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Type | Label | Description |
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Statement | ||
Theorem | rabsssn 4001* | Conditions for a restricted class abstraction to be a subset of a singleton, i.e. to be a singleton or the empty set. (Contributed by AV, 18-Apr-2019.) |
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Theorem | ralsnsg 4002* | Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.) |
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Theorem | rexsns 4003* | Restricted existential quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.) (Revised by NM, 22-Aug-2018.) |
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Theorem | rexsnsOLD 4004* | Restricted existential quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.) Obsolete as of 22-Aug-2018. Deletion blocked by r19.12snOLD 4037. Use rexsns 4003 instead. (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | ralsng 4005* | Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.) |
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Theorem | rexsng 4006* | Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.) |
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Theorem | 2ralsng 4007* | Substitution expressed in terms of two quantifications over singletons. (Contributed by AV, 22-Dec-2019.) |
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Theorem | exsnrex 4008 | There is a set being the element of a singleton if and only if there is an element of the singleton. (Contributed by Alexander van der Vekens, 1-Jan-2018.) |
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Theorem | ralsn 4009* | Convert a quantification over a singleton to a substitution. (Contributed by NM, 27-Apr-2009.) |
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Theorem | rexsn 4010* | Restricted existential quantification over a singleton. (Contributed by Jeff Madsen, 5-Jan-2011.) |
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Theorem | elpwunsn 4011 | Membership in an extension of a power class. (Contributed by NM, 26-Mar-2007.) |
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Theorem | eqoreldif 4012 | An element of a set is either equal to another element of the set or a member of the difference of the set and the singleton containing the other element. (Contributed by AV, 25-Aug-2020.) |
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Theorem | eltpg 4013 | Members of an unordered triple of classes. (Contributed by FL, 2-Feb-2014.) (Proof shortened by Mario Carneiro, 11-Feb-2015.) |
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Theorem | eldiftp 4014 | Membership in a set with three elements removed. Similar to eldifsn 4096 and eldifpr 3989. (Contributed by David A. Wheeler, 22-Jul-2017.) |
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Theorem | eltpi 4015 | A member of an unordered triple of classes is one of them. (Contributed by Mario Carneiro, 11-Feb-2015.) |
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Theorem | eltp 4016 | A member of an unordered triple of classes is one of them. Special case of Exercise 1 of [TakeutiZaring] p. 17. (Contributed by NM, 8-Apr-1994.) (Revised by Mario Carneiro, 11-Feb-2015.) |
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Theorem | dftp2 4017* | Alternate definition of unordered triple of classes. Special case of Definition 5.3 of [TakeutiZaring] p. 16. (Contributed by NM, 8-Apr-1994.) |
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Theorem | nfpr 4018 | Bound-variable hypothesis builder for unordered pairs. (Contributed by NM, 14-Nov-1995.) |
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Theorem | ifpr 4019 | Membership of a conditional operator in an unordered pair. (Contributed by NM, 17-Jun-2007.) |
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Theorem | ralprg 4020* | Convert a quantification over a pair to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) |
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Theorem | rexprg 4021* | Convert a quantification over a pair to a disjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) |
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Theorem | raltpg 4022* | Convert a quantification over a triple to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) |
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Theorem | rextpg 4023* | Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.) |
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Theorem | ralpr 4024* | Convert a quantification over a pair to a conjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.) |
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Theorem | rexpr 4025* | Convert an existential quantification over a pair to a disjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.) |
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Theorem | raltp 4026* | Convert a quantification over a triple to a conjunction. (Contributed by NM, 13-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) |
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Theorem | rextp 4027* | Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.) |
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Theorem | nfsn 4028 | Bound-variable hypothesis builder for singletons. (Contributed by NM, 14-Nov-1995.) |
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Theorem | csbsng 4029 | Distribute proper substitution through the singleton of a class. csbsng 4029 is derived from the virtual deduction proof csbsngVD 37284. (Contributed by Alan Sare, 10-Nov-2012.) |
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Theorem | csbprg 4030 | Distribute proper substitution through a pair of classes. (Contributed by Alexander van der Vekens, 4-Sep-2018.) |
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Theorem | disjsn 4031 | Intersection with the singleton of a non-member is disjoint. (Contributed by NM, 22-May-1998.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof shortened by Wolf Lammen, 30-Sep-2014.) |
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Theorem | disjsn2 4032 | Intersection of distinct singletons is disjoint. (Contributed by NM, 25-May-1998.) |
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Theorem | disjpr2 4033 | The intersection of distinct unordered pairs is disjoint. (Contributed by Alexander van der Vekens, 11-Nov-2017.) |
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Theorem | snprc 4034 | The singleton of a proper class (one that doesn't exist) is the empty set. Theorem 7.2 of [Quine] p. 48. (Contributed by NM, 21-Jun-1993.) |
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Theorem | snnzb 4035 | A singleton is nonempty iff its argument is a set. (Contributed by Scott Fenton, 8-May-2018.) |
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Theorem | r19.12sn 4036* | Special case of r19.12 2915 where its converse holds. (Contributed by NM, 19-May-2008.) (Revised by Mario Carneiro, 23-Apr-2015.) (Revised by BJ, 18-Mar-2020.) |
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Theorem | r19.12snOLD 4037* | Older form of r19.12sn 4036. Obsolete as of 18-Mar-2020. (Contributed by NM, 19-May-2008.) (Revised by Mario Carneiro, 23-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | rabsn 4038* | Condition where a restricted class abstraction is a singleton. (Contributed by NM, 28-May-2006.) |
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Theorem | rabsnifsb 4039* | A restricted class abstraction restricted to a singleton is either the empty set or the singleton itself. (Contributed by AV, 21-Jul-2019.) |
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Theorem | rabsnif 4040* | A restricted class abstraction restricted to a singleton is either the empty set or the singleton itself. (Contributed by AV, 12-Apr-2019.) (Proof shortened by AV, 21-Jul-2019.) |
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Theorem | rabrsn 4041* | A restricted class abstraction restricted to a singleton is either the empty set or the singleton itself. (Contributed by Alexander van der Vekens, 22-Dec-2017.) (Proof shortened by AV, 21-Jul-2019.) |
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Theorem | euabsn2 4042* | Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by Mario Carneiro, 14-Nov-2016.) |
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Theorem | euabsn 4043 | Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by NM, 22-Feb-2004.) |
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Theorem | reusn 4044* | A way to express restricted existential uniqueness of a wff: its restricted class abstraction is a singleton. (Contributed by NM, 30-May-2006.) (Proof shortened by Mario Carneiro, 14-Nov-2016.) |
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Theorem | absneu 4045 | Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.) |
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Theorem | rabsneu 4046 | Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.) (Revised by Mario Carneiro, 23-Dec-2016.) |
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Theorem | eusn 4047* |
Two ways to express "![]() |
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Theorem | rabsnt 4048* | Truth implied by equality of a restricted class abstraction and a singleton. (Contributed by NM, 29-May-2006.) (Proof shortened by Mario Carneiro, 23-Dec-2016.) |
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Theorem | prcom 4049 | Commutative law for unordered pairs. (Contributed by NM, 15-Jul-1993.) |
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Theorem | preq1 4050 | Equality theorem for unordered pairs. (Contributed by NM, 29-Mar-1998.) |
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Theorem | preq2 4051 | Equality theorem for unordered pairs. (Contributed by NM, 15-Jul-1993.) |
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Theorem | preq12 4052 | Equality theorem for unordered pairs. (Contributed by NM, 19-Oct-2012.) |
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Theorem | preq1i 4053 | Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.) |
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Theorem | preq2i 4054 | Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.) |
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Theorem | preq12i 4055 | Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.) |
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Theorem | preq1d 4056 | Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.) |
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Theorem | preq2d 4057 | Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.) |
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Theorem | preq12d 4058 | Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.) |
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Theorem | tpeq1 4059 | Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.) |
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Theorem | tpeq2 4060 | Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.) |
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Theorem | tpeq3 4061 | Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.) |
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Theorem | tpeq1d 4062 | Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.) |
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Theorem | tpeq2d 4063 | Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.) |
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Theorem | tpeq3d 4064 | Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.) |
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Theorem | tpeq123d 4065 | Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.) |
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Theorem | tprot 4066 | Rotation of the elements of an unordered triple. (Contributed by Alan Sare, 24-Oct-2011.) |
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Theorem | tpcoma 4067 | Swap 1st and 2nd members of an unordered triple. (Contributed by NM, 22-May-2015.) |
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Theorem | tpcomb 4068 | Swap 2nd and 3rd members of an unordered triple. (Contributed by NM, 22-May-2015.) |
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Theorem | tpass 4069 | Split off the first element of an unordered triple. (Contributed by Mario Carneiro, 5-Jan-2016.) |
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Theorem | qdass 4070 | Two ways to write an unordered quadruple. (Contributed by Mario Carneiro, 5-Jan-2016.) |
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Theorem | qdassr 4071 | Two ways to write an unordered quadruple. (Contributed by Mario Carneiro, 5-Jan-2016.) |
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Theorem | tpidm12 4072 |
Unordered triple ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | tpidm13 4073 |
Unordered triple ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | tpidm23 4074 |
Unordered triple ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | tpidm 4075 |
Unordered triple ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | tppreq3 4076 | 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.) |
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Theorem | prid1g 4077 | An unordered pair contains its first member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by Stefan Allan, 8-Nov-2008.) |
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Theorem | prid2g 4078 | An unordered pair contains its second member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by Stefan Allan, 8-Nov-2008.) |
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Theorem | prid1 4079 | An unordered pair contains its first member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 24-Jun-1993.) |
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Theorem | prid2 4080 | An unordered pair contains its second member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 5-Aug-1993.) |
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Theorem | prprc1 4081 | A proper class vanishes in an unordered pair. (Contributed by NM, 15-Jul-1993.) |
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Theorem | prprc2 4082 | A proper class vanishes in an unordered pair. (Contributed by NM, 22-Mar-2006.) |
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Theorem | prprc 4083 | An unordered pair containing two proper classes is the empty set. (Contributed by NM, 22-Mar-2006.) |
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Theorem | tpid1 4084 | One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
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Theorem | tpid2 4085 | One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
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Theorem | tpid3g 4086 | Closed theorem form of tpid3 4087. This proof was automatically generated from the virtual deduction proof tpid3gVD 37232 using a translation program. (Contributed by Alan Sare, 24-Oct-2011.) |
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Theorem | tpid3 4087 | One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
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Theorem | snnzg 4088 | The singleton of a set is not empty. (Contributed by NM, 14-Dec-2008.) |
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Theorem | snnz 4089 | The singleton of a set is not empty. (Contributed by NM, 10-Apr-1994.) |
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Theorem | prnz 4090 | A pair containing a set is not empty. (Contributed by NM, 9-Apr-1994.) |
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Theorem | prnzg 4091 | A pair containing a set is not empty. (Contributed by FL, 19-Sep-2011.) |
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Theorem | tpnz 4092 | A triplet containing a set is not empty. (Contributed by NM, 10-Apr-1994.) |
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Theorem | tpnzd 4093 | A triplet containing a set is not empty. (Contributed by Thierry Arnoux, 8-Apr-2019.) |
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Theorem | raltpd 4094* | Convert a quantification over a triple to a conjunction. (Contributed by Thierry Arnoux, 8-Apr-2019.) |
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Theorem | snss 4095 | The singleton of an element of a class is a subset of the class. Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 21-Jun-1993.) |
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Theorem | eldifsn 4096 | Membership in a set with an element removed. (Contributed by NM, 10-Oct-2007.) |
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Theorem | eldifsni 4097 | Membership in a set with an element removed. (Contributed by NM, 10-Mar-2015.) |
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Theorem | neldifsn 4098 |
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Theorem | neldifsnd 4099 |
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Theorem | rexdifsn 4100 | Restricted existential quantification over a set with an element removed. (Contributed by NM, 4-Feb-2015.) |
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