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Theorem ssdifin0 3709
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 3566 . 2  |-  ( A 
C_  ( B  \  C )  ->  ( A  i^i  C )  C_  ( ( B  \  C )  i^i  C
) )
2 incom 3533 . . 3  |-  ( ( B  \  C )  i^i  C )  =  ( C  i^i  ( B  \  C ) )
3 disjdif 3700 . . 3  |-  ( C  i^i  ( B  \  C ) )  =  (/)
42, 3eqtri 2456 . 2  |-  ( ( B  \  C )  i^i  C )  =  (/)
5 sseq0 3659 . 2  |-  ( ( ( A  i^i  C
)  C_  ( ( B  \  C )  i^i 
C )  /\  (
( B  \  C
)  i^i  C )  =  (/) )  ->  ( A  i^i  C )  =  (/) )
61, 4, 5sylancl 644 1  |-  ( A 
C_  ( B  \  C )  ->  ( A  i^i  C )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    \ cdif 3317    i^i cin 3319    C_ wss 3320   (/)c0 3628
This theorem is referenced by:  ssdifeq0  3710  marypha1lem  7438  numacn  7930  mreexexlem2d  13870  mreexexlem4d  13872  nrmsep2  17420  isnrm3  17423
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-v 2958  df-dif 3323  df-in 3327  df-ss 3334  df-nul 3629
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