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Theorem difid 3865
Description: The difference between a class and itself is the empty set. Proposition 5.15 of [TakeutiZaring] p. 20. Also Theorem 32 of [Suppes] p. 28. (Contributed by NM, 22-Apr-2004.)
Assertion
Ref Expression
difid  |-  ( A 
\  A )  =  (/)

Proof of Theorem difid
StepHypRef Expression
1 ssid 3483 . 2  |-  A  C_  A
2 ssdif0 3853 . 2  |-  ( A 
C_  A  <->  ( A  \  A )  =  (/) )
31, 2mpbi 211 1  |-  ( A 
\  A )  =  (/)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437    \ cdif 3433    C_ wss 3436   (/)c0 3761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-v 3082  df-dif 3439  df-in 3443  df-ss 3450  df-nul 3762
This theorem is referenced by:  dif0  3867  difun2  3877  diftpsn3  4138  symdifid  4378  difxp1  5281  difxp2  5282  2oconcl  7217  oev2  7237  fin1a2lem13  8850  ruclem13  14294  strle1  15221  efgi1  17371  frgpuptinv  17421  gsumdifsnd  17593  dprdsn  17669  ablfac1eulem  17705  fctop  20018  cctop  20020  topcld  20049  indiscld  20106  mretopd  20107  restcld  20187  conndisj  20430  csdfil  20908  ufinffr  20943  prdsxmslem2  21543  bcth3  22298  voliunlem3  22504  uhgra0v  25036  usgra0v  25097  cusgra1v  25188  frgra1v  25725  1vwmgra  25730  zrdivrng  26159  difres  28214  imadifxp  28215  difico  28372  0elsiga  28945  prsiga  28962  fiunelcarsg  29157  sibf0  29176  probun  29261  ballotlemfp1  29333  onint1  31115  poimirlem22  31927  poimirlem30  31935  compne  36764  fzdifsuc2  37485  dvmptfprodlem  37760  fouriercn  38037  prsal  38101  caragenuncllem  38242  carageniuncllem1  38251  caratheodorylem1  38256  uhgr0v  38997  uhgr0  38998  nbgr1vtx  39220  uvtxa01vtx0  39257  cplgr1v  39283  uhg0v  39338  gsumdifsndf  39798
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