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Theorem dif0 3641
Description: The difference between a class and the empty set. Part of Exercise 4.4 of [Stoll] p. 16. (Contributed by NM, 17-Aug-2004.)
Assertion
Ref Expression
dif0  |-  ( A 
\  (/) )  =  A

Proof of Theorem dif0
StepHypRef Expression
1 difid 3639 . . 3  |-  ( A 
\  A )  =  (/)
21difeq2i 3405 . 2  |-  ( A 
\  ( A  \  A ) )  =  ( A  \  (/) )
3 difdif 3416 . 2  |-  ( A 
\  ( A  \  A ) )  =  A
42, 3eqtr3i 2409 1  |-  ( A 
\  (/) )  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1649    \ cdif 3260   (/)c0 3571
This theorem is referenced by:  undifv  3645  dffv2  5735  2oconcl  6683  oe0m0  6700  oev2  6703  infdiffi  7545  cnfcom2lem  7591  m1bits  12879  mreexdomd  13801  efgi0  15279  vrgpinv  15328  frgpuptinv  15330  frgpnabllem1  15411  gsumval3  15441  gsumcllem  15443  dprddisj2  15524  0cld  17025  indiscld  17078  mretopd  17079  hauscmplem  17391  ptbasfi  17534  cfinfil  17846  csdfil  17847  filufint  17873  bcth3  19153  rembl  19302  volsup  19317  disjdifprg  23861  prsiga  24310  sxbrsigalem3  24416  symdif0  25392  onint1  25913
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 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rab 2658  df-v 2901  df-dif 3266  df-in 3270  df-ss 3277  df-nul 3572
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