HomeHome Metamath Proof Explorer
Theorem List (p. 32 of 314)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21444)
  Hilbert Space Explorer  Hilbert Space Explorer
(21445-22967)
  Users' Mathboxes  Users' Mathboxes
(22968-31305)
 

Theorem List for Metamath Proof Explorer - 3101-3200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremra4csbela 3101* Special case related to ra4sbc 3030. (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 3102* 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 3103* 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 3104 A more general version of cbvralf 2728 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 3105 A more general version of cbvrexf 2729 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 3106 A more general version of cbvreuv 2736 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 3107 A more general version of cbvrab 2755 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 3108* 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 3109* 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 3110 Extend class notation to include class difference (read: " A minus  B").
 class  ( A  \  B )
 
Syntaxcun 3111 Extend class notation to include union of two classes (read: " A union  B").
 class  ( A  u.  B )
 
Syntaxcin 3112 Extend class notation to include the intersection of two classes (read: " A intersect  B").
 class  ( A  i^i  B )
 
Syntaxwss 3113 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 3114 Extend wff notation with proper subclass relation.
 wff  A  C.  B
 
Theoremdifjust 3115* Soundness justification theorem for df-dif 3116. (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 3116* Define class difference, also called relative complement. Definition 5.12 of [TakeutiZaring] p. 20. For example,  ( { 1 ,  3 }  \  { 1 ,  8 } )  =  {
3 } (ex-dif 20739). Contrast this operation with union  ( A  u.  B
) (df-un 3118) and intersection  ( A  i^i  B ) (df-in 3120). 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 3117* Soundness justification theorem for df-un 3118. (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 3118* Define the union of two classes. Definition 5.6 of [TakeutiZaring] p. 16. For example,  ( { 1 ,  3 }  u.  {
1 ,  8 } )  =  { 1 ,  3 ,  8 } (ex-un 20740). Contrast this operation with difference  ( A  \  B ) (df-dif 3116) and intersection  ( A  i^i  B ) (df-in 3120). For an alternate definition in terms of class difference, requiring no dummy variables, see dfun2 3365. For union defined in terms of intersection, see dfun3 3368. (Contributed by NM, 23-Aug-1993.)
 |-  ( A  u.  B )  =  { x  |  ( x  e.  A  \/  x  e.  B ) }
 
Theoreminjust 3119* Soundness justification theorem for df-in 3120. (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 3120* Define the intersection of two classes. Definition 5.6 of [TakeutiZaring] p. 16. For example,  ( { 1 ,  3 }  i^i  { 1 ,  8 } )  =  { 1 } (ex-in 20741). Contrast this operation with union  ( A  u.  B
) (df-un 3118) and difference  ( A  \  B ) (df-dif 3116). For alternate definitions in terms of class difference, requiring no dummy variables, see dfin2 3366 and dfin4 3370. For intersection defined in terms of union, see dfin3 3369. (Contributed by NM, 29-Apr-1994.)
 |-  ( A  i^i  B )  =  { x  |  ( x  e.  A  /\  x  e.  B ) }
 
Theoremdfin5 3121* 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 3122* Alternate definition of class difference. (Contributed by NM, 25-Mar-2004.)
 |-  ( A  \  B )  =  { x  e.  A  |  -.  x  e.  B }
 
Theoremeldif 3123 Expansion of membership in a class difference. (Contributed by NM, 29-Apr-1994.)
 |-  ( A  e.  ( B  \  C )  <->  ( A  e.  B  /\  -.  A  e.  C ) )
 
Theoremeldifd 3124 If a class is in one class and not another, it is also in their difference. One-way deduction form of eldif 3123. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  B )   &    |-  ( ph  ->  -.  A  e.  C )   =>    |-  ( ph  ->  A  e.  ( B  \  C ) )
 
Theoremeldifad 3125 If a class is in the difference of two classes, it is also in the minuend. One-way deduction form of eldif 3123. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  ( B  \  C ) )   =>    |-  ( ph  ->  A  e.  B )
 
Theoremeldifbd 3126 If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif 3123. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  ( B  \  C ) )   =>    |-  ( ph  ->  -.  A  e.  C )
 
2.1.12  Subclasses and subsets
 
Definitiondf-ss 3127 Define the subclass relationship. Exercise 9 of [TakeutiZaring] p. 18. For example,  { 1 ,  2 }  C_  { 1 ,  2 ,  3 } (ex-ss 20743). Note that  A  C_  A (proved in ssid 3158). Contrast this relationship with the relationship  A  C.  B (as will be defined in df-pss 3129). For a more traditional definition, but requiring a dummy variable, see dfss2 3130. Other possible definitions are given by dfss3 3131, dfss4 3364, sspss 3236, ssequn1 3306, ssequn2 3309, sseqin2 3349, and ssdif0 3474. (Contributed by NM, 27-Apr-1994.)
 |-  ( A  C_  B  <->  ( A  i^i  B )  =  A )
 
Theoremdfss 3128 Variant of subclass definition df-ss 3127. (Contributed by NM, 3-Sep-2004.)
 |-  ( A  C_  B  <->  A  =  ( A  i^i  B ) )
 
Definitiondf-pss 3129 Define proper subclass relationship between two classes. Definition 5.9 of [TakeutiZaring] p. 17. For example,  { 1 ,  2 }  C.  {
1 ,  2 ,  3 } (ex-pss 20744). Note that  -.  A  C.  A (proved in pssirr 3237). Contrast this relationship with the relationship  A 
C_  B (as defined in df-ss 3127). Other possible definitions are given by dfpss2 3222 and dfpss3 3223. (Contributed by NM, 7-Feb-1996.)
 |-  ( A  C.  B  <->  ( A  C_  B  /\  A  =/=  B ) )
 
Theoremdfss2 3130* 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 3131* Alternate definition of subclass relationship. (Contributed by NM, 14-Oct-1999.)
 |-  ( A  C_  B  <->  A. x  e.  A  x  e.  B )
 
Theoremdfss2f 3132 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 3133 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 3134 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 3135 Membership relationships follow from a subclass relationship. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  C_  B  ->  ( C  e.  A  ->  C  e.  B ) )
 
Theoremssel2 3136 Membership relationships follow from a subclass relationship. (Contributed by NM, 7-Jun-2004.)
 |-  ( ( A  C_  B  /\  C  e.  A )  ->  C  e.  B )
 
Theoremsseli 3137 Membership inference from subclass relationship. (Contributed by NM, 5-Aug-1993.)
 |-  A  C_  B   =>    |-  ( C  e.  A  ->  C  e.  B )
 
Theoremsselii 3138 Membership inference from subclass relationship. (Contributed by NM, 31-May-1999.)
 |-  A  C_  B   &    |-  C  e.  A   =>    |-  C  e.  B
 
Theoremsseldi 3139 Membership inference from subclass relationship. (Contributed by NM, 25-Jun-2014.)
 |-  A  C_  B   &    |-  ( ph  ->  C  e.  A )   =>    |-  ( ph  ->  C  e.  B )
 
Theoremsseld 3140 Membership deduction from subclass relationship. (Contributed by NM, 15-Nov-1995.)
 |-  ( ph  ->  A  C_  B )   =>    |-  ( ph  ->  ( C  e.  A  ->  C  e.  B ) )
 
Theoremsselda 3141 Membership deduction from subclass relationship. (Contributed by NM, 26-Jun-2014.)
 |-  ( ph  ->  A  C_  B )   =>    |-  ( ( ph  /\  C  e.  A )  ->  C  e.  B )
 
Theoremsseldd 3142 Membership inference from subclass relationship. (Contributed by NM, 14-Dec-2004.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  C  e.  A )   =>    |-  ( ph  ->  C  e.  B )
 
Theoremssneld 3143 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 3144 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 3145* Inference rule based on subclass definition. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  e.  A  ->  x  e.  B )   =>    |-  A  C_  B
 
Theoremssrdv 3146* Deduction rule based on subclass definition. (Contributed by NM, 15-Nov-1995.)
 |-  ( ph  ->  ( x  e.  A  ->  x  e.  B ) )   =>    |-  ( ph  ->  A  C_  B )
 
Theoremsstr2 3147 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 3148 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 3149 Subclass transitivity inference. (Contributed by NM, 5-May-2000.)
 |-  A  C_  B   &    |-  B  C_  C   =>    |-  A  C_  C
 
Theoremsstrd 3150 Subclass transitivity deduction. (Contributed by NM, 2-Jun-2004.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  B 
 C_  C )   =>    |-  ( ph  ->  A 
 C_  C )
 
Theoremsyl5ss 3151 Subclass transitivity deduction. (Contributed by NM, 6-Feb-2014.)
 |-  A  C_  B   &    |-  ( ph  ->  B 
 C_  C )   =>    |-  ( ph  ->  A 
 C_  C )
 
Theoremsyl6ss 3152 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  ( ph  ->  A  C_  B )   &    |-  B  C_  C   =>    |-  ( ph  ->  A  C_  C )
 
Theoremsylan9ss 3153 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 3154 A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ps  ->  B 
 C_  C )   =>    |-  ( ( ps 
 /\  ph )  ->  A  C_  C )
 
Theoremeqss 3155 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 3156 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 3157 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 )
 
Theoremssid 3158 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 3159 Any class is a subclass of the universal class. (Contributed by NM, 31-Oct-1995.)
 |-  A  C_  _V
 
Theoremsseq1 3160 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 3161 Equality theorem for the subclass relationship. (Contributed by NM, 25-Jun-1998.)
 |-  ( A  =  B  ->  ( C  C_  A  <->  C 
 C_  B ) )
 
Theoremsseq12 3162 Equality theorem for the subclass relationship. (Contributed by NM, 31-May-1999.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  C_  C 
 <->  B  C_  D )
 )
 
Theoremsseq1i 3163 An equality inference for the subclass relationship. (Contributed by NM, 18-Aug-1993.)
 |-  A  =  B   =>    |-  ( A  C_  C 
 <->  B  C_  C )
 
Theoremsseq2i 3164 An equality inference for the subclass relationship. (Contributed by NM, 30-Aug-1993.)
 |-  A  =  B   =>    |-  ( C  C_  A 
 <->  C  C_  B )
 
Theoremsseq12i 3165 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 3166 An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  C_  C  <->  B  C_  C ) )
 
Theoremsseq2d 3167 An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C  C_  A  <->  C  C_  B ) )
 
Theoremsseq12d 3168 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 3169 Substitution of equality into a subclass relationship. (Contributed by NM, 16-Jul-1995.)
 |-  A  =  B   &    |-  B  C_  C   =>    |-  A  C_  C
 
Theoremeqsstr3i 3170 Substitution of equality into a subclass relationship. (Contributed by NM, 19-Oct-1999.)
 |-  B  =  A   &    |-  B  C_  C   =>    |-  A  C_  C
 
Theoremsseqtri 3171 Substitution of equality into a subclass relationship. (Contributed by NM, 28-Jul-1995.)
 |-  A  C_  B   &    |-  B  =  C   =>    |-  A  C_  C
 
Theoremsseqtr4i 3172 Substitution of equality into a subclass relationship. (Contributed by NM, 4-Apr-1995.)
 |-  A  C_  B   &    |-  C  =  B   =>    |-  A  C_  C
 
Theoremeqsstrd 3173 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  B 
 C_  C )   =>    |-  ( ph  ->  A 
 C_  C )
 
Theoremeqsstr3d 3174 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
 |-  ( ph  ->  B  =  A )   &    |-  ( ph  ->  B 
 C_  C )   =>    |-  ( ph  ->  A 
 C_  C )
 
Theoremsseqtrd 3175 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  A 
 C_  C )
 
Theoremsseqtr4d 3176 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  C  =  B )   =>    |-  ( ph  ->  A 
 C_  C )
 
Theorem3sstr3i 3177 Substitution of equality in both sides of a subclass relationship. (Contributed by NM, 13-Jan-1996.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
 |-  A  C_  B   &    |-  A  =  C   &    |-  B  =  D   =>    |-  C  C_  D
 
Theorem3sstr4i 3178 Substitution of equality in both sides of a subclass relationship. (Contributed by NM, 13-Jan-1996.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
 |-  A  C_  B   &    |-  C  =  A   &    |-  D  =  B   =>    |-  C  C_  D
 
Theorem3sstr3g 3179 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)
 |-  ( ph  ->  A  C_  B )   &    |-  A  =  C   &    |-  B  =  D   =>    |-  ( ph  ->  C  C_  D )
 
Theorem3sstr4g 3180 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
 |-  ( ph  ->  A  C_  B )   &    |-  C  =  A   &    |-  D  =  B   =>    |-  ( ph  ->  C  C_  D )
 
Theorem3sstr3d 3181 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  A  =  C )   &    |-  ( ph  ->  B  =  D )   =>    |-  ( ph  ->  C  C_  D )
 
Theorem3sstr4d 3182 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 30-Nov-1995.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  C  =  A )   &    |-  ( ph  ->  D  =  B )   =>    |-  ( ph  ->  C  C_  D )
 
Theoremsyl5eqss 3183 B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
 |-  A  =  B   &    |-  ( ph  ->  B  C_  C )   =>    |-  ( ph  ->  A  C_  C )
 
Theoremsyl5eqssr 3184 B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
 |-  B  =  A   &    |-  ( ph  ->  B  C_  C )   =>    |-  ( ph  ->  A  C_  C )
 
Theoremsyl6sseq 3185 A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
 |-  ( ph  ->  A  C_  B )   &    |-  B  =  C   =>    |-  ( ph  ->  A  C_  C )
 
Theoremsyl6sseqr 3186 A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
 |-  ( ph  ->  A  C_  B )   &    |-  C  =  B   =>    |-  ( ph  ->  A  C_  C )
 
Theoremsyl5sseq 3187 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  B  C_  A   &    |-  ( ph  ->  A  =  C )   =>    |-  ( ph  ->  B 
 C_  C )
 
Theoremsyl5sseqr 3188 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  B  C_  A   &    |-  ( ph  ->  C  =  A )   =>    |-  ( ph  ->  B 
 C_  C )
 
Theoremsyl6eqss 3189 A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  A  =  B )   &    |-  B  C_  C   =>    |-  ( ph  ->  A  C_  C )
 
Theoremsyl6eqssr 3190 A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  B  =  A )   &    |-  B  C_  C   =>    |-  ( ph  ->  A  C_  C )
 
Theoremeqimss 3191 Equality implies the subclass relation. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
 |-  ( A  =  B  ->  A  C_  B )
 
Theoremeqimss2 3192 Equality implies the subclass relation. (Contributed by NM, 23-Nov-2003.)
 |-  ( B  =  A  ->  A  C_  B )
 
Theoremeqimssi 3193 Infer subclass relationship from equality. (Contributed by NM, 6-Jan-2007.)
 |-  A  =  B   =>    |-  A  C_  B
 
Theoremeqimss2i 3194 Infer subclass relationship from equality. (Contributed by NM, 7-Jan-2007.)
 |-  A  =  B   =>    |-  B  C_  A
 
Theoremnssne1 3195 Two classes are different if they don't include the same class. (Contributed by NM, 23-Apr-2015.)
 |-  ( ( A  C_  B  /\  -.  A  C_  C )  ->  B  =/=  C )
 
Theoremnssne2 3196 Two classes are different if they are not subclasses of the same class. (Contributed by NM, 23-Apr-2015.)
 |-  ( ( A  C_  C  /\  -.  B  C_  C )  ->  A  =/=  B )
 
Theoremnss 3197* Negation of subclass relationship. Exercise 13 of [TakeutiZaring] p. 18. (Contributed by NM, 25-Feb-1996.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
 |-  ( -.  A  C_  B 
 <-> 
 E. x ( x  e.  A  /\  -.  x  e.  B )
 )
 
Theoremssralv 3198* Quantification restricted to a subclass. (Contributed by NM, 11-Mar-2006.)
 |-  ( A  C_  B  ->  ( A. x  e.  B  ph  ->  A. x  e.  A  ph ) )
 
Theoremssrexv 3199* Existential quantification restricted to a subclass. (Contributed by NM, 11-Jan-2007.)
 |-  ( A  C_  B  ->  ( E. x  e.  A  ph  ->  E. x  e.  B  ph ) )
 
Theoremralss 3200* Restricted universal quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  ( A  C_  B  ->  ( A. x  e.  A  ph  <->  A. x  e.  B  ( x  e.  A  -> 
 ph ) ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31305
  Copyright terms: Public domain < Previous  Next >