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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

Proof of Theorem difsn
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eldifsn 3839 . . 3
2 simpl 443 . . . 4
3 eleq1 2413 . . . . . . . 8
43biimpcd 215 . . . . . . 7
54necon3bd 2553 . . . . . 6
65com12 27 . . . . 5
76ancld 536 . . . 4
82, 7impbid2 195 . . 3
91, 8syl5bb 248 . 2
109eqrdv 2351 1
 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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