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Theorem ssinss1 3690
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 3682 . 2  |-  ( A  i^i  B )  C_  A
2 sstr2 3471 . 2  |-  ( ( A  i^i  B ) 
C_  A  ->  ( A  C_  C  ->  ( A  i^i  B )  C_  C ) )
31, 2ax-mp 5 1  |-  ( A 
C_  C  ->  ( A  i^i  B )  C_  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    i^i cin 3435    C_ wss 3436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-v 3083  df-in 3443  df-ss 3450
This theorem is referenced by:  inss  3691  wfrlem4  7043  wfrlem5  7044  fipwuni  7942  ssfin4  8740  distop  19997  fctop  20005  cctop  20007  ntrin  20062  innei  20127  lly1stc  20497  txcnp  20621  isfild  20859  utoptop  21235  restmetu  21571  lecmi  27240  mdslj2i  27958  mdslmd1lem1  27963  mdslmd1lem2  27964  elpwincl1  28142  pnfneige0  28752  inelcarsg  29138  ballotlemfrc  29354  ballotlemfrcOLD  29392  bnj1177  29810  bnj1311  29828  frrlem4  30511  frrlem5  30512  cldbnd  30974  neiin  30980  ontgval  31083  mblfinlem4  31893  pmodlem1  33329  pmodlem2  33330  pmod1i  33331  pmod2iN  33332  pmodl42N  33334  dochdmj1  34876  ssficl  36092  ssinss1d  37245  icccncfext  37584  fourierdlem48  37837  fourierdlem49  37838  fourierdlem113  37902  caragendifcl  38113  omelesplit  38117  carageniuncllem2  38121  carageniuncl  38122
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