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Theorem ssinss1 3397
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 3389 . 2  |-  ( A  i^i  B )  C_  A
2 sstr2 3186 . 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 3151    C_ wss 3152
This theorem is referenced by:  inss  3398  fipwuni  7179  ssfin4  7936  distop  16733  fctop  16741  cctop  16743  ntrin  16798  innei  16862  lly1stc  17222  txcnp  17314  isfild  17553  lecmi  22181  mdslj2i  22900  mdslmd1lem1  22905  mdslmd1lem2  22906  ballotlemfrc  23085  probdif  23623  wfrlem4  24259  wfrlem5  24260  frrlem4  24284  frrlem5  24285  ontgval  24870  inpws1  25042  qusp  25542  islimrs4  25582  cldbnd  26244  neiin  26250  bnj1177  29036  bnj1311  29054  pmodlem1  30035  pmodlem2  30036  pmod1i  30037  pmod2iN  30038  pmodl42N  30040  dochdmj1  31580
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-in 3159  df-ss 3166
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