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Theorem ssdisj 3614
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 3594 . . . 4  |-  ( ( B  i^i  C ) 
C_  (/)  <->  ( B  i^i  C )  =  (/) )
2 ssrin 3503 . . . . 5  |-  ( A 
C_  B  ->  ( A  i^i  C )  C_  ( B  i^i  C ) )
3 sstr2 3292 . . . . 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 210 . . 3  |-  ( A 
C_  B  ->  (
( B  i^i  C
)  =  (/)  ->  ( A  i^i  C )  C_  (/) ) )
65imp 419 . 2  |-  ( ( A  C_  B  /\  ( B  i^i  C )  =  (/) )  ->  ( A  i^i  C )  C_  (/) )
7 ss0 3595 . 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 set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    i^i cin 3256    C_ wss 3257   (/)c0 3565
This theorem is referenced by:  djudisj  5231  fimacnvdisj  5555  marypha1lem  7367  ackbij1lem16  8042  ackbij1lem18  8044  fin23lem20  8144  fin23lem30  8149  elcls3  17064  neindisj  17098  imadifxp  23875  diophren  26559
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2362
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-v 2895  df-dif 3260  df-in 3264  df-ss 3271  df-nul 3566
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