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Theorem List for Intuitionistic Logic Explorer - 3001-3100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrexss 3001* Restricted existential quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 C_
 
Theoremss2ab 3002 Class abstractions in a subclass relationship. (Contributed by NM, 3-Jul-1994.)
 {  | 
 }  C_  {  |  }
 
Theoremabss 3003* Class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)
 {  | 
 }  C_
 
Theoremssab 3004* Subclass of a class abstraction. (Contributed by NM, 16-Aug-2006.)
 C_  {  |  }
 
Theoremssabral 3005* The relation for a subclass of a class abstraction is equivalent to restricted quantification. (Contributed by NM, 6-Sep-2006.)
 C_  {  |  }
 
Theoremss2abi 3006 Inference of abstraction subclass from implication. (Contributed by NM, 31-Mar-1995.)
   =>    
 {  |  } 
 C_  {  |  }
 
Theoremss2abdv 3007* Deduction of abstraction subclass from implication. (Contributed by NM, 29-Jul-2011.)
   =>     {  |  }  C_  {  |  }
 
Theoremabssdv 3008* Deduction of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)
   =>     {  |  }  C_
 
Theoremabssi 3009* Inference of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)
   =>    
 {  |  } 
 C_
 
Theoremss2rab 3010 Restricted abstraction classes in a subclass relationship. (Contributed by NM, 30-May-1999.)
 {  |  }  C_  {  |  }
 
Theoremrabss 3011* Restricted class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)
 {  |  }  C_
 
Theoremssrab 3012* Subclass of a restricted class abstraction. (Contributed by NM, 16-Aug-2006.)
 C_  {  |  }  C_
 
Theoremssrabdv 3013* Subclass of a restricted class abstraction (deduction rule). (Contributed by NM, 31-Aug-2006.)
 C_    &       =>     C_ 
 {  |  }
 
Theoremrabssdv 3014* Subclass of a restricted class abstraction (deduction rule). (Contributed by NM, 2-Feb-2015.)
   =>     {  |  }  C_
 
Theoremss2rabdv 3015* Deduction of restricted abstraction subclass from implication. (Contributed by NM, 30-May-2006.)
   =>     {  |  }  C_ 
 {  |  }
 
Theoremss2rabi 3016 Inference of restricted abstraction subclass from implication. (Contributed by NM, 14-Oct-1999.)
   =>    
 {  |  }  C_  {  |  }
 
Theoremrabss2 3017* Subclass law for restricted abstraction. (Contributed by NM, 18-Dec-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 C_  {  |  }  C_  {  |  }
 
Theoremssab2 3018* Subclass relation for the restriction of a class abstraction. (Contributed by NM, 31-Mar-1995.)

 {  |  }  C_
 
Theoremssrab2 3019* Subclass relation for a restricted class. (Contributed by NM, 19-Mar-1997.)

 {  |  }  C_
 
Theoremssrabeq 3020* If the restricting class of a restricted class abstraction is a subset of this restricted class abstraction, it is equal to this restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.)
 V  C_  {  V  |  }  V  {  V  | 
 }
 
Theoremrabssab 3021 A restricted class is a subclass of the corresponding unrestricted class. (Contributed by Mario Carneiro, 23-Dec-2016.)

 {  |  }  C_  {  |  }
 
Theoremuniiunlem 3022* A subset relationship useful for converting union to indexed union using dfiun2 or dfiun2g and intersection to indexed intersection using dfiin2 . (Contributed by NM, 5-Oct-2006.) (Proof shortened by Mario Carneiro, 26-Sep-2015.)
 D  C  {  |  }  C_  C
 
Theoremdfpss2 3023 Alternate definition of proper subclass. (Contributed by NM, 7-Feb-1996.)
 C.  C_
 
Theoremdfpss3 3024 Alternate definition of proper subclass. (Contributed by NM, 7-Feb-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 C.  C_  C_
 
Theorempsseq1 3025 Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.)
 C.  C  C.  C
 
Theorempsseq2 3026 Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.)
 C  C.  C  C.
 
Theorempsseq1i 3027 An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
   =>     C.  C  C.  C
 
Theorempsseq2i 3028 An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
   =>     C  C.  C  C.
 
Theorempsseq12i 3029 An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
   &     C  D   =>     C.  C  C.  D
 
Theorempsseq1d 3030 An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
   =>     C.  C  C.  C
 
Theorempsseq2d 3031 An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
   =>     C  C.  C  C.
 
Theorempsseq12d 3032 An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
   &     C  D   =>     C.  C  C.  D
 
Theorempssss 3033 A proper subclass is a subclass. Theorem 10 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)
 C.  C_
 
Theorempssne 3034 Two classes in a proper subclass relationship are not equal. (Contributed by NM, 16-Feb-2015.)
 C.  =/=
 
Theorempssssd 3035 Deduce subclass from proper subclass. (Contributed by NM, 29-Feb-1996.)
 C.    =>     C_
 
Theorempssned 3036 Proper subclasses are unequal. Deduction form of pssne 3034. (Contributed by David Moews, 1-May-2017.)
 C.    =>     =/=
 
Theoremsspssr 3037 Subclass in terms of proper subclass. (Contributed by Jim Kingdon, 16-Jul-2018.)
 C.  C_
 
Theorempssirr 3038 Proper subclass is irreflexive. Theorem 7 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)
 C.
 
Theorempssn2lp 3039 Proper subclass has no 2-cycle loops. Compare Theorem 8 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 C.  C.
 
Theoremsspsstrir 3040 Two ways of stating trichotomy with respect to inclusion. (Contributed by Jim Kingdon, 16-Jul-2018.)
 C.  C.  C_  C_
 
Theoremssnpss 3041 Partial trichotomy law for subclasses. (Contributed by NM, 16-May-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 C_  C.
 
Theoremsspssn 3042 Like pssn2lp 3039 but for subset and proper subset. (Contributed by Jim Kingdon, 17-Jul-2018.)
 C_  C.
 
Theorempsstr 3043 Transitive law for proper subclass. Theorem 9 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)
 C.  C.  C  C.  C
 
Theoremsspsstr 3044 Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.)
 C_  C.  C  C.  C
 
Theorempsssstr 3045 Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.)
 C.  C_  C  C.  C
 
Theorempsstrd 3046 Proper subclass inclusion is transitive. Deduction form of psstr 3043. (Contributed by David Moews, 1-May-2017.)
 C.    &     C.  C   =>     C.  C
 
Theoremsspsstrd 3047 Transitivity involving subclass and proper subclass inclusion. Deduction form of sspsstr 3044. (Contributed by David Moews, 1-May-2017.)
 C_    &     C.  C   =>     C.  C
 
Theorempsssstrd 3048 Transitivity involving subclass and proper subclass inclusion. Deduction form of psssstr 3045. (Contributed by David Moews, 1-May-2017.)
 C.    &     C_  C   =>     C.  C
 
2.1.13  The difference, union, and intersection of two classes
 
2.1.13.1  The difference of two classes
 
Theoremdifeq1 3049 Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 \  C  \  C
 
Theoremdifeq2 3050 Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 C  \  C  \
 
Theoremdifeq12 3051 Equality theorem for class difference. (Contributed by FL, 31-Aug-2009.)
 C  D  \  C  \  D
 
Theoremdifeq1i 3052 Inference adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.)
   =>     \  C  \  C
 
Theoremdifeq2i 3053 Inference adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)
   =>     C  \  C  \
 
Theoremdifeq12i 3054 Equality inference for class difference. (Contributed by NM, 29-Aug-2004.)
   &     C  D   =>     \  C  \  D
 
Theoremdifeq1d 3055 Deduction adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.)
   =>     \  C  \  C
 
Theoremdifeq2d 3056 Deduction adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)
   =>     C  \  C  \
 
Theoremdifeq12d 3057 Equality deduction for class difference. (Contributed by FL, 29-May-2014.)
   &     C  D   =>     \  C  \  D
 
Theoremdifeqri 3058* Inference from membership to difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 C   =>     \  C
 
Theoremnfdif 3059 Bound-variable hypothesis builder for class difference. (Contributed by NM, 3-Dec-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)
 F/_   &     F/_   =>     F/_ 
 \
 
Theoremeldifi 3060 Implication of membership in a class difference. (Contributed by NM, 29-Apr-1994.)
 \  C
 
Theoremeldifn 3061 Implication of membership in a class difference. (Contributed by NM, 3-May-1994.)
 \  C  C
 
Theoremelndif 3062 A set does not belong to a class excluding it. (Contributed by NM, 27-Jun-1994.)
 C  \
 
Theoremdifdif 3063 Double class difference. Exercise 11 of [TakeutiZaring] p. 22. (Contributed by NM, 17-May-1998.)
 \  \
 
Theoremdifss 3064 Subclass relationship for class difference. Exercise 14 of [TakeutiZaring] p. 22. (Contributed by NM, 29-Apr-1994.)
 \  C_
 
Theoremdifssd 3065 A difference of two classes is contained in the minuend. Deduction form of difss 3064. (Contributed by David Moews, 1-May-2017.)
 \  C_
 
Theoremdifss2 3066 If a class is contained in a difference, it is contained in the minuend. (Contributed by David Moews, 1-May-2017.)
 C_  \  C  C_
 
Theoremdifss2d 3067 If a class is contained in a difference, it is contained in the minuend. Deduction form of difss2 3066. (Contributed by David Moews, 1-May-2017.)
 C_  \  C   =>     C_
 
Theoremssdifss 3068 Preservation of a subclass relationship by class difference. (Contributed by NM, 15-Feb-2007.)
 C_  \  C  C_
 
Theoremddifnel 3069* Double complement under universal class. The hypothesis is one way of expressing the idea that membership in is decidable. Exercise 4.10(s) of [Mendelson] p. 231, but with an additional hypothesis. For a version without a hypothesis, but which only states that is a subset of  _V  \  _V  \ , see ddifss 3169. (Contributed by Jim Kingdon, 21-Jul-2018.)
 _V  \    =>     _V  \  _V  \
 
Theoremssconb 3070 Contraposition law for subsets. (Contributed by NM, 22-Mar-1998.)
 C_  C  C_  C  C_  C  \  C_  C  \
 
Theoremsscon 3071 Contraposition law for subsets. Exercise 15 of [TakeutiZaring] p. 22. (Contributed by NM, 22-Mar-1998.)
 C_  C  \  C_  C  \
 
Theoremssdif 3072 Difference law for subsets. (Contributed by NM, 28-May-1998.)
 C_  \  C  C_  \  C
 
Theoremssdifd 3073 If is contained in , then 
\  C is contained in  \  C. Deduction form of ssdif 3072. (Contributed by David Moews, 1-May-2017.)
 C_    =>     \  C  C_  \  C
 
Theoremsscond 3074 If is contained in , then  C 
\ is contained in  C  \ . Deduction form of sscon 3071. (Contributed by David Moews, 1-May-2017.)
 C_    =>     C  \  C_  C  \
 
Theoremssdifssd 3075 If is contained in , then 
\  C is also contained in . Deduction form of ssdifss 3068. (Contributed by David Moews, 1-May-2017.)
 C_    =>     \  C  C_
 
Theoremssdif2d 3076 If is contained in and  C is contained in  D, then  \  D is contained in  \  C. Deduction form. (Contributed by David Moews, 1-May-2017.)
 C_    &     C  C_  D   =>     \  D  C_  \  C
 
Theoremraldifb 3077 Restricted universal quantification on a class difference in terms of an implication. (Contributed by Alexander van der Vekens, 3-Jan-2018.)
 e/  \
 
2.1.13.2  The union of two classes
 
Theoremelun 3078 Expansion of membership in class union. Theorem 12 of [Suppes] p. 25. (Contributed by NM, 7-Aug-1994.)
 u.  C  C
 
Theoremuneqri 3079* Inference from membership to union. (Contributed by NM, 5-Aug-1993.)
 C   =>     u.  C
 
Theoremunidm 3080 Idempotent law for union of classes. Theorem 23 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)
 u.
 
Theoremuncom 3081 Commutative law for union of classes. Exercise 6 of [TakeutiZaring] p. 17. (Contributed by NM, 25-Jun-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 u.  u.
 
Theoremequncom 3082 If a class equals the union of two other classes, then it equals the union of those two classes commuted. (Contributed by Alan Sare, 18-Feb-2012.)
 u.  C  C  u.
 
Theoremequncomi 3083 Inference form of equncom 3082. (Contributed by Alan Sare, 18-Feb-2012.)
 u.  C   =>     C  u.
 
Theoremuneq1 3084 Equality theorem for union of two classes. (Contributed by NM, 5-Aug-1993.)
 u.  C  u.  C
 
Theoremuneq2 3085 Equality theorem for the union of two classes. (Contributed by NM, 5-Aug-1993.)
 C  u.  C  u.
 
Theoremuneq12 3086 Equality theorem for union of two classes. (Contributed by NM, 29-Mar-1998.)
 C  D  u.  C  u.  D
 
Theoremuneq1i 3087 Inference adding union to the right in a class equality. (Contributed by NM, 30-Aug-1993.)
   =>     u.  C  u.  C
 
Theoremuneq2i 3088 Inference adding union to the left in a class equality. (Contributed by NM, 30-Aug-1993.)
   =>     C  u.  C  u.
 
Theoremuneq12i 3089 Equality inference for union of two classes. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
   &     C  D   =>     u.  C  u.  D
 
Theoremuneq1d 3090 Deduction adding union to the right in a class equality. (Contributed by NM, 29-Mar-1998.)
   =>     u.  C  u.  C
 
Theoremuneq2d 3091 Deduction adding union to the left in a class equality. (Contributed by NM, 29-Mar-1998.)
   =>     C  u.  C  u.
 
Theoremuneq12d 3092 Equality deduction for union of two classes. (Contributed by NM, 29-Sep-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
   &     C  D   =>     u.  C  u.  D
 
Theoremnfun 3093 Bound-variable hypothesis builder for the union of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.)
 F/_   &     F/_   =>     F/_  u.
 
Theoremunass 3094 Associative law for union of classes. Exercise 8 of [TakeutiZaring] p. 17. (Contributed by NM, 3-May-1994.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 u.  u.  C  u.  u.  C
 
Theoremun12 3095 A rearrangement of union. (Contributed by NM, 12-Aug-2004.)
 u.  u.  C  u.  u.  C
 
Theoremun23 3096 A rearrangement of union. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 u.  u.  C  u.  C  u.
 
Theoremun4 3097 A rearrangement of the union of 4 classes. (Contributed by NM, 12-Aug-2004.)
 u.  u.  C  u.  D  u.  C  u.  u.  D
 
Theoremunundi 3098 Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
 u.  u.  C  u.  u.  u.  C
 
Theoremunundir 3099 Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
 u.  u.  C  u.  C  u.  u.  C
 
Theoremssun1 3100 Subclass relationship for union of classes. Theorem 25 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)
 C_  u.
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