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Theorem sslin 3376
Description: Add left intersection to subclass relation. (Contributed by NM, 19-Oct-1999.)
Assertion
Ref Expression
sslin  |-  ( A 
C_  B  ->  ( C  i^i  A )  C_  ( C  i^i  B ) )

Proof of Theorem sslin
StepHypRef Expression
1 ssrin 3375 . 2  |-  ( A 
C_  B  ->  ( A  i^i  C )  C_  ( B  i^i  C ) )
2 incom 3342 . 2  |-  ( C  i^i  A )  =  ( A  i^i  C
)
3 incom 3342 . 2  |-  ( C  i^i  B )  =  ( B  i^i  C
)
41, 2, 33sstr4g 3213 1  |-  ( A 
C_  B  ->  ( C  i^i  A )  C_  ( C  i^i  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    i^i cin 3143    C_ wss 3144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-in 3150  df-ss 3157
This theorem is referenced by:  ss2in  3378  difdifdirss  3522  ssres2  4952  ssrnres  5089  sbthlem7  6992  ioodisj  10023  ntrss  14079  cnptoprest  14199
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