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Theorem difsn 4807
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 4795 . . 3 (𝑥 ∈ (𝐵 ∖ {𝐴}) ↔ (𝑥𝐵𝑥𝐴))
2 simpl 481 . . . 4 ((𝑥𝐵𝑥𝐴) → 𝑥𝐵)
3 nelelne 3031 . . . . 5 𝐴𝐵 → (𝑥𝐵𝑥𝐴))
43ancld 549 . . . 4 𝐴𝐵 → (𝑥𝐵 → (𝑥𝐵𝑥𝐴)))
52, 4impbid2 225 . . 3 𝐴𝐵 → ((𝑥𝐵𝑥𝐴) ↔ 𝑥𝐵))
61, 5bitrid 282 . 2 𝐴𝐵 → (𝑥 ∈ (𝐵 ∖ {𝐴}) ↔ 𝑥𝐵))
76eqrdv 2724 1 𝐴𝐵 → (𝐵 ∖ {𝐴}) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394   = wceq 1534  wcel 2099  wne 2930  cdif 3944  {csn 4633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-v 3464  df-dif 3950  df-sn 4634
This theorem is referenced by:  difsnb  4815  difsnexi  7771  domdifsn  9094  domunsncan  9112  frfi  9324  infdifsn  9702  dfn2  12539  hashgt23el  14443  clslp  23146  xrge00  32897  lindsadd  37316  lindsenlbs  37318  poimirlem2  37325  poimirlem4  37327  poimirlem6  37329  poimirlem7  37330  poimirlem8  37331  poimirlem19  37342  poimirlem23  37346  supxrmnf2  45066  infxrpnf2  45096  dvmptfprodlem  45583  hoiprodp1  46227
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