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Theorem ssdifin0 3908
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 3723 . 2  |-  ( A 
C_  ( B  \  C )  ->  ( A  i^i  C )  C_  ( ( B  \  C )  i^i  C
) )
2 incom 3691 . . 3  |-  ( ( B  \  C )  i^i  C )  =  ( C  i^i  ( B  \  C ) )
3 disjdif 3899 . . 3  |-  ( C  i^i  ( B  \  C ) )  =  (/)
42, 3eqtri 2496 . 2  |-  ( ( B  \  C )  i^i  C )  =  (/)
5 sseq0 3817 . 2  |-  ( ( ( A  i^i  C
)  C_  ( ( B  \  C )  i^i 
C )  /\  (
( B  \  C
)  i^i  C )  =  (/) )  ->  ( A  i^i  C )  =  (/) )
61, 4, 5sylancl 662 1  |-  ( A 
C_  ( B  \  C )  ->  ( A  i^i  C )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    \ cdif 3473    i^i cin 3475    C_ wss 3476   (/)c0 3785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-v 3115  df-dif 3479  df-in 3483  df-ss 3490  df-nul 3786
This theorem is referenced by:  ssdifeq0  3909  marypha1lem  7893  numacn  8430  mreexexlem2d  14900  mreexexlem4d  14902  nrmsep2  19651  isnrm3  19654
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