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Theorem List for Intuitionistic Logic Explorer - 2901-3000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcsbcomg 2901* Commutative law for double substitution into a class. (Contributed by NM, 14-Nov-2005.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  [_ A  /  x ]_
 [_ B  /  y ]_ C  =  [_ B  /  y ]_ [_ A  /  x ]_ C )
 
Theoremcsbeq2d 2902 Formula-building deduction rule for class substitution. (Contributed by NM, 22-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)
 |- 
 F/ x ph   &    |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C )
 
Theoremcsbeq2dv 2903* Formula-building deduction rule for class substitution. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)
 |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C )
 
Theoremcsbeq2i 2904 Formula-building inference rule for class substitution. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)
 |-  B  =  C   =>    |-  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C
 
Theoremcsbvarg 2905 The proper substitution of a class for setvar variable results in the class (if the class exists). (Contributed by NM, 10-Nov-2005.)
 |-  ( A  e.  V  -> 
 [_ A  /  x ]_ x  =  A )
 
Theoremsbccsbg 2906* Substitution into a wff expressed in terms of substitution into a class. (Contributed by NM, 15-Aug-2007.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  y  e.  [_ A  /  x ]_ { y  |  ph } ) )
 
Theoremsbccsb2g 2907 Substitution into a wff expressed in using substitution into a class. (Contributed by NM, 27-Nov-2005.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  A  e.  [_ A  /  x ]_ { x  |  ph } ) )
 
Theoremnfcsb1d 2908 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
 |-  ( ph  ->  F/_ x A )   =>    |-  ( ph  ->  F/_ x [_ A  /  x ]_ B )
 
Theoremnfcsb1 2909 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
 |-  F/_ x A   =>    |-  F/_ x [_ A  /  x ]_ B
 
Theoremnfcsb1v 2910* Bound-variable hypothesis builder for substitution into a class. (Contributed by NM, 17-Aug-2006.) (Revised by Mario Carneiro, 12-Oct-2016.)
 |-  F/_ x [_ A  /  x ]_ B
 
Theoremnfcsbd 2911 Deduction version of nfcsb 2912. (Contributed by NM, 21-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/_ x A )   &    |-  ( ph  ->  F/_ x B )   =>    |-  ( ph  ->  F/_ x [_ A  /  y ]_ B )
 
Theoremnfcsb 2912 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x [_ A  /  y ]_ B
 
Theoremcsbhypf 2913* Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 2620 for class substitution version. (Contributed by NM, 19-Dec-2008.)
 |-  F/_ x A   &    |-  F/_ x C   &    |-  ( x  =  A  ->  B  =  C )   =>    |-  ( y  =  A  ->  [_ y  /  x ]_ B  =  C )
 
Theoremcsbiebt 2914* Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 2918.) (Contributed by NM, 11-Nov-2005.)
 |-  ( ( A  e.  V  /\  F/_ x C ) 
 ->  ( A. x ( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C ) )
 
Theoremcsbiedf 2915* Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 13-Oct-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  F/_ x C )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  x  =  A )  ->  B  =  C )   =>    |-  ( ph  ->  [_ A  /  x ]_ B  =  C )
 
Theoremcsbieb 2916* Bidirectional conversion between an implicit class substitution hypothesis  x  =  A  ->  B  =  C and its explicit substitution equivalent. (Contributed by NM, 2-Mar-2008.)
 |-  A  e.  _V   &    |-  F/_ x C   =>    |-  ( A. x ( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C )
 
Theoremcsbiebg 2917* Bidirectional conversion between an implicit class substitution hypothesis  x  =  A  ->  B  =  C and its explicit substitution equivalent. (Contributed by NM, 24-Mar-2013.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |-  F/_ x C   =>    |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C ) )
 
Theoremcsbiegf 2918* Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 11-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
 |-  ( A  e.  V  -> 
 F/_ x C )   &    |-  ( x  =  A  ->  B  =  C )   =>    |-  ( A  e.  V  -> 
 [_ A  /  x ]_ B  =  C )
 
Theoremcsbief 2919* Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 26-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
 |-  A  e.  _V   &    |-  F/_ x C   &    |-  ( x  =  A  ->  B  =  C )   =>    |-  [_ A  /  x ]_ B  =  C
 
Theoremcsbied 2920* Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by Mario Carneiro, 13-Oct-2016.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  x  =  A ) 
 ->  B  =  C )   =>    |-  ( ph  ->  [_ A  /  x ]_ B  =  C )
 
Theoremcsbied2 2921* Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  A  =  B )   &    |-  (
 ( ph  /\  x  =  B )  ->  C  =  D )   =>    |-  ( ph  ->  [_ A  /  x ]_ C  =  D )
 
Theoremcsbie2t 2922* Conversion of implicit substitution to explicit substitution into a class (closed form of csbie2 2923). (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 13-Oct-2016.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A. x A. y ( ( x  =  A  /\  y  =  B )  ->  C  =  D )  ->  [_ A  /  x ]_ [_ B  /  y ]_ C  =  D )
 
Theoremcsbie2 2923* Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 27-Aug-2007.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( ( x  =  A  /\  y  =  B )  ->  C  =  D )   =>    |-  [_ A  /  x ]_
 [_ B  /  y ]_ C  =  D
 
Theoremcsbie2g 2924* Conversion of implicit substitution to explicit class substitution. This version of sbcie 2820 avoids a disjointness condition on  x and  A by substituting twice. (Contributed by Mario Carneiro, 11-Nov-2016.)
 |-  ( x  =  y 
 ->  B  =  C )   &    |-  ( y  =  A  ->  C  =  D )   =>    |-  ( A  e.  V  -> 
 [_ A  /  x ]_ B  =  D )
 
Theoremsbcnestgf 2925 Nest the composition of two substitutions. (Contributed by Mario Carneiro, 11-Nov-2016.)
 |-  ( ( A  e.  V  /\  A. y F/ x ph )  ->  ( [. A  /  x ].
 [. B  /  y ]. ph  <->  [. [_ A  /  x ]_ B  /  y ]. ph ) )
 
Theoremcsbnestgf 2926 Nest the composition of two substitutions. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.)
 |-  ( ( A  e.  V  /\  A. y F/_ x C )  ->  [_ A  /  x ]_ [_ B  /  y ]_ C  =  [_
 [_ A  /  x ]_ B  /  y ]_ C )
 
Theoremsbcnestg 2927* Nest the composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. [_ A  /  x ]_ B  /  y ]. ph ) )
 
Theoremcsbnestg 2928* Nest the composition of two substitutions. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.)
 |-  ( A  e.  V  -> 
 [_ A  /  x ]_
 [_ B  /  y ]_ C  =  [_ [_ A  /  x ]_ B  /  y ]_ C )
 
Theoremcsbnest1g 2929 Nest the composition of two substitutions. (Contributed by NM, 23-May-2006.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)
 |-  ( A  e.  V  -> 
 [_ A  /  x ]_
 [_ B  /  x ]_ C  =  [_ [_ A  /  x ]_ B  /  x ]_ C )
 
Theoremcsbidmg 2930* Idempotent law for class substitutions. (Contributed by NM, 1-Mar-2008.)
 |-  ( A  e.  V  -> 
 [_ A  /  x ]_
 [_ A  /  x ]_ B  =  [_ A  /  x ]_ B )
 
Theoremsbcco3g 2931* Composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.)
 |-  ( x  =  A  ->  B  =  C )   =>    |-  ( A  e.  V  ->  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. C  /  y ]. ph ) )
 
Theoremcsbco3g 2932* Composition of two class substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.)
 |-  ( x  =  A  ->  B  =  C )   =>    |-  ( A  e.  V  -> 
 [_ A  /  x ]_
 [_ B  /  y ]_ D  =  [_ C  /  y ]_ D )
 
Theoremrspcsbela 2933* Special case related to rspsbc 2868. (Contributed by NM, 10-Dec-2005.) (Proof shortened by Eric Schmidt, 17-Jan-2007.)
 |-  ( ( A  e.  B  /\  A. x  e.  B  C  e.  D )  ->  [_ A  /  x ]_ C  e.  D )
 
Theoremsbnfc2 2934* Two ways of expressing " x is (effectively) not free in  A." (Contributed by Mario Carneiro, 14-Oct-2016.)
 |-  ( F/_ x A  <->  A. y A. z [_ y  /  x ]_ A  =  [_ z  /  x ]_ A )
 
Theoremcsbabg 2935* Move substitution into a class abstraction. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  ( A  e.  V  -> 
 [_ A  /  x ]_
 { y  |  ph }  =  { y  | 
 [. A  /  x ].
 ph } )
 
Theoremcbvralcsf 2936 A more general version of cbvralf 2544 that doesn't require  A and  B to be distinct from  x or  y. Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.)
 |-  F/_ y A   &    |-  F/_ x B   &    |-  F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  A  =  B )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A. x  e.  A  ph  <->  A. y  e.  B  ps )
 
Theoremcbvrexcsf 2937 A more general version of cbvrexf 2545 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/_ y A   &    |-  F/_ x B   &    |-  F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  A  =  B )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E. x  e.  A  ph  <->  E. y  e.  B  ps )
 
Theoremcbvreucsf 2938 A more general version of cbvreuv 2552 that has no distinct variable rextrictions. Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.)
 |-  F/_ y A   &    |-  F/_ x B   &    |-  F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  A  =  B )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E! x  e.  A  ph  <->  E! y  e.  B  ps )
 
Theoremcbvrabcsf 2939 A more general version of cbvrab 2572 with no distinct variable restrictions. (Contributed by Andrew Salmon, 13-Jul-2011.)
 |-  F/_ y A   &    |-  F/_ x B   &    |-  F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  A  =  B )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  { x  e.  A  |  ph }  =  { y  e.  B  |  ps }
 
Theoremcbvralv2 2940* 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.)
 |-  ( x  =  y 
 ->  ( ps  <->  ch ) )   &    |-  ( x  =  y  ->  A  =  B )   =>    |-  ( A. x  e.  A  ps  <->  A. y  e.  B  ch )
 
Theoremcbvrexv2 2941* 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.)
 |-  ( x  =  y 
 ->  ( ps  <->  ch ) )   &    |-  ( x  =  y  ->  A  =  B )   =>    |-  ( E. x  e.  A  ps  <->  E. y  e.  B  ch )
 
2.1.11  Define basic set operations and relations
 
Syntaxcdif 2942 Extend class notation to include class difference (read: " A minus  B").
 class  ( A  \  B )
 
Syntaxcun 2943 Extend class notation to include union of two classes (read: " A union  B").
 class  ( A  u.  B )
 
Syntaxcin 2944 Extend class notation to include the intersection of two classes (read: " A intersect  B").
 class  ( A  i^i  B )
 
Syntaxwss 2945 Extend wff notation to include the subclass relation. This is read " A is a subclass of  B " or " B includes  A." When  A exists as a set, it is also read " A is a subset of  B."
 wff  A  C_  B
 
Syntaxwpss 2946 Extend wff notation with proper subclass relation.
 wff  A  C.  B
 
Theoremdifjust 2947* Soundness justification theorem for df-dif 2948. (Contributed by Rodolfo Medina, 27-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |- 
 { x  |  ( x  e.  A  /\  -.  x  e.  B ) }  =  { y  |  ( y  e.  A  /\  -.  y  e.  B ) }
 
Definitiondf-dif 2948* Define class difference, also called relative complement. Definition 5.12 of [TakeutiZaring] p. 20. Contrast this operation with union  ( A  u.  B ) (df-un 2950) and intersection  ( A  i^i  B ) (df-in 2952). 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 " A excludes  B " to mean  A  \  B. We will use " B is removed from  A " to mean  A  \  { B } i.e. the removal of an element or equivalently the exclusion of a singleton. (Contributed by NM, 29-Apr-1994.)
 |-  ( A  \  B )  =  { x  |  ( x  e.  A  /\  -.  x  e.  B ) }
 
Theoremunjust 2949* Soundness justification theorem for df-un 2950. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |- 
 { x  |  ( x  e.  A  \/  x  e.  B ) }  =  { y  |  ( y  e.  A  \/  y  e.  B ) }
 
Definitiondf-un 2950* Define the union of two classes. Definition 5.6 of [TakeutiZaring] p. 16. Contrast this operation with difference  ( A  \  B ) (df-dif 2948) and intersection  ( A  i^i  B ) (df-in 2952). (Contributed by NM, 23-Aug-1993.)
 |-  ( A  u.  B )  =  { x  |  ( x  e.  A  \/  x  e.  B ) }
 
Theoreminjust 2951* Soundness justification theorem for df-in 2952. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |- 
 { x  |  ( x  e.  A  /\  x  e.  B ) }  =  { y  |  ( y  e.  A  /\  y  e.  B ) }
 
Definitiondf-in 2952* Define the intersection of two classes. Definition 5.6 of [TakeutiZaring] p. 16. Contrast this operation with union  ( A  u.  B ) (df-un 2950) and difference  ( A  \  B ) (df-dif 2948). (Contributed by NM, 29-Apr-1994.)
 |-  ( A  i^i  B )  =  { x  |  ( x  e.  A  /\  x  e.  B ) }
 
Theoremdfin5 2953* Alternate definition for the intersection of two classes. (Contributed by NM, 6-Jul-2005.)
 |-  ( A  i^i  B )  =  { x  e.  A  |  x  e.  B }
 
Theoremdfdif2 2954* Alternate definition of class difference. (Contributed by NM, 25-Mar-2004.)
 |-  ( A  \  B )  =  { x  e.  A  |  -.  x  e.  B }
 
Theoremeldif 2955 Expansion of membership in a class difference. (Contributed by NM, 29-Apr-1994.)
 |-  ( A  e.  ( B  \  C )  <->  ( A  e.  B  /\  -.  A  e.  C ) )
 
Theoremeldifd 2956 If a class is in one class and not another, it is also in their difference. One-way deduction form of eldif 2955. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  B )   &    |-  ( ph  ->  -.  A  e.  C )   =>    |-  ( ph  ->  A  e.  ( B  \  C ) )
 
Theoremeldifad 2957 If a class is in the difference of two classes, it is also in the minuend. One-way deduction form of eldif 2955. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  ( B  \  C ) )   =>    |-  ( ph  ->  A  e.  B )
 
Theoremeldifbd 2958 If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif 2955. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  ( B  \  C ) )   =>    |-  ( ph  ->  -.  A  e.  C )
 
2.1.12  Subclasses and subsets
 
Definitiondf-ss 2959 Define the subclass relationship. Exercise 9 of [TakeutiZaring] p. 18. Note that  A  C_  A (proved in ssid 2992). Contrast this relationship with the relationship  A  C.  B (as will be defined in df-pss 2961). For a more traditional definition, but requiring a dummy variable, see dfss2 2962 (or dfss3 2963 which is similar). (Contributed by NM, 27-Apr-1994.)
 |-  ( A  C_  B  <->  ( A  i^i  B )  =  A )
 
Theoremdfss 2960 Variant of subclass definition df-ss 2959. (Contributed by NM, 3-Sep-2004.)
 |-  ( A  C_  B  <->  A  =  ( A  i^i  B ) )
 
Definitiondf-pss 2961 Define proper subclass relationship between two classes. Definition 5.9 of [TakeutiZaring] p. 17. Note that  -.  A  C.  A (proved in pssirr 3072). Contrast this relationship with the relationship  A  C_  B (as defined in df-ss 2959). Other possible definitions are given by dfpss2 3057 and dfpss3 3058. (Contributed by NM, 7-Feb-1996.)
 |-  ( A  C.  B  <->  ( A  C_  B  /\  A  =/=  B ) )
 
Theoremdfss2 2962* Alternate definition of the subclass relationship between two classes. Definition 5.9 of [TakeutiZaring] p. 17. (Contributed by NM, 8-Jan-2002.)
 |-  ( A  C_  B  <->  A. x ( x  e.  A  ->  x  e.  B ) )
 
Theoremdfss3 2963* Alternate definition of subclass relationship. (Contributed by NM, 14-Oct-1999.)
 |-  ( A  C_  B  <->  A. x  e.  A  x  e.  B )
 
Theoremdfss2f 2964 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/_ x A   &    |-  F/_ x B   =>    |-  ( A  C_  B 
 <-> 
 A. x ( x  e.  A  ->  x  e.  B ) )
 
Theoremdfss3f 2965 Equivalence for subclass relation, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 20-Mar-2004.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  ( A  C_  B 
 <-> 
 A. x  e.  A  x  e.  B )
 
Theoremnfss 2966 If  x is not free in  A and  B, it is not free in  A  C_  B. (Contributed by NM, 27-Dec-1996.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/ x  A  C_  B
 
Theoremssel 2967 Membership relationships follow from a subclass relationship. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  C_  B  ->  ( C  e.  A  ->  C  e.  B ) )
 
Theoremssel2 2968 Membership relationships follow from a subclass relationship. (Contributed by NM, 7-Jun-2004.)
 |-  ( ( A  C_  B  /\  C  e.  A )  ->  C  e.  B )
 
Theoremsseli 2969 Membership inference from subclass relationship. (Contributed by NM, 5-Aug-1993.)
 |-  A  C_  B   =>    |-  ( C  e.  A  ->  C  e.  B )
 
Theoremsselii 2970 Membership inference from subclass relationship. (Contributed by NM, 31-May-1999.)
 |-  A  C_  B   &    |-  C  e.  A   =>    |-  C  e.  B
 
Theoremsseldi 2971 Membership inference from subclass relationship. (Contributed by NM, 25-Jun-2014.)
 |-  A  C_  B   &    |-  ( ph  ->  C  e.  A )   =>    |-  ( ph  ->  C  e.  B )
 
Theoremsseld 2972 Membership deduction from subclass relationship. (Contributed by NM, 15-Nov-1995.)
 |-  ( ph  ->  A  C_  B )   =>    |-  ( ph  ->  ( C  e.  A  ->  C  e.  B ) )
 
Theoremsselda 2973 Membership deduction from subclass relationship. (Contributed by NM, 26-Jun-2014.)
 |-  ( ph  ->  A  C_  B )   =>    |-  ( ( ph  /\  C  e.  A )  ->  C  e.  B )
 
Theoremsseldd 2974 Membership inference from subclass relationship. (Contributed by NM, 14-Dec-2004.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  C  e.  A )   =>    |-  ( ph  ->  C  e.  B )
 
Theoremssneld 2975 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.)
 |-  ( ph  ->  A  C_  B )   =>    |-  ( ph  ->  ( -.  C  e.  B  ->  -.  C  e.  A ) )
 
Theoremssneldd 2976 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.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  -.  C  e.  B )   =>    |-  ( ph  ->  -.  C  e.  A )
 
Theoremssriv 2977* Inference rule based on subclass definition. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  e.  A  ->  x  e.  B )   =>    |-  A  C_  B
 
Theoremssrd 2978 Deduction rule based on subclass definition. (Contributed by Thierry Arnoux, 8-Mar-2017.)
 |- 
 F/ x ph   &    |-  F/_ x A   &    |-  F/_ x B   &    |-  ( ph  ->  ( x  e.  A  ->  x  e.  B ) )   =>    |-  ( ph  ->  A  C_  B )
 
Theoremssrdv 2979* Deduction rule based on subclass definition. (Contributed by NM, 15-Nov-1995.)
 |-  ( ph  ->  ( x  e.  A  ->  x  e.  B ) )   =>    |-  ( ph  ->  A  C_  B )
 
Theoremsstr2 2980 Transitivity of subclasses. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
 |-  ( A  C_  B  ->  ( B  C_  C  ->  A  C_  C )
 )
 
Theoremsstr 2981 Transitivity of subclasses. Theorem 6 of [Suppes] p. 23. (Contributed by NM, 5-Sep-2003.)
 |-  ( ( A  C_  B  /\  B  C_  C )  ->  A  C_  C )
 
Theoremsstri 2982 Subclass transitivity inference. (Contributed by NM, 5-May-2000.)
 |-  A  C_  B   &    |-  B  C_  C   =>    |-  A  C_  C
 
Theoremsstrd 2983 Subclass transitivity deduction. (Contributed by NM, 2-Jun-2004.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  B 
 C_  C )   =>    |-  ( ph  ->  A 
 C_  C )
 
Theoremsyl5ss 2984 Subclass transitivity deduction. (Contributed by NM, 6-Feb-2014.)
 |-  A  C_  B   &    |-  ( ph  ->  B 
 C_  C )   =>    |-  ( ph  ->  A 
 C_  C )
 
Theoremsyl6ss 2985 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  ( ph  ->  A  C_  B )   &    |-  B  C_  C   =>    |-  ( ph  ->  A  C_  C )
 
Theoremsylan9ss 2986 A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ps  ->  B 
 C_  C )   =>    |-  ( ( ph  /\ 
 ps )  ->  A  C_  C )
 
Theoremsylan9ssr 2987 A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ps  ->  B 
 C_  C )   =>    |-  ( ( ps 
 /\  ph )  ->  A  C_  C )
 
Theoremeqss 2988 The subclass relationship is antisymmetric. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
 
Theoremeqssi 2989 Infer equality from two subclass relationships. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 9-Sep-1993.)
 |-  A  C_  B   &    |-  B  C_  A   =>    |-  A  =  B
 
Theoremeqssd 2990 Equality deduction from two subclass relationships. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 27-Jun-2004.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  B 
 C_  A )   =>    |-  ( ph  ->  A  =  B )
 
Theoremeqrd 2991 Deduce equality of classes from equivalence of membership. (Contributed by Thierry Arnoux, 21-Mar-2017.)
 |- 
 F/ x ph   &    |-  F/_ x A   &    |-  F/_ x B   &    |-  ( ph  ->  ( x  e.  A  <->  x  e.  B ) )   =>    |-  ( ph  ->  A  =  B )
 
Theoremssid 2992 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.)
 |-  A  C_  A
 
Theoremssv 2993 Any class is a subclass of the universal class. (Contributed by NM, 31-Oct-1995.)
 |-  A  C_  _V
 
Theoremsseq1 2994 Equality theorem for subclasses. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
 |-  ( A  =  B  ->  ( A  C_  C  <->  B 
 C_  C ) )
 
Theoremsseq2 2995 Equality theorem for the subclass relationship. (Contributed by NM, 25-Jun-1998.)
 |-  ( A  =  B  ->  ( C  C_  A  <->  C 
 C_  B ) )
 
Theoremsseq12 2996 Equality theorem for the subclass relationship. (Contributed by NM, 31-May-1999.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  C_  C 
 <->  B  C_  D )
 )
 
Theoremsseq1i 2997 An equality inference for the subclass relationship. (Contributed by NM, 18-Aug-1993.)
 |-  A  =  B   =>    |-  ( A  C_  C 
 <->  B  C_  C )
 
Theoremsseq2i 2998 An equality inference for the subclass relationship. (Contributed by NM, 30-Aug-1993.)
 |-  A  =  B   =>    |-  ( C  C_  A 
 <->  C  C_  B )
 
Theoremsseq12i 2999 An equality inference for the subclass relationship. (Contributed by NM, 31-May-1999.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
 |-  A  =  B   &    |-  C  =  D   =>    |-  ( A  C_  C  <->  B 
 C_  D )
 
Theoremsseq1d 3000 An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  C_  C  <->  B  C_  C ) )
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