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Theorem disjdifr 4473
Description: A class and its relative complement are disjoint. (Contributed by Thierry Arnoux, 29-Nov-2023.)
Assertion
Ref Expression
disjdifr ((𝐵𝐴) ∩ 𝐴) = ∅

Proof of Theorem disjdifr
StepHypRef Expression
1 incom 4202 . 2 (𝐴 ∩ (𝐵𝐴)) = ((𝐵𝐴) ∩ 𝐴)
2 disjdif 4472 . 2 (𝐴 ∩ (𝐵𝐴)) = ∅
31, 2eqtr3i 2763 1 ((𝐵𝐴) ∩ 𝐴) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  cdif 3946  cin 3948  c0 4323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-in 3956  df-ss 3966  df-nul 4324
This theorem is referenced by:  ssdifin0  4486  fvsnun1  7180  fveqf1o  7301  f1ofvswap  7304  ralxpmap  8890  difsnen  9053  domunsn  9127  limensuci  9153  pssnn  9168  marypha1lem  9428  dif1card  10005  ackbij1lem18  10232  canthp1lem1  10647  grothprim  10829  hashgval  14293  hashun3  14344  hashfun  14397  hashbclem  14411  setsfun  17104  setsfun0  17105  setsid  17141  mreexexlem4d  17591  pwssplit1  20670  islindf4  21393  neitr  22684  regsep2  22880  restmetu  24079  volinun  25063  tdeglem4  25577  noetasuplem3  27238  noetasuplem4  27239  difeq  31756  disjdifprg  31806  tocycfvres1  32269  tocycfvres2  32270  cycpmfvlem  32271  cycpmfv3  32274  cycpmcl  32275  measunl  33214  eulerpartlemt  33370  mthmpps  34573  cldbnd  35211  poimirlem15  36503  poimirlem16  36504  poimirlem19  36507  poimirlem27  36515  selvvvval  41157  evlselvlem  41158  evlselv  41159  eldioph2lem1  41498  eldioph2lem2  41499  diophren  41551  kelac1  41805  isomenndlem  45246  seposep  47558
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