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Theorem ssdifin0 3883
Description: A subset of a difference does not intersect the subtrahend. (Contributed by Jeff Hankins, 1-Sep-2013.) (Proof shortened by Mario Carneiro, 24-Aug-2015.)
Assertion
Ref Expression
ssdifin0  |-  ( A 
C_  ( B  \  C )  ->  ( A  i^i  C )  =  (/) )

Proof of Theorem ssdifin0
StepHypRef Expression
1 ssrin 3693 . 2  |-  ( A 
C_  ( B  \  C )  ->  ( A  i^i  C )  C_  ( ( B  \  C )  i^i  C
) )
2 incom 3661 . . 3  |-  ( ( B  \  C )  i^i  C )  =  ( C  i^i  ( B  \  C ) )
3 disjdif 3873 . . 3  |-  ( C  i^i  ( B  \  C ) )  =  (/)
42, 3eqtri 2458 . 2  |-  ( ( B  \  C )  i^i  C )  =  (/)
5 sseq0 3800 . 2  |-  ( ( ( A  i^i  C
)  C_  ( ( B  \  C )  i^i 
C )  /\  (
( B  \  C
)  i^i  C )  =  (/) )  ->  ( A  i^i  C )  =  (/) )
61, 4, 5sylancl 666 1  |-  ( A 
C_  ( B  \  C )  ->  ( A  i^i  C )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    \ cdif 3439    i^i cin 3441    C_ wss 3442   (/)c0 3767
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-v 3089  df-dif 3445  df-in 3449  df-ss 3456  df-nul 3768
This theorem is referenced by:  ssdifeq0  3884  marypha1lem  7953  numacn  8478  mreexexlem2d  15502  mreexexlem4d  15504  nrmsep2  20303  isnrm3  20306
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