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Theorem nnel 2812
Description: Negation of negated membership, analogous to nne 2668. (Contributed by Alexander van der Vekens, 18-Jan-2018.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
Assertion
Ref Expression
nnel  |-  ( -.  A  e/  B  <->  A  e.  B )

Proof of Theorem nnel
StepHypRef Expression
1 df-nel 2665 . . 3  |-  ( A  e/  B  <->  -.  A  e.  B )
21bicomi 202 . 2  |-  ( -.  A  e.  B  <->  A  e/  B )
32con1bii 331 1  |-  ( -.  A  e/  B  <->  A  e.  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    e. wcel 1767    e/ wnel 2663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 185  df-nel 2665
This theorem is referenced by:  raldifsnb  4158  mpt2xopynvov0g  6943  0mnnnnn0  10829  ssnn0fi  12063  rabssnn0fi  12064  hashnfinnn0  12401  nbgranself2  24209  cusgrasizeindslem2  24247  wwlknndef  24510  wwlknfi  24511  clwwlknndef  24546  frgrawopreglem4  24821  lswn0  32037
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