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Theorem difsn 3492
 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 A B → (B ∖ {A}) = B)

Proof of Theorem difsn
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 eldifsn 3486 . . 3 (x (B ∖ {A}) ↔ (x B xA))
2 simpl 102 . . . 4 ((x B xA) → x B)
3 eleq1 2097 . . . . . . . 8 (x = A → (x BA B))
43biimpcd 148 . . . . . . 7 (x B → (x = AA B))
54necon3bd 2242 . . . . . 6 (x B → (¬ A BxA))
65com12 27 . . . . 5 A B → (x BxA))
76ancld 308 . . . 4 A B → (x B → (x B xA)))
82, 7impbid2 131 . . 3 A B → ((x B xA) ↔ x B))
91, 8syl5bb 181 . 2 A B → (x (B ∖ {A}) ↔ x B))
109eqrdv 2035 1 A B → (B ∖ {A}) = B)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   = wceq 1242   ∈ wcel 1390   ≠ wne 2201   ∖ cdif 2908  {csn 3367 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-v 2553  df-dif 2914  df-sn 3373 This theorem is referenced by:  difsnb  3497  dfn2  7950
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