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Theorem difsn 3657
 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 3650 . . 3 (𝑥 ∈ (𝐵 ∖ {𝐴}) ↔ (𝑥𝐵𝑥𝐴))
2 simpl 108 . . . 4 ((𝑥𝐵𝑥𝐴) → 𝑥𝐵)
3 eleq1 2202 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
43biimpcd 158 . . . . . . 7 (𝑥𝐵 → (𝑥 = 𝐴𝐴𝐵))
54necon3bd 2351 . . . . . 6 (𝑥𝐵 → (¬ 𝐴𝐵𝑥𝐴))
65com12 30 . . . . 5 𝐴𝐵 → (𝑥𝐵𝑥𝐴))
76ancld 323 . . . 4 𝐴𝐵 → (𝑥𝐵 → (𝑥𝐵𝑥𝐴)))
82, 7impbid2 142 . . 3 𝐴𝐵 → ((𝑥𝐵𝑥𝐴) ↔ 𝑥𝐵))
91, 8syl5bb 191 . 2 𝐴𝐵 → (𝑥 ∈ (𝐵 ∖ {𝐴}) ↔ 𝑥𝐵))
109eqrdv 2137 1 𝐴𝐵 → (𝐵 ∖ {𝐴}) = 𝐵)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   = wceq 1331   ∈ wcel 1480   ≠ wne 2308   ∖ cdif 3068  {csn 3527 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-v 2688  df-dif 3073  df-sn 3533 This theorem is referenced by:  difsnb  3663  fisseneq  6820  dfn2  8990
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