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Theorem List for Intuitionistic Logic Explorer - 3001-3100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsseq2d 3001 An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C  C_  A  <->  C  C_  B ) )
 
Theoremsseq12d 3002 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 3003 Substitution of equality into a subclass relationship. (Contributed by NM, 16-Jul-1995.)
 |-  A  =  B   &    |-  B  C_  C   =>    |-  A  C_  C
 
Theoremeqsstr3i 3004 Substitution of equality into a subclass relationship. (Contributed by NM, 19-Oct-1999.)
 |-  B  =  A   &    |-  B  C_  C   =>    |-  A  C_  C
 
Theoremsseqtri 3005 Substitution of equality into a subclass relationship. (Contributed by NM, 28-Jul-1995.)
 |-  A  C_  B   &    |-  B  =  C   =>    |-  A  C_  C
 
Theoremsseqtr4i 3006 Substitution of equality into a subclass relationship. (Contributed by NM, 4-Apr-1995.)
 |-  A  C_  B   &    |-  C  =  B   =>    |-  A  C_  C
 
Theoremeqsstrd 3007 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 3008 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 3009 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 3010 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 3011 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 3012 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 3013 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 3014 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 3015 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 3016 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 3017 B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
 |-  A  =  B   &    |-  ( ph  ->  B  C_  C )   =>    |-  ( ph  ->  A  C_  C )
 
Theoremsyl5eqssr 3018 B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
 |-  B  =  A   &    |-  ( ph  ->  B  C_  C )   =>    |-  ( ph  ->  A  C_  C )
 
Theoremsyl6sseq 3019 A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
 |-  ( ph  ->  A  C_  B )   &    |-  B  =  C   =>    |-  ( ph  ->  A  C_  C )
 
Theoremsyl6sseqr 3020 A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
 |-  ( ph  ->  A  C_  B )   &    |-  C  =  B   =>    |-  ( ph  ->  A  C_  C )
 
Theoremsyl5sseq 3021 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  B  C_  A   &    |-  ( ph  ->  A  =  C )   =>    |-  ( ph  ->  B 
 C_  C )
 
Theoremsyl5sseqr 3022 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  B  C_  A   &    |-  ( ph  ->  C  =  A )   =>    |-  ( ph  ->  B 
 C_  C )
 
Theoremsyl6eqss 3023 A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  A  =  B )   &    |-  B  C_  C   =>    |-  ( ph  ->  A  C_  C )
 
Theoremsyl6eqssr 3024 A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  B  =  A )   &    |-  B  C_  C   =>    |-  ( ph  ->  A  C_  C )
 
Theoremeqimss 3025 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 3026 Equality implies the subclass relation. (Contributed by NM, 23-Nov-2003.)
 |-  ( B  =  A  ->  A  C_  B )
 
Theoremeqimssi 3027 Infer subclass relationship from equality. (Contributed by NM, 6-Jan-2007.)
 |-  A  =  B   =>    |-  A  C_  B
 
Theoremeqimss2i 3028 Infer subclass relationship from equality. (Contributed by NM, 7-Jan-2007.)
 |-  A  =  B   =>    |-  B  C_  A
 
Theoremnssne1 3029 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 3030 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 )
 
Theoremnssr 3031* Negation of subclass relationship. One direction of Exercise 13 of [TakeutiZaring] p. 18. (Contributed by Jim Kingdon, 15-Jul-2018.)
 |-  ( E. x ( x  e.  A  /\  -.  x  e.  B ) 
 ->  -.  A  C_  B )
 
Theoremssralv 3032* 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 3033* 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 3034* 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 ) ) )
 
Theoremrexss 3035* Restricted existential quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  ( A  C_  B  ->  ( E. x  e.  A  ph  <->  E. x  e.  B  ( x  e.  A  /\  ph ) ) )
 
Theoremss2ab 3036 Class abstractions in a subclass relationship. (Contributed by NM, 3-Jul-1994.)
 |-  ( { x  |  ph
 }  C_  { x  |  ps }  <->  A. x ( ph  ->  ps ) )
 
Theoremabss 3037* Class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)
 |-  ( { x  |  ph
 }  C_  A  <->  A. x ( ph  ->  x  e.  A ) )
 
Theoremssab 3038* Subclass of a class abstraction. (Contributed by NM, 16-Aug-2006.)
 |-  ( A  C_  { x  |  ph }  <->  A. x ( x  e.  A  ->  ph )
 )
 
Theoremssabral 3039* The relation for a subclass of a class abstraction is equivalent to restricted quantification. (Contributed by NM, 6-Sep-2006.)
 |-  ( A  C_  { x  |  ph }  <->  A. x  e.  A  ph )
 
Theoremss2abi 3040 Inference of abstraction subclass from implication. (Contributed by NM, 31-Mar-1995.)
 |-  ( ph  ->  ps )   =>    |-  { x  |  ph }  C_  { x  |  ps }
 
Theoremss2abdv 3041* Deduction of abstraction subclass from implication. (Contributed by NM, 29-Jul-2011.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  { x  |  ps }  C_ 
 { x  |  ch } )
 
Theoremabssdv 3042* Deduction of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)
 |-  ( ph  ->  ( ps  ->  x  e.  A ) )   =>    |-  ( ph  ->  { x  |  ps }  C_  A )
 
Theoremabssi 3043* Inference of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)
 |-  ( ph  ->  x  e.  A )   =>    |- 
 { x  |  ph } 
 C_  A
 
Theoremss2rab 3044 Restricted abstraction classes in a subclass relationship. (Contributed by NM, 30-May-1999.)
 |-  ( { x  e.  A  |  ph }  C_  { x  e.  A  |  ps }  <->  A. x  e.  A  ( ph  ->  ps )
 )
 
Theoremrabss 3045* Restricted class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)
 |-  ( { x  e.  A  |  ph }  C_  B 
 <-> 
 A. x  e.  A  ( ph  ->  x  e.  B ) )
 
Theoremssrab 3046* Subclass of a restricted class abstraction. (Contributed by NM, 16-Aug-2006.)
 |-  ( B  C_  { x  e.  A  |  ph }  <->  ( B  C_  A  /\  A. x  e.  B  ph ) )
 
Theoremssrabdv 3047* Subclass of a restricted class abstraction (deduction rule). (Contributed by NM, 31-Aug-2006.)
 |-  ( ph  ->  B  C_  A )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  ps )   =>    |-  ( ph  ->  B  C_ 
 { x  e.  A  |  ps } )
 
Theoremrabssdv 3048* Subclass of a restricted class abstraction (deduction rule). (Contributed by NM, 2-Feb-2015.)
 |-  ( ( ph  /\  x  e.  A  /\  ps )  ->  x  e.  B )   =>    |-  ( ph  ->  { x  e.  A  |  ps }  C_  B )
 
Theoremss2rabdv 3049* Deduction of restricted abstraction subclass from implication. (Contributed by NM, 30-May-2006.)
 |-  ( ( ph  /\  x  e.  A )  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  { x  e.  A  |  ps }  C_  { x  e.  A  |  ch }
 )
 
Theoremss2rabi 3050 Inference of restricted abstraction subclass from implication. (Contributed by NM, 14-Oct-1999.)
 |-  ( x  e.  A  ->  ( ph  ->  ps )
 )   =>    |- 
 { x  e.  A  |  ph }  C_  { x  e.  A  |  ps }
 
Theoremrabss2 3051* Subclass law for restricted abstraction. (Contributed by NM, 18-Dec-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( A  C_  B  ->  { x  e.  A  |  ph }  C_  { x  e.  B  |  ph } )
 
Theoremssab2 3052* Subclass relation for the restriction of a class abstraction. (Contributed by NM, 31-Mar-1995.)
 |- 
 { x  |  ( x  e.  A  /\  ph ) }  C_  A
 
Theoremssrab2 3053* Subclass relation for a restricted class. (Contributed by NM, 19-Mar-1997.)
 |- 
 { x  e.  A  |  ph }  C_  A
 
Theoremssrabeq 3054* If the restricting class of a restricted class abstraction is a subset of this restricted class abstraction, it is equal to this restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.)
 |-  ( V  C_  { x  e.  V  |  ph }  <->  V  =  { x  e.  V  |  ph
 } )
 
Theoremrabssab 3055 A restricted class is a subclass of the corresponding unrestricted class. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |- 
 { x  e.  A  |  ph }  C_  { x  |  ph }
 
Theoremuniiunlem 3056* A subset relationship useful for converting union to indexed union using dfiun2 or dfiun2g and intersection to indexed intersection using dfiin2 . (Contributed by NM, 5-Oct-2006.) (Proof shortened by Mario Carneiro, 26-Sep-2015.)
 |-  ( A. x  e.  A  B  e.  D  ->  ( A. x  e.  A  B  e.  C  <->  { y  |  E. x  e.  A  y  =  B }  C_  C ) )
 
Theoremdfpss2 3057 Alternate definition of proper subclass. (Contributed by NM, 7-Feb-1996.)
 |-  ( A  C.  B  <->  ( A  C_  B  /\  -.  A  =  B ) )
 
Theoremdfpss3 3058 Alternate definition of proper subclass. (Contributed by NM, 7-Feb-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( A  C.  B  <->  ( A  C_  B  /\  -.  B  C_  A ) )
 
Theorempsseq1 3059 Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.)
 |-  ( A  =  B  ->  ( A  C.  C  <->  B  C.  C ) )
 
Theorempsseq2 3060 Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.)
 |-  ( A  =  B  ->  ( C  C.  A  <->  C  C.  B ) )
 
Theorempsseq1i 3061 An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
 |-  A  =  B   =>    |-  ( A  C.  C 
 <->  B  C.  C )
 
Theorempsseq2i 3062 An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
 |-  A  =  B   =>    |-  ( C  C.  A 
 <->  C  C.  B )
 
Theorempsseq12i 3063 An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
 |-  A  =  B   &    |-  C  =  D   =>    |-  ( A  C.  C  <->  B  C.  D )
 
Theorempsseq1d 3064 An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  C.  C  <->  B  C.  C ) )
 
Theorempsseq2d 3065 An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C  C.  A  <->  C  C.  B ) )
 
Theorempsseq12d 3066 An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A  C.  C  <->  B  C.  D ) )
 
Theorempssss 3067 A proper subclass is a subclass. Theorem 10 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)
 |-  ( A  C.  B  ->  A 
 C_  B )
 
Theorempssne 3068 Two classes in a proper subclass relationship are not equal. (Contributed by NM, 16-Feb-2015.)
 |-  ( A  C.  B  ->  A  =/=  B )
 
Theorempssssd 3069 Deduce subclass from proper subclass. (Contributed by NM, 29-Feb-1996.)
 |-  ( ph  ->  A  C.  B )   =>    |-  ( ph  ->  A  C_  B )
 
Theorempssned 3070 Proper subclasses are unequal. Deduction form of pssne 3068. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C.  B )   =>    |-  ( ph  ->  A  =/=  B )
 
Theoremsspssr 3071 Subclass in terms of proper subclass. (Contributed by Jim Kingdon, 16-Jul-2018.)
 |-  ( ( A  C.  B  \/  A  =  B )  ->  A  C_  B )
 
Theorempssirr 3072 Proper subclass is irreflexive. Theorem 7 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)
 |- 
 -.  A  C.  A
 
Theorempssn2lp 3073 Proper subclass has no 2-cycle loops. Compare Theorem 8 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |- 
 -.  ( A  C.  B  /\  B  C.  A )
 
Theoremsspsstrir 3074 Two ways of stating trichotomy with respect to inclusion. (Contributed by Jim Kingdon, 16-Jul-2018.)
 |-  ( ( A  C.  B  \/  A  =  B  \/  B  C.  A )  ->  ( A  C_  B  \/  B  C_  A )
 )
 
Theoremssnpss 3075 Partial trichotomy law for subclasses. (Contributed by NM, 16-May-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( A  C_  B  ->  -.  B  C.  A )
 
Theoremsspssn 3076 Like pssn2lp 3073 but for subset and proper subset. (Contributed by Jim Kingdon, 17-Jul-2018.)
 |- 
 -.  ( A  C_  B  /\  B  C.  A )
 
Theorempsstr 3077 Transitive law for proper subclass. Theorem 9 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)
 |-  ( ( A  C.  B  /\  B  C.  C ) 
 ->  A  C.  C )
 
Theoremsspsstr 3078 Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.)
 |-  ( ( A  C_  B  /\  B  C.  C ) 
 ->  A  C.  C )
 
Theorempsssstr 3079 Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.)
 |-  ( ( A  C.  B  /\  B  C_  C )  ->  A  C.  C )
 
Theorempsstrd 3080 Proper subclass inclusion is transitive. Deduction form of psstr 3077. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C.  B )   &    |-  ( ph  ->  B  C.  C )   =>    |-  ( ph  ->  A  C.  C )
 
Theoremsspsstrd 3081 Transitivity involving subclass and proper subclass inclusion. Deduction form of sspsstr 3078. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  B  C.  C )   =>    |-  ( ph  ->  A  C.  C )
 
Theorempsssstrd 3082 Transitivity involving subclass and proper subclass inclusion. Deduction form of psssstr 3079. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C.  B )   &    |-  ( ph  ->  B 
 C_  C )   =>    |-  ( ph  ->  A  C.  C )
 
2.1.13  The difference, union, and intersection of two classes
 
2.1.13.1  The difference of two classes
 
Theoremdifeq1 3083 Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( A  =  B  ->  ( A  \  C )  =  ( B  \  C ) )
 
Theoremdifeq2 3084 Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( A  =  B  ->  ( C  \  A )  =  ( C  \  B ) )
 
Theoremdifeq12 3085 Equality theorem for class difference. (Contributed by FL, 31-Aug-2009.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  \  C )  =  ( B  \  D ) )
 
Theoremdifeq1i 3086 Inference adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.)
 |-  A  =  B   =>    |-  ( A  \  C )  =  ( B  \  C )
 
Theoremdifeq2i 3087 Inference adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)
 |-  A  =  B   =>    |-  ( C  \  A )  =  ( C  \  B )
 
Theoremdifeq12i 3088 Equality inference for class difference. (Contributed by NM, 29-Aug-2004.)
 |-  A  =  B   &    |-  C  =  D   =>    |-  ( A  \  C )  =  ( B  \  D )
 
Theoremdifeq1d 3089 Deduction adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  \  C )  =  ( B  \  C ) )
 
Theoremdifeq2d 3090 Deduction adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C  \  A )  =  ( C  \  B ) )
 
Theoremdifeq12d 3091 Equality deduction for class difference. (Contributed by FL, 29-May-2014.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A  \  C )  =  ( B  \  D ) )
 
Theoremdifeqri 3092* Inference from membership to difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( ( x  e.  A  /\  -.  x  e.  B )  <->  x  e.  C )   =>    |-  ( A  \  B )  =  C
 
Theoremnfdif 3093 Bound-variable hypothesis builder for class difference. (Contributed by NM, 3-Dec-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x ( A 
 \  B )
 
Theoremeldifi 3094 Implication of membership in a class difference. (Contributed by NM, 29-Apr-1994.)
 |-  ( A  e.  ( B  \  C )  ->  A  e.  B )
 
Theoremeldifn 3095 Implication of membership in a class difference. (Contributed by NM, 3-May-1994.)
 |-  ( A  e.  ( B  \  C )  ->  -.  A  e.  C )
 
Theoremelndif 3096 A set does not belong to a class excluding it. (Contributed by NM, 27-Jun-1994.)
 |-  ( A  e.  B  ->  -.  A  e.  ( C  \  B ) )
 
Theoremdifdif 3097 Double class difference. Exercise 11 of [TakeutiZaring] p. 22. (Contributed by NM, 17-May-1998.)
 |-  ( A  \  ( B  \  A ) )  =  A
 
Theoremdifss 3098 Subclass relationship for class difference. Exercise 14 of [TakeutiZaring] p. 22. (Contributed by NM, 29-Apr-1994.)
 |-  ( A  \  B )  C_  A
 
Theoremdifssd 3099 A difference of two classes is contained in the minuend. Deduction form of difss 3098. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  ( A  \  B )  C_  A )
 
Theoremdifss2 3100 If a class is contained in a difference, it is contained in the minuend. (Contributed by David Moews, 1-May-2017.)
 |-  ( A  C_  ( B  \  C )  ->  A  C_  B )
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