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Theorem difin0 3665
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 3526 . 2  |-  ( A  i^i  B )  C_  B
2 ssdif0 3650 . 2  |-  ( ( A  i^i  B ) 
C_  B  <->  ( ( A  i^i  B )  \  B )  =  (/) )
31, 2mpbi 200 1  |-  ( ( A  i^i  B ) 
\  B )  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1649    \ cdif 3281    i^i cin 3283    C_ wss 3284   (/)c0 3592
This theorem is referenced by:  volinun  19397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-v 2922  df-dif 3287  df-in 3291  df-ss 3298  df-nul 3593
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