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

Theoremunundir 3501 Union distributes over itself. (Contributed by NM, 17-Aug-2004.)

Theoremssun1 3502 Subclass relationship for union of classes. Theorem 25 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)

Theoremssun2 3503 Subclass relationship for union of classes. (Contributed by NM, 30-Aug-1993.)

Theoremssun3 3504 Subclass law for union of classes. (Contributed by NM, 5-Aug-1993.)

Theoremssun4 3505 Subclass law for union of classes. (Contributed by NM, 14-Aug-1994.)

Theoremelun1 3506 Membership law for union of classes. (Contributed by NM, 5-Aug-1993.)

Theoremelun2 3507 Membership law for union of classes. (Contributed by NM, 30-Aug-1993.)

Theoremunss1 3508 Subclass law for union of classes. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremssequn1 3509 A relationship between subclass and union. Theorem 26 of [Suppes] p. 27. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremunss2 3510 Subclass law for union of classes. Exercise 7 of [TakeutiZaring] p. 18. (Contributed by NM, 14-Oct-1999.)

Theoremunss12 3511 Subclass law for union of classes. (Contributed by NM, 2-Jun-2004.)

Theoremssequn2 3512 A relationship between subclass and union. (Contributed by NM, 13-Jun-1994.)

Theoremunss 3513 The union of two subclasses is a subclass. Theorem 27 of [Suppes] p. 27 and its converse. (Contributed by NM, 11-Jun-2004.)

Theoremunssi 3514 An inference showing the union of two subclasses is a subclass. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremunssd 3515 A deduction showing the union of two subclasses is a subclass. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremunssad 3516 If is contained in , so is . One-way deduction form of unss 3513. Partial converse of unssd 3515. (Contributed by David Moews, 1-May-2017.)

Theoremunssbd 3517 If is contained in , so is . One-way deduction form of unss 3513. Partial converse of unssd 3515. (Contributed by David Moews, 1-May-2017.)

Theoremssun 3518 A condition that implies inclusion in the union of two classes. (Contributed by NM, 23-Nov-2003.)

Theoremrexun 3519 Restricted existential quantification over union. (Contributed by Jeff Madsen, 5-Jan-2011.)

Theoremralunb 3520 Restricted quantification over a union. (Contributed by Scott Fenton, 12-Apr-2011.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremralun 3521 Restricted quantification over union. (Contributed by Jeff Madsen, 2-Sep-2009.)

2.1.13.3  The intersection of two classes

Theoremelin 3522 Expansion of membership in an intersection of two classes. Theorem 12 of [Suppes] p. 25. (Contributed by NM, 29-Apr-1994.)

Theoremelin2 3523 Membership in a class defined as an intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.)

Theoremelin3 3524 Membership in a class defined as a ternary intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.)

Theoremincom 3525 Commutative law for intersection of classes. Exercise 7 of [TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.)

Theoremineqri 3526* Inference from membership to intersection. (Contributed by NM, 5-Aug-1993.)

Theoremineq1 3527 Equality theorem for intersection of two classes. (Contributed by NM, 14-Dec-1993.)

Theoremineq2 3528 Equality theorem for intersection of two classes. (Contributed by NM, 26-Dec-1993.)

Theoremineq12 3529 Equality theorem for intersection of two classes. (Contributed by NM, 8-May-1994.)

Theoremineq1i 3530 Equality inference for intersection of two classes. (Contributed by NM, 26-Dec-1993.)

Theoremineq2i 3531 Equality inference for intersection of two classes. (Contributed by NM, 26-Dec-1993.)

Theoremineq12i 3532 Equality inference for intersection of two classes. (Contributed by NM, 24-Jun-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)

Theoremineq1d 3533 Equality deduction for intersection of two classes. (Contributed by NM, 10-Apr-1994.)

Theoremineq2d 3534 Equality deduction for intersection of two classes. (Contributed by NM, 10-Apr-1994.)

Theoremineq12d 3535 Equality deduction for intersection of two classes. (Contributed by NM, 24-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremineqan12d 3536 Equality deduction for intersection of two classes. (Contributed by NM, 7-Feb-2007.)

Theoremdfss1 3537 A frequently-used variant of subclass definition df-ss 3326. (Contributed by NM, 10-Jan-2015.)

Theoremdfss5 3538 Another definition of subclasshood. Similar to df-ss 3326, dfss 3327, and dfss1 3537. (Contributed by David Moews, 1-May-2017.)

Theoremnfin 3539 Bound-variable hypothesis builder for the intersection of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.)

Theoremcsbing 3540 Distribute proper substitution through an intersection relation. (Contributed by Alan Sare, 22-Jul-2012.)

Theoremrabbi2dva 3541* Deduction from a wff to a restricted class abstraction. (Contributed by NM, 14-Jan-2014.)

Theoreminidm 3542 Idempotent law for intersection of classes. Theorem 15 of [Suppes] p. 26. (Contributed by NM, 5-Aug-1993.)

Theoreminass 3543 Associative law for intersection of classes. Exercise 9 of [TakeutiZaring] p. 17. (Contributed by NM, 3-May-1994.)

Theoremin12 3544 A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.)

Theoremin32 3545 A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremin13 3546 A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)

Theoremin31 3547 A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)

Theoreminrot 3548 Rotate the intersection of 3 classes. (Contributed by NM, 27-Aug-2012.)

Theoremin4 3549 Rearrangement of intersection of 4 classes. (Contributed by NM, 21-Apr-2001.)

Theoreminindi 3550 Intersection distributes over itself. (Contributed by NM, 6-May-1994.)

Theoreminindir 3551 Intersection distributes over itself. (Contributed by NM, 17-Aug-2004.)

Theoremsseqin2 3552 A relationship between subclass and intersection. Similar to Exercise 9 of [TakeutiZaring] p. 18. (Contributed by NM, 17-May-1994.)

Theoreminss1 3553 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 3554 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 3555 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 3556 An inference showing that a subclass of two classes is a subclass of their intersection. (Contributed by NM, 24-Nov-2003.)

Theoremssind 3557 A deduction showing that a subclass of two classes is a subclass of their intersection. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremssrin 3558 Add right intersection to subclass relation. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremsslin 3559 Add left intersection to subclass relation. (Contributed by NM, 19-Oct-1999.)

Theoremss2in 3560 Intersection of subclasses. (Contributed by NM, 5-May-2000.)

Theoremssinss1 3561 Intersection preserves subclass relationship. (Contributed by NM, 14-Sep-1999.)

Theoreminss 3562 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 3563 Absorption law for union. (Contributed by NM, 16-Apr-2006.)

Theoreminabs 3564 Absorption law for intersection. (Contributed by NM, 16-Apr-2006.)

Theoremnssinpss 3565 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 3566 Negation of subclass expressed in terms of proper subclass and union. (Contributed by NM, 15-Sep-2004.)

Theoremdfss4 3567 Subclass defined in terms of class difference. See comments under dfun2 3568. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremdfun2 3568 An alternate definition of the union of two classes in terms of class difference, requiring no dummy variables. Along with dfin2 3569 and dfss4 3567 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 3569 An alternate definition of the intersection of two classes in terms of class difference, requiring no dummy variables. See comments under dfun2 3568. Another version is given by dfin4 3573. (Contributed by NM, 10-Jun-2004.)

Theoremdifin 3570 Difference with intersection. Theorem 33 of [Suppes] p. 29. (Contributed by NM, 31-Mar-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremdfun3 3571 Union defined in terms of intersection (De Morgan's law). Definition of union in [Mendelson] p. 231. (Contributed by NM, 8-Jan-2002.)

Theoremdfin3 3572 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 3573 Alternate definition of the intersection of two classes. Exercise 4.10(q) of [Mendelson] p. 231. (Contributed by NM, 25-Nov-2003.)

Theoreminvdif 3574 Intersection with universal complement. Remark in [Stoll] p. 20. (Contributed by NM, 17-Aug-2004.)

Theoremindif 3575 Intersection with class difference. Theorem 34 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)

Theoremindif2 3576 Bring an intersection in and out of a class difference. (Contributed by Jeff Hankins, 15-Jul-2009.)

Theoremindif1 3577 Bring an intersection in and out of a class difference. (Contributed by Mario Carneiro, 15-May-2015.)

Theoremindifcom 3578 Commutation law for intersection and difference. (Contributed by Scott Fenton, 18-Feb-2013.)

Theoremindi 3579 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 3580 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 3581 Distributive law for intersection over union. Theorem 28 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)

Theoremundir 3582 Distributive law for union over intersection. Theorem 29 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)

Theoremunineq 3583 Infer equality from equalities of union and intersection. Exercise 20 of [Enderton] p. 32 and its converse. (Contributed by NM, 10-Aug-2004.)

Theoremuneqin 3584 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 3585 Distributive law for class difference. Theorem 39 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)

Theoremdifundir 3586 Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)

Theoremdifindi 3587 Distributive law for class difference. Theorem 40 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)

Theoremdifindir 3588 Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)

Theoremindifdir 3589 Distribute intersection over difference. (Contributed by Scott Fenton, 14-Apr-2011.)

Theoremdifdif2 3590 Set difference with a set difference. (Contributed by Thierry Arnoux, 18-Dec-2017.)

Theoremundm 3591 De Morgan's law for union. Theorem 5.2(13) of [Stoll] p. 19. (Contributed by NM, 18-Aug-2004.)

Theoremindm 3592 De Morgan's law for intersection. Theorem 5.2(13') of [Stoll] p. 19. (Contributed by NM, 18-Aug-2004.)

Theoremdifun1 3593 A relationship involving double difference and union. (Contributed by NM, 29-Aug-2004.)

Theoremundif3 3594 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 3595 Represent a set difference as an intersection with a larger difference. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremdif32 3596 Swap second and third argument of double difference. (Contributed by NM, 18-Aug-2004.)

Theoremdifabs 3597 Absorption-like law for class difference: you can remove a class only once. (Contributed by FL, 2-Aug-2009.)

Theoremsymdif1 3598 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 3599* Two ways to express symmetric difference. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremunab 3600 Union of two class abstractions. (Contributed by NM, 29-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

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