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Theorem ssdisj 3504
Description: Intersection with a subclass of a disjoint class. (Contributed by FL, 24-Jan-2007.)
Assertion
Ref Expression
ssdisj  |-  ( ( A  C_  B  /\  ( B  i^i  C )  =  (/) )  ->  ( A  i^i  C )  =  (/) )

Proof of Theorem ssdisj
StepHypRef Expression
1 ss0b 3484 . . . 4  |-  ( ( B  i^i  C ) 
C_  (/)  <->  ( B  i^i  C )  =  (/) )
2 ssrin 3394 . . . . 5  |-  ( A 
C_  B  ->  ( A  i^i  C )  C_  ( B  i^i  C ) )
3 sstr2 3186 . . . . 5  |-  ( ( A  i^i  C ) 
C_  ( B  i^i  C )  ->  ( ( B  i^i  C )  C_  (/) 
->  ( A  i^i  C
)  C_  (/) ) )
42, 3syl 15 . . . 4  |-  ( A 
C_  B  ->  (
( B  i^i  C
)  C_  (/)  ->  ( A  i^i  C )  C_  (/) ) )
51, 4syl5bir 209 . . 3  |-  ( A 
C_  B  ->  (
( B  i^i  C
)  =  (/)  ->  ( A  i^i  C )  C_  (/) ) )
65imp 418 . 2  |-  ( ( A  C_  B  /\  ( B  i^i  C )  =  (/) )  ->  ( A  i^i  C )  C_  (/) )
7 ss0 3485 . 2  |-  ( ( A  i^i  C ) 
C_  (/)  ->  ( A  i^i  C )  =  (/) )
86, 7syl 15 1  |-  ( ( A  C_  B  /\  ( B  i^i  C )  =  (/) )  ->  ( A  i^i  C )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    i^i cin 3151    C_ wss 3152   (/)c0 3455
This theorem is referenced by:  djudisj  5104  fimacnvdisj  5419  marypha1lem  7186  ackbij1lem16  7861  ackbij1lem18  7863  fin23lem20  7963  fin23lem30  7968  elcls3  16820  neindisj  16854  diophren  26896
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3456
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