Theorem sslin 3353

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 3352	. 2 |- ( A C_ B -> ( A i^i C ) C_ ( B i^i C ) )
2		incom 3319	. 2 |- ( C i^i A ) = ( A i^i C )
3		incom 3319	. 2 |- ( C i^i B ) = ( B i^i C )
4	1, 2, 3	3sstr4g 3190	1 |- ( A C_ B -> ( C i^i A ) C_ ( C i^i B ) )

Syntax hints:   -> wi 4   i^i cin 3120   C_ wss 3121

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134

This theorem is referenced by:  ss2in  3355  difdifdirss  3499  ssres2  4918  ssrnres  5053  sbthlem7  6940  ioodisj  9950  ntrss  12913  cnptoprest  13033