Theorem sslin 3348

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

Step	Hyp	Ref	Expression
1		ssrin 3347	. 2 |- ( A C_ B -> ( A i^i C ) C_ ( B i^i C ) )
2		incom 3314	. 2 |- ( C i^i A ) = ( A i^i C )
3		incom 3314	. 2 |- ( C i^i B ) = ( B i^i C )
4	1, 2, 3	3sstr4g 3185	1 |- ( A C_ B -> ( C i^i A ) C_ ( C i^i B ) )

Syntax hints:   -> wi 4   i^i cin 3115   C_ wss 3116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-in 3122  df-ss 3129

This theorem is referenced by:  ss2in  3350  difdifdirss  3493  ssres2  4911  ssrnres  5046  sbthlem7  6928  ioodisj  9929  ntrss  12759  cnptoprest  12879