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Theorem ssrin 3716
Description: Add right intersection to subclass relation. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
ssrin  |-  ( A 
C_  B  ->  ( A  i^i  C )  C_  ( B  i^i  C ) )

Proof of Theorem ssrin
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ssel 3491 . . . 4  |-  ( A 
C_  B  ->  (
x  e.  A  ->  x  e.  B )
)
21anim1d 564 . . 3  |-  ( A 
C_  B  ->  (
( x  e.  A  /\  x  e.  C
)  ->  ( x  e.  B  /\  x  e.  C ) ) )
3 elin 3680 . . 3  |-  ( x  e.  ( A  i^i  C )  <->  ( x  e.  A  /\  x  e.  C ) )
4 elin 3680 . . 3  |-  ( x  e.  ( B  i^i  C )  <->  ( x  e.  B  /\  x  e.  C ) )
52, 3, 43imtr4g 270 . 2  |-  ( A 
C_  B  ->  (
x  e.  ( A  i^i  C )  ->  x  e.  ( B  i^i  C ) ) )
65ssrdv 3503 1  |-  ( A 
C_  B  ->  ( A  i^i  C )  C_  ( B  i^i  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1762    i^i cin 3468    C_ wss 3469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-v 3108  df-in 3476  df-ss 3483
This theorem is referenced by:  sslin  3717  ss2in  3718  ssdisj  3869  ssdifin0  3901  ssres  5290  sbthlem7  7623  onsdominel  7656  phplem2  7687  infdifsn  8062  fictb  8614  fin23lem23  8695  ttukeylem2  8879  limsupgord  13244  xpsc1  14805  isacs1i  14901  rescabs  15052  lsmdisj  16488  dmdprdsplit2lem  16877  pjfval  18497  pjpm  18499  obselocv  18519  tgss  19229  neindisj2  19383  restbas  19418  neitr  19440  restcls  19441  restntr  19442  nrmsep  19617  1stcrest  19713  cldllycmp  19755  kgencn3  19787  trfbas2  20072  fclsneii  20246  fclsrest  20253  fcfnei  20264  cnextcn  20295  tsmsresOLD  20373  tsmsres  20374  trust  20460  restutopopn  20469  trcfilu  20525  metrest  20755  reperflem  21051  metdseq0  21086  iundisj2  21687  uniioombllem3  21722  ellimc3  22011  limcflf  22013  lhop1lem  22142  ppisval  23098  ppisval2  23099  ppinprm  23147  chtnprm  23149  chtwordi  23151  ppiwordi  23157  chpub  23216  chebbnd1lem1  23375  chtppilimlem1  23379  orthin  26026  3oalem6  26247  mdbr2  26877  mdslle1i  26898  mdslle2i  26899  mdslj1i  26900  mdslj2i  26901  mdsl2i  26903  mdslmd1lem1  26906  mdslmd1lem2  26907  mdslmd3i  26913  mdexchi  26916  sumdmdlem  26999  iundisj2f  27108  iundisj2fi  27256  eulerpartlemn  27946  predpredss  28813  ismblfin  29619  sstotbnd2  29860  eldioph2lem2  30285  acsfn1p  30742  sumnnodd  31127  bnj1177  33016  lcvexchlem5  33710  pnonsingN  34604  dochnoncon  36063
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