Theorem sslin 3403

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 3402	. 2 |- ( A C_ B -> ( A i^i C ) C_ ( B i^i C ) )
2		incom 3369	. 2 |- ( C i^i A ) = ( A i^i C )
3		incom 3369	. 2 |- ( C i^i B ) = ( B i^i C )
4	1, 2, 3	3sstr4g 3240	1 |- ( A C_ B -> ( C i^i A ) C_ ( C i^i B ) )

Syntax hints:   -> wi 4   i^i cin 3169   C_ wss 3170

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-in 3176  df-ss 3183

This theorem is referenced by:  ss2in  3405  difdifdirss  3549  ssres2  4995  ssrnres  5134  sbthlem7  7080  ioodisj  10135  ntrss  14666  cnptoprest  14786