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Theorem ssdifin0 3535
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 3394 . 2  |-  ( A 
C_  ( B  \  C )  ->  ( A  i^i  C )  C_  ( ( B  \  C )  i^i  C
) )
2 incom 3361 . . 3  |-  ( ( B  \  C )  i^i  C )  =  ( C  i^i  ( B  \  C ) )
3 disjdif 3526 . . 3  |-  ( C  i^i  ( B  \  C ) )  =  (/)
42, 3eqtri 2303 . 2  |-  ( ( B  \  C )  i^i  C )  =  (/)
5 sseq0 3486 . 2  |-  ( ( ( A  i^i  C
)  C_  ( ( B  \  C )  i^i 
C )  /\  (
( B  \  C
)  i^i  C )  =  (/) )  ->  ( A  i^i  C )  =  (/) )
61, 4, 5sylancl 643 1  |-  ( A 
C_  ( B  \  C )  ->  ( A  i^i  C )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    \ cdif 3149    i^i cin 3151    C_ wss 3152   (/)c0 3455
This theorem is referenced by:  ssdifeq0  3536  marypha1lem  7186  numacn  7676  mreexexlem2d  13547  mreexexlem4d  13549  nrmsep2  17084  isnrm3  17087
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3456
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