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Theorem ssdisj 3389
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 3372 . . . 4  |-  ( ( B  i^i  C ) 
C_  (/)  <->  ( B  i^i  C )  =  (/) )
2 ssrin 3271 . . . . 5  |-  ( A 
C_  B  ->  ( A  i^i  C )  C_  ( B  i^i  C ) )
3 sstr2 3074 . . . . 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 3373 . 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 1316    i^i cin 3040    C_ wss 3041   (/)c0 3333
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-dif 3043  df-in 3047  df-ss 3054  df-nul 3334
This theorem is referenced by:  djudisj  4936  fimacnvdisj  5277  unfiin  6782  hashunlem  10518
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