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Theorem List for Metamath Proof Explorer - 3201-3300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsspsstr 3201 Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.)
 |-  ( ( A  C_  B  /\  B  C.  C )  ->  A  C.  C )
 
Theorempsssstr 3202 Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.)
 |-  ( ( A  C.  B  /\  B  C_  C )  ->  A  C.  C )
 
Theoremnpss 3203 A class is not a proper subclass of another iff it satisfies a one-directional form of eqss 3115. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( -.  A  C.  B 
 <->  ( A  C_  B  ->  A  =  B ) )
 
2.1.13  The difference, union, and intersection of two classes
 
Theoremdifeq1 3204 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 3205 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 3206 Equality theorem for class difference. (Contributed by FL, 31-Aug-2009.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  \  C )  =  ( B  \  D ) )
 
Theoremdifeq1i 3207 Inference adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.)
 |-  A  =  B   =>    |-  ( A  \  C )  =  ( B  \  C )
 
Theoremdifeq2i 3208 Inference adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)
 |-  A  =  B   =>    |-  ( C  \  A )  =  ( C  \  B )
 
Theoremdifeq12i 3209 Equality inference for class difference. (Contributed by NM, 29-Aug-2004.)
 |-  A  =  B   &    |-  C  =  D   =>    |-  ( A  \  C )  =  ( B  \  D )
 
Theoremdifeq1d 3210 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 3211 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 3212 Equality deduction for class difference. (Contributed by FL, 29-May-2014.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A  \  C )  =  ( B  \  D ) )
 
Theoremdifeqri 3213* 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 3214 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 3215 Implication of membership in a class difference. (Contributed by NM, 29-Apr-1994.)
 |-  ( A  e.  ( B  \  C )  ->  A  e.  B )
 
Theoremeldifn 3216 Implication of membership in a class difference. (Contributed by NM, 3-May-1994.)
 |-  ( A  e.  ( B  \  C )  ->  -.  A  e.  C )
 
Theoremelndif 3217 A set does not belong to a class excluding it. (Contributed by NM, 27-Jun-1994.)
 |-  ( A  e.  B  ->  -.  A  e.  ( C  \  B ) )
 
Theoremneldif 3218 Implication of membership in a class difference. (Contributed by NM, 28-Jun-1994.)
 |-  ( ( A  e.  B  /\  -.  A  e.  ( B  \  C ) )  ->  A  e.  C )
 
Theoremdifdif 3219 Double class difference. Exercise 11 of [TakeutiZaring] p. 22. (Contributed by NM, 17-May-1998.)
 |-  ( A  \  ( B  \  A ) )  =  A
 
Theoremdifss 3220 Subclass relationship for class difference. Exercise 14 of [TakeutiZaring] p. 22. (Contributed by NM, 29-Apr-1994.)
 |-  ( A  \  B )  C_  A
 
Theoremssdifss 3221 Preservation of a subclass relationship by class difference. (Contributed by NM, 15-Feb-2007.)
 |-  ( A  C_  B  ->  ( A  \  C )  C_  B )
 
Theoremddif 3222 Double complement under universal class. Exercise 4.10(s) of [Mendelson] p. 231. (Contributed by NM, 8-Jan-2002.)
 |-  ( _V  \  ( _V  \  A ) )  =  A
 
Theoremssconb 3223 Contraposition law for subsets. (Contributed by NM, 22-Mar-1998.)
 |-  ( ( A  C_  C  /\  B  C_  C )  ->  ( A  C_  ( C  \  B )  <->  B  C_  ( C  \  A ) ) )
 
Theoremsscon 3224 Contraposition law for subsets. Exercise 15 of [TakeutiZaring] p. 22. (Contributed by NM, 22-Mar-1998.)
 |-  ( A  C_  B  ->  ( C  \  B )  C_  ( C  \  A ) )
 
Theoremssdif 3225 Difference law for subsets. (Contributed by NM, 28-May-1998.)
 |-  ( A  C_  B  ->  ( A  \  C )  C_  ( B  \  C ) )
 
Theoremelun 3226 Expansion of membership in class union. Theorem 12 of [Suppes] p. 25. (Contributed by NM, 7-Aug-1994.)
 |-  ( A  e.  ( B  u.  C )  <->  ( A  e.  B  \/  A  e.  C ) )
 
Theoremuneqri 3227* Inference from membership to union. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( x  e.  A  \/  x  e.  B )  <->  x  e.  C )   =>    |-  ( A  u.  B )  =  C
 
Theoremunidm 3228 Idempotent law for union of classes. Theorem 23 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  u.  A )  =  A
 
Theoremuncom 3229 Commutative law for union of classes. Exercise 6 of [TakeutiZaring] p. 17. (Contributed by NM, 25-Jun-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( A  u.  B )  =  ( B  u.  A )
 
Theoremequncom 3230 If a class equals the union of two other classes, then it equals the union of those two classes commuted. equncom 3230 was automatically derived from equncomVD 27431 using the tools program translatewithout_overwriting.cmd and minimizing. (Contributed by Alan Sare, 18-Feb-2012.)
 |-  ( A  =  ( B  u.  C )  <->  A  =  ( C  u.  B ) )
 
Theoremequncomi 3231 Inference form of equncom 3230. equncomi 3231 was automatically derived from equncomiVD 27432 using the tools program translatewithout_overwriting.cmd and minimizing. (Contributed by Alan Sare, 18-Feb-2012.)
 |-  A  =  ( B  u.  C )   =>    |-  A  =  ( C  u.  B )
 
Theoremuneq1 3232 Equality theorem for union of two classes. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  =  B  ->  ( A  u.  C )  =  ( B  u.  C ) )
 
Theoremuneq2 3233 Equality theorem for the union of two classes. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  =  B  ->  ( C  u.  A )  =  ( C  u.  B ) )
 
Theoremuneq12 3234 Equality theorem for union of two classes. (Contributed by NM, 29-Mar-1998.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  u.  C )  =  ( B  u.  D ) )
 
Theoremuneq1i 3235 Inference adding union to the right in a class equality. (Contributed by NM, 30-Aug-1993.)
 |-  A  =  B   =>    |-  ( A  u.  C )  =  ( B  u.  C )
 
Theoremuneq2i 3236 Inference adding union to the left in a class equality. (Contributed by NM, 30-Aug-1993.)
 |-  A  =  B   =>    |-  ( C  u.  A )  =  ( C  u.  B )
 
Theoremuneq12i 3237 Equality inference for union of two classes. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
 |-  A  =  B   &    |-  C  =  D   =>    |-  ( A  u.  C )  =  ( B  u.  D )
 
Theoremuneq1d 3238 Deduction adding union to the right in a class equality. (Contributed by NM, 29-Mar-1998.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  u.  C )  =  ( B  u.  C ) )
 
Theoremuneq2d 3239 Deduction adding union to the left in a class equality. (Contributed by NM, 29-Mar-1998.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C  u.  A )  =  ( C  u.  B ) )
 
Theoremuneq12d 3240 Equality deduction for union of two classes. (Contributed by NM, 29-Sep-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A  u.  C )  =  ( B  u.  D ) )
 
Theoremnfun 3241 Bound-variable hypothesis builder for the union of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x ( A  u.  B )
 
Theoremunass 3242 Associative law for union of classes. Exercise 8 of [TakeutiZaring] p. 17. (Contributed by NM, 3-May-1994.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( ( A  u.  B )  u.  C )  =  ( A  u.  ( B  u.  C ) )
 
Theoremun12 3243 A rearrangement of union. (Contributed by NM, 12-Aug-2004.)
 |-  ( A  u.  ( B  u.  C ) )  =  ( B  u.  ( A  u.  C ) )
 
Theoremun23 3244 A rearrangement of union. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( ( A  u.  B )  u.  C )  =  ( ( A  u.  C )  u.  B )
 
Theoremun4 3245 A rearrangement of the union of 4 classes. (Contributed by NM, 12-Aug-2004.)
 |-  ( ( A  u.  B )  u.  ( C  u.  D ) )  =  ( ( A  u.  C )  u.  ( B  u.  D ) )
 
Theoremunundi 3246 Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
 |-  ( A  u.  ( B  u.  C ) )  =  ( ( A  u.  B )  u.  ( A  u.  C ) )
 
Theoremunundir 3247 Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
 |-  ( ( A  u.  B )  u.  C )  =  ( ( A  u.  C )  u.  ( B  u.  C ) )
 
Theoremssun1 3248 Subclass relationship for union of classes. Theorem 25 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)
 |-  A  C_  ( A  u.  B )
 
Theoremssun2 3249 Subclass relationship for union of classes. (Contributed by NM, 30-Aug-1993.)
 |-  A  C_  ( B  u.  A )
 
Theoremssun3 3250 Subclass law for union of classes. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  C_  B  ->  A  C_  ( B  u.  C ) )
 
Theoremssun4 3251 Subclass law for union of classes. (Contributed by NM, 14-Aug-1994.)
 |-  ( A  C_  B  ->  A  C_  ( C  u.  B ) )
 
Theoremelun1 3252 Membership law for union of classes. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  e.  B  ->  A  e.  ( B  u.  C ) )
 
Theoremelun2 3253 Membership law for union of classes. (Contributed by NM, 30-Aug-1993.)
 |-  ( A  e.  B  ->  A  e.  ( C  u.  B ) )
 
Theoremunss1 3254 Subclass law for union of classes. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( A  C_  B  ->  ( A  u.  C )  C_  ( B  u.  C ) )
 
Theoremssequn1 3255 A relationship between subclass and union. Theorem 26 of [Suppes] p. 27. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( A  C_  B  <->  ( A  u.  B )  =  B )
 
Theoremunss2 3256 Subclass law for union of classes. Exercise 7 of [TakeutiZaring] p. 18. (Contributed by NM, 14-Oct-1999.)
 |-  ( A  C_  B  ->  ( C  u.  A )  C_  ( C  u.  B ) )
 
Theoremunss12 3257 Subclass law for union of classes. (Contributed by NM, 2-Jun-2004.)
 |-  ( ( A  C_  B  /\  C  C_  D )  ->  ( A  u.  C )  C_  ( B  u.  D ) )
 
Theoremssequn2 3258 A relationship between subclass and union. (Contributed by NM, 13-Jun-1994.)
 |-  ( A  C_  B  <->  ( B  u.  A )  =  B )
 
Theoremunss 3259 The union of two subclasses is a subclass. Theorem 27 of [Suppes] p. 27 and its converse. (Contributed by NM, 11-Jun-2004.)
 |-  ( ( A  C_  C  /\  B  C_  C ) 
 <->  ( A  u.  B )  C_  C )
 
Theoremunssi 3260 An inference showing the union of two subclasses is a subclass. (Contributed by Raph Levien, 10-Dec-2002.)
 |-  A  C_  C   &    |-  B  C_  C   =>    |-  ( A  u.  B )  C_  C
 
Theoremunssd 3261 A deduction showing the union of two subclasses is a subclass. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  ( ph  ->  A  C_  C )   &    |-  ( ph  ->  B 
 C_  C )   =>    |-  ( ph  ->  ( A  u.  B ) 
 C_  C )
 
Theoremssun 3262 A condition that implies inclusion in the union of two classes. (Contributed by NM, 23-Nov-2003.)
 |-  ( ( A  C_  B  \/  A  C_  C )  ->  A  C_  ( B  u.  C ) )
 
Theoremrexun 3263 Restricted existential quantification over union. (Contributed by Jeff Madsen, 5-Jan-2011.)
 |-  ( E. x  e.  ( A  u.  B ) ph  <->  ( E. x  e.  A  ph  \/  E. x  e.  B  ph ) )
 
Theoremralunb 3264 Restricted quantification over a union. (Contributed by Scott Fenton, 12-Apr-2011.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  ( A. x  e.  ( A  u.  B ) ph  <->  ( A. x  e.  A  ph  /\  A. x  e.  B  ph ) )
 
Theoremralun 3265 Restricted quantification over union. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ( A. x  e.  A  ph  /\  A. x  e.  B  ph )  ->  A. x  e.  ( A  u.  B ) ph )
 
Theoremelin 3266 Expansion of membership in an intersection of two classes. Theorem 12 of [Suppes] p. 25. (Contributed by NM, 29-Apr-1994.)
 |-  ( A  e.  ( B  i^i  C )  <->  ( A  e.  B  /\  A  e.  C ) )
 
Theoremelin2 3267 Membership in a class defined as an intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  X  =  ( B  i^i  C )   =>    |-  ( A  e.  X 
 <->  ( A  e.  B  /\  A  e.  C ) )
 
Theoremelin3 3268 Membership in a class defined as a ternary intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  X  =  ( ( B  i^i  C )  i^i  D )   =>    |-  ( A  e.  X 
 <->  ( A  e.  B  /\  A  e.  C  /\  A  e.  D )
 )
 
Theoremincom 3269 Commutative law for intersection of classes. Exercise 7 of [TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  i^i  B )  =  ( B  i^i  A )
 
Theoremineqri 3270* Inference from membership to intersection. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( x  e.  A  /\  x  e.  B )  <->  x  e.  C )   =>    |-  ( A  i^i  B )  =  C
 
Theoremineq1 3271 Equality theorem for intersection of two classes. (Contributed by NM, 14-Dec-1993.)
 |-  ( A  =  B  ->  ( A  i^i  C )  =  ( B  i^i  C ) )
 
Theoremineq2 3272 Equality theorem for intersection of two classes. (Contributed by NM, 26-Dec-1993.)
 |-  ( A  =  B  ->  ( C  i^i  A )  =  ( C  i^i  B ) )
 
Theoremineq12 3273 Equality theorem for intersection of two classes. (Contributed by NM, 8-May-1994.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  i^i  C )  =  ( B  i^i  D ) )
 
Theoremineq1i 3274 Equality inference for intersection of two classes. (Contributed by NM, 26-Dec-1993.)
 |-  A  =  B   =>    |-  ( A  i^i  C )  =  ( B  i^i  C )
 
Theoremineq2i 3275 Equality inference for intersection of two classes. (Contributed by NM, 26-Dec-1993.)
 |-  A  =  B   =>    |-  ( C  i^i  A )  =  ( C  i^i  B )
 
Theoremineq12i 3276 Equality inference for intersection of two classes. (Contributed by NM, 24-Jun-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
 |-  A  =  B   &    |-  C  =  D   =>    |-  ( A  i^i  C )  =  ( B  i^i  D )
 
Theoremineq1d 3277 Equality deduction for intersection of two classes. (Contributed by NM, 10-Apr-1994.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  i^i  C )  =  ( B  i^i  C ) )
 
Theoremineq2d 3278 Equality deduction for intersection of two classes. (Contributed by NM, 10-Apr-1994.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C  i^i  A )  =  ( C  i^i  B ) )
 
Theoremineq12d 3279 Equality deduction for intersection of two classes. (Contributed by NM, 24-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A  i^i  C )  =  ( B  i^i  D ) )
 
Theoremineqan12d 3280 Equality deduction for intersection of two classes. (Contributed by NM, 7-Feb-2007.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ps  ->  C  =  D )   =>    |-  ( ( ph  /\ 
 ps )  ->  ( A  i^i  C )  =  ( B  i^i  D ) )
 
Theoremdfss1 3281 A frequently-used variant of subclass definition df-ss 3089. (Contributed by NM, 10-Jan-2015.)
 |-  ( A  C_  B  <->  ( B  i^i  A )  =  A )
 
Theoremnfin 3282 Bound-variable hypothesis builder for the intersection of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x ( A  i^i  B )
 
Theoremcsbing 3283 Distribute proper substitution through an intersection relation. (Contributed by Alan Sare, 22-Jul-2012.)
 |-  ( A  e.  B  -> 
 [_ A  /  x ]_ ( C  i^i  D )  =  ( [_ A  /  x ]_ C  i^i  [_ A  /  x ]_ D ) )
 
Theoremrabbi2dva 3284* Deduction from a wff to a restricted class abstraction. (Contributed by NM, 14-Jan-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  ( x  e.  B  <->  ps ) )   =>    |-  ( ph  ->  ( A  i^i  B )  =  { x  e.  A  |  ps }
 )
 
Theoreminidm 3285 Idempotent law for intersection of classes. Theorem 15 of [Suppes] p. 26. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  i^i  A )  =  A
 
Theoreminass 3286 Associative law for intersection of classes. Exercise 9 of [TakeutiZaring] p. 17. (Contributed by NM, 3-May-1994.)
 |-  ( ( A  i^i  B )  i^i  C )  =  ( A  i^i  ( B  i^i  C ) )
 
Theoremin12 3287 A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.)
 |-  ( A  i^i  ( B  i^i  C ) )  =  ( B  i^i  ( A  i^i  C ) )
 
Theoremin32 3288 A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( ( A  i^i  B )  i^i  C )  =  ( ( A  i^i  C )  i^i 
 B )
 
Theoremin13 3289 A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)
 |-  ( A  i^i  ( B  i^i  C ) )  =  ( C  i^i  ( B  i^i  A ) )
 
Theoremin31 3290 A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)
 |-  ( ( A  i^i  B )  i^i  C )  =  ( ( C  i^i  B )  i^i 
 A )
 
Theoreminrot 3291 Rotate the intersection of 3 classes.. (Contributed by NM, 27-Aug-2012.)
 |-  ( ( A  i^i  B )  i^i  C )  =  ( ( C  i^i  A )  i^i 
 B )
 
Theoremin4 3292 Rearrangement of intersection of 4 classes. (Contributed by NM, 21-Apr-2001.)
 |-  ( ( A  i^i  B )  i^i  ( C  i^i  D ) )  =  ( ( A  i^i  C )  i^i  ( B  i^i  D ) )
 
Theoreminindi 3293 Intersection distributes over itself. (Contributed by NM, 6-May-1994.)
 |-  ( A  i^i  ( B  i^i  C ) )  =  ( ( A  i^i  B )  i^i  ( A  i^i  C ) )
 
Theoreminindir 3294 Intersection distributes over itself. (Contributed by NM, 17-Aug-2004.)
 |-  ( ( A  i^i  B )  i^i  C )  =  ( ( A  i^i  C )  i^i  ( B  i^i  C ) )
 
Theoremsseqin2 3295 A relationship between subclass and intersection. Similar to Exercise 9 of [TakeutiZaring] p. 18. (Contributed by NM, 17-May-1994.)
 |-  ( A  C_  B  <->  ( B  i^i  A )  =  A )
 
Theoreminss1 3296 The intersection of two classes is a subset of one of them. Part of Exercise 12 of [TakeutiZaring] p. 18. (Contributed by NM, 27-Apr-1994.)
 |-  ( A  i^i  B )  C_  A
 
Theoreminss2 3297 The intersection of two classes is a subset of one of them. Part of Exercise 12 of [TakeutiZaring] p. 18. (Contributed by NM, 27-Apr-1994.)
 |-  ( A  i^i  B )  C_  B
 
Theoremssin 3298 Subclass of intersection. Theorem 2.8(vii) of [Monk1] p. 26. (Contributed by NM, 15-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( ( A  C_  B  /\  A  C_  C ) 
 <->  A  C_  ( B  i^i  C ) )
 
Theoremssini 3299 An inference showing that the a subclass of two classes is a subclass of their intersection. (Contributed by NM, 24-Nov-2003.)
 |-  A  C_  B   &    |-  A  C_  C   =>    |-  A  C_  ( B  i^i  C )
 
Theoremssind 3300 A deduction showing that the a subclass of two classes is a subclass of their intersection. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  A 
 C_  C )   =>    |-  ( ph  ->  A 
 C_  ( B  i^i  C ) )
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