Theorem ss2in 3453

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 3450	. 2 |- ( A C_ B -> ( A i^i C ) C_ ( B i^i C ) )
2		sslin 3451	. 2 |- ( C C_ D -> ( B i^i C ) C_ ( B i^i D ) )
3	1, 2	sylan9ss 3255	1 |- ( ( A C_ B /\ C C_ D ) -> ( A i^i C ) C_ ( B i^i D ) )

Syntax hints:   -> wi 4   /\ wa 104   i^i cin 3213   C_ wss 3214

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-in 3220  df-ss 3227

This theorem is referenced by:  casefun  7389  caseinj  7393  djufun  7408  djuinj  7410  strleund  13400  strleun  13401  tgcl  15055  innei  15154  blin2  15423  vtxdfifiun  16418