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Theorem ssinss1 3571
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 3563 . 2  |-  ( A  i^i  B )  C_  A
2 sstr2 3357 . 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 3321    C_ wss 3322
This theorem is referenced by:  inss  3572  fipwuni  7433  ssfin4  8192  distop  17062  fctop  17070  cctop  17072  ntrin  17127  innei  17191  lly1stc  17561  txcnp  17654  isfild  17892  utoptop  18266  restmetu  18619  lecmi  23106  mdslj2i  23825  mdslmd1lem1  23830  mdslmd1lem2  23831  pnfneige0  24338  probdif  24680  ballotlemfrc  24786  wfrlem4  25543  wfrlem5  25544  frrlem4  25587  frrlem5  25588  ontgval  26183  mblfinlem4  26248  cldbnd  26331  neiin  26337  bnj1177  29437  bnj1311  29455  pmodlem1  30705  pmodlem2  30706  pmod1i  30707  pmod2iN  30708  pmodl42N  30710  dochdmj1  32250
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-in 3329  df-ss 3336
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