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Theorem ssdisj 3419
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 3402 . . . 4  |-  ( ( B  i^i  C ) 
C_  (/)  <->  ( B  i^i  C )  =  (/) )
2 ssrin 3301 . . . . 5  |-  ( A 
C_  B  ->  ( A  i^i  C )  C_  ( B  i^i  C ) )
3 sstr2 3104 . . . . 5  |-  ( ( A  i^i  C ) 
C_  ( B  i^i  C )  ->  ( ( B  i^i  C )  C_  (/) 
->  ( A  i^i  C
)  C_  (/) ) )
42, 3syl 14 . . . 4  |-  ( A 
C_  B  ->  (
( B  i^i  C
)  C_  (/)  ->  ( A  i^i  C )  C_  (/) ) )
51, 4syl5bir 152 . . 3  |-  ( A 
C_  B  ->  (
( B  i^i  C
)  =  (/)  ->  ( A  i^i  C )  C_  (/) ) )
65imp 123 . 2  |-  ( ( A  C_  B  /\  ( B  i^i  C )  =  (/) )  ->  ( A  i^i  C )  C_  (/) )
7 ss0 3403 . 2  |-  ( ( A  i^i  C ) 
C_  (/)  ->  ( A  i^i  C )  =  (/) )
86, 7syl 14 1  |-  ( ( A  C_  B  /\  ( B  i^i  C )  =  (/) )  ->  ( A  i^i  C )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    i^i cin 3070    C_ wss 3071   (/)c0 3363
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-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-dif 3073  df-in 3077  df-ss 3084  df-nul 3364
This theorem is referenced by:  djudisj  4966  fimacnvdisj  5307  unfiin  6814  hashunlem  10557
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