Theorem sslin 3163

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 3162	. 2 |- ( A C_ B -> ( A i^i C ) C_ ( B i^i C ) )
2		incom 3129	. 2 |- ( C i^i A ) = ( A i^i C )
3		incom 3129	. 2 |- ( C i^i B ) = ( B i^i C )
4	1, 2, 3	3sstr4g 2986	1 |- ( A C_ B -> ( C i^i A ) C_ ( C i^i B ) )

Syntax hints:   -> wi 4   i^i cin 2916   C_ wss 2917

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931

This theorem is referenced by:  ss2in  3164  difdifdirss  3307  ssres2  4638  ssrnres  4763  ioodisj  8861