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Theorem difsn 3569
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 3562 . . 3 (𝑥 ∈ (𝐵 ∖ {𝐴}) ↔ (𝑥𝐵𝑥𝐴))
2 simpl 107 . . . 4 ((𝑥𝐵𝑥𝐴) → 𝑥𝐵)
3 eleq1 2150 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
43biimpcd 157 . . . . . . 7 (𝑥𝐵 → (𝑥 = 𝐴𝐴𝐵))
54necon3bd 2298 . . . . . 6 (𝑥𝐵 → (¬ 𝐴𝐵𝑥𝐴))
65com12 30 . . . . 5 𝐴𝐵 → (𝑥𝐵𝑥𝐴))
76ancld 318 . . . 4 𝐴𝐵 → (𝑥𝐵 → (𝑥𝐵𝑥𝐴)))
82, 7impbid2 141 . . 3 𝐴𝐵 → ((𝑥𝐵𝑥𝐴) ↔ 𝑥𝐵))
91, 8syl5bb 190 . 2 𝐴𝐵 → (𝑥 ∈ (𝐵 ∖ {𝐴}) ↔ 𝑥𝐵))
109eqrdv 2086 1 𝐴𝐵 → (𝐵 ∖ {𝐴}) = 𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102   = wceq 1289  wcel 1438  wne 2255  cdif 2994  {csn 3441
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-v 2621  df-dif 2999  df-sn 3447
This theorem is referenced by:  difsnb  3575  fisseneq  6621  dfn2  8656
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