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Theorem List for Intuitionistic Logic Explorer - 3001-3100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnfcsb1d 3001 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 3002 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 3003* 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 3004 Deduction version of nfcsb 3005. (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 3005 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 3006* Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 2707 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 3007* Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 3011.) (Contributed by NM, 11-Nov-2005.)
 |-  ( ( A  e.  V  /\  F/_ x C ) 
 ->  ( A. x ( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C ) )
 
Theoremcsbiedf 3008* 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 3009* 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 3010* 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 3011* 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 3012* 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
 
Theoremcsbie 3013* Conversion of implicit substitution to explicit substitution into a class. (Contributed by AV, 2-Dec-2019.)
 |-  A  e.  _V   &    |-  ( x  =  A  ->  B  =  C )   =>    |-  [_ A  /  x ]_ B  =  C
 
Theoremcsbied 3014* 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 3015* 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 3016* Conversion of implicit substitution to explicit substitution into a class (closed form of csbie2 3017). (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 3017* 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 3018* Conversion of implicit substitution to explicit class substitution. This version of sbcie 2913 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 3019 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 3020 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 3021* 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 3022* 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 3023 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 3024* Idempotent law for class substitutions. (Contributed by NM, 1-Mar-2008.)
 |-  ( A  e.  V  -> 
 [_ A  /  x ]_
 [_ A  /  x ]_ B  =  [_ A  /  x ]_ B )
 
Theoremsbcco3g 3025* 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 3026* 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 3027* Special case related to rspsbc 2961. (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 3028* 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 3029* 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 3030 A more general version of cbvralf 2623 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 3031 A more general version of cbvrexf 2624 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 3032 A more general version of cbvreuv 2631 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 3033 A more general version of cbvrab 2656 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 3034* 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 3035* 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 3036 Extend class notation to include class difference (read: " A minus  B").
 class  ( A  \  B )
 
Syntaxcun 3037 Extend class notation to include union of two classes (read: " A union  B").
 class  ( A  u.  B )
 
Syntaxcin 3038 Extend class notation to include the intersection of two classes (read: " A intersect  B").
 class  ( A  i^i  B )
 
Syntaxwss 3039 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
 
Theoremdifjust 3040* Soundness justification theorem for df-dif 3041. (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 3041* Define class difference, also called relative complement. Definition 5.12 of [TakeutiZaring] p. 20. Contrast this operation with union  ( A  u.  B ) (df-un 3043) and intersection  ( A  i^i  B ) (df-in 3045). 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 3042* Soundness justification theorem for df-un 3043. (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 3043* Define the union of two classes. Definition 5.6 of [TakeutiZaring] p. 16. Contrast this operation with difference  ( A  \  B ) (df-dif 3041) and intersection  ( A  i^i  B ) (df-in 3045). (Contributed by NM, 23-Aug-1993.)
 |-  ( A  u.  B )  =  { x  |  ( x  e.  A  \/  x  e.  B ) }
 
Theoreminjust 3044* Soundness justification theorem for df-in 3045. (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 3045* Define the intersection of two classes. Definition 5.6 of [TakeutiZaring] p. 16. Contrast this operation with union  ( A  u.  B ) (df-un 3043) and difference  ( A  \  B ) (df-dif 3041). (Contributed by NM, 29-Apr-1994.)
 |-  ( A  i^i  B )  =  { x  |  ( x  e.  A  /\  x  e.  B ) }
 
Theoremdfin5 3046* 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 3047* Alternate definition of class difference. (Contributed by NM, 25-Mar-2004.)
 |-  ( A  \  B )  =  { x  e.  A  |  -.  x  e.  B }
 
Theoremeldif 3048 Expansion of membership in a class difference. (Contributed by NM, 29-Apr-1994.)
 |-  ( A  e.  ( B  \  C )  <->  ( A  e.  B  /\  -.  A  e.  C ) )
 
Theoremeldifd 3049 If a class is in one class and not another, it is also in their difference. One-way deduction form of eldif 3048. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  B )   &    |-  ( ph  ->  -.  A  e.  C )   =>    |-  ( ph  ->  A  e.  ( B  \  C ) )
 
Theoremeldifad 3050 If a class is in the difference of two classes, it is also in the minuend. One-way deduction form of eldif 3048. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  ( B  \  C ) )   =>    |-  ( ph  ->  A  e.  B )
 
Theoremeldifbd 3051 If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif 3048. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  ( B  \  C ) )   =>    |-  ( ph  ->  -.  A  e.  C )
 
2.1.12  Subclasses and subsets
 
Definitiondf-ss 3052 Define the subclass relationship. Exercise 9 of [TakeutiZaring] p. 18. Note that  A  C_  A (proved in ssid 3085). For a more traditional definition, but requiring a dummy variable, see dfss2 3054. Other possible definitions are given by dfss3 3055, ssequn1 3214, ssequn2 3217, and sseqin2 3263. (Contributed by NM, 27-Apr-1994.)
 |-  ( A  C_  B  <->  ( A  i^i  B )  =  A )
 
Theoremdfss 3053 Variant of subclass definition df-ss 3052. (Contributed by NM, 3-Sep-2004.)
 |-  ( A  C_  B  <->  A  =  ( A  i^i  B ) )
 
Theoremdfss2 3054* 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 3055* Alternate definition of subclass relationship. (Contributed by NM, 14-Oct-1999.)
 |-  ( A  C_  B  <->  A. x  e.  A  x  e.  B )
 
Theoremdfss2f 3056 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 3057 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 3058 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 3059 Membership relationships follow from a subclass relationship. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  C_  B  ->  ( C  e.  A  ->  C  e.  B ) )
 
Theoremssel2 3060 Membership relationships follow from a subclass relationship. (Contributed by NM, 7-Jun-2004.)
 |-  ( ( A  C_  B  /\  C  e.  A )  ->  C  e.  B )
 
Theoremsseli 3061 Membership inference from subclass relationship. (Contributed by NM, 5-Aug-1993.)
 |-  A  C_  B   =>    |-  ( C  e.  A  ->  C  e.  B )
 
Theoremsselii 3062 Membership inference from subclass relationship. (Contributed by NM, 31-May-1999.)
 |-  A  C_  B   &    |-  C  e.  A   =>    |-  C  e.  B
 
Theoremsseldi 3063 Membership inference from subclass relationship. (Contributed by NM, 25-Jun-2014.)
 |-  A  C_  B   &    |-  ( ph  ->  C  e.  A )   =>    |-  ( ph  ->  C  e.  B )
 
Theoremsseld 3064 Membership deduction from subclass relationship. (Contributed by NM, 15-Nov-1995.)
 |-  ( ph  ->  A  C_  B )   =>    |-  ( ph  ->  ( C  e.  A  ->  C  e.  B ) )
 
Theoremsselda 3065 Membership deduction from subclass relationship. (Contributed by NM, 26-Jun-2014.)
 |-  ( ph  ->  A  C_  B )   =>    |-  ( ( ph  /\  C  e.  A )  ->  C  e.  B )
 
Theoremsseldd 3066 Membership inference from subclass relationship. (Contributed by NM, 14-Dec-2004.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  C  e.  A )   =>    |-  ( ph  ->  C  e.  B )
 
Theoremssneld 3067 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 3068 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 3069* Inference based on subclass definition. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  e.  A  ->  x  e.  B )   =>    |-  A  C_  B
 
Theoremssrd 3070 Deduction 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 3071* Deduction based on subclass definition. (Contributed by NM, 15-Nov-1995.)
 |-  ( ph  ->  ( x  e.  A  ->  x  e.  B ) )   =>    |-  ( ph  ->  A  C_  B )
 
Theoremsstr2 3072 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 3073 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 3074 Subclass transitivity inference. (Contributed by NM, 5-May-2000.)
 |-  A  C_  B   &    |-  B  C_  C   =>    |-  A  C_  C
 
Theoremsstrd 3075 Subclass transitivity deduction. (Contributed by NM, 2-Jun-2004.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  B 
 C_  C )   =>    |-  ( ph  ->  A 
 C_  C )
 
Theoremsstrid 3076 Subclass transitivity deduction. (Contributed by NM, 6-Feb-2014.)
 |-  A  C_  B   &    |-  ( ph  ->  B 
 C_  C )   =>    |-  ( ph  ->  A 
 C_  C )
 
Theoremsyl6ss 3077 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  ( ph  ->  A  C_  B )   &    |-  B  C_  C   =>    |-  ( ph  ->  A  C_  C )
 
Theoremsylan9ss 3078 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 3079 A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ps  ->  B 
 C_  C )   =>    |-  ( ( ps 
 /\  ph )  ->  A  C_  C )
 
Theoremeqss 3080 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 3081 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 3082 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 3083 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 )
 
Theoremeqelssd 3084* Equality deduction from subclass relationship and membership. (Contributed by AV, 21-Aug-2022.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  x  e.  A )   =>    |-  ( ph  ->  A  =  B )
 
Theoremssid 3085 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
 
Theoremssidd 3086 Weakening of ssid 3085. (Contributed by BJ, 1-Sep-2022.)
 |-  ( ph  ->  A  C_  A )
 
Theoremssv 3087 Any class is a subclass of the universal class. (Contributed by NM, 31-Oct-1995.)
 |-  A  C_  _V
 
Theoremsseq1 3088 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 3089 Equality theorem for the subclass relationship. (Contributed by NM, 25-Jun-1998.)
 |-  ( A  =  B  ->  ( C  C_  A  <->  C 
 C_  B ) )
 
Theoremsseq12 3090 Equality theorem for the subclass relationship. (Contributed by NM, 31-May-1999.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  C_  C 
 <->  B  C_  D )
 )
 
Theoremsseq1i 3091 An equality inference for the subclass relationship. (Contributed by NM, 18-Aug-1993.)
 |-  A  =  B   =>    |-  ( A  C_  C 
 <->  B  C_  C )
 
Theoremsseq2i 3092 An equality inference for the subclass relationship. (Contributed by NM, 30-Aug-1993.)
 |-  A  =  B   =>    |-  ( C  C_  A 
 <->  C  C_  B )
 
Theoremsseq12i 3093 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 3094 An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  C_  C  <->  B  C_  C ) )
 
Theoremsseq2d 3095 An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C  C_  A  <->  C  C_  B ) )
 
Theoremsseq12d 3096 An equality deduction for the subclass relationship. (Contributed by NM, 31-May-1999.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A  C_  C  <->  B  C_  D ) )
 
Theoremeqsstri 3097 Substitution of equality into a subclass relationship. (Contributed by NM, 16-Jul-1995.)
 |-  A  =  B   &    |-  B  C_  C   =>    |-  A  C_  C
 
Theoremeqsstrri 3098 Substitution of equality into a subclass relationship. (Contributed by NM, 19-Oct-1999.)
 |-  B  =  A   &    |-  B  C_  C   =>    |-  A  C_  C
 
Theoremsseqtri 3099 Substitution of equality into a subclass relationship. (Contributed by NM, 28-Jul-1995.)
 |-  A  C_  B   &    |-  B  =  C   =>    |-  A  C_  C
 
Theoremsseqtrri 3100 Substitution of equality into a subclass relationship. (Contributed by NM, 4-Apr-1995.)
 |-  A  C_  B   &    |-  C  =  B   =>    |-  A  C_  C
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