Theorem ss2in 3309

Description: Intersection of subclasses. (Contributed by NM, 5-May-2000.)

Assertion

Ref	Expression
ss2in	|- ( ( A C_ B /\ C C_ D ) -> ( A i^i C ) C_ ( B i^i D ) )

Proof of Theorem ss2in

Step	Hyp	Ref	Expression
1		ssrin 3306	. 2 |- ( A C_ B -> ( A i^i C ) C_ ( B i^i C ) )
2		sslin 3307	. 2 |- ( C C_ D -> ( B i^i C ) C_ ( B i^i D ) )
3	1, 2	sylan9ss 3115	1 |- ( ( A C_ B /\ C C_ D ) -> ( A i^i C ) C_ ( B i^i D ) )

Syntax hints:   -> wi 4   /\ wa 103   i^i cin 3075   C_ wss 3076

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-in 3082  df-ss 3089

This theorem is referenced by:  casefun  6978  caseinj  6982  djufun  6997  djuinj  6999  strleund  12086  strleun  12087  tgcl  12272  innei  12371  blin2  12640