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Theorem List for Intuitionistic Logic Explorer - 2901-3000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcsbabg 2901* Move substitution into a class abstraction. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 V  [_  ]_
 {  |  }  {  |  [.  ].
 }
 
Theoremcbvralcsf 2902 A more general version of cbvralf 2521 that doesn't require and to be distinct from or . Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.)
 F/_   &     F/_   &     F/   &     F/   &       &       =>   
 
Theoremcbvrexcsf 2903 A more general version of cbvrexf 2522 that has no distinct variable restrictions. Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.) (Proof shortened by Mario Carneiro, 7-Dec-2014.)
 F/_   &     F/_   &     F/   &     F/   &       &       =>   
 
Theoremcbvreucsf 2904 A more general version of cbvreuv 2529 that has no distinct variable rextrictions. Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.)
 F/_   &     F/_   &     F/   &     F/   &       &       =>   
 
Theoremcbvrabcsf 2905 A more general version of cbvrab 2549 with no distinct variable restrictions. (Contributed by Andrew Salmon, 13-Jul-2011.)
 F/_   &     F/_   &     F/   &     F/   &       &       =>     {  |  }  {  |  }
 
Theoremcbvralv2 2906* Rule used to change the bound variable in a restricted universal quantifier with implicit substitution which also changes the quantifier domain. (Contributed by David Moews, 1-May-2017.)
   &       =>   
 
Theoremcbvrexv2 2907* Rule used to change the bound variable in a restricted existential quantifier with implicit substitution which also changes the quantifier domain. (Contributed by David Moews, 1-May-2017.)
   &       =>   
 
2.1.11  Define basic set operations and relations
 
Syntaxcdif 2908 Extend class notation to include class difference (read: " minus ").
 \
 
Syntaxcun 2909 Extend class notation to include union of two classes (read: " union ").
 u.
 
Syntaxcin 2910 Extend class notation to include the intersection of two classes (read: " intersect ").
 i^i
 
Syntaxwss 2911 Extend wff notation to include the subclass relation. This is read " is a subclass of " or " includes ." When exists as a set, it is also read " is a subset of ."
 C_
 
Syntaxwpss 2912 Extend wff notation with proper subclass relation.
 C.
 
Theoremdifjust 2913* Soundness justification theorem for df-dif 2914. (Contributed by Rodolfo Medina, 27-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

 {  |  }  {  |  }
 
Definitiondf-dif 2914* Define class difference, also called relative complement. Definition 5.12 of [TakeutiZaring] p. 20. Contrast this operation with union  u. (df-un 2916) and intersection  i^i (df-in 2918). Several notations are used in the literature; we chose the  \ convention used in Definition 5.3 of [Eisenberg] p. 67 instead of the more common minus sign to reserve the latter for later use in, e.g., arithmetic. We will use the terminology " excludes " to mean  \ . We will use " is removed from " to mean  \  { } i.e. the removal of an element or equivalently the exclusion of a singleton. (Contributed by NM, 29-Apr-1994.)
 \  {  |  }
 
Theoremunjust 2915* Soundness justification theorem for df-un 2916. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

 {  |  }  {  |  }
 
Definitiondf-un 2916* Define the union of two classes. Definition 5.6 of [TakeutiZaring] p. 16. Contrast this operation with difference  \ (df-dif 2914) and intersection  i^i (df-in 2918). (Contributed by NM, 23-Aug-1993.)
 u.  {  |  }
 
Theoreminjust 2917* Soundness justification theorem for df-in 2918. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

 {  |  }  {  |  }
 
Definitiondf-in 2918* Define the intersection of two classes. Definition 5.6 of [TakeutiZaring] p. 16. Contrast this operation with union  u. (df-un 2916) and difference  \ (df-dif 2914). (Contributed by NM, 29-Apr-1994.)
 i^i  {  |  }
 
Theoremdfin5 2919* Alternate definition for the intersection of two classes. (Contributed by NM, 6-Jul-2005.)
 i^i  {  |  }
 
Theoremdfdif2 2920* Alternate definition of class difference. (Contributed by NM, 25-Mar-2004.)
 \  {  |  }
 
Theoremeldif 2921 Expansion of membership in a class difference. (Contributed by NM, 29-Apr-1994.)
 \  C  C
 
Theoremeldifd 2922 If a class is in one class and not another, it is also in their difference. One-way deduction form of eldif 2921. (Contributed by David Moews, 1-May-2017.)
   &     C   =>     \  C
 
Theoremeldifad 2923 If a class is in the difference of two classes, it is also in the minuend. One-way deduction form of eldif 2921. (Contributed by David Moews, 1-May-2017.)
 \  C   =>   
 
Theoremeldifbd 2924 If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif 2921. (Contributed by David Moews, 1-May-2017.)
 \  C   =>     C
 
2.1.12  Subclasses and subsets
 
Definitiondf-ss 2925 Define the subclass relationship. Exercise 9 of [TakeutiZaring] p. 18. Note that  C_ (proved in ssid 2958). Contrast this relationship with the relationship  C. (as will be defined in df-pss 2927). For a more traditional definition, but requiring a dummy variable, see dfss2 2928 (or dfss3 2929 which is similar). (Contributed by NM, 27-Apr-1994.)
 C_  i^i
 
Theoremdfss 2926 Variant of subclass definition df-ss 2925. (Contributed by NM, 3-Sep-2004.)
 C_  i^i
 
Definitiondf-pss 2927 Define proper subclass relationship between two classes. Definition 5.9 of [TakeutiZaring] p. 17. Note that  C. (proved in pssirr 3038). Contrast this relationship with the relationship  C_ (as defined in df-ss 2925). Other possible definitions are given by dfpss2 3023 and dfpss3 3024. (Contributed by NM, 7-Feb-1996.)
 C.  C_  =/=
 
Theoremdfss2 2928* Alternate definition of the subclass relationship between two classes. Definition 5.9 of [TakeutiZaring] p. 17. (Contributed by NM, 8-Jan-2002.)
 C_
 
Theoremdfss3 2929* Alternate definition of subclass relationship. (Contributed by NM, 14-Oct-1999.)
 C_
 
Theoremdfss2f 2930 Equivalence for subclass relation, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 3-Jul-1994.) (Revised by Andrew Salmon, 27-Aug-2011.)
 F/_   &     F/_   =>     C_
 
Theoremdfss3f 2931 Equivalence for subclass relation, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 20-Mar-2004.)
 F/_   &     F/_   =>     C_
 
Theoremnfss 2932 If is not free in and , it is not free in  C_ . (Contributed by NM, 27-Dec-1996.)
 F/_   &     F/_   =>     F/  C_
 
Theoremssel 2933 Membership relationships follow from a subclass relationship. (Contributed by NM, 5-Aug-1993.)
 C_  C  C
 
Theoremssel2 2934 Membership relationships follow from a subclass relationship. (Contributed by NM, 7-Jun-2004.)
 C_  C  C
 
Theoremsseli 2935 Membership inference from subclass relationship. (Contributed by NM, 5-Aug-1993.)
 C_    =>     C  C
 
Theoremsselii 2936 Membership inference from subclass relationship. (Contributed by NM, 31-May-1999.)
 C_    &     C    =>     C
 
Theoremsseldi 2937 Membership inference from subclass relationship. (Contributed by NM, 25-Jun-2014.)
 C_    &     C    =>     C
 
Theoremsseld 2938 Membership deduction from subclass relationship. (Contributed by NM, 15-Nov-1995.)
 C_    =>     C  C
 
Theoremsselda 2939 Membership deduction from subclass relationship. (Contributed by NM, 26-Jun-2014.)
 C_    =>     C  C
 
Theoremsseldd 2940 Membership inference from subclass relationship. (Contributed by NM, 14-Dec-2004.)
 C_    &     C    =>     C
 
Theoremssneld 2941 If a class is not in another class, it is also not in a subclass of that class. Deduction form. (Contributed by David Moews, 1-May-2017.)
 C_    =>     C  C
 
Theoremssneldd 2942 If an element is not in a class, it is also not in a subclass of that class. Deduction form. (Contributed by David Moews, 1-May-2017.)
 C_    &     C    =>     C
 
Theoremssriv 2943* Inference rule based on subclass definition. (Contributed by NM, 5-Aug-1993.)
   =>     C_
 
Theoremssrd 2944 Deduction rule based on subclass definition. (Contributed by Thierry Arnoux, 8-Mar-2017.)

 F/   &     F/_   &     F/_   &       =>     C_
 
Theoremssrdv 2945* Deduction rule based on subclass definition. (Contributed by NM, 15-Nov-1995.)
   =>     C_
 
Theoremsstr2 2946 Transitivity of subclasses. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
 C_  C_  C  C_  C
 
Theoremsstr 2947 Transitivity of subclasses. Theorem 6 of [Suppes] p. 23. (Contributed by NM, 5-Sep-2003.)
 C_  C_  C  C_  C
 
Theoremsstri 2948 Subclass transitivity inference. (Contributed by NM, 5-May-2000.)
 C_    &     C_  C   =>     C_  C
 
Theoremsstrd 2949 Subclass transitivity deduction. (Contributed by NM, 2-Jun-2004.)
 C_    &     C_  C   =>     C_  C
 
Theoremsyl5ss 2950 Subclass transitivity deduction. (Contributed by NM, 6-Feb-2014.)
 C_    &     C_  C   =>     C_  C
 
Theoremsyl6ss 2951 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 C_    &     C_  C   =>     C_  C
 
Theoremsylan9ss 2952 A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
 C_    &     C_  C   =>     C_  C
 
Theoremsylan9ssr 2953 A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.)
 C_    &     C_  C   =>     C_  C
 
Theoremeqss 2954 The subclass relationship is antisymmetric. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 5-Aug-1993.)
 C_  C_
 
Theoremeqssi 2955 Infer equality from two subclass relationships. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 9-Sep-1993.)
 C_    &     C_    =>   
 
Theoremeqssd 2956 Equality deduction from two subclass relationships. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 27-Jun-2004.)
 C_    &     C_    =>   
 
Theoremeqrd 2957 Deduce equality of classes from equivalence of membership. (Contributed by Thierry Arnoux, 21-Mar-2017.)

 F/   &     F/_   &     F/_   &       =>   
 
Theoremssid 2958 Any class is a subclass of itself. Exercise 10 of [TakeutiZaring] p. 18. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
 C_
 
Theoremssv 2959 Any class is a subclass of the universal class. (Contributed by NM, 31-Oct-1995.)
 C_  _V
 
Theoremsseq1 2960 Equality theorem for subclasses. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
 C_  C  C_  C
 
Theoremsseq2 2961 Equality theorem for the subclass relationship. (Contributed by NM, 25-Jun-1998.)
 C  C_  C  C_
 
Theoremsseq12 2962 Equality theorem for the subclass relationship. (Contributed by NM, 31-May-1999.)
 C  D  C_  C  C_  D
 
Theoremsseq1i 2963 An equality inference for the subclass relationship. (Contributed by NM, 18-Aug-1993.)
   =>     C_  C  C_  C
 
Theoremsseq2i 2964 An equality inference for the subclass relationship. (Contributed by NM, 30-Aug-1993.)
   =>     C  C_  C  C_
 
Theoremsseq12i 2965 An equality inference for the subclass relationship. (Contributed by NM, 31-May-1999.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
   &     C  D   =>     C_  C  C_  D
 
Theoremsseq1d 2966 An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)
   =>     C_  C  C_  C
 
Theoremsseq2d 2967 An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)
   =>     C  C_  C  C_
 
Theoremsseq12d 2968 An equality deduction for the subclass relationship. (Contributed by NM, 31-May-1999.)
   &     C  D   =>     C_  C  C_  D
 
Theoremeqsstri 2969 Substitution of equality into a subclass relationship. (Contributed by NM, 16-Jul-1995.)
   &     C_  C   =>     C_  C
 
Theoremeqsstr3i 2970 Substitution of equality into a subclass relationship. (Contributed by NM, 19-Oct-1999.)
   &     C_  C   =>     C_  C
 
Theoremsseqtri 2971 Substitution of equality into a subclass relationship. (Contributed by NM, 28-Jul-1995.)
 C_    &     C   =>     C_  C
 
Theoremsseqtr4i 2972 Substitution of equality into a subclass relationship. (Contributed by NM, 4-Apr-1995.)
 C_    &     C    =>     C_  C
 
Theoremeqsstrd 2973 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
   &     C_  C   =>     C_  C
 
Theoremeqsstr3d 2974 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
   &     C_  C   =>     C_  C
 
Theoremsseqtrd 2975 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
 C_    &     C   =>     C_  C
 
Theoremsseqtr4d 2976 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
 C_    &     C    =>     C_  C
 
Theorem3sstr3i 2977 Substitution of equality in both sides of a subclass relationship. (Contributed by NM, 13-Jan-1996.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
 C_    &     C   &     D   =>     C  C_  D
 
Theorem3sstr4i 2978 Substitution of equality in both sides of a subclass relationship. (Contributed by NM, 13-Jan-1996.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
 C_    &     C    &     D    =>     C  C_  D
 
Theorem3sstr3g 2979 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)
 C_    &     C   &     D   =>     C  C_  D
 
Theorem3sstr4g 2980 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
 C_    &     C    &     D    =>     C  C_  D
 
Theorem3sstr3d 2981 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)
 C_    &     C   &     D   =>     C  C_  D
 
Theorem3sstr4d 2982 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 30-Nov-1995.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
 C_    &     C    &     D    =>     C  C_  D
 
Theoremsyl5eqss 2983 B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
   &     C_  C   =>     C_  C
 
Theoremsyl5eqssr 2984 B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
   &     C_  C   =>     C_  C
 
Theoremsyl6sseq 2985 A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
 C_    &     C   =>     C_  C
 
Theoremsyl6sseqr 2986 A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
 C_    &     C    =>     C_  C
 
Theoremsyl5sseq 2987 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 C_    &     C   =>     C_  C
 
Theoremsyl5sseqr 2988 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 C_    &     C    =>     C_  C
 
Theoremsyl6eqss 2989 A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)
   &     C_  C   =>     C_  C
 
Theoremsyl6eqssr 2990 A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)
   &     C_  C   =>     C_  C
 
Theoremeqimss 2991 Equality implies the subclass relation. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
 C_
 
Theoremeqimss2 2992 Equality implies the subclass relation. (Contributed by NM, 23-Nov-2003.)
 C_
 
Theoremeqimssi 2993 Infer subclass relationship from equality. (Contributed by NM, 6-Jan-2007.)
   =>     C_
 
Theoremeqimss2i 2994 Infer subclass relationship from equality. (Contributed by NM, 7-Jan-2007.)
   =>     C_
 
Theoremnssne1 2995 Two classes are different if they don't include the same class. (Contributed by NM, 23-Apr-2015.)
 C_  C_  C  =/=  C
 
Theoremnssne2 2996 Two classes are different if they are not subclasses of the same class. (Contributed by NM, 23-Apr-2015.)
 C_  C  C_  C  =/=
 
Theoremnssr 2997* Negation of subclass relationship. One direction of Exercise 13 of [TakeutiZaring] p. 18. (Contributed by Jim Kingdon, 15-Jul-2018.)
 C_
 
Theoremssralv 2998* Quantification restricted to a subclass. (Contributed by NM, 11-Mar-2006.)
 C_
 
Theoremssrexv 2999* Existential quantification restricted to a subclass. (Contributed by NM, 11-Jan-2007.)
 C_
 
Theoremralss 3000* Restricted universal quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 C_
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