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Theorem difsn 4297
Description: An element not in a set can be removed without affecting the set. (Contributed by NM, 16-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
difsn 𝐴𝐵 → (𝐵 ∖ {𝐴}) = 𝐵)

Proof of Theorem difsn
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eldifsn 4287 . . 3 (𝑥 ∈ (𝐵 ∖ {𝐴}) ↔ (𝑥𝐵𝑥𝐴))
2 simpl 473 . . . 4 ((𝑥𝐵𝑥𝐴) → 𝑥𝐵)
3 nelelne 2888 . . . . 5 𝐴𝐵 → (𝑥𝐵𝑥𝐴))
43ancld 575 . . . 4 𝐴𝐵 → (𝑥𝐵 → (𝑥𝐵𝑥𝐴)))
52, 4impbid2 216 . . 3 𝐴𝐵 → ((𝑥𝐵𝑥𝐴) ↔ 𝑥𝐵))
61, 5syl5bb 272 . 2 𝐴𝐵 → (𝑥 ∈ (𝐵 ∖ {𝐴}) ↔ 𝑥𝐵))
76eqrdv 2619 1 𝐴𝐵 → (𝐵 ∖ {𝐴}) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1480  wcel 1987  wne 2790  cdif 3552  {csn 4148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-v 3188  df-dif 3558  df-sn 4149
This theorem is referenced by:  difsnb  4306  difsnexi  6919  domdifsn  7987  domunsncan  8004  frfi  8149  infdifsn  8498  dfn2  11249  clslp  20862  xrge00  29471  lindsenlbs  33036  poimirlem2  33043  poimirlem4  33045  poimirlem6  33047  poimirlem7  33048  poimirlem8  33049  poimirlem19  33060  poimirlem23  33064  dvmptfprodlem  39465  hoiprodp1  40109
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