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Theorem ssinss1 3410
Description: Intersection preserves subclass relationship. (Contributed by NM, 14-Sep-1999.)
Assertion
Ref Expression
ssinss1  |-  ( A 
C_  C  ->  ( A  i^i  B )  C_  C )

Proof of Theorem ssinss1
StepHypRef Expression
1 inss1 3402 . 2  |-  ( A  i^i  B )  C_  A
2 sstr2 3199 . 2  |-  ( ( A  i^i  B ) 
C_  A  ->  ( A  C_  C  ->  ( A  i^i  B )  C_  C ) )
31, 2ax-mp 8 1  |-  ( A 
C_  C  ->  ( A  i^i  B )  C_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    i^i cin 3164    C_ wss 3165
This theorem is referenced by:  inss  3411  fipwuni  7195  ssfin4  7952  distop  16749  fctop  16757  cctop  16759  ntrin  16814  innei  16878  lly1stc  17238  txcnp  17330  isfild  17569  lecmi  22197  mdslj2i  22916  mdslmd1lem1  22921  mdslmd1lem2  22922  ballotlemfrc  23101  probdif  23638  wfrlem4  24330  wfrlem5  24331  frrlem4  24355  frrlem5  24356  ontgval  24942  inpws1  25145  qusp  25645  islimrs4  25685  cldbnd  26347  neiin  26353  bnj1177  29352  bnj1311  29370  pmodlem1  30657  pmodlem2  30658  pmod1i  30659  pmod2iN  30660  pmodl42N  30662  dochdmj1  32202
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-in 3172  df-ss 3179
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