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Theorem List for Metamath Proof Explorer - 3501-3600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremineq1d 3501 Equality deduction for intersection of two classes. (Contributed by NM, 10-Apr-1994.)

Theoremineq2d 3502 Equality deduction for intersection of two classes. (Contributed by NM, 10-Apr-1994.)

Theoremineq12d 3503 Equality deduction for intersection of two classes. (Contributed by NM, 24-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremineqan12d 3504 Equality deduction for intersection of two classes. (Contributed by NM, 7-Feb-2007.)

Theoremdfss1 3505 A frequently-used variant of subclass definition df-ss 3294. (Contributed by NM, 10-Jan-2015.)

Theoremdfss5 3506 Another definition of subclasshood. Similar to df-ss 3294, dfss 3295, and dfss1 3505. (Contributed by David Moews, 1-May-2017.)

Theoremnfin 3507 Bound-variable hypothesis builder for the intersection of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.)

Theoremcsbing 3508 Distribute proper substitution through an intersection relation. (Contributed by Alan Sare, 22-Jul-2012.)

Theoremrabbi2dva 3509* Deduction from a wff to a restricted class abstraction. (Contributed by NM, 14-Jan-2014.)

Theoreminidm 3510 Idempotent law for intersection of classes. Theorem 15 of [Suppes] p. 26. (Contributed by NM, 5-Aug-1993.)

Theoreminass 3511 Associative law for intersection of classes. Exercise 9 of [TakeutiZaring] p. 17. (Contributed by NM, 3-May-1994.)

Theoremin12 3512 A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.)

Theoremin32 3513 A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremin13 3514 A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)

Theoremin31 3515 A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)

Theoreminrot 3516 Rotate the intersection of 3 classes. (Contributed by NM, 27-Aug-2012.)

Theoremin4 3517 Rearrangement of intersection of 4 classes. (Contributed by NM, 21-Apr-2001.)

Theoreminindi 3518 Intersection distributes over itself. (Contributed by NM, 6-May-1994.)

Theoreminindir 3519 Intersection distributes over itself. (Contributed by NM, 17-Aug-2004.)

Theoremsseqin2 3520 A relationship between subclass and intersection. Similar to Exercise 9 of [TakeutiZaring] p. 18. (Contributed by NM, 17-May-1994.)

Theoreminss1 3521 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.)

Theoreminss2 3522 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.)

Theoremssin 3523 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.)

Theoremssini 3524 An inference showing that a subclass of two classes is a subclass of their intersection. (Contributed by NM, 24-Nov-2003.)

Theoremssind 3525 A deduction showing that a subclass of two classes is a subclass of their intersection. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremssrin 3526 Add right intersection to subclass relation. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremsslin 3527 Add left intersection to subclass relation. (Contributed by NM, 19-Oct-1999.)

Theoremss2in 3528 Intersection of subclasses. (Contributed by NM, 5-May-2000.)

Theoremssinss1 3529 Intersection preserves subclass relationship. (Contributed by NM, 14-Sep-1999.)

Theoreminss 3530 Inclusion of an intersection of two classes. (Contributed by NM, 30-Oct-2014.)

2.1.13.4  Combinations of difference, union, and intersection of two classes

Theoremunabs 3531 Absorption law for union. (Contributed by NM, 16-Apr-2006.)

Theoreminabs 3532 Absorption law for intersection. (Contributed by NM, 16-Apr-2006.)

Theoremnssinpss 3533 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.)

Theoremnsspssun 3534 Negation of subclass expressed in terms of proper subclass and union. (Contributed by NM, 15-Sep-2004.)

Theoremdfss4 3535 Subclass defined in terms of class difference. See comments under dfun2 3536. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremdfun2 3536 An alternate definition of the union of two classes in terms of class difference, requiring no dummy variables. Along with dfin2 3537 and dfss4 3535 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.)

Theoremdfin2 3537 An alternate definition of the intersection of two classes in terms of class difference, requiring no dummy variables. See comments under dfun2 3536. Another version is given by dfin4 3541. (Contributed by NM, 10-Jun-2004.)

Theoremdifin 3538 Difference with intersection. Theorem 33 of [Suppes] p. 29. (Contributed by NM, 31-Mar-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremdfun3 3539 Union defined in terms of intersection (De Morgan's law). Definition of union in [Mendelson] p. 231. (Contributed by NM, 8-Jan-2002.)

Theoremdfin3 3540 Intersection defined in terms of union (De Morgan's law. Similar to Exercise 4.10(n) of [Mendelson] p. 231. (Contributed by NM, 8-Jan-2002.)

Theoremdfin4 3541 Alternate definition of the intersection of two classes. Exercise 4.10(q) of [Mendelson] p. 231. (Contributed by NM, 25-Nov-2003.)

Theoreminvdif 3542 Intersection with universal complement. Remark in [Stoll] p. 20. (Contributed by NM, 17-Aug-2004.)

Theoremindif 3543 Intersection with class difference. Theorem 34 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)

Theoremindif2 3544 Bring an intersection in and out of a class difference. (Contributed by Jeff Hankins, 15-Jul-2009.)

Theoremindif1 3545 Bring an intersection in and out of a class difference. (Contributed by Mario Carneiro, 15-May-2015.)

Theoremindifcom 3546 Commutation law for intersection and difference. (Contributed by Scott Fenton, 18-Feb-2013.)

Theoremindi 3547 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.)

Theoremundi 3548 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.)

Theoremindir 3549 Distributive law for intersection over union. Theorem 28 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)

Theoremundir 3550 Distributive law for union over intersection. Theorem 29 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)

Theoremunineq 3551 Infer equality from equalities of union and intersection. Exercise 20 of [Enderton] p. 32 and its converse. (Contributed by NM, 10-Aug-2004.)

Theoremuneqin 3552 Equality of union and intersection implies equality of their arguments. (Contributed by NM, 16-Apr-2006.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremdifundi 3553 Distributive law for class difference. Theorem 39 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)

Theoremdifundir 3554 Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)

Theoremdifindi 3555 Distributive law for class difference. Theorem 40 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)

Theoremdifindir 3556 Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)

Theoremindifdir 3557 Distribute intersection over difference. (Contributed by Scott Fenton, 14-Apr-2011.)

Theoremdifdif2 3558 Set difference with a set difference. (Contributed by Thierry Arnoux, 18-Dec-2017.)

Theoremundm 3559 De Morgan's law for union. Theorem 5.2(13) of [Stoll] p. 19. (Contributed by NM, 18-Aug-2004.)

Theoremindm 3560 De Morgan's law for intersection. Theorem 5.2(13') of [Stoll] p. 19. (Contributed by NM, 18-Aug-2004.)

Theoremdifun1 3561 A relationship involving double difference and union. (Contributed by NM, 29-Aug-2004.)

Theoremundif3 3562 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.)

Theoremdifin2 3563 Represent a set difference as an intersection with a larger difference. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremdif32 3564 Swap second and third argument of double difference. (Contributed by NM, 18-Aug-2004.)

Theoremdifabs 3565 Absorption-like law for class difference: you can remove a class only once. (Contributed by FL, 2-Aug-2009.)

Theoremsymdif1 3566 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.)

2.1.13.5  Class abstractions with difference, union, and intersection of two classes

Theoremsymdif2 3567* Two ways to express symmetric difference. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremunab 3568 Union of two class abstractions. (Contributed by NM, 29-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoreminab 3569 Intersection of two class abstractions. (Contributed by NM, 29-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremdifab 3570 Difference of two class abstractions. (Contributed by NM, 23-Oct-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremnotab 3571 A class builder defined by a negation. (Contributed by FL, 18-Sep-2010.)

Theoremunrab 3572 Union of two restricted class abstractions. (Contributed by NM, 25-Mar-2004.)

Theoreminrab 3573 Intersection of two restricted class abstractions. (Contributed by NM, 1-Sep-2006.)

Theoreminrab2 3574* Intersection with a restricted class abstraction. (Contributed by NM, 19-Nov-2007.)

Theoremdifrab 3575 Difference of two restricted class abstractions. (Contributed by NM, 23-Oct-2004.)

Theoremdfrab2 3576* Alternate definition of restricted class abstraction. (Contributed by NM, 20-Sep-2003.)

Theoremdfrab3 3577* Alternate definition of restricted class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)

Theoremnotrab 3578* Complementation of restricted class abstractions. (Contributed by Mario Carneiro, 3-Sep-2015.)

Theoremdfrab3ss 3579* Restricted class abstraction with a common superset. (Contributed by Stefan O'Rear, 12-Sep-2015.) (Proof shortened by Mario Carneiro, 8-Nov-2015.)

Theoremrabun2 3580 Abstraction restricted to a union. (Contributed by Stefan O'Rear, 5-Feb-2015.)

2.1.13.6  Restricted uniqueness with difference, union, and intersection

-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-

Theoremreuss2 3581* Transfer uniqueness to a smaller subclass. (Contributed by NM, 20-Oct-2005.)

Theoremreuss 3582* Transfer uniqueness to a smaller subclass. (Contributed by NM, 21-Aug-1999.)

Theoremreuun1 3583* Transfer uniqueness to a smaller class. (Contributed by NM, 21-Oct-2005.)

Theoremreuun2 3584* Transfer uniqueness to a smaller or larger class. (Contributed by NM, 21-Oct-2005.)

Theoremreupick 3585* Restricted uniqueness "picks" a member of a subclass. (Contributed by NM, 21-Aug-1999.)

Theoremreupick3 3586* Restricted uniqueness "picks" a member of a subclass. (Contributed by Mario Carneiro, 19-Nov-2016.)

Theoremreupick2 3587* Restricted uniqueness "picks" a member of a subclass. (Contributed by Mario Carneiro, 15-Dec-2013.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)

2.1.14  The empty set

Syntaxc0 3588 Extend class notation to include the empty set.

Definitiondf-nul 3589 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 3590. (Contributed by NM, 5-Aug-1993.)

Theoremdfnul2 3590 Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring] p. 20. (Contributed by NM, 26-Dec-1996.)

Theoremdfnul3 3591 Alternate definition of the empty set. (Contributed by NM, 25-Mar-2004.)

Theoremnoel 3592 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.)

Theoremn0i 3593 If a set has elements, it is not empty. (Contributed by NM, 31-Dec-1993.)

Theoremne0i 3594 If a set has elements, it is not empty. (Contributed by NM, 31-Dec-1993.)

Theoremvn0 3595 The universal class is not equal to the empty set. (Contributed by NM, 11-Sep-2008.)

Theoremn0f 3596 A nonempty class has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. This version of n0 3597 requires only that not be free in, rather than not occur in, . (Contributed by NM, 17-Oct-2003.)

Theoremn0 3597* A nonempty class has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. (Contributed by NM, 29-Sep-2006.)

Theoremneq0 3598* A nonempty class has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. (Contributed by NM, 5-Aug-1993.)

Theoremreximdva0 3599* Restricted existence deduced from non-empty class. (Contributed by NM, 1-Feb-2012.)

Theoremn0moeu 3600* A case of equivalence of "at most one" and "only one". (Contributed by FL, 6-Dec-2010.)

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