Theorem ss2in 3435

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 3432	. 2 |- ( A C_ B -> ( A i^i C ) C_ ( B i^i C ) )
2		sslin 3433	. 2 |- ( C C_ D -> ( B i^i C ) C_ ( B i^i D ) )
3	1, 2	sylan9ss 3240	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 3199   C_ wss 3200

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-in 3206  df-ss 3213

This theorem is referenced by:  casefun  7283  caseinj  7287  djufun  7302  djuinj  7304  strleund  13185  strleun  13186  tgcl  14787  innei  14886  blin2  15155  vtxdfifiun  16147