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Theorem dif0 3747
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 3745 . . 3  |-  ( A 
\  A )  =  (/)
21difeq2i 3469 . 2  |-  ( A 
\  ( A  \  A ) )  =  ( A  \  (/) )
3 difdif 3480 . 2  |-  ( A 
\  ( A  \  A ) )  =  A
42, 3eqtr3i 2463 1  |-  ( A 
\  (/) )  =  A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1369    \ cdif 3323   (/)c0 3635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rab 2722  df-v 2972  df-dif 3329  df-in 3333  df-ss 3340  df-nul 3636
This theorem is referenced by:  unvdif  3751  iinvdif  4240  dffv2  5762  2oconcl  6941  oe0m0  6958  oev2  6961  infdiffi  7861  cnfcom2lem  7932  cnfcom2lemOLD  7940  m1bits  13634  mreexdomd  14585  efgi0  16215  vrgpinv  16264  frgpuptinv  16266  frgpnabllem1  16349  gsumval3OLD  16380  gsumval3  16383  gsumcllem  16384  gsumcllemOLD  16385  dprddisj2  16535  dprddisj2OLD  16536  0cld  18640  indiscld  18693  mretopd  18694  hauscmplem  19007  ptbasfi  19152  cfinfil  19464  csdfil  19465  filufint  19491  bcth3  20840  rembl  21020  volsup  21035  disjdifprg  25917  prsiga  26572  sxbrsigalem3  26685  symdif0  27853  onint1  28293  bj-disjdif  32443
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