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Theorem difsnid 4775
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 uncom 4118 . 2 ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = ({𝐵} ∪ (𝐴 ∖ {𝐵}))
2 snssi 4773 . . 3 (𝐵𝐴 → {𝐵} ⊆ 𝐴)
3 undif 4446 . . 3 ({𝐵} ⊆ 𝐴 ↔ ({𝐵} ∪ (𝐴 ∖ {𝐵})) = 𝐴)
42, 3sylib 217 . 2 (𝐵𝐴 → ({𝐵} ∪ (𝐴 ∖ {𝐵})) = 𝐴)
51, 4eqtrid 2789 1 (𝐵𝐴 → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  cdif 3912  cun 3913  wss 3915  {csn 4591
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 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-sn 4592
This theorem is referenced by:  fnsnsplit  7135  fsnunf2  7137  difsnexi  7700  difsnen  9004  enfixsn  9032  pssnn  9119  phplem2OLD  9169  pssnnOLD  9216  dif1ennnALT  9228  frfi  9239  xpfiOLD  9269  dif1card  9953  hashgt23el  14331  hashfun  14344  fprodfvdvdsd  16223  prmdvdsprmo  16921  mreexexlem4d  17534  symgextf1  19210  symgextfo  19211  symgfixf1  19226  gsumdifsnd  19745  gsummgp0  20039  islindf4  21260  scmatf1  21896  gsummatr01  22024  tdeglem4  25440  tdeglem4OLD  25441  finsumvtxdg2sstep  28539  dfconngr1  29174  satfv1lem  33996  bj-raldifsn  35600  lindsadd  36100  lindsenlbs  36102  poimirlem25  36132  poimirlem27  36134  hdmap14lem4a  40363  hdmap14lem13  40372  supxrmnf2  43742  infxrpnf2  43772  fsumnncl  43887  hoidmv1lelem2  44907  gsumdifsndf  46189  mgpsumunsn  46511
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