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Theorem List for Metamath Proof Explorer - 3401-3500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnssinpss 3401 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 3402 Negation of subclass expressed in terms of proper subclass and union. (Contributed by NM, 15-Sep-2004.)

Theoremdfss4 3403 Subclass defined in terms of class difference. See comments under dfun2 3404. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremdfun2 3404 An alternate definition of the union of two classes in terms of class difference, requiring no dummy variables. Along with dfin2 3405 and dfss4 3403 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 3405 An alternate definition of the intersection of two classes in terms of class difference, requiring no dummy variables. See comments under dfun2 3404. Another version is given by dfin4 3409. (Contributed by NM, 10-Jun-2004.)

Theoremdifin 3406 Difference with intersection. Theorem 33 of [Suppes] p. 29. (Contributed by NM, 31-Mar-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremdfun3 3407 Union defined in terms of intersection (DeMorgan's law). Definition of union in [Mendelson] p. 231. (Contributed by NM, 8-Jan-2002.)

Theoremdfin3 3408 Intersection defined in terms of union (DeMorgan's law. Similar to Exercise 4.10(n) of [Mendelson] p. 231. (Contributed by NM, 8-Jan-2002.)

Theoremdfin4 3409 Alternate definition of the intersection of two classes. Exercise 4.10(q) of [Mendelson] p. 231. (Contributed by NM, 25-Nov-2003.)

Theoreminvdif 3410 Intersection with universal complement. Remark in [Stoll] p. 20. (Contributed by NM, 17-Aug-2004.)

Theoremindif 3411 Intersection with class difference. Theorem 34 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)

Theoremindif2 3412 Bring an intersection in and out of a class difference. (Contributed by Jeff Hankins, 15-Jul-2009.)

Theoremindif1 3413 Bring an intersection in and out of a class difference. (Contributed by Mario Carneiro, 15-May-2015.)

Theoremindifcom 3414 Commutation law for intersection and difference. (Contributed by Scott Fenton, 18-Feb-2013.)

Theoremindi 3415 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 3416 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 3417 Distributive law for intersection over union. Theorem 28 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)

Theoremundir 3418 Distributive law for union over intersection. Theorem 29 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)

Theoremunineq 3419 Infer equality from equalities of union and intersection. Exercise 20 of [Enderton] p. 32 and its converse. (Contributed by NM, 10-Aug-2004.)

Theoremuneqin 3420 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 3421 Distributive law for class difference. Theorem 39 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)

Theoremdifundir 3422 Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)

Theoremdifindi 3423 Distributive law for class difference. Theorem 40 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)

Theoremdifindir 3424 Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)

Theoremindifdir 3425 Distribute intersection over difference. (Contributed by Scott Fenton, 14-Apr-2011.)

Theoremundm 3426 DeMorgan's law for union. Theorem 5.2(13) of [Stoll] p. 19. (Contributed by NM, 18-Aug-2004.)

Theoremindm 3427 DeMorgan's law for intersection. Theorem 5.2(13') of [Stoll] p. 19. (Contributed by NM, 18-Aug-2004.)

Theoremdifun1 3428 A relationship involving double difference and union. (Contributed by NM, 29-Aug-2004.)

Theoremundif3 3429 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 3430 Represent a set difference as an intersection with a larger difference. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremdif32 3431 Swap second and third argument of double difference. (Contributed by NM, 18-Aug-2004.)

Theoremdifabs 3432 Absorption-like law for class difference: you can remove a class only once. (Contributed by FL, 2-Aug-2009.)

Theoremsymdif1 3433 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.)

Theoremsymdif2 3434* Two ways to express symmetric difference. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremunab 3435 Union of two class abstractions. (Contributed by NM, 29-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoreminab 3436 Intersection of two class abstractions. (Contributed by NM, 29-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremdifab 3437 Difference of two class abstractions. (Contributed by NM, 23-Oct-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremnotab 3438 A class builder defined by a negation. (Contributed by FL, 18-Sep-2010.)

Theoremunrab 3439 Union of two restricted class abstractions. (Contributed by NM, 25-Mar-2004.)

Theoreminrab 3440 Intersection of two restricted class abstractions. (Contributed by NM, 1-Sep-2006.)

Theoreminrab2 3441* Intersection with a restricted class abstraction. (Contributed by NM, 19-Nov-2007.)

Theoremdifrab 3442 Difference of two restricted class abstractions. (Contributed by NM, 23-Oct-2004.)

Theoremdfrab2 3443* Alternate definition of restricted class abstraction. (Contributed by NM, 20-Sep-2003.)

Theoremdfrab3 3444* Alternate definition of restricted class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)

Theoremnotrab 3445* Complementation of a restricted class abstractions. (Contributed by Mario Carneiro, 3-Sep-2015.)

Theoremdfrab3ss 3446* Restricted class abstraction with a common superset. (Contributed by Stefan O'Rear, 12-Sep-2015.) (Proof shortened by Mario Carneiro, 8-Nov-2015.)

Theoremrabun2 3447 Abstraction restricted to a union. (Contributed by Stefan O'Rear, 5-Feb-2015.)

Theoremreuss2 3448* Transfer uniqueness to a smaller subclass. (Contributed by NM, 20-Oct-2005.)

Theoremreuss 3449* Transfer uniqueness to a smaller subclass. (Contributed by NM, 21-Aug-1999.)

Theoremreuun1 3450* Transfer uniqueness to a smaller class. (Contributed by NM, 21-Oct-2005.)

Theoremreuun2 3451* Transfer uniqueness to a smaller or larger class. (Contributed by NM, 21-Oct-2005.)

Theoremreupick 3452* Restricted uniqueness "picks" a member of a subclass. (Contributed by NM, 21-Aug-1999.)

Theoremreupick3 3453* Restricted uniqueness "picks" a member of a subclass. (Contributed by Mario Carneiro, 19-Nov-2016.)

Theoremreupick2 3454* 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 3455 Extend class notation to include the empty set.

Definitiondf-nul 3456 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 3457. (Contributed by NM, 5-Aug-1993.)

Theoremdfnul2 3457 Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring] p. 20. (Contributed by NM, 26-Dec-1996.)

Theoremdfnul3 3458 Alternate definition of the empty set.. (Contributed by NM, 25-Mar-2004.)

Theoremnoel 3459 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 3460 If a set has elements, it is not empty. (Contributed by NM, 31-Dec-1993.)

Theoremne0i 3461 If a set has elements, it is not empty. (Contributed by NM, 31-Dec-1993.)

Theoremvn0 3462 The universal class is not equal to the empty set. (Contributed by NM, 11-Sep-2008.)

Theoremn0f 3463 A nonempty class has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. This version of n0 3464 requires only that not be free in, rather than not occur in, . (Contributed by NM, 17-Oct-2003.)

Theoremn0 3464* A nonempty class has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. (Contributed by NM, 29-Sep-2006.)

Theoremneq0 3465* A nonempty class has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. (Contributed by NM, 5-Aug-1993.)

Theoremreximdva0 3466* Restricted existence deduced from non-empty class. (Contributed by NM, 1-Feb-2012.)

Theoremn0moeu 3467* A case of equivalence of "at most one" and "only one". (Contributed by FL, 6-Dec-2010.)

Theoremrex0 3468 Vacuous existential quantification is false. (Contributed by NM, 15-Oct-2003.)

Theoremeq0 3469* The empty set has no elements. Theorem 2 of [Suppes] p. 22. (Contributed by NM, 29-Aug-1993.)

Theoremeqv 3470* The universe contains every set. (Contributed by NM, 11-Sep-2006.)

Theorem0el 3471* Membership of the empty set in another class. (Contributed by NM, 29-Jun-2004.)

Theoremabvor0 3472* 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.)

Theoremabn0 3473 Nonempty class abstraction. (Contributed by NM, 26-Dec-1996.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)

Theoremrabn0 3474 Non-empty restricted class abstraction. (Contributed by NM, 29-Aug-1999.)

Theoremrab0 3475 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.)

Theoremrabeq0 3476 Condition for a restricted class abstraction to be empty. (Contributed by Jeff Madsen, 7-Jun-2010.)

Theoremrabxm 3477* Law of excluded middle, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.)

Theoremrabnc 3478* Law of noncontradiction, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.)

Theoremun0 3479 The union of a class with the empty set is itself. Theorem 24 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)

Theoremin0 3480 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.)

Theoreminv1 3481 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.)

Theoremunv 3482 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.)

Theorem0ss 3483 The null set is a subset of any class. Part of Exercise 1 of [TakeutiZaring] p. 22. (Contributed by NM, 5-Aug-1993.)

Theoremss0b 3484 Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23 and its converse. (Contributed by NM, 17-Sep-2003.)

Theoremss0 3485 Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23. (Contributed by NM, 13-Aug-1994.)

Theoremsseq0 3486 A subclass of an empty class is empty. (Contributed by NM, 7-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremssn0 3487 A class with a nonempty subclass is nonempty. (Contributed by NM, 17-Feb-2007.)

Theoremabf 3488 A class builder with a false argument is empty. (Contributed by NM, 20-Jan-2012.)

Theoremeq0rdv 3489* Deduction rule for equality to the empty set. (Contributed by NM, 11-Jul-2014.)

Theoremun00 3490 Two classes are empty iff their union is empty. (Contributed by NM, 11-Aug-2004.)

Theoremvss 3491 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.)

Theorem0pss 3492 The null set is a proper subset of any non-empty set. (Contributed by NM, 27-Feb-1996.)

Theoremnpss0 3493 No set is a proper subset of the empty set. (Contributed by NM, 17-Jun-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theorempssv 3494 Any non-universal class is a proper subclass of the universal class. (Contributed by NM, 17-May-1998.)

Theoremdisj 3495* Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 17-Feb-2004.)

Theoremdisjr 3496* Two ways of saying that two classes are disjoint. (Contributed by Jeff Madsen, 19-Jun-2011.)

Theoremdisj1 3497* Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 19-Aug-1993.)

Theoremreldisj 3498 Two ways of saying that two classes are disjoint, using the complement of relative to a universe . (Contributed by NM, 15-Feb-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremdisj3 3499 Two ways of saying that two classes are disjoint. (Contributed by NM, 19-May-1998.)

Theoremdisjne 3500 Members of disjoint sets are not equal. (Contributed by NM, 28-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

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