |  -.  ph
 }  =  ( _V  \  { x  |  ph } )
 
Theoremunrab 3398 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 3399 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 3400* 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 }
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