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Theorem ssdisj 3837
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 3776 . . . 4  |-  ( ( B  i^i  C ) 
C_  (/)  <->  ( B  i^i  C )  =  (/) )
2 ssrin 3684 . . . . 5  |-  ( A 
C_  B  ->  ( A  i^i  C )  C_  ( B  i^i  C ) )
3 sstr2 3472 . . . . 5  |-  ( ( A  i^i  C ) 
C_  ( B  i^i  C )  ->  ( ( B  i^i  C )  C_  (/) 
->  ( A  i^i  C
)  C_  (/) ) )
42, 3syl 16 . . . 4  |-  ( A 
C_  B  ->  (
( B  i^i  C
)  C_  (/)  ->  ( A  i^i  C )  C_  (/) ) )
51, 4syl5bir 218 . . 3  |-  ( A 
C_  B  ->  (
( B  i^i  C
)  =  (/)  ->  ( A  i^i  C )  C_  (/) ) )
65imp 429 . 2  |-  ( ( A  C_  B  /\  ( B  i^i  C )  =  (/) )  ->  ( A  i^i  C )  C_  (/) )
7 ss0 3777 . 2  |-  ( ( A  i^i  C ) 
C_  (/)  ->  ( A  i^i  C )  =  (/) )
86, 7syl 16 1  |-  ( ( A  C_  B  /\  ( B  i^i  C )  =  (/) )  ->  ( A  i^i  C )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    i^i cin 3436    C_ wss 3437   (/)c0 3746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3080  df-dif 3440  df-in 3444  df-ss 3451  df-nul 3747
This theorem is referenced by:  djudisj  5374  fimacnvdisj  5698  marypha1lem  7795  ackbij1lem16  8516  ackbij1lem18  8518  fin23lem20  8618  fin23lem30  8623  elcls3  18820  neindisj  18854  imadifxp  26091  diophren  29301
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