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

Theoremdifsn 3501 An element not in a set can be removed without affecting the set. (Contributed by NM, 16-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremdifprsnss 3502 Removal of a singleton from an unordered pair. (Contributed by NM, 16-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremdifprsn1 3503 Removal of a singleton from an unordered pair. (Contributed by Thierry Arnoux, 4-Feb-2017.)

Theoremdifprsn2 3504 Removal of a singleton from an unordered pair. (Contributed by Alexander van der Vekens, 5-Oct-2017.)

Theoremdiftpsn3 3505 Removal of a singleton from an unordered triple. (Contributed by Alexander van der Vekens, 5-Oct-2017.)

Theoremdifsnb 3506 equals if and only if is not a member of . Generalization of difsn 3501. (Contributed by David Moews, 1-May-2017.)

Theoremdifsnpssim 3507 is a proper subclass of if is a member of . In classical logic, the converse holds as well. (Contributed by Jim Kingdon, 9-Aug-2018.)

Theoremsnssi 3508 The singleton of an element of a class is a subset of the class. (Contributed by NM, 6-Jun-1994.)

Theoremsnssd 3509 The singleton of an element of a class is a subset of the class (deduction rule). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremdifsnss 3510 If we remove a single element from a class then put it back in, we end up with a subset of the original class. If equality is decidable, we can replace subset with equality as seen in nndifsnid 6080. (Contributed by Jim Kingdon, 10-Aug-2018.)

Theorempw0 3511 Compute the power set of the empty set. Theorem 89 of [Suppes] p. 47. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremsnsspr1 3512 A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 27-Aug-2004.)

Theoremsnsspr2 3513 A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 2-May-2009.)

Theoremsnsstp1 3514 A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)

Theoremsnsstp2 3515 A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)

Theoremsnsstp3 3516 A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)

Theoremprsstp12 3517 A pair is a subset of an unordered triple containing its members. (Contributed by Jim Kingdon, 11-Aug-2018.)

Theoremprsstp13 3518 A pair is a subset of an unordered triple containing its members. (Contributed by Jim Kingdon, 11-Aug-2018.)

Theoremprsstp23 3519 A pair is a subset of an unordered triple containing its members. (Contributed by Jim Kingdon, 11-Aug-2018.)

Theoremprss 3520 A pair of elements of a class is a subset of the class. Theorem 7.5 of [Quine] p. 49. (Contributed by NM, 30-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremprssg 3521 A pair of elements of a class is a subset of the class. Theorem 7.5 of [Quine] p. 49. (Contributed by NM, 22-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremprssi 3522 A pair of elements of a class is a subset of the class. (Contributed by NM, 16-Jan-2015.)

Theoremprsspwg 3523 An unordered pair belongs to the power class of a class iff each member belongs to the class. (Contributed by Thierry Arnoux, 3-Oct-2016.) (Revised by NM, 18-Jan-2018.)

Theoremsssnr 3524 Empty set and the singleton itself are subsets of a singleton. (Contributed by Jim Kingdon, 10-Aug-2018.)

Theoremsssnm 3525* The inhabited subset of a singleton. (Contributed by Jim Kingdon, 10-Aug-2018.)

Theoremeqsnm 3526* Two ways to express that an inhabited set equals a singleton. (Contributed by Jim Kingdon, 11-Aug-2018.)

Theoremssprr 3527 The subsets of a pair. (Contributed by Jim Kingdon, 11-Aug-2018.)

Theoremsstpr 3528 The subsets of a triple. (Contributed by Jim Kingdon, 11-Aug-2018.)

Theoremtpss 3529 A triplet of elements of a class is a subset of the class. (Contributed by NM, 9-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremtpssi 3530 A triple of elements of a class is a subset of the class. (Contributed by Alexander van der Vekens, 1-Feb-2018.)

Theoremsneqr 3531 If the singletons of two sets are equal, the two sets are equal. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 27-Aug-1993.)

Theoremsnsssn 3532 If a singleton is a subset of another, their members are equal. (Contributed by NM, 28-May-2006.)

Theoremsneqrg 3533 Closed form of sneqr 3531. (Contributed by Scott Fenton, 1-Apr-2011.)

Theoremsneqbg 3534 Two singletons of sets are equal iff their elements are equal. (Contributed by Scott Fenton, 16-Apr-2012.)

Theoremsnsspw 3535 The singleton of a class is a subset of its power class. (Contributed by NM, 5-Aug-1993.)

Theoremprsspw 3536 An unordered pair belongs to the power class of a class iff each member belongs to the class. (Contributed by NM, 10-Dec-2003.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theorempreqr1g 3537 Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal. Closed form of preqr1 3539. (Contributed by Jim Kingdon, 21-Sep-2018.)

Theorempreqr2g 3538 Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the second elements are equal. Closed form of preqr2 3540. (Contributed by Jim Kingdon, 21-Sep-2018.)

Theorempreqr1 3539 Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal. (Contributed by NM, 18-Oct-1995.)

Theorempreqr2 3540 Reverse equality lemma for unordered pairs. If two unordered pairs have the same first element, the second elements are equal. (Contributed by NM, 5-Aug-1993.)

Theorempreq12b 3541 Equality relationship for two unordered pairs. (Contributed by NM, 17-Oct-1996.)

Theoremprel12 3542 Equality of two unordered pairs. (Contributed by NM, 17-Oct-1996.)

Theoremopthpr 3543 A way to represent ordered pairs using unordered pairs with distinct members. (Contributed by NM, 27-Mar-2007.)

Theorempreq12bg 3544 Closed form of preq12b 3541. (Contributed by Scott Fenton, 28-Mar-2014.)

Theoremprneimg 3545 Two pairs are not equal if at least one element of the first pair is not contained in the second pair. (Contributed by Alexander van der Vekens, 13-Aug-2017.)

Theorempreqsn 3546 Equivalence for a pair equal to a singleton. (Contributed by NM, 3-Jun-2008.)

Theoremdfopg 3547 Value of the ordered pair when the arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremdfop 3548 Value of an ordered pair when the arguments are sets, with the conclusion corresponding to Kuratowski's original definition. (Contributed by NM, 25-Jun-1998.)

Theoremopeq1 3549 Equality theorem for ordered pairs. (Contributed by NM, 25-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremopeq2 3550 Equality theorem for ordered pairs. (Contributed by NM, 25-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremopeq12 3551 Equality theorem for ordered pairs. (Contributed by NM, 28-May-1995.)

Theoremopeq1i 3552 Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)

Theoremopeq2i 3553 Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)

Theoremopeq12i 3554 Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)

Theoremopeq1d 3555 Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)

Theoremopeq2d 3556 Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)

Theoremopeq12d 3557 Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremoteq1 3558 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)

Theoremoteq2 3559 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)

Theoremoteq3 3560 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)

Theoremoteq1d 3561 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)

Theoremoteq2d 3562 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)

Theoremoteq3d 3563 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)

Theoremoteq123d 3564 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)

Theoremnfop 3565 Bound-variable hypothesis builder for ordered pairs. (Contributed by NM, 14-Nov-1995.)

Theoremnfopd 3566 Deduction version of bound-variable hypothesis builder nfop 3565. This shows how the deduction version of a not-free theorem such as nfop 3565 can be created from the corresponding not-free inference theorem. (Contributed by NM, 4-Feb-2008.)

Theoremopid 3567 The ordered pair in Kuratowski's representation. (Contributed by FL, 28-Dec-2011.)

Theoremralunsn 3568* Restricted quantification over the union of a set and a singleton, using implicit substitution. (Contributed by Paul Chapman, 17-Nov-2012.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theorem2ralunsn 3569* Double restricted quantification over the union of a set and a singleton, using implicit substitution. (Contributed by Paul Chapman, 17-Nov-2012.)

Theoremopprc 3570 Expansion of an ordered pair when either member is a proper class. (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremopprc1 3571 Expansion of an ordered pair when the first member is a proper class. See also opprc 3570. (Contributed by NM, 10-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremopprc2 3572 Expansion of an ordered pair when the second member is a proper class. See also opprc 3570. (Contributed by NM, 15-Nov-1994.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremoprcl 3573 If an ordered pair has an element, then its arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.)

Theorempwsnss 3574 The power set of a singleton. (Contributed by Jim Kingdon, 12-Aug-2018.)

Theorempwpw0ss 3575 Compute the power set of the power set of the empty set. (See pw0 3511 for the power set of the empty set.) Theorem 90 of [Suppes] p. 48 (but with subset in place of equality). (Contributed by Jim Kingdon, 12-Aug-2018.)

Theorempwprss 3576 The power set of an unordered pair. (Contributed by Jim Kingdon, 13-Aug-2018.)

Theorempwtpss 3577 The power set of an unordered triple. (Contributed by Jim Kingdon, 13-Aug-2018.)

Theorempwpwpw0ss 3578 Compute the power set of the power set of the power set of the empty set. (See also pw0 3511 and pwpw0ss 3575.) (Contributed by Jim Kingdon, 13-Aug-2018.)

Theorempwv 3579 The power class of the universe is the universe. Exercise 4.12(d) of [Mendelson] p. 235. (Contributed by NM, 14-Sep-2003.)

2.1.18  The union of a class

Syntaxcuni 3580 Extend class notation to include the union of a class (read: 'union ')

Definitiondf-uni 3581* Define the union of a class i.e. the collection of all members of the members of the class. Definition 5.5 of [TakeutiZaring] p. 16. For example, { { 1 , 3 } , { 1 , 8 } } = { 1 , 3 , 8 } . This is similar to the union of two classes df-un 2922. (Contributed by NM, 23-Aug-1993.)

Theoremdfuni2 3582* Alternate definition of class union. (Contributed by NM, 28-Jun-1998.)

Theoremeluni 3583* Membership in class union. (Contributed by NM, 22-May-1994.)

Theoremeluni2 3584* Membership in class union. Restricted quantifier version. (Contributed by NM, 31-Aug-1999.)

Theoremelunii 3585 Membership in class union. (Contributed by NM, 24-Mar-1995.)

Theoremnfuni 3586 Bound-variable hypothesis builder for union. (Contributed by NM, 30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremnfunid 3587 Deduction version of nfuni 3586. (Contributed by NM, 18-Feb-2013.)

Theoremcsbunig 3588 Distribute proper substitution through the union of a class. (Contributed by Alan Sare, 10-Nov-2012.)

Theoremunieq 3589 Equality theorem for class union. Exercise 15 of [TakeutiZaring] p. 18. (Contributed by NM, 10-Aug-1993.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremunieqi 3590 Inference of equality of two class unions. (Contributed by NM, 30-Aug-1993.)

Theoremunieqd 3591 Deduction of equality of two class unions. (Contributed by NM, 21-Apr-1995.)

Theoremeluniab 3592* Membership in union of a class abstraction. (Contributed by NM, 11-Aug-1994.) (Revised by Mario Carneiro, 14-Nov-2016.)

Theoremelunirab 3593* Membership in union of a class abstraction. (Contributed by NM, 4-Oct-2006.)

Theoremunipr 3594 The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16. (Contributed by NM, 23-Aug-1993.)

Theoremuniprg 3595 The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16. (Contributed by NM, 25-Aug-2006.)

Theoremunisn 3596 A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53. (Contributed by NM, 30-Aug-1993.)

Theoremunisng 3597 A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53. (Contributed by NM, 13-Aug-2002.)

Theoremdfnfc2 3598* An alternative statement of the effective freeness of a class , when it is a set. (Contributed by Mario Carneiro, 14-Oct-2016.)

Theoremuniun 3599 The class union of the union of two classes. Theorem 8.3 of [Quine] p. 53. (Contributed by NM, 20-Aug-1993.)

Theoremuniin 3600 The class union of the intersection of two classes. Exercise 4.12(n) of [Mendelson] p. 235. (Contributed by NM, 4-Dec-2003.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

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