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Theorem difsnid 4815
Description: If we remove a single element from a class then put it back in, we end up with the original class. (Contributed by NM, 2-Oct-2006.)
Assertion
Ref Expression
difsnid (𝐵𝐴 → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = 𝐴)

Proof of Theorem difsnid
StepHypRef Expression
1 snssi 4813 . 2 (𝐵𝐴 → {𝐵} ⊆ 𝐴)
2 undifr 4489 . 2 ({𝐵} ⊆ 𝐴 ↔ ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = 𝐴)
31, 2sylib 218 1 (𝐵𝐴 → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  cdif 3960  cun 3961  wss 3963  {csn 4631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-sn 4632
This theorem is referenced by:  fnsnsplit  7204  fsnunf2  7206  difsnexi  7780  difsnen  9092  enfixsn  9120  pssnn  9207  phplem2OLD  9253  dif1ennnALT  9309  frfi  9319  xpfiOLD  9357  dif1card  10048  hashgt23el  14460  hashfun  14473  fprodfvdvdsd  16368  prmdvdsprmo  17076  mreexexlem4d  17692  symgextf1  19454  symgextfo  19455  symgfixf1  19470  gsumdifsnd  19994  gsummgp0  20332  islindf4  21876  scmatf1  22553  gsummatr01  22681  tdeglem4  26114  finsumvtxdg2sstep  29582  dfconngr1  30217  fmptunsnop  32715  satfv1lem  35347  bj-raldifsn  37083  lindsadd  37600  lindsenlbs  37602  poimirlem25  37632  poimirlem27  37634  hdmap14lem4a  41854  hdmap14lem13  41863  supxrmnf2  45383  infxrpnf2  45413  fsumnncl  45528  hoidmv1lelem2  46548  gsumdifsndf  48025  mgpsumunsn  48206
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