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Theorem difsn 3845
 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 3839 . . 3 (x (B {A}) ↔ (x B xA))
2 simpl 443 . . . 4 ((x B xA) → x B)
3 eleq1 2413 . . . . . . . 8 (x = A → (x BA B))
43biimpcd 215 . . . . . . 7 (x B → (x = AA B))
54necon3bd 2553 . . . . . 6 (x B → (¬ A BxA))
65com12 27 . . . . 5 A B → (x BxA))
76ancld 536 . . . 4 A B → (x B → (x B xA)))
82, 7impbid2 195 . . 3 A B → ((x B xA) ↔ x B))
91, 8syl5bb 248 . 2 A B → (x (B {A}) ↔ x B))
109eqrdv 2351 1 A B → (B {A}) = B)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ≠ wne 2516   ∖ cdif 3206  {csn 3737 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-sn 3741 This theorem is referenced by:  difsnb  3850  adj11  3889  nnsucelrlem3  4426  nnsucelr  4428  ssfin  4470
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