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Theorem difin0 3433
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 3297 . 2  |-  ( A  i^i  B )  C_  B
2 ssdif0 3420 . 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 1619    \ cdif 3075    i^i cin 3077    C_ wss 3078   (/)c0 3362
This theorem is referenced by:  volinun  18735
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-v 2729  df-dif 3081  df-in 3085  df-ss 3089  df-nul 3363
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