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Theorem difsn 3542
 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 3535 . . 3 (𝑥 ∈ (𝐵 ∖ {𝐴}) ↔ (𝑥𝐵𝑥𝐴))
2 simpl 107 . . . 4 ((𝑥𝐵𝑥𝐴) → 𝑥𝐵)
3 eleq1 2145 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
43biimpcd 157 . . . . . . 7 (𝑥𝐵 → (𝑥 = 𝐴𝐴𝐵))
54necon3bd 2292 . . . . . 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 2081 1 𝐴𝐵 → (𝐵 ∖ {𝐴}) = 𝐵)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   = wceq 1285   ∈ wcel 1434   ≠ wne 2249   ∖ cdif 2979  {csn 3416 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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-v 2612  df-dif 2984  df-sn 3422 This theorem is referenced by:  difsnb  3548  fisseneq  6474  dfn2  8420
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