Type | Label | Description |
Statement |
|
Theorem | nfcsb1 3101 |
Bound-variable hypothesis builder for substitution into a class.
(Contributed by Mario Carneiro, 12-Oct-2016.)
|
      ![]_ ]_](_urbrack.gif)  |
|
Theorem | nfcsb1v 3102* |
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 3103* |
Version of nfsbcd 2994 with a disjoint variable condition.
(Contributed by
NM, 23-Nov-2005.) (Revised by Gino Giotto, 10-Jan-2024.)
|
               ![]. ].](_drbrack.gif)   |
|
Theorem | nfcsbd 3104 |
Deduction version of nfcsb 3106. (Contributed by NM, 21-Nov-2005.)
(Revised by Mario Carneiro, 12-Oct-2016.)
|
          
    ![]_ ]_](_urbrack.gif)   |
|
Theorem | nfcsbw 3105* |
Bound-variable hypothesis builder for substitution into a class.
Version of nfcsb 3106 with a disjoint variable condition.
(Contributed by
Mario Carneiro, 12-Oct-2016.) (Revised by Gino Giotto, 10-Jan-2024.)
|
        ![]_ ]_](_urbrack.gif)  |
|
Theorem | nfcsb 3106 |
Bound-variable hypothesis builder for substitution into a class.
(Contributed by Mario Carneiro, 12-Oct-2016.)
|
        ![]_ ]_](_urbrack.gif)  |
|
Theorem | csbhypf 3107* |
Introduce an explicit substitution into an implicit substitution
hypothesis. See sbhypf 2798 for class substitution version. (Contributed
by NM, 19-Dec-2008.)
|
    
 
  ![]_ ]_](_urbrack.gif)   |
|
Theorem | csbiebt 3108* |
Conversion of implicit substitution to explicit substitution into a
class. (Closed theorem version of csbiegf 3112.) (Contributed by NM,
11-Nov-2005.)
|
            ![]_ ]_](_urbrack.gif)    |
|
Theorem | csbiedf 3109* |
Conversion of implicit substitution to explicit substitution into a
class. (Contributed by Mario Carneiro, 13-Oct-2016.)
|
               ![]_ ]_](_urbrack.gif)   |
|
Theorem | csbieb 3110* |
Bidirectional conversion between an implicit class substitution
hypothesis and its explicit substitution equivalent.
(Contributed by NM, 2-Mar-2008.)
|
         ![]_ ]_](_urbrack.gif)   |
|
Theorem | csbiebg 3111* |
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 3112* |
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 3113* |
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 3114* |
Conversion of implicit substitution to explicit substitution into a
class. (Contributed by AV, 2-Dec-2019.)
|

   ![]_ ]_](_urbrack.gif)  |
|
Theorem | csbied 3115* |
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 3116* |
Conversion of implicit substitution to explicit class substitution,
deduction form. (Contributed by Mario Carneiro, 2-Jan-2017.)
|
     
     ![]_ ]_](_urbrack.gif)   |
|
Theorem | csbie2t 3117* |
Conversion of implicit substitution to explicit substitution into a
class (closed form of csbie2 3118). (Contributed by NM, 3-Sep-2007.)
(Revised by Mario Carneiro, 13-Oct-2016.)
|
           ![]_ ]_](_urbrack.gif)   ![]_ ]_](_urbrack.gif)   |
|
Theorem | csbie2 3118* |
Conversion of implicit substitution to explicit substitution into a
class. (Contributed by NM, 27-Aug-2007.)
|
    
 ![]_ ]_](_urbrack.gif)   ![]_ ]_](_urbrack.gif)
 |
|
Theorem | csbie2g 3119* |
Conversion of implicit substitution to explicit class substitution.
This version of sbcie 3009 avoids a disjointness condition on and
by
substituting twice. (Contributed by Mario Carneiro,
11-Nov-2016.)
|
  
 
  ![]_ ]_](_urbrack.gif)   |
|
Theorem | sbcnestgf 3120 |
Nest the composition of two substitutions. (Contributed by Mario
Carneiro, 11-Nov-2016.)
|
          ![]. ].](_drbrack.gif)   ![]. ].](_drbrack.gif)    ![]_ ]_](_urbrack.gif)  ![]. ].](_drbrack.gif)    |
|
Theorem | csbnestgf 3121 |
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 3122* |
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 3123* |
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 3124 |
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 3125* |
Idempotent law for class substitutions. (Contributed by NM,
1-Mar-2008.)
|
   ![]_ ]_](_urbrack.gif)   ![]_ ]_](_urbrack.gif)   ![]_ ]_](_urbrack.gif)   |
|
Theorem | sbcco3g 3126* |
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 3127* |
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 3128* |
Special case related to rspsbc 3057. (Contributed by NM, 10-Dec-2005.)
(Proof shortened by Eric Schmidt, 17-Jan-2007.)
|
      ![]_ ]_](_urbrack.gif)   |
|
Theorem | sbnfc2 3129* |
Two ways of expressing " is (effectively) not free in ."
(Contributed by Mario Carneiro, 14-Oct-2016.)
|
       
 ![]_ ]_](_urbrack.gif)   ![]_ ]_](_urbrack.gif)   |
|
Theorem | csbabg 3130* |
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 3131 |
A more general version of cbvralf 2707 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 3132 |
A more general version of cbvrexf 2708 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 3133 |
A more general version of cbvreuv 2717 that has no distinct variable
rextrictions. Changes bound variables using implicit substitution.
(Contributed by Andrew Salmon, 13-Jul-2011.)
|
        
 
    
   |
|
Theorem | cbvrabcsf 3134 |
A more general version of cbvrab 2747 with no distinct variable
restrictions. (Contributed by Andrew Salmon, 13-Jul-2011.)
|
        
 
     
  |
|
Theorem | cbvralv2 3135* |
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 3136* |
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 3137* |
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 3138 |
Extend class notation to include class difference (read: " minus
").
|

  |
|
Syntax | cun 3139 |
Extend class notation to include union of two classes (read: "
union ").
|

  |
|
Syntax | cin 3140 |
Extend class notation to include the intersection of two classes (read:
" intersect
").
|

  |
|
Syntax | wss 3141 |
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 3142* |
Soundness justification theorem for df-dif 3143. (Contributed by Rodolfo
Medina, 27-Apr-2010.) (Proof shortened by Andrew Salmon,
9-Jul-2011.)
|
         |
|
Definition | df-dif 3143* |
Define class difference, also called relative complement. Definition
5.12 of [TakeutiZaring] p. 20.
Contrast this operation with union
  (df-un 3145) and intersection   (df-in 3147).
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 3144* |
Soundness justification theorem for df-un 3145. (Contributed by Rodolfo
Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon,
9-Jul-2011.)
|
 
 
 
   |
|
Definition | df-un 3145* |
Define the union of two classes. Definition 5.6 of [TakeutiZaring]
p. 16. Contrast this operation with difference  
(df-dif 3143) and intersection   (df-in 3147). (Contributed
by NM, 23-Aug-1993.)
|
 
 
   |
|
Theorem | injust 3146* |
Soundness justification theorem for df-in 3147. (Contributed by Rodolfo
Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon,
9-Jul-2011.)
|
 
 
 
   |
|
Definition | df-in 3147* |
Define the intersection of two classes. Definition 5.6 of
[TakeutiZaring] p. 16. Contrast
this operation with union
  (df-un 3145) and difference   (df-dif 3143).
(Contributed by NM, 29-Apr-1994.)
|
 
 
   |
|
Theorem | dfin5 3148* |
Alternate definition for the intersection of two classes. (Contributed
by NM, 6-Jul-2005.)
|
 
   |
|
Theorem | dfdif2 3149* |
Alternate definition of class difference. (Contributed by NM,
25-Mar-2004.)
|
 
   |
|
Theorem | eldif 3150 |
Expansion of membership in a class difference. (Contributed by NM,
29-Apr-1994.)
|
       |
|
Theorem | eldifd 3151 |
If a class is in one class and not another, it is also in their
difference. One-way deduction form of eldif 3150. (Contributed by David
Moews, 1-May-2017.)
|
    

   |
|
Theorem | eldifad 3152 |
If a class is in the difference of two classes, it is also in the
minuend. One-way deduction form of eldif 3150. (Contributed by David
Moews, 1-May-2017.)
|
       |
|
Theorem | eldifbd 3153 |
If a class is in the difference of two classes, it is not in the
subtrahend. One-way deduction form of eldif 3150. (Contributed by David
Moews, 1-May-2017.)
|
       |
|
2.1.12 Subclasses and subsets
|
|
Definition | df-ss 3154 |
Define the subclass relationship. Exercise 9 of [TakeutiZaring] p. 18.
Note that (proved in ssid 3187). For a more traditional
definition, but requiring a dummy variable, see dfss2 3156. Other possible
definitions are given by dfss3 3157, ssequn1 3317, ssequn2 3320, and sseqin2 3366.
(Contributed by NM, 27-Apr-1994.)
|
     |
|
Theorem | dfss 3155 |
Variant of subclass definition df-ss 3154. (Contributed by NM,
3-Sep-2004.)
|
     |
|
Theorem | dfss2 3156* |
Alternate definition of the subclass relationship between two classes.
Definition 5.9 of [TakeutiZaring]
p. 17. (Contributed by NM,
8-Jan-2002.)
|
   
   |
|
Theorem | dfss3 3157* |
Alternate definition of subclass relationship. (Contributed by NM,
14-Oct-1999.)
|
    |
|
Theorem | dfss2f 3158 |
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 3159 |
Equivalence for subclass relation, using bound-variable hypotheses
instead of distinct variable conditions. (Contributed by NM,
20-Mar-2004.)
|
     
  |
|
Theorem | nfss 3160 |
If is not free in and , it is not free in .
(Contributed by NM, 27-Dec-1996.)
|
      |
|
Theorem | ssel 3161 |
Membership relationships follow from a subclass relationship.
(Contributed by NM, 5-Aug-1993.)
|
     |
|
Theorem | ssel2 3162 |
Membership relationships follow from a subclass relationship.
(Contributed by NM, 7-Jun-2004.)
|
     |
|
Theorem | sseli 3163 |
Membership inference from subclass relationship. (Contributed by NM,
5-Aug-1993.)
|
   |
|
Theorem | sselii 3164 |
Membership inference from subclass relationship. (Contributed by NM,
31-May-1999.)
|
 |
|
Theorem | sselid 3165 |
Membership inference from subclass relationship. (Contributed by NM,
25-Jun-2014.)
|
     |
|
Theorem | sseld 3166 |
Membership deduction from subclass relationship. (Contributed by NM,
15-Nov-1995.)
|
   
   |
|
Theorem | sselda 3167 |
Membership deduction from subclass relationship. (Contributed by NM,
26-Jun-2014.)
|
       |
|
Theorem | sseldd 3168 |
Membership inference from subclass relationship. (Contributed by NM,
14-Dec-2004.)
|
       |
|
Theorem | ssneld 3169 |
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 3170 |
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 3171* |
Inference based on subclass definition. (Contributed by NM,
5-Aug-1993.)
|
   |
|
Theorem | ssrd 3172 |
Deduction based on subclass definition. (Contributed by Thierry Arnoux,
8-Mar-2017.)
|
       
  
  |
|
Theorem | ssrdv 3173* |
Deduction based on subclass definition. (Contributed by NM,
15-Nov-1995.)
|
 
  
  |
|
Theorem | sstr2 3174 |
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 3175 |
Transitivity of subclasses. Theorem 6 of [Suppes] p. 23. (Contributed by
NM, 5-Sep-2003.)
|
     |
|
Theorem | sstri 3176 |
Subclass transitivity inference. (Contributed by NM, 5-May-2000.)
|
 |
|
Theorem | sstrd 3177 |
Subclass transitivity deduction. (Contributed by NM, 2-Jun-2004.)
|
       |
|
Theorem | sstrid 3178 |
Subclass transitivity deduction. (Contributed by NM, 6-Feb-2014.)
|
     |
|
Theorem | sstrdi 3179 |
Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim,
3-Jun-2011.)
|
     |
|
Theorem | sylan9ss 3180 |
A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.)
(Proof shortened by Andrew Salmon, 14-Jun-2011.)
|
         |
|
Theorem | sylan9ssr 3181 |
A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.)
|
         |
|
Theorem | eqss 3182 |
The subclass relationship is antisymmetric. Compare Theorem 4 of
[Suppes] p. 22. (Contributed by NM,
5-Aug-1993.)
|
 
   |
|
Theorem | eqssi 3183 |
Infer equality from two subclass relationships. Compare Theorem 4 of
[Suppes] p. 22. (Contributed by NM,
9-Sep-1993.)
|
 |
|
Theorem | eqssd 3184 |
Equality deduction from two subclass relationships. Compare Theorem 4
of [Suppes] p. 22. (Contributed by NM,
27-Jun-2004.)
|
       |
|
Theorem | eqrd 3185 |
Deduce equality of classes from equivalence of membership. (Contributed
by Thierry Arnoux, 21-Mar-2017.)
|
       
     |
|
Theorem | eqelssd 3186* |
Equality deduction from subclass relationship and membership.
(Contributed by AV, 21-Aug-2022.)
|
    
    |
|
Theorem | ssid 3187 |
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 3188 |
Weakening of ssid 3187. (Contributed by BJ, 1-Sep-2022.)
|
   |
|
Theorem | ssv 3189 |
Any class is a subclass of the universal class. (Contributed by NM,
31-Oct-1995.)
|
 |
|
Theorem | sseq1 3190 |
Equality theorem for subclasses. (Contributed by NM, 5-Aug-1993.) (Proof
shortened by Andrew Salmon, 21-Jun-2011.)
|
     |
|
Theorem | sseq2 3191 |
Equality theorem for the subclass relationship. (Contributed by NM,
25-Jun-1998.)
|
     |
|
Theorem | sseq12 3192 |
Equality theorem for the subclass relationship. (Contributed by NM,
31-May-1999.)
|
   
   |
|
Theorem | sseq1i 3193 |
An equality inference for the subclass relationship. (Contributed by
NM, 18-Aug-1993.)
|

  |
|
Theorem | sseq2i 3194 |
An equality inference for the subclass relationship. (Contributed by
NM, 30-Aug-1993.)
|

  |
|
Theorem | sseq12i 3195 |
An equality inference for the subclass relationship. (Contributed by
NM, 31-May-1999.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
|
   |
|
Theorem | sseq1d 3196 |
An equality deduction for the subclass relationship. (Contributed by
NM, 14-Aug-1994.)
|
   
   |
|
Theorem | sseq2d 3197 |
An equality deduction for the subclass relationship. (Contributed by
NM, 14-Aug-1994.)
|
   
   |
|
Theorem | sseq12d 3198 |
An equality deduction for the subclass relationship. (Contributed by
NM, 31-May-1999.)
|
     
   |
|
Theorem | eqsstri 3199 |
Substitution of equality into a subclass relationship. (Contributed by
NM, 16-Jul-1995.)
|
 |
|
Theorem | eqsstrri 3200 |
Substitution of equality into a subclass relationship. (Contributed by
NM, 19-Oct-1999.)
|
 |