Type | Label | Description |
Statement |
|
Theorem | sbccsbg 3101* |
Substitution into a wff expressed in terms of substitution into a class.
(Contributed by NM, 15-Aug-2007.)
|
    ![]. ].](_drbrack.gif)
  ![]_ ]_](_urbrack.gif)      |
|
Theorem | sbccsb2g 3102 |
Substitution into a wff expressed in using substitution into a class.
(Contributed by NM, 27-Nov-2005.)
|
    ![]. ].](_drbrack.gif)
  ![]_ ]_](_urbrack.gif)      |
|
Theorem | nfcsb1d 3103 |
Bound-variable hypothesis builder for substitution into a class.
(Contributed by Mario Carneiro, 12-Oct-2016.)
|
         ![]_ ]_](_urbrack.gif)   |
|
Theorem | nfcsb1 3104 |
Bound-variable hypothesis builder for substitution into a class.
(Contributed by Mario Carneiro, 12-Oct-2016.)
|
      ![]_ ]_](_urbrack.gif)  |
|
Theorem | nfcsb1v 3105* |
Bound-variable hypothesis builder for substitution into a class.
(Contributed by NM, 17-Aug-2006.) (Revised by Mario Carneiro,
12-Oct-2016.)
|
    ![]_ ]_](_urbrack.gif)  |
|
Theorem | nfsbcdw 3106* |
Version of nfsbcd 2997 with a disjoint variable condition.
(Contributed by
NM, 23-Nov-2005.) (Revised by Gino Giotto, 10-Jan-2024.)
|
               ![]. ].](_drbrack.gif)   |
|
Theorem | nfcsbd 3107 |
Deduction version of nfcsb 3109. (Contributed by NM, 21-Nov-2005.)
(Revised by Mario Carneiro, 12-Oct-2016.)
|
          
    ![]_ ]_](_urbrack.gif)   |
|
Theorem | nfcsbw 3108* |
Bound-variable hypothesis builder for substitution into a class.
Version of nfcsb 3109 with a disjoint variable condition.
(Contributed by
Mario Carneiro, 12-Oct-2016.) (Revised by Gino Giotto, 10-Jan-2024.)
|
        ![]_ ]_](_urbrack.gif)  |
|
Theorem | nfcsb 3109 |
Bound-variable hypothesis builder for substitution into a class.
(Contributed by Mario Carneiro, 12-Oct-2016.)
|
        ![]_ ]_](_urbrack.gif)  |
|
Theorem | csbhypf 3110* |
Introduce an explicit substitution into an implicit substitution
hypothesis. See sbhypf 2801 for class substitution version. (Contributed
by NM, 19-Dec-2008.)
|
    
 
  ![]_ ]_](_urbrack.gif)   |
|
Theorem | csbiebt 3111* |
Conversion of implicit substitution to explicit substitution into a
class. (Closed theorem version of csbiegf 3115.) (Contributed by NM,
11-Nov-2005.)
|
            ![]_ ]_](_urbrack.gif)    |
|
Theorem | csbiedf 3112* |
Conversion of implicit substitution to explicit substitution into a
class. (Contributed by Mario Carneiro, 13-Oct-2016.)
|
               ![]_ ]_](_urbrack.gif)   |
|
Theorem | csbieb 3113* |
Bidirectional conversion between an implicit class substitution
hypothesis and its explicit substitution equivalent.
(Contributed by NM, 2-Mar-2008.)
|
         ![]_ ]_](_urbrack.gif)   |
|
Theorem | csbiebg 3114* |
Bidirectional conversion between an implicit class substitution
hypothesis and its explicit substitution equivalent.
(Contributed by NM, 24-Mar-2013.) (Revised by Mario Carneiro,
11-Dec-2016.)
|
          ![]_ ]_](_urbrack.gif)    |
|
Theorem | csbiegf 3115* |
Conversion of implicit substitution to explicit substitution into a
class. (Contributed by NM, 11-Nov-2005.) (Revised by Mario Carneiro,
13-Oct-2016.)
|
    
 
  ![]_ ]_](_urbrack.gif)   |
|
Theorem | csbief 3116* |
Conversion of implicit substitution to explicit substitution into a
class. (Contributed by NM, 26-Nov-2005.) (Revised by Mario Carneiro,
13-Oct-2016.)
|
      ![]_ ]_](_urbrack.gif)
 |
|
Theorem | csbie 3117* |
Conversion of implicit substitution to explicit substitution into a
class. (Contributed by AV, 2-Dec-2019.)
|

   ![]_ ]_](_urbrack.gif)  |
|
Theorem | csbied 3118* |
Conversion of implicit substitution to explicit substitution into a
class. (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by Mario
Carneiro, 13-Oct-2016.)
|
      
  ![]_ ]_](_urbrack.gif)   |
|
Theorem | csbied2 3119* |
Conversion of implicit substitution to explicit class substitution,
deduction form. (Contributed by Mario Carneiro, 2-Jan-2017.)
|
     
     ![]_ ]_](_urbrack.gif)   |
|
Theorem | csbie2t 3120* |
Conversion of implicit substitution to explicit substitution into a
class (closed form of csbie2 3121). (Contributed by NM, 3-Sep-2007.)
(Revised by Mario Carneiro, 13-Oct-2016.)
|
           ![]_ ]_](_urbrack.gif)   ![]_ ]_](_urbrack.gif)   |
|
Theorem | csbie2 3121* |
Conversion of implicit substitution to explicit substitution into a
class. (Contributed by NM, 27-Aug-2007.)
|
    
 ![]_ ]_](_urbrack.gif)   ![]_ ]_](_urbrack.gif)
 |
|
Theorem | csbie2g 3122* |
Conversion of implicit substitution to explicit class substitution.
This version of sbcie 3012 avoids a disjointness condition on and
by
substituting twice. (Contributed by Mario Carneiro,
11-Nov-2016.)
|
  
 
  ![]_ ]_](_urbrack.gif)   |
|
Theorem | sbcnestgf 3123 |
Nest the composition of two substitutions. (Contributed by Mario
Carneiro, 11-Nov-2016.)
|
          ![]. ].](_drbrack.gif)   ![]. ].](_drbrack.gif)    ![]_ ]_](_urbrack.gif)  ![]. ].](_drbrack.gif)    |
|
Theorem | csbnestgf 3124 |
Nest the composition of two substitutions. (Contributed by NM,
23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.)
|
         ![]_ ]_](_urbrack.gif)   ![]_ ]_](_urbrack.gif)    ![]_ ]_](_urbrack.gif)  ![]_ ]_](_urbrack.gif)   |
|
Theorem | sbcnestg 3125* |
Nest the composition of two substitutions. (Contributed by NM,
27-Nov-2005.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)
|
    ![]. ].](_drbrack.gif)   ![]. ].](_drbrack.gif)
   ![]_ ]_](_urbrack.gif)  ![]. ].](_drbrack.gif)    |
|
Theorem | csbnestg 3126* |
Nest the composition of two substitutions. (Contributed by NM,
23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.)
|
   ![]_ ]_](_urbrack.gif)   ![]_ ]_](_urbrack.gif)
   ![]_ ]_](_urbrack.gif)  ![]_ ]_](_urbrack.gif)   |
|
Theorem | csbnest1g 3127 |
Nest the composition of two substitutions. (Contributed by NM,
23-May-2006.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)
|
   ![]_ ]_](_urbrack.gif)   ![]_ ]_](_urbrack.gif)    ![]_ ]_](_urbrack.gif)  ![]_ ]_](_urbrack.gif)   |
|
Theorem | csbidmg 3128* |
Idempotent law for class substitutions. (Contributed by NM,
1-Mar-2008.)
|
   ![]_ ]_](_urbrack.gif)   ![]_ ]_](_urbrack.gif)   ![]_ ]_](_urbrack.gif)   |
|
Theorem | sbcco3g 3129* |
Composition of two substitutions. (Contributed by NM, 27-Nov-2005.)
(Revised by Mario Carneiro, 11-Nov-2016.)
|
  
   ![]. ].](_drbrack.gif)   ![]. ].](_drbrack.gif)
  ![]. ].](_drbrack.gif)    |
|
Theorem | csbco3g 3130* |
Composition of two class substitutions. (Contributed by NM,
27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.)
|
  
  ![]_ ]_](_urbrack.gif)   ![]_ ]_](_urbrack.gif)
  ![]_ ]_](_urbrack.gif)   |
|
Theorem | rspcsbela 3131* |
Special case related to rspsbc 3060. (Contributed by NM, 10-Dec-2005.)
(Proof shortened by Eric Schmidt, 17-Jan-2007.)
|
      ![]_ ]_](_urbrack.gif)   |
|
Theorem | sbnfc2 3132* |
Two ways of expressing " is (effectively) not free in ."
(Contributed by Mario Carneiro, 14-Oct-2016.)
|
       
 ![]_ ]_](_urbrack.gif)   ![]_ ]_](_urbrack.gif)   |
|
Theorem | csbabg 3133* |
Move substitution into a class abstraction. (Contributed by NM,
13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|
   ![]_ ]_](_urbrack.gif)      ![]. ].](_drbrack.gif)    |
|
Theorem | cbvralcsf 3134 |
A more general version of cbvralf 2710 that doesn't require and
to be distinct from or . Changes
bound variables using
implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.)
|
        
 
    
   |
|
Theorem | cbvrexcsf 3135 |
A more general version of cbvrexf 2711 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.)
|
        
 
    
   |
|
Theorem | cbvreucsf 3136 |
A more general version of cbvreuv 2720 that has no distinct variable
rextrictions. Changes bound variables using implicit substitution.
(Contributed by Andrew Salmon, 13-Jul-2011.)
|
        
 
    
   |
|
Theorem | cbvrabcsf 3137 |
A more general version of cbvrab 2750 with no distinct variable
restrictions. (Contributed by Andrew Salmon, 13-Jul-2011.)
|
        
 
     
  |
|
Theorem | cbvralv2 3138* |
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.)
|
    
      |
|
Theorem | cbvrexv2 3139* |
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.)
|
    
      |
|
Theorem | rspc2vd 3140* |
Deduction version of 2-variable restricted specialization, using
implicit substitution. Notice that the class for the second set
variable may
depend on the first set variable .
(Contributed by AV, 29-Mar-2021.)
|
                       |
|
2.1.11 Define basic set operations and
relations
|
|
Syntax | cdif 3141 |
Extend class notation to include class difference (read: " minus
").
|

  |
|
Syntax | cun 3142 |
Extend class notation to include union of two classes (read: "
union ").
|

  |
|
Syntax | cin 3143 |
Extend class notation to include the intersection of two classes (read:
" intersect
").
|

  |
|
Syntax | wss 3144 |
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 ".
|
 |
|
Theorem | difjust 3145* |
Soundness justification theorem for df-dif 3146. (Contributed by Rodolfo
Medina, 27-Apr-2010.) (Proof shortened by Andrew Salmon,
9-Jul-2011.)
|
         |
|
Definition | df-dif 3146* |
Define class difference, also called relative complement. Definition
5.12 of [TakeutiZaring] p. 20.
Contrast this operation with union
  (df-un 3148) and intersection   (df-in 3150).
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.)
|
 
     |
|
Theorem | unjust 3147* |
Soundness justification theorem for df-un 3148. (Contributed by Rodolfo
Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon,
9-Jul-2011.)
|
 
 
 
   |
|
Definition | df-un 3148* |
Define the union of two classes. Definition 5.6 of [TakeutiZaring]
p. 16. Contrast this operation with difference  
(df-dif 3146) and intersection   (df-in 3150). (Contributed
by NM, 23-Aug-1993.)
|
 
 
   |
|
Theorem | injust 3149* |
Soundness justification theorem for df-in 3150. (Contributed by Rodolfo
Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon,
9-Jul-2011.)
|
 
 
 
   |
|
Definition | df-in 3150* |
Define the intersection of two classes. Definition 5.6 of
[TakeutiZaring] p. 16. Contrast
this operation with union
  (df-un 3148) and difference   (df-dif 3146).
(Contributed by NM, 29-Apr-1994.)
|
 
 
   |
|
Theorem | dfin5 3151* |
Alternate definition for the intersection of two classes. (Contributed
by NM, 6-Jul-2005.)
|
 
   |
|
Theorem | dfdif2 3152* |
Alternate definition of class difference. (Contributed by NM,
25-Mar-2004.)
|
 
   |
|
Theorem | eldif 3153 |
Expansion of membership in a class difference. (Contributed by NM,
29-Apr-1994.)
|
       |
|
Theorem | eldifd 3154 |
If a class is in one class and not another, it is also in their
difference. One-way deduction form of eldif 3153. (Contributed by David
Moews, 1-May-2017.)
|
    

   |
|
Theorem | eldifad 3155 |
If a class is in the difference of two classes, it is also in the
minuend. One-way deduction form of eldif 3153. (Contributed by David
Moews, 1-May-2017.)
|
       |
|
Theorem | eldifbd 3156 |
If a class is in the difference of two classes, it is not in the
subtrahend. One-way deduction form of eldif 3153. (Contributed by David
Moews, 1-May-2017.)
|
       |
|
2.1.12 Subclasses and subsets
|
|
Definition | df-ss 3157 |
Define the subclass relationship. Exercise 9 of [TakeutiZaring] p. 18.
Note that (proved in ssid 3190). For a more traditional
definition, but requiring a dummy variable, see dfss2 3159. Other possible
definitions are given by dfss3 3160, ssequn1 3320, ssequn2 3323, and sseqin2 3369.
(Contributed by NM, 27-Apr-1994.)
|
     |
|
Theorem | dfss 3158 |
Variant of subclass definition df-ss 3157. (Contributed by NM,
3-Sep-2004.)
|
     |
|
Theorem | dfss2 3159* |
Alternate definition of the subclass relationship between two classes.
Definition 5.9 of [TakeutiZaring]
p. 17. (Contributed by NM,
8-Jan-2002.)
|
   
   |
|
Theorem | dfss3 3160* |
Alternate definition of subclass relationship. (Contributed by NM,
14-Oct-1999.)
|
    |
|
Theorem | dfss2f 3161 |
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.)
|
           |
|
Theorem | dfss3f 3162 |
Equivalence for subclass relation, using bound-variable hypotheses
instead of distinct variable conditions. (Contributed by NM,
20-Mar-2004.)
|
     
  |
|
Theorem | nfss 3163 |
If is not free in and , it is not free in .
(Contributed by NM, 27-Dec-1996.)
|
      |
|
Theorem | ssel 3164 |
Membership relationships follow from a subclass relationship.
(Contributed by NM, 5-Aug-1993.)
|
     |
|
Theorem | ssel2 3165 |
Membership relationships follow from a subclass relationship.
(Contributed by NM, 7-Jun-2004.)
|
     |
|
Theorem | sseli 3166 |
Membership inference from subclass relationship. (Contributed by NM,
5-Aug-1993.)
|
   |
|
Theorem | sselii 3167 |
Membership inference from subclass relationship. (Contributed by NM,
31-May-1999.)
|
 |
|
Theorem | sselid 3168 |
Membership inference from subclass relationship. (Contributed by NM,
25-Jun-2014.)
|
     |
|
Theorem | sseld 3169 |
Membership deduction from subclass relationship. (Contributed by NM,
15-Nov-1995.)
|
   
   |
|
Theorem | sselda 3170 |
Membership deduction from subclass relationship. (Contributed by NM,
26-Jun-2014.)
|
       |
|
Theorem | sseldd 3171 |
Membership inference from subclass relationship. (Contributed by NM,
14-Dec-2004.)
|
       |
|
Theorem | ssneld 3172 |
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.)
|
       |
|
Theorem | ssneldd 3173 |
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.)
|
    
  |
|
Theorem | ssriv 3174* |
Inference based on subclass definition. (Contributed by NM,
5-Aug-1993.)
|
   |
|
Theorem | ssrd 3175 |
Deduction based on subclass definition. (Contributed by Thierry Arnoux,
8-Mar-2017.)
|
       
  
  |
|
Theorem | ssrdv 3176* |
Deduction based on subclass definition. (Contributed by NM,
15-Nov-1995.)
|
 
  
  |
|
Theorem | sstr2 3177 |
Transitivity of subclasses. Exercise 5 of [TakeutiZaring] p. 17.
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon,
14-Jun-2011.)
|
 
   |
|
Theorem | sstr 3178 |
Transitivity of subclasses. Theorem 6 of [Suppes] p. 23. (Contributed by
NM, 5-Sep-2003.)
|
     |
|
Theorem | sstri 3179 |
Subclass transitivity inference. (Contributed by NM, 5-May-2000.)
|
 |
|
Theorem | sstrd 3180 |
Subclass transitivity deduction. (Contributed by NM, 2-Jun-2004.)
|
       |
|
Theorem | sstrid 3181 |
Subclass transitivity deduction. (Contributed by NM, 6-Feb-2014.)
|
     |
|
Theorem | sstrdi 3182 |
Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim,
3-Jun-2011.)
|
     |
|
Theorem | sylan9ss 3183 |
A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.)
(Proof shortened by Andrew Salmon, 14-Jun-2011.)
|
         |
|
Theorem | sylan9ssr 3184 |
A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.)
|
         |
|
Theorem | eqss 3185 |
The subclass relationship is antisymmetric. Compare Theorem 4 of
[Suppes] p. 22. (Contributed by NM,
5-Aug-1993.)
|
 
   |
|
Theorem | eqssi 3186 |
Infer equality from two subclass relationships. Compare Theorem 4 of
[Suppes] p. 22. (Contributed by NM,
9-Sep-1993.)
|
 |
|
Theorem | eqssd 3187 |
Equality deduction from two subclass relationships. Compare Theorem 4
of [Suppes] p. 22. (Contributed by NM,
27-Jun-2004.)
|
       |
|
Theorem | eqrd 3188 |
Deduce equality of classes from equivalence of membership. (Contributed
by Thierry Arnoux, 21-Mar-2017.)
|
       
     |
|
Theorem | eqelssd 3189* |
Equality deduction from subclass relationship and membership.
(Contributed by AV, 21-Aug-2022.)
|
    
    |
|
Theorem | ssid 3190 |
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.)
|
 |
|
Theorem | ssidd 3191 |
Weakening of ssid 3190. (Contributed by BJ, 1-Sep-2022.)
|
   |
|
Theorem | ssv 3192 |
Any class is a subclass of the universal class. (Contributed by NM,
31-Oct-1995.)
|
 |
|
Theorem | sseq1 3193 |
Equality theorem for subclasses. (Contributed by NM, 5-Aug-1993.) (Proof
shortened by Andrew Salmon, 21-Jun-2011.)
|
     |
|
Theorem | sseq2 3194 |
Equality theorem for the subclass relationship. (Contributed by NM,
25-Jun-1998.)
|
     |
|
Theorem | sseq12 3195 |
Equality theorem for the subclass relationship. (Contributed by NM,
31-May-1999.)
|
   
   |
|
Theorem | sseq1i 3196 |
An equality inference for the subclass relationship. (Contributed by
NM, 18-Aug-1993.)
|

  |
|
Theorem | sseq2i 3197 |
An equality inference for the subclass relationship. (Contributed by
NM, 30-Aug-1993.)
|

  |
|
Theorem | sseq12i 3198 |
An equality inference for the subclass relationship. (Contributed by
NM, 31-May-1999.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
|
   |
|
Theorem | sseq1d 3199 |
An equality deduction for the subclass relationship. (Contributed by
NM, 14-Aug-1994.)
|
   
   |
|
Theorem | sseq2d 3200 |
An equality deduction for the subclass relationship. (Contributed by
NM, 14-Aug-1994.)
|
   
   |