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Theorem difsn 4745
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 4734 . . 3 (𝑥 ∈ (𝐵 ∖ {𝐴}) ↔ (𝑥𝐵𝑥𝐴))
2 simpl 483 . . . 4 ((𝑥𝐵𝑥𝐴) → 𝑥𝐵)
3 nelelne 3040 . . . . 5 𝐴𝐵 → (𝑥𝐵𝑥𝐴))
43ancld 551 . . . 4 𝐴𝐵 → (𝑥𝐵 → (𝑥𝐵𝑥𝐴)))
52, 4impbid2 225 . . 3 𝐴𝐵 → ((𝑥𝐵𝑥𝐴) ↔ 𝑥𝐵))
61, 5bitrid 282 . 2 𝐴𝐵 → (𝑥 ∈ (𝐵 ∖ {𝐴}) ↔ 𝑥𝐵))
76eqrdv 2734 1 𝐴𝐵 → (𝐵 ∖ {𝐴}) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1540  wcel 2105  wne 2940  cdif 3895  {csn 4573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2941  df-v 3443  df-dif 3901  df-sn 4574
This theorem is referenced by:  difsnb  4753  difsnexi  7673  domdifsn  8919  domunsncan  8937  frfi  9153  infdifsn  9514  dfn2  12347  hashgt23el  14239  clslp  22405  xrge00  31582  lindsadd  35875  lindsenlbs  35877  poimirlem2  35884  poimirlem4  35886  poimirlem6  35888  poimirlem7  35889  poimirlem8  35890  poimirlem19  35901  poimirlem23  35905  supxrmnf2  43308  infxrpnf2  43338  dvmptfprodlem  43821  hoiprodp1  44463
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