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Theorem difin0 3404
Description: The difference of a class from its intersection is empty. Theorem 37 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
difin0  |-  ( ( A  i^i  B ) 
\  B )  =  (/)

Proof of Theorem difin0
StepHypRef Expression
1 inss2 3265 . 2  |-  ( A  i^i  B )  C_  B
2 ssdif0im 3395 . 2  |-  ( ( A  i^i  B ) 
C_  B  ->  (
( A  i^i  B
)  \  B )  =  (/) )
31, 2ax-mp 5 1  |-  ( ( A  i^i  B ) 
\  B )  =  (/)
Colors of variables: wff set class
Syntax hints:    = 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: (None)
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