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Theorem ssdifin0 3412
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 3269 . 2  |-  ( A 
C_  ( B  \  C )  ->  ( A  i^i  C )  C_  ( ( B  \  C )  i^i  C
) )
2 incom 3236 . . 3  |-  ( ( B  \  C )  i^i  C )  =  ( C  i^i  ( B  \  C ) )
3 disjdif 3403 . . 3  |-  ( C  i^i  ( B  \  C ) )  =  (/)
42, 3eqtri 2136 . 2  |-  ( ( B  \  C )  i^i  C )  =  (/)
5 sseq0 3372 . 2  |-  ( ( ( A  i^i  C
)  C_  ( ( B  \  C )  i^i 
C )  /\  (
( B  \  C
)  i^i  C )  =  (/) )  ->  ( A  i^i  C )  =  (/) )
61, 4, 5sylancl 407 1  |-  ( A 
C_  ( B  \  C )  ->  ( A  i^i  C )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1314    \ cdif 3036    i^i cin 3038    C_ wss 3039   (/)c0 3331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-dif 3041  df-in 3045  df-ss 3052  df-nul 3332
This theorem is referenced by:  ssdifeq0  3413
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