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Theorem dif0 3690
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 3688 . . 3  |-  ( A 
\  A )  =  (/)
21difeq2i 3454 . 2  |-  ( A 
\  ( A  \  A ) )  =  ( A  \  (/) )
3 difdif 3465 . 2  |-  ( A 
\  ( A  \  A ) )  =  A
42, 3eqtr3i 2457 1  |-  ( A 
\  (/) )  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1652    \ cdif 3309   (/)c0 3620
This theorem is referenced by:  undifv  3694  dffv2  5788  2oconcl  6739  oe0m0  6756  oev2  6759  infdiffi  7604  cnfcom2lem  7650  m1bits  12944  mreexdomd  13866  efgi0  15344  vrgpinv  15393  frgpuptinv  15395  frgpnabllem1  15476  gsumval3  15506  gsumcllem  15508  dprddisj2  15589  0cld  17094  indiscld  17147  mretopd  17148  hauscmplem  17461  ptbasfi  17605  cfinfil  17917  csdfil  17918  filufint  17944  bcth3  19276  rembl  19427  volsup  19442  disjdifprg  24009  prsiga  24506  sxbrsigalem3  24614  symdif0  25661  onint1  26191
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rab 2706  df-v 2950  df-dif 3315  df-in 3319  df-ss 3326  df-nul 3621
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