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Theorem difin0 3725
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 3547 . 2  |-  ( A  i^i  B )  C_  B
2 ssdif0 3710 . 2  |-  ( ( A  i^i  B ) 
C_  B  <->  ( ( A  i^i  B )  \  B )  =  (/) )
31, 2mpbi 201 1  |-  ( ( A  i^i  B ) 
\  B )  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1653    \ cdif 3303    i^i cin 3305    C_ wss 3306   (/)c0 3613
This theorem is referenced by:  volinun  19471
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-v 2964  df-dif 3309  df-in 3313  df-ss 3320  df-nul 3614
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