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Theorem sslin 3210
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 3209 . 2  |-  ( A 
C_  B  ->  ( A  i^i  C )  C_  ( B  i^i  C ) )
2 incom 3176 . 2  |-  ( C  i^i  A )  =  ( A  i^i  C
)
3 incom 3176 . 2  |-  ( C  i^i  B )  =  ( B  i^i  C
)
41, 2, 33sstr4g 3051 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 2983    C_ wss 2984
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-in 2990  df-ss 2997
This theorem is referenced by:  ss2in  3211  difdifdirss  3348  ssres2  4696  ssrnres  4826  ioodisj  9304
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