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Theorem difsn 3475
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 3469 . . 3 (x (B ∖ {A}) ↔ (x B xA))
2 simpl 102 . . . 4 ((x B xA) → x B)
3 eleq1 2082 . . . . . . . 8 (x = A → (x BA B))
43biimpcd 148 . . . . . . 7 (x B → (x = AA B))
54necon3bd 2226 . . . . . 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 2020 1 A B → (B ∖ {A}) = B)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   = wceq 1228   wcel 1374  wne 2186  cdif 2891  {csn 3350
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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-v 2537  df-dif 2897  df-sn 3356
This theorem is referenced by:  difsnb  3480
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