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Theorem difsn 3652
 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 3645 . . 3
2 simpl 108 . . . 4
3 eleq1 2200 . . . . . . . 8
43biimpcd 158 . . . . . . 7
54necon3bd 2349 . . . . . 6
65com12 30 . . . . 5
76ancld 323 . . . 4
82, 7impbid2 142 . . 3
91, 8syl5bb 191 . 2
109eqrdv 2135 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 103   wceq 1331   wcel 1480   wne 2306   cdif 3063  csn 3522 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 2119 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-v 2683  df-dif 3068  df-sn 3528 This theorem is referenced by:  difsnb  3658  fisseneq  6813  dfn2  8983
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