Theorem sslin 3266

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 3265	. 2 |- ( A C_ B -> ( A i^i C ) C_ ( B i^i C ) )
2		incom 3232	. 2 |- ( C i^i A ) = ( A i^i C )
3		incom 3232	. 2 |- ( C i^i B ) = ( B i^i C )
4	1, 2, 3	3sstr4g 3104	1 |- ( A C_ B -> ( C i^i A ) C_ ( C i^i B ) )

Syntax hints:   -> wi 4   i^i cin 3034   C_ wss 3035

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095

This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-v 2657  df-in 3041  df-ss 3048

This theorem is referenced by:  ss2in  3268  difdifdirss  3411  ssres2  4802  ssrnres  4937  sbthlem7  6800  ioodisj  9662  ntrss  12124  cnptoprest  12243