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

Theoremtppreq3 3501 An unordered triple is an unordered pair if one of its elements is identical with another element. (Contributed by Alexander van der Vekens, 6-Oct-2017.)

Theoremprid1g 3502 An unordered pair contains its first member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by Stefan Allan, 8-Nov-2008.)

Theoremprid2g 3503 An unordered pair contains its second member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by Stefan Allan, 8-Nov-2008.)

Theoremprid1 3504 An unordered pair contains its first member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 5-Aug-1993.)

Theoremprid2 3505 An unordered pair contains its second member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 5-Aug-1993.)

Theoremprprc1 3506 A proper class vanishes in an unordered pair. (Contributed by NM, 5-Aug-1993.)

Theoremprprc2 3507 A proper class vanishes in an unordered pair. (Contributed by NM, 22-Mar-2006.)

Theoremprprc 3508 An unordered pair containing two proper classes is the empty set. (Contributed by NM, 22-Mar-2006.)

Theoremtpid1 3509 One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremtpid2 3510 One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremtpid3g 3511 Closed theorem form of tpid3 3512. (Contributed by Alan Sare, 24-Oct-2011.)

Theoremtpid3 3512 One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremsnnzg 3513 The singleton of a set is not empty. (Contributed by NM, 14-Dec-2008.)

Theoremsnmg 3514* The singleton of a set is inhabited. (Contributed by Jim Kingdon, 11-Aug-2018.)

Theoremsnnz 3515 The singleton of a set is not empty. (Contributed by NM, 10-Apr-1994.)

Theoremsnm 3516* The singleton of a set is inhabited. (Contributed by Jim Kingdon, 11-Aug-2018.)

Theoremprmg 3517* A pair containing a set is inhabited. (Contributed by Jim Kingdon, 21-Sep-2018.)

Theoremprnz 3518 A pair containing a set is not empty. (Contributed by NM, 9-Apr-1994.)

Theoremprm 3519* A pair containing a set is inhabited. (Contributed by Jim Kingdon, 21-Sep-2018.)

Theoremprnzg 3520 A pair containing a set is not empty. (Contributed by FL, 19-Sep-2011.)

Theoremtpnz 3521 A triplet containing a set is not empty. (Contributed by NM, 10-Apr-1994.)

Theoremsnss 3522 The singleton of an element of a class is a subset of the class. Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 5-Aug-1993.)

Theoremeldifsn 3523 Membership in a set with an element removed. (Contributed by NM, 10-Oct-2007.)

Theoremeldifsni 3524 Membership in a set with an element removed. (Contributed by NM, 10-Mar-2015.)

Theoremneldifsn 3525 is not in . (Contributed by David Moews, 1-May-2017.)

Theoremneldifsnd 3526 is not in . Deduction form. (Contributed by David Moews, 1-May-2017.)

Theoremrexdifsn 3527 Restricted existential quantification over a set with an element removed. (Contributed by NM, 4-Feb-2015.)

Theoremsnssg 3528 The singleton of an element of a class is a subset of the class. Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 22-Jul-2001.)

Theoremdifsn 3529 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 3530 Removal of a singleton from an unordered pair. (Contributed by NM, 16-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremdifprsn1 3531 Removal of a singleton from an unordered pair. (Contributed by Thierry Arnoux, 4-Feb-2017.)

Theoremdifprsn2 3532 Removal of a singleton from an unordered pair. (Contributed by Alexander van der Vekens, 5-Oct-2017.)

Theoremdiftpsn3 3533 Removal of a singleton from an unordered triple. (Contributed by Alexander van der Vekens, 5-Oct-2017.)

Theoremdifsnb 3534 equals if and only if is not a member of . Generalization of difsn 3529. (Contributed by David Moews, 1-May-2017.)

Theoremdifsnpssim 3535 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 3536 The singleton of an element of a class is a subset of the class. (Contributed by NM, 6-Jun-1994.)

Theoremsnssd 3537 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 3538 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 6111. (Contributed by Jim Kingdon, 10-Aug-2018.)

Theorempw0 3539 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 3540 A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 27-Aug-2004.)

Theoremsnsspr2 3541 A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 2-May-2009.)

Theoremsnsstp1 3542 A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)

Theoremsnsstp2 3543 A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)

Theoremsnsstp3 3544 A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)

Theoremprsstp12 3545 A pair is a subset of an unordered triple containing its members. (Contributed by Jim Kingdon, 11-Aug-2018.)

Theoremprsstp13 3546 A pair is a subset of an unordered triple containing its members. (Contributed by Jim Kingdon, 11-Aug-2018.)

Theoremprsstp23 3547 A pair is a subset of an unordered triple containing its members. (Contributed by Jim Kingdon, 11-Aug-2018.)

Theoremprss 3548 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 3549 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 3550 A pair of elements of a class is a subset of the class. (Contributed by NM, 16-Jan-2015.)

Theoremprsspwg 3551 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 3552 Empty set and the singleton itself are subsets of a singleton. (Contributed by Jim Kingdon, 10-Aug-2018.)

Theoremsssnm 3553* The inhabited subset of a singleton. (Contributed by Jim Kingdon, 10-Aug-2018.)

Theoremeqsnm 3554* Two ways to express that an inhabited set equals a singleton. (Contributed by Jim Kingdon, 11-Aug-2018.)

Theoremssprr 3555 The subsets of a pair. (Contributed by Jim Kingdon, 11-Aug-2018.)

Theoremsstpr 3556 The subsets of a triple. (Contributed by Jim Kingdon, 11-Aug-2018.)

Theoremtpss 3557 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 3558 A triple of elements of a class is a subset of the class. (Contributed by Alexander van der Vekens, 1-Feb-2018.)

Theoremsneqr 3559 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 3560 If a singleton is a subset of another, their members are equal. (Contributed by NM, 28-May-2006.)

Theoremsneqrg 3561 Closed form of sneqr 3559. (Contributed by Scott Fenton, 1-Apr-2011.)

Theoremsneqbg 3562 Two singletons of sets are equal iff their elements are equal. (Contributed by Scott Fenton, 16-Apr-2012.)

Theoremsnsspw 3563 The singleton of a class is a subset of its power class. (Contributed by NM, 5-Aug-1993.)

Theoremprsspw 3564 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 3565 Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal. Closed form of preqr1 3567. (Contributed by Jim Kingdon, 21-Sep-2018.)

Theorempreqr2g 3566 Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the second elements are equal. Closed form of preqr2 3568. (Contributed by Jim Kingdon, 21-Sep-2018.)

Theorempreqr1 3567 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 3568 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 3569 Equality relationship for two unordered pairs. (Contributed by NM, 17-Oct-1996.)

Theoremprel12 3570 Equality of two unordered pairs. (Contributed by NM, 17-Oct-1996.)

Theoremopthpr 3571 A way to represent ordered pairs using unordered pairs with distinct members. (Contributed by NM, 27-Mar-2007.)

Theorempreq12bg 3572 Closed form of preq12b 3569. (Contributed by Scott Fenton, 28-Mar-2014.)

Theoremprneimg 3573 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 3574 Equivalence for a pair equal to a singleton. (Contributed by NM, 3-Jun-2008.)

Theoremdfopg 3575 Value of the ordered pair when the arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremdfop 3576 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 3577 Equality theorem for ordered pairs. (Contributed by NM, 25-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremopeq2 3578 Equality theorem for ordered pairs. (Contributed by NM, 25-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremopeq12 3579 Equality theorem for ordered pairs. (Contributed by NM, 28-May-1995.)

Theoremopeq1i 3580 Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)

Theoremopeq2i 3581 Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)

Theoremopeq12i 3582 Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)

Theoremopeq1d 3583 Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)

Theoremopeq2d 3584 Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)

Theoremopeq12d 3585 Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremoteq1 3586 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)

Theoremoteq2 3587 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)

Theoremoteq3 3588 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)

Theoremoteq1d 3589 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)

Theoremoteq2d 3590 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)

Theoremoteq3d 3591 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)

Theoremoteq123d 3592 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)

Theoremnfop 3593 Bound-variable hypothesis builder for ordered pairs. (Contributed by NM, 14-Nov-1995.)

Theoremnfopd 3594 Deduction version of bound-variable hypothesis builder nfop 3593. This shows how the deduction version of a not-free theorem such as nfop 3593 can be created from the corresponding not-free inference theorem. (Contributed by NM, 4-Feb-2008.)

Theoremopid 3595 The ordered pair in Kuratowski's representation. (Contributed by FL, 28-Dec-2011.)

Theoremralunsn 3596* 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 3597* Double restricted quantification over the union of a set and a singleton, using implicit substitution. (Contributed by Paul Chapman, 17-Nov-2012.)

Theoremopprc 3598 Expansion of an ordered pair when either member is a proper class. (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremopprc1 3599 Expansion of an ordered pair when the first member is a proper class. See also opprc 3598. (Contributed by NM, 10-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremopprc2 3600 Expansion of an ordered pair when the second member is a proper class. See also opprc 3598. (Contributed by NM, 15-Nov-1994.) (Revised by Mario Carneiro, 26-Apr-2015.)

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